article
stringlengths 8.52k
116k
| abstract
stringlengths 247
3.09k
|
|---|---|
tungsten ( w ) is one of the most important constituents of tokamak reactor walls @xcite .
additionally , it radiates strongly over almost all ionisation stages . for example , the most intense emission lines of w ions @xcite are from w xxii to w l in the vuv to the soft x - ray region , covering an electron temperature range from about 0.5 to 5.0 kev .
similarly , ptterich et al .
@xcite have predicted emission features from w lxi to w lxix in the 0.10.15 nm , 1.84.0 nm and around 8 nm ranges .
however , to assess radiation loss and for modelling plasmas , atomic data ( including energy levels and oscillator strengths or radiative decay rates ) are required for many of the w ions .
their need for atomic data for several ions , including those of w , has increased significantly due to the developing iter project .
therefore , several groups of people are actively engaged in producing atomic data .
early calculations for a number of w ions ( w xxxviii to w xlviii ) were performed by fournier @xcite .
he adopted a relativistic atomic structure code , but reported only limited results for energy levels and oscillator strengths ( @xmath0-values ) .
a thorough critical compilation of experimental , theoretical and analytical energy levels of w ions ( w iii through w lxxiv ) has been undertaken by kramida and shirai @xcite and has been further reviewed by kramida @xcite .
these energy levels , along with some spectral lines , are also available on the nist ( national institute of standards and technology ) website at http://www.nist.gov/pml/data/asd.cfm .
recently , spectra in the euv wavelength range ( 420 nm ) have been measured by ralchenko et al .
@xcite , for a number of w ions , namely w lv to w lxiv .
similarly , clementson et al .
@xcite have discussed spectroscopy of many w ions ( w xlvii to w lxxii ) . on the other side , calculations have been performed for several w ions , such as by quinet @xcite for w xlviii to w lxii .
although he adopted the grasp code for the calculations , his reported results for energy levels and radiative rates ( @xmath1-values ) are confined to forbidden lines within the 3p@xmath2 and 3d@xmath2 configurations . however , for the modelling of plasmas , atomic data among a wider range of levels / transitions are preferred .
therefore , we have already reported such data for two w ions , namely w xl @xcite and w lviii @xcite . in this paper , we extend our work to eight other w ions , s - like ( w lix ) to f - like ( w lxvi ) . as in our earlier research @xcite and those of others @xcite ,
we have adopted the fully relativistic multi - configuration dirac - fock ( mcdf ) atomic structure code @xcite , better known as the general - purpose relativistic atomic structure package ( grasp ) @xcite .
this code is based on the @xmath3 coupling scheme , includes higher - order relativistic corrections arising from the breit interaction and qed ( quantum electrodynamics ) effects , and is suitable for the heavy ions considered here .
however , this original version @xcite has undergone several revisions , such as by @xcite , and the one employed here ( and by many other workers ) has been revised by dr .
p. h. norrington , and is freely available at http://web.am.qub.ac.uk / darc/.
extensive configuration interaction ( ci ) has been incorporated in grasp , as described below for each ion , and for the optimisation of the orbitals the option of ` extended average level ' ( eal ) , in which a weighted ( proportional to 2@xmath4 + 1 ) trace of the hamiltonian matrix is minimised , has been adopted .
the grasp code has a few other choices for optimisation , such as average level ( al ) and extended optimal level ( eol ) .
however , in general , the results obtained with the al option are comparable with those of eal as already discussed and demonstrated by us for several other ions , such as those of kr @xcite and xe @xcite .
similarly , the eol option may provide slightly more accurate data for a few predefined levels , but is only useful if the experimental energies are known , which is not the case for a majority of the levels of the ions studied here .
clementson and beiersdorfer @xcite have measured wavelengths for 3 lines of w lix .
they also calculated these with two different codes , i.e. grasp and fac ( flexible atomic code ) , and there is no ( major ) discrepancy among the results . for modelling purposes , feldman et al .
@xcite calculated atomic data for many w ions , including w lix , but did not report the data .
furthermore , they used a simple model consisting of the 3s@xmath53p@xmath6 , 3s3p@xmath7 , 3s@xmath53p@xmath83d , and 3p@xmath9 configurations , generating 48 levels in total . for our work ,
we have performed two sets of calculations using the grasp code . in the first ( grasp1 ) we have included 2762 levels of the all possible combinations of the @xmath10 = 3 orbitals , i.e. 18 configurations in number .
the second ( grasp2 ) involves an additional 28 configurations , which are [ 3s@xmath53p@xmath8 , 3s@xmath53p@xmath53d , 3s3p@xmath6 , 3s3p@xmath83d , 3s@xmath53p3d@xmath5 , 3s3p@xmath53d@xmath5 , and 3p@xmath7]4@xmath12 .
these 46 configurations generate 12 652 levels in total . in table
a we compare the energies obtained from both models , but for only the lowest 20 levels .
differences between the two sets of energies are less than 0.025 ryd and the inclusion of larger ci in the grasp2 calculations has lowered the energies for most of the levels .
therefore , it is necessary to assess the effect of further ci on the energy levels . for this
we have adopted the fac code of gu @xcite , which is also fully relativistic and is available from the website https://www - amdis.iaea.org / fac/. this code is comparatively more efficient to run and generally yields results similar to those obtained with other atomic structure codes , as has already been demonstrated in several of our earlier papers
see for example aggarwal et al . @xcite .
with fac we have also performed two sets of calculations , i.e. fac1 : includes the same 2762 levels as in grasp1 , and fac2 : also includes levels of the 3@xmath134@xmath12 configurations , generating 38 694 levels in total .
energies obtained from both these models are also listed in table a for comparison .
discrepancies between the grasp1 and fac1 energies are up to 0.15 ryd ( see level 13 ) , in spite of including the _ same _ ci .
this is because of the differences in the algorithms of the codes and also in calculating the central potentials .
additionally , the energies obtained from fac are generally lower for most levels .
however , inclusion of additional ci in the fac2 calculations further lowers the energies , but only up to 0.02 ryd for some of the levels .
therefore , it may be reasonable to say that the inclusion of ci in our grasp2 calculations is sufficient to calculate accurate results , but differences with fac2 remain of up to 0.15 ryd .
the nist compilation is only for a few levels of w lix , which are mostly based on the experimental and theoretical work of clementson et al .
however , these energies are not very accurate as indicated on their website , and many levels are also missing from the compilation .
nevertheless , in table a we have included their energies for comparison .
unfortunately , differences between their compiled energies and our ( any of the ) calculations are up to 0.4 ryd for some of the levels , such as 1820 .
therefore , there may be scope to improve upon our calculated energies but the ( in)accuracy can not definitely be determined by the limited comparison shown in table a. our calculated energies from grasp2 are listed in table 1 along with those from fac2 for the lowest 220 levels , which belong to the @xmath14 3 configurations . beyond these , the levels of the @xmath10 = 4 configurations start mixing . discrepancies between the two sets of energies are smaller than 0.4 ryd ( @xmath15 0.5% ) for a majority of levels and the orderings are different only in a few instances , such as 70/71 and 151/152 .
we also note that some differences may be because of a mismatch between the two sets of energies , as it is not always possible to perfectly match these due to their different notations .
also note that the @xmath16 designations of the levels listed in table 1 are not always unambiguous , and a few of these can be ( inter)changed with varying amounts of ci , codes , and authors preferences .
this is inevitable in any calculation because of the strong mixing among some of the levels . as examples
, we list the lowest 20 levels in table b. for some , such as 1 , 2 , 10 , and 12 , there is a clear dominance of one vector ( level ) and hence there is no scope for ambiguity .
however , for others , such as 39 , several vectors ( levels ) dominate and therefore it is not straightforward to designate such levels . for example
, the eigenvector for level 19 is dominant in 19 but is also significant in 4 .
however , the eigenvector for level 105 is dominant in both levels 4 and 105 ( not listed in table b ) .
finally , it may be noted that the degeneracy among the levels of w lix is very large
see for example levels 3 , 5 , 9 , 19 , and 32 of 3s@xmath53p@xmath83d @xmath7d@xmath17 , which are separated by up to @xmath1830 ryd . for the ground state energy
the breit and qed contributions are 28.7 and 21.7 ryd , respectively , although they amount to only @xmath180.1% . for this ion
we have also performed two calculations with grasp using different levels of ci , i.e. grasp1 : includes 1313 levels of the 15 @xmath10 = 3 configurations , which are 3s@xmath53p@xmath8 , 3s@xmath53p@xmath53d , 3s3p@xmath6 , 3s@xmath53p3d@xmath5 , 3s3p@xmath83d , 3s3p@xmath53d@xmath5 , 3p@xmath7 , 3p@xmath63d , 3s@xmath53d@xmath8 , 3p@xmath83d@xmath5 , 3s3p3d@xmath8 , 3p@xmath53d@xmath8 , 3s3d@xmath6 , 3p3d@xmath6 , and 3d@xmath7 . in the other calculation ( grasp2 ) , a further 20 configurations of [ 3s@xmath53p@xmath5 , 3s3p@xmath8 , 3s@xmath53p3d , 3s3p@xmath53d , and 3s@xmath53d@xmath5]4@xmath12 are included , generating in total 3533 levels .
similarly , two calculations with fac are performed , i.e. fac1 with the same ci as in grasp2 , and fac2 , which also includes all possible combinations of 3@xmath204@xmath12 , generating 14 608 levels in total .
energies for the lowest 220 levels from both grasp2 and fac2 are listed in table 2 .
these levels belong to the first 8 configurations listed above . for the higher - lying levels , those of @xmath10 = 4 intermix with @xmath10 = 3 . in table
c we compare our energies for the lowest 25 levels of w lx from grasp1 , grasp2 , fac1 , and fac2 with the nist compilation .
ci for w lx is not as important as for w lix , because differences between our grasp1 and grasp2 energies are smaller than 0.02 ryd .
similarly , discrepancies between the fac1 and fac2 energies are less than 0.03 ryd .
however , differences between the grasp2 and fac2 energies are up to 0.3 ryd for some levels , for reasons already explained in section 2.1 .
the nist compilation is only for the lowest 25 levels , listed in table c , and our grasp2 energies are ( generally ) lower by up to 0.3 ryd
see for example , levels 13 , 17 and 22 .
similar differences remain between the nist and fac2 energies , and therefore are not due to a lack of ci .
however , it is worth emphasising that the compiled energies of nist are mostly based on interpolation / extrapolation and hence are likely not very accurate .
more importantly , there are differences in the designations of a few levels , particularly the ground state , which is ( 3s@xmath53p@xmath8 ) @xmath5d@xmath21 in our work , but @xmath5p@xmath21 in nist .
this is a highly mixed level and the eigenvector for @xmath5p@xmath21 dominates in both levels 1 and 25 see table d in which eigenvectors for the lowest 25 are listed .
however , we have preferred to designate the lower ( ground ) level as @xmath5d@xmath21 , because the placings of @xmath5d@xmath22 and @xmath5p@xmath23 ( levels 5 and 6 ) are unambiguous . there may be similar differences in designations with other calculations because of the very high mixing among some of the levels of w lx .
as for other w ions , we have performed two calculations each with the grasp and fac codes to assess the effect of ci .
these are grasp1 : 518 levels of 12 configurations [ 3s@xmath53p@xmath5 , 3s3p@xmath8 , 3s@xmath53p3d , 3s3p@xmath53d , 3p@xmath6 , 3s@xmath53d@xmath5 , 3p@xmath83d , 3s3p3d@xmath5 , 3p@xmath53d@xmath5 , 3s3d@xmath8 , 3p3d@xmath8 , and 3d@xmath6 ] ; grasp2 : 4364 levels of 48 configurations , the additional 36 are [ 3s@xmath53p , 3s3p@xmath5 , 3s@xmath53d , 3s3p3d , 3p@xmath8 , 3p@xmath53d , 3s3d@xmath5 , 3p3d@xmath5 , and 3d@xmath8]4@xmath12 ; fac1 : 9798 levels of 3 * 4 , 3 * 3 4 * 1 and 3 * 4 5 * 1 ; and finally fac2 : which includes 27 122 levels in total , the additional ones arising from 3 * 3 6 * 1 and 3 * 2 4 * 2 configurations .
energies obtained from these calculations are compared in table e with the nist compilation for the lowest 21 levels of w lxi , which are the only ones in common . as for other ions ,
the ci is not very important for this ion , because the grasp1 and grasp2 energies agree within to 0.02 ryd , and the fac1 and fac2 energies show no appreciable differences .
similarly , the agreement between our grasp2 and fac2 energies is better than 0.2 ryd
see levels 1215 . however , as for other ions , the differences with the nist compilation are larger , up to 0.4 ryd see level 9 for example .
again , the nist energies are not very accurate and therefore such differences are not surprising .
an important difference between our calculations and the nist compilation is the designation for level 4 , i.e. ( 3s3p@xmath8 ) @xmath7s@xmath26 which is @xmath8p@xmath26 ( 64 ) in the latter .
both these levels are highly mixed , as may be seen from the eigenvectors listed in table f for the lowest 21 levels _ plus _ the remaining two of the 3s3p@xmath8 configuration , i.e. @xmath8p@xmath26 and @xmath27p@xmath28 . our recommended energies for the lowest 215 levels of w lxi are listed in table 3 from the grasp2 and fac2 calculations .
these levels belong to the @xmath10 = 3 configurations and beyond these those of @xmath10 = 4 intermix .
finally , there are no major differences in the orderings of the two sets of level energies . for w lxii the experimental energies are also as sparse as for other w ions . however , two sets of theoretical energy levels @xcite are available in the literature .
safronova and safronova @xcite adopted a relativistic many - body perturbation theory ( rmbpt ) and reported energies for the lowest 40 levels belonging to the 3s@xmath53p , 3s3p@xmath5 , 3s@xmath53d , 3s3p3d , 3p@xmath8 , and 3p@xmath53d configurations .
in addition , s. aggarwal et al .
@xcite have calculated energies for the lowest 148 levels of the 3s@xmath53p , 3s3p@xmath5 , 3s@xmath53d , 3s3p3d , 3p@xmath8 , 3p@xmath53d , 3s3d@xmath5 , 3p3d@xmath5 , and 3d@xmath8 ( nine ) configurations , adopting the same version of the grasp code as in the present work .
the rmbpt energies @xcite are closer to the nist compilation and in general are lower than those of s. aggarwal et al .
by up to 0.4 ryd
see table 2 of @xcite .
we have performed several sets of calculations with the grasp code but mention only three here , namely : grasp1 , which includes the basic 148 levels of the 9 configurations listed above ; grasp2 , which considers an additional 776 ( total 924 ) levels of the [ 3s3p , 3s3d , 3p3d , 3s@xmath5 , 3p@xmath5 , and 3d@xmath5]4@xmath12 ( 24 ) configurations ; and finally grasp3 which includes a further 1079 levels ( total 2003 ) of the 30 additional configurations , i.e. [ 3s3p , 3s3d , 3p3d , 3s@xmath5 , 3p@xmath5 , and 3d@xmath5]5@xmath12
. s. aggarwal et al .
@xcite included ci among 35 configurations , which are the basic 9 of grasp1 _ plus _ another 26 , i.e. 3s3p4@xmath12 , 3s3d4@xmath12 , 3p3d4@xmath12 , 3s@xmath54@xmath12 , 3p@xmath54@xmath12 ( except 3p@xmath54d ) , 3p4@xmath29 ( except 3p4p@xmath5 ) , and 3d4@xmath29 .
it is not clear why they overlooked configurations such as : 3p@xmath54d , 3p4p@xmath5 , 3s4@xmath29 , and 3@xmath124@xmath30 .
in addition , their 35 configurations generate 1007 levels in total ( see table 1 of @xcite ) whereas they mention only 894 , and therefore there is an anomaly of 113 levels .
however , we stress that ( particularly ) the omission of the 3p@xmath54d and 3p4p@xmath5 configurations does not affect the energies or the corresponding lifetimes , as already discussed by one of us @xcite .
more importantly , levels of the 3@xmath124@xmath29 configurations lie at energies well above those of our grasp3 calculations , and hence are omitted from our work .
this has been confirmed by our larger calculation with 75 configurations and 2393 levels .
for the same reason we preferred not to include the 4@xmath29 configurations for the calculations of energy levels for other w ions . a complete set of energies for all 148 levels ( of the grasp1 calculations )
are listed in table 4 from grasp3 and fac2 ( see below ) .
we note that levels from all other configurations clearly lie _ above _ these 148 and hence there is no intermixing . as with grasp
, we have also performed several calculations with fac , but focus on only two , i.e. fac1 : includes the same 2003 levels as in grasp3 , and fac2 : contains 12 139 levels in total , the additional ones arising from the 3 * 2 6 * 1 , 3 * 1 4 * 2 , 3 * 1 5 * 2 and 3 * 1 6 * 2 configurations . in table
g we compare our energies from grasp2 , grasp3 , fac1 , and fac2 with those of nist for the lowest 21 levels , which are in common . also included in this table
are the results of safronova and safronova @xcite from rmbpt .
the corresponding data of s. aggarwal et al .
@xcite are not considered because they are similar to our grasp2 calculations and have already been discussed previously @xcite . although a considerably large ci has been included in our calculations , it does not appear to be too important for w lxii , because the grasp2 and grasp3 ( and fac1 and fac2 ) energies are practically identical .
therefore , the discrepancies between the grasp and fac energies ( up to 0.4 ryd , particularly for level 21 ) are not due to different levels of ci but because of the computational and theoretical dissimilarities in the codes . nevertheless ,
although the nist energies are not claimed to be very accurate , their agreements with those from fac and rmbpt are better ( within 0.1 ryd ) than with grasp . regarding all the 148 levels in table 4 , the differences between the grasp and fac energies are up to 0.4 ryd for some ( see levels 77 upwards in the table ) .
finally , as for other w ions , configuration mixing is strong for w lxii also and therefore there is always a possibility of ( inter)change of level designations listed in table 4 .
for the 21 levels listed in table g , their designations and orderings are the same between nist and our calculations , but differ with those of s. aggarwal et al .
@xcite for some , such as levels 10 and 68 , i.e. ( 3p@xmath8 ) @xmath5d@xmath21 and @xmath5p@xmath21 , which are reversed by them .
these two levels ( and many more ) have strong mixing , as may be seen from table h in which we list the eigenvectors for the lowest 21 levels plus 68 , i.e. 3p@xmath8 @xmath5p@xmath21 . similarly , there is a _ disagreement _ for most level designations between our work and nist with those of safronova and safronova @xcite . for this ion , earlier calculations for energy levels are by safronova and safronova @xcite using the rmbpt method for the lowest 35 levels of the 3s@xmath5 , 3s3p , 3p@xmath5 , 3s3d , 3p3d , and 3d@xmath5 configurations , whereas the nist compilation is only for 9 levels see table i. as for other ions we have performed several sets of calculations with grasp and fac and here we only state our final results . for the grasp calculations
we have considered 58 configurations , which are 3@xmath29 , 3s3p , 3s3d , 3p3d , 3@xmath124@xmath12 , 4@xmath29 , 4@xmath30 , 3@xmath125@xmath12 , and 3@xmath126@xmath12 ( except 6h ) , while for fac we include 991 levels , the additional ones arising from 3@xmath127@xmath12 and 4@xmath125@xmath12 .
however , levels of the 4@xmath29 , 4@xmath30 and 4@xmath125@xmath12 configurations mostly lie above those of 3@xmath127@xmath12 and can therefore be neglected .
energy levels from both calculations are listed in table 5 for the lowest 210 levels . in table
i a comparison is shown for the lowest 35 levels with the nist compilation and the rmbpt calculations @xcite .
as for w lxii , the fac and rmbpt energies agree closely with each other as well as with nist , but our grasp energies are higher by up to 0.3 ryd for many levels .
similarly , mixing for the levels is strong for a few as shown in table j for the lowest 35 see in particular levels 22 , 25 and 34 . for this ion
we have gradually increased the number of orbitals to perform grasp calculations for up to 1235 levels .
the configurations included are 2p@xmath9@xmath31 with @xmath14 7 and @xmath32 4 , 2p@xmath73@xmath30 , 2p@xmath73@xmath29 , 2p@xmath74@xmath30 , 2p@xmath74@xmath29 , and 2p@xmath73@xmath124@xmath12 .
however , we note that the levels of 2p@xmath9@xmath31 lie _ below _ those of the other configurations . for this reason we only list the lowest 30 levels in table k , all belonging to 2p@xmath9@xmath31 .
however , with fac we have performed comparatively larger calculations for up to @xmath10 = 20 and all possible values of @xmath12 , i.e. 1592 levels in total .
these results are also listed in table k along with those of nist , which are confined to the @xmath14 5 levels .
the nist energies differ with fac by up to 0.26 ryd for some levels ( see 20 ) , but discrepancies are smaller than 0.15 ryd with those with grasp .
again , the differences between the grasp and fac energies are not because of different levels of ci , but due to methodological variations .
it has not been possible to include higher 2p@xmath9@xmath31 configurations in our grasp calculations , but since the fac energies have been obtained ( as stated above ) in table 6 we list these for the lowest 396 levels , all belonging to 2p@xmath9@xmath31 with @xmath14 20
. this will be helpful for future comparisons .
finally , unlike the other w ions discussed above , there is no ( strong ) mixing and/or ambiguity for the designation of the 2p@xmath9@xmath31 levels listed in tables k and 6 .
safronova et al .
@xcite have reported energies for 242 levels of w lxiv from three independent codes , namely rmbpt , hullac ( hebrew university lawrence livermore atomic code @xcite ) and the atomic structure code of r.d .
cowan available at http://das101.isan.troitsk.ru/cowan.htm .
although nist energies for this ion are only available for a few levels , as already seen in table k , their rmbpt results are closest to the measurements .
additionally , based on the comparisons made for other w ions , their rmbpt energies should be the most accurate .
nevertheless , the rmbpt energy for level 2 ( 2p@xmath73s @xmath8p@xmath26 ) differs by 1.3% and 6.4% with those from hullac and cowan , respectively .
corresponding differences for the remaining levels are up to 0.3% and 1% , respectively .
only the lowest 5 levels of table k are common with their work , as the remaining 237 belong to the 2p@xmath73@xmath30 configurations .
therefore , our listed energies in table 6 supplement their data . the nist compilation of energies for this ion is limited to only 10 levels of the 2p@xmath73@xmath12 configurations
. however , vilkas et al .
@xcite have reported energies for 141 levels of the 2p@xmath9 , ( 2s2p@xmath9)3@xmath12 , 4@xmath12 , 5@xmath12 ( except 5 g ) , and ( 2p@xmath7 ) 3@xmath12 , 4@xmath12 , 5@xmath12 ( except 5 g ) configurations . for their calculations they adopted the relativistic multi - reference many - body mller - plesset ( mrmp ) perturbation theory , and included ci up to the @xmath10 = 5 orbitals .
we have included the same configurations for our calculations with grasp , which generate 157 levels in total because we have also considered the 5 g orbital .
however , in table 7 we list energies for only the lowest 121 , because beyond this the levels of the 2s2p@xmath96@xmath12 configurations start mixing in the same way as of 2s2p@xmath95 g with those of 2s2p@xmath94@xmath12 see levels 9299 in the table .
additionally , we have performed larger calculations with fac with up to 1147 levels , belonging to the 2 * 8 , ( 2 * 7 ) 3 * 1 , 4 * 1 , 5 * 1 , 6 * 1 , 7 * 1 , and 2 * 6 3 * 2 configurations .
these results are also listed in table 7 for comparison .
differences between the grasp and fac energies are up to 0.5 ryd ( 0.07% ) for some levels , but the level orderings are almost identical . similarly , there is no difference in level orderings with the mrmp calculations @xcite and the energies differ only by less than 0.6 ryd ( 0.06% ) with grasp
see levels 63 and 7783 .
therefore , overall there is no ( significant ) discrepancy between the three independent calculations .
however , in general the fac energies are lower than those from grasp for a majority of levels , whereas those of mrmp are higher . in table
l , we compare energies with the nist compilation for only the _ common _ levels .
there is no uniform pattern for ( dis)agreement between the theoretical and experimental energies . in general , the mrmp energies are closer to those of nist whereas those from fac differ the most .
unfortunately , these comparisons are not sufficient for accuracy determination , particularly when the nist energies are not based on direct measurements .
finally , as for most w ions , for w lxv also there is a strong mixing for some levels and therefore the level designations listed in table 7 can vary , although the mrmp calculations @xcite have the same labels as in our work .
nevertheless , in table m we list the eigenvectors for the lowest 33 levels , which include all of the nist compilation .
note particularly the mixing for levels 24 , 25 and 31 . for this ion
we have performed a series of calculations with grasp with gradually increasing ci and our final set includes 501 levels of 38 configurations , which are : 2s@xmath52p@xmath7 , 2s2p@xmath9 , ( 2s@xmath52p@xmath6 , 2s2p@xmath7 , 2p@xmath9)3@xmath12 , 4@xmath12 , 5@xmath12 .
similarly , calculations with fac have been performed for up to 1113 levels from the 2 * 7 and ( 2 * 6 ) 3 * 1 , 4 * 1 , 5 * 1 , 6 * 1 , 7 * 1 configurations .
these levels span an energy range of up to 1360 ryd .
opening the 1s shell gives rise to levels above 5000 ryd and therefore has not been included in the calculations .
energies from both of these calculations are listed in table 8 for the lowest 150 levels , because beyond this the levels of the @xmath10 = 5 configurations start mixing .
however , the listed levels include all of the @xmath10 = 3 configurations .
differences between the two sets of energies are up to 0.5 ryd for some levels , except three ( 145147 ) for which the discrepancies are slightly larger , up to 0.7 ryd . the level orderings are also the same for a majority of levels , but slightly differ in a few instances , such as for 93112 .
nist listings are available for only two levels , namely 2s@xmath52p@xmath7 @xmath5p@xmath23 and 2s2p@xmath9 @xmath5s@xmath33 , and the energy for the latter is lower by 0.5 ryd than the theoretical results .
no other similar theoretical energies are available for this ion for comparison purposes .
finally , this ion is no exception for level mixing and examples of this are listed in table n for the lowest 48 levels see in particular 13 , 15 , 40 , and 42 .
apart from energy levels , calculations have been made for absorption oscillator strengths ( @xmath0-values , dimensionless ) , radiative rates ( @xmath1-values , s@xmath34 ) and line strengths ( @xmath35-values , in atomic units , 1 a.u .
= 6.460@xmath3610@xmath37 cm@xmath5 esu@xmath5 ) .
however , @xmath0- and @xmath1-values for all types of transition ( @xmath39 ) are connected by the following expression : @xmath40 where @xmath41 and @xmath42 are the electron mass and charge , respectively , @xmath43 the velocity of light , @xmath44 the transition wavelength in @xmath45 , and @xmath46 and @xmath47 the statistical weights of the lower @xmath48 and upper @xmath4 levels , respectively . similarly , @xmath0- and
@xmath1-values are related to @xmath35 by the standard equations given in @xcite . in tables
916 we present results for energies ( wavelengths , @xmath44 in @xmath49 ) , @xmath1- , @xmath0- and @xmath35- values for electric dipole ( e1 ) transitions in w ions , which have been obtained with the grasp code . for other types of transitions , namely magnetic dipole ( m1 ) , electric quadrupole ( e2 ) , and magnetic quadrupole ( m2 ) ,
only the @xmath1-values are listed , because the corresponding results for @xmath0- or @xmath35-values can be obtained using eqs .
( 1 - 5 ) given in @xcite . additionally
, we have also listed the ratio ( r ) of the velocity ( coulomb gauge ) and length ( babushkin gauge ) forms which often ( but not necessarily ) give an indication of the accuracy .
the _ indices _ used to represent the lower and upper levels of a transition are defined in tables 18 .
furthermore , only a limited range of transitions are listed in tables 916 , but full tables are available online in the electronic version . for the w ions considered here ,
existing @xmath1- ( or @xmath0- ) values are available mostly for three ions , i.e. al - like w lxii @xcite , mg - like w lxiii @xcite and na - like w lxiv @xcite .
therefore , we confine our comparisons to these three ions . in table o we compare the @xmath0-values for common e1 transitions with the results of safronova and safronova @xcite .
both sets of data agree very well for all transitions .
similarly , for a few weak transitions ( @xmath0 @xmath18 10@xmath50 ) , such as 122 , 23 and 1419 , the ratio r is up to 1.7 and is closer to unity for the comparatively strong transitions .
similar comparison with their results for transitions in w lxiii is shown in table p. for the common transitions listed here , r is unity for all , and @xmath0-values agree closely for most with only a few exceptions , such as 2032 , 2130 and 2634 for which discrepancies are a factor of two . however , we note that the @xmath0- ( or @xmath1- ) values of @xcite are only for a small number of transitions whereas our results listed in tables 12 and 13 cover a much wider range .
vilkas et al .
@xcite have listed @xmath1-values for some ( not all ) transitions of w lxv and in table
q we compare their results with our calculations with grasp , but only from the lowest three to higher excited levels .
additionally we have listed the @xmath0-values to indicate the strength of transitions . as for other w ions
, r is also listed for these transitions and is within a few percent of unity , irrespective of the @xmath0-value .
there are no appreciable differences between the two sets of @xmath1-values and discrepancies , if any , are ( generally ) within @xmath1820% .
the comparisons of @xmath1- ( @xmath0- ) values discussed above are only for a subset of transitions . considering a wider range , for a majority of strong transitions ( @xmath0 @xmath51 0.01 )
r is often within 20% of unity , as already seen in tables o , p and q. however , there are ( as always ) some exceptions .
for example , there are only six transitions of w lxiii with @xmath0 @xmath52 0.01 for which r is up to 1.6 , namely 148166 ( @xmath0 = 0.011 , r = 1.3 ) , 158173 ( @xmath0 = 0.021 , r = 1.3 ) , 160174 ( @xmath0 = 0.028 , r = 1.6 ) , 161175 ( @xmath0 = 0.025 , r = 1.4 ) , 162176 ( @xmath0 = 0.027 , r = 1.4 ) , and 163177 ( @xmath0 = 0.029 , r = 1.6 ) . therefore , based on this and other comparisons
already discussed , our assessment of accuracy for the @xmath0-values for a majority of strong transitions is @xmath1820% . finally ,
for much weaker transitions ( often with @xmath0 @xmath53 10@xmath50 ) , r can be several orders of magnitude and it is very difficult to assess the accuracy of the @xmath0-values because results are often much more variable with ci and/or codes .
generally , such transitions do not make an appreciable contribution to plasma modelling and their results are mostly required for completeness .
the lifetime @xmath54 of a level @xmath4 is given by 1.0/@xmath55@xmath56 and the summation includes @xmath1-values from all types of transitions , i.e. e1 , e2 , m1 , and m2 . since this is a measurable quantity it helps to assess the accuracy of @xmath1-values , particularly when a single ( type of ) transition dominates .
unfortunately , to our knowledge no measurements of @xmath54 are available for the levels of the w ions considered here , but in tables 18 we list our calculated results .
previous theoretical results are available for two ions , i.e. w lxii @xcite and w lxv @xcite .
unfortunately , the @xmath54 of s. aggarwal et al . @xcite
contain large errors , by up to 14 orders of magnitude , for over 90% of the levels of w lxii and bear no relationship to the @xmath1-values , as already discussed @xcite . for w lxv , the reported @xmath54 of vilkas et al .
@xcite are included in table 7 , and there is no significant discrepancy for any level .
energy levels and radiative rates for e1 , e2 , m1 , and m2 transitions are reported for eight w ions ( w lix to w lxvi ) .
a large number of levels are considered for each ion and the data sets reported here are significantly larger than available in the literature . for our calculations
the grasp code has been adopted , although fac has also been utilised for the determination of energy levels to assess the importance of ci , larger than that considered in grasp .
it is concluded that ci beyond a certain level does not appreciably improve the level energies .
differences between the grasp and fac energies , and the available experimental and theoretical values , are often smaller than 0.5 ryd , or equivalently the listed energy levels for all w ions are assessed to be accurate to better than 1% , but scope remains for improvement .
a similar assessment of accuracy for the corresponding @xmath1-values is not feasible , mainly because of the paucity of other comparable results .
however , for strong transitions ( with large @xmath0-values ) , the accuracy for @xmath1-values and lifetimes may be @xmath1820%
. lifetimes for these levels are also listed although no measurements are currently available in the literature .
however , previous theoretical values are available for most levels of w lxv and there is no discrepancy with our work .
kma is thankful to awe aldermaston for financial support .
owing to space limitations , only parts of tables 916 are presented here , the full tables being made available as supplemental material in conjunction with the electronic publication of this work .
supplementary data associated with this article can be found , in the online version , at doi : nn.nnnn / j.adt.2016.nn.nnn . 999 t. ptterich , r. neu , r. dux , a.d .
whiteford , m.g . omullane and the asdex upgrade team , plasma phys .
fusion 50 ( 2008 ) 085016 .
k. fournier , at .
data nucl .
data tables 68 ( 1998 ) 1 .
kramida , t. shirai , at .
data nucl .
data tables 95 ( 2009 ) 305 + 1051 .
a. kramida , can j. phys .
93 ( 2015 ) 487 .
y. ralchenko , i.n .
draganic , j.n .
tan , j.d .
gillaspy , j.m .
poneroy , j. reader , u. feldman , g.e .
holland , j. phys .
b41 ( 2008 ) 021003 .
j. clementson , p. beiersdorfer , g.v .
brown , m.f .
gu , h. lundberg , y. podpaly , and e. trbert .
can . j. phys . * 89 * , 571 ( 2011 ) .
p. quinet , j. phys .
b44 ( 2011 ) 195007 .
aggarwal , f.p .
keenan , can j. phys . 92 ( 2014 )
aggarwal , f.p .
keenan , at .
data nucl .
data tables 100 ( 2014 ) 1399 .
aggarwal , f.p .
keenan , can j. phys . 92 ( 2014 )
aggarwal , f.p .
keenan , at .
data nucl .
data tables 100 ( 2014 ) 1603 .
s. aggarwal , a.k.s .
jha , i. khatri , n. singh , m. mohan , chin .
b24 ( 2015 ) 053201 .
i.p . grant , b.j .
mckenzie , p.h .
norrington , d.f .
mayers , n.c .
pyper , comput .
commun . 21 ( 1980 ) 207 .
dyall , i.p .
grant , c.t .
johnson , f.a .
parpia , e.p .
plummer , comput .
55 ( 1989 ) 425 .
p. jnsson , x. he , c.f .
fischer , i.p .
grant , comput .
177 ( 2007 ) 597 .
p. jnsson , g. gaigalas , j. biero , c.f .
fischer , i.p .
grant , comput .
commun . 184
( 2013 ) 2197 .
aggarwal , f.p .
keenan , k.d .
lawson , at .
data nucl .
data tables 94 ( 2008 ) 323 .
aggarwal , f.p .
keenan , k.d .
lawson , at .
data nucl .
data tables 96 ( 2010 ) 123 .
j. clementson , p. beiersdorfer , phys .
a81 ( 2010 ) 052509 .
u. feldman , j.f .
seely , e. landi , yu . ralchenko , nucl .
fusion 48 ( 2008 ) 045004 .
86 ( 2008 ) 675 .
aggarwal , v. tayal , g.p .
gupta , f.p .
keenan , at .
data nucl .
data tables 93 ( 2007 ) 615 .
safronova , a.s .
safronova , j. phys .
b 43 ( 2010 ) 074026 . k.m .
aggarwal , chin .
b24 ( 2015 ) 103201 .
safronova , a.s .
safronova , p. beiersdorfer , at .
data nucl .
data tables 95 ( 2009 ) 751 .
a. bar - shalom , m. klapisch , j. oreg , j. quant .
spectrosc . radiat .
transfer 71 ( 2001 ) 169 .
vilkas , j.m .
l ' opez - encarnaci ' on , y. ishikawa , at .
data nucl .
data tables 94 ( 2008 ) 50 . @rlllrrrr@ index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + + index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + 1 & 3s@xmath53p@xmath6 & @xmath8p@xmath57 & 00.0000 & 0.0000 & 0.0000 & 0.00000 & 0.0000 + 2 & 3s@xmath53p@xmath6 & @xmath27s@xmath58 & 01.394 & 1.4629 & 1.4691 & 1.46006 & 1.4675 + 3 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath59 & 17.2585 & 17.3063 & 17.2920 & 17.23239 & 17.2200 + 4 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8p@xmath60 & 17.852 & 17.9536 & 17.9413 & 17.85940 & 17.8689 + 5 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath61 & 17.852 & 17.9342 & 17.9213 & 17.87821 & 17.8489 + 6 & 3s@xmath53p@xmath8(@xmath5p)3d & @xmath8f@xmath62 & 17.9173 & 17.9812 & 17.9637 & 17.90613 & 17.8913 + 7 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath27g@xmath63 & 23.4589 & 23.4945 & 23.4758 & 23.42838 & 23.4095 + 8 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8d@xmath59 & 23.94 & 23.9785 & 23.9637 & 23.91315 & 23.8975 + 9 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath62 & 25.29 & 25.4172 & 25.3923 & 25.35176 & 25.3264 + 10 & 3s@xmath53p@xmath6 & @xmath8p@xmath64 & & 25.4932 & 25.4899 & 25.52002 & 25.5134 + 11 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8p@xmath61 & 25.96 & 26.2537 & 26.2281 & 26.18601 & 26.1616 + 12 & 3s@xmath53p@xmath6 & @xmath27d@xmath57 & & 26.2096 & 26.2037 & 26.23460 & 26.2258 + 13 & 3s@xmath53p@xmath5(@xmath27s)3d@xmath5(@xmath8f ) & @xmath8f@xmath57 & & 35.3490 & 35.3341 & 35.19957 & 35.1818 + 14 & 3s@xmath53p@xmath5(@xmath8p)3d@xmath5(@xmath8p ) & @xmath7d@xmath58 & & 37.1186 & 37.1256 & 36.96614 & 36.9719 + 15 & 3s3p@xmath7 & @xmath8p@xmath59 & 39.0268 & 39.1209 & 39.1426 & 39.03447 & 39.0438 + 16 & 3s3p@xmath7 & @xmath27p@xmath61 & 40.62 & 40.6371 & 40.6465 & 40.56564 & 40.5641 + 17 & 3s@xmath53p@xmath5(@xmath27s)3d@xmath5(@xmath8f ) & @xmath8f@xmath65 & & 41.8457 & 41.8247 & 41.70542 & 41.6775 + 18 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8f@xmath59 & 42.43 & 42.0799 & 42.0634 & 42.03196 & 42.0150 + 19 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath60 & 42.53 & 42.1816 & 42.1679 & 42.13380 & 42.1196 + 20 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8d@xmath61 & 42.60 & 42.2933 & 42.2833 & 42.23549 & 42.2231 + + nist : http://www.nist.gov/pml/data/asd.cfm + grasp1 : present results with the grasp code from 18 configurations and 2762 levels + grasp2 : present results with the grasp code from 46 configurations and 12 652 levels + fac1 : present results with the fac code from 2762 levels + fac2 : present results with the fac code from 38 694 levels + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 3s@xmath53p@xmath6 & @xmath8p@xmath57 & 0.67 ( 1)+0.31 ( 12 ) + 2 & 3s@xmath53p@xmath6 & @xmath27s@xmath58 & 0.36 ( 46)+0.64 ( 2 ) + 3 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath59 & 0.16 ( 3)+0.08 ( 41)+0.19 ( 18)+0.06 ( 45)+0.10 ( 26)+0.23(108)+0.12(132 ) + 4 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8p@xmath60 & 0.26 ( 19)+0.14 ( 4)+0.12 ( 37)+0.48(105 ) + 5 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath61 & 0.24 ( 5)+0.14 ( 20)+0.10 ( 43)+0.18 ( 30)+0.27(107 ) + 6 & 3s@xmath53p@xmath8(@xmath5p)3d & @xmath8f@xmath62 & 0.13 ( 9)+0.12(106)+0.21 ( 22)+0.05 ( 44)+0.27 ( 6)+0.06(136)+0.14 ( 40 ) + 7 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath27g@xmath63 & 0.26 ( 32)+0.14 ( 25)+0.10 ( 7)+0.48(130 ) + 8 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8d@xmath59 & 0.10 ( 3)+0.14 ( 41)+0.16 ( 8)+0.07 ( 28)+0.04(108)+0.22 ( 39)+0.19(132 ) + 9 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath62 & 0.14 ( 9)+0.09(106)+0.06 ( 22)+0.17 ( 27)+0.30(136)+0.17 ( 40 ) + 10 & 3s@xmath53p@xmath6 & @xmath8p@xmath64 & 0.98 ( 10 ) + 11 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8p@xmath61 & 0.22 ( 35)+0.14 ( 11)+0.08 ( 29)+0.09 ( 30)+0.14(107)+0.24(147 ) + 12 & 3s@xmath53p@xmath6 & @xmath27d@xmath57 & 0.31 ( 1)+0.67 ( 12 ) + 13 & 3s@xmath53p@xmath5(@xmath27s)3d@xmath5(@xmath8f ) & @xmath8f@xmath57 & 0.23 ( 59)+0.06 ( 88)+0.10 ( 96)+0.05(285)+0.05(350)+0.08 ( 38)+0.06(277)+0.22 ( 13)+0.07(325 ) + 14 & 3s@xmath53p@xmath5(@xmath8p)3d@xmath5(@xmath8p ) & @xmath7d@xmath58 & 0.28 ( 14)+0.16 ( 85)+0.04(114)+0.21 ( 42)+0.21(371)+0.10(342 ) + 15 & 3s3p@xmath7 & @xmath8p@xmath59 & 0.07 ( 28)+0.85 ( 15 ) + 16 & 3s3p@xmath7 & @xmath27p@xmath61 & 0.12 ( 5)+0.05 ( 11)+0.11 ( 43)+0.24 ( 86)+0.36 ( 16 ) + 17 & 3s@xmath53p@xmath5(@xmath27s)3d@xmath5(@xmath8f ) & @xmath8f@xmath65 & 0.23 ( 57)+0.16(221)+0.05(262)+0.08 ( 87)+0.10(139)+0.05(282)+0.30 ( 17 ) + 18 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8f@xmath59 & 0.29 ( 3)+0.10 ( 41)+0.40 ( 18)+0.13 ( 8)+0.04 ( 26 ) + 19 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath60 & 0.53 ( 19)+0.21 ( 4)+0.24 ( 37 ) + 20 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8d@xmath61 & 0.32 ( 5)+0.05 ( 35)+0.40 ( 20)+0.07 ( 11)+0.06 ( 86)+0.05 ( 16 ) + + @rlllrrrr@ index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + + index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + 1 & 3s@xmath53p@xmath8 & @xmath5d@xmath66 & 00.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 + 2 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath67 & 16.8821 & 16.9403 & 16.9357 & 16.8671 & 16.8529 + 3 & 3s@xmath53p@xmath5(@xmath27s)3d & @xmath5d@xmath68 & 23.903 & 23.8688 & 23.8613 & 23.8040 & 23.7839 + 4 & 3s@xmath53p@xmath8 & @xmath6s@xmath66 & 25.060 & 25.0648 & 25.0619 & 25.0905 & 25.0865 + 5 & 3s@xmath53p@xmath8 & @xmath5d@xmath69 & 25.9556 & 25.9490 & 25.9411 & 25.9727 & 25.9639 + 6 & 3s@xmath53p@xmath8 & @xmath5p@xmath70 & 27.019 & 27.0845 & 27.0831 & 27.1065 & 27.1049 + 7 & 3s3p@xmath6 & @xmath6p@xmath68 & 37.9315 & 37.9970 & 38.0129 & 37.9087 & 37.9134 + 8 & 3s3p@xmath6 & @xmath5p@xmath67 & 40.242 & 40.1880 & 40.1973 & 40.1132 & 40.1082 + 9 & 3s3p@xmath6 & @xmath5s@xmath71 & 40.205 & 40.3454 & 40.3636 & 40.2582 & 40.2632 + 10 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath68 & 42.01 & 41.8139 & 41.8076 & 41.7643 & 41.7479 + 11 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6d@xmath71 & 42.16 & 42.0470 & 42.0473 & 41.9973 & 41.9888 + 12 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6d@xmath67 & 42.24 & 42.0827 & 42.0820 & 42.0263 & 42.0159 + 13 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5g@xmath72 & 42.97 & 42.7366 & 42.7255 & 42.6856 & 42.6644 + 14 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5d@xmath68 & 44.848 & 44.9721 & 44.9564 & 44.9235 & 44.8978 + 15 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5p@xmath71 & 45.51 & 45.7510 & 45.7334 & 45.6952 & 45.6683 + 16 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5d@xmath67 & 45.6572 & 45.8196 & 45.8091 & 45.7563 & 45.7341 + 17 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath72 & 47.96 & 47.7759 & 47.7717 & 47.7352 & 47.7188 + 18 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5p@xmath67 & 48.90 & 48.7073 & 48.7034 & 48.6656 & 48.6501 + 19 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5g@xmath73 & 48.98 & 48.7536 & 48.7404 & 48.7119 & 48.6861 + 20 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5f@xmath68 & 49.19 & 48.9732 & 48.9665 & 48.9329 & 48.9141 + 21 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6p@xmath68 & 50.74 & 50.5135 & 50.4973 & 50.4698 & 50.4412 + 22 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5f@xmath72 & 50.87 & 50.5992 & 50.5800 & 50.5583 & 50.5262 + 23 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5d@xmath67 & 51.38 & 51.2398 & 51.2202 & 51.1970 & 51.1682 + 24 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5s@xmath71 & 51.67 & 51.5331 & 51.5132 & 51.4871 & 51.4582 + 25 & 3s@xmath53p@xmath8 & @xmath5p@xmath66 & 52.18 & 52.2859 & 52.2799 & 52.3345 & 52.3257
+ + nist : http://www.nist.gov/pml/data/asd.cfm + grasp1 : present results with the grasp code from 15 configurations and 1313 levels + grasp2 : present results with the grasp code from 35 configurations and 3533 levels + fac1 : present results with the fac code from 1313 levels + fac2 : present results with the fac code from 14 608 levels + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 3s@xmath53p@xmath8 & @xmath5d@xmath66 & 0.25 ( 4)+0.27 ( 1)+0.48 ( 25 ) + 2 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath67 & 0.34 ( 2)+0.12 ( 12)+0.10 ( 23)+0.11 ( 18)+0.31 ( 60 ) + 3 & 3s@xmath53p@xmath5(@xmath27s)3d & @xmath5d@xmath68 & 0.17 ( 10)+0.20 ( 57)+0.14 ( 21)+0.15 ( 20)+0.30 ( 3 ) + 4 & 3s@xmath53p@xmath8 & @xmath6s@xmath66 & 0.55 ( 4)+0.45 ( 1 ) + 5 & 3s@xmath53p@xmath8 & @xmath5d@xmath69 & 1.00 ( 5 ) + 6 & 3s@xmath53p@xmath8 & @xmath5p@xmath70 & 0.98 ( 6 ) + 7 & 3s3p@xmath6 & @xmath6p@xmath68 & 0.66 ( 7)+0.27 ( 45 ) + 8 & 3s3p@xmath6 & @xmath5p@xmath67 & 0.08 ( 2)+0.06 ( 58)+0.11 ( 18)+0.06 ( 16)+0.11 ( 41)+0.32 ( 8)+0.24 ( 51 ) + 9 & 3s3p@xmath6 & @xmath5s@xmath71 & 0.05 ( 59)+0.07 ( 24)+0.24(137)+0.07 ( 48)+0.53 ( 9 ) + 10 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath68 & 0.46 ( 10)+0.16 ( 20)+0.29 ( 74 ) + 11 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6d@xmath71 & 0.79 ( 11)+0.04 ( 59)+0.14 ( 70 ) + 12 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6d@xmath67 & 0.32 ( 2)+0.28 ( 12)+0.12 ( 18)+0.12 ( 16)+0.05 ( 73)+0.04 ( 51 ) + 13 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5g@xmath72 & 0.18 ( 17)+0.14 ( 56)+0.53 ( 13)+0.12 ( 22 ) + 14 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5d@xmath68 & 0.10 ( 10)+0.10 ( 57)+0.12 ( 21)+0.30 ( 14)+0.24 ( 74)+0.06 ( 72 ) + 15 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5p@xmath71 & 0.30 ( 59)+0.35 ( 15)+0.18 ( 24)+0.08 ( 9 ) + 16 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5d@xmath67 & 0.22 ( 58)+0.08 ( 18)+0.26 ( 16)+0.14 ( 73)+0.13 ( 8)+0.07 ( 51 ) + 17 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath72 & 0.37 ( 17)+0.48 ( 67)+0.07 ( 56)+0.07 ( 22 ) + 18 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5p@xmath67 & 0.16 ( 12)+0.23 ( 58)+0.05 ( 23)+0.25 ( 18)+0.06 ( 16)+0.20 ( 73 ) + 19 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5g@xmath73 & 0.37 ( 69)+0.62 ( 19 ) + 20 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5f@xmath68 & 0.04 ( 10)+0.22 ( 57)+0.07 ( 21)+0.32 ( 20)+0.04 ( 74)+0.25 ( 72 ) + 21 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6p@xmath68 & 0.35 ( 21)+0.10 ( 20)+0.14 ( 14)+0.08 ( 74)+0.27 ( 72 ) + 22 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5f@xmath72 & 0.17 ( 17)+0.05 ( 67)+0.18 ( 56)+0.15 ( 13)+0.44 ( 22 ) + 23 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5d@xmath67 & 0.05 ( 2)+0.11 ( 12)+0.46 ( 23)+0.14 ( 16)+0.22 ( 73 ) + 24 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5s@xmath71 & 0.31 ( 70)+0.23 ( 15)+0.36 ( 24 ) + 25 & 3s@xmath53p@xmath8 & @xmath5p@xmath66 & 0.19 ( 4)+0.28 ( 1)+0.50 ( 25 ) + + @rlllrrrr@ index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + + index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + 1 & 3s@xmath53p@xmath5 & @xmath8p@xmath58 & 00.0000 & 00.0000 & 0.0000 & 0.0000 & 0.0000 + 2 & 3s@xmath53p@xmath5 & @xmath8p@xmath64 & 25.5337 & 25.5392 & 25.5395 & 25.5650 & 25.5675 + 3 & 3s@xmath53p@xmath5 & @xmath27d@xmath57 & 26.2946 & 26.2402 & 26.2351 & 26.2574 & 26.2587 + 4 & 3s3p@xmath8 & @xmath7s@xmath59 & 38.2094 & 38.0581 & 38.0606 & 37.9699 & 37.9709 + 5 & 3s3p@xmath8 & @xmath8d@xmath61 & 39.9800 & 40.1129 & 40.1071 & 40.0211 & 40.0210 + 6 & 3s@xmath53p3d & @xmath8f@xmath59 & & 41.7903 & 41.7744 & 41.7215 & 41.7222 + 7 & 3s@xmath53p3d & @xmath8d@xmath61 & & 45.1086 & 45.0826 & 45.0158 & 45.0129 + 8 & 3s@xmath53p3d & @xmath8p@xmath59 & 49.57 & 49.1925 & 49.1738 & 49.1246 & 49.1247 + 9 & 3s@xmath53p3d & @xmath8f@xmath62 & 49.65 & 49.2768 & 49.2540 & 49.2059 & 49.2052 + 10 & 3s@xmath53p@xmath5 & @xmath8p@xmath57 & 52.27 & 52.3039 & 52.2993 & 52.3463 & 52.3487 + 11 & 3s@xmath53p@xmath5 & @xmath27s@xmath58 & 53.71 & 53.8224 & 53.8202 & 53.8658 & 53.8642 + 12 & 3s3p@xmath5(@xmath6p)3d & @xmath7f@xmath64 & & 54.6247 & 54.6092 & 54.4384 & 54.4378 + 13 & 3s3p@xmath5(@xmath6p)3d & @xmath8p@xmath57 & & 55.1766 & 55.1588 & 54.9863 & 54.9850 + 14 & 3s3p@xmath5(@xmath6p)3d & @xmath7p@xmath65 & & 61.6560 & 61.6353 & 61.4690 & 61.4679 + 15 & 3s3p@xmath5(@xmath6p)3d & @xmath8f@xmath57 & & 62.7077 & 62.6817 & 62.5113 & 62.5082 + 16 & 3s3p@xmath8 & @xmath8d@xmath59 & 63.01 & 62.9640 & 62.9654 & 62.9020 & 62.9054 + 17 & 3s3p@xmath8 & @xmath8d@xmath62 & 64.46 & 64.3393 & 64.3357 & 64.2689 & 64.2712 + 18 & 3s3p@xmath8 & @xmath8p@xmath60 & 65.29 & 65.1798 & 65.1832 & 65.1238 & 65.1237 + 19 & 3s3p@xmath8 & @xmath8p@xmath61 & 66.52 & 66.2880 & 66.2870 & 66.2268 & 66.2275 + 20 & 3s3p@xmath8 & @xmath27d@xmath59 & 66.51 & 66.2776 & 66.2699 & 66.2141 & 66.2160 + 21 & 3s3p@xmath8 & @xmath8s@xmath61 & 67.24 & 67.1867 & 67.1794 & 67.1047 & 67.1036
+ + nist : http://www.nist.gov/pml/data/asd.cfm + grasp1 : present results with the grasp code from 12 configurations and 518 levels + grasp2 : present results with the grasp code from 48 configurations and 4364 levels + fac1 : present results with the fac code from 9798 levels + fac2 : present results with the fac code from 27 122 levels + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 3s@xmath53p@xmath5 & @xmath8p@xmath58 & 0.69 ( 1)+0.31 ( 11 ) + 2 & 3s@xmath53p@xmath5 & @xmath8p@xmath64 & 1.00 ( 2 ) + 3 & 3s@xmath53p@xmath5 & @xmath27d@xmath57 & 0.35 ( 10)+0.64 ( 3 ) + 4 & 3s3p@xmath8 & @xmath7s@xmath59 & 0.27 ( 4)+0.16 ( 16)+0.08 ( 20)+0.46 ( 64 ) + 5 & 3s3p@xmath8 & @xmath8d@xmath61 & 0.14 ( 21)+0.28 ( 5)+0.16 ( 19)+0.24 ( 69)+0.09 ( 7)+0.05 ( 29 ) + 6 & 3s@xmath53p3d & @xmath8f@xmath59 & 0.74 ( 6)+0.19 ( 27 ) + 7 & 3s@xmath53p3d & @xmath8d@xmath61 & 0.04 ( 21)+0.04 ( 5)+0.05 ( 69)+0.44 ( 7)+0.14 ( 24)+0.25 ( 29 ) + 8 & 3s@xmath53p3d & @xmath8p@xmath59 & 0.34 ( 23)+0.45 ( 8)+0.14 ( 27 ) + 9 & 3s@xmath53p3d & @xmath8f@xmath62 & 0.50 ( 9)+0.22 ( 28)+0.26 ( 22 ) + 10 & 3s@xmath53p@xmath5 & @xmath8p@xmath57 & 0.64 ( 10)+0.35 ( 3 ) + 11 & 3s@xmath53p@xmath5 & @xmath27s@xmath58 & 0.30 ( 1)+0.67 ( 11 ) + 12 & 3s3p@xmath5(@xmath6p)3d & @xmath7f@xmath64 & 0.38 ( 12)+0.07 ( 32)+0.05 ( 57)+0.08 ( 42)+0.05(137)+0.30(110 ) + 13 & 3s3p@xmath5(@xmath6p)3d & @xmath8p@xmath57 & 0.18 ( 30)+0.11(105)+0.08(134)+0.12 ( 13)+0.12 ( 45)+0.21(113)+0.09(135 ) + 14 & 3s3p@xmath5(@xmath6p)3d & @xmath7p@xmath65 & 0.13 ( 33)+0.19(106)+0.15 ( 14)+0.04 ( 49)+0.08 ( 60)+0.05(116)+0.30(132 ) + 15 & 3s3p@xmath5(@xmath6p)3d & @xmath8f@xmath57 & 0.09 ( 30)+0.05(105)+0.25 ( 15)+0.07(134)+0.08(119)+0.07 ( 63)+0.09(113)+0.21(135 ) + 16 & 3s3p@xmath8 & @xmath8d@xmath59 & 0.45 ( 4)+0.49 ( 16 ) + 17 & 3s3p@xmath8 & @xmath8d@xmath62 & 0.94 ( 17 ) + 18 & 3s3p@xmath8 & @xmath8p@xmath60 & 0.86 ( 18)+0.14 ( 25 ) + 19 & 3s3p@xmath8 & @xmath8p@xmath61 & 0.18 ( 21)+0.20 ( 5)+0.34 ( 19)+0.12 ( 69)+0.04 ( 7)+0.12 ( 24 ) + 20 & 3s3p@xmath8 & @xmath27d@xmath59 & 0.08 ( 4)+0.08 ( 16)+0.55 ( 20)+0.07 ( 6)+0.12 ( 23)+0.08 ( 27 ) + 21 & 3s3p@xmath8 & @xmath8s@xmath61 & 0.32 ( 21)+0.20 ( 5)+0.26 ( 19)+0.18 ( 69 ) + ... + 64 & 3s3p@xmath8 & @xmath8p@xmath59 & 0.17 ( 4)+0.16 ( 16)+0.14 ( 20)+0.50 ( 64 ) + 69 & 3s3p@xmath8 & @xmath27p@xmath61 & 0.23 ( 21)+0.23 ( 5)+0.14 ( 19)+0.35 ( 69 ) + + @rlllrrrrr@ index & configuration & level & nist & grasp2 & grasp3 & fac1 & fac2 & rmbpt + + + + index & configuration & level & nist & grasp2 & grasp3 & fac1 & fac2 & rmbpt + + + 1 & 3s@xmath53p & @xmath5p@xmath70 & 00.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 + 2 & 3s3p@xmath5 & @xmath6p@xmath71 & 12.3076 & 12.4425 & 12.4422 & 12.3220 & 12.3204 & 12.3018 + 3 & 3s@xmath53p & @xmath5p@xmath66 & 26.7311 & 26.7061 & 26.7060 & 26.7306 & 26.7314 & 26.7056 + 4 & 3s3p@xmath5 & @xmath6p@xmath67 & 36.7742 & 36.8823 & 36.8835 & 36.7935 & 36.7959 & 36.7790 + 5 & 3s3p@xmath5 & @xmath5d@xmath68 & 38.109 & 38.2380 & 38.2375 & 38.1447 & 38.1457 & 38.1110 + 6 & 3s3p@xmath5 & @xmath5d@xmath67 & 39.6875 & 39.7869 & 39.7857 & 39.7028 & 39.7033 & 39.6637 + 7 & 3s3p@xmath5 & @xmath5p@xmath71 & 40.4238 & 40.5824 & 40.5806 & 40.4826 & 40.4819 & 40.4024 + 8 & 3s@xmath53d & @xmath5d@xmath67 & 43.9039 & 44.0171 & 44.0103 & 43.9482 & 43.9448 & 43.8775 + 9 & 3s@xmath53d & @xmath5d@xmath68 & 49.263 & 49.3268 & 49.3215 & 49.2778 & 49.2759 & 49.2626 + 10 & 3p@xmath8 & @xmath5d@xmath66 & & 51.9171 & 51.9167 & 51.7504 & 51.7496 & 51.7270 + 11 & 3s3p3d & @xmath6f@xmath66 & & 52.9289 & 52.9278 & 52.7221 & 52.7218 & 52.6746 + 12 & 3s3p3d & @xmath6f@xmath69 & & 53.7715 & 53.7680 & 53.5988 & 53.5989 & 53.5817 + 13 & 3s3p3d & @xmath6d@xmath70 & & 55.7589 & 55.7524 & 55.5868 & 55.5849 & 55.5323 + 14 & 3s3p3d & @xmath6d@xmath66 & & 56.2688 & 56.2632 & 56.0908 & 56.0883 & 56.0220 + 15 & 3s3p3d & @xmath6p@xmath69 & & 59.7857 & 59.7835 & 59.6243 & 59.6251 & 59.6240 + 16 & 3s3p3d & @xmath6f@xmath74 & & 60.8471 & 60.8418 & 60.6804 & 60.6797 & 60.6611 + 17 & 3s3p(@xmath8p)3d & @xmath5f@xmath69 & & 61.6177 & 61.6112 & 61.4480 & 61.4457 & 61.4177 + 18 & 3s3p(@xmath8p)3d & @xmath5d@xmath66 & & 62.0001 & 61.9931 & 61.8237 & 61.8206 & 61.7773 + 19 & 3s3p@xmath5 & @xmath6p@xmath68 & 64.372 & 64.4986 & 64.4980 & 64.4306 & 64.4321 & 64.3709 + 20 & 3s3p@xmath5 & @xmath5s@xmath71 & 67.115 & 67.2861 & 67.2849 & 67.2135 & 67.2094 & 67.1156 + 21 & 3p@xmath5(@xmath8p)3d & @xmath6f@xmath67 & 67.479 & 67.8062 & 67.8002 & 67.3757 & 67.3714 & 67.4809
+ + nist : http://www.nist.gov/pml/data/asd.cfm + grasp2 : present results with the grasp code from 33 configurations and 928 levels + grasp3 : present results with the grasp code from 63 configurations and 2003 levels + fac1 : present results with the fac code from 2003 levels + fac2 : present results with the fac code from + rmbpt : earlier results of safronova and safronova @xcite + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 3s@xmath53p & @xmath5p@xmath70 & 1.00 ( 1 ) + 2 & 3s3p@xmath5 & @xmath6p@xmath71 & 0.52 ( 2)+0.17 ( 7)+0.30 ( 20 ) + 3 & 3s@xmath53p & @xmath5p@xmath66 & 0.98 ( 3 ) + 4 & 3s3p@xmath5 & @xmath6p@xmath67 & 0.88 ( 4)+0.08 ( 6 ) + 5 & 3s3p@xmath5 & @xmath5d@xmath68 & 0.37 ( 19)+0.61 ( 5 ) + 6 & 3s3p@xmath5 & @xmath5d@xmath67 & 0.28 ( 22)+0.49 ( 6)+0.18 ( 8) + 7 & 3s3p@xmath5 & @xmath5p@xmath71 & 0.29 ( 2)+0.69 ( 7 ) + 8 & 3s@xmath53d & @xmath5d@xmath67 & 0.08 ( 22)+0.12 ( 6)+0.79 ( 8) + 9 & 3s@xmath53d & @xmath5d@xmath68 & 0.96 ( 9 ) + 10 & 3p@xmath8 & @xmath5d@xmath66 & 0.23 ( 11)+0.11 ( 18)+0.06 ( 37)+0.12 ( 25)+0.16 ( 10)+0.26 ( 68 ) + 11 & 3s3p3d & @xmath6f@xmath66 & 0.55 ( 11)+0.10 ( 31)+0.07 ( 25)+0.08 ( 10)+0.14 ( 68 ) + 12 & 3s3p3d & @xmath6f@xmath69 & 0.42 ( 12)+0.07 ( 17)+0.18 ( 35)+0.24 ( 32 ) + 13 & 3s3p3d & @xmath6d@xmath70 & 0.58 ( 13)+0.06 ( 28)+0.10 ( 39)+0.26 ( 33 ) + 14 & 3s3p3d & @xmath6d@xmath66 & 0.32 ( 14)+0.18 ( 27)+0.20 ( 37)+0.13 ( 31)+0.05 ( 40)+0.04 ( 68 ) + 15 & 3s3p3d & @xmath6p@xmath69 & 0.11 ( 12)+0.31 ( 29)+0.38 ( 15)+0.07 ( 17)+0.08 ( 35 ) + 16 & 3s3p3d & @xmath6f@xmath74 & 0.42 ( 16)+0.23 ( 30)+0.08 ( 36)+0.26 ( 38 ) + 17 & 3s3p(@xmath8p)3d & @xmath5f@xmath69 & 0.13 ( 12)+0.12 ( 15)+0.46 ( 17)+0.05 ( 32)+0.21 ( 41 ) + 18 & 3s3p(@xmath8p)3d & @xmath5d@xmath66 & 0.11 ( 14)+0.05 ( 27)+0.35 ( 18)+0.16 ( 37)+0.20 ( 40 ) + 19 & 3s3p@xmath5 & @xmath6p@xmath68 & 0.61 ( 19)+0.38 ( 5 ) + 20 & 3s3p@xmath5 & @xmath5s@xmath71 & 0.18 ( 2)+0.14 ( 7)+0.67 ( 20 ) + 21 & 3p@xmath5(@xmath8p)3d & @xmath6f@xmath67 & 0.06 ( 22)+0.04 ( 6)+0.30 ( 21)+0.10 ( 43)+0.08 ( 59)+0.10 ( 99)+0.27 ( 93 ) + ... + 68 & 3p@xmath8 & @xmath5p@xmath66 & 0.19 ( 25)+0.28 ( 10)+0.50 ( 68 ) + + @rlllrrr@ index & configuration & level & nist & grasp & fac & rmbpt + + + + index & configuration & level & nist & grasp & fac & rmbpt + + + 1 & 3s@xmath5 & @xmath27s@xmath58 & 00.0000 & 0.0000 & 0.0000 & 0.0000 + 2 & 3s3p & @xmath8p@xmath60 & 10.261 & 10.3595 & 10.2414 & 10.2650 + 3 & 3s3p & @xmath8p@xmath61 & 11.4036 & 11.5247 & 11.4028 & 11.4104 + 4 & 3p@xmath5 & @xmath8p@xmath58 & & 24.7520 & 24.5032 & 24.4911 + 5 & 3s3p & @xmath8p@xmath59 & 37.398 & 37.4521 & 37.3609 & 37.3992 + 6 & 3s3p & @xmath27p@xmath61 & 40.0821 & 40.2273 & 40.1296 & 40.1225 + 7 & 3p@xmath5 & @xmath27d@xmath57 & & 50.6187 & 50.4140 & 50.4418 + 8 & 3p@xmath5 & @xmath8p@xmath64 & & 50.7934 & 50.5757 & 50.5885 + 9 & 3s3d & @xmath8d@xmath64 & 53.100 & 53.2554 & 53.0898 & 53.0968 + 10 & 3s3d & @xmath8d@xmath57 & 54.0418 & 54.2279 & 54.0506 & 54.0421 + 11 & 3s3d & @xmath8d@xmath65 & 59.214 & 59.3590 & 59.1988 & 59.2129 + 12 & 3s3d & @xmath27d@xmath57 & 60.490 & 60.6497 & 60.4812 & 60.4926 + 13 & 3p3d & @xmath8f@xmath59 & & 64.4267 & 64.1328 & 64.1616 + 14 & 3p3d & @xmath8d@xmath61 & & 67.0032 & 66.7097 & 66.6958 + 15 & 3p3d & @xmath8p@xmath59 & & 71.9267 & 71.6393 & 71.6688 + 16 & 3p3d & @xmath8f@xmath62 & & 72.1483 & 71.8606 & 71.8875 + 17 & 3p@xmath5 & @xmath8p@xmath57 & & 78.4319 & 78.2413 & 78.2584 + 18 & 3p@xmath5 & @xmath27s@xmath58 & & 79.8512 & 79.6567 & 79.6637 + 19 & 3p3d & @xmath8d@xmath59 & & 92.4948 & 92.2324 & 92.2537 + 20 & 3p3d & @xmath8p@xmath60 & & 93.2392 & 92.9761 & 92.9927 + 21 & 3p3d & @xmath8p@xmath61 & & 93.2818 & 93.0183 & 93.0311 + 22 & 3p3d & @xmath27f@xmath62 & & 93.2541 & 92.9898 & 92.9946 + 23 & 3p3d & @xmath8f@xmath63 & & 97.7310 & 97.4746 & 97.5357 + 24 & 3p3d & @xmath27d@xmath59 & & 98.6141 & 98.3571 & 98.4030 + 25 & 3p3d & @xmath8d@xmath62 & & 100.0066 & 99.7469 & 99.7687 + 26 & 3p3d & @xmath27p@xmath61 & & 100.9429 & 100.6812 & 100.6989 + 27 & 3d@xmath5 & @xmath8f@xmath57 & & 107.5330 & 107.1956 & 107.2079 + 28 & 3d@xmath5 & @xmath8p@xmath58 & & 109.5116 & 109.1710 & 109.1670 + 29 & 3d@xmath5 & @xmath8f@xmath65 & & 113.3167 & 112.9856 & 113.0263 + 30 & 3d@xmath5 & @xmath8p@xmath57 & & 114.2228 & 113.8902 & 113.9201 + 31 & 3d@xmath5 & @xmath27g@xmath75 & & 114.3842 & 114.0519 & 114.0755 + 32 & 3d@xmath5 & @xmath8p@xmath64 & & 114.6216 & 114.2887 & 114.3189 + 33 & 3d@xmath5 & @xmath8f@xmath75 & & 119.7747 & 119.4490 & 119.5071 + 34 & 3d@xmath5 & @xmath27d@xmath57 & & 120.5499 & 120.2230 & 120.2754 + 35 & 3d@xmath5 & @xmath27s@xmath58 & & 122.6951 & 122.3631 & 122.3938
+ + nist : http://www.nist.gov/pml/data/asd.cfm + grasp : present results with the grasp code from 58 configurations and 509 levels + fac : present results with the fac code from 991 levels + rmbpt : earlier results of safronova and safronova @xcite + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 3s@xmath5 & @xmath27s@xmath58 & 1.00 ( 1 ) + 2 & 3s3p & @xmath8p@xmath60 & 1.00 ( 2 ) + 3 & 3s3p & @xmath8p@xmath61 & 0.72 ( 3)+0.27 ( 6 ) + 4 & 3p@xmath5 & @xmath8p@xmath58 & 0.69 ( 4)+0.30 ( 18 ) + 5 & 3s3p & @xmath8p@xmath59 & 1.00 ( 5 ) + 6 & 3s3p & @xmath27p@xmath61 & 0.27 ( 3)+0.72 ( 6 ) + 7 & 3p@xmath5 & @xmath27d@xmath57 & 0.28 ( 17)+0.56 ( 7)+0.05 ( 10)+0.12 ( 12 ) + 8 & 3p@xmath5 & @xmath8p@xmath64 & 1.00 ( 8) + 9 & 3s3d & @xmath8d@xmath64 & 1.00 ( 9 ) + 10 & 3s3d & @xmath8d@xmath57 & 0.06 ( 17)+0.07 ( 7)+0.69 ( 10)+0.18 ( 12 ) + 11 & 3s3d & @xmath8d@xmath65 & 1.00 ( 11 ) + 12 & 3s3d & @xmath27d@xmath57 & 0.26 ( 10)+0.69 ( 12 ) + 13 & 3p3d & @xmath8f@xmath59 & 0.76 ( 13)+0.20 ( 24 ) + 14 & 3p3d & @xmath8d@xmath61 & 0.53 ( 14)+0.18 ( 21)+0.29 ( 26 ) + 15 & 3p3d & @xmath8p@xmath59 & 0.35 ( 19)+0.48 ( 15)+0.14 ( 24 ) + 16 & 3p3d & @xmath8f@xmath62 & 0.52 ( 16)+0.22 ( 25)+0.26 ( 22 ) + 17 & 3p@xmath5 & @xmath8p@xmath57 & 0.64 ( 17)+0.36 ( 7 ) + 18 & 3p@xmath5 & @xmath27s@xmath58 & 0.31 ( 4)+0.69 ( 18 ) + 19 & 3p3d & @xmath8d@xmath59 & 0.19 ( 13)+0.52 ( 19)+0.04 ( 15)+0.25 ( 24 ) + 20 & 3p3d & @xmath8p@xmath60 & 1.00 ( 20 ) + 21 & 3p3d & @xmath8p@xmath61 & 0.36 ( 14)+0.59 ( 21)+0.04 ( 26 ) + 22 & 3p3d & @xmath27f@xmath62 & 0.48 ( 16)+0.18 ( 25)+0.35 ( 22 ) + 23 & 3p3d & @xmath8f@xmath63 & 1.00 ( 23 ) + 24 & 3p3d & @xmath27d@xmath59 & 0.11 ( 19)+0.46 ( 15)+0.40 ( 24 ) + 25 & 3p3d & @xmath8d@xmath62 & 0.61 ( 25)+0.38 ( 22 ) + 26 & 3p3d & @xmath27p@xmath61 & 0.11 ( 14)+0.23 ( 21)+0.66 ( 26 ) + 27 & 3d@xmath5 & @xmath8f@xmath57 & 0.74 ( 27)+0.23 ( 34 ) + 28 & 3d@xmath5 & @xmath8p@xmath58 & 0.71 ( 28)+0.30 ( 35 ) + 29 & 3d@xmath5 & @xmath8f@xmath65 & 1.00 ( 29 ) + 30 & 3d@xmath5 & @xmath8p@xmath57 & 0.21 ( 27)+0.49 ( 30)+0.30 ( 34 ) + 31 & 3d@xmath5 & @xmath27g@xmath75 & 0.29 ( 33)+0.71 ( 31 ) + 32 & 3d@xmath5 & @xmath8p@xmath64 & 1.00 ( 32 ) + 33 & 3d@xmath5 & @xmath8f@xmath75 & 0.71 ( 33)+0.29 ( 31 ) + 34 & 3d@xmath5 & @xmath27d@xmath57 & 0.06 ( 27)+0.48 ( 30)+0.46 ( 34 ) + 35 & 3d@xmath5 & @xmath27s@xmath58 & 0.30 ( 28)+0.69 ( 35 ) + + @rlllrrr@ index & configuration & level & nist & grasp & fac & @xmath54 ( s ) + + + + index & configuration & level & nist & grasp & fac & @xmath54 ( s ) + + + 1 & 2p@xmath93s & @xmath5s@xmath76 & 000.0 & 0.0000 & 0.0000 & ........ + 2 & 2p@xmath93p & @xmath5p@xmath23 & 011.7280 & 11.8989 & 11.7457 & 2.218@xmath7711 + 3 & 2p@xmath93p & @xmath5p@xmath21 & 039.1890 & 39.3365 & 39.2218 & 5.664@xmath7713 + 4 & 2p@xmath93d & @xmath5d@xmath78 & 052.9692 & 53.1127 & 52.9352 & 6.986@xmath7713 + 5 & 2p@xmath93d & @xmath5d@xmath79 & 059.2105 & 59.3372 & 59.1730 & 4.987@xmath7712 + 6 & 2p@xmath94s & @xmath5s@xmath76 & 239.12 & 239.0661 & 238.9973 & 1.501@xmath7714 + 7 & 2p@xmath94p & @xmath5p@xmath23 & 243.92 & 243.9788 & 243.8505 & 1.267@xmath7714 + 8 & 2p@xmath94p & @xmath5p@xmath21 & 255.18 & 255.2154 & 255.0981 & 2.010@xmath7714 + 9 & 2p@xmath94d & @xmath5d@xmath78 & 260.37 & 260.4510 & 260.3002 & 8.821@xmath7715 + 10 & 2p@xmath94d & @xmath5d@xmath79 & 263.09 & 263.1426 & 262.9954 & 8.466@xmath7715 + 11 & 2p@xmath94f & @xmath5f@xmath22 & 265.94 & 265.8618 & 265.7361 & 4.087@xmath7715 + 12 & 2p@xmath94f & @xmath5f@xmath80 & 267.12 & 267.0446 & 266.9176 & 4.198@xmath7715 + 13 & 2p@xmath95s & @xmath5s@xmath76 & & 345.5593 & 345.3305 & 1.888@xmath7714 + 14 & 2p@xmath95p & @xmath5p@xmath23 & & 348.0234 & 347.7664 & 1.600@xmath7714 + 15 & 2p@xmath95p & @xmath5p@xmath21 & & 353.6728 & 353.4209 & 2.398@xmath7714 + 16 & 2p@xmath95d & @xmath5d@xmath78 & & 356.2383 & 355.9695 & 1.189@xmath7714 + 17 & 2p@xmath95d & @xmath5d@xmath79 & 357.54 & 357.6240 & 357.3573 & 1.168@xmath7714 + 18 & 2p@xmath95f & @xmath5f@xmath22 & 358.84 & 358.9640 & 358.7180 & 7.736@xmath7715 + 19 & 2p@xmath95f & @xmath5f@xmath80 & 359.46 & 359.5722 & 359.3256 & 7.962@xmath7715 + 20 & 2p@xmath95 g & @xmath5g@xmath81 & 359.77 & 359.7585 & 359.5057 & 1.361@xmath7714 + 21 & 2p@xmath95 g & @xmath5g@xmath82 & 360.11 & 360.1191 & 359.8662 & 1.378@xmath7714 + 22 & 2p@xmath96s & @xmath5s@xmath76 & & 401.9007 & 401.5572 & 2.653@xmath7714 + 23 & 2p@xmath96p & @xmath5p@xmath23 & & 403.3052 & 402.9603 & 2.262@xmath7714 + 24 & 2p@xmath96p & @xmath5p@xmath21 & & 406.5339 & 406.2203 & 3.266@xmath7714 + 25 & 2p@xmath96d & @xmath5d@xmath78 & & 407.9823 & 407.6849 & 1.742@xmath7714 + 26 & 2p@xmath96d & @xmath5d@xmath79 & & 408.7855 & 408.5064 & 1.735@xmath7714 + 27 & 2p@xmath96f & @xmath5f@xmath22 & & 409.5470 & 409.2872 & 1.310@xmath7714 + 28 & 2p@xmath96f & @xmath5f@xmath80 & & 409.8998 & 409.6444 & 1.351@xmath7714 + 29 & 2p@xmath96 g & @xmath5g@xmath81 & & 410.0204 & 409.7698 & 2.328@xmath7714 + 30 & 2p@xmath96 g & @xmath5g@xmath82 & & 410.2293 & 409.9788 & 2.357@xmath7714
+ + nist : http://www.nist.gov/pml/data/asd.cfm + grasp : present results with the grasp code from 50 configurations and 1235 levels + fac : present results with the fac code from 1592 levels + @rllrrrr@ index@xmath83 & configuration & level & nist & grasp & fac & mrmp + + + + index@xmath83 & configuration & level & nist & grasp & fac & mrmp + + + 1 & 2s@xmath52p@xmath9 & @xmath27s@xmath58 & 0.000 & 0.0000 & 0.0000 & 0.0000 + 3 & 2s@xmath52p@xmath73s & @xmath27p@xmath61 & 610.640 & 610.2292 & 610.1423 & 610.5354 + 9 & 2s@xmath52p@xmath73p & @xmath27s@xmath58 & 653.859 & 653.7288 & 653.5037 & 653.7409 + 11 & 2s@xmath52p@xmath73d & @xmath8p@xmath61 & 661.507 & 660.9754 & 660.7169 & 661.1325 + 17 & 2s@xmath52p@xmath73d & @xmath27p@xmath61 & 670.246 & 670.5722 & 670.2893 & 670.6958 + 19 & 2s@xmath52p@xmath73s & @xmath8p@xmath61 & 711.936 & 711.7088 & 711.6628 & 712.0517 + 21 & 2s@xmath52p@xmath73p & @xmath8p@xmath58 & 726.088 & 725.8494 & 725.6751 & 725.9370 + 27 & 2s2p@xmath93p & @xmath8p@xmath61 & 758.302 & 758.6381 & 758.2086 & 758.3025 + 29 & 2s@xmath52p@xmath73d & @xmath8d@xmath61 & 765.027 & 764.8414 & 764.5743 & 764.9308 + 33 & 2s2p@xmath93p & @xmath27p@xmath61 & 786.651 & 787.2457 & 786.8073 & 786.8504 + + @xmath84 : see table 7 for definition of all levels + nist : http://www.nist.gov/pml/data/asd.cfm + grasp : present results with the grasp code from 25 configurations and 157 levels + fac : present results with the fac code from 1147 levels + mrmp : earlier calculations of vilkas et al
. @xcite @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 2s@xmath52p@xmath9 & @xmath27s@xmath58 & 1.00 ( 1 ) + 2 & 2s@xmath52p@xmath73s & @xmath8p@xmath59 & 1.00 ( 2 ) + 3 & 2s@xmath52p@xmath73s & @xmath27p@xmath61 & 0.34 ( 19)+0.66 ( 3 ) + 4 & 2s@xmath52p@xmath73p & @xmath8p@xmath64 & 0.09 ( 20)+0.49 ( 4)+0.31 ( 25)+0.10 ( 6 ) + 5 & 2s@xmath52p@xmath73p & @xmath8d@xmath57 & 0.50 ( 5)+0.17 ( 8)+0.34 ( 24 ) + 6 & 2s@xmath52p@xmath73p & @xmath27p@xmath64 & 0.08 ( 20)+0.36 ( 25)+0.56 ( 6 ) + 7 & 2s@xmath52p@xmath73p & @xmath8d@xmath65 & 1.00 ( 7 ) + 8 & 2s@xmath52p@xmath73p & @xmath8p@xmath57 & 0.67 ( 8)+0.34 ( 24 ) + 9 & 2s@xmath52p@xmath73p & @xmath27s@xmath58 & 0.37 ( 21)+0.62 ( 9 ) + 10 & 2s@xmath52p@xmath73d & @xmath8p@xmath60 & 1.00 ( 10 ) + 11 & 2s@xmath52p@xmath73d & @xmath8p@xmath61 & 0.32 ( 29)+0.66 ( 11 ) + 12 & 2s@xmath52p@xmath73d & @xmath8f@xmath62 & 0.53 ( 12)+0.07 ( 16)+0.40 ( 31 ) + 13 & 2s@xmath52p@xmath73d & @xmath8d@xmath59 & 0.18 ( 28)+0.55 ( 13)+0.10 ( 30)+0.18 ( 15 ) + 14 & 2s@xmath52p@xmath73d & @xmath8f@xmath63 & 1.00 ( 14 ) + 15 & 2s@xmath52p@xmath73d & @xmath27d@xmath59 & 0.04 ( 28)+0.06 ( 13)+0.41 ( 30)+0.49 ( 15 ) + 16 & 2s@xmath52p@xmath73d & @xmath8d@xmath62 & 0.71 ( 16)+0.27 ( 31 ) + 17 & 2s@xmath52p@xmath73d & @xmath27p@xmath61 & 0.18 ( 29)+0.18 ( 11)+0.62 ( 17 ) + 18 & 2s@xmath52p@xmath73s & @xmath8p@xmath60 & 1.00 ( 18 ) + 19 & 2s@xmath52p@xmath73s & @xmath8p@xmath61 & 0.66 ( 19)+0.34 ( 3 ) + 20 & 2s@xmath52p@xmath73p & @xmath8d@xmath64 & 0.74 ( 20)+0.23 ( 6 ) + 21 & 2s@xmath52p@xmath73p & @xmath8p@xmath58 & 0.62 ( 21)+0.37 ( 9 ) + 22 & 2s2p@xmath93s & @xmath8s@xmath64 & 0.08 ( 4)+0.86 ( 22 ) + 23 & 2s2p@xmath93s & @xmath27s@xmath58 & 1.00 ( 23 ) + 24 & 2s@xmath52p@xmath73p & @xmath27d@xmath57 & 0.49 ( 5)+0.17 ( 8)+0.34 ( 24 ) + 25 & 2s@xmath52p@xmath73p & @xmath8s@xmath64 & 0.07 ( 20)+0.42 ( 4)+0.26 ( 25)+0.10 ( 6)+0.14 ( 22 ) + 26 & 2s2p@xmath93p & @xmath8p@xmath60 & 1.00 ( 26 ) + 27 & 2s2p@xmath93p & @xmath8p@xmath61 & 0.66 ( 27)+0.31 ( 33 ) + 28 & 2s@xmath52p@xmath73d & @xmath8f@xmath59 & 0.74 ( 28)+0.20 ( 15 ) + 29 & 2s@xmath52p@xmath73d & @xmath8d@xmath61 & 0.48 ( 29)+0.15 ( 11)+0.35 ( 17 ) + 30 & 2s@xmath52p@xmath73d & @xmath8p@xmath59 & 0.36 ( 13)+0.48 ( 30)+0.14 ( 15 ) + 31 & 2s@xmath52p@xmath73d & @xmath27f@xmath62 & 0.44 ( 12)+0.22 ( 16)+0.34 ( 31 ) + 32 & 2s2p@xmath93p & @xmath8p@xmath59 & 1.00 ( 32 ) + 33 & 2s2p@xmath93p & @xmath27p@xmath61 & 0.32 ( 27)+0.67 ( 33 ) + + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 2s@xmath52p@xmath7 & @xmath5p@xmath85 & 1.00 ( 1 ) + 2 & 2s@xmath52p@xmath7 & @xmath5p@xmath86 & 1.00 ( 2 ) + 3 & 2s2p@xmath9 & @xmath5s@xmath87 & 1.00 ( 3 ) + 4 & 2s@xmath52p@xmath63s & @xmath6p@xmath88 & 0.69 ( 4)+0.31 ( 28 ) + 5 & 2s@xmath52p@xmath63s & @xmath5p@xmath89 & 0.12 ( 26)+0.56 ( 5)+0.32 ( 29 ) + 6 & 2s@xmath52p@xmath63s & @xmath5s@xmath87 & 0.23 ( 86)+0.12 ( 27)+0.66 ( 6 ) + 7 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6p@xmath85 & 0.07 ( 31)+0.31 ( 7)+0.18 ( 92)+0.09 ( 13)+0.16 ( 43)+0.17 ( 33 ) + 8 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath5d@xmath90 & 0.27 ( 38)+0.12 ( 10)+0.29 ( 8)+0.25 ( 32)+0.07 ( 45 ) + 9 & 2s@xmath52p@xmath6(@xmath27s)3p & @xmath5p@xmath86 & 0.19 ( 87)+0.04 ( 30)+0.08 ( 40)+0.66 ( 9 ) + 10 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6p@xmath90 & 0.40 ( 10)+0.27 ( 8)+0.07 ( 32)+0.24 ( 45 ) + 11 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath5s@xmath86 & 0.09 ( 30)+0.18 ( 40)+0.37 ( 11)+0.32 ( 46 ) + 12 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6d@xmath91 & 0.67 ( 12)+0.32 ( 41 ) + 13 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath5p@xmath85 & 0.24 ( 92)+0.15 ( 39)+0.21 ( 13)+0.13 ( 43)+0.12 ( 33)+0.15 ( 14 ) + 14 & 2s@xmath52p@xmath6(@xmath27s)3p & @xmath5p@xmath85 & 0.13 ( 31)+0.12 ( 7)+0.16 ( 13)+0.04 ( 43)+0.50 ( 14 ) + 15 & 2s@xmath52p@xmath6(@xmath27d)3d & @xmath5p@xmath89 & 0.32 ( 50)+0.23 ( 58)+0.07 ( 62)+0.22 ( 55)+0.08 ( 15 ) + 16 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath6d@xmath88 & 0.13 ( 51)+0.35 ( 16)+0.06(106)+0.10 ( 23)+0.22 ( 54)+0.10 ( 60 ) + 17 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath6p@xmath87 & 0.12 ( 49)+0.50 ( 17)+0.05 ( 22)+0.22 ( 63)+0.10 ( 53 ) + 18 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath5f@xmath92 & 0.28 ( 56)+0.06 ( 20)+0.34 ( 18)+0.28 ( 52 ) + 19 & 2s@xmath52p@xmath6(@xmath27s)3d & @xmath5d@xmath89 & 0.17(101)+0.07 ( 58)+0.07 ( 62)+0.62 ( 19 ) + 20 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath6d@xmath92 & 0.06 ( 56)+0.41 ( 20)+0.22 ( 18)+0.28 ( 61 ) + 21 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath6f@xmath93 & 0.67 ( 21)+0.32 ( 59 ) + 22 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath5p@xmath87 & 0.05 ( 49)+0.12 ( 17)+0.50 ( 22)+0.10 ( 63)+0.22 ( 53 ) + 23 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath5d@xmath88 & 0.05 ( 16)+0.22(106)+0.14 ( 57)+0.26 ( 23)+0.09 ( 54)+0.21 ( 60 ) + 24 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath5p@xmath89 & 0.18 ( 58)+0.19 ( 62)+0.26 ( 24)+0.10 ( 55)+0.21 ( 15 ) + 25 & 2s@xmath52p@xmath6(@xmath27s)3d & @xmath5d@xmath88 & 0.10 ( 51)+0.11 ( 16)+0.04 ( 57)+0.09 ( 23)+0.61 ( 25 ) + 26 & 2s@xmath52p@xmath63s & @xmath6p@xmath89 & 0.86 ( 26)+0.12 ( 5 ) + 27 & 2s@xmath52p@xmath63s & @xmath5p@xmath87 & 0.34 ( 86)+0.67 ( 27 ) + 28 & 2s@xmath52p@xmath63s & @xmath5d@xmath88 & 0.31 ( 4)+0.67 ( 28 ) + 29 & 2s@xmath52p@xmath63s & @xmath5d@xmath89 & 0.32 ( 5)+0.66 ( 29 ) + 30 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6p@xmath86 & 0.27 ( 87)+0.52 ( 30)+0.21 ( 11 ) + 31 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6d@xmath85 & 0.56 ( 31)+0.14 ( 39)+0.13 ( 13)+0.07 ( 43)+0.07 ( 33 ) + 32 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5f@xmath90 & 0.12 ( 38)+0.05 ( 10)+0.14 ( 8)+0.53 ( 32)+0.15 ( 45 ) + 33 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5p@xmath85 & 0.12 ( 7)+0.13 ( 92)+0.18 ( 13)+0.25 ( 43)+0.29 ( 33 ) + 34 & 2s2p@xmath7(@xmath8p)3s & @xmath6p@xmath90 & 0.96 ( 34 ) + 35 & 2s2p@xmath7(@xmath8p)3s & @xmath5p@xmath85 & 0.27 ( 89)+0.67 ( 35 ) + 36 & 2s2p@xmath7(@xmath27p)3s & @xmath5p@xmath86 & 0.05 ( 30)+0.06 ( 11)+0.06 ( 46)+0.10 ( 88)+0.20 ( 90)+0.52 ( 36 ) + 37 & 2s2p@xmath7(@xmath27p)3s & @xmath5p@xmath85 & 0.04 ( 92)+0.16 ( 89)+0.22 ( 35)+0.52 ( 37 ) + 38 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6d@xmath90 & 0.55 ( 38)+0.28 ( 10)+0.14 ( 8) + 39 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath5d@xmath85 & 0.24 ( 7)+0.16 ( 92)+0.46 ( 39 ) + 40 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath5p@xmath86 & 0.09 ( 87)+0.23 ( 30)+0.23 ( 40)+0.19 ( 11)+0.10 ( 46)+0.10 ( 36 ) + 41 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5f@xmath91 & 0.32 ( 12)+0.67 ( 41 ) + 42 & 2s2p@xmath7(@xmath8p)3p & @xmath6s@xmath89 & 0.11 ( 95)+0.44(130)+0.29 ( 42)+0.13 ( 65 ) + 43 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5d@xmath85 & 0.16 ( 92)+0.15 ( 13)+0.29 ( 43)+0.27 ( 33)+0.04 ( 89)+0.05 ( 37 ) + 44 & 2s2p@xmath7(@xmath8p)3p & @xmath5d@xmath88 & 0.40(133)+0.17 ( 66)+0.44 ( 44 ) + 45 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5d@xmath90 & 0.14 ( 10)+0.15 ( 8)+0.14 ( 32)+0.52 ( 45 ) + 46 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5p@xmath86 & 0.07 ( 87)+0.34 ( 40)+0.05 ( 11)+0.52 ( 46 ) + 47 & 2s2p@xmath7(@xmath27p)3p & @xmath5p@xmath87 & 0.10 ( 91)+0.18 ( 99)+0.08 ( 70)+0.40 ( 47)+0.22(135 ) + 48 & 2s2p@xmath7(@xmath27p)3p & @xmath5d@xmath89 & 0.21 ( 95)+0.05(134)+0.08 ( 65)+0.52 ( 48)+0.10 ( 69 ) + + @rrlll@ i & j & rmbpt & grasp & r + + + + i & j & rmbpt & grasp & r + + + 1 & 2 & 3.17@xmath772 & 3.17@xmath772 & 9.8@xmath771 + 1 & 4 & 2.07@xmath773 & 2.07@xmath773 & 1.0@xmath940 + 1 & 6 & 1.05@xmath771 & 1.05@xmath771 & 1.0@xmath940 + 1 & 8 & 4.99@xmath771 & 4.99@xmath771 & 1.0@xmath940 + 1 & 20 & 2.07@xmath774 & 2.07@xmath774 & 1.3@xmath940 + 1 & 22 & 1.22@xmath774 & 1.22@xmath774 & 7.0@xmath771 + 2 & 3 & 3.17@xmath774 & 3.23@xmath774 & 1.7@xmath940 + 3 & 4 & 2.48@xmath773 & 2.48@xmath773 & 1.0@xmath940 + 3 & 5 & 2.01@xmath772 & 2.01@xmath772 & 1.1@xmath940 + 3 & 9 & 1.07@xmath772 & 1.07@xmath771 & 1.1@xmath940 + 3 & 19 & 9.34@xmath772 & 9.34@xmath772 & 1.0@xmath940 + 3 & 20 & 7.73@xmath772 & 7.73@xmath772 & 9.9@xmath771 + 3 & 22 & 2.69@xmath771 & 2.69@xmath771 & 1.0@xmath940 + 5 & 17 & 1.27@xmath772 & 1.65@xmath772 & 1.0@xmath940 + 12 & 22 & 4.23@xmath773 & 4.23@xmath773 & 9.0@xmath771 + 14 & 19 & 1.06@xmath774 & 1.06@xmath774 & 1.4@xmath940 + 19 & 29 & 1.38@xmath774 & 1.38@xmath774 & 7.7@xmath771 + 19 & 31 & 2.43@xmath774 & 2.43@xmath774 & 1.2@xmath940 + 19 & 35 & 5.52@xmath772 & 5.52@xmath772 & 1.0@xmath940 + 19 & 36 & 9.69@xmath772 & 9.69@xmath772 & 1.1@xmath940 + 19 & 37 & 1.16@xmath772 & 1.16@xmath772 & 1.0@xmath940 + 19 & 38 & 6.93@xmath772 & 6.93@xmath772 & 1.1@xmath940 + 19 & 40 & 3.91@xmath773 & 3.91@xmath773 & 9.1@xmath771 + 22 & 27 & 1.14@xmath774 & 1.14@xmath774 & 9.1@xmath771 + 22 & 28 & 1.03@xmath773 & 1.02@xmath773 & 1.0@xmath940 + 22 & 33 & 1.01@xmath773 & 1.01@xmath773 & 1.3@xmath940 + 22 & 35 & 1.87@xmath773 & 1.87@xmath773 & 1.2@xmath940 + 22 & 37 & 1.97@xmath772 & 1.97@xmath772 & 1.1@xmath940 + 22 & 39 & 3.71@xmath772 & 3.71@xmath773 & 9.9@xmath771 + 22 & 40 & 9.91@xmath773 & 9.91@xmath773 & 1.0@xmath940 + + rmbpt : earlier results of safronova and safronova @xcite + grasp : present results with the grasp code from 63 configurations and 2003 levels + r : ratio of velocity and lrength forms of @xmath0-values + @rrlll@ i & j & rmbpt & grasp & r + + + + i & j & rmbpt & grasp & r + + + 1 & 6 & 5.97@xmath771 & 6.08@xmath771 & 1.0@xmath940 + 2 & 8 & 2.76@xmath771 & 2.81@xmath771 & 1.0@xmath940 + 2 & 9 & 2.23@xmath771 & 2.27@xmath771 & 1.0@xmath940 + 3 & 4 & 3.11@xmath772 & 3.18@xmath772 & 1.0@xmath940 + 3 & 7 & 6.43@xmath772 & 6.45@xmath772 & 1.0@xmath940 + 3 & 8 & 5.27@xmath772 & 5.36@xmath772 & 1.0@xmath940 + 3 & 9 & 3.61@xmath772 & 3.67@xmath772 & 1.0@xmath940 + 3 & 10 & 3.12@xmath771 & 3.19@xmath771 & 1.0@xmath940 + 3 & 12 & 1.85@xmath772 & 1.85@xmath772 & 1.0@xmath940 + 4 & 14 & 4.36@xmath771 & 4.43@xmath771 & 1.0@xmath940 + 5 & 11 & 9.10@xmath772 & 9.27@xmath772 & 1.1@xmath940 + 5 & 17 & 1.36@xmath771 & 1.39@xmath771 & 1.0@xmath940 + 6 & 17 & 2.63@xmath771 & 2.67@xmath771 & 1.0@xmath940 + 6 & 18 & 9.49@xmath772 & 9.66@xmath772 & 1.0@xmath940 + 7 & 16 & 5.51@xmath772 & 5.63@xmath772 & 1.1@xmath940 + 7 & 19 & 1.03@xmath771 & 1.06@xmath771 & 1.0@xmath940 + 7 & 22 & 3.45@xmath772 & 3.44@xmath772 & 1.1@xmath940 + 8 & 15 & 9.74@xmath772 & 9.90@xmath772 & 1.1@xmath940 + 8 & 19 & 8.62@xmath772 & 8.76@xmath772 & 1.0@xmath940 + 8 & 20 & 3.39@xmath772 & 3.45@xmath772 & 1.0@xmath940 + 8 & 21 & 9.14@xmath772 & 9.31@xmath772 & 1.0@xmath940 + 9 & 19 & 1.19@xmath771 & 1.21@xmath771 & 1.0@xmath940 + 9 & 20 & 5.15@xmath772 & 5.24@xmath772 & 1.0@xmath940 + 9 & 21 & 1.16@xmath771 & 1.18@xmath771 & 1.0@xmath940 + 10 & 14 & 2.57@xmath772 & 2.19@xmath772 & 1.1@xmath940 + 10 & 19 & 2.20@xmath772 & 2.20@xmath772 & 1.0@xmath940 + 10 & 22 & 2.97@xmath771 & 3.04@xmath771 & 1.0@xmath940 + 11 & 23 & 1.77@xmath771 & 1.79@xmath771 & 1.0@xmath940 + 11 & 24 & 2.50@xmath772 & 2.53@xmath772 & 1.0@xmath940 + 11 & 25 & 7.50@xmath772 & 7.65@xmath772 & 1.0@xmath940 + 12 & 24 & 1.17@xmath771 & 1.19@xmath771 & 1.0@xmath940 + 12 & 25 & 7.35@xmath772 & 7.47@xmath772 & 1.0@xmath940 + 12 & 26 & 8.57@xmath772 & 8.71@xmath772 & 1.0@xmath940 + 13 & 27 & 1.09@xmath771 & 1.11@xmath771 & 1.0@xmath940 + 14 & 27 & 1.05@xmath771 & 1.78@xmath771 & 1.0@xmath940 + 14 & 28 & 7.59@xmath772 & 7.72@xmath772 & 1.0@xmath940 + 15 & 29 & 6.22@xmath772 & 6.33@xmath772 & 1.0@xmath940 + 15 & 30 & 8.59@xmath772 & 8.73@xmath772 & 1.0@xmath940 + 15 & 32 & 6.76@xmath772 & 6.87@xmath772 & 1.0@xmath940 + 16 & 29 & 6.51@xmath772 & 6.61@xmath772 & 1.0@xmath940 + 16 & 30 & 1.87@xmath772 & 1.90@xmath772 & 1.0@xmath940 + 16 & 31 & 1.09@xmath771 & 1.12@xmath771 & 1.0@xmath940 + 17 & 25 & 1.66@xmath771 & 1.69@xmath771 & 1.1@xmath940 + 18 & 26 & 1.98@xmath771 & 2.02@xmath771 & 1.1@xmath940 + 20 & 32 & 1.95@xmath771 & 1.06@xmath771 & 1.1@xmath940 + 21 & 30 & 3.50@xmath772 & 6.40@xmath772 & 1.1@xmath940 + 21 & 32 & 3.31@xmath772 & 3.35@xmath772 & 1.1@xmath940 + 22 & 31 & 1.22@xmath771 & 9.22@xmath772 & 1.1@xmath940 + 23 & 33 & 3.03@xmath772 & 3.11@xmath772 & 1.1@xmath940 + 24 & 34 & 8.49@xmath772 & 8.64@xmath772 & 1.1@xmath940 + 25 & 33 & 9.07@xmath772 & 1.24@xmath771 & 1.1@xmath940 + 26 & 34 & 3.78@xmath772 & 5.90@xmath772 & 1.1@xmath940 + 26 & 35 & 5.39@xmath772 & 5.39@xmath772 & 1.1@xmath940 + + rmbpt : earlier results of safronova and safronova @xcite + grasp : present results with the grasp code from 58 configurations and 509 levels + r : ratio of velocity and length forms of @xmath0-values + @rrrrrr@ i & j & mrmp & grasp & f ( grasp ) & r + + + + i & j & mrmp & grasp & f ( grasp ) & r + + + 1 & 3 & 1.206@xmath9414 & 1.5309@xmath9414 & 1.5354@xmath771 & 1.0@xmath770 + 1 & 11 & 6.551@xmath9413 & 8.2270@xmath9413 & 7.0330@xmath772 & 9.8@xmath771 + 1 & 17 & 2.613@xmath9415 & 2.8077@xmath9415 & 2.3320@xmath940 & 9.8@xmath771 + 1 & 19 & 2.694@xmath9413 & 3.8180@xmath9413 & 2.8152@xmath772 & 9.9@xmath771 + 1 & 27 & 6.243@xmath9414 & 7.7623@xmath9414 & 5.0372@xmath771 & 1.0@xmath770 + 1 & 29 & 1.227@xmath9415 & 1.3590@xmath9415 & 8.6763@xmath771 & 9.8@xmath771 + 1 & 33 & 3.350@xmath9414 & 4.4021@xmath9414 & 2.6529@xmath771 & 1.0@xmath770 + 1 & 39 & 4.193@xmath9413 & 5.3006@xmath9413 & 2.7134@xmath772 & 9.6@xmath771 + 1 & 53 & 1.021@xmath9415 & 1.0126@xmath9415 & 4.8992@xmath771 & 9.7@xmath771 + 1 & 83 & 2.365@xmath9414 & 2.3392@xmath9414 & 9.1872@xmath772 & 9.7@xmath771 + 1 & 101 & 8.690@xmath9414 & 8.9169@xmath9414 & 3.4857@xmath771 & 9.7@xmath771 + 1 & 111 & 1.118@xmath9414 & 1.3923@xmath9414 & 5.2373@xmath772 & 9.9@xmath771 + 1 & 113 & 1.850@xmath9414 & 2.2983@xmath9414 & 8.4470@xmath772 & 9.9@xmath771 + 1 & 129 & 2.953@xmath9414 & 3.0675@xmath9414 & 9.9018@xmath772 & 9.7@xmath771 + 1 & 143 & 6.038@xmath9413 & 7.5371@xmath9413 & 2.3136@xmath772 & 9.9@xmath771 + 1 & 145 & 9.992@xmath9413 & 1.2558@xmath9414 & 3.8142@xmath772 & 9.8@xmath771 + 2 & 4 & 2.917@xmath9410 & 3.0224@xmath9410 & 1.8434@xmath772 & 9.2@xmath771 + 2 & 6 & 2.395@xmath9411 & 2.4114@xmath9411 & 1.1881@xmath772 & 9.8@xmath771 + 2 & 7 & 1.669@xmath9412 & 1.6814@xmath9412 & 1.9361@xmath771 & 1.1@xmath770 + 2 & 8 & 9.092@xmath9411 & 9.1611@xmath9411 & 7.1771@xmath772 & 9.7@xmath771 + 2 & 22 & 1.580@xmath9413 & 1.5536@xmath9413 & 6.1622@xmath772 & 9.5@xmath771 + 2 & 40 & 4.805@xmath9413 & 5.4832@xmath9413 & 6.6094@xmath772 & 1.0@xmath770 + 2 & 41 & 2.845@xmath9413 & 3.2448@xmath9413 & 6.5169@xmath772 & 1.0@xmath770 + 2 & 42 & 3.484@xmath9413 & 4.0782@xmath9413 & 1.0485@xmath771 & 9.9@xmath771 + 2 & 68 & 2.616@xmath9413 & 2.9775@xmath9413 & 1.7587@xmath772 & 1.0@xmath770 + 2 & 71 & 2.101@xmath9413 & 2.4627@xmath9413 & 3.2866@xmath772 & 9.8@xmath771 + 3 & 6 & 1.342@xmath9412 & 1.3518@xmath9412 & 1.1467@xmath771 & 1.0@xmath770 + 3 & 8 & 8.614@xmath9411 & 8.6757@xmath9411 & 1.1693@xmath771 & 1.0@xmath770 + 3 & 9 & 2.411@xmath9412 & 2.4572@xmath9412 & 5.3889@xmath772 & 9.5@xmath771 + 3 & 23 & 2.171@xmath9413 & 2.1941@xmath9413 & 4.6553@xmath772 & 9.5@xmath771 + 3 & 41 & 2.894@xmath9413 & 3.2979@xmath9413 & 1.1095@xmath771 & 9.9@xmath771 + 3 & 43 & 2.956@xmath9413 & 3.4575@xmath9413 & 6.3790@xmath772 & 9.9@xmath771 + 3 & 45 & 3.309@xmath9413 & 3.8940@xmath9413 & 2.3629@xmath772 & 1.0@xmath770 + 3 & 73 & 2.062@xmath9413 & 2.4239@xmath9413 & 7.6916@xmath773 & 1.0@xmath770 + + mrmp : earlier results of vilkas et al . @xcite + grasp : present results with the grasp code from 25 configurations and 157 levels + r : ratio of velocity and lrength forms of @xmath0-values +
[ cols= " < , < " , ]
|
calculations of energy levels , radiative rates and lifetimes are reported for eight ions of tungsten , i.e. s - like ( w lix ) to f - like ( w lxvi ) .
a large number of levels has been considered for each ion and extensive configuration interaction has been included among a range of configurations . for the calculations , the general - purpose relativistic atomic structure package ( grasp ) has been adopted , and radiative rates ( as well as oscillator strengths and line strengths ) are listed for all e1 , e2 , m1 , and m2 transitions of the ions .
comparisons have been made with earlier available experimental and theoretical energies , although these are limited to only a few levels for most ions .
therefore for additional accuracy assessments , particularly for energy levels , analogous calculations have been performed with the flexible atomic code ( fac ) .
+ + _ received _ : 28 september 2015 , _ accepted _ : 12 february 2016 s - like to f - like tungsten ions , energy levels , radiative rates , oscillator strengths , line strengths , lifetimes
|
the understanding of the implications of non - markovianity and the reasons for its occurrence are still largely elusive . yet
, they are stimulating a growing interest in light of their potential impact on many disciplines , from quantum information and nano - technology up to quantum biology @xcite . an important contribution to
this quest came from the formulation of quantitative measures of the degree of non - markovianity of a process @xcite .
in general , these tools address different _ features _ of non - markovianity , from the lack of divisibility of a map @xcite to the ability of the environment to reciprocate the information transfer from the system .
this process occurs unidirectionally in a markovian dynamics @xcite , while the re - focusing of information on the system is the signature of memory effects , as verified in all - optical set - ups @xcite .
the handiness of such instruments has recently triggered the analysis of non - markovianity in quantum many - body systems such as quantum spin chains @xcite or impurity - embedded ultra - cold atomic systems @xcite and in excitation - transfer processes in photosynthetic complexes @xcite .
while these studies relate non - markovian features to the critical behavior of a quantum many - body system @xcite , they also provide a promising arena where the roots for non - markovianity can be researched in physically motivated contexts . in this paper
we explore the competition between two profoundly different mechanisms in a simple open quantum model that is relevant for the physics of nitrogen - vacancy centers in diamonds @xcite and molecular nanomagnets @xcite .
specifically , we address the interplay between the dynamics induced on a two - level system by its coherent interaction with other ( environmental ) spins , and the markovian process describing the relaxation of the latter .
one would expect that , when such memoryless dissipative coupling determines the shortest dynamical timescale of the system , markovianity should emerge preponderantly , especially as the number of environmental spins increases .
indeed , one could imagine that a sort of `` markovianity - mixing '' property would hold as a result of the increasing difficulty to re - build the coherence of the system when many decoherence channels are open .
quite strikingly , we show that this is not generally true . in order to do this using a physically relevant model , general enough to encompass
the unexpected features that we would like to highlight , we consider a spin - star configuration whose peripheral sites are coupled to _ rigid _ boson environments , assumed to induce a memoryless dissipative dynamics . while certainly not exhausting the possible scenarios that can be tackled , our choice is illustrative since the degree of non - markovianity ( as defined in ref .
@xcite ) _ can actually increase _ with the number of peripheral spins , while stronger interactions with the boson baths only affect its rate of growth .
the features of the system at hand are quite complex and a rich non - markovianity phase diagram emerges , spanning degrees of memory - keeping effects all the way down to zero values .
this can be exploited to qualitatively modify the character of the dynamics by engineering its features via accessible control parameters such as the detuning between the central and the outer spins . in turn , this opens up the possibility to implement qubit - state preparation protocols in an open - system scenario that exploits non - markovinity , along the lines of refs .
@xcite and beyond the well - established markovian dissipative framework @xcite . in the following ,
we first present the model and its solution in the simplest terms in sec . [ model ] , while the microscopic description and more sophisticated solution method are presented in the appendices .
we then proceed to the analysis of the non - markovianity of the dynamics in sec .
[ nonmasec ] and sec .
[ timenonma ] .
some concluding remarks are given in sec .
[ conclu ] .
the physical set - up that we describe is sketched in fig .
[ bloch_graph ] a , which shows a central spin ( labelled @xmath0 ) coupled to @xmath1 outer spins , with bonds along the branch of a star .
each environmental spin is further coupled to a local boson reservoir .
the evolution of the central spin is ruled by the master equation @xmath2+\sum_{j=1}^n\hat{\mathcal{l}}_j[\rho(t)]\}\ ] ] with @xmath3 the density matrix of the whole system .
each lindblad superoperator @xmath4 describes local dissipation at temperature @xmath5 ( the same for all the baths ) as @xcite @xmath6 where @xmath7 describes the effective coupling of each external spin to its thermal reservoir , populated by @xmath8 excitations ( @xmath9 , where @xmath10 is the boltzmann constant ) . in what follows
, we will consider the peripheral spins to be initially prepared in @xmath11 . *
( a ) * interacts with @xmath1 peripheral spins , each affected by its own local environment .
( b ) evolution of states @xmath12 ( red trajectory ) and @xmath13 ( blue one ) for a star with @xmath14 peripheral sites .
the bloch spheres in the left ( right ) column correspond to the isotropic ( anisotropic ) spin - spin coupling .
the top ( bottom ) row is for the resonant ( off - resonant at @xmath15 ) case with @xmath16 and @xmath17 . in the isotropic cases ,
the final state of spin @xmath0 is pure , while for @xmath18 it is mixed .
@xmath19 prevents the intersections of the trajectories , which are the dynamical points at which the trace distance is strictly null.,width=321 ] to solve the master equation , we use the damping basis @xcite made out of tensor products of eigenoperators of @xmath4 . in this basis , the density matrix of the system reads @xmath20 where @xmath21 , @xmath22 , @xmath23 and @xmath24 are right eigenoperators of @xmath4 with eigenvalues @xmath25 .
the set of operators @xmath26 is composed of the tensor product of @xmath1 damping - basis elements , one for each peripheral spin . due to the symmetry of the hamiltonian ,
if @xmath27 and @xmath28 consist of the same elements of the damping bases ( although differing for their order ) , the respective coefficients must satisfy @xmath29 .
this simple observation allows us to reduce the number of relevant operators from @xmath30 to @xmath31 . with the help of the single - spin dual damping basis @xmath32 , made of left eigenoperators of @xmath33 s , and using the orthogonality condition @xmath34=\delta_{kk'}\delta_{jl}$ ] , we find @xmath35 with @xmath36\rbrace{+}\lambda_m\delta_{rn}\delta_{ms}$ ] and @xmath37 . by calling @xmath38 , the state of the spin star at time
@xmath39 is @xmath40 tracing over the degrees of freedom of the peripheral spins , we find @xmath41 this gives the exact solution for the dynamics of the central spin , valid for any @xmath1 once the expressions for @xmath42 are taken . with this at hand , in the next section we evaluate the amount non - markovianity of the time evolution .
to quantify the degree of non - markovianity of the dynamical evolution of the central spin described in eq .
( [ exactsol ] ) , we employ the measure put forward in ref .
@xcite , which is based on the idea that memory effects can be characterized by the information flowing out of the open system @xmath0 and quantified in terms of the trace distance @xmath43=\text{tr } |\rho_{0,1 } ( t ) -\rho_{0,2 } ( t)|/2 $ ] between any two of its states @xmath44 .
the trace distance quantifies the distinguishability of two states and leads to measure non - markovianity as @xmath45 , \label{nonmarkovianity}\]]where @xmath46 is the union of the intervals where @xmath47 . to provide a general assessment of the dynamics of spin @xmath0
, we consider the coupling with the external spins to be described by the anisotropic xy model @xmath48,\ ] ] where @xmath49 is an anisotropy parameter and @xmath50 is the spin - spin coupling strength . for isotropic coupling ( @xmath51 ) and zero temperature ,
we obtain a simple scaling law @xcite : for any @xmath52 @xmath53 is obtained from the expression valid for @xmath54 with the re - definition @xmath55 .
this enables the analytic optimization over the input states entering @xmath56 . by calling @xmath57
, we have @xmath58{=}\sqrt{\delta \rho^{00}(t ) @xmath59 and we have introduced @xmath60[{(g+i\delta)\sinh(zt)+z\cosh(zt)}]/{2z}$ ] , @xmath61 and the energy mismatch @xmath62 between the central and outer spins .
the maximum in eq . is achieved for the pure states @xmath63 with @xmath64 . here , @xmath65 are the angles that identify the respective bloch vector .
@xmath56 is optimized by equatorial antipodal states ( _ i.e. _ states with @xmath66 and @xmath67 ) . in [ appb ] , we provide an alternative analytic approach to the evolution of spin @xmath0 and the dependence of the trace distance on such angles .
the trajectories described on the bloch sphere by the evolved states are shown in fig .
[ bloch_graph ] ( b ) [ top row , left - most sphere ] where we see that the states tend to intersect , giving @xmath68 . for @xmath19 ,
the states that optimize the measure of non - markovianity are those with @xmath69 ( the phases being immaterial ) as shown in fig .
[ trd6 ] ( b ) .
interestingly , non - zero values of @xmath70 hinder the intersections of the state trajectories [ cf .
[ bloch_graph ] ( b ) ] .
however , this does not prevent the dynamics to become markovian at proper working points , as we show later on .
the evolution of spin @xmath0 can be characterized using @xmath56 .
when the peripheral spins are detached from their respective baths , any information seeded in the central site undergoes coherent oscillations from the center to the periphery of the star and back . for @xmath71 and peripheral spins prepared in @xmath72 , the dynamics induced by @xmath73 with @xmath74 is ( strongly ) non - markovian at all times @xcite . in our case
, the interaction of the outer spins with their environments radically modifies this picture . as an example , in fig .
[ trd6 ] ( a ) we plot the trace distance for the optimal states at @xmath75 .
we ramp up the spin - bath interaction strength @xmath7 , at set values of the intra - star coupling @xmath50 , looking for the influences that an explicitly markovian mechanism has on the degree of non - markovianity that arises from the dynamical environment to which particle @xmath0 is exposed .
we find a non - monotonic behavior of the trace distance that results in non - markovianity .
the quantitative features of @xmath76 depend on the actual strength of the markovian process : as @xmath7 increases , the revivals of the trace distance become less pronounced . as @xmath56 depends on the number of temporal regions where @xmath77 , fig .
[ trd6 ] ( a ) tells us that @xmath56 decreases as @xmath7 increases , thus showing that , at resonance , a strong influence from the rigid environmental baths over the peripheral spins is sufficient to make the whole process markovian .
for @xmath78 peripheral spins with @xmath79 and @xmath80 ( dot - dashed line ) , @xmath16 ( dashed line ) and @xmath81 ( solid line ) . as the relaxation time becomes shorter , the revivals of @xmath82 are suppressed as a result of a reduction of information back - flow from the baths .
( b ) @xmath56 against @xmath70 for @xmath83 .
the two lines correspond to @xmath84 ( solid blue curve ) and @xmath85 ( dashed red curve ) , which are the optimal states in different detuning regions : @xmath56 is the topmost curve in each region .
there is a finite window of detunings ( light - shadowed region marked as m ) where @xmath86 [ nm marks regions where @xmath87 .
inset : @xmath56 against @xmath70 for @xmath88 and 1.5 ( from top to bottom curve ) . , width=264 ] this
is expected as the excitations distributed to the peripheral spins by spin @xmath0 find the _ sink _ embodied by the baths . the reduced ability to feed back information sets @xmath86 . however , the general picture is more involved : it is sufficient to move to the off - resonant case to face a rather rich _ phase diagram _ of non - markovianity . fig .
[ trd6 ] ( b ) considers the case of coupling mechanisms such that @xmath83 and explores the effect that an energy mismatch between spin @xmath0 and the peripheral sites has on @xmath56 .
we find two ranges of values of @xmath70 for which @xmath86 , symmetrically with respect to @xmath89 . in between and beyond such regions
, @xmath90 behaves quite distinctively : at resonance , the measure of non - markovianity achieves a global maximum ( equatorial states realize the maximum upon which @xmath56 depends ) . for larger detunings ,
@xmath56 changes slowly with @xmath70 ( @xmath91 being the optimal states ) .
clearly , the trend followed by @xmath90 also depends on @xmath92 : small values of @xmath92 push the dynamics towards strong non - markovianity , regardless of @xmath70 , as many coherent oscillations occur between site @xmath0 and the periphery before the initial excitation is lost into the environments . at the same time , the range of detunings for which @xmath86 increases with @xmath7 [ cf .
inset of fig .
[ trd6 ] ( b ) ] .
however non - markovianity persists , both on and off resonance , even when @xmath7 becomes the largest parameter .
this demonstrates an effective control of the degree of non - markovianity of the dynamics undergone by spin @xmath0 , which can be tuned by both the energy mismatch between the outer spins and the central one , @xmath70 , and the intra - star coupling strength .
, @xmath50 . against @xmath1 and @xmath70 for @xmath93 and @xmath94 .
differently from @xmath17 , except for a small range of values , the detuning has no effect on the _ character _ of the dynamics of spin @xmath0 .
strikingly , @xmath56 grows with @xmath1 ( almost linearly for @xmath95 ) .
( b ) analytic behavior of @xmath96 versus @xmath1 for @xmath51 , @xmath75 , @xmath97 at @xmath17 ( @xmath98 ) .
inset : we present the case corresponding to @xmath99 [ other parameters as in panel ( b)].,width=264 ] our discussions so far were restricted to the isotropic coupling at zero temperature , @xmath100 . when the peripheral spins interact with baths populated by @xmath101 thermal excitations , the markovianity regions disappear .
this is seen in fig .
[ nm - ani ] ( a ) where we show a typical case of the behavior of @xmath96 against @xmath70 and @xmath1 .
the anisotropy of the intra - star coupling is crucial for the determination of the dynamics : for @xmath102 the pair of states that maximize @xmath56 changes with the number of peripheral spins . a numerical search for the optimal states can be performed , leading to quite surprising results concerning the scaling of @xmath56 with the size of the spin environment .
intuitively , one would conclude that , as @xmath1 grows , the dynamics of spin @xmath0 will be pushed towards markovianity .
this is not the case : as shown in fig .
[ nm - ani ] ( a ) , @xmath56 _ increases _ with @xmath1 if @xmath79 , regardless of @xmath70 .
this shows that the non - markovian character resists such markovianity - enforcing mechanisms and , counter - intuitively , overcomes them .
we have checked this behavior for the exact analytical expression obtained at @xmath103 [ cf .
[ nm - ani ] ( b ) ] .
the picture somehow changes for @xmath104 : @xmath56 decreases with the growing dimension of the star .
however , even for @xmath95 the non - markovian character is preserved and @xmath56 achieves a non - null quasi - asymptotic value .
the non - markovianity measure gives an integral characterization of the dynamics .
more details on the time dependence of the system - environment information - exchange process is obtained by considering the ratio of in - flowing to out - flowing information , up to a given value @xmath105 of the evolution time . to this end
, we define @xmath106 , where the in - flow [ out - flow ] @xmath107 [ @xmath108 is defined as [ minus ] the integral of @xmath109 , over the time intervals in which it is positive ( negative ) , but only up to @xmath105 . to evaluate these quantities explicitly , we chose as input states the same @xmath110 that optimize the non - markovianity measure @xmath111 .
the ratio @xmath112 gives the fraction of the lost information that returns to the system within @xmath105 , and its behavior is quite different in the various dynamical regimes that we have identified so far . in fig .
[ seigrafici ] , @xmath113 is shown for three values of @xmath70 corresponding to the three regions of fig . [ trd6 ] ( b ) .
the diverse evolutions of @xmath112 signal qualitatively different dynamical behaviors of the system , depending on both the detuning and the anisotropy parameter . at short times
, @xmath112 is always zero ( information has to flow out of the system before it can come back ) , while its first peak is determined by the first revival of the trace distance [ see fig . [ trd6 ] ( a ) ] .
then , its features become strongly dependent on @xmath70 . at long times and at resonance , where a maximum of @xmath56 is found for @xmath79 , information oscillates between the star and spin @xmath0 and @xmath114 [ cf . fig .
[ seigrafici ] ( a ) ] .
the overall dynamics is non - markovian also for the case of fig .
[ seigrafici ] ( c ) , where the time behavior of @xmath115 is shown for a large detuning . in this case , however , @xmath113 decays to zero at long times .
thus , the regions of non - markovianity in fig .
[ trd6 ] ( b ) correspond to different behaviors : near resonance , a fraction of information comes back to the system , different input states remain distinguishable even at long times and thus no equilibrium state is found . for large detunings ,
non - markovianity is built up at short times , while different input states converge towards a long - time equilibrium .
on the other hand , for intermediate values of the detuning [ _ i.e. _ for @xmath70 in the markovianity region of fig .
[ trd6 ] ( b ) ] and @xmath79 , there is no back - flow . even for @xmath104 ,
the fraction of information that comes back is quite small .
the picture changes when @xmath50 increases , the evolution becoming increasingly non - markovian and the role of the anisotropy being fully reversed : @xmath79 implies a larger @xmath113 , persisting for longer times at resonance . versus @xmath49 for a star of @xmath116 sites at @xmath17 , with @xmath117 ( left plots ) and @xmath16 ( right plots ) for three different values of the detunig : @xmath118 for the plots ( a ) and ( d ) , @xmath119 for ( b ) and ( e ) , while @xmath120 for ( c ) and ( f ) . ]
we have used a measure of non - markovianity to show the possibility to control the dynamics of an open quantum system coupled to many independent decohering channels .
we have highlighted the key role played by the detuning and the degree of anisotropy of the system - environment coupling : both can be used to explore a rich non - markovianity phase diagram , where qualitatively different scaling laws with the number of decoherence channels are found . the ability to switch from a markovian to a non - markovian regime by means of a local parameter
could be used to prepare a quantum system in a desired state : indeed the markovian character of processes can be employed for state engineering and information manipulation @xcite . on other hand , while the formation of a steady entangled state is supported by non - markovianity , a purely markovian dynamics produces separable steady states @xcite .
we acknowledge financial support from the uk epsrc ( ep / g004759/1 ) .
sl thanks the centre for theoretical atomic , molecular and optical physics for hospitality during the early stages of this work .
99 s. f. huelga , .
rivas , and m. b. plenio , phys .
. lett . * 108 * , 160402 ( 2012 ) .
f. caruso , a. w. chin , a. datta , s. f. huelga , and m. b. plenio , phys .
a * 81 * , 062346 ( 2010 ) ; m. b. plenio and s. f. huelga , new j. phys .
* 10 * , 113019 ( 2008 ) ; m. mohseni _
et al_. , j.chem . phys . * 129 * , 174106 ( 2008 ) ; a. olaya - castro , c .
f. lee , f. f. olsen and n. f. johnson , phys .
b * 78 * , 085115 ( 2008 ) ; p. rebentrost _ et al_. , new j. phys . * 11 * 033003 ( 2009 ) ; f. caruso _ et al_. , j. chem . phys . * 131 * , 105106 ( 2009 ) ; m. thorwart _
et al_. , chem .
phys . lett . *
478 * , 234 ( 2009 ) ; p. rebentrost , r. chakraborty , and a. aspuru - guzik , j. chem .
* 131 * , 184102 ( 2009 ) ; s. lorenzo , f. plastina , and m. paternostro , phys . rev .
a * 84 * , 032124 ( 2011 ) .
p. rebentrost and a. aspuru - guzik , j. chem
. phys . * 134 * , 101103 ( 2011 ) .
m. m. wolf , j. eisert , t. s. cubitt , and j. i. cirac , phys .
* 101 * , 150402 ( 2008 ) .
rivas , s. f. huelga , and m. b. plenio , phys .
lett . * 105 * , 050403 ( 2010 ) .
breuer , e. m. laine , and j. piilo , phys .
lett . * 103 * , 210401 ( 2009 ) .
lu , x. wang , and c.p .
sun , phys .
a. * 82 * , 042103 ( 2010 ) . b .- h . liu _ et al_. , nature phys .
* 7 * , 931 ( 2011 ) ; a. chiuri , _ et al .
_ , sci . rep .
* 2 * , 968 ( 2012 ) .
t. j. g. apollaro , c. di franco , f. plastina , and m. paternostro , phys .
a * 83 * , 032103 ( 2011 ) .
p. haikka , s. mcendoo , g. de chiara , g. m. palma and s. maniscalco , phys .
a * 84 * , 031602(r ) ( 2011 ) .
p. haikka , j. goold , s. mcendoo , f. plastina and s. maniscalco , phys .
a * 85 * , 060101(r ) ( 2012 ) .
f. reinhard _
et al . _ ,
* 108 * , 200402 ( 2012 ) .
a. ardavan _
et al_. , phys .
lett . * 98 * , 057201 ( 2007 ) ; a. candini _
et al_. , phys .
lett . * 104 * , 037203 ( 2010 ) .
h. p. breuer and f. petruccione , _ the theory of open quantum systems _
( oxford university press , oxford , 2007 ) .
h. p. breuer , d. burgarth and f. petruccione , phys .
b * 70 * , 045323 ( 2004 ) . h. j. briegel , and b. g. englert , phys . rev .
a * 47 * , 3311 - 3329 ( 1993 ) .
s. f. huelga , .
rivas , and m. b. plenio , phys .
* 108 * , 160402 ( 2012 ) .
b. kraus , _ et al .
a * 78 * , 042307 ( 2008 ) .
f. verstraete , m. m. wolf , and j. i. cirac , nature phys .
* 5 * , 633 ( 2009 ) .
the total hamiltonian of the spin - star system @xmath121 consists of a few contributions .
the first one is the system s free energy [ we take units such that @xmath122 throughout the paper ] @xmath123 , describing the free evolution of @xmath124 spin-@xmath125 particles [ here , @xmath126 is the @xmath127-pauli matrix of spin @xmath128 ( @xmath129 ) ] , each with transition frequency @xmath130 between spin states @xmath131 and @xmath132 . the second term in @xmath133
describes the energy of @xmath1 sets of @xmath134 harmonic modes ( one set per peripheral spin of the star ) with creation ( annihilation ) operators @xmath135 ( @xmath136 ) which satisfy the commutation relations
@xmath137 = \delta_{kk'}\delta_{jl}$ ] .
the central and peripheral spins are coupled by @xmath138 , whose explicit form will be specified later on .
each peripheral spin interacts with its own bath as @xmath139 , where @xmath140 with @xmath141 .
we assume that the their local bath induce a markovian dynamics of the peripheral spins and take uniform couplings , so that the evolution of the central spin is ruled by @xmath142+\sum_{j=1}^n\hat{\mathcal{l}}_j[\rho(t)]\}\ ] ]
here we provide an alternative solution to the dynamics of the system . in the interaction picture with respect to @xmath133 the schrdinger equation reads @xmath143 where the interaction hamiltonian is given by @xmath144 with @xmath145 the operator @xmath146 counts the number of excitations in the system and commutes with the total hamiltonian @xmath147 , so that any initial state of the form @xmath148 evolves after time @xmath39 into the state @xmath149 where the state @xmath150 denotes the product state @xmath151 and @xmath152 for the sites on the star ; @xmath153 is the vacuum state of all the reservoirs , and @xmath154 the state with one particle in mode @xmath155 in the @xmath128th reservoir .
+ the amplitude @xmath156 is constant in time because of @xmath157 .
+ substituting eq . into the schrdinger equation one finds @xmath158 we assume in the following that @xmath159 .
this means that the two level systems on the star are in the @xmath160 state and that each environment is in the vacuum state initially .
+ the total initial state is given by the product state @xmath161 formally integrating eq . and substituting into eq .
one obtains the system for the amplitude @xmath162 , @xmath163 we can define the kernels @xmath164 describing the two - point correlation function of each reservoir , which are the fourier transform of the respective environmental spectral density @xmath165 for the moment , we do not make any restrictive hypothesis on the form of @xmath166 , so that our results will be valid for an environment with a generic spectral density . in order to solve the system above it
is convenient to pass in the laplace domain : @xmath167&=&\;c_1(0)-i j
\sum_{j=1}^n \tilde{c_j}[s+i(\epsilon_0-\epsilon_j)]\\ & s \tilde{c_j}[s]&=&-i j \tilde{c_1}[s - i(\epsilon_0-\epsilon_j)]- \tilde{c_j}[s]\tilde{f}_j[s ] \end{aligned}\end{cases } \label{eq-22}\ ] ] solving the second of eq . [ eq-22 ] respect to @xmath168$],assuming that all the reservoirs are the same ( @xmath169 ) , and substituting in the first we get @xmath170=c_1(0)\dfrac{s - i\delta - f[s - i(\epsilon_0-\epsilon)]}{s^2-is(\epsilon_0-\epsilon)-i s f[s - i(\epsilon_0-\epsilon)]+j^2 n}\label{c1s}\nonumber\ ] ] where @xmath171 ( @xmath172 ) . to specify the model , but still retaining a general enough description , we consider a lorentzian spectral density for each bath ( which gives rise to an exponentially decaying correlation function ) : @xmath173 here @xmath174 is the detuning of the center frequency of the bath @xmath175 and the frequency of the two - level system @xmath130 , the parameter @xmath49 defines the spectral width of the environment , which is associated with the reservoir correlation time by the relation @xmath176 and the parameter @xmath7 is related to the relaxation time scale @xmath177 by the relation @xmath178 .
+ we will consider @xmath179 , and in this case we may distinguish between the markovian and the non - markovian regimes ( for the dynamics of the environmental spins themselves ) using the ratio of @xmath7 and @xmath49 : @xmath180 gives a markovian regime and @xmath181 corresponds to non - markovian regime .
+ substituting in eq .
[ c1s ] and anti - transforming we have @xmath182 with @xmath183}{(\alpha_1-\alpha_2)(\alpha_2-\alpha_3)(\alpha_1-\alpha_3 ) } , \end{aligned}\label{gt}\ ] ] where @xmath184 , @xmath185 and for @xmath186 . here ,
@xmath187 s are the roots of the equation @xmath188 already at this point , it is evident how the only effects of increasing @xmath1 is to redefine the coupling constant @xmath50 .
the solution of the schrdinger equation of the total system with initial states of the form lyes in the sector of the hilbert space corresponding to zero or one excitations .
+ we can construct the exact dynamical map describing the time - evolution of the reduced density matrix of the central spin which is given by @xmath189 where @xmath190 for @xmath191 . using eq . and eq . we find @xmath192 the optimization of the initial states in eq .
( [ nonmarkovianity ] ) obtains the maximally possible non - markovianity of a particular quantum evolution . + in our case , the maximization is achieved by pure states , thus we choose as initial states for eq .
[ statoiniziale ] @xmath193 where we used the fact that , since @xmath147 is invariant under rotations along z - axis , the maximum is obtained for @xmath194 .
|
we study the interplay between forgetful and memory - keeping evolution enforced on a two - level system by a multi - spin environment whose elements are coupled to local bosonic baths .
contrarily to the expectation that any non - markovian effect would be _ buried _ by the forgetful mechanism induced by the spin - bath coupling , one can actually induce a full markovian - to - non - markovian transition of the two - level system s dynamics , controllable by parameters such as the mismatch between the energy of the two - level system and of the spin environment . for a symmetric coupling
, the amount of non - markovianity surprisingly grows with the number of decoherence channels .
|
the classical theory of force - free motions of rigid particle systems has a long history in connection with investigation in mathematics and mechanics . from the differential calculus , in the case of translational particle motions in euclidean space ( newton axioms of mechanics ) , to the riemann geometry .
hence , the definition of geodesics has immediately concerned with force - free particle motions on surfaces .
the introduction of riemannian manifolds and the geometry of their geodesics were motivated by the mechanics of constrained particle systems . in the last few years
, the study of force - free motion systems on riemannian manifolds of constant sectional curvature has attracted the interest of several authors @xcite , @xcite , @xcite , @xcite , @xcite and @xcite . in particular ,
in @xcite the authors define the notion of a pendulum on a surface of constant gaussian curvature @xmath0 and they study the motion of a mass at a fixed distance from a pivot .
so , a pendulum problem on a surface of constant curvature is defined as a pivot point and a mass connected to that point by a rigid massless rod of fixed length @xmath1 .
it is assumed that the pivot is constrained to move along some fixed curve with prescribed motion .
the rod provides the only force on the mass in order to keep the mass at the fixed distance from the pivot .
no torque is applied to the rod , ( fig .
1 ) . in @xcite
it is studied the pendulum problem when the pivot moves along a geodesic path , and the space is a surface of constant ( not zero ) curvature .
it is considered the surface immersed in @xmath2 or @xmath3 and it is obtained the differential motion equation by newtonian procedure doing a laborious calculation . moreover , the cases @xmath4 and @xmath5 are necessary to come from different forms . in this work
, we deal with the pendulum problem from an analytic point of view .
this procedure allows approach simultaneously the cases of positive and negative curvature and also of zero curvature ( as a limit case ) , with very simple computations .
the key of our study is a lagragian approach , which has a 1-dimensional configuration space .
as adirect consequence the internal curvature force is conservative and its potential function is easily calculated .
moreover , this method can be used to another related problems , as the elastic pendulum , or the quantum pendulum .
mainly , this work concerns the case in which the pivot moves along a geodesic .
let @xmath6 be the angle between the rod and the motion direction of the pivot .
we will assume the following convention : if the constant curvature of the space is @xmath4 , @xmath7 if the constant curvature of the space is @xmath5 , @xmath8 the main result is the following : using the previous result and geometric arguments , we obtain the motion equation of the system when the pivot accelerates along a geodesic ( see section 4 ) .
further developements are discussed in section 5 .
it is well known that a complete , simply - connected riemannian @xmath9-manifold with constant sectional curvature is isometric to one of model spaces @xmath10 , @xmath11 or @xmath12 ( see @xcite ) .
let @xmath13 be the 2-dimensional sphere of radius @xmath14 , endowed with the metric induced from @xmath15 , where @xmath16 consider the coordinate system @xmath17 in @xmath18 with @xmath19 and @xmath20 , being @xmath21 in this coordinate system the metric is given by @xmath22 and the non - zero christoffel symbols are @xmath23 and @xmath24 .
let s consider also the hyperbolic plane @xmath25 , where @xmath26 and @xmath27 is the induced metric from the lorentz - minkowski spacetime @xmath3 .
let @xmath28 be the coordinate system in @xmath29 with @xmath30 and @xmath31 , where @xmath32 the metric is given by @xmath33 and the non - zero christoffel symbols are @xmath34 and @xmath35 .
firstly , we deal with the rigid pendulum problem in the case that the pivot moves along a geodesic of its space , with constant speed . we will do analogously the study for @xmath4 and @xmath5 . note that in the case @xmath4 we must require that @xmath36 to guarantee that the mass and the rod are on the same side of the geodesic line .
suppose that the pivot moves along the geodesic @xmath37 with constant speed .
let s take a reference associated to the pivot .
denote by @xmath6 the angle between the rigid rod and the direction of pivot motion .
the lagrangian of the system has two components , on the one hand , the kinetic energy of the mass @xmath38 , and on the other hand , the potential energy @xmath39 , which is due to the curvature of space , i.e. , @xmath40 if the space is flat .
consider the isosceles geodesic triangle formed in an infinitesimal temporal interval , by the rod and the arc @xmath41 traced by the mass ( fig .
2 ) .
the sine theorem for spherical geometry allows us to write @xmath42 thus @xmath43 therefore , if the particle has mass @xmath44 , the kinetic energy is given by @xmath45 consider the coordinate system @xmath17 with the metric @xmath46 and suppose that the pivot moves along the geodesic @xmath37 with constant speed @xmath47 . obtain us the mass acceleration due to the curvature , so suppose that the mass moves along the curve @xmath48 its acceleration is given by @xmath49 as consequence , the external force that we must be to apply on the rod will be @xmath50 and the curvature potential energy @xmath39 satisfies @xmath51 , where @xmath52 . hence , @xmath53 where we have taken @xmath54 as origin energy . again , using the sine theorem ( fig .
3 ) we obtain @xmath55 thus @xmath56 now , we can enunciate the following theorem , suppose that the pivot of pendulum moves with constant speed @xmath47 along a geodesic on a surface with constant curvature @xmath4 .
let @xmath57 the angle at the time @xmath58 between the rigid rod and the direction of pivot motion .
then the lagrangian of the system is @xmath59 therefore , the motion differential equation is given by @xmath60 if we do @xmath61 , @xmath62 and @xmath63 obtaining the lagrangian function of the pendulum moving in the euclidean plane , when the pivot moves along a straigh line with constant speed .
consider us now , the pivot moving along the geodesic @xmath64 , with constant speed @xmath47 .
analogously to the spherical case , we take a reference joint with the pivot and we denote @xmath6 the angle between the rigid rod and the direction of pivot motion . if we consider the isosceles hyperbolic differential triangle ( fig .
2 ) , and making use of the corresponding theorems of the hyperbolic geometry , we can to conclude that the kinetic energy of the system is given by @xmath65 reasoning as the in spherical case , we consider that the pivot moves along the geodesic @xmath64 . if the mass moves along the curve @xmath66 we can to calculate the curvature potential function , @xmath67 where we have taken @xmath68 as origin energy . again , using the sine theorem in the hyperbolic case , we obtain @xmath69 thus , @xmath70 suppose that the pivot of the pendulum moves with constant speed @xmath47 along a geodesic on a surface with constant curvature @xmath5 .
let @xmath57 the angle at the time @xmath58 between the rigid rod and the direction of pivot motion .
then the lagrangian of the system is given by @xmath71 therefore , the motion differential equation is @xmath72 with our convention , we can to enunciate ( compare with @xcite , theorem a ) : suppose that the pivot of pendulum moves with constant speed @xmath47 along a geodesic on a surface with constant curvature @xmath0 .
let @xmath57 the angle at the time @xmath58 between the rigid rod and the direction of pivot motion .
then the lagrangian of the system is given by @xmath73 therefore , the motion differential equation is @xmath74 suppose @xmath4 and consider the motion equation ( [ motion1 ] ) . making the change @xmath75 , we obtain the differential equation @xmath76 which is the equation of planar pendulum of length @xmath77 in euclidean space subject to a constant gravitational field of magnitude @xmath78 .
its stable and unstable equilibria at @xmath79 and @xmath80 , correspond to the stable @xmath81 and unstable @xmath82 equilibria of the pendulum on the spherical surface . on the other hand ,
if @xmath5 , making @xmath83 in the motion equation ( [ motion ] ) , we obtain , in similar form , that @xmath82 is stable equilibria and @xmath81 is unstable equilibria .
moreover , it is well known that the equation ( [ planar ] ) can be exactly solved in term of a elliptic integral of first kind ( see @xcite , for instance ) , which can not be evaluated in a closed form . a first approximation to this problem in arbitrary constant curvature is given for small oscillations around the stable equilibria points of this physical system .
that is , for small @xmath84 , @xmath85 , and the equation of motion is approximated by @xmath86 observe that the previous equation represents a simple harmonic oscillator of frequence @xmath87 . since the hamiltonian function @xmath88 is time
independent , it is an integral on phase space and it represents the system energy . using the standard notation @xmath89 in the phase space , we can write @xmath90 the legendre transformations allow us to give @xmath91 moreover , it is easy to see that @xmath88 is constant if and only if @xmath92 is constant ( compare with ( * ? ? ? * prop .
suppose now that the pivot moves along a geodesic path with lineal acceleration @xmath93 .
we come to compute the acceleration induced by the accelerated pivot at the mass making use of a geometric argument .
we follow the following steps : \a ) translate to the mass the pivot acceleration @xmath93 .
\b ) transform this acceleration in angular acceleration between the rod and the geodesic path traced by the pivot .
\c ) project on the orthogonal direction to the rod .
indeed , consider the geodesic triangle of the figure 4 .
the translated acceleration to the mass is given by @xmath94 and the angular acceleration by @xmath95 finally , in order to project , it is enough to multiply by @xmath96 @xmath97 on another hand , taking into account the previous triangle , from the pythagoras and cosine theorem in non - euclidean geometries , we have @xmath98 thus @xmath99 . as a consequence ,
the searched acceleration is @xmath100 so , we prove in a different approach the result ( ( * ? ? ?
* theorem d ) ) assume that the pivot moves with speed @xmath101 along a geodesic on a surface with constant curvature @xmath0 .
let @xmath57 the angle that the rigid rod makes with the direction of the motion of the pivot .
then the pendulum satisfies the differential equation @xmath102 where @xmath103
as a first application , consider us the case of a pendulum whose pivot moves along a geodesic on a surface of constant curvature with constant speed @xmath47 and whose rod is elastic .
let @xmath104 be the elastic constant of the rod , let @xmath1 be the length of the rod and let @xmath105 be the elongation of the rod at the time @xmath58 .
our analytical approach allows us to initiate the study of dynamical system in similar form as the rigid case .
so , it is not difficult to see that its lagrangian function is given by @xmath106 secondly , we will make an approximation to the quantum system .
denote @xmath107 .
then , ( [ hamiltonian ] ) can be written , for @xmath108 , in terms of @xmath6 and @xmath109 as @xmath110 observe that the same expression , up an additive term , is obtained for @xmath5 .
therefore , a suitable time independient schrodinger equation for this system can be described as @xmath111 where @xmath112 is a complex function representing the wave function of the mass particle in this physical system .
observe that it is expected that the energy should take quantized values .
in fact , if it is assumed that @xmath113 is confined in a region where @xmath6 is close to @xmath68 , the previous equation , in this first approximation , corresponds to the quantum harmonic oscillator , whose energy is found to be @xmath114 observe that , within this simplification , in the quantum of energy , @xmath115 , appears the speed of the rod and the curvature of the ambient space .
the authors are partially supported by the spanish mec - feder grant mtm2010 - 18099 .
|
the dynamics of force free motion of pendulums on surfaces of constant gaussian curvature is addressed when the pivot moves along a geodesic obtaining the lagragian of the system . as a application it is possible the study of elastic and quantum pendulums .
_ ams classification scheme numbers _ : 70h03 , 53a35 , 53b20 .
_ keywords _ : dynamical systems , lagrangian and hamiltonian mechanics , semi - riemannian geometry , constant curvature , pendulum .
|
( hereafter ) is a 33ms radio pulsar discovered in the green bank telescope ( gbt ) 350 mhz drift - scan pulsar survey @xcite . with a dispersion measure of 3.27pc@xmath5
, it appeared to be one of the closest pulsars to the earth .
further observations showed was in a binary system with an orbital period of 2.45days and a minimum companion mass of about 1@xmath6 .
this sort of system straddles the line between potential companion types .
it could be a double - neutron star ( dns ) , of which there are only roughly 12 and whose study is crucial to understanding the formation of sources of khz gravitational waves ( e.g. , * ? ? ?
* ) and testing general relativity ( e.g. , * ? ? ?
, it could be a pulsar with a massive white dwarf companion a so - called `` intermediate - mass binary pulsar '' ( imbp)that descended from a binary with a more massive companion than in traditional systems with pulsars and low - mass white dwarfs @xcite .
imbp systems are rare , with fewer than 20 known , and massive white dwarfs are themselves rare , with fewer than 8% of the white dwarfs ( wds ) from optical surveys having masses above @xmath7 @xcite .
understanding the formation and evolution of imbp systems provides a crucial piece in our understanding of binary evolution and pulsar recycling , and helps delineate evolutionary paths between low - mass nss and high - mass white dwarfs @xcite .
@xcite used very long baseline interferometry astrometry to measure the parallax of with exquisite precision .
they find a distance of @xmath0pc ( it is the second closest binary pulsar system and one of the closest nss of any type ) .
the astrometric data also suggested an edge - on orbit , opening up the possibility of a measurement of the shapiro delay @xcite , which gives two post - keplerian @xcite parameters for the system and hence determines the component masses ( e.g. , * ? ? ? * ) .
here we present the detailed timing analysis of the system , including the measurement of the shapiro delay and the determination of the masses ( ) .
we then present deep optical and near - infrared searches for the companion to ( ) , which we use to constrain models of its formation and evolution ( ) .
we find that the system almost certainly must be an imbp system , but that we do not detect the companion , constraining it to be one of the coolest white dwarfs ever observed .
unlike some sources where temperature inferences are highly dependent on white dwarf model atmospheres ( e.g. , * ? ? ?
* ) , this measurement is robust , given the small uncertainties on the mass and ( especially ) distance .
we conclude in .
radio observations of to measure the shapiro delay occurred in the last week of 2011 may with the 100 m robert c. byrd gbt .
we had a 6hr observation taken around superior conjunction of the binary system augmented by five 2hr observations at each of the other five shapiro extrema , all using the green bank ultimate pulsar processing instrument ( guppi ; @xcite ) .
the 800mhz of bandwidth centered at 1500mhz in two orthogonal polarizations was separated into 512 nyquist - sampled frequency channels of width 1.5625mhz via a polyphase filter bank .
these channels , sampled at 8-bits , provided full polarization information and an effective time resolution of 0.64@xmath8s .
each channel was coherently dedispersed at the nominal dispersion measure ( dm ) of the pulsar ( 3.27761pc@xmath9 at the time , although we later refined this measurement ) .
each observing session was broken into 30-minute observations of separated by 60s calibration scans of the extragalactic radio source 3c 190 .
the calibration scans were taken in the same mode as the pulsar observations , but also included a 25hz noise diode inserted into the receiver . ) .
the position angle of the linear polarization is given in the upper panel . as is the case with most msps
, the polarization position angle variations do not permit a rotating vector model fit , so we can not constrain the emission geometry .
[ fig : profile ] , scaledwidth=40.0% ] data reduction was performed using the ` psrchive ` package @xcite .
flux calibration used the on- and off - source scans of 3c 190 .
this was followed by removal of radio frequency interference by the psrzap utility .
the calibrated pulse profile determined from the long observation covering conjunction is given in .
the data were aligned in time using the best ephemeris ( below ) , divided into 16 frequency channels , and re - fit for dispersion measure and rotation measure using a bootstrap error analysis .
we found that the period - averaged flux density varied by a factor of a few over the course of long observations due to scintillation , with an average of 12mjy at 1500mhz .
individual times - of - arrival ( toas ) were measured from the folded total - intensity profiles using the frequency domain algorithm in ` psrchive ` @xcite .
a template was created by fitting three gaussians to the summed pulse profile . from these gaussian components
, we created a noise - free template with the phase of the fundamental component in the frequency domain rotated to zero .
the observations were divided into 2minute segments , with one toa measured for each segment . note that since interstellar scintillation caused the flux to vary considerably , there was a proportional change in the toa precision that varied over the data set .
these data were combined with previous data taken for the discovery observations of @xcite to produce a timing model .
we used the `` dd '' model @xcite in ` tempo ` , which incorporates the shapiro delay .
the astrometric data for this model were taken from @xcite , and we used the de421 jpl ephemeris @xcite .
timing fits with no shapiro delay were statistically unacceptable , with an rms residual of @xmath10s ( @xmath11 for 931 degrees - of - freedom ) , and a clear shapiro delay signature was obvious in the residuals ( ) . with the shapiro delay included in the fit the rms residual was 4.2@xmath8s ( @xmath12 for 929 degrees - of - freedom ) , with no obvious remaining structure in the residuals ( varying the astrometric parameters within the uncertainties from @xcite changed the timing results by @xmath13 ) .
the shapiro delay determines the inclination of the orbit and the companion mass ; this is then combined with the binary mass function to determine the pulsar s mass .
due to the combination of several different and much less precise observing modes from earlier monitoring with the high - precision shapiro delay campaign , we estimated the timing parameters with a bootstrap error analysis .
we give the full timing results , with 1-@xmath14 error estimates from the bootstrap analysis , in .
timing residuals for , using the new data from this paper ( blue : mjd 55,60055,921 ) and older data ( gray ) , as a function of orbital phase ( true anomaly plus longitude of periastron ) .
top : residuals computed from the best - fit model without shapiro delay ( the rms residual is @xmath10s ) .
middle : residuals computed including shapiro delay .
the red curve is the best - fit shapiro delay profile .
bottom : residuals computed relative to the best - fit model including shapiro delay ( the rms residual is @xmath15s ) .
conjunction is at a phase of 0.25 . in all panels
the left axis shows the residuals in @xmath8s , while the right axis shows the residuals in milliperiods .
note the different @xmath16-axis scales.,scaledwidth=50.0% ] our data consist of high - quality coherently dedispersed data from an intensive 1 week campaign and a few other epochs .
the remainder of the data were both less precise and less uniform , with a wider range of observation frequency and instrumental setup .
this makes it difficult ( if not impossible ) to robustly constrain long - term secular changes like periastron precession ( @xmath17 ; @xcite ) . nonetheless , we tried a fit with @xmath17 fixed to the value predicted by general relativity ( @xmath18 ) .
the resulting fit was good , with the rms decreasing to 3.8@xmath8s .
the pulsar and companion masses each increased by about 1@xmath14 compared to the values in .
given the small eccentricity and inhomogeneous data set with large gaps we do not believe that fitting for @xmath17 is viable at this time , but encourage further long - term monitoring of this system to establish its secular behavior .
l c + spin period ( s ) & 0.032817859053065(3 ) + period derivative ( ss@xmath19 ) & @xmath20 + dispersion measure ( pc@xmath9 ) & 3.2842(6 ) + rotation measure ( radm@xmath21 ) & + 2.6(1 ) + reference epoch ( mjd ) & 55743 + right ascension ( j2000 ) & 22:22:05.969101(1 ) + declination ( j2000 ) & @xmath22:37:15.72441(4 ) + r.a . proper motion ( mas@xmath23 ) & 44.73(4 ) + dec proper motion ( mas@xmath23 ) & @xmath245.68(6 ) + parallax ( mas ) & @xmath25 + position epoch ( mjd ) & 55743 + span of timing data ( mjd ) & 5500555922 + number of toas & 943 + rms residual ( @xmath8s ) & 4.2 + + orbital period ( days ) & 2.4457599929(3 ) + projected semi - major axis ( lt - s ) & 10.8480276(12 ) + epoch of periastron ( mjd ) & 55742.13242(0 ) + orbital eccentricity & @xmath26 + longitude of periastron ( deg ) & 119.778(12 ) + mass function ( @xmath6 ) & 0.22907971(8 ) + @xmath27 & 0.9985(3 ) + companion mass ( @xmath6 ) & 1.05(6 ) + + distance ( pc ) & 267.3@xmath28 + transverse velocity ( km@xmath29 ) & 57.1@xmath30 + orbital inclination @xmath31 ( deg ) & 86.8(4 ) + shklovskii period derivative ( @xmath32 ) & @xmath33 + intrinsic period derivative ( @xmath32 ) & @xmath34 + surface magnetic field ( @xmath35 gauss ) & 0.719 + spin - down luminosity ( @xmath36 ) & 1.72 + characteristic age ( gyr ) & 33.8 + pulsar mass ( @xmath6 ) & 1.20(14 ) + flux density at 1500mhz ( mjy ) & 12 + we observed the position of at optical and near - infrared wavelengths , as listed in .
the deepest keck observations used the red side of the low - resolution imaging spectrometer ( lris ; @xcite ) on the 10 m keck i telescope .
the data were reduced using standard procedures in ` iraf ` , subtracting the bias , dividing by flatfields , and combining the individual exposures .
the seeing was about @xmath37 in the combined @xmath38 image , and @xmath39 in the combined @xmath40 image .
we computed an astrometric solution fitting for a shift and separate scales and rotations along each axis ( i.e. , a six - parameter fit ) using 100 non - saturated stars identified from the sloan digital sky survey ( sdss ) data release 10 ( dr10 ; @xcite ) , giving rms residuals of @xmath41 in each coordinate .
we did photometric calibration relative to sdss photometry , identifying 23 well - detected , well - separated , non - saturated stars , and transforming from the sdss filter set to johnson cousins using the appropriate transformation equations .
the zero - point uncertainty was @xmath42mag , although there are systematic uncertainties coming from our filter transformations .
we see no object at the position of the pulsar ( ) ; the closest object is about @xmath43 from the position of the pulsar ( about @xmath44 away ) and appears extended ( @xmath45 and statistical position uncertainties of @xmath46 in each coordinate ) .
we determined the 3@xmath14 upper limits using ` sextractor ` @xcite to determine the magnitude that gave a 0.3mag uncertainty ( verified with fake - star tests ) , which we give in .
we observed in @xmath47-band with the goodman spectrograph @xcite on the 4.1 m southern astrophysical research ( soar ) telescope over two nights in 2013 july .
all exposures were dithered and binned by a factor of two in both dimensions .
the frames were bias - subtracted and flattened with a dome flat .
we then used a median of the data ( having masked the scattered - light halos of three saturated stars ) from the second night constructed without registration to create a sky flat , which we smoothed with a @xmath48pixel boxcar filter .
this corrects for larger - scale brightness variations .
cosmic rays were interpolated on individual exposures using the ` lacosmic ` routine @xcite .
the seeing varied considerably over the course of the observations , going from @xmath49 to @xmath43 .
we then shifted each exposure by an integer number of pixels for registration and summed them .
the final summed image has an effective seeing of @xmath50 and a total exposure time of 2.6hr .
the photometric zero - point was again computed relative to the sdss dr10 data , using 31 stars .
the astrometric solution was done using six 30s exposures through http://astrometry.net @xcite . as with the keck data , we see no object at the position of the pulsar ( ) and
give a 3@xmath14 upper limit in .
l c c c c c soar / goodman & 2013 jul 2 & @xmath47 & @xmath51 & 26.4 & 19.2 + soar / goodman & 2013 jul 3 & @xmath47 & @xmath52 & & + keck
i / lris(red ) & 2013 aug 4 & @xmath38 & @xmath53 & 26.3 & 19.1 + keck
i / lris(red ) & 2013 aug 4 & @xmath40 & @xmath53 & 26.0 & 18.9 + keck ii / nirc2 & 2013 oct 12 & @xmath54 & @xmath55 & 21.0 & 13.9 + while they were taken through different filters and with very different instruments / resolutions , we tried combining the keck @xmath38-band and soar @xmath47-band images using ` swarp ` @xcite .
we still see no source at the position of the pulsar .
the data are sufficiently different that a limiting flux is difficult to compute , but it could be as much as 0.3mag fainter than the limits in .
the near - infrared observations come from the nirc2 camera square field of view . ] on the 10 m keck ii telescope , and used the laser guide star adaptive optics ( ao ) system @xcite .
the data were taken through thin clouds and the ao corrections were not optimal , resulting in a delivered image quality of @xmath41 fwhm .
the images were reduced using a custom pipeline implemented with ` python ` and ` pyraf ` using dark frames and dome - flats .
a sky fringe frame was created by combining dithered images of multiple targets with the bright stars masked .
we used ` sextractor ` @xcite for the preliminary detection and masking of stars .
the fringe frame was subtracted from the flat - fielded data after being scaled to the appropriate sky background level . before coadding the frames ,
each frame was corrected for optical distortion using a distortion solution measured for nirc2 .
a faint glare has been visible in the lower right ( south - west ) corner of the nirc2 wide camera images starting in 2009 august .
the shape and amplitude of the glare vary with telescope orientation , resisting correction through surface fitting or modeling .
instead we masked the glare using a triangular region .
there was no independent photometric calibration that night , and only a single star is visible on the co - added image .
to determine a photometric zero - point , we used photometry for that star from the sdss dr10 .
we then employed the empirical main - sequence color relations from @xcite , inferring the @xmath56 color from the observed @xmath57 color ( we ignore differences between @xmath58 and @xmath54 filters ) . for this star ( sdss [email protected] )
we infer a spectral type of k2.5 and predict @xmath59 . we expect zero - point uncertainties of @xmath60mag or so based on comparison of the other sdss colors to those predicted using @xcite .
again we see no object at the position of the pulsar , and give 3@xmath14 upper limits in .
since we do not detect the optical counterpart of the companion , the first inference is that the companion could be a low - mass ns .
it would be the lowest mass ns known @xcite , although it is only a roughly 23 @xmath14 excursion from the mean of the companions in dns systems @xcite : rare , given the @xmath61dns systems , but not impossible . in that case , its eccentricity of @xmath62 would be a factor of @xmath63 lower than any other dns system ( has the lowest eccentricity of @xmath64 , although this may be an ns wd system ; @xcite ; van leeuwen et al .
2014 , , submitted ) . in
we show the eccentricity versus component masses for all dns and ns wd systems with well - determined masses .
in fact there are three ns wd systems with higher eccentricities : , which was likely not recycled @xcite ; , which has had its eccentricity increased by dynamical interactions @xcite ; and ( likely an imbp , with the eccentricity the result of unstable mass transfer ; @xcite ) .
the normal formation scenario for a dns involves two core - collapse supernova explosions , with the eccentricity the result of the second explosion and its kick , and no final mass - transfer phase to circularize the orbit ( e.g. , * ? ? ?
in contrast , formation via an electron - capture supernova ( ecs ; @xcite ) could result in a significantly lower ns mass ( @xcite ; @xcite ) along with a lower supernova kick @xcite .
has a low transverse velocity ( @xmath65 ) , although higher than some systems thought to be the products of ecss ( given the age of the system , this velocity may be more related to motion in the galactic potential than birth conditions ) .
this may reflect the velocity dispersion of the progenitor systems .
however , the contrast between and other systems thought to be the results of ecss ( e.g. , psr j1906 + 0746 or ; @xcite ) is extreme , with the ratio of eccentricities above 200 as mentioned previously . in a scenario without a kick we can place an upper limit on the amount of material that could have been ejected by the explosion to @xmath66 ( with @xmath67 the pulsar mass and @xmath68 the current companion mass ; e.g. , * ? ? ?
this is a much tighter bound than in any of the other systems proposed for this mechanism , and difficult to reconcile with the change in binding energy needed to collapse to a ns @xmath69 ( with @xmath70 km for an ns ) , presumably released as neutrinos ( e.g. , * ? ? ?
* ) : this leads to the horizontal line in , above which all confirmed dns systems are found . in order to have a dns system with such a low eccentricity
, we need to invoke increasingly exotic ( and perhaps implausible ) evolutionary scenarios .
for instance , if the system began as a hierarchical triple @xcite , then the inner components could have formed a standard eccentric dns system early on
. later evolution of the outer member could have led to a circum - binary accretion disk that would have worked to circularize the inner system , after which the outer object would have exploded or otherwise been ejected from the system .
the other possible scenario is that the companion could be a massive wd , making the system an imbp .
its orbital eccentricity is somewhat high compared to most low - mass binary pulsars of similar periods ( based on @xcite ) , but not nearly as high as a dns , consistent with an imbp classification @xcite .
it falls in the locus of other co wds in the `` corbet '' ( binary period versus spin period ) diagram in @xcite . the pulsar mass is lower than most pulsar
wd binaries , but is consistent with the short orbital - period imbp discussed by @xcite which may indicate a similar formation mechanism involving a common envelope @xcite .
however , as a wd it would be extremely faint : far fainter than any of the optical companions to imbps currently known @xcite or indeed any wd companion to a millisecond pulsar ( msp ) with a similar mass @xcite ; it is perhaps the faintest wd ever observed . with the apparent magnitude limits from
, we can compute absolute magnitude limits in each band .
we use the distance @xmath71pc @xcite , and we estimate the extinction to be @xmath72mag from @xcite . in terms of bolometric luminosity the most constraining limit ends up coming from the @xmath38-band data , where we limit @xmath73 ( the @xmath47-band limit of @xmath74 is very similar , given slight differences in bolometric correction ) . for comparison , the companion to with a median companion mass of @xmath75 has @xmath76 @xcite . in
we plot the absolute magnitude against mass for pulsar+wd systems as well as select cool wds with parallax distances : even compared to the observed truncation of the cooling sequence in old halo globular clusters like ngc 6397 @xcite or m4 @xcite , the putative companion is far fainter : at the distance of ngc 6397 , our limit of @xmath73 translates to an apparent magnitude of @xmath77 , compared to @xmath78 , or @xmath79 for the coolest wds seen in ngc 6397 .
some of the difference comes from the change in radius : a @xmath80 wd has a radius about 65% of that of a typical @xmath81 wd , leading to a 1mag change in brightness at the same effective temperature .
but the difference in is more like 2.5mag , so the companion to must also be cooler than the known thick disk / halo wds . beyond the absolute magnitude , which is directly computable from observable quantities , we can limit the radius / temperature of a putative wd by using our @xmath38-band absolute magnitude limit to constrain the bolometric luminosity .
this is more complicated , as it involves atmosphere calculations in an uncertain and poorly tested regime , but it should be reasonably reliable .
we use the synthetic photometry and evolutionary models from @xcite and @xcite for h and he atmospheres , respectively . ] . for isolated wds pure he atmospheres
can be largely excluded because of bondi - hoyle accretion from the ism @xcite , and even small amounts of hydrogen mixed into the helium can cause near - infrared flux deficiencies like pure hydrogen ( see below ; @xcite ) . however , the binary orbit and msp wind in this system could have inhibited such accretion and therefore a he atmosphere is possible . in any case
a pure he atmosphere will serve as a limiting case compared to the h models .
these models are used to convert the absolute magnitude limits into temperature limits , so for simplicity we use the @xmath80 models ( differences in bolometric corrections as a function of mass are small , @xmath82mag ) .
the most constraining limit is again from the @xmath38-band data , where @xmath73 implies @xmath83k ( see ) for a h atmosphere .
the he - atmosphere models do not extend to sufficiently cool temperatures but stop at @xmath84k with @xmath85 . at lower temperatures the details of the atmospheric physics are rather uncertain , but a blackbody is likely an acceptable approximation ( p. bergeron , 2014 , private communication ) . with a he atmosphere
an effective temperature @xmath86k would be required ( ) . the h limits are more constraining since more of the flux appears in the optical regime rather than the near - infrared a consequence of collisionally induced absorption by molecular h@xmath87 @xcite .
these limits change slightly with mass given the small but finite mass uncertainties , since the radius would change with mass : going to the @xmath88 h model we can constrain @xmath89k ( at our nominal mass of @xmath90 the radius of a c / o wd is about @xmath91 , and it scales as @xmath92 ) . as inferred from
, the companion to would be far cooler than any known wd from other surveys ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) , where the coolest objects tend to have @xmath93k .
however , we can not exclude such a very cool wd on age grounds .
wd cooling curves , which start out having more massive objects warmer at the same age , eventually cross to have more massive objects cooler at the same age (; this is also visible in ) .
this is because massive wds crystallize earlier , at a higher @xmath94 ( but at a similar internal temperature ) , at which point the faster debye cooling takes over @xcite .
cooling ages for these models may not be reliable , as the impacts of state changes , sedimentation , and chemical processes are not precisely known , and the atmospheres are not trivial to calculate @xcite .
but we believe conservatively that the cooling age is close to 10gyr , almost certainly @xmath95gyr . in we
show example cooling curves , computed for thin and thick da atmospheres and c / o wds ( likely irradiation is a negligible perturbation to the wds surface temperature , given the measured spin - down luminosity of the pulsar ) . for the model closest to the best - fit mass of we would infer that the true age is near 9gyr , with the possible range from 612gyr .
the upper limit provided by the pulsar s characteristic spin - down age ( 34gyr after correction for the shklovskii effect [ @xcite ] ) is not constraining ; the assumption that the pulsar s initial spin period is much shorter than the current spin period is clearly not valid .
instead , we take as our upper limit to the age that of the milky way s halo ( @xmath96gyr ; @xcite ) minus the @xmath97myr required for the main - sequence lifetime of a @xmath98 progenitor @xcite , although this does not really exclude any models .
such an age would , however , imply a lower limit to the ( re-)birth period of about 25ms , assuming spin - down with a braking index @xmath99 ( magnetic dipole radiation ) .
we note that the cooling models in may not be the only solution for this progenitor : changing the wd composition ( likely it is below the transition to o / ne / mg wds based on @xcite , although binary evolution could change that ; also see @xcite ) or atmosphere ( helium , carbon , etc ) could lead to different solutions , and to draw robust conclusions we need to explore a wider range of models with better observational constraints .
there are also considerable complications and uncertainties in models for these temperatures : for instance , the models of ( * ? ? ?
* the basti database ) give rather different ages as @xmath94 never drops below @xmath100k for @xmath80 models , even for ages of @xmath101gyr , while @xcite and @xcite do have @xmath102 models go below 4000k ( note that the models in @xcite are primarily o / ne rather than c / o ) .
however , we believe the @xmath94 upper limits to be more robust , as they do tend to agree between different calculations . while extreme , the companion to may not be especially unique .
similar ultra - cool wds are presumably present in globular clusters and in the field even if they are often too faint to identify on their own .
individual ultra - cool wds can be identified but only if very nearby , like the two objects in @xcite at @xmath103pc . if we correct roughly for the different progenitor masses between the @xcite systems and @xcite and use a @xcite initial mass function , we would estimate @xmath104 massive wds of a similar age within 300pc , which is of the same order as the luminosity function from ( * ? ? ?
* also see @xcite ) extrapolated and @xcite say that at most a few percent of wds are lost off the faint end of the luminosity function . ] to @xmath105 .
instead , binary systems are the best way to identify cold wds ( e.g. , * ? ? ?
* ) , which is effectively the technique used here .
but even in binary systems where we know that a source is present , the systems will often be too distant for good constraints ( i.e. , psr j1454@xmath245846 in ; @xcite ) .
we still require a fortuitously nearby system for useful observations .
the occurrence of a nearby massive wd like the companion to is reasonably consistent with expectations based on the observed binary population : there are five pulsar binaries from the atnf pulsar catalog @xcite within 300pc , and the other four have low - mass he wd companions .
this @xmath106 ratio is similar to that for co wd compared to he wd companions in the whole atnf catalog ( also see @xcite ) , and the pulsars spin - down ages appear to have similar distributions for both companion types .
finally , we can ask whether an ns is the most likely companion to an ultra - cool wd .
most binaries are assumed to have mass ratios near one ( @xcite , but see @xcite ) , but a binary composed of two ultra - cool wds would be just as hard to detect optically as a single object .
if the companion were a lower - mass wd or a main - sequence star the binary could be visible , although it would require spectroscopic follow - up to identify the companion and in the absence of _ gaia _ this has not been done for the majority of stars within a few hundred pc .
so the situation of , with an ns companion , is reasonably plausible as the initial mass ratio would have been close to one and the chances of companion follow - up and identification after discovery of the pulsar are high .
we have determined an accurate mass for the partially recycled pulsar and its companion ; the latter is value consistent with both an ns and a wd . despite not finding the companion in a deep optical / near - infrared search
, we reject a dns explanation as the binary system shows evidence of circularization requiring mass transfer after the last supernova .
instead the companion is likely a high - mass wd . using the extremely precise distance determination from @xcite
, we can set a robust limit of @xmath107 .
this implies an very old and cool wd : fainter than all other pulsar companions by a factor of about 100 , and fainter than the lower - mass `` ultra - cool '' wd in the solar neighborhood by a factor of about four .
converting this limit to a temperature depends somewhat on the assumed mass and composition , but we believe an effective temperature limit of @xmath108k is a robust upper limit .
for such an object to not be older than the milky way requires that it have already entered the faster debye cooling regime , i.e. , that it already crystallized ( also see @xcite ) . future searches , if they can detect the companion to , will be a unique probe of the very late stages of wd evolution , with a well - determined mass and radius that are not usually available for studies of such objects .
we thank an anonymous referee for useful suggestions , and t. tauris , m. van kerkwijk , i. stairs , p. bergeron and r. oshaughnessy for helpful discussions .
is supported by the national science foundation grant ast-1312822 .
m.a.m . and d.r.l .
are supported by wvepscor , the nsf pire program , and the research corporation for scientific advancement .
jrb acknowledges support from wvepscor , the national radio astronomy observatory , the national science foundation ( ast 0907967 ) , and the smithsonian astrophysical observatory ( chandra proposal 12400736 ) .
a.t.d . was supported by an nwo veni fellowship .
some of the data presented herein were obtained at the w. m. keck observatory , which is operated as a scientific partnership among the california institute of technology , the university of california and the national aeronautics and space administration .
the observatory was made possible by the generous financial support of the w. m. keck foundation .
the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community .
we are most fortunate to have the opportunity to conduct observations from this mountain . based on observations obtained at the southern astrophysical research ( soar ) telescope , which is a joint project of the ministrio da cincia , tecnologia , e inovao ( mcti ) da repblica federativa do brasil , the u.s
. national optical astronomy observatory ( noao ) , the university of north carolina at chapel hill ( unc ) , and michigan state university ( msu ) . funding for sdss - iii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , and the u.s .
department of energy office of science .
the sdss - iii web site is http://www.sdss3.org/. we made extensive use of simbad , ads , and astropy ( http://www.astropy.org ; @xcite ) . ,
i. h. 2010 , in iau symp .
261 , relativity in fundamental astronomy : dynamics , reference frames , and data analysis , ed .
s. a. klioner , p. k. seidelmann , & m. h. soffel , ( cambridge : cambridge univ . press ) , 218
|
the recycled pulsar is one of the closest known neutron stars , with a parallax distance of @xmath0pc and an edge - on orbit .
we measure the shapiro delay in the system through pulsar timing with the green bank telescope , deriving a low pulsar mass ( @xmath1 ) and a high companion mass ( @xmath2 ) consistent with either a low - mass neutron star or a high - mass white dwarf .
we can largely reject the neutron star hypothesis on the basis of the system s extremely low eccentricity ( @xmath3)too low to have been the product of two supernovae under normal circumstances .
however , despite deep optical and near - infrared searches with soar and the keck telescopes we have not discovered the optical counterpart of the system .
this is consistent with the white dwarf hypothesis only if the effective temperature is @xmath4k , a limit that is robust to distance , mass , and atmosphere uncertainties .
this would make the companion to one of the coolest white dwarfs ever observed .
for the implied age to be consistent with the age of the milky way requires the white dwarf to have already crystallized and entered the faster debye - cooling regime .
|
a convex polyhedron @xmath0 is the intersection of half - spaces of the @xmath1-dimensional euclidean space @xmath2 .
we are interested in numerical calculation of the probability @xmath3 , where @xmath4 is a random vector distributed as a @xmath1-dimensional normal distribution with mean @xmath5 and covariance matrix @xmath6 .
when @xmath0 is the orthant @xmath7 in @xmath2 , the probability is called the orthant probability and can be regarded as a function of @xmath5 and @xmath6 . by the inclusion exclusion identity given in @xcite and @xcite , the probability content of the convex polyhedron can be written as a linear combination of the orthant probabilities .
for this reason , the literature includes many discussions of methods for evaluating the orthant probabilities .
for example , @xcite proposed a method based on recursive integration , and @xcite and @xcite proposed the use of the randomized quasi - monte carlo procedure . for details , see @xcite . in @xcite , the probability content of a general convex polyhedron
was evaluated by calculating the orthant probabilities .
the motivation of our study is to evaluate the probability content of a convex polyhedron by a completely different and novel approach that uses the holonomic gradient method ( hgm ) proposed in @xcite .
the hgm , which is based on the theory and algorithms of @xmath8-modules , is a method for numerically calculating definite integrals .
it can be applied to a broad class of problems .
in fact , various applications of hgm have been proposed , for example , @xcite , @xcite , and @xcite . in order to apply hgm
, we need to regard the probability @xmath9 as a function and provide an explicit pfaffian equation for it . in the case where @xmath0 is the orthant and @xmath10 , the orthant probability is a function of @xmath6 and schlfli gave a recurrence formula for it in @xcite . in @xcite , plackett generalized schlfli s result for the case where @xmath0 is the orthant and @xmath11 . in @xcite
, we provided a holonomic system and a pfaffian equation that is associated with the orthant probability .
our pfaffian equation corresponds with the reduction formula in @xcite and @xcite . in this paper
, we generalize our previous results @xcite and give a recurrence formula as a pfaffian equation for the case of a general convex polyhedron .
let @xmath12 be a cholesky decomposition of the covariance matrix @xmath6 , and let @xmath13 be a random vector that is @xmath1-variate normally distributed with mean vector @xmath14 and for which the covariance matrix is the identity matrix .
then , we have @xmath15 , and the set @xmath16 is also a convex polyhedron .
hence , it is enough to consider the case in which the mean vector of @xmath4 is @xmath14 and the covariance matrix of @xmath4 is the identity matrix . under this assumption
, the probability @xmath3 can be written as @xmath17 the polyhedron @xmath0 can be written as @xmath18 where @xmath19 and @xmath20 are real numbers .
we wish to study this integral with the hgm , and as a first step , we will assume that the convex polyhedron is in the `` general position ; '' the precise definition of `` general position '' will be given in section [ sec3 ] .
let @xmath21 be a @xmath22 matrix , and let @xmath23 be a vector with length @xmath24 .
we are interested in the analytic properties of the function @xmath25 which is defined on a neighborhood of @xmath26 . here
, we denote by @xmath27 the heaviside function .
note that this function is an interesting specialization of the one studied by aomoto in @xcite and aomoto , kita , orlik , and terao in @xcite .
for the meaning of the specialization , see remark [ rem:1 ] , below . in this paper , we provide a holonomic system and a pfaffian equation associated with this function .
the pfaffian equation is required by the hgm for @xmath28 . in order to explicitly provide the holonomic system
, we decompose the function by the inclusion exclusion identity associated with the polyhedron @xmath0 .
we also show that the holonomic rank of the system is equal to the number of nonempty faces of the polyhedron @xmath0 .
this pfaffian equation is a generalization of the recursion formula given by plackett @xcite .
in addition , the singular locus of the pfaffian equation is compatible with that of the schlfli function given in @xcite .
this paper is constructed as follows . in section [ sec2 ]
, we will give a brief explanation of holonomic modules and pfaffian equations .
in section [ sec3 ] , we provide an analytic continuation of the function @xmath28 . in section [ sec:6 ] ,
we prove an existence of an open neiborhood of a given point in general position . in section [ sec4 ]
, we give a system of linear partial differential equations with polynomial coefficients for the function @xmath28 and show that the system induces a holonomic module . in section [ sec5 ] , we show that the holonomic rank of the module is equal to the number of nonempty faces of @xmath0 and explicitly provide the pfaffian equation associated with the module .
before starting the main discussion , we briefly review holonomic modules and pfaffian equations . for a comprehensive presentation , see @xcite and the references cited therein . we denote by @xmath29 the ring of differential operators of @xmath24 variables @xmath30 with polynomial coefficients . here
, we put @xmath31 .
let us consider a system of linear partial differential equations @xmath32 for unknown functions @xmath33 .
let @xmath34 be the free @xmath35-module with the basis @xmath36 , and let @xmath37 be a @xmath35-submodule of @xmath34 generated by @xmath38 .
note that the basis @xmath36 is an arbitrary set and @xmath39 is not a function .
we denote by @xmath40 the quotient module @xmath41 .
the set consisting of the holomorphic functions on a domain @xmath42 forms a left @xmath35-module @xmath43 . for a morphism @xmath44 of left @xmath35-modules , the functions @xmath45 satisfy system .
for this reason , we call a vector - valued function @xmath46 on @xmath47 a solution of @xmath40 when there is a morphism of @xmath35-modules @xmath48 such that @xmath49 . 0 in order to define the notion of the holonomic modules , we need the sheaf @xmath50 of rings of linear partial differential operators with analytic coefficients .
let @xmath51 and @xmath52 .
for a differential operator @xmath53 , we define the order @xmath54 and the principal symbol @xmath55 of @xmath0 as @xmath56\end{aligned}\ ] ] where @xmath57 denotes the sheaf of holomorphic functions of @xmath58 .
similarly , for a element @xmath59 of @xmath60 , we also define the order @xmath54 and the principal symbol @xmath55 of @xmath0 as @xmath61)^m.\end{aligned}\ ] ] here , @xmath62 and @xmath36 denotes a basis of @xmath60 and that of @xmath63)^m$ ] .
the @xmath64$]-module @xmath65 generated by @xmath66 is called the _
graded module _ of @xmath67 .
the support of the quotient module @xmath68)^m/\overline{\mathcal n}$ ] is called the _ characteristic variety _ of @xmath69 , and denoted by @xmath70 . 0
we denote by @xmath71 $ ] a polynomial ring of @xmath72 variables @xmath73 . for a differential operator @xmath74 in @xmath35
, we define the order @xmath54 and the _ principal symbol _
@xmath55 of @xmath0 as @xmath75.\end{aligned}\ ] ] similarly , for a element @xmath76 of @xmath34 , we also define the order @xmath54 and the principal symbol @xmath55 of @xmath0 as @xmath77)^m.\end{aligned}\ ] ] here , @xmath78)^m$ ] is the free @xmath71$]-module with a basis @xmath79 and @xmath80 . by the theory of the grbner basis in weyl algebra , the _
characteristic variety _
@xmath81 of @xmath40 can be computed explicitly . for details , see @xcite . according to the bernstein inequality ,
the krull dimension of the characteristic variety is not less than @xmath24 ( see , e.g. , @xcite ) .
when the dimension of @xmath81 is equal to @xmath24 , the @xmath35-module @xmath40 is said to be _
when the system of differential equations induces a holonomic @xmath35-module , we call a holonomic system .
we denote by @xmath82 the ring of differential operators of @xmath24 variables @xmath30 with rational function coefficients .
the left @xmath35-module @xmath83}m$ ] is a left @xmath82-module , where @xmath84 is the field of rational functions . when the module @xmath40 is holonomic , @xmath83}m$ ] as a linear space over @xmath84 has finite dimension .
this value is called the holonomic rank of @xmath40 , and we denote it by @xmath85 .
let @xmath86 be the holonomic rank of @xmath40 , and let @xmath87 be a basis of @xmath83}m$ ] as a linear space over @xmath84 .
then , there exist rational functions @xmath88 such that @xmath89 in @xmath83}m$ ] .
moreover , the matrices @xmath90 satisfy the integrability condition @xmath91 we call equation a pfaffian equation associated with the holonomic module @xmath40 .
the union of the zero sets of the denominators of the elements of @xmath92 s is called the singular locus of the pfaffian equation .
note that a pfaffian equation associated with @xmath40 depends on the choice of the basis of @xmath83}m$ ] , and it is not unique .
0 when a vector valued function @xmath93 satisfies a holonomic system , there is a pfaffian equation associated with the holonomic module defined by .
note that a basis @xmath94 of @xmath83}m$ ] can be regarded as a function .
when a path @xmath95\rightarrow \mathbf c^n$ ] which does not through the singular locus of the pfaffian equation and the value @xmath96 at @xmath97 are given , we can evaluate @xmath98 as following .
firstly , we have an ordinary differential equation for the function @xmath99 from the pfaffian equation . next , we have the value of @xmath98 by solving the ordinary differential equation numerically with the initial value @xmath100 .
since each @xmath39 can be written as a linear combination of @xmath94 s with rational function coefficient , we can evaluate @xmath101 from the value of @xmath98 .
the method evaluating @xmath102 by the above procedure is called holonomic gradient method .
in this section , we show that the function @xmath28 in can be regarded as a real analytic function . since the heaviside function @xmath103 is the hyperfunction defined by @xmath104 , we can expect the function @xmath28 to be expressed in terms of a logarithmic function .
however , we can not find a suitable @xmath1-simplex for the @xmath1-form obtained by replacing @xmath105 in the integrand of with @xmath106 . in order to overcome this difficulty
, we use a decomposition of @xmath28 , which will be given in , and show that @xmath28 can be written as a linear combination of complex integrals .
this implies that @xmath28 is a real analytic function .
first , let us review some notions of polyhedra . in the remainder of this paper
, we will assume that @xmath1 and @xmath24 are positive integers .
a subset @xmath107 is called a _ half - space _ if @xmath27 can be written as @xmath108 for some @xmath109 .
a _ polyhedron _ is a finite intersection of half - spaces .
an inequality @xmath110 is called _ valid _ for a polyhedron @xmath0 if all the points of @xmath0 satisfy the inequality .
we call a subset @xmath111 an _ affine subspace _ if @xmath112 can be written as an intersection of hyperplanes . for a subset @xmath111 ,
the affine hull of @xmath112 is the smallest affine subspace which contains @xmath112 , and we denote it by @xmath113 .
the _ dimension _ of a polyhedron @xmath0 is the dimension of @xmath114 , and we denote it by @xmath115 . for a polyhedron @xmath0 and an inequality @xmath110 which is valid for @xmath0 , the intersection @xmath116
is called a _ face _ of @xmath0 .
the dimension of a face @xmath117 is the dimension of @xmath118 .
a _ facet _ of a polyhedron @xmath0 is a face of @xmath0 whose dimension equals to @xmath119 . for details ,
see @xcite .
in order to describe the combinatorial structure of a polyhedron , we use the notion of the abstract simplicial complex @xcite .
let @xmath120 be a set consisting of subsets of @xmath121:=\{1 , \ , 2 , \dots , n\}$ ] .
we call @xmath120 an abstract simplicial complex when @xmath122 and @xmath123 implies @xmath124 .
let @xmath120 and @xmath125 be two abstract simplicial complexes .
we say that @xmath120 is _ equal _ to @xmath125 and denote @xmath126 when @xmath120 and @xmath125 are equal as sets .
we say that @xmath120 and @xmath125 are equivalent and denote @xmath127 when there is a bijection @xmath128 such that @xmath129 if and only if @xmath130 .
let @xmath131 be a polyhedron , and let @xmath132 be all the facets of @xmath0 . for each facet
@xmath133 , there is a unique half - space @xmath134 that satisfies @xmath135 and @xmath136 ( see , e.g. , exercise 2.14(iv ) of lecture 2 in @xcite ) .
we call @xmath137 _ the family of the bounding half - spaces for the polyhedron @xmath0 .
_ the _ nerve _ of @xmath138 is the abstract simplicial complex defined by @xmath139 : f_j \neq \emptyset \ } , \quad \left(f_j:= \bigcap_{j\in j } f_j\right).\ ] ] we also call @xmath120 the _ abstract simplicial complex associated with the polyhedron @xmath0_. when @xmath120 is an abstract simplicial complex associated with a polyhedron @xmath0 , we have @xmath140 for any @xmath141 $ ] .
next , we introduce the notion of a polyhedron in the `` general position . ''
since we need to consider information for points at infinity , we will use the idea of `` homogenization . ''
the _ homogenization _ @xmath142 of a half - space @xmath143 is defined as @xmath144 for a family of half - spaces @xmath145 , we call @xmath146 the _ homogenization _ of @xmath147 . here , we put @xmath148 .
we say that a family of half - spaces @xmath149 ( or its homogenization @xmath150 ) is _ in general position _
when , for @xmath151 , @xmath152 is a @xmath153-dimensional cone ( i.e. , the affine hull of the cone is @xmath153-dimensional affine space ) or @xmath154 .
this is somewhat analogous to the `` general position '' for hyperplane arrangements in ( * ? ? ?
section 9 ) , but we emphasize that they are different ( see example [ ex:2 ] ) .
the polyhedron @xmath155 is _ in general position _ when the family @xmath147 of the bounding half - spaces of @xmath0 is in general position .
[ ex:2 ] let @xmath156 .
we define @xmath157 by @xmath158 then , the family of half - spaces @xmath159 is in general position . however , the family of half - spaces @xmath160 is not in general position .
in fact , the homogenization of @xmath147 and @xmath161 can be written as @xmath162 where @xmath163 calculating @xmath164 for each @xmath165 , we can show that @xmath166 is in general position .
for example , the set @xmath167 is a @xmath168-dimensional cone , and the set @xmath169 is equal to @xmath170 . on the other hand ,
the family @xmath171 is not in general position since the dimension of @xmath172 is not equal to @xmath173 .
hence , the polyhedron @xmath174 in figure [ fig:1](a ) is in general position , but the polyhedron @xmath175 in figure [ fig:1](b ) is not in general position .
we note that the hyperplane arrangement @xmath176 is not in general position in the sense of @xcite , but @xmath147 is in general position .
let @xmath177 . using the notation in example [ ex:2 ] , the family of the half - space @xmath178 , which is shown in figure [ fig:1](c ) , is not in general position .
however , the polyhedron @xmath179 is in general position since the family of the bounding half - spaces is @xmath180 . [ cols="^,^,^ " , ] note that our definition of general position is more restrictive than that of @xcite , and it is less restrictive than that of @xcite . for example , the polyhedron @xmath175 in example [ ex:2 ] is in general position by the definition in @xcite ; and the polyhedron @xmath174 in example [ ex:2 ] is not in general position by the definition in @xcite . let @xmath131 be a polyhedron .
suppose the family of bounding half - spaces for @xmath0 is given by @xmath181 we denote by @xmath182 the @xmath22 matrix @xmath183 , and by @xmath184 the vector @xmath185 .
let @xmath133 be the intersection of @xmath0 and the hyperplane @xmath186 .
the sets @xmath132 are all of the facets of @xmath0 .
let @xmath120 be the nerve of @xmath187 , which is the abstract simplicial complex of @xmath0 .
edelsbrunner showed the inclusion
exclusion identity for the indicator function of a polyhedron ( * ? ? ?
* lemma 5.1 . ) .
[ edelsbrunner ] if @xmath120 is the abstract simplicial complex associated with a polyhedron @xmath0 , then the indicator function of @xmath0 can be written as @xmath188 for the polyhedron @xmath174 in example [ ex:2 ] , the inclusion exclusion identity can be written as follows : @xmath189 the first term of the right - hand side corresponds to the empty set . with the heaviside function @xmath27
, the edelsbrunner s identity can be written as @xmath190 under the general position assumption , this identity can be generalized as follows .
[ ptb:8 ] in the notation above , if the polyhedron @xmath0 is in general position , then there exists a neighborhood @xmath47 of @xmath191 such that the equation @xmath192 holds for all @xmath193 .
our proof of theorem [ ptb:8 ] is technical , and it is independent from the other parts of this paper ; thus , it is given in the appendix .
consider @xmath24 polynomials @xmath194 with variables @xmath195 , and let @xmath196 for @xmath197 .
note that @xmath198 .
we put @xmath199 by theorem [ ptb:8 ] , the function @xmath28 in can be decomposed as @xmath200 on a neighborhood of @xmath191 if the polyhedron @xmath0 is in general position . in order to give analytic continuations of the function @xmath28 , it is enough to consider @xmath201 . for @xmath202
, let @xmath203 be an @xmath204 matrix , where @xmath205 is the number of the elements in @xmath117 and @xmath206 this is a submatrix of the gram matrix of @xmath21 .
the matrices @xmath203 are symmetric and positive semidefinite .
since the function @xmath201 can be written as @xmath207 we can expect that @xmath201 is written by a @xmath1-simplex and the @xmath1-form @xmath208 in fact , we can find a suitable @xmath1-simplex and thus have proposition [ prop : c - int ] . [
lem : c - int ] if the polyhedron @xmath0 is in general position , then for @xmath197 , the value of @xmath209 can be written as @xmath210 here , for @xmath211 , @xmath212 is a smooth map from @xmath2 to @xmath213 .
we suppose the multivalued function @xmath214 satisfies @xmath215 and the branch cut is @xmath216 .
let @xmath217 be the number of elements in @xmath117 . by the general position assumption ,
@xmath217 is not greater than @xmath1 .
we denote by @xmath218 all of the elements of @xmath117 .
since the polyhedron @xmath0 is in general position , the vectors @xmath219 are linearly independent .
consequently , the determinant @xmath220 is not zero .
let @xmath221 be an orthonormal basis of the orthogonal complement of the subspace @xmath222 .
we denote the vector @xmath223 by @xmath224 .
the matrix @xmath225 is regular , and we set @xmath226 . without loss of generality , we can assume @xmath227 . under this assumption , we have @xmath228 .
let @xmath229 be a positive bounded function on @xmath230 , i.e. , @xmath231 and @xmath232 .
for a vector @xmath233 , we define @xmath234 by @xmath235 here , we put @xmath236 for @xmath237 . by the coordinate transformation @xmath238 , the integral
can be written as @xmath239 where @xmath240 calculating this integral recursively , we have @xmath241 where @xmath242 . by the coordinate transformation @xmath243 , the above integral is @xmath244 which is equal to @xmath209
. moreover , we have the following proposition .
[ prop : c - int ] let @xmath245 be a domain @xmath246 .
the function @xmath201 can be written as @xmath247 on a connected open neighborhood of @xmath248 in @xmath245 . here
, @xmath249 denotes the integral path @xmath212 in lemma [ lem : c - int ] .
and we suppose the multivalued function @xmath214 satisfies @xmath215 and the branch cut is @xmath216 . since @xmath250 , by arguments similar to those in the proof of lemma [ lem : c - int ] , @xmath201 can be written as @xmath251 the matrix @xmath252 and the integral path @xmath253 can be constructed similarly .
we need to show that the above integral is equal to .
there is a smooth path @xmath254)$ ] in the general linear group of degree @xmath1 over @xmath255 such that @xmath256 and @xmath257 .
the homotopy between @xmath258 and @xmath253 is given by @xmath259 consequently , the value of the integral on the right - hand side of does not change when we change the integral path with @xmath253 .
the function @xmath28 is a real analytic function , and it has an analytic continuation along every path in @xmath260 .
in this section , we explicitly give a system of differential equations for the function @xmath261 and show that the system is holonomic .
note that the function @xmath262 is defined by . since the equation @xmath263 holds on a neighborhood of @xmath191 , a holonomic system for @xmath28 can be given as the integration module of a holonomic module associated with @xmath264 for more about the integration module , see @xcite and @xcite . for @xmath265 $ ]
, we define a hyperfunction @xmath266 by @xmath267 note that @xmath268 .
[ lem:2 ] if @xmath269 $ ] is not an element of @xmath270 , then we have @xmath271 for @xmath197 .
consequently , we have @xmath272 . since @xmath120 is an abstract simplicial complex
, we have @xmath273 .
take @xmath274 , then we have @xmath275 , since @xmath276 .
we now provide a system of differential equations for the @xmath266 s .
let @xmath277 be a vector whose elements are functions indexed by the set @xmath120 .
let us consider the system defined by the following : @xmath278 where @xmath279 for @xmath280 .
[ lem:1 ] let @xmath281 for @xmath197
. then the function @xmath102 satisfies equations , , , and . when @xmath282 , it is obvious that equations , , , and hold , since @xmath283 .
suppose @xmath117 is not the empty set .
we first check equation .
since @xmath284 for @xmath285 , we have @xmath286 here , we apply the chain rule for hyperfunctions .
the equation above implies .
we now show equation .
when @xmath285 , both sides of are equal to @xmath14 .
when @xmath287 , we have @xmath288 by the chain rule .
equation holds by lemma [ lem:2 ] . for @xmath289 , we have @xmath290 this implies . by
, we have @xmath291 by this equation and lemma [ lem:1 ] , we have the following proposition .
[ ann_chip ] the vector - valued function @xmath292 satisfies equations , , , and .
next , we show that the system defined by , , , and for @xmath293 is a holonomic system . in the remainder of this paper
, we will frequently use the following rings : @xmath294&:= \mathbf c[a_{ij } , b_j , x_i , \xi_{a_{ij } } , \xi_{b_j } , \xi_{x_i}:\ , 1\leq i\leq d ,
\ , 1\leq j\leq n].\\\end{aligned}\ ] ] we also use the free modules @xmath295 , and @xmath296^{|\mathcal f|}$ ] , whose basis is @xmath297 .
[ prop : chip ] let @xmath298 be the sub left @xmath299-module of @xmath300 generated by @xmath301 then , the quotient module @xmath302 is holonomic . the principal symbols of , , , and
are the following : @xmath303 for @xmath122 , let @xmath304 be an algebraic variety defined by @xmath305 by ( * ? ? ?
* proposition 1 ) , the union @xmath306 includes @xmath81 . since the rank of the jacobian matrix of and is @xmath307 , the krull dimension of @xmath304 is equal to @xmath307 .
hence , the dimension of @xmath81 is not greater than @xmath307 .
let @xmath308 be a @xmath299-submodule of @xmath300 generated by , , , and @xmath309 proposition [ prop : chip ] implies that the quotient module @xmath310 is a holonomic module .
moreover , the function @xmath311 is a solution of @xmath308 ( see @xcite ) . by lemma [ lem:1 ] ,
the function @xmath312 is also a solution of @xmath308 for @xmath197 . calculating the integration module of @xmath308 with respect to @xmath58
, we have the following theorem .
[ th : hol ] let @xmath37 be the sub left @xmath313-module of @xmath314 generated by @xmath315 then the quotient module @xmath316 is isomorphic to the integration module @xmath317 .
consequently , @xmath40 is a holonomic module .
we denote by @xmath318 the canonical morphism from @xmath314 to the integration module @xmath317 .
let @xmath319 .
since we have @xmath320 the @xmath299-module @xmath308 is generated by , , and @xmath321 consequently , we have @xmath322 .
next , we show the opposed inclusion @xmath323 . regarding @xmath324 as a subset of @xmath325 ,
the left @xmath313-module @xmath326 includes @xmath327 .
since the module @xmath308 is generated by , , , and , the module @xmath326 can be written as @xmath328 here , we denote by @xmath329 the differential operators in , , and . the left @xmath313-module is equal to @xmath330 note that the first term of is different from that of .
in fact , for any @xmath331 and the differential operator in , , and , we have @xmath332 these equations imply that for any differential operator @xmath333 in , , and , the operator @xmath334 is an element of @xmath335 . consequently , module includes @xmath336 by induction on the multi - index @xmath337 , module includes @xmath338 for any @xmath337 .
hence , module includes .
the opposite inclusion is obvious .
we denote by @xmath339 the differential operators in , , and . by and
, the left @xmath313-module @xmath340 is equal to module .
obviously , this module is equal to the left @xmath313-module @xmath341 where @xmath342 .
note that the module @xmath326 is equal to . since @xmath327 is a subset of the intersection of @xmath314 and ,
we have @xmath343 let @xmath344 .
then the element @xmath0 can be written as @xmath345 , where @xmath346 and @xmath347 .
the element @xmath348 is an element of the @xmath299-module @xmath349 .
let @xmath350 be a lex order which satisfies @xmath351 for @xmath352 and any monomial @xmath353 .
since the grbner basis for @xmath354 with respect to @xmath350 is given by @xmath355 the leading term of @xmath356 must be divided by some @xmath357 . since @xmath358 , we have @xmath359 .
therefore , we have @xmath360
let @xmath40 be the module derived in theorem [ th : hol ] . in this section , we will evaluate the holonomic rank of @xmath40 and derive a pfaffian equation associated with @xmath40 .
the following lemma gives a lower bound of the holonomic rank .
[ lem : liearly - independent ] the real analytic functions @xmath201 , where @xmath117 runs over the abstract simplicial complex @xmath120 associated with @xmath0 , are linearly independent solutions of @xmath40 .
consequently , the holonomic rank of @xmath40 is not less than the number of the nonempty faces of @xmath0 . by proposition [ prop : c - int ] and theorem [ th : hol ] ,
it is obvious that the function @xmath361 is a solution of @xmath40 .
suppose a matrix @xmath362 and a vector @xmath363 satisfy equation .
note that @xmath19 and @xmath20 are real numbers .
let @xmath364 be a sufficiently small neighborhood of @xmath365 .
note that the function @xmath361 is defined on @xmath366 , by proposition [ prop : c - int ] .
we prove the linear independence of these functions .
suppose @xmath367 and @xmath368 we denote @xmath369 by @xmath370 .
suppose @xmath371 for some complex numbers @xmath372 .
take an arbitrary @xmath373 such that @xmath374 , and suppose @xmath375 .
it is enough to show that @xmath376 .
note that @xmath377 for @xmath378 , since @xmath379 .
define @xmath380 and @xmath381 as @xmath382 since there is an element of @xmath383 which is not included in @xmath384 for @xmath385 , we have @xmath386 for all @xmath387 . by the lebesgue convergence theorem , we have @xmath388
since we have @xmath389 for all @xmath387 , again by the lebesgue convergence theorem , we have @xmath390 . by the assumption of the induction ,
we have @xmath391 taking the limit of both sides as @xmath392 , we have @xmath393 .
the holonomic rank of @xmath40 is equal to the number of nonempty faces of @xmath0 , i.e. , @xmath394 in addition , a pfaffian equation associated with @xmath40 is given by @xmath395 here , @xmath396 is the inverse matrix of @xmath203 in .
note that the right - hand side of can be reduced by and . we have and by and , respectively . by and
, the right - hand side of can be written as @xmath397 consequently , we have . by , , and ,
the module @xmath398 } m$ ] is spanned by @xmath399 as a linear space over @xmath400 , and we have @xmath401 . by this inequality and lemma [ lem : liearly - independent ] , we have @xmath402 . [ rem:1 ]
we note that the heaviside function @xmath105 is equal to @xmath403 as a schwartz distribution .
thus we may expect that our integral representation is a `` specialization '' of the integral considered in the cohomology groups in @xcite .
however , we have no rigorous understanding of this limiting procedure of the twisted cohomology .
an interesting observation is that the holonomic rank of @xmath40 can be smaller than the dimension of the twisted cohomology group @xmath404 , where @xmath405 and @xmath406 is a connection defined in @xcite . in our case , the dimension of the twisted cohomology is @xmath407 . the number of faces of the polyhedron in general position with @xmath24 facets can be smaller than this value
. * acknowledgements . *
the author wishes to thank nobuki takayama for useful discussions during the preparation of this paper , and akimichi takemura for the suggestion that the holonomic gradient method can be applied to the evaluation of the normal probability of convex polyhedra .
10 k. aomoto
. analytic structure of schlfli function .
, 68:116 , 1977 .
k. aomoto .
configurations and invariant gauss - manin connections of integrals i. , 5(2):249287 , 1982 .
k. aomoto and m. kita . .
springer , new york , 2011 .
k. aomoto , m. kita , p. orlik , and h. terao . twisted de rham cohomology groups of logarithmic forms .
, 128:119152 , 1997 .
j. e. bjrk . .
north - holland , new york , 1979 .
h. edelsbrunner .
the union of balls and its dual shape .
, 13:415440 , 1995 .
h. i. gassmann .
multivariate normal probabilities : implementing an old idea of plackett s .
, 12(3):731752 , 2003 .
a. genz .
numerical computation of multivariate normal probabilities .
, 1:141150 , 1992 .
a. genz and f. bretz . .
lecture notes in statistics .
springer - verlag , heidelberg , 2009 .
a. genz , f. bretz , t. miwa , x. mi , f. leisch , f. scheipl , and t. hothorn . , 2013 .
r package version 0.9 - 9996 .
h. hashiguchi , y. numata , n. takayama , and a. takemura .
holonomic gradient method for the distribution function of the largest root of wishart matrix .
, 117:296312 , 2013 .
t. hibi , k. nishiyama , and n. takayama .
pfaffian systems of @xmath408-hypergeometric equations .
http://arxiv.org/abs/1212.6103 , 2012 .
t. koyama and a. takemura .
calculation of orthant probabilities by the holonomic gradient method .
http://arxiv.org/abs/1211.6822 , 2012 .
s. kuriki , t. miwa , and a. j. hayter .
abstract tubes associated with perturbed polyhedra with applications to multidimensional normal probability computations . in t. hibi , editor , _ harmony of grbner bases and the modern industrial society _ , pages 169183 , singapore , 2012 .
world scientific .
a. miwa , j. hayter , and s. kuriki .
the evaluation of general non - centered orthant probabilities .
65:223234 , 2003 . d. q. naiman and h. p. wynn .
abstract tubes , improved inclusion - exclusion identities and inequalities and importance sampling . , 25(5):19541983 , 1997 .
h. nakayama , k. nishiyama , m. noro , k. ohara , t. sei , n. takayama , and a. takemura .
holonomic gradient descent and its application to the fisher - bingham integral .
, 47:639658 , 2011 .
t. oaku .
computation of the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients .
, 11:485497 , 1994 .
t. oaku .
algorithms for b - functions , restrictions , and algebraic local cohomology groups of d - modules .
, 19:61105 , 1997 .
t. oaku , y. shiraki , and n. takayama .
algorithms for d - modules and numerical analysis . in z.
li and w. sit , editors , _
computer mathematics _ , pages 2339 , river edge , 2003 .
world scientific .
r. l. plackett . a reduction formula for normal multivariate integrals .
, 41(3/4):351360 , 1954 . l. schlfli . on the multiple integral @xmath409
whose limits are @xmath410 and @xmath411 .
, 2:261301 , 1858 .
t. sei , h. shibata , a. takemura , k. ohara , and n. takayama .
properties and applications of fisher distribution on the rotation group . , 116:440455 , 2013 . g. m. ziegler . .
springer - verlag , new york , 1995 .
in this appendix , we prove theorem [ ptb:8 ] . for this purpose ,
we need to present some notation and some lemmas , most importantly , theorem [ ptb:5 ] .
the argument in the proof of lemma [ ptb:1 ] is also important , since analogous arguments will appear repeatedly in the proof of theorem [ ptb:5 ] .
let @xmath412 be a matrix where we denote by @xmath413 the @xmath414-th column vector of @xmath21 .
put @xmath415 . for a matrix @xmath21 , we put @xmath416 ) , \\ \mathcal h(a ) & : = \left\{h_1(a ) , \dots , h_n(a)\right\ } , \\
p(a ) & : = \bigcap_{j=1}^n h_j(a ) , \\
f_j(a ) & : = \partial h_j(a ) \cap p(a)\quad ( j\in [ n ] ) , \\ f_j(a)&:= \bigcap_{j\in j } f_j(a)\quad ( j\subset [ n ] ) , \\
\mathcal f(a ) & : = \left\{j\subset[n ] : f_j(a)\neq \emptyset \right\ } .\end{aligned}\ ] ] note that @xmath417 is not necessarily a facet of @xmath418 , and @xmath419 is not necessarily equivalent to the abstract simplicial complex associated with @xmath418 . for this difficulty ,
we need the notion of families of half - spaces in general position .
in fact , in lemma [ ptb:4 ] , we will show that the abstract simplicial complex associated with @xmath418 is equivalent to @xmath419 under the general position assumption , which is required by the proof of theorem [ ptb:8 ] . in order to consider combinatorial structures at the point at infinity
, we introduce the following notion : for the abstract simplicial complex @xmath120 associated with a polyhedron , the _ homogenization _ of @xmath120 is the abstract simplicial complex defined by @xmath420 since @xmath421 , we have @xmath422 for @xmath141 $ ] .
note that @xmath423 does not hold in general .
the following are the `` homogenization '' of the notations given in the previous paragraph . for a matrix @xmath21
, we put @xmath424 for all @xmath151 , @xmath425 includes the zero vector .
analogous to the case of the abstract simplicial complex associated with polyhedra , we have @xmath426 . for a family of half - spaces in general position
, we have the following lemma , which is required by the proof of lemma [ ptb:4 ] . [ ptb:1 ]
suppose @xmath427 is in general position .
let @xmath265 $ ] .
if @xmath428 , then the set @xmath429 is a @xmath430-dimensional face of @xmath418 . if @xmath431 , then the set @xmath429 is empty .
in particular , @xmath418 is a @xmath1-dimensional polyhedron . if @xmath431 , it is obvious that @xmath432 .
let us consider the case where @xmath428 . put @xmath433 . since @xmath427 is in general position , we have @xmath434 or @xmath435 for any @xmath436 .
in fact , the equality of contradicts the assumption about the dimension when @xmath437 .
however , we have @xmath437 , since @xmath428 implies @xmath438 .
hence , condition holds for any @xmath436 . for @xmath436 ,
take @xmath439 .
then the affine combination @xmath440 is an element of @xmath441 where @xmath442 denotes the interior of @xmath112 .
since @xmath443 , we have @xmath444 let @xmath58 be an element of this set , and let @xmath445 be an open ball centered at @xmath58 whose closure is included in @xmath446 .
let @xmath447 be an arbitrary point in @xmath448 , and let @xmath449 be an intersection point of @xmath450 and the line between @xmath58 and @xmath447 .
since @xmath451 and @xmath447 can be written as an affine combination of @xmath58 and @xmath449 , we have @xmath452 by the arbitrariness of @xmath447 , we have @xmath453 analogously , we have @xmath454 since @xmath455 is in general position , the left - hand side of is a @xmath153-dimensional cone . consequently , the vectors @xmath456 are linearly independent . moreover , the vectors @xmath457 are linearly independent . in fact , if there are @xmath458 such that @xmath459 , then we have @xmath460 for @xmath461 hence , the dimension of @xmath462 is equal to @xmath430 . put the notion in proposition 1.9(farkas lemma ) in @xcite as @xmath471 , i\in[n]}\in \mathbf r^{m\times d } , & z & = ( a_{01 } , \dots , a_{0n})^\top \in \mathbf r^m , \\
\mathbf a_0 & = ( -c_1 , \dots , -c_d ) \in \mathbf r^d , & z_0 & = c_1\\ \mathbf c & = ( \lambda_1 , \dots , \lambda_n).\end{aligned}\ ] ] note that we assume @xmath473 and @xmath418 is a @xmath1-dimensional polyhedron .
let @xmath117 be a facet of @xmath418 ; then there is an inequality @xmath474 valid for @xmath418 such that @xmath475 since @xmath418 is not empty , condition ( ii ) in the farkas lemma does not hold . by the farkas lemma ,
there exist @xmath468)$ ] such that @xmath476 moreover , there is a unique @xmath477 which is greater than @xmath14 .
in fact , if there is not such a @xmath414 , then we have @xmath478 .
this implies @xmath479 or @xmath282 .
this is a contradiction .
[ ptb:5 ] let @xmath131 be a polyhedron in general position , let @xmath24 be the number of facets of @xmath0 , and let @xmath120 be the abstract simplicial complex associated with @xmath0 .
then , the set @xmath493 is an open set of @xmath494 .
* step 1 * .
first , we define @xmath304 . for @xmath497 ,
let @xmath304 consist of @xmath498 such that the vectors @xmath456 are linearly independent and the set includes a nonzero element .
for @xmath151 such that @xmath499 , let @xmath304 be the intersection of @xmath500 and @xmath501 suppose @xmath503 .
let @xmath504 ; then we have @xmath505 by the definition of @xmath47 . when @xmath506 , the set is equal to @xmath425 and includes a nonzero element .
when @xmath507 , by an argument analogous to that in the proof of lemma [ ptb:1 ] , we can show that the set includes a nonzero element . consequently , equation holds .
since the left - hand side of is a @xmath153-dimensional affine subspace , the vectors @xmath456 are linearly independent .
hence , we have @xmath508 .
suppose @xmath509 .
for @xmath504 , it is obvious that @xmath434 . by the assumption for @xmath120
, we have @xmath422 for arbitrary @xmath141 $ ] . by the argument in the previous paragraph
, @xmath504 implies @xmath510 .
hence , we have @xmath511 .
* step 3 * .
we show that the left - hand side of includes the right - hand side .
suppose a matrix @xmath21 is included in the right - hand side of .
let @xmath265 $ ] .
when @xmath122 , @xmath512 implies that the set includes a nonzero element .
hence , we have @xmath513 . when @xmath514 , @xmath515 implies @xmath516 .
consequently , we have @xmath517 .
next , we show that @xmath427 is in general position .
it is enough to show that @xmath425 is a @xmath153-dimensional cone or that it is equal to @xmath154 for @xmath502 .
when @xmath503 , set is not empty and equation holds . since the vectors @xmath456 are linearly independent , the right - hand side of is a @xmath153-dimensional affine subspace .
when @xmath518 , we have @xmath519 . hence , @xmath427 is in general position
. therefore , equation holds .
suppose @xmath520 .
let @xmath515 .
since the vectors @xmath521 are linearly independent , there is a subset @xmath522 $ ] such that @xmath523 and the @xmath524 matrix @xmath525 has the inverse matrix @xmath526 .
let @xmath527 be a bijection , then @xmath528 defines a coordinate transformation . by this coordinate transformation ,
the condition that set includes a nonzero element can be written as follows : there exists @xmath529 such that @xmath530 let @xmath531 be the set consisting of @xmath21 such that the matrix @xmath525 is invertible and there exists @xmath532 satisfying condition .
then , we have @xmath533 , \\ |i|=|j| } } v_{ij}.\ ] ] in equation , the value of @xmath534 is determined uniquely by the value of the other variables .
we can regard as a condition for the variables @xmath535 .
the set @xmath531 is the projection of the following open subset of @xmath536 : the set consisting of @xmath537 that satisfies and in which @xmath525 is invertible .
this implies that @xmath538 is open .
consequently , @xmath304 is an open set .
suppose @xmath539 .
there is the isomorphism between the set and @xmath540 . here
, @xmath541 denotes the @xmath24-th direct product of a @xmath1-dimensional sphere @xmath542 .
since @xmath304 is closed under the transformation which multiplies column vectors by positive numbers , @xmath543 the image of @xmath304 under the isomorphism is @xmath544 .
it is enough to show @xmath545 is an open set . since the projection @xmath546 is a continuous map from a compact set to a hausdorff space , it is a closed map .
the set @xmath547 is the projection of the following closed set of @xmath548 : the intersection of @xmath548 and the subset of @xmath549 consisting of @xmath550 satisfying @xmath551 hence , @xmath545 is open . with the notation of theorem [ ptb:5 ] ,
it is enough to show that for @xmath504 , the abstract simplicial complex associated with @xmath418 is equivalent to @xmath120 .
let @xmath504 . by lemma [ ptb:4 ] ,
the abstract simplicial complex associated with @xmath418 is equivalent to @xmath419 . by corollary [ ptb:11 ] , @xmath517 implies @xmath552 .
|
the probability content of a convex polyhedron with a multivariate normal distribution can be regarded as a real analytic function .
we give a system of linear partial differential equations with polynomial coefficients for the function and show that the system induces a holonomic module .
the rank of the holonomic module is equal to the number of nonempty faces of the convex polyhedron , and we provide an explicit pfaffian equation ( an integrable connection ) that is associated with the holonomic module .
these are generalizations of results for the schlfli function that were given by aomoto . + _
keywords_. convex polyhedron , inclusion exclusion identity , holonomic modules , holonomic rank , pfaffian equation + msc classes : 16s32 , 62h10
|
a couple of decades ago it was found that radio emission is observed by about three times more often from double galaxies and members of groups of galaxies than from single , isolated galaxies ( tovmassian @xcite ; sulentic @xcite ) .
it is widely accepted that galactic nuclear activity , one of the manifestations of which is a radio emission , is a result of interaction between galaxies . events of interaction should take place much more often in dense environments of compact groups ( cgs ) of galaxies .
therefore it was expected that some of member galaxies in cgs should be relatively strong radio emitters .
the first systematic search for radio emission from hickson compact groups ( hcgs ) , revealed a radio emission at 18 cm above a flux limit of 1.5 mjy from 41 out of 88 observed member galaxies ( menon & hickson @xcite ) .
it was found ( tovmassian & shahbazian @xcite ) that in groups of galaxies the radio emission is most often observed from the optically first - ranked galaxies .
menon ( @xcite ) showed that the radio emission in hcgs is almost always associated with the first - ranked _ elliptical _
galaxies , while the radio detected _ spirals _ are uniformly distributed among the three brightest members of a group .
prior to hcgs the lists of the so called shakhbasian compact groups of compact galaxies ( scgcgs ) were published ( shakhbazian @xcite , baier & tiersch @xcite , and references therein ) .
it is the largest and a relatively homogeneous sample of galaxy groups known up to now .
shakhbazian s cgs were found on the poss prints by an eye search .
they were identified according to the following criteria : - the groups consist of 5 - 15 members ; - the apparent magnitude of the individual galaxies is between @xmath1 and @xmath2 ; - the groups are compact , i.e. the distances between galaxies are only 3 - 5 times the diameters of galaxies ; - nearly all member galaxies are extremly red , at most 1 - 2 blue galaxies in a group is an exception ; - members of groups are compact ( relatively high surface brightness , borders not diffuse ) ; - the groups are isolated .
shcgs originally were called compact groups of _ compact _ galaxies , because the images of most of the constituent galaxies in these groups seemed very compact .
later observations of these groups with high angular resolution revealed that member galaxies mostly are of e and s0 types .
the fraction of early type galaxies is 77% in scgs compared to 51% in hcgs and 40% in field galaxies ( tiersch et al .
member galaxies in these groups are significantly redder than the field galaxies of the same morphological type .
the difference in b - v and v - r is by @xmath3 redder compared to galaxies of the rc3 .
anyhow , there were found emission - line , and even seyfert galaxies among shakhbazian groups members .
since , as it was found out , members of shakhbazian groups turned to be not `` compact '' , these groups in recent papers were called shakhbazian compact groups of galaxies ( scggs ) ( tovmassian et al .
@xcite , hereafter tmtst , @xcite ) , or in analogy with hcgs , - shakhbazian compact groups ( shcgs ) ( tovmassian et al .
@xcite ) . galaxies in shcgs are relatively weak , because they are at least by three times further than hcgs ( tmtst ) .
mainly for this reason , and probably for very unusual supposed composition the shcgs attracted little attention until recently .
the redshifts of only about 70 relatively bright and nearby shcgs have been measured so far ( robinson & wampler @xcite ; arp et al .
@xcite ; mirzoyan et al .
@xcite ; amirkhanian & egikian @xcite ; amirkhanian @xcite ; kodaira et al .
@xcite , @xcite ; kodaira & sekiguchi @xcite ; lynds et al .
@xcite ; del olmo & moles @xcite ; tiersch et al .
@xcite ) .
the densities of galaxies in shcgs approaches to @xmath4 mpc@xmath5 .
hence the interaction and merging processes should be frequent in them . as a consequence
some shcgs could be a radio and fir emitters .
however due to larger distances and different morphological content ( tiersh et al .
@xcite ) we could expect smaller rate of radio and fir detections from shcgs in comparison with hcgs .
the results of the search for fir emission of shcgs were published recently by tmtst . in this paper
we present the results of a search for radio emission from shcgs .
three hundred seventy seven groups were included in the original lists of shcgs .
the positions of the group centers were given with accuracies of about @xmath6 .
for identification of radio sources with shcgs we used accurate coordinates of member galaxies measured afterwards from the digitized sky survey ( dss ) by stoll et al .
( @xcite , and references therein ) .
positions of four groups , shcg 206 , 241 , 301 and 353 , were largely incorrect in original lists , and they were not found during accurate position measurements . it was revealed also that the group shcg 214 coincides with shcg 252 , and the group shcg 340 consists of two groups , shcg 340a and shcg 340b .
spectral observations showed that five groups , shcg 12 , 13 , 78 , 146 , and 180 , consist mainly of stars .
thus we searched for a radio emission at the positions of 367 shcgs .
for looking for a radio emission from shcgs we first examined the nrao vla sky survey ( nvss ) ( condon et al .
@xcite ) at 1.4 ghz at the positions of all 367 shcgs .
generally an area @xmath7 centered on each group was investigated .
the first survey ( white et al .
@xcite ) , made at the same frequency , was used in addition for those 175 shcgs that are located in the area of the sky , covered by the survey until 1st november 1998 .
the errors of the radio position measurements in the used surveys are small enough . for the nvss sources they vary from @xmath8 for sources stronger than 15 mjy , to @xmath9 at the survey flux limit at @xmath10 mjy ( condon et al .
@xcite ) . for the first radio sources the errors are smaller : @xmath11 for sources with flux densities @xmath12 mjy , and @xmath13 at the survey treshold of 1 mjy ( white et al .
three hundred fifty three nvss radio sources were found within boundaries of 179 shcgs .
fifty three of these sources were registered also in the first .
seven more sources were found only in the first .
the more correct positions and fluxes on the latter radio sources from the first were used in the farther discussion .
high positional accuracies of the used radio surveys allowed to identify with high confidence some of the found radio sources with certain galaxies in dense environments of shcgs .
the positions of these radio sources coincide appreciably well , generally within @xmath14 , with corresponding objects in shcgs .
the probability of the reality of radio identifications was estimated by the `` likelihood ratio '' ( lr ) ( de ruiter et al .
it was assumed that the density of background radio sources is @xmath15 per @xmath16 for high galactic latitudes ( cohen et al .
@xcite ) the identification was considered as reliable if @xmath17 .
this may not be very correct in this case , since we identify radio sources in dense groups of galaxies .
for this reason we inspected each identification on the dss images .
ninety three of the found 353 radio sources were identified with certain objects in 74 shcgs .
the list of identifications is presented in table 1 .
the shakhbazian designation of the group and the number of the galaxy ( as labelled in the original shcg finding charts ) , considered to be the radio emitter , is given in the first row of column 1 .
the ra and dec ( j2000.0 ) of the galaxy are given in columns 2 and 3 of the first row . in the second row
the designation of the identified radio source ( column 1 ) , its coordinates ( column 2 and 3 ) , and the flux density ( column 4 ) are given . in five cases
( shcg 054 , 163 , 298 , 347 , and 352 ) the nvss radio source is located between two nearby galaxies , and it was not possible to determine which of them is the radio emitter .
it could not be excluded , however , that both galaxies are radio emitters .
two or more radio emitting galaxies were found in thirteen shcgs .
radio sources found within boundaries of nine shcgs : 051 , 057 , 120 , 203 , 248 , 250 , 330 , 346 , and 352 , coincide appreciably well with the positions of relatively weak objects , not considered as member galaxies of corresponding groups in the original shcg lists .
these identifications should be considered as tentative .
though the mentioned objects could , however , be members of corresponding groups .
future spectral observations of these weak objects may clarify whether they are really members of the corresponding groups .
two hundred sixty radio sources detected within the boundaries of 99 shcgs were not identified with any visible object .
they may be just background radio sources .
_ shcg 001.01_. the central brightest galaxy of s0 type .
there is an emission line at 3727 in the spectra of this galaxy ( robinson , & wampler @xcite ) .
_ shcg 003.01_. the central brightest galaxy .
the group is known also as vv 153 .
_ shcg 007.01_. the central brightest galaxy .
_ shcg 009.07_. the galaxy is located at the edge of the group .
_ shcg 010.01_. the central brightest spiral galaxy .
_ shcg 011.04_. one of the brightest galaxies .
_ shcg 016.01_. the brightest galaxy , spiral . as a typical radio galaxy it has two radio emitting lobes on two sides of the galaxy , and a weaker component coinciding with the galaxy itself ( fig .
radio lobes are at a projected distance of about 10 kpc from the galaxy .
the redshift of the group , known also as i zw 167 and arp 330 , is 0.02913 ( amirkhanian @xcite ) .
the fir source in tmtst was tentatively identified with galaxy no .
3 , which according to stoll et al .
( @xcite ) is an hii region .
the consideration of fig . 2 in tmtst shows that both galaxies , no . 3 and also no .
1 with radio emission , may be fir emitters .
fir observations with better angular resolution may clarify the situation .
_ shcg 018.02_. one of the bright galaxies .
_ shcg 021.01_. the dominant galaxy of the group with @xmath18 .
two other radio sources , first j234646.6 - 014417 and first j234647.6 - 014414 seem to be ejected from the brightest galaxy ( fig .
_ shcg 023.02_. one of the bright galaxies .
_ shcg 024.02_. the brightest galaxy , @xmath19 .
_ shcg 029.03_. the brightest galaxy .
_ shcg 033.03_. the 2-d by brightness ( @xmath20 ) galaxy in the group .
it is a spiral .
_ shcg 040.01_. the dominant cd galaxy ( struble & rood @xcite ) . the group is known also abell cluster a0193 , z=0.0498 ( struble & rood @xcite ) .
_ shcg 041.01_. the brightest galaxy of the group .
_ shcg 042.11_. a weak galaxy , membership to the group should be proved spectroscopically . the brightest galaxy no . 2 , of possibly s0 type ,
is located nearby .
_ shcg 051.01_. the brightest galaxy , located at the periphery of the group . _ shcg 051.anon_. a weak object that could possibly be a member of the group .
it is not excluded , however , that the radio source may be just a projected one .
_ shcg 053.01_. the brightest galaxy of the group , that is known also as abell cluster a1050 . _ shcg 053.04_. one of the bright galaxies .
_ shcg 054.06_. one of galaxies of the group . its membership to the group
should be proved by spectral observations .
the group is known also as abell cluster a1067 .
_ shcg 054.09/10_. a pair weak of galaxies .
_ shcg 057.01_. the brightest galaxy .
_ shcg 057.02_. probable identification . _ shcg 062.01_. the brightest galaxy , located at the periphery of the group .
more stronger radio source first j112552.1 + 382153 is located very close to the first one .
it may , probably be ejection from the galaxy shcg 062.01 , or a projected source .
_ shcg 065.07_. a galaxy located at the periphery of the group , known also as abell 1284 .
the fir source found here is the galaxy no .
27 in the southern part of the group ( tmtst ) .
_ shcg 074.08_. relatively weak galaxy , that is very close to the brighter galaxy no .
the fir source was identified with galaxy no .
9 ( tmtst ) .
the center of the fir source is located almost between galaxies no . 8 and 9 .
hence both galaxies may be fir emitters .
_ shcg 083.01_. the brightest galaxy , @xmath21 . _ shcg 104.03_. galaxy is located at about @xmath22 to the south from the brightest galaxy no .
the latter was identified as the fir source ( tmtst ) .
it is not excluded , however , that just galaxy no .
3 may really be a fir emitter .
_ shcg 120.anon_. the central brightest radio source first j110431.2 + 355157 coincides with a weak object between galaxies no .
4 and 6 , located at the periphery of the group . two weaker radio sources first j110432.6 + 355212 and first j110433.1 + 355222 seem to be radio lobes ejected from the central object ( fig . 1c ) .
the group is known also as a cluster a1151 .
the belonging of the object to the group should be checked by spectral observations .
the fir emitter , located just here , was considered as an uncertain identification in mtst .
apparently the same object may really be a fir emitter .
_ shcg 141.01_. the brightest galaxy , @xmath23 .
_ shcg 149.01_. one of the brightest galaxies in the central region of the group .
_ shcg 163.01/03_. brightest galaxies ( probably interacting ) in the center of the group .
_ shcg 166.02_. one of the bright galaxies of the group , known also as a cluster a2247 .
_ shcg 168.06_. one of the bright galaxies of the group .
the fir source was identified with the brightest galaxy no .
1 ( tmtst ) . _
shcg 177.01_. the brightest galaxy . _ shcg 181.04_. the galaxy no
. 4 is located nearby to the brightest galaxy no .
shcg 182.01_. the brightest galaxy , located at the end of the elongated group .
_ shcg 186.01_. one of three , probably interacting brightest galaxies .
_ shcg 194.01_. the dominant galaxy .
_ shcg 199.01_. the dominant galaxy .
_ shcg 199.03_. the second by brightness .
_ shcg 202.03_. the brightest galaxy .
_ shcg 203.anon_. a pair of relatively strong radio sources is identified with very weak object in the north part of the group ( fig .
the pair was not considered as a member of the group in the original shcg lists .
spectral observations are needed to clarify the case .
_ shcg 205.03_. one of the brightest galaxies .
_ shcg 209.01_. the brightest galaxy located in the center of the group .
_ shcg 219.01_. radio source first j145233.5 + 275751 is identified with the brightest galaxy .
it seem to be in interaction with galaxy no .
2 . galaxy no .
1 has halo or spiral arms .
the group is known also as a cluster a1984 .
two other radio sources , first j145231.6 + 275807 and first j145234.2 + 275751 , seem to be ejected from galaxy no . 1 ( fig .
this galaxy and the galaxy no .
1 are in interaction . _
shcg 234.04_. relatively weak galaxy , located in the central part of the group .
its membership to the group should be proved by spectral observations .
_ shcg 248.04_. the the brightest galaxy of the group . in tmtst
the fir source was identified with the same galaxy . _
shcg 248.anon_. relatively bright galaxy located at about @xmath6 to the west from the group .
it may be a member of the same group .
the consideration of the isophotes of the fir source ( tmtst ) shows that both this galaxy and the galaxy no .
4 of the group may be fir emitters .
_ shcg 250.anon_. weak object that could possibly be a member of the group .
it is not excluded , however , that the radio source may be just a projected one to this group . _
shcg 273.06_. relatively bright galaxy , what , probably , is in interaction with galaxy no .
the fir source was identified with galaxy no .
3 ( tmtst ) .
the finding of the radio emission from this galaxy suggests that just it may be the fir source .
the errors of the iras positional measurements in this case are @xmath24 .
_ shcg 279.04_. relatively weak galaxy , @xmath25 . probable identification . _
shcg 282.04_. galaxy of s0 type , that is apparently interacting with the brightest galaxy no . 1 of sbc type . _ shcg 289.03_. e type galaxy , that apparently interacts with galaxies no . 1 ( of e / s0 type ) and 4 . galaxy no .
1 is the brightest galaxy , @xmath26 , and galaxy no .
3 is the third , being a little bit weaker , @xmath27 .
_ shcg 298.02/06_. a pair of apparently interacting bright galaxies .
the stellar magnitudes are equal to @xmath28 and @xmath29 respectively .
_ shcg 303.03_. bright galaxy in the center of the group , apparently is interacting with another bright galaxy no .
shcg 309.07_. the brightest galaxy , @xmath30 .
it seems to be interacting with galaxy no .
. _ shcg 312.10_. the dominant galaxy .
it seems to be in interaction with galaxy no .
9 , and has a couple of dwarf satellites .
_ shcg 317.01_. the first survey shows very complex structure ( fig .
1f ) of the radio source identified with the brightest galaxy of the group ( @xmath31 ) .
it has several components connected with bridges .
the overall size of the radio complex is about 100 kpc .
the redshift of the group is 0.0434 ( tiersch et al .
, in preparation ) .
_ shcg 320.09_. one the bright galaxies , @xmath32 .
_ shcg 329.03_. the brightest galaxy of the group , @xmath33 .
_ shcg 330.anon_. a weak object that could possibly be a member of the group .
it is not excluded , however , that the radio source may be just a projected one . _
shcg 331.07_. the brightest galaxy , @xmath34 .
the fir source was identified with the same galaxy ( tmtst ) .
_ shcg 335.02_. the 2-d by brightness galaxy , @xmath35 . _
shcg 339.01_. one of the three brightest galaxies .
_ shcg 340b.05_. one of the brightest galaxies .
_ shcg 344.07_. galaxy of sab type .
seems to be in interaction with another , relatively weak galaxy .
the fir source was identified in tmtst with the brightest galaxy no . 1 of s0 type
. the reconsideration of the fir isophotes here with taking into account the errors of the iras positional measurements in this case ( @xmath36 ) allows to suggest that galaxy no . 7
may also be a fir emitter .
_ shcg 346.anon_. a weak object , that could possibly be a member of the group .
it is not excluded , however , that the radio source may be just a projected one . _ shcg 347.01_. the brightest galaxy , located at the end of the elongated group .
_ shcg 347.03/04_. the radio source is located between galaxies no . 3 and 4 , and could be identified with one of them . _
shcg 348.02_. one of two brightest galaxies in the group .
_ shcg 351.06_. the dominant spiral galaxy ( ugc 06212 ) .
it is located at the periphery of the group . in tmtst
the fir source was identified with the same galaxy .
_ shcg 352.anon_. the position of the radio source coincides well with the weak object at about @xmath37 to the s - w from the brightest galaxy no .
shcg 359.01_. the brightest galaxy .
_ shcg 360.01_. the brightest galaxy of s0 type .
this very compact group , is known also as the cluster a2113 .
_ shcg 362.01/04_. the radio source is located at about @xmath38 south of galaxy no.4 , and could be identified with it , or with the brightest galaxy no .
1 , located nearby . these two galaxies seems to be interacting .
the group is known also as iii zw 108 . _
shcg 371.02_. the group is very dense .
the radio source is identified with one of the three central bright galaxies of the group , nos . 2 , 3 or 4 .
the latter is the brightest in the group . in tmtst
the fir source was tentatively identified with galaxy no .
2 . however , the optical spectrum of these galaxy is a normal one with absorption lines , while the spectrum of galaxy no .
4 has emission lines .
( the results of the spectral observations of this group will be published elsewhere ) . on the basis of spectral data we assume that galaxy no .
4 may also be a fir emitter .
_ shcg 372.03_. one of the bright galaxies
. _ shcg 376.04_. the dominant galaxy .
the fir source was identified with the same galaxy ( tmtst ) .
spectral observations of this group ( the results of which will be published elsewhere ) show that this galaxy , and also some others in the group , have emission lines .
a radio emission , as we already mentioned , is most often observed from the optically first - ranked galaxies or other brightest members of groups of galaxies ( tovmassian & shakhbazian @xcite , menon @xcite ) .
we identified most of radio sources detected in shcgs ( 64 ) with the brightest or one of the bright galaxies in corresponding groups . only in nineteen cases
the identified objects are not the brightest members .
if the found regularity ( tovmassian & shakhbazian @xcite , menon @xcite ) holds in the case of shcgs as well , then some of the latter nineteen identifications with weaker members of groups may not be correct .
the membership of these objects to the corresponding groups should be checked by spectral observations . due to large distances of shcgs from us
the angular sizes of member galaxies are generally small enough .
for this reason the angular resolution of the first ( @xmath39 ) and , especially of the nvss ( @xmath40 ) does not allow in most cases to distinguish whether the observed radio radiation is emitted from the nuclear region of the galaxy , or from its disk .
hence we assumed that the measured flux refers to the disc .
however , in some cases the high angular resolution of the first survey allowed , to resolve detected radio sources .
radio sources , identified with shcg 016.01 , 021.01 , 219.02 , and 317.01 have composite structure ( fig .
some of them consist of two lobes located diametrically in two sides of the parent galaxy , that is characteristic to classical radio galaxies . in the case of shcg 016.01 and 219.01
the radio sources are of fr ii type .
radio sources identified with shcg 120.anon and 203.anon also seem to consist of two components . in shcg 016.01 , 120.anon , and 317.01 the radio emission of the central galaxy itself is also observed .
the radio source in shcg 317 is a very complex one ( fig .
radio sources detected within shcg areas were cross - identified with sources of the cats ( astrophysical catalogs support system ) database ( verkhodanov et al .
@xcite ) , that unifies 200 radio astronomical catalogs , including the texas catalog at 365 mhz ( douglas et al .
@xcite ) , 6c at 151 mhz ( baldwin et el .
@xcite ) , high sensitivity wens at 327 mhz ( rengelink et al . , @xcite ) , and others .
we used the task _ match _ in the identification circle of a @xmath41 radius .
we found that twenty two of the detected radio sources have been observed at least at two frequencies .
the spectra of these radio sources are presented in fig .
2 . the spectral indices of these sources were determined . for determination of spectral indices we used the least square methods to fit the obtained data sample for construction of the spectra .
the spectral indices are presented in tab .
most of the spectra have normal slopes .
the radio source , identified with shcg 248.04 have very unusual , too steep spectrum .
it is possible that the flux density of this source at 365 mhz is overestimated .
the spectra of three sources , shcg 041.01 , 051.04 and 163.01/03 are inverted .
redshifts of only a handful shcgs have been measured until recently ( robinson & wampler @xcite ; arp et al .
@xcite ; mirzoyan et al .
@xcite ; amirkhanian & egikian @xcite ; amirkhanian @xcite ; kodaira et al .
@xcite , 1990 ; kodaira & sekiguchi @xcite ; lynds et al .
@xcite ; del olmo & moles @xcite ) .
the redshifts of only 37 shcgs with detected radio sources are known .
most of these redshifts are yet unpublished ( tiersch et al . , @xcite ) .
we derived the radio luminosities of these sources at 1.4 ghz by assuming @xmath42 @xmath43 and @xmath44 .
the mean redshift of the group members , if available , was used in calculations .
the derived radio luminosities are presented in table 3 .
we compared radio luminosities of galaxies in shcgs with that of in hcgs ( fig .
3 ) . for drawing fig .
3 we used total fluxes of 56 hcg spiral galaxies ( menon @xcite ) at 1415 mhz , and also 34 more hcg galaxies identified with the nvss and first radio sources by us .
the list of the latter galaxies is presented in table 4 .
the consideration of fig .
3 shows that radio sources in shcgs are more powerful .
indeed , the radio luminosities of more than half of hcgs located at redshifts @xmath45 are less than 22.0 @xmath46 , while only one out of eleven shcgs at the same distances has such low radio luminosity .
it is seen also that most of powerful shcg radio sources are located at larger distances , where no hcgs were found .
since redshifts are known for only a limited sample of shcgs , it is not yet possible to study the radio luminosity function of shcg galaxies .
such study may be done after compilition of the program initieted by tiersch et al .
( @xcite ) on the spectral study of shcgs .
the available data allow , anyhow , to compare the radio and fir emission of hcgs and shcgs . for comparison of the radio and fir emission abilities of shcgs and hcgs
we draw the graph log @xmath47-log @xmath48 ( fig .
the 60 @xmath0 m iras band was by far the most sensitive for the detection of extragalactic objects ( see , for example , @xcite , @xcite ) . for star forming galaxies a very close connection between two apparently unrelated physical mechanisms , the thermal emission from a dust and the synchrotron radio emission from relitivistic electrons ,
was found ( dickey & salpeter @xcite , helou , sofier & rowan - robinson @xcite , hickson et al .
@xcite , helou & bicay ( @xcite ) .
it was shown that the ratio of the fir and radio fluxes of starburst galaxies is almost constant .
if the ratio of the fir and radio fluxes of galaxies in the considered cgs is also constant , as it is in the star forming galaxies , then due to different distances from us , and also due to differences in the emitted fluxes , the cg galaxies should be distributed on fig .
4 along a diagonal line . for drawing fig .
4 we used : - thirty four spiral hcg galaxies from menon s list ( @xcite ) , and also twelve hcg galaxies from tab .
4 ( this paper ) , the fir emission of which at 60 @xmath0 m was mesured by allam et al .
( @xcite ) .
nine galaxies from table 4 also are spirals . - eleven shcg galaxies with detected radio and fir emission ( tmtst ) .
these are galaxies in groups shcg 016 , 074 , 104 , 120 , 168 , 248 , 273 , 331 , 344 , 371 , and 376 .
- markarian - starburst ( sb ) and markarian - seyfert ( sy ) galaxies .
the fir ( at 60 @xmath0 m ) and radio fluxes of markarian galaxies are taken from bicay et al .
( @xcite ) .
the high accuracies of radio positional measurements allowed to identify with high confidence the detected radio source with a certain galaxy in the corresponding cg .
the situation is not the same in the case of the iras fir observations .
absolute positions provided by the iras are accurate up to @xmath49 ( within @xmath50 ) in the in - scan and @xmath51 in the cross - scan directions respectively . in the case of hcgs , that are nearer to us and
have relatively larger angular dimensions in comparison with shcgs , in most cases ( 37 out of 47 ) it was possible to identify the detected fir source with a certain galaxy in the corresponding group ( allam et al .
@xcite ) . the fir sources detected in shcgs ( tmtst ) are in general weaker than those in hcgs .
as a result their positional measurement accuracies reach usually values of about @xmath22 to @xmath52 in the in - scan and the cross - scan directions respectively . for this reason , and also for the relatively smaller angular sizes of shcgs it was not possible to determine with certainty which galaxy in a dense group
is the fir emitter ( tmtst ) . as a probable one the brightest member of the group
was usually mentioned . for construction of fig .
4 we attributed the measured fir flux eather to a single galaxy or to two nearby galaxies ( see subsection 3.2 ) . this uncertainty does not however influence the made conclusions .
the consideration of fig .
4 shows that most of hcg galaxies ( 30 out 47 , i.e. @xmath53 ) are located along and somewhat lower of the arbitrarily drawn diagonal dashed line .
these galaxies thus obey the correlation between the thermal emission from a dust and the synchrotron radio emission from relitivistic electrons found for star forming galaxies , and hence they also are sb galaxies .
galaxies that are lower of the dashed line apparently have relatively stronger fir emission . along the same dashed line
are distributed , as it was expected , also most of markarian sb galaxies .
seventeen hcg galaxies ( @xmath54 ) are higher of the dashed line .
it means that they have stronger radio emission than sb galaxies .
it is remarkable that two of these galaxies , hcg 92c and hcg 96a , are seyferts , and three others , hcg 56b , hcg 68a and hcg 68b , are e and s0 type galaxies .
the situation is different for shcg galaxies .
nine of them ( 82% ) are located above the arbitrarily drawn diagonal line . above this line
are located also twelve out of thirteen ( @xmath55 ) markarian - seyfert galaxies .
hence most of shcg galaxies are not sb galaxies .
if the identification of the fir source with a radio emitting galaxy is not correct then the galaxy with detected radio emission and thus with smaller fir emission would move on fig .
4 to the left and hence would be located even higher of the dashed line .
the blending of a few probable fir sources in dense groups would have the same effect .
it is worth to note that those hcg and shcg galaxies for which either fir or radio emission was detected would also have different locations on the log@xmath47-log@xmath48 graph .
there are 73 shcg galaxies , fluxes of which at 1.4 ghz exceed 1 jy , but only upper limits of fluxes at 60 @xmath0 m were determined ( tmtst ) .
these galaxies would apparently be located on fig .
4 higher of the dashed line .
at the same time there are only fifteen shcg galaxies with determined fir fluxes ( tmtst ) , and radio fluxes lower than 1 jy detection limit ( this paper ) .
the latter would be located lower of the dashed line . in the case of hcg galaxies
the situation is vice versa .
there are 34 hcg galaxies with 1.4 ghz fluxes exceeding the 1 mjy limit ( menon @xcite , and present paper ) , and the fir fluxes lower than the limiting value ( allam et al .
these galaxies would be located on fig .
4 upper of the dashed line .
much more hcg galaxies , 65 , with measured fir fluxes and upper limits of radio fluxes , would be located below the dashed line , in the lower part of fig .
4 . thus in the case of galaxies with eather one of the fluxes ( eather 60 @xmath0 m or 1.4 ghz ) measured , we see the same trend : most shcg galaxies would be located upper of the dashed line on fig .
meanwhile hcg galaxies would be located mainly below this line .
sulentic & de mello rabaca ( @xcite ) claimed that the fir sources in hcgs are likely the combined contribution of two or more members . if to assume that in all hickson s groups with detected fir emission , the fir emitters are in reality two galaxies with equal fluxes , then the corresponding points on fig .
4 should move to the left by 0.3 , and still they would be located lower than shcg galaxies . moreover , if the same is valid also for the more dense shcgs , then the corresponding positions of shcg galaxies on fig . 4 should also move to the left , and the found trend would certainly not be altered .
hence , one may conclude that galaxies in shcgs are relatively stronger radio emitters , while hcg galaxies are stronger fir emitters .
it means that physical conditions in shcgs are somehow favorable for triggering the agns with relatively strong synchrotron emission of relativistic electrons , while in hcgs the conditions are favorable for formation of sb galaxies with relatively strong thermal dust emission
. what could be the reason for such difference ?
cgs are generally very dense formations . according to n - body simulations ( carnevali et al .
@xcite , barnes @xcite , @xcite , mamon @xcite , zheng et al .
@xcite ) the member galaxies at high density environments of cgs should interact and merge in one large galaxy in about @xmath56 years .
for this reason the very existence of cgs has been questioned by some authors ( walke & mamon @xcite , mamon @xcite , @xcite , hernquist et al .
meanwhile hickson & rood ( @xcite ) , mendes de oliveira & giraud ( @xcite ) , mendes de oliveira ( @xcite ) , oleak et al .
( @xcite ) and recently tovmassian et al . (
@xcite ) presented firm evidences on the reality of cgs .
for explaining the existence of cgs governato et al .
( @xcite ) proposed a second generation merger scenario , according to which cgs permanently aggregate new members from their surrounding areas .
such scenario could be valid since , according to rood & williams ( @xcite ) , vennik et al .
( @xcite ) and ramella et al .
( @xcite ) , most of hcgs are associated with loose groups of galaxies .
shcgs and hcgs differ from each other by the number of members and by morphological content .
if hcgs contain generally four - five members , shcgs have up to fifteen members .
the relative number of e or s0 type galaxies in hcgs compose only @xmath57 of all members ( hickson et al .
@xcite ) , while shggs are more rich in early type galaxies .
about 77% member galaxies in the latter are of e and s0 type ( tiersch et al .
@xcite ) . for this reason shcgs
are considered as more evolved systems in comparison with hcgs . since almost half of hcg galaxies are spirals with sufficient amount of gas and dust , the gravitational interaction between them and with a suggested newcomer galaxy may trigger starburst processes in interacting galaxies .
the fir emission is characteristic to such events .
meanwhile in the more evolved shcgs most galaxies are of e and s0 type , and the number of spirals is relatively small . the gas was pushed out from them during previous interaction processes between galaxies , and filled the intergalactic space .
the spirals here should have shed also their dust content .
the presence of intergalactic gas in cgs was proved by the detection of diffuse x - ray emission ( ebeling et al .
@xcite , pildis et al .
@xcite , saracco & ciliegi @xcite , tiersch et al .
@xcite ) from some of the groups . due to the interaction with a newcomer galaxy to the group
this intergalactic gas may be falling in the form of cooling flows directly into the center of the preferentially dominant early type galaxy in the group . in the result of this an active nucleus of the galaxy with sufficiently strong radio emission
may be formed . due to small amount or even absence of dust in such galaxies
their fir emission would be very weak or even completely absent .
three hundred fifty three nvss ( condon et al .
@xcite ) radio sources are found within boundaries of 179 shcgs .
sixty sources were registered also in the first ( white et al .
@xcite ) survey , of which seven sources - only in the first .
the comparison of the radio and fir fluxes of galaxies in hcgs and shcgs showed that the latter are more stronger radio emitters , while hcg galaxies are generally more stronger fir emitters .
it is suggested that the reason for this may be that shcgs are more evolved in comparison with hcgs , and galaxies in them do not have enough dust for attributing the fir emission .
on the other hand the conditions in shcgs are more favorable for formation of agn with relatively stronger radio emission .
hmt acknowledge the support of the deutsche forschungsgemeinschaft dfg project no .
444mex 112/2/98 .
vhch was partially supported by conacyt research grant 28499-e .
ovv is grateful to the russian foundation for basic researches for the cats database support ( grant no 96 - 07 - 89075 ) .
ht is grateful to the government of the land brandenburg and the deutsche forschungsgemeinschaft dfg project no .
ti 215/6 - 3 for the support of this work .
the digitized sky surveys were produced at the space telescope science institute under u.s .
government grant nag w-2166 .
the images of these surveys are based on photographic data obtained using the oschin schmidt telescope on palomar mountain and the uk schmidt telescope .
verkhodanov , o.v . ,
trushkin s.a . , andernach h. , chernenkov , v.n .
1997 , `` the cats database to operate with astrophysical catalogs '' , in `` astronomical data with astrophysical catalogs . in ' '
astronomical data analysis software and systems vi " , editors : gareth hunt and h. e. payne , asp conference series , v. 125 , p.322 - 325 .
& 10 55 05.7 & + 40 27 30 nvss j105506 + 402726 & 10 55 06.18 & + 40 27 26.7 & 3.9 & 11 15 23.4 & + 53 41 23 first j1115 + 534122 & 11 15 23.504 & + 53 41 22.28 & 1.32 & 11 05 53.8 & + 39 46 58 first j110553.7 + 394654 & 11 05 53.766 & + 39 46 54.81 & 39.93 & 13 24 01.5 & + 19 03 21 nvss j132401 + 190320 & 13 24 01.74 & + 19 03 20.2 & 4.2 & 14 10 48.1 & + 46 15 58 first j141448.1 + 461557 & 14 10 48.176 & + 46 15 57.45 & 11.78 & 14 11 01.7 & + 44 42 15 first j111411 + 444214 & 14 11 01.826 & + 44 42 14.34 & 1.72 & 16 49 11.3 & + 53 25 12 first j164911.4 + 532510 & 16 49 11.496 & + 53 25 10.91 & 23.43 first j164910.5 + 532507 & 16 49 10.398 & + 53 25 07.31 & 62.14 first j164912.5 + 532514 & 16 49 12.582 & + 53 25 14.56 & 53.58 & 08 53 37.2 & + 79 09 17 nvss j085339 + 790915 & 08 53 39.73 & + 79 09 15.4 & 3.8 & 23 46 48.6 & -01 44 16 first j234648.6 - 014416 & 23 46 48.633 & -01 44 16.75 & 64.52 first j234646.6 - 014417 & 23 46 46.680 & -01 44 17.41 & 13.70 first j234647.6 - 014414 & 23 46 47.641 & -01 44 14.56 & 6.72 & 16 10 03.2 & + 52 14 52 first j161003 + 521450 & 16 10 03.560 & + 52 14 50.33 & 1.15 & 23 46 56.2 & -00 52 27 first j234656.3 - 005227 & 23 46 56.352 & -00 52 27.21 & 2.19 & 16 08 45.1 & + 52 26 17 first j160845.1 + 522616 & 16 08 45.191 & + 52 26 13.1 & 2.48 & 01 03 42.7 & -01 08 13 first j010342.5 - 010813 & 01 03 42.521 & -01 08 13.10 & 3.80 & 01 25 07.6 & + 08 41 59 nvss j012510 + 084224 & 01 25 07.87 & + 08 41 59.2 & 33.4 & 01 29 00.6 & + 07 40 40 nvss j012900 + 074042 & 01 29 00.61 & + 07 40 42.1 & 36.8 & 01 30 44.6 & + 07 50 09 nvss j013044 + 075004 & 01 30 44.65 & + 07 50 04.4 & 3.3 & 10 30 44.6 & + 39 12 45 first j103044.7 + 391245 & 10 30 44.619 & + 39 12 44.28 & 14.2 & 10 30 44.1 & + 39 10 29 + first j103044.0 + 391028 & 10 30 44.196 & + 39
10 29.73 & 5.64 + _ 053.01 _ & 10 36 46.7 & + 44 49 48 first j103646.4 + 444946 & 10 36 46.496 & + 44 49 46.39 & 2.74 & 10 36 52.8 & + 44 48 21 first j103653.0 + 444818 & 10 36 53.027 & + 44 48 18.19 & 35.82 & 10 36 45.7 & + 44 49 54 first j103645.9 + 444955 & 10 36 45.988 & + 44 49 55.16 & 3.86 & 10 40 37.2 & + 40 12 58 first j104037.3 + 401257 & 10 40 37.320 & + 40 12 57.92 & 1.16 & 10 40 27.1 & + 40 13 41 & 10 40 27.0 & + 40 13 48 nvss j104026 + 401345 & 10 40 26.92 & + 40 13 45.9 & 11.2 & 10 45 26.7 & + 49 31 08 first j104527.2 + 493106 & 10 45 27.266 & + 49 31 06.27 & 20.01 & 10 45 26.1 & + 49 31 25 first j104526.4 + 493116 & 10 45 26.492 & + 49 31 16.98 & 29.92 & 11 25 53.2 & + 38 22 04 first j112553.3 + 382201 & 11 25 53.305 & + 38 22 01.38 & 28.95 first j112552.1 + 382153 & 11 25 52.121 & + 38 21 53.30 & 30.32 & 11 30 50.6 & + 35 04 15 first j113050.5 + 350415 & 11 30 50.578 & + 35 04 15.03 & 4.79 & 14 20 57.3 & + 43 02 54 first j142057.5 + 430250 & 14 20 57.562 & + 43 02 50.45 & 2.39 & 23 26 08.8 & -01 43 30 first j232608.8 - 014329 & 23 26 08.867 & -01 43 29.83 & 5.849 & 09 27 13.5 & + 52 58 33 first j092713 + 525832 & 09 27 13.561 & + 52 58 32.58 & 2.08 & 11 04 31.3 & + 35 52 08 first j110431.2 + 355157 & 11 04 31.227 & + 35 51 57.70 & 3.72 first j110432.6 + 355212 & 11 04 32.605 & + 35 52 12.54 & 71.46 first j110433.1 + 355222 & 11 04 33.137 & + 35 52 22.14 & 9.01 & 01 04 22.2 & -01 33 27 first j010422.2 - 013326 & 01 04 22.203 & -01 33 26.54 & 12.67 & 15 21 03.9 & + 75 04 20 & 15 21 04.8 & + 75 04 12 nvss j152105 + 750421 & 15 21 05.13 & + 75 04 21.5 & 7.1 & 18 28 03.2 & + 83 06 07 nvss j182806 + 830605 & 18 28 06.27 & + 83 06 05.8 & 169.5 & 01 57 36.3 & + 29 38 56 nvss j015736 + 293853 & 01 57 36.50 & + 29 38 53.5 & 43.4 & 08 28 00.0 & + 28 15 37 nvss j082800 + 285134 & 08 28 00.28 & + 28 15 34.1 & 10.0 & 08 38 23 & + 29 45 22 first j083823.2 + 294521 & 08 38 23.271 & + 29 45 21.67 & 3.92 & 09 22 51.2 & + 28 55 53 first j092251.1 + 285552 & 09 22 51.148 & + 28 55 52.89 & 5.17 & 11 03 06.7 & + 27 48 25 first j110306 + 274823 & 11 03 06.800 & + 27 48 23.19 & 1.13 & 11 35 22.7 & + 30 43 43 first j113522.8 + 304343 & 11 35 22.855 & + 30 43 43.14 & 1.45 & 11 35 21.3 & + 30 42 58 first j113521.4 + 304257 & 11 35 21.480 & + 30 42 57.20 & 0.77 & 12 19 51.8 & + 28 25 18 first j121951.6 + 282521 & 12 19 51.668 & + 28 25 21.51 & 8.00 & 12 29 02.7 & + 27 27 11 first j122902.3 + 272702 & 12 29 02.387 & + 27 27 02.36 & 179.56 first j122902.7 + 272731 & 12 29 02.770 & + 27 27 31.58 & 234.00 & 12 35 19.0 & + 27 34 41 nvss j123519 + 273438 & 12 35 19.05 & + 27 34 38.4 & 3.84 & 13 10 38.9 & + 31 44 20 first j131039.0 + 314417 & 13 10 39.094 & + 31 44 17.01 & 1.88 & 14 33 35.4 & + 26 41 54 first j143335 + 264153 & 14 33 35.463 & + 26 41 53.91 & 1.65 & 14 52 33.4 & + 27 57 51 first j145233.5 + 275751 & 14 52 33.543 & + 27 57 51.72 & 31.38 first j145231.6 + 275807 & 14 52 31.680 & + 27 58 07.84 & 48.27 first j145234.2 + 275751 & 14 52 34.297 & + 27 57 51.86 & 40.66 & 10 48 25.0 & + 36 15 46 first j104825.0 + 361547 & 10 48 25.004 & + 36 15 47.71 & 9.09 & 13 12 15.6 & + 36 11 10 first j131216.0 + 361108 & 13 12 16.040 & + 36 11 08.36 & 2.47 & 13 12 10.0 & + 36 11 14 first j131210.1 + 361112 & 13 12 10.191 & + 36 11 12.45 & 0.91 & 13 34 47.0 & + 33 08 57 first j133447.0 + 330857 & 13 34 47.004 & + 33 08 57.16 & 7.78 & 13 56 19.0 & + 35 11 20 first j135619 + 351119 & 13 56 19.184 & + 35 11 19.85 & 1.35 & 00 13 54.6 & -08 38 49 first j001354 - 083850 & 00 13 54.851 & -08 38 50.83 & 1.08 & 02 52 35.5 & -13 06 15 nvss
j025235 - 130617 & 02 52 35.85 & -13 06 17.6 & 9.7 & 10 52 53.1 & -11 00 21 nvss j105252 - 110020 & 10 52 52.89 & -11 00 20.7 & 75.8 & 13 58 10.6 & -12 53 03 & nvss j135810 - 125306 & 13 58 10.67 & -12 53 07.0 & 5.8 & 22 12 44.5 & -13 40 26 & 22 12 44.8 & -13 40 33 nvss j221244 - 134026 & 22 12 44.84 & -13 40 26.9 & 15.4 & 23 17 33.1 & -09 05 32 first j231733.2 - 090532 & 23 17 33.297 & -09 05 32.74 & 5.04 & 00 51 19.5 & -07 24 39 nvss j005119 - 072440 & 00 51 19.63 & -07 24 40.1 & 6.3 & 01 03 30.0 & -03 32 30 nvss j010330 - 033239 & 01 03 30.97 & -03 32 39.3 & 3.7 & 02 10 53.3 & -06 33 33 first j021052.5 - 063343 & 02 10 52.527 & -06 33 43.22 & 97.0 first
j021053.6 - 063333 & 02 10 53.615 & -06 33 33.59 & 176.52 first
j021054.2 - 063344 & 02 10 54.215 & -06 33 44.21 & 96.57 first
j021050.1 - 063336 & 02 10 50.189 & -06 33 36.38 & 132.83 & 11 14 46.8 & -06 21 36 nvss j111446 - 062137 & 11 14 46.59 & -06 21 37.3 & 4.1 & 14 37 12.0 & -03 45 51 nvss j143712 - 034547 & 14 37 12.02 & -03 45 47.2 & 8.1 & 15 14 22.9 & -09 36 07 nvss j151423 - 093603 & 15 14 23.36 & -09 36 03.6 & 6.7 & 22 25 26.8 & -02 47 02 nvss j222526 - 024702 & 22 25 26.76 &
-02 47 02.4 & 42.6 & 23 23 37.4 & -07 24 00 nvss j232337 - 072357 & 23 23 37.04 & -07 23 57.3 & 4.6 & 00 42 24.6 & + 20 22 05 nvss j004224 + 202204 & 00 42 24.38 & + 20 22 04.5 & 3.3 & 08 47 35.8 & + 03 42 01 nvss j084736 + 034159 & 08 47 36.07 & + 03 41 59.9 & 4.5 & 09 15 12.4 & + 05 14 23 nvss j091512 + 051426 & 09 15 12.20 & + 05 14 26.4 & 9.3 & 09 17 28.6 & + 07 42 31nvss j091729 + 074233 & 09 17 29.10 & + 07 42 33.2 & 8.0 & 09 17 34.0 & + 07 41 10 + _ 347.04 _ & 09 17 34.3 & + 07 41 22 + nvss j091729 + 074233 & 09 17 34.15 & + 07 41 16.0 & 18.3 + _ 348.02 _ & 09 26 29.3 & + 03 26 17 nvss j092629 + 032617 & 09 26 29.41 & + 03 26 17.7 & 10.8 & 11 10 24.7 & + 04 49 47 nvss j111024 + 044945 & 11 10 24.63 & + 04 49 45.3 & 11.93 & 11 21 31.5 & + 02 53 02nvss j112131 + 025303 & 11 21 31.61 & + 02 53 03.5 & 5.2 & 14 29 54.4 & + 18 50 07 nvss j142954 + 185008 & 14 29 54.19 & + 18 50 08.6 & 2.6 & 15 41 26.5 & + 04 43 56 nvss j154126 + 044355 & 15 41 26.54 & + 04 43 55.8 & 27.2 & 23 32 36.4 & + 19 22 27 + _ 362.01 _ & 23 32 37.1 & + 19 22 33 + nvss j 233236 + 192215 & 23 32 36.82 & + 19 22 15.4 & 3.6 + _ 370.03 _ & 09 50 20.5 & + 23 16 54 nvss j095029 + 231655 & 09 50 20.470 & + 23 16 55.54 & 1.26 & 11 43 33.1 & + 21 53 50 nvss j114333 + 215406 & 11 43 33.28 & + 21 54 06.1 & 4.7 & 11 46 49.5 & + 24 08 22 first j114649.5 + 240821 & 11 46 49.539 & + 24 08 21.43 & 2.7 & 13 56 35.7 & + 23 21 37 first j135635.7 + 232135 & 13 56 35.734 & + 23 21 35.95 & 4.59 007.01 & -0.01 010.01 & -1.46 016.01 & -0.58 041.01 & 0.27 051.01 & 0.79 053.04 & -0.12 054.06 & -0.75 054.09/10 & -0.52 057.01 & -0.77 062.01 & -0.45 065.07 & -1.00 120.anon & -0.73 163.01/03 & 0.26 168.06 & -0.72 177.01 & -0.63 182.01 & -1.01 203.anon & -0.71 219.01 & -0.42 234.04 & -0.48 248.04 & -3.44 250.anon & -1.04 317.01 & -0.67 001.01 & 0.1168 & 23.36 + 016.01 & 0.0301 & 23.74 + 021.01 & 0.0773 & 24.34 + 029.03 & 0.0346 & 22.11 + 033.03 & 0.0337 & 22.14 + 040.01 & 0.0486 & 23.53 + 041.01 & 0.0900 & 24.11 + 083.01 & 0.0970 & 23.38 + 166.02 & 0.0396 & 24.04 + 168.06 & 0.1262 & 25.07
+ 181.01 & 0.0917 & 23.56 + 202.03 & 0.0262 & 22.38 + 205.03 & 0.0932 & 23.16 + 218.02 & 0.0947 & 22.81 + 248.04 & 0.2712 & 23.90 + 248.anon & 0.2712 & 23.46 + 254.03 & 0.0638 & 22.38 + 282.04 & 0.1428 & 24.83 + 289.03 & 0.0706 & 23.10 + 298.02/06 & 0.1692 & 24.29 + 309.07 & 0.0892 & 23.34 + 312.10 & 0.0733 & 22.94 + 317.01 & 0.0434 & 24.61 + 330.anon & 0.1078 & 23.53 + 331.07 & 0.0534 & 23.72 + 335.02 & 0.0875 & 23.18 + 340b.05 & 0.1045 & 23.19 + 344.07 & 0.0774 & 23.03 + 346.anon & 0.1349 & 23.87 + 348.02 & 0.0894 & 23.57 + 351.06 & 0.0290 & 22.64 + 352.anon & 0.0490 & 22.73 + 359.01 & 0.0328 & 22.08 + 360.01 & 0.1082 & 24.14 + 362.01/04 & 0.02291 & 21.91 + 371.02 & 0.1301 & 23.55 + 376.04 & 0.0660 & 22.93 _ 2b _ & ci nvss b002843 + 081155 & 13.4 & sc nvss b003143 - 214248 & 43.5 & sab nvss b003619 + 064718 & 5.7 & s0 nvss b004656 + 231800 & 4.1 & e2 nvss b020502 + 015639 & 5.1 & sc nvss b024258 - 175503 & 4.7 & sab nvss b024316 - 175400 & 2.9 & sb first j032043.2 - 010008 & 5.41 & e5 nvss b042457 - 102607 & 4.4 & e7 first j091339.4 + 295934 & 25.58 & e3 nvss b101924 + 180522 & 4.5 & sc nvss b103527 - 265140 & 15.5 & s0 first j112230.0 + 241646 & 7.44 & sbbc nvss b112613 + 210419 & 21.7 & sb0 first j113240.2 + 525701 & 25.81 & s0 first j113235.2 + 525650 & 2.76 & sb0a nvss b113918 + 103451 & 3.3 & e2 first j120307.2 + 514030 & 50.89 & s0a first j121218.8 + 291046 & 1.45 & sbc first j121231.0 + 291006 & 37.01 & e3 nvss
b125029 - 085604 & 5.4 & s0 nvss
b132306 - 033526 & 2.3 & e3 nvss
b132703 - 291520 & 3.5 & e2 nvss b132705 - 291357 & 3.8 & s0 first j135326.6 + 401658 & 38.30 & e2 first j135326.7 + 401809 & 7.99 & sbc first j141057.2 + 252949 & 3.21 & sb first j141102.5 + 253110 & 8.40 & e1 nvss b151710 + 210435 & 16.3 & sbb nvss b154805 + 682219 & 7.0 & e0
nvss b155659 + 205344 & 10.2 & e2 nvss b164644 + 775541 & 21.2 & e2 nvss b194859 - 305716 & 19.6 & e2 nvss b194849 - 305644 & 8.1 & e1 nvss b231444 + 182606 & 30.1
|
three hundred fifty three radio sources from the nrao vla sky survey ( nvss ) ( condon et al .
@xcite ) and the first survey ( white et al .
@xcite ) , are detected in the areas of 179 shakhbazian compact groups ( shcgs ) of galaxies .
ninety three of them are identified with galaxies in 74 shcgs .
six radio sources have complex structure .
the radio spectra of 22 sources are determined .
radio luminosities of galaxies in shcgs are in general higher than that of galaxies in hickson compact groups ( hcgs ) .
the comparison of radio ( at 1.4 ghz ) and fir ( at 60 @xmath0 m ) fluxes of shcg galaxies with that of hcg galaxies shows that galaxies in shcgs are relatively stronger emitters at radio wavelengths , while galaxies in hcgs have relatively stronger fir emission .
the reasons of such difference is discussed .
|
modern lattice qcd simulations are mostly based on direct evaluation of the path integral of the theory . such approach , while being very general and efficient for many applications , suffers from a number of problems , most notable of which are the sign problem at finite chemical potential , critical slowing down at small quark masses and large finite - volume effects as well as small signal - to - noise ratio in the analysis of excited states .
these problems are inherent to standard monte - carlo simulations and can not be efficiently solved by simply increasing the computation power , since the required computing time quickly increases ( in the worst cases , exponentially ) with the required precision . such situation makes it tempting to devise alternative simulation algorithms for non - abelian lattice gauge theories .
one of the efficient alternative numerical methods is the so - called diagrammatic monte - carlo , a method based on stochastic summation of all the terms in the strong- or weak - coupling expansion of the observable of interest @xcite .
such a method in some cases allows one to reduce or avoid completely the sign problem in the original path integral , and does not suffer from finite - volume effects .
furthermore , one can construct algorithms which yield particular correlation functions in terms of probability distributions of some random variables , which greatly facilitates the analysis of excited states @xcite .
this is the idea of the `` worm '' algorithm by prokofev and svistunov @xcite , in which the probability distribution of the positions @xmath2 , @xmath3 of `` head '' and the `` tail '' of the worm yields the two - point green function @xmath4 .
diagrammatic monte - carlo and the `` worm '' algorithm have been successfully applied to a number of statistical models with discrete symmetry groups such as the ising model , the xy model and unitary fermi gas and showed practically no critical slowing down near quantum phase transitions
. however , application of such methods to lattice field theories with continuous field variables ( such as two - dimensional @xmath1 and @xmath5 sigma - models , abelian gauge theories and the @xmath6 theory ) resulted so far in quite complicated and model - dependent algorithms @xcite . a generalization of such algorithms to @xmath7 sigma - models or to non - abelian gauge theories is still not found .
these algorithms are in essence based on the strong - coupling expansion , and while their applicability is not limited by the strong - coupling regime , one can expect that algorithms based on the weak - coupling expansion might show better performance near the continuum limit .
typically , the weak - coupling expansion in such lattice theories is either quite complicated or non - convergent . up to now
, divergent behavior of the weak - coupling perturbative expansions strongly limits the applicability of diagrammatic monte - carlo to field theories with continuous field variables . in a recent paper
@xcite a method was proposed to construct convergent series which approximate the non - analytic path integrals with desired precision .
this method , however , is difficult to generalize to physically interesting field theories such as non - abelian lattice gauge theories .
another way to obtain convergent series while preserving important physical properties of the theory is to sum over diagrams with certain topology only .
this corresponds to the large-@xmath0 limit in quantum field theories and matrix models , that is , the limit of infinite dimensionality of an internal symmetry group , such as @xmath1 or @xmath7 . for such theories ,
each feynman diagram acquires a factor @xmath8 , where @xmath9 is the euler character of this diagram @xcite . in the limit @xmath10 , the contribution of planar diagrams with @xmath11 dominates , and the sum over all planar diagrams typically has a finite convergence radius @xcite . in this paper
we describe a stochastic method for summing over all planar diagrams in large-@xmath0 quantum field theories .
the method is based on stochastic solution of schwinger - dyson equations , so that the correlators of field variables are obtained as stationary probability distributions of certain random variables . in this way
we implement the idea of importance sampling , so that numerically small observables correspond to unlikely events .
these probability distributions are sampled by the so - called nonlinear random processes .
in contrast to conventional markov chains , stationary probability distributions of such random processes satisfy nonlinear equations , and hence they can be called `` nonlinear random processes '' or `` nonlinear markov chains '' in the terminology of @xcite .
factorization of single - trace operators in the large-@xmath0 limit of quantum field theories corresponds to the phenomena of `` chaos propagation '' in random processes @xcite . while in the diagrammatic monte - carlo and in the `` worm '' algorithm the diagrams
are stored in computer memory as a whole and are updated in such a way that the detailed balance condition is satisfied at each step , the method described in this paper works only with external lines .
in contrast to the standard metropolis algorithm , one should not know explicitly the weight of each diagram , and the transition probabilities do not satisfy any detailed balance condition . unlike the quite popular `` numerical functional methods '' in continuum gauge theories ( see @xcite for a review ) , the proposed method does not require any truncation of the hierarchy of schwinger - dyson equations , and work only with singlet operators w.r.t .
the internal symmetry group .
another distinct feature is that the computational complexity of the method does not depend on @xmath0 , while the standard monte - carlo , the functional methods and the `` worm '' algorithm all require infinite computational resources in the limit @xmath10 .
this feature might be advantageous for numerical checks of the predictions of the holographic models which are dual to large-@xmath0 quantum field theories @xcite .
in section [ sec : sdeq_general ] we analyze the general structure of schwinger - dyson equations in large-@xmath0 quantum field theories on the example of a scalar matrix - valued field theory .
when large-@xmath0 factorization is taken into account , schwinger - dyson equations become nonlinear equations with infinitely many unknowns . in section [ sec : recursive_process ] we describe nonlinear random processes of recursive type @xcite which can be used to stochastically solve such equations . in section
[ sec : sds_stochastic_solution ] we apply such random processes to solve schwinger - dyson equations in several large-@xmath0 theories . in subsection
[ subsec : phi4_general ] we consider the scalar matrix - valued field theory , for which the perturbative expansion yields the conventional feynman diagrams in momentum space . in subsection
[ subsec : matrix_model ] this solution is compared with the exact solution of the simplest quantum field theory in zero dimensions , that is , the hermitian matrix model @xcite .
the convergence of such solution and the strength of the sign problem is discussed . in subsection
[ subsec : weingarten ] we consider the weingarten model @xcite and demonstrate how the proposed method can be used to simulate random surfaces on the hypercubic lattice . in this case , our method reproduces an ensemble of open , rather than closed , random surfaces , with critical behavior which is quite different from those of the closed planar random surfaces .
since the structure of schwinger - dyson equations in the weingarten model is similar to the loop equations in large-@xmath0 non - abelian lattice gauge theories @xcite , studying this model might be helpful for further extensions of the present approach to non - abelian gauge theories . while the method described in section [ sec : recursive_process ] works well for non - compact field variables , for field theories with compact field variables , such as nonlinear sigma - models or non - abelian lattice gauge theories ,
a straightforward stochastic interpretation of schwinger - dyson equations is only possible in the strong coupling limit . in the weak - coupling limit one
expects the field correlators to contain both the perturbative part in the coupling constant @xmath12 as well as nonperturbative corrections of the form @xmath13 with some constant @xmath14 .
moreover , perturbative expansion in powers of @xmath12 typically results in asymptotic series , and nonperturbative corrections appear as a result of resummation of such series @xcite . in section
[ sec : rps_with_mem ] we show how such nonperturbative corrections can be taken into account by a further relaxation of the markov property of the random process .
the basic idea is to absorb the divergent part of the series into a self - consistent redefinition of the expansion parameter .
these redefined parameters play the role of nonperturbative `` condensates '' @xcite .
it turns out that the redefined expansion parameters can be estimated with increasing precision from the previous history of the random process which solves the schwinger - dyson equations , thus leading to the emergence of the `` memory '' of the random process .
the approach of the redefined parameters to their self - consistent values is reminiscent somehow of the renormalization - group flow @xcite .
such dependence on the previous history makes the random process essentially non - markovian , so that the stationary probability distribution also satisfies some nonlinear equation .
we illustrate this idea on the example of @xmath1 sigma - model in two dimensions , which is equivalent to a bosonic random walk with a self - consistent mass .
random process which simulates this model has the `` memory '' but no `` recursive '' structure .
presumably , in order to sum up both perturbative and non - perturbative corrections which arise in non - abelian lattice gauge theories or @xmath15 sigma - models , one should devise the `` recursive '' nonlinear random process ( which would sum up perturbative corrections ) with memory ( which would generate nonperturbative quantities in a renormalization - group - like way ) .
finally , in the concluding section we summarize the present work and discuss its extension to non - abelian lattice gauge theories in the limit of large @xmath0 .
in order to analyze the general structure of schwinger - dyson equations for large-@xmath0 quantum field theories , let us first consider the theory of a hermitian @xmath16 matrix - valued field @xmath17 with the following lagrangian : @xmath18 } = n { { \rm tr } \ , } \phi { \left ( x \right ) } \ , { \left ( m^2 - \delta \right ) } \ , \phi { \left ( x \right ) } + \frac{n \lambda}{4 } { { \rm tr } \ , } \phi^{4 } { \left ( x \right ) } .\end{aligned}\ ] ] this theory is most convenient to illustrate the method described in this paper , since its perturbative expansion leads to conventional feynman diagrams in the momentum space .
since this theory should be somehow regularized , let us assume from the very beginning that the action ( [ phi4_field_action ] ) is defined on the euclidean hypercubic @xmath19-dimensional lattice with total volume @xmath20 in lattice units .
thus , the coordinates @xmath2 take integer values and @xmath21 is the lattice laplacian ( for definiteness , with periodic boundary conditions ) .
schwinger - dyson equations for a theory with the action ( [ phi4_field_action ] ) read @xcite : @xmath22 where the single - trace correlators are @xmath23 , @xmath24 is the laplacian acting on @xmath25 and we have already taken into account the factorization property in the limit @xmath10 @xcite : @xmath26 these equations hold for any argument of the correlators , but the resulting system is redundant , and it is sufficient to consider only those schwinger - dyson equations which were obtained by the variation of the fields at @xmath25 .
it is convenient now to go to the momentum representation , introducing the green functions in momentum space @xmath27 . in order to keep all expressions as symmetric as possible , we do not separate the factor @xmath28 in @xmath29 explicitly .
this condition will be automatically satisfied by the nonlinear random process which we describe in subsection [ subsec : phi4_general ] .
the equations ( [ phi4_sds_original_n2 ] ) , ( [ phi4_sds_original ] ) in the momentum representation are : @xmath30 where @xmath31 is the free scalar propagator on the hypercubic lattice .
all momenta are assumed to lie in the first brillouin zone @xmath32 and are added modulo @xmath33 .
the structure of these equations is schematically illustrated on fig .
[ fig : phi4_sds_general ] , where dashed blobs denote the green functions @xmath29 and empty blobs denote @xmath34 . ) .
dashed blobs denote the green functions @xmath29 and empty blobs denote the free propagator @xmath34.,title="fig:",width=226 ] + thus we have obtained an infinite system of quadratic functional equations for the set of functions @xmath29 with @xmath35 .
such structure is common for large-@xmath0 quantum field theories : schwinger - dyson equations are quadratic equations for infinite set of unknown variables . in the case of scalar matrix field theory considered here ,
the unknown variables are the functions of the sequences of momenta @xmath36 for any even @xmath37 . in the case of lattice gauge theories or string theories
schwinger - dyson equations are most naturally formulated in terms of the wilson loops , which are the functions defined on the discrete space of closed loops on the lattice @xcite . in this case , the equations are also quadratic w.r.t .
the wilson loops .
for @xmath1 sigma - model , schwinger - dyson equations are also quadratic equations which involve only the two - point function ( see section [ sec : rps_with_mem ] ) .
typically , systems of equations with infinitely many unknowns can be efficiently solved by stochastic methods .
it is advantageous to estimate the value of each unknown variable as a probability of observing some state of a random process . in this case
the unknowns with numerically small values correspond to unlikely events , and the set of infinitely many unknown variables is automatically truncated to a set of unknowns with sufficiently large values .
such methods are well - known mainly in the context of kinetic equations @xcite .
recently they were also discussed in the context of probabilistic programming @xcite . in the next section
we describe a discrete - time , discrete - space method of such type which is in our opinion most suitable for solving the schwinger - dyson equations in the large-@xmath0 limit .
we consider nonlinear equations of the following form : @xmath38 where @xmath2 , @xmath3 , @xmath39 , @xmath40 are the elements of some space @xmath41 and @xmath42 with @xmath43 denotes summation or integration over all the elements of this space .
we also assume that the functions @xmath44 , @xmath45 and @xmath46 satisfy the inequalities @xmath47 for any @xmath39 , @xmath40 .
we would like to find a stochastic process for which @xmath48 is proportional to the probability of the occurrence of the element @xmath2 in some configuration space .
obviously , ordinary markov process with configuration space @xmath41 can not solve such a problem , since stationary distributions of markov processes obey linear equations . in order to solve the nonlinear equation ( [ random_process_eq ] ) , one can ,
for example , extend somehow the configuration space .
extensions of markov processes with stationary probability distributions which obey nonlinear equations have been considered recently in @xcite . in this section
we concentrate on random processes similar to recursive markov chains of @xcite .
the basic idea is that at any time one can leave the current chain and start a new one , then returning back to the old chain at some time .
the initial state of a newly created chain depends on the states of older chains .
thus one has not a single markov chain , but rather an infinite stack of chains .
the random process which we describe below will be similar to these recursive markov chains , but instead of referring to `` recursion '' we will explicitly introduce the underlying stack structure .
here we first consider the equations ( [ random_process_eq ] ) with the coefficients @xmath44 , @xmath45 and @xmath46 being all positive , and in appendix [ sec : arbitrary_coefficients ] we generalize to coefficients with arbitrary signs or complex phases .
consider an extended configuration space which consists of ordered sequences @xmath49 for arbitrary @xmath50 , with @xmath51 .
it is illustrative to interpret such configuration space as a stack of elements of the space @xmath41 , so that @xmath52 is at the top of the stack .
the desired random process can be specified by the following prescriptions . at each discrete time step
do one of the following : create : : : with probability @xmath44 create new element @xmath43 and push it to the stack .
evolve : : : with probability @xmath53 pop the element @xmath3 from the stack and push the element @xmath2 to the stack .
join : : : with probability @xmath46 consecutively pop two elements @xmath39 , @xmath40 from the stack and push a single element @xmath2 to the stack .
restart : : : with probability @xmath54 , where @xmath39 , @xmath40 are the two topmost elements in the stack , empty the stack and push a single element @xmath43 into it , with probability distribution proportional to @xmath44 .
the last action is also the procedure used to initialize the random process .
the `` evolve '' action is just the evolution of a single markov chain at the top of the stack , with transition probabilities proportional to @xmath53 .
the condition ( [ probability_ineq ] ) and the positivity requirement ensures that @xmath44 , @xmath53 and @xmath46 can be interpreted as probabilities .
consider now an equation for the stationary probability distribution of such a markov chain .
it has a general form @xmath55 , where @xmath56 is a stationary probability of the occurrence of a state @xmath57 and @xmath58 is the transition probability between the states @xmath59 and @xmath57 .
let @xmath60 be the stationary probability to find the elements @xmath61 in the stack .
this probability distribution function is obviously normalized to unity : @xmath62 the equation for the stationary probability distribution in our case reads : @xmath63 where @xmath64 and @xmath65 . by a direct substitution
one can check that there is a factorized solution for @xmath60 : @xmath66 where @xmath48 obeys exactly the equation ( [ random_process_eq ] ) and @xmath67 obeys the following inhomogeneous linear equation : @xmath68 thus , for any equation of the form ( [ random_process_eq ] ) with positive coefficients which satisfy ( [ probability_ineq ] ) , there is a random process whose stationary distribution encodes the solution of this equation as in ( [ random_process_factorized_pd ] ) .
the factorization of the stationary probability distribution of random processes with such an infinite configuration space is known as the `` propagation of chaos '' in random processes and was discovered for classical kinetic equations by mckean , vlasov and kac @xcite . comparing the equation ( [ random_process_eq ] ) with schwinger - dyson equations ( [ phi4_sds_original_n2 ] ) , ( [ phi4_sds_original ] ) , ( [ phi4_sds_momentum_n2 ] ) and ( [ phi4_sds_momentum ] ) , we conclude that this property corresponds to the factorization of single - trace operators in large-@xmath0 quantum field theories .
it is interesting that time reversal of the random process described above leads to the so - called branching random process @xcite , which has quite different properties .
this is due to the fact that for such random processes there is no detailed balance condition , and hence no time reversal symmetry .
we do not consider here a subtle mathematical question of the existence of solutions to equation ( [ random_process_eq ] ) , since in our case it is ensured by the physical applications of this equation .
finally , let us describe a practical procedure for finding @xmath48 by simulating the random process described above . by standard statistical methods
, one should sample the probability distribution @xmath69 of the topmost element in the stack ( provided there is more than one element in it , otherwise we estimate @xmath67 rather than @xmath48 , see ( [ random_process_factorized_pd ] ) ) . from ( [ random_process_factorized_pd ] ) , we get @xmath70 , with @xmath71 . it should be stressed that @xmath48 is not normalized to unity , but rather satisfies the inequality @xmath72 . the value of the normalization constant @xmath73 can be also easily found numerically , since the probability to find @xmath74 elements in the stack decreases as @xmath75 for @xmath76 .
after presenting the general method in section [ sec : recursive_process ] , we are ready to describe a stochastic numerical solution of schwinger - dyson equations ( [ phi4_sds_momentum_n2 ] ) , ( [ phi4_sds_momentum ] ) . for simplicity ,
let us assume that the coupling constant @xmath77 in ( [ phi4_field_action ] ) is negative .
this allows us to apply directly the results of section [ sec : recursive_process ] , where all the coefficients in ( [ random_process_eq ] ) are assumed to be positive . in the case of positive @xmath77 ,
additional sign variables for each sequence of momenta can be easily introduced following appendix [ sec : arbitrary_coefficients ] .
this will be done in the next subsection for the hermitian matrix model .
note that while at finite @xmath0 the theory with negative coupling constant is not defined and the correlators are non - analytic in @xmath77 @xcite , in the leading order in @xmath0 perturbative series converge even when the coupling is negative , but not exceeding some critical value @xcite .
correspondingly , in the planar approximation the correlators are analytic in @xmath77
. the space @xmath41 in ( [ random_process_eq ] ) should be the space of ordered sequences ( of any size ) of momenta @xmath36 , correspondingly , the extended configuration space is a stack which contains such sequences .
it is convenient also to introduce two normalization constants @xmath78 and @xmath14 , so that the functions @xmath79 which will be estimated stochastically are defined as @xmath80 where @xmath20 is again the total volume of space . the constant @xmath14 can be thought of as the renormalization constant for the one - particle wave functions , and @xmath78 - as the overall wavefunction normalization . in terms of the functions
@xmath81 the schwinger - dyson equations ( [ phi4_sds_momentum_n2 ] ) and ( [ phi4_sds_momentum ] ) read : @xmath82 comparing the schwinger - dyson equations ( [ phi4_sds_momentum_n2_rescaled ] ) , ( [ phi4_sds_momentum_rescaled ] ) with the general equation ( [ random_process_eq ] ) , we arrive at the random process which stochastically solves these equations .
this random process is specified by the following probabilistic choice of actions at each discrete time step : create : : : with probability @xmath83 push a new sequence of momenta @xmath84 to the stack .
add : : : with probability @xmath85 modify the topmost sequence of momenta @xmath36 in the stack by adding a pair of momenta @xmath84 either as @xmath86 or @xmath87 .
create vertex : : : with probability @xmath88 replace the topmost sequence @xmath89 in the stack by @xmath90 .
this action can only be performed if the topmost sequence contains more than two elements .
join : : : with probability @xmath91 pop the two sequences @xmath36 , @xmath92 from the stack ( provided there are more than two elements in it ) and join them into a single sequence as @xmath93 .
push the result to the stack .
restart : : : otherwise restart with a stack containing a sequence @xmath94 , @xmath95 being distributed with the probability proportional to @xmath34 since the momenta are always added to the stack in pairs which sum up to zero , for all sequences in the stack the total sum of all momenta in the sequence is always zero .
the @xmath96 factors in ( [ phi4_sds_momentum_n2_rescaled ] ) , ( [ phi4_sds_momentum_rescaled ] ) ensure that the probability distributions of the newly created momenta can be normalized to unity .
let us check whether the inequalities ( [ probability_ineq ] ) are satisfied for such a process , that is , whether the total probability of all possible actions does not exceed unity .
for the free propagator @xmath34 one has the inequalities @xmath97 and @xmath98 .
the total probability of all possible actions can be then estimated as @xmath99 .
clearly , for sufficiently small @xmath100 this estimate can be always made smaller than unity by increasing @xmath14 and @xmath78 . in subsections [ subsec :
matrix_model ] and [ subsec : weingarten ] we will analyze such bounds on coupling constants in more details for hermitian matrix model and for the weingarten model .
since the constructed process involves no permutations , one can trace the history of each momenta in the stack - from creation to joining into a vertex or a `` restart operation '' . by drawing all the momenta in stack as points on the vertical lines of some two - dimensional grid and connecting the corresponding points along the horizontal lines , all planar diagrams of the theory ( with an arbitrary number of external lines ) can be obtained .
note also that the number of vertices in planar diagrams drawn by this random process can not exceed the number of time steps from the previous `` restart '' action .
thus in order to maximize the mean order of diagrams which are summed up in some fixed number of time steps , it is advantageous to maximally reduce the rate of `` restart '' events , that is , to saturate the inequalities ( [ probability_ineq ] ) .
one could also try to devise a random process which would solve the schwinger - dyson equations ( [ phi4_sds_original_n2 ] ) , ( [ phi4_sds_original ] ) directly in physical space - time , rather than in the momentum space .
the configuration space of such a process would be the stack of sequences of points @xmath49 .
as compared to the algorithm in the momentum space , there would be an additional choice of moving the last point @xmath52 in the topmost sequence to adjacent lattice sites , with the probability proportional to the hopping parameter @xmath101 .
this would correspond to drawing the worldlines of virtual and real particles by bosonic random walks .
interestingly , such worldlines can be mapped onto the string worldsheets in simplicial string theory @xcite .
as well , the creation of a new interaction vertex would only be possible if three such random walks would intersect in one point .
however , this is an unlikely event , with probability going to zero in the continuum limit .
thus , solving the schwinger - dyson equations directly in the coordinate representation would lead to a less efficient numerical algorithm .
note that for the theory ( [ phi4_field_action ] ) at finite @xmath0 the schwinger - dyson equations are linear equations , which are , however , defined on much larger functional space : the set of unknown functions includes also expectation values of multi - trace operators , such as @xmath102 .
one can try to solve these linear equations by interpreting them as the equations for the stationary probability distribution of a markov process .
the configuration space of such a process should be a space of sequences of the form @xmath103 , thus encoding the expectation values of all multi - trace operators .
however , such a straightforward procedure leads to non - normalizable transition probabilities , indicating that the series which one tries to sum up are divergent .
only when the terms subleading in @xmath104 are omitted from the schwinger - dyson equations , they can be interpreted as stochastic equations . at the same time
, we obtain the markov process on the extended configuration space described in section [ sec : recursive_process ] , which we interpret as the stack of sequences .
the property of the `` propagation of chaos '' @xcite ensures large-@xmath0 factorization of single - trace operators ( see equation ( [ random_process_factorized_pd ] ) ) .
we are thus led to the random process of recursive type @xcite .
to check the considerations of the previous subsection , let us consider the theory ( [ phi4_field_action ] ) in zero dimensions , that is , the hermitian matrix model with the following partition function : @xmath105 the green functions now depend only on one integer @xmath74 : @xmath106 .
the schwinger - dyson equations ( [ phi4_sds_original_n2 ] ) , ( [ phi4_sds_original ] ) also take a very simple form : @xmath107 here we will assume that the coupling constant @xmath77 can be both positive and negative , in order to illustrate the method described in appendix [ sec : arbitrary_coefficients ] . let us again define the `` renormalized '' green functions @xmath108 as @xmath109 . in the case of arbitrary sign of @xmath77 , the configuration space of the random process
should be the stack which contains integer positive numbers and additional sign variables .
following appendix [ sec : arbitrary_coefficients ] , we introduce the variables @xmath110 and @xmath111 which are proportional to the probabilities to find the elements @xmath112 or @xmath113 at the top of the stack ( provided the stack contains more than one element )
. then @xmath114 .
we thus arrive at the following random process for stochastic evaluation of @xmath115 . at each discrete time step one
performs at random one of the following actions : * with probability @xmath116 add new element @xmath117 to the stack .
* with probability @xmath118 increase the topmost element in the stack by @xmath119 and do not change its sign . * with probability @xmath120 decrease the topmost element in the stack by @xmath119 ( if it is greater than one ) and multiply its sign by the sign of @xmath77 . * with
probability @xmath121 pop the two elements @xmath122 and @xmath123 from the stack ( provided there are more than two elements ) and push the element @xmath124 to the stack .
* otherwise empty the stack and push into it a single element @xmath117 .
note that for positive @xmath77 elements with the minus sign are not generated , so that @xmath125 and the random process automatically reduces to the one described in section [ sec : recursive_process ] .
the inequalities ( [ probability_ineq ] ) read now : @xmath126 as discussed in subsection [ subsec : phi4_general ] , in order to increase the efficiency of the algorithm it is advantageous to saturate this inequality .
it is easy to see that at the same time we saturate the upper bound on the absolute value of the coupling constant @xmath77 . maximizing this upper bound with respect to @xmath78 and @xmath14
, we see that @xmath100 can not exceed the value @xmath127 , where @xmath128 is the convergence radius of the planar perturbative expansion which can be found from the exact solution of the matrix model ( [ matrix_model_def ] ) @xcite .
thus the described random process covers only some finite subrange of coupling constants for which the model ( [ matrix_model_def ] ) is defined .
it is easy to understand the origin of this limitation : in fact the random process described above simulates an ensemble of diagrams with an arbitrary number of external legs , with the weight of each diagram being proportional to @xmath129 , where @xmath130 is the number of vertices .
the number of open diagrams with a given number of vertices is obviously larger than the number of closed diagrams , hence the sums over open diagrams have smaller convergence radius .
+ for @xmath131 , there is a continuous set of @xmath78 , @xmath14 which saturate the inequality ( [ brezin_process_inequalities ] ) .
one can , for example , fix @xmath14 and express @xmath78 as a function of @xmath77 : @xmath132 we call the solution with the minus sign in front of the square root `` branch 1 '' and the other solution `` branch 2 '' . + on fig .
[ fig : brezin_moments ] we plot the green functions @xmath133 evaluated using the described random process as functions of @xmath77 up to @xmath134 .
these results were obtained after @xmath135 discrete time steps at fixed @xmath136 and with @xmath78 given by the `` branch 1 '' of ( [ nn_vs_lambda ] ) .
the error bars are smaller than the symbols on the plot .
solid lines are the exact results for @xmath133 in the planar approximation , obtained using the saddle point method @xcite . + autocorrelation time and
mean stack size for the described random process are plotted on figs .
[ fig : autocorrelation_time ] and [ fig : mean_stack_size ] , respectively , as the functions of the coupling constant @xmath77 .
the observable used to define the autocorrelation time was the sum of all numbers in the stack .
first , we note that `` branch 1 '' is more advantageous for simulations , since with larger mean stack size one can gain more statistics .
however , in this case the autocorrelation time is also larger .
interestingly , for this branch both the autocorrelation time and the mean stack size have maximum near @xmath137 rather than near the `` critical point '' of the random process @xmath138 . for `` branch 2 '' ,
these quantities increase slowly towards @xmath138 .
+ in order to characterize the strength of the sign problem , we consider the quantity @xmath139 @xmath140 if the random process generates only elements with the plus sign and @xmath141 if the numbers of pluses and minuses exactly cancel . in practice , it is advantageous to have as large @xmath142 as possible , so that the difference @xmath143 can be estimated with maximal precision .
@xmath142 are plotted on fig .
[ fig : sign_problem_severity ] as the function of @xmath77 for @xmath144 . for @xmath145 ,
@xmath146 decreases with @xmath77 and @xmath74 .
the sign cancelation is thus moderate for @xmath147 ( @xmath148 ) and becomes more and more important for higher - order correlators - @xmath149 at @xmath150 is close to zero .
it is interesting that @xmath142 are almost equal for two different choices of @xmath78 in ( [ nn_vs_lambda ] ) .
thus there are no indications of severe critical slowing down in the whole range of possible coupling constants @xmath151 .
the sign problem is also moderate for low - order correlators , but becomes more severe for higher - order correlators .
it could be extremely interesting to extend the applicability of the described random process up to @xmath152 while preserving these attractive features of the algorithm .
weingarten model @xcite is a lattice field theory which in the large-@xmath0 limit reproduces the sum over all closed surfaces with genus one on the hypercubic lattice .
the action for each surface is proportional to its area , thus the model can be considered as a lattice regularization of bosonic strings with nambu - goto action .
although this model does not have a nontrivial continuum limit for any space dimensionality @xcite , the structure of the functional integral and of the schwinger - dyson equations in this model are similar to those in large-@xmath0 non - abelian lattice gauge theory , and the analysis of this model might be helpful for the extension of the approach described here to non - abelian gauge theories . in order to derive the schwinger - dyson equations , it is convenient to consider the reduced weingarten model @xcite , which in the large - n limit is equivalent to the original model , similarly to the eguchi - kawai model for non - abelian lattice gauge theory . it can be shown that in contrast to reduced lattice gauge theories , for the reduced weingarten model additional twisting is not necessary @xcite . ) in the weingarten model ( [ weingarten_model_def ] ) .
these equations should hold for any link ( marked by a thick line ) which belongs to the loop.,title="fig:",width=226 ] + the reduced model is defined by an integral over complex @xmath153 matrices @xmath154 with @xmath155 : @xmath156 if one treats the second term in the exponent in ( [ weingarten_model_def ] ) as a perturbation and expands @xmath157 in powers of @xmath158 , the resulting sum over planar diagrams is equivalent to the sum over all possible closed surfaces of genus one on the lattice with the weight @xmath159 , where @xmath160 is the area of each surface . a basic observable in this model
is the sum over all planar surfaces which are bounded by some closed loop @xmath161 .
the loop @xmath161 can be uniquely specified by a sequence @xmath162 , where @xmath163 s take the values @xmath164 . in order to reconstruct the loop @xmath161 from the sequence
, one should start from an arbitrary point on a hypercubical lattice and move along one link in the direction @xmath165 , forward if @xmath165 is positive and backward if @xmath165 is negative . from this new position one should similarly move in the direction @xmath166 , and so on . from the diagrammatic expansion one
can see that such a sum over surfaces is given by the following correlator : @xmath167 where one takes the conjugate variable @xmath168 if @xmath169 is negative .
this observable is similar to wilson loop in lattice gauge theory , but , unlike the wilson loop , it does not have a `` zigzag symmetry '' @xcite : passing a link forward and immediately backward changes the value of the wilson loop . in the large-@xmath0 limit the single - loop observables factorize , which allows us to obtain a closed set of schwinger - dyson equations for @xmath170 @xcite : @xmath171 these equations should hold for any lattice link @xmath172 belonging to the loop @xmath161 , but the resulting system of equations is redundant , and it is sufficient to consider only one link @xmath165 on the loop . the equations ( [ weingarten_loop_equations ] ) are schematically illustrated on fig .
[ fig : weingarten_loop_equations ] , where the link @xmath165 is marked by a thick line .
we see that the equations ( [ weingarten_loop_equations ] ) again take the form similar to ( [ random_process_eq ] ) .
let us now define the `` renormalized '' observable @xmath173 by rescaling @xmath170 by the factors @xmath78 and @xmath174 as @xmath175 .
one can interpret the factor @xmath176 as the mass attached to the boundaries of random surfaces , somewhat like the bare quark mass in qcd .
the equations ( [ weingarten_loop_equations ] ) then take the following form : @xmath177 let us now devise a random process of the type described in section [ sec : recursive_process ] , which solves stochastically these equations .
the configuration space is now a stack which contains closed loops , that is , sequences of indices @xmath178 .
the desired random process is defined by the following possible actions at each discrete time step : create a new loop : : : with probability @xmath179 create a new elementary loop @xmath180 , where @xmath178 is random ( either positive or negative ) . join loops : : : with probability @xmath181 pop the two loops @xmath182 , @xmath183 from the stack and form a new loop @xmath161 by joining the loops @xmath184 , @xmath185 with a link in the random direction @xmath163 ( either positive or negative ) : @xmath186 .
this action can only be performed if there are more than two loops in the stack .
flatten loop : : : if the three links in the end of the sequence on the top of the stack form a boundary of the plaquette , that is , if the topmost loop has the form @xmath187 for some @xmath163 and @xmath188 , replace these three links by a single link in the direction @xmath188 with probability @xmath189 : @xmath190 .
append to loop : : : with probability @xmath191 append a pair @xmath192 , where @xmath163 is random ( either positive or negative ) , to the topmost sequence @xmath162 in the stack as @xmath193 or @xmath194 .
the probabilities of these two choices are equal .
restart : : : otherwise start with a stack containing an elementary random loop @xmath180 , where @xmath178 is chosen randomly .
again assuming that the sum of the probabilities of all possible actions is equal to one and the probability of `` restart '' events is minimized , we obtain an equation relating @xmath158 , @xmath78 and @xmath174 : @xmath195 maximization with respect to @xmath78 yields the relation between @xmath174 and @xmath158 : @xmath196 we call the solution with the minus sign in front of the square root `` branch 1 '' and the other solution `` branch 2 ''
. + the value of @xmath158 in ( [ weingarten_q_vs_beta ] ) can not exceed the critical value @xmath197 . as we have already seen on the example of the matrix model
, this critical value does not necessarily coincide with the true critical point @xmath198 at which the sum over planar surfaces diverges .
indeed , @xmath199 does not exceed the lower bound @xmath200 obtained in @xcite , and is significantly lower than the critical values obtained numerically in @xcite .
in fact , for @xmath201 all the observables are still dominated by the lowest - order perturbative contributions .
expectation values of the observables @xmath202 ( @xmath203 loop ) and @xmath204 ( @xmath205 loop ) , which were obtained after @xmath206 iterations of the random process described above ( with @xmath174 given by `` branch 1 '' of ( [ weingarten_q_vs_beta ] ) ) , are plotted on fig .
[ fig : wgtn_w1x1_w1x0 ] as the functions of the coupling constant @xmath158 .
solid line corresponds to the first two terms in the perturbative expansion : @xmath207 , @xmath208 . within statistical errors
, one sees only the lowest - order perturbative contributions .
it should be stressed that the proposed random process implements stochastic summation of diagrams of _ all _ orders , but due to the smallness of @xmath209 , a very large computational time is required to see the contributions of higher - order terms .
+ note that when the coupling constant @xmath158 tends to zero ( that is , the `` bare string tension '' of the random surfaces tends to infinity ) and @xmath174 lies between the two solutions of ( [ weingarten_q_vs_beta ] ) , the above random process describes just the growth of `` branched polymers '' , whose branches are bosonic random walks and hence correspond to particles rather than `` strings '' .
these branches consist of loops in which every lattice link is passed twice and which hence sweep out zero area . taking the limit @xmath210 in ( [ weingarten_q_vs_beta ] )
, we find that the minimal value of @xmath174 is @xmath211 . in order to understand this critical value
we first note that in the limit @xmath210 the observables @xmath170 are all equal to one if the links @xmath212 form a loop which sweeps out zero area and zero otherwise .
the probabilities to encounter such loops in the described random process is hence proportional to @xmath213 .
simple examples of such loops , which can be also thought of as the random tree - like graphs on the lattice , are shown on fig .
[ fig : branched_polymers_illustrated ] . now imagine adding to some loop @xmath95 links stemming from some lattice site .
since the loop includes each link twice , the probability decreases by @xmath214 .
the number of possible configurations of @xmath95 links is @xmath215 since each of @xmath95 links can point along any of @xmath19 directions both forward and backward .
an additional factor of @xmath216 appears since the zero - area loops pass twice through each point and the new links can be inserted between the links pointing either forward or backward ( for example , compare the configurations on the left and on the right of fig .
[ fig : branched_polymers_illustrated ] ) .
finally , one can add any number @xmath217 of branches to any point belonging to the branched polymer . at the criticality , adding any number of random links to some configuration should not change its overall weight .
therefore , the change of the weight due to the added links times the number of ways to add them should be equal to unity .
we are thus led to the following equation for @xmath218 : @xmath219 or @xmath220 .
thus , in the limit @xmath210 we indeed reproduce branched polymers with the correct critical behavior . at nonzero @xmath158 ,
deviation from trivial branched polymer configurations can be characterized by the rate of the `` flatten loop '' events .
indeed , since the probability of @xmath74 such events is proportional to @xmath221 , such sequence of events corresponds to a random surface ( which is in general open ) consisting of @xmath74 lattice plaquettes , plus some number of random trees .
one can therefore think of the described random process as of the process of drawing random loops which sweep out random planar surfaces .
the average rate of `` flattening '' events is plotted on fig .
[ fig : wgtn_mfr ] on the right as a function of the coupling constant @xmath158 for different dimensions @xmath19 and for different choices of @xmath174 in ( [ weingarten_q_vs_beta ] ) .
one can see that in the whole range of coupling constants @xmath222 the rate of flattening events is numerically very small . on the other hand ,
the number of links in the loops , as well as the number of loops stored in the stack , are quite large .
mean stack size and mean length of the topmost loop in the stack are plotted on fig .
[ fig : wgtn_mss_mll ] as a function of the coupling constant @xmath158 for different dimensions @xmath19 and for different choices of @xmath174 in ( [ weingarten_q_vs_beta ] ) .
one can conclude therefore that the `` branched polymers '' actually dominate in the properties of the described random process .
critical behavior of these random trees is universal for any dimension @xmath19 , that is why such observables as the mean stack size or the mean loop length , which are mainly sensitive to the length of loops rather than to the area of random surfaces , practically do not depend on space dimensionality .
while the closed planar surfaces in the vicinity of the true critical point @xmath198 of the weingarten model are also dominated by `` branched polymers '' @xcite , in our ensemble of open random surfaces this dominance can be thought of as the manifestation of the tachyonic instability of open , rather than closed , strings .
the fact that the critical coupling @xmath199 in our case is smaller than the true critical point @xmath198 can be explained by the fact that the number of open surfaces with a given area is obviously larger than the number of closed surfaces with the same area .
the true critical coupling constant @xmath198 can be quite easily found by a very simple re - weighting procedure , which will be described in details in a separate publication .
in section [ sec : sds_stochastic_solution ] we have described a stochastic method for the solution of the schwinger - dyson equations for theories with noncompact variables .
this method works only at small coupling constants and implements stochastic summation of perturbative series . for theories with compact variables , such as nonlinear @xmath223-models or lattice gauge theories ,
the structure of the schwinger - dyson equations is such that the described method can be straightforwardly applied only in the strong coupling regime , where one can stochastically sum all terms in the strong - coupling expansion .
however , the continuum limit of such theories typically corresponds to the weak - coupling limit .
an additional complication is that for physically interesting theories the observables can not be expressed as convergent power series in the small coupling constant @xmath12 , but rather contain non - analytic part which is typically of the form @xmath13 with some constant @xmath14 @xcite . in this section
we point out one possible way to deal with this problem .
the basic idea is to absorb non - perturbative corrections into some self - consistent redefinition of the expansion parameter @xcite .
recently , a similar resummation method was also considered in @xcite . solving the self - consistency condition leads to the concept of a nonlinear random process with memory @xcite , in which all previous history of the process is used to estimate the value of the self - consistent expansion parameter .
let us illustrate this idea on the simplest example of @xmath1 sigma model in the limit of large @xmath0 .
the model is defined by the following path integral over unit @xmath0-component vectors @xmath224 living on the sites of the @xmath19-dimensional hypercubic lattice : @xmath225 where summation goes over all neighboring lattice sites . despite its simplicity , this model in @xmath226 dimensions is asymptotically free and has a mass gap which depends nonperturbatively on the coupling constant @xmath77 .
schwinger - dyson equations in this theory can be written in terms of the two - point function @xmath227 , @xmath228 as : @xmath229 clearly , these equations have the structure similar to ( [ random_process_eq ] ) , but the inequalities ( [ probability_ineq ] ) are satisfied only for sufficiently large @xmath77 , that is , in the strong - coupling regime .
therefore , the continuum limit at @xmath230 can not be reached by the method described in section [ sec : sds_stochastic_solution ] .
let us , however , rewrite the equation ( [ sigma_model_sd ] ) as @xmath231 and introduce the `` hopping parameter '' @xmath232 now the equation ( [ sigma_model_sd ] ) in the form ( [ sigma_model_sd_as_rw ] ) looks like the equation for the free massive scalar propagator on the lattice with the mass @xmath233 in lattice units .
note that in the weak - coupling limit @xmath230 @xmath234 , @xmath235 and we approach the continuum limit .
let us now solve the equation ( [ sigma_model_sd_as_rw ] ) stochastically , assuming that @xmath236 is proportional to the stationary probability distribution @xmath48 of some random process : @xmath237 , @xmath238 . from ( [ sigma_model_sd_as_rw ] )
we get @xmath239 .
the equation ( [ sigma_model_sd_as_rw ] ) now looks as @xmath240 combining this equation with the definition ( [ sigma_model_kappa_def ] ) , it is easy to show that @xmath241 obeys the following self - consistency condition : @xmath242 the equation ( [ sigma_model_rw_eq ] ) has the form ( [ random_process_eq ] ) without the nonlinear term and thus can be interpreted as the equation for the stationary probability distribution of the position of an ordinary bosonic random walk , defined by the following possible actions at each discrete time step : move : : : with probability @xmath243 move along the random unit lattice vector @xmath244 .
restart : : : with probability @xmath245 start again at the origin @xmath246 .
this ensures that @xmath247 and hence @xmath241 never exceeds its critical value @xmath248 .
therefore @xmath236 and @xmath48 can be expanded in powers of @xmath241 .
thus we have defined a new expansion parameter @xmath241 , which should obey the self - consistency equation ( [ sigma_model_kappa_return_prob ] ) , and obtained a well - defined convergent expansion , namely , the sum over all paths on the lattice with the weight @xmath249 , where @xmath250 is the length of the path . note that the quantity @xmath251 in fact plays the role similar to the gluon condensate ( which is expressed in terms of the mean plaquette in lattice theory ) in non - abelian gauge theory : one can absorb all the divergences into the self - consistent definition of condensates @xcite .
the final step in the construction of the nonlinear random process which solves the equations ( [ sigma_model_sd ] ) is the solution of the self - consistency equation ( [ sigma_model_kappa_return_prob ] ) .
one possible solution is to use the iterations @xmath252 here @xmath253 is the return probability of a bosonic random walk with hopping parameter @xmath241 . in practice
, one should simulate the bosonic random walk at fixed @xmath254 for some number @xmath255 of discrete time steps , and then estimate @xmath256 as @xmath257 , where @xmath258 is the number of discrete time steps spent at @xmath246 . from ( [ sigma_model_iterations ] ) one then gets @xmath259 , and the process is repeated until the value of @xmath241 stabilizes with sufficient numerical precision .
we call such algorithm `` algorithm a '' .
one can also consider an ultimate case , for which the return probability is updated and estimated as @xmath260 every time the point @xmath246 is reached . now
@xmath261 is the time from the start of the random process and @xmath258 is the number of time steps spent at @xmath246 .
such algorithm will be called `` algorithm b '' .
mathematically , such random processes are not markov processes , since the transition probabilities at each next step depend ( via @xmath262 ) on the behavior of the process at all previous time steps .
stationary probability distributions of such processes obey nonlinear equations ( such as ( [ sigma_model_sd ] ) ) @xcite , and and hence they are also called nonlinear random processes . as an interesting side remark ,
let us discuss such a theory at finite temperature , which is described by a bosonic random walk on the cylinder .
clearly , an ordinary bosonic random walk does not feel this compactification of space , and its stationary probability distribution is just a periodic linear combination of the corresponding distribution in infinite space .
such behavior can not lead to any nonlinear finite - temperature effects such as phase transitions . on the other hand , if the parameters of the random walk depend on the return probability , as in ( [ sigma_model_kappa_return_prob ] ) , there is a nonlinear feedback mechanism since in the compactified space the returns are more likely .
thus finite temperature indeed affects the local behavior of the random walker with memory and might lead to interesting critical phenomena .
+ in order to illustrate such a stochastic solution of the equations ( [ sigma_model_sd ] ) , we consider the case @xmath226 . in two dimensions
the model ( [ sigma_model_pf ] ) is asymptotically free , and one can introduce the lattice spacing by fixing the value of mass in physical units ( we set @xmath263 ) : @xmath264 .
the process of convergence of the lattice spacing to its exact value is illustrated on fig .
[ fig : sigma_model_convergence ] for both the algorithms `` a '' and `` b '' . for algorithm `` a '' we have used @xmath265 .
the algorithm `` a '' converges much faster than the algorithm `` b '' .
the values of lattice spacing obtained using both algorithms are compared with the exact solution on fig .
[ fig : spacing_vs_lambda ] . in agreement with asymptotic freedom ,
lattice spacing quickly decreases with @xmath77 .
again , algorithm `` a '' yields more precise results in the same number of time steps .
in this paper we have presented numerical strategies for the stochastic summation and re - summation of perturbative expansions in large-@xmath0 quantum field theories .
our basic approach was to interpret the schwinger - dyson equations as the equations for the stationary probability distribution of some random process .
since schwinger - dyson equations in such theories are nonlinear equations , we had to use so - called nonlinear random processes , rather than ordinary markov processes whose stationary probability distributions always obey linear equations .
it is interesting to note that since the configuration spaces of random processes described in this paper are discrete , their numerical implementation require floating - point operations only for the random choice of actions . thus such algorithms can be potentially much faster than the standard monte - carlo simulations based on floating - point arithmetic , and can be advantageous for machines based on gpus .
our final goal is to extend the presented approach to non - abelian lattice gauge theories .
however , in this case direct stochastic interpretation of schwinger - dyson equations is only possible at strong coupling , while the continuum limit of such theories corresponds to the weak - coupling limit . in section [ sec : rps_with_mem ] , we have discussed a way to access the weak - coupling limit , which , however , was implemented numerically only for @xmath1 sigma - model at large @xmath0 .
the basic idea is to absorb the divergences into a self - consistent redefinition of the expansion parameter and solve the self - consistency conditions using random processes with memory . in some sense
, @xmath1 sigma - model can be thought of as the bosonic random walk in its own condensate , and the approach to the self - consistent value of mass gap ( see fig . [
fig : sigma_model_convergence ] ) - as a renormalization - group flow . for non - abelian gauge theories the redefined expansion parameters can emerge as the lagrange multipliers for the `` zigzag symmetry '' of the qcd string and should also satisfy some self - consistency conditions @xcite .
zigzag symmetry means that when one adds a line which is passed forward and backward to the boundary of the fluctuating string , the amplitudes should not change . in lattice gauge theory ,
this condition is equivalent to the unitarity of the link variables @xmath266 , which is similar to the condition @xmath267 in @xmath1 sigma - model .
these redefined parameters can be also related to the gluon condensate @xcite . by analogy with the sigma - model
, one can think that non - abelian gauge theories are similar to strings moving in some self - consistent condensates .
such a picture is also close to the idea of holographic ads / cft duality for non - abelian gauge theories , where the dual string lives in some self - consistent gravitational background , and the parameters of this background can be related to gluon condensates in gauge theory @xcite .
in fact , the requirement that the metric of the holographic background approaches that of the ads space - time ensures the zigzag symmetry of the strings which end on the ads boundary @xcite . in view of these qualitative considerations ,
our hope is that the loop equations in non - abelian gauge theories can be solved stochastically by a random process similar to the one which was devised for the weingarten model of random surfaces ( see subsection [ subsec : weingarten ] ) , but with some self - consistent choice of parameters , which might be implemented as the `` memory '' in the random process . among other possible applications of the presented method one can think of the solution of schwinger - dyson equations in continuum gauge theories , combined with the renormalization group methods @xcite , numerical analysis of quantum gravity models described by various matrix models , and numerical solution of hydrodynamical equations @xcite .
it should be noted here that several attempts at the stochastic solution of the loop equations in large-@xmath0 gauge theories have been already described in the literature quite a long time ago @xcite .
these algorithms were , in essence , based on the so - called branching random processes , so that the wilson loop @xmath268 is proportional to the probability of transition from the initial loop configuration @xmath161 to the empty configuration with no loops . in particular , in contrast to the algorithm described in subsection [ subsec : weingarten ] , where one of the basic steps is to join loops , in the algorithms described in @xcite the basic step was to split a self - intersecting loop into two loops .
as a result , these algorithms did not implement the importance sampling and were not able to produce any sensible results for the four - dimensional gauge theory . generally , branching random processes similar to those considered in @xcite can be obtained from the `` recursive '' nonlinear random process described in this paper by time reversal . however , since such processes do not satisfy any detailed balance condition , they are not invariant under this operation , and lead to very different numerical algorithms .
i am grateful to drs .
m. i. polikarpov , yu .
m. makeenko , a. s. gorsky , n. v. prokofev and i. ya .
arefeva for interesting and stimulating discussions .
i d like also to thank drs .
f. bruckmann and a. schaefer for their kind hospitality at the university of regensburg , where a part of this work was written .
this work was partly supported by grants rfbr 09 - 02 - 00338-a , rfbr 08 - 02 - 00661-a , a grant for the leading scientific schools nsh-6260.2010.2 , by the federal special - purpose programme `` personnel '' of the russian ministry of science and education , and by personal grants from the `` dynasty '' foundation and from the fair - russia research center ( frrc ) .
the random process described in section [ sec : recursive_process ] was devised under the assumption that the coefficients @xmath44 , @xmath269 and @xmath46 are real and positive for any @xmath2 , @xmath39 and @xmath40 . in this appendix
we show how the solution of the equation ( [ random_process_eq ] ) with arbitrary signs or complex phases on the r.h.s . can be reduced to the solution of another equation of the form ( [ random_process_eq ] ) with all positive coefficients , provided the inequalities ( [ probability_ineq ] )
are satisfied .
we begin by discussing the case of real but non - positive coefficients in details , and finally sketch the extension to complex - valued coefficients . to this end ,
let us represent the sign - alternating coefficients in ( [ random_process_eq ] ) as : @xmath270 where @xmath271 , @xmath272 and @xmath273 are all positive and also obey the following inequality : @xmath274 for any @xmath39 , @xmath40 .
obviously , these inequalities can be satisfied if the inequalities ( [ probability_ineq ] ) are satisfied .
let us now introduce two functions @xmath275 and @xmath276 , which satisfy the following equations : it is now easy to check that the difference @xmath278 satisfies the equation ( [ random_process_eq ] ) . on the other hand , the equations ( [ sign_alt_repr_eq ] ) again
have the form of ( [ random_process_eq ] ) with all positive coefficients , but with the configuration space @xmath279 being the direct product @xmath280 .
in other words , each variable @xmath2 now in addition carries the `` sign '' , @xmath281 or @xmath282 , which can be written as @xmath283 or @xmath284 . basing on the results presented in section [ sec : recursive_process ] , one can devise the random process which solves these equations : create : : : with probability @xmath271 create new element @xmath285 and push it to the stack .
evolve : : : with probability @xmath286 pop the element @xmath287 from the stack and push the element @xmath288 to the stack , with probability @xmath289 do the same but flip the sign of @xmath3 .
join : : : with probability @xmath290 consecutively pop two elements @xmath291 , @xmath292 from the stack and push a single element @xmath293 to the stack .
that is , two pluses or two minuses associated with @xmath3 s give @xmath283 , but one plus and one minus give @xmath284 .
with probability @xmath294 do the same , but flip the resulting sign , that is , push the element @xmath295 to the stack .
restart : : : otherwise , empty the stack and push a single element @xmath285 into it , with probability proportional to @xmath271 . as in section
[ sec : recursive_process ] , @xmath275 and @xmath276 are proportional to probabilities to find the elements @xmath283 or @xmath284 on the top of the stack , provided there is more than one element in it .
the extension of this construction to complex - valued coefficients is quite straightforward .
we represent the coefficients in ( [ random_process_eq ] ) as @xmath296 , where @xmath297 is real and positive , and similarly for the other coefficients .
the configuration space @xmath279 becomes the direct product @xmath298 , where @xmath299 is the unit circle in the complex plane .
the stack now contains the pairs @xmath300 , with @xmath301 } $ ] being the complex phase . the function @xmath48 is estimated as @xmath302 , where @xmath303 is the probability to find the element @xmath300 at the top of the stack . in the random process
, new elements are created with probability distribution proportional to @xmath297 , and in the `` evolve '' and the `` join '' actions the phases of the elements and the coefficients in ( [ random_process_eq ] ) are added modulo @xmath33 , similarly to signs . note that this solution does not in general have the property of importance sampling .
indeed , one can have some @xmath2 for which @xmath48 is numerically very small , but the random process can spend an almost equal large amount of time in the states @xmath283 and @xmath284 , so that numerically large @xmath275 and @xmath276 nearly cancel . whether this occurs or not depends on the particular system of equations and on the particular unknown variables , but potentially this feature can make numerical simulations less efficient .
|
we propose a stochastic method for solving schwinger - dyson equations in large-@xmath0 quantum field theories .
expectation values of single - trace operators are sampled by stationary probability distributions of the so - called nonlinear random processes .
the set of all histories of such processes corresponds to the set of all planar diagrams in the perturbative expansions of the expectation values of singlet operators .
we illustrate the method on the examples of the matrix - valued scalar field theory and the weingarten model of random planar surfaces on the lattice . for theories with compact field variables , such as sigma - models or non - abelian lattice gauge theories ,
the method does not converge in the physically most interesting weak - coupling limit . in this case
one can absorb the divergences into a self - consistent redefinition of expansion parameters .
stochastic solution of the self - consistency conditions can be implemented as a `` memory '' of the random process , so that some parameters of the process are estimated from its previous history .
we illustrate this idea on the example of two - dimensional @xmath1 sigma - model .
extension to non - abelian lattice gauge theories is discussed .
|
the question of neutrino mass is one of the most profound in modern particle physics .
most plausible models of neutrino mass solve the puzzle of why neutrino masses are so small by introducing a new scale at high energy , and precision studies of neutrino physics therefore hold the potential to investigate physics at scales beyond those reachable in current accelerator experiments .
they also make the study of the possible majorana nature of neutrinos possible ( see @xcite for a thorough discussion of this ) .
while the neutrino mass differences have now been measured at about 10% precision by oscillation experiments ( see e.g. @xcite ) the absolute mass scale remains unknown and inaccessible to oscillation experiments .
there are , however , several possible paths to measuring the absolute neutrino mass .
the kinematical effect of neutrino mass can be probed either via its effect on the beta decay spectrum or via its effect on cosmological structure formation . if neutrinos are majorana particles a different possibility is to search for neutrinoless double beta decay because the transition probability for this process is proportional to the neutrino mass squared . in the past year
there have been several papers discussing how to unify the data analysis for the various approaches @xcite .
this is a non - trivial issue , given that completely different physics is involved and that the three probes are actually sensitive to three distinct observables . here
we present a new markov chain monte carlo global analysis of neutrino parameters using both cosmological and experimental data .
the analysis software is based on the cosmomc markov chain monte carlo ( mcmc ) package for cosmological parameter estimation @xcite , appropriately modified to incorporate all parameters related to neutrino physics .
this approach uses bayesian inference instead of the frequentist method commonly used in particle physics .
the approach is somewhat similar to the mcmc technique developed in @xcite to constrain mssm parameters .
however , a key difference is that here we keep the full cosmological parameter estimation which allows for a closer study of the interplay between neutrino data and cosmological parameter estimation . in section
ii we describe the methodology used and in section iii we present the main results for various different assumptions about present and future data , as well as different parameter spaces .
finally we present our conclusions in section iv .
the mcmc bayesian inference approach has been described in detail for instance in @xcite . based on assumed priors on each parameter
it samples the likelihood function using the markov chain monte carlo method and from that the posterior credible intervals for all parameters can be calculated . before running the markov chains it is therefore necessary to specify both the parameters to be used and the priors on all parameters . for the neutrino physics part we have used the mass of the lightest eigenstate @xmath0 ( @xmath1 for the normal hierachy , @xmath2 for the inverted hierarchy ) , the two mass differences @xmath3 , @xmath4 and the three mixing angles @xmath5 , @xmath6 , @xmath7 .
we also assume that neutrinos are majorana particles so that there are two additional majorana phases @xmath8 , @xmath9 , which together with the mass differences and mixing angles specify the observables related to absolute neutrino mass ( assuming only active neutrinos ) . in total there are then 8 parameters related to the neutrino sector .
there are three separate observables related to the three different types of probes .
cosmology is only sensitive to the neutrino mass , and until the accuracy reaches the 0.05 ev level only to the sum of neutrino masses ( see @xcite for a thorough discussion of this ) @xmath10 in terms of the parameters used in cosmomc this corresponds to @xmath0 , @xmath11 and @xmath12 .
these are therefore the only parameters which are regarded as `` slow '' in the sense that a change in one of them requires recalculation of the transfer function for cosmological perturbations . at the projected level of accuracy of katrin
the change in the electron energy spectrum can be described using a single effective mass parameter which is essentially the incoherent sum ( see e.g.@xcite ) @xmath13 the parameter actually measured in such experiments is in fact @xmath14 which , being a fit - parameter , can be positive or negative when measured .
conversely , the effective mass measured in neutrinoless double beta decay is the coherent sum @xcite @xmath15 which allows for phase cancelation .
the actual parameter measured in any neutrinoless double beta decay experiment is the half - life @xmath16 which is related to @xmath17 via the relation @xcite @xmath18 where @xmath19 is a phase - space factor and @xmath20 the nuclear matrix element squared . in principle
the mcmc code should use the measured @xmath16 and the calculated matrix element ( both including uncertainties ) as parameters instead of @xmath21 .
however , for simplicity we assume a gaussian error on @xmath21 with the estimated error on the matrix element from @xcite .
in the following we have used cosmological parameters consistent with the `` vanilla '' @xmath22cdm model : @xmath23 , the physical baryon density , @xmath24 , the physical cdm density , @xmath25 , the amplitude of primordial fluctuations , @xmath26 the scalar spectral index , @xmath27 , the optical depth to reionization , and @xmath28 , the hubble parameter .
spatial flatness has been assumed so that @xmath29 . in total
there are then 6 parameters related to the cosmological model . for the purpose of parameter estimation
the publicly available cosmomc markov chain monte carlo has been modified to perform parameter estimation in this 14-dimensional parameter space .
cosmomc has been set to use the fast / slow parameter scheme . in table
[ table : priors ] we give the list of parameters as well as their priors .
note that we also include the two parameters @xmath30 , the dark energy equation of state , and @xmath31 , the running of the scalar spectral index , both of which are discussed in section [ sec : extended ] .
lcll parameter & prior & & fast / slow @xmath32 & -2 - 0 & top hat & slow + @xmath11 & @xmath33 ev@xmath34 & top hat & slow + @xmath12 & @xmath35 ev@xmath34 & top hat & slow + @xmath36 & 0 - 1 & top hat & fast + @xmath37 & 0 - 1 & top hat & fast + @xmath38 & 0 - 1 & top hat & fast + @xmath8 & 0-@xmath39 & top hat & fast + @xmath9 & 0-@xmath39 & top hat & fast + @xmath23 & 0.005 - 0.1 & top hat & slow + @xmath24 & 0.01 - 0.99 & top hat & slow + @xmath27 & 0.01 - 0.8 & top hat & slow + @xmath26 & 0.5 - 1.5 & top hat & fast + @xmath40 & 2.4 - 4 & top hat & fast + @xmath41 & 0.3 - 1 & top hat & slow + @xmath42 & -2 -0 & top hat & slow + @xmath43 & -0.2 - 0.2 & top hat & fast for the observables related to neutrino oscillation data we make the simple assumption of gaussian errors , given by the combination of different experiments .
we note that this assumption can easily be changed in the code and replaced with the full likelihood calculation from experimental data .
we use the same constraints as in @xcite , given by @xmath44 with all errors being @xmath45 .
the assumption of gaussian errors does not significantly alter any of our results .
based on the approach described in the previous section we have calculated the present bound on neutrino properties using various combinations of data sets from cosmology , tritium decay and neutrinoless double beta decay respectively .
cosmological constraints on @xmath46 have been calculated by many different authors for various assumptions about parameters and using different data sets ( see e.g.@xcite ) . here
we present just one particular example , which is exactly the same as used in @xcite .
we use the wmap cmb temperature and polarisation data @xcite , the sdss - lrg and 2df large scale structure data @xcite , the sdss - lrg baryon acoustic oscillation data @xcite , and the sni - a data set compiled in @xcite .
details about the cosmological data can be found in @xcite . using only the cosmological data we find a bound of @xmath47 ev for the minimal @xmath22cdm model .
we note that this is slightly lower than the 0.6 ev found in @xcite for the same model and data .
this is worth noticing because the prior on @xmath0 is logarithmic in the present study while it was linear in @xcite .
a logarithmic prior on @xmath0 tends to favour small @xmath0 values because of the large parameter space volume at negative @xmath48 and therefore shifts the allowed region slightly down .
this phenomenon is an integral part of bayesian inference because a prior probability distribution needs to be specified . in frequentist statistics
this problem does not occur and the result does not depend on any priors . it should be noted that in the limit of gaussian statistics the two methods yield exactly the same result .
the phenomenon has been recently been studied in the context of neutrino properties .
for example it was shown in @xcite that bayesian inference and likelihood maximisation give very different results for cosmological parameters such as the radiation density as long as the likelihood function is non - gaussian . as more data is added and the likelihood function approaches a gaussian the two methods converge .
the question of bayesian versus frequentist statistics was studied in @xcite in the context of katrin .
for example the difference between a linear and a logarithmic prior on @xmath49 was investigated and found to have some ( not crucial ) effect . in conclusion
, assumptions about priors will have an effect on the posterior distributions as long as the likelihood function is non - gaussian which is the case for parameters which are not extremely well constrained .
cosmo+katrin+gerda normal hierarchy & & & @xmath50 & @xmath51@xmath52 ( ev ) & @xmath53 & @xmath54 & @xmath55 & @xmath56 @xmath46 ( ev ) & @xmath57 & @xmath58 & @xmath59 & @xmath60 @xmath49/@xmath46 & @xmath61 & @xmath62 & @xmath63 & @xmath64 @xmath21 ( ev ) & @xmath65 & @xmath66 & @xmath67 & @xmath68 @xmath24 & @xmath69 & @xmath70 & @xmath71 & @xmath72 @xmath26 & @xmath73 & @xmath74 & @xmath75 & @xmath75 inverted hierarchy & & & @xmath76 & @xmath51@xmath77 ( ev ) & @xmath78 & @xmath79 & @xmath80 & @xmath81 @xmath46 ( ev ) & @xmath82 & @xmath83 & @xmath84 & @xmath85 @xmath49/@xmath46 & @xmath86 & @xmath87 & @xmath88 & @xmath89 @xmath21 ( ev ) & @xmath90 & @xmath91 & @xmath92 & @xmath93 @xmath24 & @xmath94 & @xmath94 & @xmath71 & @xmath71 @xmath26 & @xmath95 & @xmath73 & @xmath96 & @xmath97 normal hierarchy & & & @xmath77 ( ev ) & x & x & x & @xmath98 @xmath46 ( ev ) & x & x & x & @xmath99 @xmath49/@xmath46 & x & x & x & @xmath100 @xmath21 ( ev ) & x & x & x & @xmath101 @xmath24 & x & x & x & @xmath102 @xmath26 & x & x & x & @xmath103 inverted hierarchy & & & @xmath52 ( ev ) & x & x & x & @xmath104 @xmath46 ( ev ) & x & x & x & @xmath105 @xmath49/@xmath46 & x & x & x & @xmath106 @xmath21 ( ev ) & x & x & x & @xmath107 @xmath24 & x & x & x & @xmath108 @xmath26 & x & x & x & @xmath109 the upper bound on the effective neutrino mass provided by the heidelberg - moscow ( hm ) experiment provides an additional and comparable constraint on the absolute neutrino mass scale @xcite .
we use the constraint @xmath110 based on the nuclear matrix element calculation in @xcite . note that this mass range is more restrictive than what was used in @xcite because the theoretically predicted half - life has been corrected downwards in @xcite compared to @xcite .
we stress again that the conversion of half - life to effective mass @xmath21 depends strongly on the nuclear matrix element and that the bound used here could turn out to be too restrictive .
as can be seen from table [ table : parameters ] , adding the hm data does shift the allowed range on @xmath46 and @xmath21 down .
since the best fit cosmological model in any case has @xmath111 it has no influence on other parameters such as @xmath112 and @xmath26 .
note that we have not derived any cosmological constraint based on the claimed positive evidence from heidelberg - moscow @xcite .
using the same assumptions as above on the nuclear matrix element the claimed evidence translates roughly into @xmath113 ( 90% c.l . ) .
to get a better idea about the future interplay between the three different methods for measuring the absolute mass scale we have performed similar likelihood analyses for the presently available cosmological data together with forecasts for the katrin beta decay experiment @xcite and the gerda neutrinoless double beta decay experiment @xcite . for katrin
we assume a gaussian @xmath114 error on @xmath115 of @xmath116 , roughly in accordance with what was used in @xcite . for the gerda
neutrinoless double beta decay experiment we assume a gaussian error on @xmath117 of 0.01 ev@xmath34 , corresponding roughly to gerda phase 2 @xcite .
we note that other neutrinoless double beta decay experiments such as majorana and cuore @xcite will reach roughly the same sensitivity in a broadly comparable time - frame ( see e.g.@xcite ) . using these very rough experimental characteristics of the two experiments
we have proceeded to calculate constraints on neutrino parameters using two different assumptions : + * ( a ) * in this case we assume no positive detection from either experiment so that the best fit values are @xmath118 , + * ( b ) * here we assume a positive detection from both experiments , @xmath119 ev@xmath34 and @xmath120 ev@xmath34 .
+ the last case would for instance be realised in a model with normal hierarchy , @xmath121 . for both cases we perform parameter estimation assuming both normal and inverted hierarchy . in the first case ( a ) we note that katrin alone would significantly tighten the cosmological constraint on @xmath46 and gerda
would improve this even further .
@xmath21 is , as expected , mainly constrained by adding the gerda data .
the second case ( b ) is more interesting from the perspective of combining data sets .
the best fit values both correspond to roughly @xmath45 evidence for non - zero @xmath49 and @xmath21 respectively .
however , with the combined data @xmath122 is excluded at roughly @xmath123 , likewise @xmath124 is excluded at a similar significance .
this exercise clearly shows the advantage of analysing all neutrino parameters in this global way , instead of simply adding constraints .
it should be noted that since @xmath125 is the best fit to present cosmological data the case ( b ) has a best - fit @xmath126 which is higher than case ( a ) by @xmath127 , i.e. it gives a slightly ( not substantially ) worse fit to cosmological data . in figs .
[ fig : like1 ] and [ fig : like2 ] we show the likelihood contours for cases ( a ) and ( b ) for the assumption of normal hierarchy . in both cases we have used the present uncertainties on the parameters of the mixing matrix ( specified in eq . [
eq : mixing ] ) which means that the non - trivial behaviour for small @xmath128 can not be resolved . given future
improved constraints from reactor or long baseline experiments this region will look significantly different ( see e.g.@xcite for a thorough discussion of this point ) . to complete this section
we have done a parameter study of cases ( a ) and ( b ) using only katrin and gerda data , excluding cosmological constraints .
the results of this can be seen in table [ table : parameter2 ] . for case
( a ) where there is no detection from either experiment the combination of cosmological data with katrin and gerda slightly strengthens the bound on parameters , but since they center on the same best fit value there is no marked difference when cosmological data is added . however , this changes completely when case ( b ) is studied . here ,
katrin and gerda data prefer a higher value for @xmath46 and the combination of all three data sets significantly shift the allowed range for all of the neutrino parameters .
we note that other cosmological parameters such as @xmath24 and @xmath26 are not affected in any way when the minimal @xmath22cdm model is assumed .
this conclusion does not hold when larger cosmological parameter sets are used , a point discussed in the next subsection .
@xmath52 ( ev ) & @xmath129 & @xmath130 @xmath131 ( ev ) & @xmath132 & @xmath133 @xmath49/@xmath46 & @xmath134 & @xmath135 @xmath21 ( ev ) & @xmath136 & @xmath137 normal hierarchy & @xmath52 ( ev ) & @xmath138 & @xmath139 @xmath46 ( ev ) & @xmath99 & @xmath140 @xmath49/@xmath46 & @xmath141 & @xmath142 @xmath17 ( ev ) & @xmath143 & @xmath144 hierarchy & @xmath145 ev , @xmath146 ev @xmath52 ( ev ) & @xmath147 @xmath46 ( ev ) & @xmath148 @xmath49/@xmath46 & @xmath149 @xmath150 ( ev ) & @xmath151 @xmath30 & @xmath152 in order to illustrate the relation between neutrino experiments and cosmological parameter estimation we have performed the same analysis as before , but now adding two additional cosmological parameters to the fit : @xmath30 , the dark energy equation of state , and @xmath31 , the running of the scalar spectral index ( giving a total of 16 parameters in the mcmc analysis ) .
particularly @xmath30 is known to be degenerate with @xmath46 and therefore any independent information on @xmath46 from experiments is potentially important for dark energy physics .
this particular degeneracy has been studied quite extensively in recent literature .
the most recent example is @xcite where the impact of a positive katrin detection of @xmath49 on the estimation of @xmath30 is discussed . in fig .
[ fig : deg1 ] we show the degeneracy between @xmath46 and @xmath30 for our case ( b ) , assuming normal hierarchy .
the corresponding numbers are shown in table [ table : parameter3 ] .
the results confirm previous findings , i.e. that a strongly negative equation of state for dark energy can be compensated by increasing the neutrino mass @xcite .
this also means that the allowed region of @xmath30 for case ( b ) is shifted to more negative values , in this case only marginally allowing a cosmological constant ( the 1d 95% credible interval is @xmath153 ) .
the present result compares well with what is obtained in @xcite , although the assumed best fit values are slightly different .
note also that our treatment of cosmological data is slightly different from @xcite because we use the full bao correlation function instead of the @xmath154 functional parametrisation . for comparison , in fig .
[ fig : deg2 ] we show the @xmath155 degeneracy for cosmological data only . in this case
we find the result @xmath156 ev and @xmath157 , both at 95% c.l .
, a result completely consistent with what was found in @xcite for the same model and data , but using maximisation instead of marginalisation . note also that in this extended model the best fit @xmath126 increases by 2.6 compared to the case where only cosmological data is used ( compared to 3.1 in the smaller parameter space discussed above ) , i.e. in the extended model the inconsistency between cosmology and the assumed positive detection from katrin and gerda is less pronounced .
the exercise carried out in this subsection clearly illustrates why cosmological bounds on neutrino properties are model dependent .
note that this degeneracy would be even stronger if only cmb data is considered .
a detailed neutrino parameter estimation study has been carried out using the markov chain monte carlo technique with the goal of unifying the various techniques for measuring the absolute neutrino mass scale .
the mcmc technique is extremely powerful in this regard and allows for a very fast scanning many - dimensional likelihood spaces . in the concrete example here
we have used 8 parameters describing the properties of light , active majorana neutrinos , and 6 further parameters which specify the cosmology .
we find that for present data the combination of cosmological data with the upper limit on @xmath21 from heidelberg - moscow slightly improves the existing cosmological bound on the sum of neutrino masses .
more interestingly we have studied the interplay between various future constraints from cosmology , tritium decay and neutrinoless double beta decay . if all probes come up with a negative result the addition of data sets does not yield any radically new information .
however , we have also studied an example in which the upcoming katrin and gerda experiments are both assumed to provide tentative evidence for neutrino mass . in this case
the combination of all three types of data allows for a much stronger constraint on neutrino properties than otherwise allowed .
finally we have also studied how experimental data from tritium decay or neutrinoless double beta decay can help in cosmological parameter estimation , particularly concerning the dark energy equation of state .
it should be noted that in the present analysis only presently available cosmological data has been used . in the same time frame as katrin and gerda new cosmological data will become available and is likely to improve the cosmological neutrino mass bound significantly ( see @xcite for a non - exhaustive list ) . in the somewhat longer term cosmological constraints can be potentially be pushed below 0.1 ev sensitivity to @xmath46 . at the same time
neutrinoless double beta decay experiments will have equally improved sensitivity and it will very likely be possible to determine the absolute neutrino mass as well as the nature of the mass hierarchy . in conclusion , the combination of cosmological data with experimental neutrino data in a global analysis will be extremely useful in the future , when more precise experimental data becomes available .
use of computing resources from the danish center for scientific computing ( dcsc ) is acknowledged .
use of the cosmomc package @xcite is acknowledged .
amand fssler is thanked for discussions on the effective neutrino mass in neutrinoless double beta decay .
r. n. mohapatra , `` physics of neutrino mass , '' econf * c040802 * , l011 ( 2004 ) [ new j. phys .
* 6 * , 82 ( 2004 ) ] [ arxiv : hep - ph/0411131 ] .
r. n. mohapatra _ et al .
_ , `` theory of neutrinos , '' arxiv : hep - ph/0412099 .
r. n. mohapatra and a. y. smirnov , `` neutrino mass and new physics , '' ann
. rev .
nucl . part .
sci . * 56 * , 569 ( 2006 ) [ arxiv : hep - ph/0603118 ] .
m. maltoni , t. schwetz , m. a. tortola and j. w. f. valle , `` status of global fits to neutrino oscillations , '' new j. phys .
* 6 * , 122 ( 2004 ) [ arxiv : hep - ph/0405172 ] .
g. l. fogli _ et al .
_ , `` observables sensitive to absolute neutrino masses : a reappraisal after wmap-3y and first minos results , '' phys .
d * 75 * , 053001 ( 2007 ) [ arxiv : hep - ph/0608060 ] . o. host , o. lahav , f. b. abdalla and k. eitel , `` forecasting neutrino masses from combining katrin and the cmb : frequentist and bayesian analyses , '' arxiv:0709.1317 [ hep - ph ] .
a. lewis and s. bridle , `` cosmological parameters from cmb and other data : a monte - carlo approach , '' phys . rev .
d * 66 * ( 2002 ) 103511 [ arxiv : astro - ph/0205436 ] a. lewis , homepage , http://cosmologist.info r. r. de austri , r. trotta and l. roszkowski , `` a markov chain monte carlo analysis of the cmssm , '' jhep * 0605 * , 002 ( 2006 ) [ arxiv : hep - ph/0602028 ] .
n. christensen , r. meyer , l. knox and b. luey , `` bayesian methods for cosmological parameter estimation from cosmic microwave background measurements , '' class .
grav . * 18 * , 2677 ( 2001 ) [ arxiv : astro - ph/0103134 ] .
j. lesgourgues , s. pastor and l. perotto , `` probing neutrino masses with future galaxy redshift surveys , '' phys
. rev .
d * 70 * , 045016 ( 2004 ) [ arxiv : hep - ph/0403296 ] .
j. lesgourgues and s. pastor , `` massive neutrinos and cosmology , '' phys .
* 429 * ( 2006 ) 307 [ arxiv : astro - ph/0603494 ] .
s. s. masood , s. nasri , j. schechter , m. a. tortola , j. w. f. valle and c. weinheimer , `` exact relativistic beta decay endpoint spectrum , '' arxiv:0706.0897 [ hep - ph ] .
c. aalseth _
et al . _ ,
`` neutrinoless double beta decay and direct searches for neutrino mass , '' arxiv : hep - ph/0412300 . s. m. bilenky and s. t. petcov , `` massive neutrinos and neutrino oscillations , '' rev .
* 59 * , 671 ( 1987 ) [ erratum - ibid . *
61 * , 169 ( 1989 ) ] .
v. a. rodin , a. faessler , f. simkovic and p. vogel , `` assessment of uncertainties in qrpa 0nu beta beta - decay nuclear matrix elements , '' nucl .
a * 766 * , 107 ( 2006 ) .
v. a. rodin , a. faessler , f. simkovic and p. vogel , `` erratum : assessment of uncertainties in qrpa @xmath158-decay nuclear matrix elements [ nucl .
phys . a 766 , 107 ( 2006 ) ] , '' nucl . phys .
a * 766 * , 107 ( 2006 ) [ arxiv:0706.4304 [ nucl - th ] ] .
a. faessler , talk at the path to neutrino mass workshop , aarhus ,
september 2007 ( http://astroparticle.phys.au.dk ) c. zunckel and p. g. ferreira , `` conservative estimates of the mass of the neutrino from cosmology , '' arxiv : astro - ph/0610597 .
m. cirelli and a. strumia , `` cosmology of neutrinos and extra light particles after wmap3 , '' jcap * 0612 * ( 2006 ) 013 [ arxiv : astro - ph/0607086 ] .
a. goobar , s. hannestad , e. mortsell and h. tu , `` a new bound on the neutrino mass from the sdss baryon acoustic peak , '' jcap * 0606 * ( 2006 ) 019 [ arxiv : astro - ph/0602155 ] . j. r. kristiansen , h. k. eriksen and o. elgaroy , `` revised wmap constraints on neutrino masses and other extensions of the minimal lambda cdm model , '' phys . rev .
d * 74 * , 123005 ( 2006 ) .
u. seljak , a. slosar and p. mcdonald , `` cosmological parameters from combining the lyman - alpha forest with cmb , galaxy clustering and sn constraints , '' jcap * 0610 * , 014 ( 2006 ) [ arxiv : astro - ph/0604335 ]
. s. hannestad , `` neutrino masses and the number of neutrino species from wmap and 2dfgrs , '' jcap * 0305 * , 004 ( 2003 ) [ arxiv : astro - ph/0303076 ] .
s. hannestad , `` primordial neutrinos , '' ann . rev .
nucl . part .
* 56 * ( 2006 ) 137 [ arxiv : hep - ph/0602058 ] .
s. hannestad , a. mirizzi , g. g. raffelt and y. y. y. wong , `` cosmological constraints on neutrino plus axion hot dark matter , '' jcap * 0708 * , 015 ( 2007 ) [ arxiv:0706.4198 [ astro - ph ] ] .
d. n. spergel _ et al .
_ , `` wilkinson microwave anisotropy probe ( wmap ) three year results : implications for cosmology , '' astrophys
. j. suppl .
* 170 * ( 2007 ) 377 [ arxiv : astro - ph/0603449 ] .
g. hinshaw _ et al .
_ , `` three - year wilkinson microwave anisotropy probe ( wmap ) observations : temperature analysis , '' astrophys .
j. suppl .
* 170 * ( 2007 ) 288 [ arxiv : astro - ph/0603451 ] .
l. page _ et al .
_ , `` three year wilkinson microwave anisotropy probe ( wmap ) observations : polarization analysis , '' astrophys .
j. suppl .
* 170 * ( 2007 ) 335 [ arxiv : astro - ph/0603450 ] .
w. j. percival _ et al .
_ , `` the shape of the sdss dr5 galaxy power spectrum , '' astrophys . j. * 657 * ( 2007 ) 645 [ arxiv : astro - ph/0608636 ] .
m. tegmark _ et al .
_ , `` cosmological constraints from the sdss luminous red galaxies , '' phys .
d * 74 * ( 2006 ) 123507 [ arxiv : astro - ph/0608632 ] .
s. cole _ et al . _
[ 2dfgrs collaboration ] , `` the 2df galaxy redshift survey : power - spectrum analysis of the final dataset and cosmological implications , '' mon . not .
* 362 * ( 2005 ) 505 [ arxiv : astro - ph/0501174 ] .
d. j. eisenstein _ et al .
_ [ sdss collaboration ] , `` detection of the baryon acoustic peak in the large - scale correlation function of sdss luminous red galaxies , '' astrophys .
j. * 633 * ( 2005 ) 560 [ arxiv : astro - ph/0501171 ] ; see also http://cmb.as.arizona.edu/@xmath159eisenste/acousticpeak t. m. davis _ et al .
_ , `` scrutinizing exotic cosmological models using essence supernova data combined with other cosmological probes , '' arxiv : astro - ph/0701510 .
j. hamann , s. hannestad , g. g. raffelt and y. y. y. wong , `` observational bounds on the cosmic radiation density , '' jcap * 0708 * , 021 ( 2007 ) [ arxiv:0705.0440 [ astro - ph ] ] . h. v. klapdor - kleingrothaus
_ et al . _
, `` latest results from the heidelberg - moscow double - beta - decay experiment , '' eur .
j. a * 12 * , 147 ( 2001 ) [ arxiv : hep - ph/0103062 ] . h. v. klapdor - kleingrothaus , a. dietz , h. l. harney and i. v. krivosheina , `` evidence for neutrinoless double beta decay , '' mod .
a * 16 * , 2409 ( 2001 ) [ arxiv : hep - ph/0201231 ] . h. v. klapdor - kleingrothaus , i. v. krivosheina , a. dietz and o. chkvorets , search for neutrinoless double beta decay with enriched ge-76 in gran sasso phys .
b * 586 * , 198 ( 2004 ) [ arxiv : hep - ph/0404088 ] .
h. volker klapdor - kleingrothaus , `` first evidence for neutrinoless double beta decay - and world status of double beta experiments , '' arxiv : hep - ph/0512263 .
s. schonert _ et al . _
[ gerda collaboration ] , `` the germanium detector array ( gerda ) for the search of neutrinoless beta beta decays of ge-76 at lngs , '' nucl .
suppl . * 145 * , 242 ( 2005 ) .
j. r. kristiansen and o. elgaroy , `` cosmological implications of the katrin experiment , '' arxiv:0709.4152 [ astro - ph ] .
r. ardito _ et al .
_ , `` cuore : a cryogenic underground observatory for rare events , '' arxiv : hep - ex/0501010 .
r. gaitskell _ et al . _
[ majorana collaboration ] , `` white paper on the majorana zero - neutrino double - beta decay experiment , '' arxiv : nucl - ex/0311013 .
s. pascoli , s. t. petcov and t. schwetz , `` the absolute neutrino mass scale , neutrino mass spectrum , majorana cp - violation and neutrinoless double - beta decay , '' nucl .
b * 734 * , 24 ( 2006 ) [ arxiv : hep - ph/0505226 ] .
m. lindner , a. merle and w. rodejohann , `` improved limit on theta(13 ) and implications for neutrino masses in neutrino - less double beta decay and cosmology , '' phys .
d * 73 * , 053005 ( 2006 ) [ arxiv : hep - ph/0512143 ] .
s. hannestad , `` neutrino masses and the dark energy equation of state : relaxing the cosmological neutrino mass bound , '' phys .
lett . * 95 * , 221301 ( 2005 ) [ arxiv : astro - ph/0505551 ] .
a. de la macorra , a. melchiorri , p. serra and r. bean , `` the impact of neutrino masses on the determination of dark energy properties , '' astropart .
* 27 * , 406 ( 2007 ) [ arxiv : astro - ph/0608351 ] .
j. q. xia , h. li , g. b. zhao and x. zhang , `` probing for the cosmological parameters with planck measurement , ''
arxiv:0708.1111 [ astro - ph ] . s. gratton , a. lewis and g. efstathiou , `` prospects for constraining neutrino mass using planck and lyman - alpha forest data , '' arxiv:0705.3100 [ astro - ph ] . s. hannestad and y. y. y. wong , `` neutrino mass from future high redshift galaxy surveys : sensitivity and detection threshold , '' jcap * 0707 * , 004 ( 2007 ) [ arxiv : astro - ph/0703031 ] .
l. perotto , j. lesgourgues , s. hannestad , h. tu and y. y. y. wong , `` probing cosmological parameters with the cmb : forecasts from full monte carlo simulations , '' jcap * 0610 * , 013 ( 2006 ) [ arxiv : astro - ph/0606227 ] .
m. takada , e. komatsu and t. futamase , `` cosmology with high - redshift galaxy survey : neutrino mass and inflation , '' phys . rev .
d * 73 * , 083520 ( 2006 ) [ arxiv : astro - ph/0512374 ] .
j. lesgourgues , l. perotto , s. pastor and m. piat , `` probing neutrino masses with cmb lensing extraction , '' phys . rev .
d * 73 * , 045021 ( 2006 ) [ arxiv : astro - ph/0511735 ] .
s. wang , z. haiman , w. hu , j. khoury and m. may , `` weighing neutrinos with galaxy cluster surveys , '' phys . rev
. lett .
* 95 * , 011302 ( 2005 ) [ arxiv : astro - ph/0505390 ] .
y. s. song and l. knox , `` dark energy tomography , '' arxiv : astro - ph/0312175 .
s. hannestad , `` can cosmology detect hierarchical neutrino masses ?
, '' phys .
d * 67 * , 085017 ( 2003 ) [ arxiv : astro - ph/0211106 ] .
|
we present a markov chain monte carlo global analysis of neutrino parameters using both cosmological and experimental data .
results are presented for the combination of all presently available data from oscillation experiments , cosmology , and neutrinoless double beta decay .
in addition we explicitly study the interplay between cosmological , tritium decay and neutrinoless double beta decay data in determining the neutrino mass parameters .
we furthermore discuss how the inference of non - neutrino cosmological parameters can benefit from future neutrino mass experiments such as the katrin tritium decay experiment or neutrinoless double beta decay experiments .
|
recent discoveries of photoinduced metallic phases in several mott insulators@xcite made us consider the strongly correlated electron physics from a new point of view .
however , the optical properties of photoinduced carriers in mott insulators are not well understood even at low excitation densities .
understanding these optical properties is a prerequisite to understanding the incipient creation of metallic phases .
drude response by itinerant carriers is observed in the case of band semiconductors with low excited - carrier density ( 10@xmath2@xmath310@xmath4 @xmath5)@xcite .
the comparison between the optical properties of photoinduced carriers at low excitation densities in mott insulators and those in band semiconductors would be important in gaining deeper insight into strongly correlated electron physics .
the detailed nature of various carrier conductions , exhibiting drude or hopping conduction , can be well characterized in the terahertz ( thz ) regime@xcite .
thz time - domain spectroscopy ( thz - tds ) is a powerful tool for analysing terahertz conductivity @xmath6 ( @xmath7 ) .
the remarkable advantage of thz - tds is its simultaneous determination of both the real and imaginary parts of @xmath6 , without using the kramers - kronig transformation@xcite .
the coherent nature of the thz pulse is also utilized to investigate photoinduced @xmath6 by , for instance , optical - pump thz - probe ( optp ) studies .
recent progress in thz technologies@xcite and the methodology of analysis@xcite in optp experiments enable evaluation of transient @xmath6 in many substances , such as semiconductors , high - t@xmath8 superconductors , liquids and organic materials@xcite .
because the optp method essentially detects non - equilibrium processes , such as surface recombination and carrier diffusion , the time dependence of a complicated spatial carrier distribution must be considered@xcite .
this difficulty is avoided by using thin - film samples@xcite where the optical pump pulse penetrates , and by analysing the photoinduced phase as a layer with a homogeneous @xmath6@xcite .
furthermore the extraction of @xmath6 , varying quickly compared with the pulse width of thz probe pulse , only seems possible within some restricted conditions@xcite .
therefore , to analyse photoinduced @xmath6 for a wide variety of mott insulators , free from the restrictions in sample preparation and analysis , we initially examined a nearly equilibrium state of photoexcited bulk material with a longer energy - relaxation time , @xmath9 .
the photoexcitation of a material with longer @xmath9 creates a quasi - equilibrium state averaging over various non - equilibrium processes , making it easy to obtain the optical constants of the highest carrier - concentration region in the material .
our analysis also required extracting @xmath6 of materials with inhomogeneous carrier distributions in a more rigorous manner .
one good candidate for accomplishing this is the transfer - matrix method , which expresses inhomogeneous carrier distribution by a multi - layer system .
this method has been briefly commented on in the literature@xcite .
the promise that the transfer - matrix method can incorporate inhomogeneity is seen in the analyses of reflectivity in optical pump - probe studies@xcite .
although it is a versatile method , its effectiveness has not been thoroughly discussed , especially in thz - tds studies . in this study , we found that @xmath9 of a mott insulator ytio@xmath0 with mott gap of approximately 1 ev@xcite is 1.5 ms at 1.47 ev photoexcitation .
we characterized the photoinduced @xmath6 by comparing it with that of a band semiconductor si with a bandgap of 1.1 ev@xcite .
we also present a more detailed discussion of the transfer - matrix method .
single - crystalline samples of ytio@xmath0 , with the orthorhombic perovskite gdfeo@xmath0-type structure , were grown by the floating zone method@xcite .
the si sample was commercial high - resistivity si .
photoconductivity was measured to assess @xmath9 .
the light emitted from a multimode continuous wave ( cw ) laser diode ( ld ) with photon energy of 1.47 ev was modulated and used for illuminating the sample under an electric field of about 0.3 kv / cm .
the photocurrent @xmath10 flowing through a 100 @xmath11 resistance , connected in series with the sample , was lock - in detected .
optp experiment was performed by a transmission thz - tds system described in detail elsewhere@xcite .
the thicknesses of platelet samples are 420 @xmath1 m for ytio@xmath0 and 512 @xmath1 m for si .
the optical pulses were generated by a mode - locked ti - sapphire laser with a repetition rate of 76 mhz and central wavelength of 800 nm .
both the thz emitter and detector were low - temperature grown gaas photoconductive antennas .
si lenses were attached to the antennas to enhance the emission power and collection efficiency of thz pulses .
the thz spectral range in this experiment was between 0.5 and 8 mev .
the thz - wave - emission sides of the samples were photoexcited by the multi - mode cw ld with an incident angle of 45@xmath12 .
the pump - beam power was 0.8 w for si and 1.2 w for ytio@xmath0 . for ytio@xmath0 ,
the polarization of thz electric field @xmath13 is parallel to the _ b_-axis .
the beam diameter of the ld light was about 8 mm and larger than that of the thz probe pulse , which is energy dependent ( e.g. 2 mm at 2 mev and 1 mm at 4 mev ) .
the fluence rate was 1.6 w/@xmath14 for si and 2.4 w/@xmath14 for ytio@xmath0 , respectively .
the temperature rise resulting from the thermalisation by photoexcitation@xcite is estimated not to exceed 1 k. optp measurements were also performed with the 1.55 ev optical - pump pulses split from the ti - sapphire laser to investigate the nature of conduction of carriers induced by light with photon energy larger than that of the cw ld .
this experiment studied the anisotropy of photoinduced @xmath6 in ytio@xmath0 .
the optical - pump pulse power was 230 mw .
the optical - pump pulse beam diameter was about 2 mm ( fluence : 96 nj/@xmath14 ) - slightly smaller than that of the thz probe pulse below 2 mev .
the @xmath13 was applied along either the @xmath15- or @xmath16-axis , maintaining the polarization of the optical - pump pulse electric field @xmath17 parallel to the @xmath15- or @xmath16-axis .
all measurements were performed at room temperature .
figure 1 shows the modulation - frequency dependence of @xmath10 .
increasing the modulation frequency causes @xmath10 to decrease , according to @xmath18 , where @xmath19 is the proportional coefficient .
the obtained @xmath19 and @xmath9 are listed in the figure . longer @xmath9 ( @xmath3 15 @xmath1s for si and @xmath3 1.5 ms for ytio@xmath0 )
indicates that the samples are in quasi - equilibrium photoinduced state during the thz - tds measurements .
@xmath9 of ytio@xmath0 is almost excitation - intensity independent ; this implies that a thermal effect does not dominate the relaxation process .
although the direction of @xmath10 and of electric field of the cw ld pump are not identified accurately in ytio@xmath0 , a huge anisotropic @xmath9 that depends on the direction of @xmath10 and the polarization of the excitation light is not anticipated .
the optp results shown later support this ( see fig .
the temporal evolution of @xmath13 transmitted through si and ytio@xmath0 are shown in fig .
2(a ) and 2(b ) , respectively , with and without ld excitations ( 1.47 ev ) .
photoexcitation attenuates both thz waves implying thz wave absorption is by the photoinduced carriers .
thz energy dependence of transmission @xmath20 and phase shift @xmath21 are obtained by fourier transformation of thz waves , as shown in fig .
3(a ) and 3(b ) .
they are calculated with the equations @xmath22 and @xmath23 , where @xmath24 and @xmath25 are the fourier transformed amplitude and phase , with and without excitation , respectively .
thz wave absorption by photoinduced carriers is responsible for @xmath20 decreasing below 1 for both samples , but the two exhibit different energy dependencies . as the thz energy increases , @xmath20 of si approaches 1 , while that of ytio@xmath0 gradually decreases .
the opposite sign of @xmath21 of the two samples strongly indicates different fundamental conduction mechanisms of the photoinduced carriers .
negative ( positive ) @xmath26 roughly means the refractive index is reduced ( increased ) , compared with an unexcited state , which influences the negative ( positive ) real part of the dielectric constant of photoinduced carriers . as explained below
, these results indicate a metallic nature below a plasma frequency in si , and a localized nature , such as hopping carriers , in ytio@xmath0 . before showing the photoinduced @xmath6 of si and ytio@xmath0
, we mention the detailed procedure of the transfer - matrix method . a spatially inhomogeneous distribution of photoinduced carriers is initially regarded as exponentially decaying .
subsequently a non - exponentially decaying distribution is introduced .
the photoinduced phase with an exponentially decaying carrier distribution is divided into many thin slabs ( see fig . 4 ) .
each slab is supposed to have a uniform complex refractive index @xmath27 whose value is set to reproduce the exponential decay of the photoinduced carrier concentration .
the transfer - matrix of each slab is described by @xmath28 where @xmath29 is the number index of the slab , @xmath30 , @xmath31 is the incident thz wavelength in vacuum , and @xmath32 is the slab thickness@xcite .
the @xmath27 of each slab is calculated from the complex dielectric constant @xmath33 using , @xmath34 where @xmath35 is @xmath33 without excitation , @xmath36 , the parameter to be optimized in this analysis , is @xmath33 resulting from carriers at the photoexcited surface of the sample , @xmath37 ( @xmath38 ) is the depth from the photoexcited surface into the sample along the thz wave propagation , and @xmath39 is the optical penetration depth . for si ,
frequency independent @xmath35@xcite of 11.7 is used . for ytio@xmath0 , @xmath35
is experimentally determined by the thz - tds measurement and is weakly energy dependent ( e.g. 16.5 + 0.4@xmath40 at 2 mev and 17 + 0.4@xmath40 at 4 mev ) .
@xmath39 of si at 1.47 ev is determined to be 8.4 @xmath1 m using the absorption coefficient from an optical data handbook@xcite .
that of ytio@xmath0 at 1.47 ev was calculated to be 0.22 @xmath1 m from the reported reflectivity spectra@xcite ( 0.05 - 40 ev ) combined with the kramers - kronig transformation . then the total matrix @xmath41 is described as @xmath42 where @xmath43 is the total number of photoexcited slabs .
finally , the thz complex transmission is given by @xmath44 where @xmath45 and @xmath46 mean the thz wave transmission with and without excitation , respectively , @xmath47 and @xmath48 .
the fitting of experimental @xmath20 and @xmath21 following the above mentioned procedure provides @xmath6 resulting from carriers at the photoexcited surface through @xmath49 where @xmath50 is the vacuum permittivity .
note that the convergence of transmission is checked carefully by decreasing the thickness or by increasing the number of slabs .
a thickness of the photoexcited phase ( @xmath51 , @xmath43=100 ) 5 to 10 times thicker than @xmath39 is typically employed .
figure 5 shows photoinduced @xmath6 of si and ytio@xmath0 .
@xmath6 of si can be interpreted by the drude model as @xmath52 where @xmath53 is the carrier density , @xmath1 is the mobility , @xmath54 is the carrier collision rate and @xmath55 is the effective mass .
the photoexcitation introduces both electrons and holes ; therefore , the tentatively assigned @xmath55 value is 0.26@xmath56 for electrons and 0.37@xmath56 for holes@xcite , where @xmath56 is the free - electron mass .
hereafter , @xmath1 of each carrier is denoted as @xmath57 for electrons and @xmath58 for holes .
we have considered the following two cases , neither of which can be excluded at the present stage .
one is the two - carrier model of electrons and holes .
the other takes only electrons into consideration , assuming that holes with heavy @xmath55 do not contribute to @xmath6 .
the solid lines in fig .
5(a ) represent the calculated @xmath6 for the two - carrier model and the broken lines represent the electron - only model .
the curves are in agreement with the experimental @xmath59 .
this suggests that the itinerant carriers are certainly photogenerated in si . the obtained @xmath57 and @xmath58 are 2410 ( @xmath60210 ) @xmath14/vs and 500 ( @xmath6090 ) @xmath14/vs for the two - carrier model , and
@xmath57 is 1820 ( @xmath60100 ) @xmath14/vs for the other model .
they are roughly consistent with the literature values@xcite , but it is to be noted that @xmath57 in both models might be larger than the predicted ones
. the ambiguity of @xmath55 may be responsible for this deviation .
the most striking feature in @xmath6 of ytio@xmath0 is the negative @xmath61 .
it suggests an existence of localized carriers@xcite , which is very different from si . the localization may arise from the on - site strong coulomb interaction between 3d electrons in ytio@xmath0 .
to explain @xmath6 , we used the empirical jonscher law@xcite , which expresses @xmath6 of many materials with hopping carriers .
the jonscher law is given by@xcite @xmath62 where @xmath63 is the dc conductivity , @xmath64 the proportional coefficient and @xmath65 is restricted between 0 and 1 .
as shown in fig .
5(b ) , the solid curves from the jonscher law seem to agree with experimental @xmath6 . in the solid curves , @xmath65 , @xmath63 and @xmath64
are 0.95 , 235 ( @xmath6010 ) @xmath66@xmath67 and 2.40 ( @xmath600.05)@xmath6810@xmath69 @xmath66@xmath67s@xmath70 , respectively .
the allowed @xmath65 ranges from 0.91 to 0.99 , and corresponding @xmath63 and @xmath64 are 210 ( @xmath6010 ) @xmath66@xmath67 and 1.41 ( @xmath600.03)@xmath6810@xmath71 @xmath66@xmath67s@xmath72 , and 260 ( @xmath6010 ) @xmath66@xmath67 and 1.43 ( @xmath600.03)@xmath6810@xmath73 @xmath66@xmath67s@xmath74 , respectively .
note that @xmath6 can be also fitted by a two - component model , such as the drude - lorentz .
the estimated photoinduced carrier number at the surface layer is about 0.015 per ti site .
photoexcited ytio@xmath0 with the derived carrier density would be equivalent to chemically hole - doped y@xmath75ca@xmath76tio@xmath0 with @xmath77 much less than 0.1 given in ref . 36 .
the @xmath78 spectrum of y@xmath75ca@xmath76tio@xmath0 in this composition region is very different than a drude response .
therefore , it would be difficult to expect a drude component to exist .
clarifying this point might require broadband spectroscopic information obtained under photoexcitation or the temperature dependence of @xmath6 . since long relaxation times , @xmath9 ,
are observed in both samples , a diffusion or a surface - recombination process , making the carrier distribution a non - exponential decay type , must be considered , and the analysis method modified .
the carrier number @xmath79 along the thz wave propagation in a quasi - equilibrium state is obtained using a one - dimensional diffusion equation@xcite as follows : @xmath80 where @xmath81 depends on the time @xmath82 and the position @xmath83 along the thz wave propagation , and @xmath84 is the @xmath85-function .
@xmath86 is the diffusion coefficient and given by @xmath87 where @xmath88 is equal to @xmath89 , @xmath90 is the boltzmann constant and @xmath91 is the sample temperature equal to 300 k. the solution@xcite of eq .
( [ equ:1-diffusion ] ) is @xmath92\right .
\nonumber \\ & & \left.-\frac{v_{s}}{\frac{d}{d_{p}}-v_{s}}f\left(v_{s}\sqrt{\frac{t}{d}}+\frac{z}{2\sqrt{dt}}\right)\right\}\exp\left(-\frac{t}{\tau}\right ) , \label{equ : diff - sol}\end{aligned}\ ] ] where @xmath93 is the surface recombination velocity and @xmath94 is related to the error function by @xmath95erf@xmath96 .
the carrier number in the quasi - equilibrium state requires the integration of @xmath81 with respect to @xmath82 , @xmath97 therefore , with the assumption of a conduction model and the knowledge of @xmath79 determined by appropriate @xmath98 and @xmath93 , @xmath99 can be calculated using eq .
( [ equ : trans ] ) . in this case , eq .
( [ equ : epsilon ] ) is replaced by @xmath100 where @xmath101 and @xmath102 are @xmath79 and @xmath33 of the highest carrier - concentration layer , respectively .
after the determination of @xmath98 of si (= 346 @xmath14/vs ) using the literature values@xcite , @xmath93 is varied between 1@xmath6810@xmath103 cm / s and 1@xmath6810@xmath104 cm / s .
representative @xmath79 normalized at @xmath101 are shown in the inset of fig .
the @xmath101 is observed around 1@xmath32 @xmath1 m . assuming that both electrons and holes obeying the drude conductivity are responsible for @xmath6 , @xmath99 is confirmed as being consistent with experimental data for both @xmath79 ( see fig .
the estimated @xmath53 is 5.2 ( @xmath600.3)@xmath6810@xmath4 @xmath5 and is comparable to that obtained by the previous model .
this indicates that , at the highest carrier - concentration layer almost the same @xmath53 can be obtained , irrespective of the carrier distribution decay type . for ytio@xmath0 ,
both @xmath93 and @xmath98 are unknown parameters .
the wide range sweep of @xmath93 and @xmath98 gives various @xmath79 curves as depicted in the inset of fig .
6(b ) with peak positions around 0.1 @xmath1 m . for each @xmath79
, the experimental @xmath99 is well reproduced by the jonscher law , where @xmath65 is restricted within the same range obtained in fig .
5(b ) ( [email protected] ) .
typical examples are shown in fig .
6(b ) with @xmath65 of 0.95 . the other parameters ( @xmath63 in @xmath66@xmath67 and @xmath64 in @xmath66@xmath67s@xmath70 ) for the dotted - solid , solid and broken lines are 125(@xmath6010 ) and 1.40(@xmath600.05)@xmath6810@xmath69 , 125(@xmath6010 ) and 1.2(@xmath600.1)@xmath6810@xmath69 , and 90(@xmath6010 ) and 1.0(@xmath600.1)@xmath6810@xmath69 , respectively . @xmath78 and @xmath105 calculated from the parameters are half to two - thirds of those in fig .
thus , for ytio@xmath0 , the @xmath6 extracted from the model with exponentially - decaying carrier distribution roughly represents the highest carrier - concentration layer in the model using eq .
( [ equ : new - epsilon ] ) .
the photoinduced carrier number at the highest carrier - concentration layer in ytio@xmath0 is calculated as 0.015 per ti site@xcite .
it can be proposed , therefore , that a phase with localized carriers would emerge initially at the photogeneration of the metallic phase in mott insulators .
photoexcitation creates both electrons and holes , which differs from chemical doping , and a comparison of @xmath6 between photoexcited ytio@xmath0 and a hole - doped y@xmath75ca@xmath76tio@xmath0 is discussed .
the absolute value of @xmath78 of photoexcited state might be much larger than that of corresponding y@xmath75ca@xmath76tio@xmath0 if the extrapolation of @xmath78 in y@xmath75ca@xmath76tio@xmath0 is carried out toward the thz energy .
the preservation of spectral weight implies that the localization energy of photoinduced carriers would be much lower than for holes in y@xmath75ca@xmath76tio@xmath0 , even if both holes and electrons contribute to @xmath6 in photoexcited ytio@xmath0 .
as it is not clear that the large difference in localization energy originates from only holes in such a low - carrier system , it is plausible that electrons with small localization energies also contribute to @xmath6 .
therefore @xmath6 of photoexcited ytio@xmath0 would be supported by bound electrons as well as holes . in a halogen - bridged ni one - dimensional chain compound [ ni(chxn)@xmath106br]br@xmath106 ( chxn = cyclohexanediamine ) , which is compared with ytio@xmath0 composed of a three - dimensional ti network , the localized @xmath107 is determined at a lower excitation density@xcite . despite being in a different energy region ,
carrier localization in photoexcited mott insulators at low excitation densities may be the general phenomenon , irrespective of the dimensionality .
the fact that one - dimensional mott insulators , such as [ ni(chxn)@xmath106br]br@xmath106@xcite and sr@xmath106cuo@xmath0@xcite , exhibit @xmath9 in the order of pico - seconds may suggest that dimensionality is a decisive factor of @xmath9 .
figure 7 shows @xmath20 and @xmath21 obtained by optp experiments using a femtosecond - pulse laser ( 1.55 ev ) for ytio@xmath0 . since the period of optical - pump arrival time ( 13 ns ) is much shorter than @xmath9 , the photoinduced carriers are also in quasi - equilibrium state . in both polarizations of @xmath17
, it is found that the degree of variation from the unexcited state in thz wave amplitude and phase is larger for @xmath108 within the measured thz energy range .
this implies that the absolute values of @xmath107 and @xmath61 for @xmath108 are larger than those for @xmath109 .
the anisotropy would reflect the crystal symmetry of ytio@xmath0 or the 3@xmath32-orbital state at ti site . comparing with the optp results to those using the cw ld
, the @xmath6 does not seem to depend strongly on the optical - photon energy .
we have optically characterized photoinduced carriers of mott insulator ytio@xmath0 at low excitation densities in the thz regime by optp measurements , and compared the experimental results with those for band semiconductor si .
the @xmath9 of the photoinduced carriers in ytio@xmath0 is about 1.5 ms .
the inhomogeneous carrier distribution along the thz wave propagation can be treated accurately using the transfer - matrix method .
this method successfully determined @xmath6 of the highest carrier - concentration layer under the quasi - equilibrium states .
ytio@xmath0 shows localized @xmath59 , possibly with the jonscher law , whereas si exhibits the drude response .
anisotropic @xmath6 in ytio@xmath0 is determined .
our study demonstrates that localized carriers might play an important role in the incipient formation of metallic phases in photoexcited mott insulators .
although the exact origin of the localization in ytio@xmath0 remains an open question , thz - tds under photoexcitation with another photon energy or for another mott insulator might provide the answer .
we note here that a preliminary thz - tds experiment of ytio@xmath0 excited by a cw ld of 1.9 ev also leads to localized @xmath6 .
this work was supported by casio science foundation , and strategic information and communications r&d promotion programme of ministry of public management , home affairs , posts and telecommunications .
99 k. miyano , t. tanaka , y. tomioka , and y. tokura , phys .
lett . * 78 * , 4257 ( 1997 ) .
a. cavalleri , cs .
tth , c. w. siders , j. a. squier , f. rksi , p. forget , and j. c. kieffer , phys .
lett . * 87 * , 237401 ( 2001 ) .
s. iwai , m. ono , a. maeda , h. matsuzaki , h. kishida , h. okamoto , and y. tokura , phys .
lett . * 91 * , 057401 ( 2003 ) .
n. tajima , j. fujisawa , n. naka , t. ishihara , r. kato , y. nishio , and k. kajita , j. phys .
74 * , 511 ( 2005 ) .
m. chollet , l. guerin , n. uchida , s. fukaya , h. shimoda , t. ishikawa , k. matsuda , t. hasegawa , a. ota , h. yamochi , g. saito , r. tazaki , s. adachi , and s. koshihara , science * 307 * , 86 ( 2005 ) .
l. perfetti , p. a. loukakos , m. lisowski , u. bovensiepen , h. berger , s. biermann , p. s. cornaglia , a. georges , and m. wolf , phys .
97 * , 067402 ( 2006 ) .
s. e. ralph , y. chen , j. woodall , and d. mcinturff , phys .
b * 54 * , 5568 ( 1996 ) . m. c. beard , g. m. turner , and c. a. schmuttenmaer , phys .
b * 62 * , 15764 ( 2000 ) .
j. shan , f. wang , e. knoesel , m. bonn , and t. f. heinz , phys .
* 90 * , 247401 ( 2003 ) .
g. l. dakovski , b. kubera , s. lan , and j. shan , j. opt .
b * 23 * , 139 ( 2006 )
. m. van exter and d. grischkowsky , appl .
lett . * 56 * , 1694 ( 1990 ) .
n. katzenellenbogen and d. grischkowsky , appl .
. lett . * 61 * , 840 ( 1992 ) .
jeon and d. grischkowsky , phys .
78 * , 1106 ( 1997 ) . h. harimochi , j. kitagawa , m. ishizaka , y. kadoya , m. yamanishi , s. matsuishi , and h. hosono , phys .
b * 70 * , 193104 ( 2004 ) .
g. grner , _ millimeter and submillimeter wave spectroscopy of solids _
( springer , berlin , 1998 ) .
d. mittleman , _ sensing with terahertz radiation _ ( springer , berlin , 2003 ) .
k. sakai , _ terahertz optoelectronics _
( springer , berlin , 2005 ) .
j. t. kindt and c. a. schmuttenmaer , j. chem .
* 110 * , 8589 ( 1999 ) .
h. nmec , f. kadlec , and p. kuel , j. chem . phys . * 117 * , 8454 ( 2002 ) . h. k. nienhuys and v. sundstrm , phys .
b * 71 * , 235110 ( 2005 ) .
r. d. averitt , g. rodriguez , j. l. w. siders , s. a. trugman , and a. j. taylor , j. opt .
b * 17 * , 327 ( 2000 ) .
r. huber , f. tauser , a. brodschelm , m. bichler , g. abstreiter , and a. leitenstorfer , nature * 414 * , 286 ( 2001 ) .
r. d. averitt , a. i. lobad , c. kwon , s. a. trugman , v. k. thorsm@xmath110lle , and a. j. taylor , phys .
lett . * 87 * , 017401 ( 2001 ) .
m. c. beard , g. m. turner , and c. a. schmuttenmaer , j. appl .
phys . * 90 * , 5915 ( 2001 ) .
e. knoesel , m. bonn , j. shan , and t. f. heinz , phys .
lett . * 86 * , 340 ( 2001 ) .
f. a. hegmann , r. r. tykwinski , k. p. h. lui , j. e. bullock , and j. e. anthony , phys .
lett . * 89 * , 227403 ( 2002 ) . g. m. turner , m. c. beard , and c. a. schmuttenmaer , j. phys .
b * 106 * , 11716 ( 2002 ) .
j. demsar , r. d. averitt , a. j. taylor , v. v. kabanov , w. n. kang , h. j. kim , e. m. choi , and s. i. lee , phys .
* 91 * , 267002 ( 2003 ) .
r. a. kaindl , m. a. carnahan , d. hgele , r. lvenlch , and d. s. chemla , nature * 423 * , 734 ( 2003 ) .
e. hendry , f. wang , j. shan , t. f. heinz , and m. bonn , phys .
b * 69 * , 081101 ( 2004 ) .
t. kampfrath , l. perfetti , f. schapper , c. frischkorn , and m. wolf , phys .
lett . * 95 * , 187403 ( 2005 ) .
m. i. gallant and h. m. van driel , phys .
b * 26 * , 2133 ( 1982 ) .
m. schall and p. u. jepsen , opt .
lett . * 25 * , 13 ( 2000 ) .
h. m. ma , y. x. liu , y. fei , and f. m. li , j. appl .
phys . * 65 * , 5031 ( 1989 ) .
h. okamoto , y. ishige , s. tanaka , h. kishida , s. iwai , and y. tokura , phys .
b * 70 * , 165202 ( 2004 ) . y. taguchi , y. tokura , t. arima , and f. inaba , phys .
b * 48 * , 511 ( 1993 ) . s. m. sze , _ physics of semiconductor devices _
( wiley - interscience , new york , 1981 ) .
m. tsubota , f. iga , t. nakano , k. uchihira , s. kura , m. takemura , y. bando , k. umeo , t. takabatake , e. nishibori , m. takata , m. sakata , k. kato , and y. ohishi , j. phys .
. jpn . * 72 * , 3182 ( 2003 ) .
j. kitagawa , m. ishizaka , y. kadoya , s. matsuishi , and h. hosono , j. phys .
* 75 * , 084715 ( 2006 ) .
j. kitagawa , y. kadoya , m. tsubota , f. iga and t. takabatake , j. magn .
magn . mat . * 310 * 913 ( 2007 ) .
the temperature rise @xmath111 under an equilibrium condition is calculated as follows .
if all of the excitation power @xmath112 is transformed into heat in a sample , @xmath111 is related to @xmath112 by @xmath113 where @xmath114 is the sample length , @xmath115 is the illuminated area@xcite of 0.5 @xmath14 and @xmath116 is the thermal conductivity of the sample ( 150 w / mk for si@xcite ) . with the lack of @xmath116 data for ytio@xmath0 ,
we employ underestimated @xmath11710 w / mk .
m. born and e. wolf , _ principles of optics , 6th ed . _ ( pergamon , new york , 1980 ) .
h. harimochi , j. kitagawa , y. kadoya , and m .yamanishi , jpn . j. appl .
phys . * 43 * , 7320 ( 2004 ) .
m. van exter and d. grischkowsky , phys .
b * 41 * , 12140 ( 1990 ) .
e. d. palik , _ handbook of optical constants of solids _ ( academic press , new york , 1985 ) .
y. okimoto , t. katsufuji , y. okada , t. arima , and y. tokura , phys .
b * 51 * , 9581 ( 1995 )
. d. g. cooke , a. n. macdonald , a. hryciw , j. wang , q. li , a. meldrum , and f. a. hegmann , phys .
b * 73 * , 193311 ( 2006 ) .
a. k. jonscher , nature ( london ) * 267 * , 673 ( 1977 ) .
s. r. elliott , adv .
phys . * 36 * , 135 ( 1987 ) .
j. vaitkus , phys .
status solidi a * 34 * , 769 ( 1976 ) .
the photoinduced carrier density @xmath53 at @xmath112 is tentatively expressed by @xmath118 where the first term indicates the carrier accumulation resulting from longer @xmath9 at photon energy @xmath119 of 1.47 ev , the second one is the fresnel loss ( @xmath3 0.8 for ytio@xmath0 and @xmath3 0.7 for si ) .
@xmath120 means the effective @xmath39 defined by @xmath121/@xmath101 and is equal to 13 @xmath1 m for si and 0.55 @xmath1 m for ytio@xmath0 .
this equation is initially applied to the experimental result of si ( fig .
6(a ) ) with @xmath1225.2@xmath6810@xmath4 @xmath5 , in which @xmath115 is determined to be 0.5 @xmath14 . using the @xmath115 ,
the carrier number of ytio@xmath0 can be derived .
t. ogasawara , m. ashida , n. motoyama , h. eisaki , s. uchida , y. tokura , h. ghosh , a. shukla , s. mazumdar , and m. kuwata - gonokami , phys .
lett . * 85 * , 2204 ( 2000 ) . transmitted through ( a ) si and ( b ) ytio@xmath0 with ( solid lines ) and without ( broken lines ) excitations ( 1.47 ev ) .
the polarization of @xmath13 is parallel to the @xmath15-axis in ytio@xmath0 .
the excitation power is 0.8 w for si and 1.2 w for ytio@xmath0 . ] .
the total number of thin slabs is @xmath43 , and the transfer - matrix of each slab is expressed as @xmath124 ( @xmath125 ) .
the depth from the excited surface into the sample along the thz wave propagation is denoted as @xmath37 . ] at the photoexcited surface .
the real and the imaginary part of conductivity correspond to @xmath107 and @xmath61 , respectively .
the solid and broken curves in ( a ) are @xmath6 calculated by the drude models with @xmath53=6.6 ( @xmath600.3)@xmath6810@xmath4 @xmath5 , @xmath57=2410 ( @xmath60210 ) @xmath14/vs and @xmath58=500(@xmath6090 ) @xmath14/vs for the solid curves ( two - carrier model of electrons and holes ) , and with @xmath53=9.3(@xmath600.4)@xmath6810@xmath4 @xmath5 and @xmath57=1820 ( @xmath60100 ) @xmath14/vs for the broken ones ( only electrons under consideration ) , respectively . in the two - carrier model , the same @xmath53
is assumed for each carrier .
the solid curves in ( b ) are calculated @xmath6 using the jonscher law with @xmath126235 ( @xmath6010 ) @xmath66@xmath67 , @xmath1272.40 ( @xmath600.05)@xmath6810@xmath69 @xmath66@xmath67s@xmath70 and @xmath1280.95 .
] analysed by the model with non - exponential carrier - distribution decay .
the evaluated parameters @xmath93 and @xmath98 are listed in the figure . in ( a ) , drude conductivity characterized by @xmath1225.2 ( @xmath600.3)@xmath6810@xmath4 @xmath5 and the literature values of @xmath1291500 @xmath14/vs and @xmath130450 @xmath14/vs is assumed .
the conduction model employed in ( b ) is the jonscher law ( @xmath65=0.95 ) , where @xmath63 in @xmath66@xmath67 and @xmath64 in @xmath66@xmath67s@xmath70 are 125 ( @xmath6010 ) and 1.40 ( @xmath600.05)@xmath6810@xmath69 , 125 ( @xmath6010 ) and 1.2 ( @xmath600.1)@xmath6810@xmath69 , and 90 ( @xmath6010 ) and 1.0 ( @xmath600.1)@xmath6810@xmath69 for the dotted - solid , solid and broken curves , respectively .
the insets of ( a ) and ( b ) show @xmath83 dependencies of the carrier numbers normalized at @xmath101 . ]
|
we performed optical - pump terahertz - probe measurements of a mott insulator ytio@xmath0 and a band semiconductor si using a laser diode ( 1.47 ev ) and a femtosecond pulse laser ( 1.55 ev ) .
both samples possess long energy - relaxation times ( 1.5 ms for ytio@xmath0 and 15 @xmath1s for si ) ; therefore , it is possible to extract terahertz complex conductivities of photoinduced carriers under equilibrium .
we observed highly contrasting behavior - drude conductivity in si and localized conductivity possibly obeying the jonscher law in ytio@xmath0 .
the carrier number at the highest carrier - concentration layer in ytio@xmath0 is estimated to be 0.015 per ti site .
anisotropic conductivity of ytio@xmath0 is determined .
our study indicates that localized carriers might play an important role in the incipient formation of photoinduced metallic phases in mott insulators .
in addition , this study shows that the transfer - matrix method is effective for extracting an optical constant of a sample with a spatially inhomogeneous carrier distribution .
|
traffic flow is one of the most interesting phenomena of many - body systems which may be controlled by basic dynamical equation .
recent developments in study of traffic flow has brought a renewed interest in microscopic approaches , such as optimal velocity model ( ov - model ) @xcite , which is a new version of car following model @xcite , cellular automaton models @xcite , coupled map lattice models @xcite and fluid dynamical models @xcite .
the ov - model , among others , has especially attracted interest because it provides us with a possibility of unified understanding of both free and congested traffic flows from common basic dynamical equations . unlike traditional car following models
, it introduces optimal velocity function @xmath5 as a desirable velocity depending on headway distance @xmath1 .
the basic equation of ov - model for a series of vehicles on a circuit of length @xmath6 is , @xmath7 where @xmath8 denotes the position of the @xmath9-th vehicle , @xmath10 headway and @xmath11 the total vehicle number .
the constant parameter @xmath4 is the sensitivity .
a driver accelerates ( or decelerates ) his vehicle in proportion to the difference between his velocity and the optimal velocity @xmath5 .
as easily noticed , a homogeneous flow is a solution to eq .
( [ eq : ovm0 ] ) .
in such a flow , vehicles have a common headway @xmath12 , which is the inverse of the vehicle density .
stability of homogeneous flows is analyzed within a linear approximation@xcite ; it is stable for @xmath13 and unstable for @xmath14 .
the critical value is found to be @xmath15 . in order to demonstrate that the ov - model describes `` spontaneous generation of congestion '' ,
numerical simulations were made using eq .
( [ eq : ovm0 ] ) .
it was found that for @xmath16 , i.e. , if the density is above the critical value , a slightly perturbed homogeneous flow develops to a congested flow after enough time .
the congested flow consists of alternating two distinct regions ; congested regions ( high density ) , and smoothly moving regions or free regions ( low density ) . in this way
the traffic congestion occurs spontaneously in the ov - model .
this phenomenon can be understood as a sort of phase transition from a homogeneous flow state to a congested flow state @xcite .
a remarkable feature of the well - established congested flow is that the velocity of the @xmath9-th vehicle @xmath17 has the same time dependence as that of the preceding ( @xmath18-th ) vehicle except a certain time delay @xmath0 .
it is also found that the global pattern moves backward with a velocity @xmath3 .
this kind of behavior of vehicles may be called `` repetitive pattern formation '' .
it leads to formation of a closed trajectory ( `` limit cycle '' ) on @xmath1-@xmath2 plane , along which representative points for all the vehicles move one after another .
the convergence of vehicles trajectories to a closed trajectory signals the congestion in a traffic flow . therefore the determination of the closed trajectory is one of the most important subjects to understand congested flows .
however this has been done mostly in computer simulations .
the purpose of the present paper is to obtain this closed trajectory directly by an analytical method .
actually it has been found that , in the vicinity of the critical point for the congested flow , equation ( [ eq : ovm0 ] ) can be reduced to the modified kdv equation by the dynamical reduction method @xcite . in this paper
we propose a new analytical approach to the pattern formation , which may be applicable to any congested flow .
we argue that , once the repetitive pattern is formed , the coupled car following equations reduce to a single difference - differential equation ( rondo equation ) for a universal function ( rondo function ) . a rondo function determines a closed trajectory on @xmath1-@xmath2 plane . to make the rondo equation tractable , we have simplified our question on the following two points : firstly we have assumed that ov - functions are piece - wise linear ; secondly we have concentrated our attention to an asymptotic trajectory , which is the key concept to be explained in the next section .
we would like to stress that our method does not lose its generality by making the above assumption on ov - functions : an ov - function to be obtained from real data may be approximated by a piece - wise linear function . with the above simplifications , we have solved the rondo equation for each model and given an asymptotic trajectory on the @xmath1-@xmath2 plane .
our result clearly tells us that , once an ov - function and the sensitivity @xmath4 are given , an asymptotic trajectory is uniquely determined ; this then implies that the parameters @xmath0 and @xmath3 for a collective motion of vehicles are given as a function of @xmath4 .
therefore our approach provides us with a method to determine @xmath4 dependence of the global parameters @xmath0 and @xmath3 .
this paper is organized as follows .
next section summarizes the main results obtained from numerical simulations of the ov - model , with emphasis on the pattern formation in a congested flow .
the concept of an asymptotic trajectory is explained in this section . as will become clear in later sections ,
an asymptotic trajectory is a very important concept to understand the ov - model .
we derive the difference - differential equation and present a general strategy on how to solve it in section [ sec : rondo approach ] .
this part summarizes the central idea of this paper . in order to demonstrate how the rondo approach works , we analytically solve , in section [ sec : piece - wise linear ] , some simple models with piece - wise linear ov - functions .
our first model has been investigated by sugiyama and yamada@xcite . here
we solve this model in the context of the rondo approach .
although an asymptotic trajectory is very close to a real trajectory observed in a computer simulation , it is not exactly the same to the latter .
we describe some aspects of real trajectories based on our knowledge of the asymptotic one in section [ sec : trajectories around cusps ] .
the final section is devoted to summary and discussions .
let us recollect what we have learned with numerical simulations of an ov - model @xcite .
suppose a simulation is performed with a given ov - function and a fixed sensitivity @xmath4 .
after a congested flow is well - established , typical features of the repetitive behavior can be observed in the following two figures .
= 8.2 cm = 8.0 cm figs .
[ fig : intro.traje.eps](a ) and ( b ) show that vehicles move in alternating regions of free and congested flows .
it is recognized that every vehicle behaves in the same manner as its preceding one with a certain time delay @xmath0 : as a result , the congested region moves backward with the velocity @xmath3 .
once the location of any vehicle , say , the @xmath9-th vehicle , is given as a function of @xmath19 , we may reproduce the pattern in fig .
[ fig : intro.traje.eps](a ) by plotting a series of functions shifted in time and position by @xmath0 and @xmath20 appropriately . therefore we expect that a congested flow may be completely determined by a function of @xmath19 and global parameters @xmath0 and @xmath20 .
the precise specifications of our approach to this repetitive behavior will be explained in the next section .
[ fig : intro.traje.eps](b ) clearly shows there exists a `` limit cycle '' on the @xmath1-@xmath2 plane , a closed curve with two cusps at points c @xmath21 and f @xmath22 , both of which are on the ov - function . at these cusp points ,
we find @xmath23 , which means that homogeneous flows with such headways and velocities are linearly stable .
representative points for all vehicles move on this `` limit cycle '' in the same direction as anti - clockwise .
it follows from the conservation of flow , @xmath24 that the straight line connecting c and f has the slope @xmath25 and intersects with the vertical axis at @xmath26 . in the rest of this section
, we would like to explain the concept of an asymptotic trajectory on the @xmath1-@xmath2 plane .
suppose that a vehicle passes through two congested regions different in size on a circuit .
then the representative point moves on curves as shown in fig .
[ fig : c - point.eps ] .
each trajectory may not form an actual cusp , rather it will form a round shaped tip .
also we find that the larger the congested region , the sharper the shape of the tip : the minimum velocity of the vehicle is smaller for a longer congestion .
it is rather easy to imagine that for a very short congestion the vehicle can not decelerate itself enough to reach the velocity appropriate for a longer congested region . if we plot minimum velocities for longer and longer congested regions , we would find a limiting value for the minimum velocities .
this value must be realized for an infinitely long congested region . with a similar argument
we find the limiting value for maximum velocities corresponding to an infinitely long free region .
we may imagine the following extreme situation : a vehicle starting from an infinitely long free ( congested ) region goes toward an infinitely long congested ( free ) region .
the trajectory for this limiting situation will be called as a decelerating ( an accelerating ) asymptotic trajectory . combining them we would find a closed curve with two real cusps on the ov - function .
the duration for a vehicle to stay in a congested region would obviously get longer for a larger congested region .
for an asymptotic trajectory it becomes infinite .
this does fit to our linear analysis since the behavior of a vehicle is controlled by exponential functions in time . in section [ sec : trajectories around cusps ] , we shall see that exponential functions determine a curve near a cusp .
we begin our description of the rondo approach with two basic assumptions : 1 .
velocities of the @xmath9-th and @xmath27-th vehicles have exactly the same time dependence if a certain time delay @xmath0 is taken into account , @xmath28 .
the pattern of traffic flow moves backward with a constant velocity @xmath3 .
the above described properties are expressed as , @xmath29 all the vehicles behavior is represented with a single universal function @xmath30 ; for the wave equation . ] @xmath31 with @xmath9-th vehicle s headway given as @xmath32 @xmath11 coupled car following equations ( [ eq : ovm0 ] ) are reduced to a single difference - differential equation for @xmath33 , @xmath34 in the following it will be called the rondo equation . in this paper , we will seek the rondo function @xmath33 for the asymptotic trajectory . before studying concrete models ,
let us consider its generic properties . since the position and the velocity of vehicles are obviously continuous in time , @xmath33 is a continuously differentiable function .
an asymptotic trajectory connects the points f and c , each of which corresponds to an infinitely long free or congested region ( an approximately homogeneous flow ) satisfying the stability condition mentioned in section [ sec : pattern formation ]
. therefore @xmath33 should be homogeneous flows asymptotically in the infinite past and future : @xmath35 const . as @xmath36 .
an asymptotic trajectory interpolates two stable solutions of the equation ( [ eq : ovm0 ] ) . in this sense @xmath33
may be regarded as a `` kink solution ''
. like ov - models studied in earlier papers , each model in the next section has an ov - function which is symmetric with respect to a point , s@xmath37 .
so we assume this property in the following arguments and quote our result in appendix [ ap - sec : symmetric ] .
two end points of an asymptotic trajectory , c@xmath21 and f@xmath22 , are symmetric with respect to s. three points c , f and s are on a straight line with a slope @xmath25 and an intersection @xmath26 .
note that once the slope is given , the intersection is uniquely determined since the point s must be on the line .
as shown in appendix [ ap - sec : symmetric ] , accelerating and decelerating asymptotic trajectories are symmetric with respect to s. therefore it is sufficient to study one of them ; in the rest of this paper , we will take an decelerating asymptotic trajectory . here
we summarize conditions which should be satisfied by the function @xmath33 : 1 .
@xmath33 and @xmath38 are continuous for any @xmath19 ( @xmath39 ) ; 2 .
@xmath40 and @xmath41 ; 3 .
@xmath42 and @xmath43 , where @xmath44 and @xmath45 .
we would like to explain a way to solve the rondo equation , which contains an ov - function and a sensitivity @xmath4 as well as @xmath0 and @xmath3 associated the pattern formation .
( 1 ) first we give the parameter @xmath0 . by drawing a straight line through the point
s with the slope @xmath25 , we find the intersection @xmath26 and the points c and f. ( 2 ) now @xmath4 is the only free parameter of the rondo equation .
if we could solve the equation , we would obtain a one - parameter family of ( or @xmath4-dependent ) rondo functions .
( 3 ) among this family , the right rondo function is selected by requiring that it connects the points c and f. this condition also determines a unique value for @xmath4 .
accordingly we find the @xmath4-dependence of @xmath0 .
we consider here a class of models with piece - wise linear ov - functions .
the rondo equation is now linearized for all regions of @xmath1 , and therefore exactly solvable .
the first model has the step function for the ov - function , @xmath46 this ov - model has been solved in ref .
@xcite . here
we explain how this model can be solved in our rondo approach . in this model ,
the rondo equation is given by @xmath47 where @xmath48 is the heviside function .
in the motion corresponding to a decelerating asymptotic trajectory , the representative point for a vehicle moves from region ii into region i. let us take the time @xmath19 such that the point moves into the region i at @xmath49 , which implies @xmath50 .
the equation of motion is @xmath51 the general solutions for two regions are @xmath52 the integration constants are determined as @xmath53 and @xmath54 from requirements that @xmath55 be a continuous function and asymptotically constant for @xmath56 .
the continuous function @xmath33 is @xmath57 here we choose the origin of position coordinate so that @xmath58 . the function @xmath33 with its relation to the headway
, @xmath59 , completely determines a decelerating asymptotic trajectory on the @xmath1-@xmath2 plane . from the condition 3 ) in the section [ sec : rondo approach ]
, the asymptotic trajectory connects symmetric points on the ov - function .
this gives us a condition @xmath60 , @xmath61 .
\label{eq : si}\ ] ] this may be expressed as the transcendental equation for @xmath62 @xmath63 since the @xmath64 is found to be the constant ( 1.59362 .. ) , we obtain @xmath65 this gives us the @xmath4-dependence of @xmath0 , which was first obtained in ref .
@xcite .
it is instructive to see a relation between the function @xmath33 and the decelerating trajectory depicted in fig .
[ fig : step.eps ] . two curves in fig .
[ fig : step.eps](b ) correspond to the @xmath18-th and @xmath9-th vehicles locations . at @xmath49 the @xmath9-th vehicle s representative point moves into the region
i and @xmath33 is described by an exponential function . before that time
, the function @xmath33 is linear in @xmath19 .
the curve for the @xmath18-th vehicle changes from the linear to the exponential behavior at @xmath66 .
it is given via a parallel displacement by the vector @xmath67 from the curve @xmath33 .
for the time @xmath68 , curves are two parallel straight lines , which implies the headway of the @xmath9-th vehicle does not change till that time from the infinite past .
the @xmath9-th vehicle has the constant velocity @xmath69 for @xmath70 .
this tells us that the @xmath9-th vehicle is in the free region for @xmath68 , indicated by the point f in fig .
[ fig : step.eps](a ) .
it is also easy to observe that at @xmath66 the @xmath18-th vehicle starts to decelerate . as a result
the headway of the @xmath9-th vehicle decreases ; at @xmath49 it reaches the value @xmath71 and the @xmath9-th vehicle starts to decelerate itself . in this model , the points f and c
are both characterized as points which are reached in the infinite future or past . as a traffic flow
, we are describing a soliton like solution connecting a half infinite vehicles , running with the velocity @xmath72 and the headway @xmath73 , and another half infinite vehicles going into the congested region associated with the point c. in the step function model , the @xmath1-dependence of the rondo equation is too simple ; @xmath0-dependence is not explicit .
so we would like to consider a slightly improved model , whose ov - function shown in fig . [ fig : piovf.eps ] has a finite slope .
thus we call this model as the single slope function model .
the ov - function is characterized by the following parameters : @xmath74 , the slope ; @xmath69 , the maximum velocity ; @xmath71 , the headway for optimal velocity @xmath75 . from the linear analysis in the paper @xcite ,
we know that a homogeneous flow becomes unstable and a congested flow is expected for @xmath76 .
this condition is assumed in our analysis here .
the ov - function has sharp bends at @xmath77 which divide @xmath1 into three regions , i , ii and iii , as indicated in fig .
[ fig : piovf.eps ] .
we would like to find a rondo function @xmath33 for a decelerating asymptotic trajectory , along which headway of a vehicle monotonically decreases from @xmath78 to @xmath79 .
we assume it reaches @xmath80 at @xmath81 and @xmath82 at @xmath49 .
the rondo equation takes of the form , @xmath83 subject to the conditions , @xmath84 here @xmath85 .
note that the time @xmath86 is not a free parameter .
rather , it should be determined from ( [ eq : t =- tau ] ) via solving the rondo equation . for regions
i and iii , the rondo equation becomes the same as that in the step function model .
therefore , we obtain @xmath87 our purpose in this subsection is to describe a method to find the rondo function in the region ii which correctly interpolates those in ( [ eq : sab ] ) . to this end , we may introduce a series of functions for each time interval : @xmath88 the first function @xmath89 and the last one @xmath90 are for the region i and iii , respectively .
let us find @xmath91 ( for @xmath92 ) .
it follows from ( [ eq : sab ] ) that @xmath93 where we fix again @xmath94 .
the @xmath9-th vehicle s headway is then given by @xmath95 since the condition ( [ eq : t=0 ] ) determines the constant @xmath96 @xmath97 we find the rondo function for @xmath92 to be @xmath98 this leads to a linear relation between the velocity and the headway , @xmath99 in the @xmath100 limit , we find that @xmath101 and @xmath102 .
now we consider @xmath103 ( for @xmath104 ) . in this time interval ,
the rondo equation for @xmath103 is expressed as @xmath105 in terms of the rondo function for the region i , @xmath106 . by using the differential operator , @xmath107
we may rewrite the above equation as @xmath108 the general solution may be written as a sum of a particular solution and the solution to the homogeneous equation @xmath109 .
it is easy to see that @xmath110 is a particular solution , since the function @xmath111 given in ( [ eq : fi ] ) satisfies @xmath112 the exponents @xmath113 for a homogeneous solution are @xmath114 a general solution is given as @xmath115 the constants @xmath116 and @xmath117 are determined as @xmath118 and @xmath119 , by the requirement that @xmath120 and @xmath103 must be continuous up to the first derivative at @xmath49 . therefore @xmath121 and the headway and the velocity for the @xmath9-th vehicle is given as @xmath122 we would now like to give general formula for @xmath123 ( for @xmath124 $ ] ) with @xmath125 .
suppose that @xmath123 for @xmath126 is known to us and we are trying to find @xmath127 for @xmath128 .
@xmath127 satisfies the second order linear differential equation , @xmath129 with boundary conditions ; @xmath130 while @xmath123 satisfies a similar equation , @xmath131 the function @xmath123 describes the behavior of the @xmath9-th vehicle only for @xmath132 .
however we will find it useful to define the function by the relation ( [ eq : fk ] ) even outside the interval @xmath133 .
the function used outside the interval will be denoted as @xmath134 .
the difference of ( [ eq : fk+1 ] ) and ( [ eq : fk ] ) gives us the following equation for @xmath135 $ ] , @xmath136 from ( [ eq : f1 ] ) and ( [ eq : f0 ] ) we find @xmath137 for @xmath138 $ ] . by using the function @xmath139 defined in the relation , @xmath140 ( [ eq : diff - eq ] ) and ( [ eq : diff - eq - k0 ] )
are rewritten into the following equations for @xmath141 , @xmath142 note that @xmath143 defined in ( [ eq : g1 ] ) satisfies @xmath144 .
the conditions ( [ eq : connect - cond ] ) become @xmath145 and @xmath146 ( for @xmath147 ) . the rondo function @xmath33 for the region
ii is given as a sum of @xmath139 , @xmath148 for @xmath149 .
in the appendix [ ap - sec : general formula ] we will give general solutions to differential equations ( [ eq : gk ] ) and ( [ eq : g1 ] ) .
@xmath150 from the appendix , @xmath151 is consistent to ( [ eq : sol - f1 ] ) .
similarly @xmath152 is given as @xmath153 so @xmath154 for @xmath155 $ ] becomes @xmath156 while the headway for the @xmath9-th vehicle is @xmath157 the general formula in appendix [ ap - sec : general formula ] may be used further to generate @xmath158 , needed to describe the trajectory in the region ii .
let us consider the region iii . at @xmath81 , the function ( [ eq : fii ] ) for the region ii must be continuously connected to @xmath90 including their first derivatives .
this condition yields @xmath90 for @xmath159 as @xmath160 the continuity condition for the first derivative requires also that @xmath161 this completes our construction of an asymptotic trajectory .
now let us find the @xmath4-dependence of @xmath0 for the present model .
the time @xmath86 may be determined by the condition ( [ eq : t =- tau ] ) , @xmath162 ; then eq .
( [ eq : dot - fii ] ) gives us a relation between @xmath4 and @xmath0 . by using ( [ eq :
fii ] ) for @xmath163 , concrete expressions of ( [ eq : t =- tau ] ) and ( [ eq : dot - fii ] ) may be obtained . for @xmath164 ,
@xmath163 is simply given by @xmath165 and the above conditions are expressed as : @xmath166 the requirement of symmetry , @xmath167 , mentioned in section [ sec : rondo approach ] becomes @xmath168 .
this relation and eq .
( [ eq : delta x_a ] ) allow us to express @xmath169 as , @xmath170 finally , we reach to the coupled equations which determine @xmath0 and @xmath171 for given slope @xmath74 and sensitivity @xmath4 : @xmath172=\omega e^{\frac a2\tau},\end{array } \right .
\label{eq : a - t}\ ] ] where @xmath173 is given in eq . ( [ eq : omega ] ) . for @xmath174 , eq . ( [ eq : a - t ] ) is solved numerically to give the @xmath4-dependence of @xmath0 and @xmath171 as shown in fig .
[ fig : at.eps ] . since we used @xmath103 for @xmath175 , eq .
( [ eq : a - t ] ) is valid only for @xmath176 ( @xmath171 coincides with @xmath0 when @xmath177 at @xmath178 ) .
we observe in fig .
[ fig : at.eps ] that @xmath0 behaves as @xmath179 for small @xmath4 : this implies that , for congested flows to be formed , the delay @xmath0 must be larger for less sensitive drivers . in the limit of @xmath180 ,
the present model reduces to the step function model , in which @xmath181 and @xmath182 .
this may be confirmed with eq .
( [ eq : a - t ] ) since @xmath183 reaches a constant , @xmath184 , and @xmath185 behaves like @xmath186 when @xmath187 goes to zero . ) may be rewritten in terms of rescaled variables @xmath183 , @xmath185 and @xmath187 . ]
+ in order to see the validity of our approach , let us compare our results and trajectories obtained by simulations . in fig .
[ fig:1-slope.eps ] , thick curves show parts of the asymptotic trajectories , to be determined by the function @xmath103 , for @xmath188 , @xmath189 and @xmath190 .
these curves are to be compared with numerical simulations shown as thin curves .
the function @xmath103 is enough to give an asymptotic trajectory for @xmath191 , as mentioned above .
it is expected that when @xmath4 gets closer to its critical value ( @xmath192 , in this case ) , we need functions @xmath123 with higher @xmath193 to form an entire trajectory
. there appears a flat trajectory again in the region iii . as in the step function model
, it takes time @xmath0 for a vehicle to move on the flat trajectory and there is only one vehicle traveling on this interval .
the ov - function for the single slope model has flat regions i and iii like the step function model .
for those regions the rondo equation does not depend on @xmath194 : a vehicle does not react to the motion of the preceding one .
this motivates us to consider a more realistic model with an ov - function which has non - zero gradient for any headway .
the ov - function has a slope @xmath195 in regions i and iii , and @xmath196 in region ii ( see fig .
[ fig : wovf.eps ] ) : @xmath197 & \delta x_{\rm a}\le\delta x\le\delta x_{\rm b}&{\rm ( region\ ii)}\\ f_1\left[\delta x+(\kappa^{-1}-1 ) ( \delta x_{\rm b}-\delta x_{\rm a})\right ] & \delta x_{\rm b } \le \delta x & { \rm ( region\ iii)}. \end{array}\right.\ ] ] where @xmath198 .
obviously this function is symmetric around @xmath199 , where @xmath71 is the middle point of the region ii .
here we should note that the sensitivity @xmath4 must satisfy @xmath200 for generation of the congestion in this model , since the homogeneous flow is expected to be linearly unstable only in the region ii . as in the previous subsection , we assume that the headway reaches @xmath80 at @xmath81 and @xmath82 at @xmath49 .
the solution of the rondo equation ( [ eq : zt - eq ] ) must satisfy the conditions ( [ eq : t=0 ] ) , ( [ eq : t =- tau ] ) .
the rondo function @xmath33 for three regions will be denoted as follows ; @xmath201 for @xmath91 ( @xmath92 ) , the rondo equation is given by @xmath202 where @xmath203 to find a solution to the homogeneous equation @xmath204 , we use the ansatz @xmath205 which gives an equation for the exponent @xmath113 , @xmath206 as long as the condition @xmath207 holds , there are two real solutions : the negative @xmath208 and the positive one @xmath209 . because of the asymptotic behavior , @xmath210 as @xmath211 , only the exponent @xmath212 is relevant to the function @xmath91 . adding a particular solution of ( [ eq : df1 ] )
, we obtain the solution subject to conditions ( [ eq : t=0 ] ) and @xmath58 as @xmath213 by calculating @xmath214 and @xmath215 from ( [ eq : ff0 ] ) , @xmath216 we obtain a linear trajectory given by @xmath217 in the region ii the function @xmath218 is divided into @xmath123 s for @xmath219 $ ] as was done in the previous subsection .
we may now study the rondo equation , @xmath220 where @xmath111 and @xmath221 is @xmath222 with @xmath195 replaced by @xmath196 : @xmath223 . in terms of @xmath221 , equation ( [ eq : df1 ] )
becomes @xmath224 where @xmath214 is given by ( [ eq : dx0 ] ) .
the basic technique in the last subsection may be used to solve ( [ eq : df2 ] ) with a slight modification .
let us define @xmath139 s with @xmath225 , which satisfy the equations , @xmath226 where @xmath227 .
the boundary conditions are @xmath228 for @xmath229 . in solving these
, we may use again the formula given in the appendix [ ap - sec : general formula ] .
once we find the series of @xmath139 , we obtain the function @xmath218 by the relation ( [ eq : fii ] ) .
further we can determine the time @xmath86 by the condition ( [ eq : t =- tau ] ) , @xmath230 . in the region iii ( @xmath159 )
, the rondo equation take the form , @xmath231 where @xmath232 is @xmath233 for @xmath234 and @xmath235 for @xmath236 .
the solution must be continuously connected to @xmath218 including the first derivative at @xmath81 .
( [ eq : fiii ] ) can be solved by the same manner as in the region ii , though homogeneous solutions to the equation ( [ eq : fiii ] ) include the hyperbolic functions instead of the trigonometric functions .
the logic at the end of section 3 may be used to get a better perspective on what we have discussed up to now , and it will leads us to find the @xmath4-dependence of @xmath0 .
first , with a given slope @xmath25 , we draw the straight line through the point s. the value of @xmath26 and the coordinates of c and f are given in terms of @xmath0 and the parameters in the ov - function , @xmath237 @xmath238 and @xmath239 in solving the rondo equation , we have introduced a parameter @xmath171 determined by eq .
( [ eq : t =- tau ] ) . using this @xmath171 and @xmath3
expressed as eq .
( [ eq : vb ] ) , we obtain one parameter family of ( @xmath4-dependent ) solution to the rondo equation .
then , we find an appropriate value of @xmath4 for a given @xmath0 by the requirement that the asymptotic trajectory connects c and f : @xmath240 . in fig .
[ fig : w.eps ] , we show trajectories from our analytic study and a computer simulation for the double slope model .
we have chosen a particular value for @xmath4 so that @xmath241 and we used the rondo function for @xmath242 to draw the part of the decelerating asymptotic trajectory .
clearly the rondo function reproduces the trajectory obtained via a computer simulation . by the procedure described in this subsection ,
we may easily obtain the remaining part of the asymptotic trajectory .
when this is carried out , we expect that it reaches to the point f in the infinite past . in the following
we will give another argument to support this expectation .
the rondo equation for @xmath243 in the region iii may be solved with exponential functions plus a particular solution , as for the region i : the function @xmath90 is the sum of a term linear in @xmath19 and exponential functions . in the limit @xmath244 ,
only the linear term survives expressing that vehicles have the velocity for a free region , @xmath245 ; the exponents satisfy eq .
( [ eq : gamma ] ) and have positive real parts so that exponential functions vanish as @xmath246
. it would be appropriate to explain how solutions to eq .
( [ eq : gamma ] ) are distributed on the complex @xmath113-plane . without going into the details , here we only quote features relevant to our arguments .
there is only one solution with negative real part , it actually a real solution @xmath247 .
other solutions have positive real parts , which are relevant when @xmath248 .
there is only one real solution , @xmath249 ; there are complex pair solutions with their real parts larger than @xmath249 .
since @xmath249 has the smallest positive real part , it dominates among exponential functions when @xmath246 .
thus for very large negative @xmath19 , the function @xmath90 may be approximated by the exponential functions with @xmath249 . in the region
i , we found that the decelerating asymptotic trajectory is linear on the @xmath1-@xmath2 plane owing to the exponential term in @xmath91 .
similarly , the approximated @xmath90 define a line on the @xmath1-@xmath2 plane , starting the point f as shown in fig .
[ fig : w.eps ] .
we observe that this line is actually the tangent line to the trajectory at the point f. the absence of the flat region is directly related to our observation that the point f in reached only in the infinite past .
if we consider the limit to have a flat region , @xmath250 , we only have a negative solution @xmath251 .
when we would like to consider the ov - model applied for a realistic situation , the relevant ov - function may be approximately realized by a piece - wise linear function .
since it is unlikely that the function has a flat region , the above feature of the double slope model must be generic .
in the last section , we discussed asymptotic trajectories in models with piece - wise linear ov - functions .
the asymptotic trajectories can be realized only when the number of vehicles becomes infinite . in computer simulations in refs.@xcite , a finite number of vehicles run around a circuit ; a vehicle goes through all the free and congested regions in a finite time .
the rondo equation probably has solutions even for such situations . though we have not worked out how to obtain entire trajectories for such vehicles , we are able to discuss parts of the trajectories around the points c or f. we are going to discuss this subject in this section . to make our explanation concrete , we take a trajectory around the end point c. in a congested region , all the vehicles have almost the same velocity .
when a vehicle is about to reach a congested region , its velocity would be slightly different from @xmath252 and the function @xmath33 may be written as @xmath253 .
we take a linear approximation for an ov - function @xmath5 around the point ( @xmath254 ) , @xmath255 since @xmath256 , the rondo equation ( [ eq : zt - eq ] ) becomes a linear difference - differential equation for @xmath257 , @xmath258 for the ansatz @xmath259 , we find an equation for the exponent @xmath113 , @xmath260 as long as @xmath261 , there are two real solutions : the negative , @xmath212 , and the positive one , @xmath262 .. see the discussion in the last section . ] trajectories considered here cross ( at @xmath49 ) the ov - function at points slightly different from c : we denote their coordinates by @xmath263 .
we find the solution for @xmath33 as , @xmath264 we may find @xmath214 and @xmath265 from this solution @xmath266 by eliminating the time @xmath19 , we find the equation for the trajectory around the point c , @xmath267 ^{\gamma_{\rm in}}\kern-1 mm \left[\frac{(a+\gamma_{\rm out})(v - v_{\rm c } ) -af_{\rm c}(\delta x -\delta x_{\rm c})}{\gamma_{\rm out}\delta v } \right]^{\gamma_{\rm out}}\kern-6mm=1 .
\label{eq : linear_traj}\ ] ] this curve shown in fig .
[ fig : linearized.eps ] has two asymptotes through c with slopes @xmath268 and @xmath269 , which correspond to the decelerating and accelerating asymptotic trajectories , respectively .
two asymptotes of ( [ eq : linear_traj ] ) divide the @xmath1-@xmath2 plane into four areas . in fig .
[ fig : flow.eps ] , curves are shown for solutions to eq .
( [ eq : linear_eq ] ) with various initial conditions : the point c is a saddle point .
the linear analysis applies to the point f as well , so curves in the left - lower region would describe the behavior of vehicles close to a free region .
the condition , @xmath270 , helps us to evaluate @xmath271 , the time interval a vehicle would spend around the point c ; @xmath272 the time @xmath271 is related to the length of a congested region @xmath273 and the number of vehicles in this region @xmath274 as follows , @xmath275 therefore the size of a congested region is larger for smaller @xmath276 .
we have investigated the repetitive pattern formation observed in a computer simulation from an analytical point of view .
the rondo approach was proposed to describe the repetitive pattern .
in addition to @xmath4 and @xmath5 , the difference - differential equation for the rondo function @xmath33 contains two macroscopic parameters , @xmath0 and @xmath3 , which specify the motion of a global pattern . in this paper
we mainly paid our attention to rondo functions for asymptotic trajectories .
the rondo equation was solved for three simple models with piece - wise linear ov - functions . in order
to determine the @xmath4-dependence of @xmath0 and @xmath3 , we gave analytic expressions for rondo functions .
we would like to emphasize that the concept of asymptotic trajectory plays a key role to determine the @xmath4-dependence of @xmath0 and @xmath3 . as a first step to understand more realistic situations ,
we have studied some trajectories around cusps . in the following we discuss three questions related to the rondo approach :
( 1 ) extension to ov - models with asymmetric ov - functions ; ( 2 ) more on realistic trajectories ; ( 3 ) possibility to find the rondo function _ forward _ in time . studying more realistic models
, we may encounter an asymmetric ov - function .
we explain how our procedure developed in this paper may be extended to such situations . even with an asymmetric ov - function
, we may define the concept of accelerating and decelerating asymptotic trajectories .
when the symmetry is absent , accelerating and decelerating asymptotic trajectories are not related each other and must be found independently .
the condition that both of them share the same end points c and f will determine the @xmath4-dependence of @xmath0 .
even though we have mainly studied asymptotic trajectories , the rondo equation itself must be applicable for any repetitive motion of vehicles .
on the other hand , as we have observed for piece - wise linear models , the difference between asymptotic trajectories and the results of computer simulations are very small .
therefore whether we would like to obtain a realistic trajectory out of the rondo approach or not is very much dependent of our purpose .
the rondo equation gives us a functional relation which may be written as , @xmath277 , \label{eq : functional}\ ] ] with three parameters @xmath4 , @xmath3 and @xmath0 .
when we know the function @xmath33 for the time interval @xmath278 $ ] , there are two ways to use the above equation : ( 1 ) substituting this on the r.h.s . , we find @xmath33 for @xmath279 $ ] ; ( 2 ) the same information may be used on the l.h.s . to have a differential equation for @xmath33 on the interval
@xmath280 $ ] .
although the former sounds much easier , we have been able to use the rondo equation only in the latter manner .
here we would explain the reason why it was so .
now let us consider , for concrete , a decelerating asymptotic trajectory in the double slope model .
in order to use eq .
( [ eq : functional ] ) in the approach ( 1 ) , we need the rondo function describing a part of the asymptotic trajectory coming out of f ; the rest of the asymptotic trajectory may be obtained just by differentiating the initial function repeatedly .
so the initial rondo function is of vital importance .
since the ov - function is symmetric , the transcendental equation ( [ eq : gamma ] ) may be used to find the initial rondo function .
let us remember how solutions are distributed .
there are only two real solutions , @xmath281 and @xmath282 , and infinitely many complex solutions whose real parts are larger than @xmath249 .
the initial rondo function is a linear combination of infinitely many exponential functions with @xmath283 .
therefore it contains infinitely many coefficients , which must be determined to give an asymptotic trajectory with properties described in section 3 . to find an asymptotic trajectory in this way
, we probably need some new techniques .
in this paper we have considered the repetitive pattern in traffic flow .
such repetitive structure is also observed in various phenomena , and we believe that our approach may be helpful to understand them .
* appendix *
in this paper we have used the following property of an asymptotic trajectory : @xmath21 and @xmath22 are at symmetric positions for ov - function symmetric around a point s. here we would like to give a proof of the above claim for an ov - function symmetric with respect to the point s. suppose an ov - function function and a sensitivity are given .
our rondo equation contains two parameters @xmath0 and @xmath3 ; @xmath284 if we could find a solution @xmath33 for the equation , this means that , for the ov - function and the sensitivity @xmath4 , corresponding pattern with the delay @xmath0 and the backward velocity @xmath3 may be realized .
the ov - function is taken to be an odd function around the point s@xmath285 .
this assumption is expressed with an odd function @xmath286 as follows , @xmath287 by putting this form to the rondo equation , it now looks like , @xmath288 we assume that a solution to the rondo equation has been found .
it gives a trajectory on the @xmath1-@xmath2 plane , whose coordinate we denote as @xmath289 .
they are expressed with the function @xmath33 as follows , @xmath290 by using the fact that @xmath291 is odd , it is easily shown that @xmath292 given below satisfies the rondo equation as well .
@xmath293 where @xmath294 and @xmath295 .
it is also easy to see that @xmath289 and @xmath292 are symmetric with respect to the point s. therefore if the former defines a trajectory from a free to a congested region , the latter defines that for opposite direction .
here we emphasize that two trajectories have different backward velocities but with the same delay time @xmath0 . in computer simulations , we observe that a pattern of a congested flow is characterized with two parameters @xmath0 and @xmath3 ; both regions , connecting free to congested or congested to free , moves with the same backward velocity @xmath3 .
so two trajectories connecting free and congested regions must have the same parameters .
this must be also true for an asymptotic trajectory
. therefore @xmath296 must be equal to @xmath3 itself .
this implies that two trajectories discussed above form a closed trajectory .
thus we may conclude the following : 1 ) solutions expressed by @xmath33 and @xmath297 satisfy the rondo equation with the same parameters @xmath0 and @xmath3 ; 2 ) the two points on the ov - function connected by trajectories are symmetric with respect to the point s ; 3 ) the straight line through the two points includes the point s.
in the following we will give a general formula for the second order linear differential equation , @xmath298 the solution for the eq .
( [ eq : ap.g0 ] ) is @xmath299 in terms of the function @xmath300 defined as follows , @xmath301 the initial conditions at @xmath49 are expressed as @xmath302 from the eqs .
( [ eq : ap.gk ] ) , ( [ eq : ap.g0 ] ) and @xmath303 , the equation to determine @xmath300 is @xmath304 the initial value problem with ( [ eq : ap.gk.init ] ) and ( [ eq : ap.gk ] ) may be solved by the functions @xmath305}(2\theta)^m \left(\begin{array}{c}\sin\theta\\\cos\theta\end{array}\right)_m \label{eq : ap.sol - gk}\ ] ] here @xmath306 and @xmath307 $ ] is the maximum integer which does not exceed @xmath308 .
we give functions for @xmath309 explicitly .
@xmath310 in this article , we also use another series of solutions @xmath311 of equations ( [ eq : ap.gk ] ) with initial condition ; @xmath312 it is easily shown that @xmath313 is given by @xmath314 where the second equality is valid only for @xmath147 .
we also give @xmath313 for @xmath309 explicitly .
|
in the optimal velocity model proposed as a new version of car following model , it has been found that a congested flow is generated spontaneously from a homogeneous flow for a certain range of the traffic density .
a well - established congested flow obtained in a numerical simulation shows a remarkable repetitive property such that the velocity of a vehicle evolves exactly in the same way as that of its preceding one except a time delay @xmath0 .
this leads to a global pattern formation in time development of vehicles motion , and gives rise to a closed trajectory on @xmath1-@xmath2 ( headway - velocity ) plane connecting congested and free flow points . to obtain the closed trajectory analytically
, we propose a new approach to the pattern formation , which makes it possible to reduce the coupled car following equations to a single difference - differential equation ( rondo equation ) . to demonstrate our approach
, we employ a class of linear models which are exactly solvable .
we also introduce the concept of `` asymptotic trajectory '' to determine @xmath0 and @xmath3 ( the backward velocity of the pattern ) , the global parameters associated with vehicles collective motion in a congested flow , in terms of parameters such as the sensitivity @xmath4 , which appeared in the original coupled equations .
-15 mm -2 mm
|
according to all hydro - dynamical simulation for structure formation run in the framework of a @xmath8cdm cosmology , large concentrations of galaxies are the best tracers of the filamentary web of dark - matter that our local universe is made of .
embedded into these filaments of shining ordinary matter ( stars in galaxies ) , should be hidden the , still to be found , largest reservoir of baryons in the local universe : the so called warm - hot intergalactic medium ( whim : e.g cen & ostriker , 2006 ) .
this , metal enriched ( through galaxy - igm feedback ) , otherwise primordial , medium should have temperatures in the range logt@xmath9 ( in k ) , and very low baryon densities @xmath10 @xmath11 . at such temperatures , h
and he are mostly fully ionized ( and so very difficult to detect ) , and the most abundant metal , oxygen , is mainly present in its stable he - like form : ovii .
the strongest bound - bound transition of the ovii ion is the k@xmath1 at @xmath12 , and should thus imprint absorption lines in the soft x - ray spectra of background objects whose lines of sight cross one or more whim filaments between us and the target .
however , these lines are expected to be extremely weak .
expected ovii column densities along a random line of sight crossing a whim filament , are n@xmath13_few_@xmath14 @xmath7 ( e.g. cen & fang , 2006 ) , giving rise to redshifted ovii k@xmath1 absorption lines with ew@xmath15 m . for these reasons detecting the whim
has proven to be very challenging .
the few detections so far , for the majority of the whim at logt@xmath16 , are either highly controversial ( e.g. nicastro et al .
, 2005a , b ) , or single - line and low statistical significance ( e.g. mathur , weinberg & chen , 2003 , nicastro , 2010 , zappacosta , 2012 ) .
perhaps the only exception that seemed to have gathered the largest consensus ( e.g. tananbaum et al . ,
2014 ) despite the large - compared to typical whim expectations - ew and associated ovii column density reported by the authors ( ew=@xmath17 m ) , is the proposed detection of a single - line ( ovii k@xmath1 ) whim filament at the redshift of the sculptor wall , reported by buote et al .
( 2009 : hereinafter b09 ) and fang et al .
( 2010 : hereinafter f10 ) along the line of sight to the blazar h 2356 - 309 .
recently , the same authors reported yet another evidence for a new ovii k@xmath1 whim absorber , again at @xmath18 , along the line of sight to the blazar mkn 501 ( ren , fang & buote , 2014 : hereinafter rfb14 ) .
indeed , the ovii k@xmath1 resonant line shifts to @xmath19 at @xmath20 , a redshift consistent with both the @xmath21 and @xmath22 intervals at which the sculptor wall and the hercules supercluster cross the lines of sight to h 2356 - 309 and mkn 501 , respectively . for this reason
b09 , f10 and rfb14 , identified the lines detected in the letg and rgs spectra of these two blazars at @xmath23 ( the average and maximum semi - dispersion of the two measurements in the letg and rgs spectra of h 2356 - 309 , respectively : f10 ) and @xmath24 ( rfb14 ) , as redshifted ovii k@xmath1 imprinted by two whim filaments permeating the two large scale structures of galaxies , the sculptor wall and the hercules superstructures . in a companion paper ( nicastro et al .
, 2015 : hereinafter n15 ) , we present a systematic study of the cold and mildly ionized interstellar ( ism ) and circum - galactic ( cgm ) medium of our galaxy through the modeling of the oi k@xmath1 ( @xmath25 ) , oii k@xmath1 ( @xmath26 ) and oii k@xmath3 ( @xmath27 ) absorption lines imprinted by our galaxy s ism / cgm in the spectra of two distinct samples of galactic and extragalactic sources . in particular , the oii k@xmath3 transition , firstly identified by gatuzz et al . , ( 2013a , b ) , is the weak ( oscillator strength @xmath28 , behar , private communication ) @xmath29 inner shell transition of the n - like ( 7 electrons ) ion of oxygen .
this line is hinted at in 9 of the 20 galactic x - ray binary ( xrb ) spectra and 8 out of the 29 agn spectra of the n15 sample , and has an average rest - frame wavelength of @xmath27 .
the 8 agns of the n15 sample , in whose spectra the oii k@xmath3 absorption line is hinted at , include the two blazars h 2356 - 309 ( @xmath30 , jones et al . , 2009 ) and mkn 501 ( @xmath31 , falco et al . , 2000 ) ,
for which the same line had been instead identified as intervening @xmath0 ovii k@xmath1 absorption tracing whim filaments ( b09 , f10 and rfb14 ) . here
we re - analyze all the available high resolution x - ray spectra of the two sightlines towards h 2356 - 309 and mkn 501 and demonstrate that , even for these two sightlines , the most likely identification of the @xmath32 line is indeed that of a @xmath2 oii k@xmath3 transition imprinted by a large amount of low - ionization metal medium ( limm ) that permeates the halo of our galaxy , at large distances from the galactic plane and perhaps up to the galaxy s cgm ( n15 ) .
this , at least for the line of sight to h 2356 - 309 ( the only one for which both the k@xmath1 and k@xmath3 transitions of oii are detectables ) does not completely rule out a possible contribution , at exactly the same wavelengths , by redshifted ovii k@xmath1 , but dramatically limits it to an ovii column density more than ten times lower than that claimed by b09 and f10 .
we reduced and analyzed archival hrc - letg and rgs data of the two blazars h 2356 - 309 and mkn 501 .
the data of h 2356 - 309 include 11 hrc - letg observations and 9 xmm - rgs observations , while mkn 501 has only 6 xmm - rgs observations .
all data were reduced with the latest versions of the xmm-_newton _ and _ chandra _ data reduction and analysis softwares ( `` science analysis system '' , gabriel et al . , 2004 - sas - v. 13.5.0 , and `` chandra interactive analysis of observation '' , fruscione & siemiginowska , 1999 - ciao - v. 4.6.1 ) and calibrations ( automatically set according to the given observation , for xmm-_newton _ data , and caldb v. 4.6.2 for _ chandra _ ) .
we followed the appropriate xmm-_newton _ and _ chandra _ data reduction / analysis threads and documentation to extract `` cleaned '' ( excluding periods of high background ) letg and rgs spectra and responses for all observations and co - added their spectra to maximize the snre ( * we also checked , a - posteriori , that all the results of our fitting procedures to the co - added spectra of h 2356 - 309 and mkn 501 where confirmed by simultaneous fitting of the individual spectra of the two sources * ) .
the final spectra of our two targets total 193 ks ( mkn 501 , xmm - rgs ) , 649 ks ( h 2356 - 309 , xmm - rgs ) and 466 ks ( h 2356 - 309 , hrc - letg )
. we used the fitting package _ sherpa _
, in ciao ( freeman , doe & siemignowska , 1999 ) to perform spectral fitting of the rgs and letg spectra of our targets , in the 6 - 30 ( rgs ) and 6 - 50 ( letg ) intervals , with the main aim to disentangle local ( i.e. @xmath2 ) absorption from our own galaxy s ism / cgm , from intervening absorption from putative whim filaments along the lines of sight to our two targets . following the procedure described in n15 , we first modeled the continuum of our targets with a power - law ( model _ xspowerlaw _ in _ sherpa _ ) attenuated by neutral absorption ( the xspec - native model _ xstbabs _ in _ sherpa _ : wilms , allen & mccray , 2000 ) . the best - fitting pure - continuum statistics for our three spectra are @xmath33 ( letg spectrum of h 2356 - 309 ) , @xmath34 ( rgs1+rgs2 spectra of h 2356 - 309 ) and @xmath35 ( rgs1+rgs2 spectra of mkn 501 ) .
figure [ res ] shows the 21.5 - 24 portions of the best - fitting pure - continuum model residuals ( in standard deviations ) of the letg and rgs spectra of h 2356 - 309 and mkn 501 . in the rgs panels , the green segments mark the spectral intervals where the presence of rgs instrumental features ( bad pixels ) hamper the search for unresolved absorption lines .
once a reasonable description of the broad - band continuum was obtained , we then proceded to search for , and model , the possible presence of unresolved ( fwhm@xmath36 km s@xmath37 and fwhm@xmath38 km s@xmath37 at @xmath39 , for the rgs and the letg , respectively ) absorption lines in the 18 - 24 band . to first identify candidate unresolved absorption lines and then confirm their identification in this spectral range we followed the procedure described in nicastro et al .
this procedure yielded the results summarized in table 1 ( line ids are also labeled in figure [ res ] ) .
for all the lines , we report only statistical errors at the the 1@xmath40 significance level .
@xmath0 i d + & width & & + in & in m & & + + @xmath41 & @xmath42 & oi k@xmath1 & oiv k@xmath1 + @xmath43 & @xmath44 & oii k@xmath1 & n / a + @xmath45 & @xmath46 & oiv k@xmath1 & ovi k@xmath1 + @xmath47 & @xmath48 & oii k@xmath3 & ovii k@xmath1 + 22.02 ( frozen ) & @xmath49 ( at 3@xmath40 ) & ovi k@xmath1 & n / a + @xmath50 & @xmath51 & ovii k@xmath1 & n / a + @xmath5219.21 ( frozen ) & @xmath53 ( at 3@xmath40 ) & n / a & ovii k@xmath3 + + @xmath54 & @xmath55 & oi k@xmath1 & oiv k@xmath1 + @xmath56 & @xmath57 & oii k@xmath3 & ovii k@xmath1 + @xmath58 & @xmath59 & ovi k@xmath1 & n / a + @xmath60 & @xmath61 & ovii k@xmath1 n / a + @xmath5219.25 ( frozen ) & @xmath62 ( at 3@xmath40 ) & n / a & ovii k@xmath3 + + @xmath63 & @xmath64 & oi k@xmath1 & oiv k@xmath1 + @xmath65 & @xmath61 & oii k@xmath3 & ovii k@xmath1 + @xmath66 & @xmath67 & ovi k@xmath1 & n / a + @xmath68 & @xmath69 & ovii k@xmath1 n / a + @xmath5219.26 ( frozen ) & @xmath70 ( at 3@xmath40 ) & n / a & ovii k@xmath3 + + @xmath52 for each spectrum , ovii k@xmath3 3@xmath40 upper limits are given at the redshift of the putative ovii k@xmath1 whim line , as derived from the best - fitting position of the line at @xmath71
. the 18 - 24 letg spectrum of h 2356 - 309 shows the presence of 5 unresolved absorption lines , which can all be identified as @xmath2 local transitions imprinted by a wide ionization range of oxygen ions in our galaxy s multi - phase ism . in particular , the letg data of h 2356 - 309 hint at the oi k@xmath1 ( 2.5@xmath40 ) , oii k@xmath1 ( 2.3@xmath40 ) and k@xmath3 ( 2@xmath40 ) , oiv k@xmath1 ( 2.9@xmath40 ) and ovii k@xmath1 ( 2.3@xmath40 ) transitions ( fig .
[ res ] , top panel ) .
the rgs data of h 2356 - 309 confirm the presence of the local oi k@xmath1 , oii k@xmath3 and ovii k@xmath1 absorption lines ( detected at statistical significances of 3.4@xmath40 , 2.8@xmath40 and 2.8@xmath40 , respectively ) and hint at the presence of a ovi k@xmath1 line ( only 1.6@xmath40 ) , but can not * safely * detect either the oii k@xmath1 or the oiv k@xmath1 lines , due to the presence of instrumental features at the relevant wavelengths ( fig .
[ res ] , middle panel , green horizontal segments ) . .
two of these lines , the @xmath2 ovii k@xmath1 and the line at @xmath72 ( letg ) or @xmath56 ( rgs ) ( table 1 ) were also reported by b09 and f10 ( who use the same letg data that we use here ) , who identified the second of these lines as an ovii k@xmath1 absorber at @xmath73 imprinted by an intervening whim filament at a redshift consistent with that where the sculptor wall concentration of galaxies crosses our line of sight to h 2356 - 309 .
the line at @xmath74 detected in the letg spectrum ( not detectable in the rgs spectrum because of an overlapping instrumental feature ) and identifiable with local oiv k@xmath1 absorption , was not reported instead by f15 , though the line is clearly visible in the 20 - 23.5 portion of the spectrum that fn15 show in their fig .
4 . the 18 - 24 letg spectrum of mkn 501 hints at the presence of the same 4 unresolved absorption lines that are detected in the rgs spectrum of h 2356 - 309 , and that can be identified as @xmath2 local transitions imprinted by oi k@xmath1 ( 4.2@xmath40 ) , oii k@xmath3 ( 2.8@xmath40 ) , ovi k@xmath1 ( 2.2@xmath40 ) and ovii k@xmath1 ( 3.2@xmath40 ) transitions ( fig . [ res ] , bottom panel ) . as for the rgs spectrum of h 2356 - 309
, the rgs spectrum of mkn 501 can not detect either the local oii k@xmath1 or oiv k@xmath1 lines , due to the presence of instrumental features at the relevant wavelengths ( fig .
[ res ] , bottom panel , green horizontal segments ) . also in the case of the rgs spectrum of mkn 501 , the @xmath2 ovii k@xmath1 line and the line at @xmath75 ( rgs ) ( table 1 ) were already reported by rfb14 ( using the same rgs data that we use here ) , who , again , identified the second of these lines as an ovii k@xmath1 absorber at @xmath76 ( consistent with the source systemic redshift ) imprinted by an intervening whim filament at a redshift consistent with that where the hercules supercluster concentration of galaxies crosses our line of sight to mkn 501 .
from the possible identifications listed in columns 3 and 4 of table 1 , it is evident that 3 of the 6 different lines detected ( or hinted at ) in the letg and rgs spectra of h 2356 - 309 and mkn 501 have unambiguous identifications .
these are , the @xmath2 ovii k@xmath1 ( letg and rgs ) , ovi k@xmath1 ( rgs only in h 2356 - 309 , where also letg is available , but consistent with the 30 m 3@xmath40 letg ew upper limit at that position ) and oii k@xmath1 ( letg only , because the rgs spectrum is blocked by an instrumental feature at those wavelengths ) , and can only be imprinted by a multiphase ism / cgm , with one or more high - ionization component producing ovi and ovii absorption and a low - ionization component producing oii ( and oi ) absorption . however , the 3 remaining lines have ambiguous identification
. they could all be imprinted by a multi - phase mildly ionized medium in the galaxy s ism / cgm ( with dominant oi - iv ions ) , but could also be identifiable with @xmath0 transitions imprinted by hotter medium , dominated by ovi - vii ions .
this is because the ovii k@xmath1 , ovi k@xmath1 and oiv k@xmath1 transitions , at @xmath0 overlap with the rest frame wavelengths of the oii k@xmath3 , oiv k@xmath1 and oi k@xmath1 transitions , respectively , expected to be imprinted by the galaxy s low - ionization and medium - ionization metal mediums ( limm - e.g. n15 and references therein - and mimm ) . to discriminate between these two possibilities for these 3 absorption lines ,
we make use of two different versions of our photoionized absorber spectral engine ( phase , krongold et al . ,
2003 ) code : one photo - ionized , for the ism / cgm of our galaxy ( _ galabs _ model hereinafter : see n15 for details ) , needed to model at least the three lines with unambiguous ism / cgm identification , and one hybridly - ionized and optimized for the whim ( _ whimabs _ model , hereinafter : see , e.g. , zappacosta et al . , 2010 ) , to attempt to model at least part of the three controversial and ambiguous lines .
the letg and rgs data of h 2356 - 309 have snre=13 ( letg ) and 26 ( rgs ) at 22 , much larger , when combined , than the snre=24 of the rgs spectrum of mkn 501 .
moreover , the combined letg and rgs spectrum of h 2356 - 309 , detects 6 different lines ( 5 in the letg and 3 in the rgs , 2 of which in common with the letg ) , compared to the 4 lines detected in the lower s / n rgs spectrum of mkn 501 .
h 2356 - 309 is therefore the best suited target to test our physically self - consistent ism / cgm versus whim models . to verify whether all the 6 different lines detected ( or hint at ) in the letg and rgs spectra of h 2356 - 309 could be modeled by absorption due to our own galaxy multi - phase ism / cgm
, we followed the procedure we used in n15 .
namely , we model simultaneously the 6 - 50 letg and the 6 - 30 rgs1 and rgs2 data of h 2356 - 309 with a continuum model consisting of a powerlaw ( native xspec model _ xspowerlaw _ in _ sherpa _ ) attenuated by the column of neutral galactic gas ( xspec native model _ xstbabs _ in _ sherpa _ , and add to each of the three spectra ( letg , rgs1 and rgs2 ) a number of _ galabs _ components till all the lines detected at single - line statistical significance @xmath77 could be adequately modeled . for each
_ galabs _ component we link all parameters to a single variable value in the 3 spectra , except the redshift that is allowed to vary by @xmath78 km s@xmath37 from spectrum to spectrum , to allow for line misalignment within instrument resolution elements .
this procedure required three different _ galabs _ components : ( a ) a low - ionization component , with a temperature of t@xmath79 k and a * poorly constrained * doppler parameter @xmath80 km s@xmath37 ( the limm : dubbed warm ionized metal medium in n15 ) , that models well the lines at @xmath81 and 23.35 , identified as oi k@xmath1 ( best - fitting limm ew@xmath82 m versus best - fitting gaussian @xmath83ew@xmath84 ) and oii k@xmath1 ( ew@xmath85 m versus ew@xmath86 m ) transitions , but is able to reproduce only half of the @xmath87 line , as oii k@xmath3 ( ew@xmath88 m versus @xmath83ew@xmath89 m ) ; ( b ) a warmer mildly ionized ism / cgm component with t@xmath90 k and b@xmath91 km s@xmath37 ( the mimm ) , modeling less than half of the @xmath92 line with the oiv k@xmath1 transitions ( ew@xmath93 m versus ew@xmath94 m ) ; ( c ) a hot ism / cgm component , with t@xmath95 k and b@xmath91 km s@xmath37 ( hereinafter high - ionization metal medium : himm ) , that models about 2/3 of the @xmath96 line , identified as ovii k@xmath1 ( ew@xmath97 m versus @xmath83ew@xmath98 m ) , but fails to model the ovi k@xmath1 line hinted at a single - line statistical significance of only @xmath99 in the rgs1 spectrum .
[ bfmodel ] shows the 21 - 24 portions of the letg spectrum of h 2356 - 309 , with superimposed the best - fitting ism / cgm model ( _ model - b _ ) including the limm , the mimm and the himm ( top panel ) .
the ews of the 5 lines modeled by the three _ galabs _ components , are consistent , within their 1@xmath40 uncertainties , with the best - fitting ews obtained with _ model - a _ and listed in table 1 , but the best - fitting values of the oii k@xmath3 , oiv k@xmath1 and ovii k@xmath1 are all lower than the those obtained with _ model - a _ ( single - line gaussian fit ) .
this could simply reflect the fact that the photo - ionization - equilibrium models that we use are not a sufficiently accurate description of the gas physics , but it might also indicate that additional components are needed .
for example , for the unmodeled ovi k@xmath1 line at @xmath100 and the not fully modeled @xmath2 ovii k@xmath1 , a fourth , ism / cgm phase could be present , with a degree of ionization in between that of the mimm and the himm , while for the partly unmodeled @xmath2 oii k@xmath3 oiv k@xmath1 an extragalactic intervening whim component may be needed , with temperature such to be dominated by ovi and ovii ions .
we conclude that the 3-phase ism / cgm _ model - b _ reproduces in a statistically satisfactory way the combined letg and rgs data of h 2356 - 309 , but leaves some room for ( statistically not required : @xmath101 for 5332 dof ) intervening @xmath0 whim absorption , filling in the 1/3 missing line opacity at the wavelengths of the @xmath2 oii k@xmath3 and the half missing opacity at the wavelengths of the @xmath2 oiv k@xmath1 , with the @xmath0 ovii k@xmath1 and ovi k@xmath1 transitions , respectively . to verify the need for an additional whim component
, we added a _
whimabs_ hybrid - ionization component to our best - fit ism / cgm 3-phase _ model - b _ , and re - fitted the data leaving the temperature , equivalent hydrogen column density , and doppler parameter of the gas , free to vary and constraining its redshift to vary only between @xmath102 ( _ model - c _ ) .
this yielded a negligible improve in statistics ( only @xmath103 for 3 additional degree of freedom over the initial 115 , in the 21 - 24 spectral interval ) , a best - fitting redshift of the putative whim filament of @xmath104 , a temperature t@xmath105 k , a doppler parameter @xmath106 km s@xmath37 and an oxygen column density of n@xmath107 @xmath7 , in much better agreement with theoretical expectations for the whim than the implausible large oxygen column reported by f15 ( n@xmath108 @xmath7 for their best fitting @xmath109 km s@xmath37 ) .
the _ whimabs _ component fills in the missing opacities at the wavelengths of the @xmath2 oii k@xmath3 and oiv k@xmath1 transitions , with @xmath20 ovii k@xmath1 and ovi k@xmath1 absorption , and the best - fitting _ model - c _ molds well the 5 absorption lines detected in the letg+rgs spectrum of h 2356 - 309 ( fig .
[ bfmodel ] , bottom panel ) .
table 2 lists , for these 5 lines , the best - fitting _ model - c _ ews , together with the corresponding best - fitting _ model - a _ ews already reported in table 1 .
.*comparison between _ model - c _ and _ model - a _ absorption line ews , for the h 2356 - 309 sightline . * [ cols="^,^,^,^ " , ] oii absorption is ubiquitously detected through its strongest k@xmath1 transition , along virtually all galactic and extragalactic lines of sight of the n15 samples .
the 8 times weaker k@xmath3 transition is hinted at in 9 out of the 20 lines of sight and 9 out of 21 extragalactic lines of sight of the n15 samples .
this absorption line is detected with average equivalent widths @xmath110 m ( down to a 3@xmath40 detectability threshold of @xmath111 m ) and @xmath112 m ( down to a looser - because of the lower s / n of the agn spectra - 3@xmath40 detectability threshold of 80 m ) , for the galactic and extragalactic sources , respectively ( n15 ) .
the n15 sample of extragalactic sources includes the two blazars h 2356 - 309 and mkn 501 , whose letg and rgs data we also reanalyze here . in our re - analysis of all the high - resolution x - ray spectra available for these two targets , for the absorption line at @xmath113 , we measure @xmath114 and @xmath61 m , for h 2356 - 309 and mkn 501 , respectively , fully consistent with the average ew@xmath115 measured in our samples of galactic and/or extragalactic lines of sight in n15 . based on this comparison ,
we conclude that the @xmath113 absorption line observed in the spectra of h 2356 - 309 and mkn 501 is likely to be produced entirely by oii k@xmath3 absorption .
in this work we reanalyze all the available high - resolution x - ray spectra of the two blazars h 2356 - 309 and mkn 501 , for which claims of intervening @xmath0 whim absorptions were presented by b09 , f10 and rfb14 .
we demonstrate that the presence of metal absorption from the diffuse limm in the galaxy s ism / cgm , casts serious doubts on these two ovii k@xmath1 whim detection claims ( b09 , f10 , rfb14 ) .
in particular , we show that the most likely identification for this putative @xmath116 ovii k@xmath1 whim line , is instead that of the k@xmath3 transition of oii , imprinted by the galaxy s limm ( n15 ) . for the case of h 2356 - 309 , for which we dispose of combined _ chandra _ letg and xmm-_newton _ rgs data that allow us to detect 5 absorption lines between 21 - 24 , at single - line significance @xmath117
, we conclude that any redshifted ( @xmath20 ) ovii k@xmath1 whim contribution to the unavoidable presence of an oii k@xmath3 absorber at @xmath118 , must be lower than n@xmath119 @xmath7 ( for @xmath120 km s@xmath37 ) , in much better agreement with theoretical expectations for the whim , than the implausible large oxygen column reported by f15 ( n@xmath108 @xmath7 for their best fitting @xmath121 km s@xmath37 ) .
we thank the anonymous referee for the useful comments that helped improving the paper .
fn and fs acknowledge support from inaf - prin grant 1.05.01.98.10 .
buote , d.a .
et al . , 2009 ,
apj , 695 , 1351 : b09 cen , r. & ostriker , j.p . , 2006 , 650 , 560 fang , t. et al . , 2010 , apj , 714 , 1715 : f10 falco , e. et al . , 2000 , `` the updated zwicky catalog ( uzc ) '' , vol 1 . , p. 1
freeman , p. , doe s. & siemiginowska a. , 1999 , spie , 4477 , 76 fruscione a. & siemiginowska a. , 1999 stin , 9906596 gabriel c. et al . , 2004 ,
aspc , 314 , 759 gatuzz , e. et al . , 2013 ,
apj , 768 , 60 ( 2013a ) gatuzz , e. et al . , 2013 ,
apj , 778 , 83 ( 2013b ) gupta , a. et al . , 2012 ,
apj , 756 , l8 jones , d.h .
et al . , 2009 , `` the 6df galaxy survey data release 3 '' , vol . 1 , p. 1
krongold , y. et al . , 2003 , apj , 597 , 832 mathur , s. , weinberg , d.h .
& chen , x. , 2003 , apj , 582 , 82 nicastro , f. et al . ,
2015 , mnras , submitted : n15 nicastro , f. et al .
, 2005 , nature , 433 , 495 ( 2005a ) nicastro et al .
, 2005 , apj , 629 , 700 ( 2005b ) nicastro , f. et al . , 2010 ,
apj , 715 , 854 ren , b. , fang , t. & buote , d.a . , 2014 ,
apj , 782 , l6 : rfb14 tananbaum , h. et al . , 2014 ,
rpph , volume 77 , issue 6 , article i d .
066902 : rxiv:1405.7847 wilms , j. , allen , a. , & mccray , r. , 2000 , apj , 542 , 914 zappacosta l. , nicastro , f. , krongold , y. & maiolino , r. , 2012 , apj , 753 , 137 zappacosta , l. et al .
, 2010 , apj , 717 , 74
|
in this letter we demonstrate that the two claims of @xmath0 ovii k@xmath1 absorption lines from warm hot intergalactic medium ( whim ) along the lines of sight to the blazars h 2356 - 309 ( buote et al . , 2009 ; fang et al . , 2010 ) and mkn 501 ( ren , fang & buote , 2014 ) are likely misidentifications of the @xmath2 oii k@xmath3 line produced by a diffuse low - ionization metal medium in the galaxy s interstellar and circum - galactic mediums .
we perform detailed modeling of all the available high signal - to - noise chandra letg and xmm - newton rgs spectra of h 2356 - 309 and mkn 501 and demonstrate that the @xmath0 whim absorption along these two sightlines is statistically not required .
our results , however , do not rule out a small contribution from the @xmath0 ovii k@xmath1 absorber along the line of sight to h 2356 - 309 . in our model
the temperature of the putative @xmath4 whim filament is t@xmath5 k and the ovii column density is n@xmath6 @xmath7 , twenty times smaller than the ovii column density previously reported , and now more consistent with the expectations from cosmological hydrodynamical simulations .
absorption lines , ism , galaxy , whim
|
there is a well - known problem ( problem 5.43 ) in @xcite that asks the reader to show that if a charged particle starts at the center of a circular ( of radius @xmath0 ) , radially - symmetric , flux - free magnetic field region , it will exit the region ( if it exits ) perpendicular to the circular boundary . this is an exercise in angular momentum conservation , and its ultimate utility resides in running the problem backwards : if you shoot a particle into a region with this special magnetic field , it will hit the center provided it enters perpendicular to the circular boundary of the region .
our interest in the problem begins with the determination of the critical velocity that allows the particle to escape the field region at all .
since there is no traditional potential energy barrier to go over , it is not immediately obvious what sets the minimum escape " speed here . after we determine the condition for escape , highlighting the role of a gauge - fixed magnetic vector potential in classical mechanics , we turn to particle trajectories in quantum mechanics . from schrdinger s equation in a region with magnetic vector potential @xmath1
, we can establish that the expectation value of position satisfies the following ode ( see @xcite problem 4.59 , for example @xcite ) : @xmath2 where @xmath3 is the canonical momentum .
if the magnetic field was constant , this would reduce to @xmath4 and the expectation value @xmath5 would be directly comparable to the classical velocity . for magnetic fields that are _
not _ constant , the right - hand side of ( [ expp ] ) defines an exotic effective force , one which is very different from @xmath6 ( as we shall see ) . as an equation of motion
, we do nt know what to expect for @xmath7 from ( [ expp ] ) .
indeed , we shall see that the expectation value of position is quite different from the classical position vector for these magnetic trajectories , and there are other differences as well . if the equation of motion for the expectation value of position was ( [ wrongexpp ] ) , we would expect the speed " ( the magnitude of @xmath5 here ) to be constant , just as it is classically .
but the effective force on the right of ( [ expp ] ) does not lead to a constant magnitude for the expectation value of velocity .
there are also similarities between the classical trajectories and the position expectation value of quantum mechanical solutions we will use a numerical solution of schrdinger s equation to show that the expectation value of kinetic energy is constant ( as it should be for motion in a magnetic field ) , and we can also establish that certain geometric properties of the quantum mechanical trajectory are shared with the classical trajectory : the particle exits the field region perpendicular to the boundary , and we can get both bound " motion , and escape " trajectories . the difference between the trajectory - based speed " , @xmath8 and the kinetic energy speed " , @xmath9 is the main distinction between the classical and quantum mechanical trajectories , but it is a significant difference .
the lagrangian for a particle moving in the presence of a magnetic field is : @xmath10 where @xmath1 is the magnetic vector potential .
the canonical momentum is then @xmath11 .
the legendre transform of the lagrangian defines the hamiltonian : @xmath12 we know that the hamiltonian is conserved , and that the speed of the particle is also a constant of the motion ( typical of motion in magnetic fields , which do no work ) . for our target problem , the magnetic field points in the @xmath13 direction , and we re interested in motion occurring in the @xmath14 plane ( we will set the initial velocity to lie in this plane ) . the magnetic vector potential takes the general form : @xmath15 ( its magnitude depends only on @xmath16 , similar to the magnetic field itself ) . in polar coordinates ,
the hamiltonian is @xmath17,\ ] ] and we can immediately identify the conserved @xmath18 ( angular momentum ) from the equation of motion : @xmath19 . the magnetic vector potential acts as a momentum , and we have to be careful to separate the velocity portion of the canonical momentum , @xmath20 ( with its constant magnitude ) , from the potential part . in order to untangle the two , at least initially , we ll take @xmath21 , and give the particle initial speed @xmath22 ( in the @xmath23 direction ) .
since we are starting at the origin , we ll pick the constant @xmath24 ( to avoid the @xmath25 and @xmath26 that would appear in @xmath27 otherwise ) . under these simplifying ( but reasonable ) assumptions ,
the initial value of the hamiltonian is : @xmath28 just the kinetic energy of the particle at @xmath29 . at any other time
, we have @xmath30,\ ] ] so that @xmath31 because of the form of @xmath1 ( which points in the @xmath32 direction ) , the radial momentum is @xmath33 , and we can solve ( [ psva ] ) for @xmath34 , @xmath35 the value of @xmath34 can not be imaginary ( when @xmath36 , all of the motion occurs in the @xmath32 direction ) , and so this relation provides precisely the desired escape speed " if a particle is to exit the field region , it must have @xmath37 where @xmath38 is the maximum vector potential magnitude over the domain .
what do we make of the fact that if we take @xmath22 less than this escape speed , there will be imaginary values for @xmath34 ?
the particle never gets to those regions when @xmath36 , the particle turns around , so that all of the motion will occur within a circle of radius @xmath39 defined by the value of @xmath40 at which @xmath41 .
the escape speed in ( [ escape ] ) uses the maximum value of @xmath40 in order to overcome all such constraining circles .
the escape speed depends on the magnitude of the vector potential , but the vector potential has gauge freedom , how do we know that the maximum height " is being pinned down to a unique value ?
so far , we have required that @xmath15 , appropriate for a radially symmetric magnetic field pointing in the @xmath42 direction , in coulomb gauge .
we also took @xmath21 in order to set the initial particle angular momentum to zero . for flux - free fields ( over the domain of the disk of radius @xmath0 )
, there is an additional requirement : @xmath43 and for our form for @xmath1 , this reads @xmath44 which means that @xmath45 .
what could we add to @xmath1 that preserves these basic requirements ?
the gradient of a function @xmath46 could be added to @xmath1 , @xmath47 , yielding the same magnetic field .
but , if we are to remain in coulomb gauge , @xmath46 must be a harmonic function , @xmath48 .
the flux - free boundary condition imposes the additional requirement that @xmath46 is independent of @xmath49 ( else we ca nt get @xmath50 at @xmath0 for all @xmath49 ) , so we are left with @xmath51 for constant @xmath52 and @xmath53 , which will not allow us to set the boundary condition at @xmath54 ( @xmath55 ) unless @xmath56 .
so in this case , the gauge is fully fixed , and that s what allows us to unambiguously identify an escape speed .
we can also use this @xmath45 requirement to solve the original problem posed in @xcite from ( [ dots ] ) , we have , at @xmath0 : @xmath57 , so that all of the velocity is in the @xmath58 direction , with none of it in the @xmath32 direction , the particle exits the region radially .
as a model flux - free , radial magnetic field , confined to the region @xmath59 , take @xmath60 this linear magnetic field is the simplest we can pick that can be made flux - free .
the potential that satisfies the requirements @xmath61 , and whose curl matches @xmath62 is @xmath63 predictably quadratic in @xmath16 .
the first term in the parentheses represents a constant magnetic field of magnitude @xmath64 . here
, we can determine the escape speed , @xmath65 , analytically
the maximum of the potential occurs at @xmath66 where the magnitude is @xmath67 . for initial speeds less than this , we will get bound trajectories , and for initial speeds greater than this , the particle will exit the field region perpendicular to the boundary .
examples are shown in figure [ fig : exe ] , in which we plot two bound trajectories together with their bounding circles ( of radius @xmath39 obtained by solving @xmath41 for @xmath39 ) , and the trajectory for a particle that escapes .
these trajectories were generated using a standard fourth - order runge - kutta solver . for @xmath39 ) and is shown in green .
for the bottom plot , @xmath22 is above the escape speed , and the particle exits perpendicular to the boundary.,width=240 ]
on the quantum mechanical side , we start with the same hamiltonian ( [ hofa ] ) in schrdinger s equation @xmath68 where we understand that @xmath69 . writing out schrdinger s equation with the momentum substitution in place , @xmath70 = i \ , \hbar \ , \frac{{\partial}\psi}{{\partial}t},\ ] ] let @xmath71 , @xmath72 , @xmath73 , where the barred variables are dimensionless , then @xmath74 = i \ , \frac{{\partial}\psi}{{\partial}\bar t}\ ] ] for @xmath75 , and where @xmath76 is a dimensionless variable that allows us to set the magnitude of the vector potential .
our starting point will be a gaussian centered at the origin with initial momentum expectation value @xmath77 normalized and written in cartesian coordinates : @xmath78 where @xmath52 is a parameter that tells us how sharply peaked the gaussian is the standard deviation for this initial gaussian is @xmath79 .
using @xmath80 , @xmath81 , @xmath82 , @xmath83 , the initial wavefunction can be written in terms of the dimensionless variables , @xmath84 with @xmath85 from above .
we can , finally , introduce the dimensionless wave function : @xmath86 , where the initial @xmath87 is just the above with the factor of @xmath88 removed .
we ll use a norm - preserving modification of crank - nicolson , developed in @xcite .
the idea is to use finite difference to generate forward and backward euler methods ( as with the usual crank - nicolson , see , for example @xcite ) but in a way that preserves the hermiticity of the discrete hamiltonian . to define the elements of the method , introduce a grid in ( the dimensionless ) @xmath89 and @xmath90 : @xmath91 and @xmath92 for constant spacing @xmath93 .
we ll also discretize in time , @xmath94 .
let @xmath95 , with @xmath96 and similarly for @xmath97 ( the magnetic vector potential is time - independent here ) .
using finite difference approximations to the derivatives in ( [ dimproblem ] ) , with a forward euler approximation for the temporal derivative gives @xmath98 .
\end{aligned}\ ] ] suppose our spatial grid has @xmath99 points in both the @xmath89 and @xmath90 directions , then we can embed the spatial values of @xmath100 , at time level @xmath101 , in a vector of length @xmath102 : @xmath103 so that given the @xmath89 and @xmath90 grid locations , @xmath104 and @xmath105 ( respectively ) , the index in @xmath106 is : @xmath107 . using these spatial vectors ,
the euler update above can be written as a matrix - vector product , defining @xmath108 from the details of the right - hand - side of ( [ method ] ) : @xmath109 from its definition , @xmath110 , it is hermitian by construction . similarly , backwards
euler takes the form : @xmath111 and the crank - nicolson method is then defined by @xmath112 this method is norm - preserving , and can be used with our initial wave function , projected onto the grid , to develop the @xmath113 update : @xmath114^{-1 } \ , \left [ { { \mathbb i } } + \frac{1}{2 } \ , i \ , \delta \bar t \
, { { \mathbb h}}\right ] \right)}^n \ , { \bm \psi}^0.\ ] ] implicit in the method is that the wave function must be zero at the boundary of the numerical domain ( that allows us to set the values of @xmath115 at the boundaries , the @xmath116 and @xmath117 points , in ( [ method ] ) ) our problem is immersed in an infinite square box in cartesian coordinates .
we chose to make the spatial grid with @xmath118 points in each direction , extending from @xmath119 to @xmath120 ( in dimensionless length ) .
our ( dimensionless ) time step was @xmath121 , and we took @xmath122 in the initial wave function that tells us roughly how many steps it would take to get the position expectation value of a free gaussian to hit the edge of the domain : @xmath123 steps . in order to probe the behavior inside the field region , we took @xmath124 so that a free gaussian s position expectation value would leave the region in @xmath125 time steps . to choose @xmath126 , note that the standard deviation for a free gaussian is @xmath127 and we would like the rate of spreading to be small compared to the expectation value of momentum , so that roughly : @xmath128 .
the initial expectation value of momentum is numerically determined ( even though @xmath129 is specified , we may or may not capture it numerically ) , and that determination is sensitive to the choice of @xmath126 if the initial gaussian is too sharply peaked , there will not be enough representation on the grid to numerically integrate the expectation value accurately .
we found that @xmath130 led to @xmath131 , an initial error of @xmath132 ( given that @xmath133 ) due to : 1 .
the finite difference approximation to the derivative ( needed to approximate @xmath134 ) , and 2 .
the use of a simple box - sum to approximate the expectation value integrals .
the choice @xmath130 also localized the particle inside the field region the probability of finding the particle within the circle of radius @xmath135 was , numerically , @xmath136 at @xmath137 . with these choices in place , we used the crank - nicolson method described above to move the initial gaussian forward in time with @xmath138 .
the method preserved norm very well the difference between the max and min total probability over the time of numerical solution was @xmath139 .
after running for @xmath140 steps , the expectation value of position indicated that the particle had left the field region , and a plot of that exit is shown in figure [ fig : exit ] .
the velocity vector at exit makes an angle of @xmath141 ( radians ) with @xmath32 at the location of exit , so the velocity vector is roughly perpendicular to the boundary , with an error of @xmath142 ( equivalent in size to the initial error in the expectation value of momentum ) .
the expectation value of energy @xmath143 ( calculated numerically using finite differences for derivatives and a simple box sum for the integration ) has max - minus - min value of @xmath144 over the first @xmath145 times steps , so that energy is conserved well here .
we also calculate the expectation value of the particle s velocity : @xmath146 ( using finite difference to approximate the time - derivative ) , and from that we can compute the speed " of the particle ( the magnitude of @xmath5 ) that is also shown in figure [ fig : exit ] .
the speed is not constant , but difference over the range in question is still within @xmath147 , so it is not clear if this is just the original error or if the speed is truly fluctuating . .
the speed " of the particle as a function of time is shown below .
, width=288 ] from the expectation value of position , we can also generate @xmath148 using finite difference for the temporal derivative , and we can compare that with the effective force defined by the right - hand - side of ( [ expp ] ) . we can also establish that the effective force defined by the right - hand - side of ( [ wrongexpp ] ) ( namely @xmath149 ) is not the one generating the motion here by computing it explicitly in figure [ fig : allots ] , we plot the left - hand - side of ( [ expp ] ) as a function of time ( the curve shown in the plot connects the tips of these force vectors ) , together with the effective forces from the right - hand - sides of ( [ expp ] ) and ( [ wrongexpp ] ) .
( and represents the left - hand side of ( [ expp ] ) ) , the dashed line is the expectation value found on the right - hand side of ( [ expp ] ) , and the dotted line is the value of @xmath6.,width=384 ] it is clear that while the correspondence between the left and right - hand sides of ( [ expp ] ) ( the solid and dashed lines in figure [ fig : allots ] ) is not perfect , the effective force defined by ( [ expp ] ) is far closer to governing the dynamics of @xmath7 than the effective force defined by ( [ wrongexpp ] ) .
to exhibit bound " behavior , we raise the height of the magnetic barrier " , taking @xmath150 and leaving everything else the same .
the resulting trajectory is shown in the top panel of figure [ fig : bounded ] ( here we take @xmath151 steps ) this time , the speed " of the particle is _ not _ constant ( shown in the lower panel of figure [ fig : bounded ] ) , yet the energy remained constant to within @xmath152 ( meaning the difference of the maximum value and minimum value of energy over the time - scales shown in the position expectation value plot ) .
this is fundamentally different behavior than the classical case and comes from the fact that the notion of speed " in quantum mechanics has two different interpretations there is the magnitude of the expectation value of velocity , @xmath8 which is not constant , and alternatively @xmath9 which is constant . in classical mechanics , there is no distinction to be made . is shown above , with the speed below.,width=288 ] we can once again compare the forces " defined by ( [ expp ] ) and ( [ wrongexpp ] ) in the trapped case those are shown in figure [ fig : trapforce ] , and again we see that the left and right - hand sides of ( [ expp ] ) are better matched than the left - hand side of ( [ expp ] ) and the fictitious @xmath6 ( the right - hand side of ( [ wrongexpp ] ) ) .
these force expectation values introduce additional error , above and beyond the discretization error in the crank - nicolson method itself , because of the approximations to both derivatives and integrals needed to evaluate them , so we do nt expect perfect matches . ) , the dashed line is computed using the right - hand side of ( [ expp ] ) . in the bottom figure ,
the solid line is again the left - hand side of ( [ expp ] ) the dotted line is @xmath6 , the right - hand side of ( [ wrongexpp ] ) .
the two cases have been separated here for clarity.,width=384 ] while there are numerical errors associated both with the crank - nicolson method and the calculation of expectation values , there is an implicit physical difference between the quantum mechanical problem and the classical one .
our numerical method required that the wave function vanish at the edges of our square domain , we put an infinite square well around the domain to keep the particle localized .
there is no such constraining force in the classical problem nor would the constraining force play much of a role there if we confined the classical trajectory to live in a box of side length @xmath153 ( where @xmath0 is the radius of the field region ) , and we considered trapped motion , the boundary would never be probed .
the quantum mechanical effect of the boundary is very different there is non - zero probability of finding the particle outside the magnetic field region , even for cases in which the expectation value of position remains inside the field region , and that external " portion of the wave function reflects off of the boundary . because our expectation values are integrated over the entire domain , those boundary effects get transmitted to the dynamics of the expectation value .
we have attempted to minimize this contribution to our problem by placing the boundaries far away , and keeping the initial gaussian localized within the field region but the boundaries do put a bound on how long we expect to be able to compare the classical and quantum trajectories .
the motion of particles in the presence of magnetic fields is complicated
few closed - form solutions exist , and while we can say quite a bit about the behavior of particles moving in magnetic fields , the trajectories themselves require numerical solution , even classically .
the situation is worse quantum mechanically even constant magnetic fields prove difficult to handle solving schrdinger s equation for such fields , starting from a reasonable initial wave function ( like gaussian ) is not possible analytically . in this paper
, we use numerical methods to study the motion of particles in magnetic fields , both classical trajectories ( solved using runge - kutta methods ) and quantum ones using a modification of crank - nicolson .
we started by looking at the classical problem of particle motion , first showing that for radially symmetric flux - free fields , particles will escape the circular field region provided their initial speed is larger than the escape " speed set by the magnetic vector potential .
we generated some trajectories for both trapped " and escape " behavior numerically to verify that the escape speed matches its theoretical prediction .
that prediction relied on a completely gauge - fixed magnetic vector potential in coulomb gauge it would be interesting to explore the effect of other gauge choices . on the quantum mechanical side
, we extended crank - nicolson to handle magnetic fields while retaining the norm - preservation of the method . using a linear , flux - free magnetic field
, we verified that the behavior of the position expectation value matches the classical trajectories in the following ways : 1 .
particles exit perpendicular to the boundary of the field region , and 2
. trajectories can remain inside the field region or escape , depending on the relation of the initial momentum to the field strength @xcite .
we also verified that the expectation value of energy remains constant , agreeing with the classical result , and yet classically , energy conservation _ means _ that the speed of the particle is constant ( since the only energy is kinetic ) for the quantum mechanical particle ,
however , the speed @xmath154 is not constant , even though the energy is ( so that @xmath9 is constant ) . in the case of a uniform magnetic field
, our classical intuition can be used to predict the behavior of quantum mechanical expectation values , basically because the dynamical variable @xmath5 appears in ( [ wrongexpp ] ) just as @xmath155 appears in the lorentz force law ( and indeed , we recover circular motion with predictable radius and constant speed using our initial gaussian and a constant magnetic field for @xmath156 solved using our numerical crank - nicolson method ) . for the more complicated flux - free magnetic field considered here , our classical intuition does not help us , because the effective force on the right - hand side of ( [ expp ] ) involves @xmath157 and @xmath62 inside the expectation value roughly speaking , we are looking at an effective force of the form @xmath158 rather than @xmath6 , _ different _ effective forces , leading to demonstrably different dynamics . in the context of ehrenfest s theorem ( see , for example @xcite
, the informal statement is that quantum mechanical expectation values obey classical laws " ) , while ( [ expp ] ) does give us a classical law " like newton s second law , the force on the right is unfamiliar , and not directly comparable to the lorentz force law
. it would be interesting to try to generate a classical analogue to the quantum mechanical effective force in ( [ expp ] ) so that a direct comparison of the classical ( under the influence of a modified effective " force ) and quantum trajectories was possible @xcite .
18 david j. griffiths , _ introduction to electrodynamics _
( 4th ed . , pearson , 2013 ) .
david j. griffiths , _ introduction to quantum mechanics _
( 2nd ed . ,
pearson , 2005 ) .
this is an example of ehrenfest s theorem .
the proof of this type of identity comes directly from calculating the expectation value of velocity and taking the time - derivative of it , then employing schrdinger s equation to remove the time - derivatives of @xmath159 that naturally appear .
katherine newton , _ bohmian mechanics and the aharonov - bohm effect : a computational approach _ ( reed college senior thesis , 2015 ) .
a. goldberg , h. m. schey & j. l. schwartz ,
computer - generated motion pictures of one - dimensional quantum - mechanical transmission and reflection phenomena , " _ am .
_ * 35 * , 177186 , 1967 .
the method is also described in : + william h. press , saul a. teukolsky , william t. vetterling & brian p. flannery , _ numerical recipes : the art of scientific computing _ , 3rd ed . , cambridge university press , 2007 , and + j. franklin , _ computational methods for physics _ , cambridge university press , 2013 .
although unlike the classical motion , it is difficult to determine , given the magnitude of @xmath62 , exactly what initial momentum will lead to bound motion or escape .
it would be interesting to estimate the
escape speed " for the quantum mechanical problem , although this is much more difficult than its classical analogue because of the different behaviors of @xmath9 , @xmath160 and @xmath161 itself , which is not just the kinetic piece for a gaussian initial wave function ( @xmath161 also depends on @xmath52 in ( [ initialgaussian ] ) ) .
there should be a transition from quantum mechanical behavior to classical in some regime ( of angular momentum , say ) , allowing for a convergence of our two effective forces " , and the details of that convergence would be interesting to probe .
|
we study the motion of a particle in a particular magnetic field configuration both classically and quantum mechanically . for flux - free radially symmetric magnetic fields defined on circular regions ,
we establish that particle escape speeds depend , classically , on a gauge - fixed magnetic vector potential , and demonstrate some trajectories associated with this special type of magnetic field .
then we show that some of the geometric features of the classical trajectory ( perpendicular exit from the field region , trapped and escape behavior ) are reproduced quantum mechanically using a numerical method that extends the norm - preserving crank - nicolson method to problems involving magnetic fields . while there are similarities between the classical trajectory and the position expectation value of the quantum mechanical solution , there are also differences , and we demonstrate some of these .
|
numerical simulations of an asymptotically free field theory on a lattice provide information about continuum physics when they are performed at values of the correlation length which lay within the scaling window .
this region is defined by the inequalities @xmath4 where @xmath5 , @xmath6 and @xmath7 are the lattice spacing , the correlation length and the lattice size respectively . to this window
it corresponds another scaling window in terms of the bare coupling @xmath8 . to understand the relationship between both regions , it is enough to recall that @xmath9 where @xmath10 is proportional to some power of @xmath8 . in this expression
, @xmath11 is the renormalization group independent mass parameter .
this parameter depends on the regularization used , thus the scaling window in terms of the bare coupling can be shifted towards the region of lower couplings if we use a lattice regularization in which @xmath11 is small .
gauge theories regularized on a random lattice present a @xmath11 parameter some orders of magnitude smaller than on a regular square lattice @xcite .
therefore , the simulations on a random lattice can be performed at lower values of the bare coupling .
this fact may make the simulations on random lattices more advantageous than on regular square lattices . the physical signal in the monte carlo data can be masked by the presence of pertubative expansions related to mixings , perturbative tails and finite renormalizations of composite operators defined on the lattice .
if the expansion parameter is smaller ( and the perturbative coefficients do not get larger ) , these perturbative terms may be better controlled and the non - perturbative physical signal be more clearly seen .
also the non - universal terms in the scaling function could be less relevant , thus expediting the asymptotic scaling .
the computation of the perturbative coefficients on random lattices might seem rather involved .
however , the higher degree of rotational invariance on a random lattice should help in this respect @xcite . in order to address these prospects on a physically interesting theory , like qcd , in reference
@xcite one of us started to study some features concerning another asymptotically free field theory , the @xmath0 non - linear @xmath1-model in two dimensions .
the action for this model in the continuum is @xmath12 with the constraint @xmath13 for all @xmath14 . in reference
@xcite we chose this theory because of its properties in common with qcd .
indeed , it is an asymptotically free field theory @xcite and presents a topological content @xcite for @xmath15 .
moreover , its mass gap is exactly known @xcite .
it was shown that the first coefficients of the renormalization group functions @xmath16 and @xmath17 for this model regularized on a random lattice are universal and that the @xmath11 parameter depends on the degree of randomness @xmath18 ( see below ) used in the construction of the random lattice .
the present work has several aims .
firstly , we want to verify the @xmath18 dependence of the @xmath11 parameter and support the scenario of a common continuum limit , the same than that of the theory defined in eq .
( [ action ] ) . for this purpose
, the theory with @xmath15 , 4 and 8 has been simulated on both regular and random lattices to extract the topological susceptibility and mass gap .
we have also succesfully used the cooling method @xcite on a random lattice to extract the topological content of the theory .
the @xmath15 value was chosen to study both the topological susceptibility and the mass gap , while the higher values of @xmath19 were used in order to have a better asymptotic scaling @xcite on the mass gap data .
another reason to study the topological charge on random lattices is that some sites on these lattices can group together forming clusters with a typical size less than one lattice spacing .
this geometry might allow the existence of very small instantons which could hardly survive on regular lattices .
we have always used lattices large compared to the correlation length at our @xmath20 , with @xmath21 , 300 and 400 .
we have never averaged the results of the simulations among different random lattices because the previous lattice sizes are large enough to include an implicit average among several subregions of the lattice .
the updating has been performed by using a cluster algorithm @xcite adapted to the random lattice .
another purpose of the present work is to analyse whether the simulations on a random lattice improve the asymptotic scaling or not .
actually , this is easy to check with the mass gap data because the value of @xmath22 is known @xcite . as it was shown in @xcite , @xmath23 for the @xmath1-model .
hence , it is not expected any dramatic shift of the scaling window towards small values of the bare coupling . to construct the random lattice
, we followed the procedure of t. d. lee at al .
the only new ingredient is the introduction of a degree of randomness @xmath18 .
the sites of the lattice are the centers of hard spheres , the radius of which is @xmath24 .
these hard spheres are randomly located . at small values of @xmath18 ,
the lattice is locally less random as we will see in section 2 .
there is a @xmath25 at which all distributions ( link lengths , distance between neighbouring sites , plaquette areas , etc . )
become dirac deltas .
the plan of the paper is as follows . in section 2 ,
we explain how to construct a random lattice and discuss in detail both the @xmath18 dependence and some geometrical properties like average link lengths and average distance between neighbouring sites .
we also give a geometrical argument in favour of a unique random lattice , once @xmath18 is fixed , as the number of sites gets larger . in section 3
we explain the simulation algorithm used . in section 4
we recall the scaling and @xmath20 functions for the non - linear @xmath1 model .
we also explain the procedure followed to extract both the mass gap and the topological susceptibility . a detailed description of the cooling method is also given . in section 5
we show and analyse the monte carlo data for the physical observables described in section 4 .
the conclusions are presented in section 6 .
we define a random lattice as a set of @xmath26 points located at random on a volume @xmath27 with the condition that there are no two sites closer than @xmath28 where @xmath5 is the lattice spacing and @xmath18 is a parameter to be fixed . these sites are placed with periodic boundary conditions .
thus , we consider the 2-dimensional lattice as a torus . in order to simulate a field theory on this geometry
, we must define a net of connections throughout the lattice , linking neighbouring sites . in the first part of the section
, we will explain how to place the sites on the lattice and in the second part we will review the triangularization process which is used to construct the links and plaquettes .
finally , we will give some properties of the random lattices .
if @xmath26 sites are placed on a 2-dimensional volume @xmath27 , we define the lattice spacing @xmath5 as @xmath29 .
the sites of our random lattice are the centers of hard spheres ( actually discs ) , the radii of which are @xmath24 .
thus , the sites are not closer than @xmath28 . fixing the parameter @xmath30 $ ] yields random lattices of different degrees of randomness as we will see .
the value of @xmath25 can be determined with the following argument . for large values of @xmath18 , the discs are small and they can be placed loosely on the volume @xmath27 .
then , the distributions of link lengths , distances between sites , etc .
display a wide dispersion of values .
instead , for small @xmath18 there is less play in placing the discs and consequently the distributions show a smaller variance . on a regular square lattice
, the hard discs will be placed as in figure @xmath31 . in this case ,
but also when @xmath32 there can be a bit of dispersion in the distributions characterizing the geometry of the lattice . for example , if we shift one column as shown in figure @xmath33 then the distance @xmath34 from any site @xmath35 to its closest neighbour still satisfies @xmath36 .
the link lengths are no longer all equal to @xmath5 . in particular , the links joining the shifted sites with their non - shifted neighbours are strictly longer than @xmath5 .
thus , a lattice constructed with @xmath37 need not be regular .
the usual regular square lattice is only a particular case of the lattice with @xmath32 . from these considerations
, one can conclude that the smallest @xmath18 is reached when the hard discs are packed tight as in figure 2 , forming an hexagonal lattice . in this case , the hard nature of the discs prevent the sites from shifting or moving from their positions in the figure .
therefore , all link lengths , distances between sites , etc .
are the same . stated in other words ,
the corresponding distributions are dirac deltas . as a consequence
, the corresponding @xmath18 must be the smallest allowed .
the volume occupied by a regular hexagonal lattice with @xmath38 sites is @xmath39 , hence the parameter @xmath18 is @xmath40 .
an explicit recipe for putting the @xmath26 sites on the volume of the lattice is the following .
the first site must be put at random in any position on the volume @xmath27 .
now , imagine we have already placed @xmath41 sites , then , by using the random generator , we propose the coordinates of the @xmath42-th site and check that its distance to all the @xmath43 previously accepted sites is less than @xmath28 .
if the check is satisfied , we accept this site . if not , we reject it and repeat the process by proposing another @xmath42-th site .
this procedure is repeated until all @xmath26 sites have been placed on the lattice .
clearly , the number of proposed sites @xmath44 satisfies @xmath45 . in figure 3
we plot the experimental ratio @xmath46 as a function of @xmath18 obtained during the construction of lattices with @xmath47 and @xmath48 sites . from this plot ,
it is clear that the construction of random lattices with small @xmath18 is quite time consuming .
one can speed up the check about distances by dividing the volume @xmath27 into boxes , large enough to contain a few sites on average .
then , the check for a proposed site is performed in the box it belongs to and in its neighbouring boxes . however , even with this improvement , creating lattices with @xmath49 is almost impossible with our recipe .
the smallest value of @xmath18 we have used in our simulations is @xmath50 .
the ratio @xmath46 depends only on @xmath18 , not on @xmath26 .
figure 3 confirms this statement .
let us derive a theoretical expression for the curve in figure 3 for large @xmath18 .
if a site can not have neighbours closer than a distance @xmath28 , then once @xmath43 sites have already been placed , the area left free for putting new sites is @xmath51 .
this expression does not take into account the fact that these circles of radius @xmath28 can overlap ( with the constraint of leaving their centers , the sites , not covered ) . for large @xmath18 ,
the overlaps are less frequent and the previous formula is adequate .
therefore , the probability for a proposed point to be accepted after having put @xmath43 sites is @xmath52 now , using @xmath53 , we conclude that the total number of proposed sites divided by @xmath26 is @xmath54 for large enough @xmath26 , the sum becomes an integral giving the ratio @xmath46 as a function of @xmath18 @xmath55 this expression , as previously stated , is independent of @xmath26 .
it works well for @xmath56 .
once the lattice volume @xmath27 has been filled with the @xmath26 sites for some chosen value of @xmath18 , we proceed to the triangularization process .
we follow the method of reference @xcite .
it consists in joining sets of three sites to form a triangle with the only condition that the circle circumscribed by these three points does not contain any other site .
the sides of that triangle are the links joining the three sites and the triangle itself is a plaquette .
this construction is unique and fills the whole lattice with no overlapping among the triangles @xcite .
we also impose periodic boundary conditions in the triangularization process . in figure 4
we show two @xmath57 sites random lattices , with @xmath58 and @xmath59 it is useful to define also the dual lattice .
its dual sites are the centers of the above - mentioned circumscribed circles .
it is clear that any link is the common side of two triangles .
thus , every link of the random lattice must be surrounded by two dual sites .
the line joining these two dual sites is called the dual link .
therefore , every link is associated with a dual link .
let us call @xmath60 the @xmath61-component of the link vector that points from the site @xmath35 to the site @xmath62 .
the length of this link is @xmath63 .
the length of the corresponding dual link is @xmath64 . throughout this work
, we will often make use of the matrix @xmath65 defined as @xmath66 we introduce another vector , @xmath67 which is twice the vector joining the center of the vector link @xmath68 with the center of the associated dual link @xmath69 .
another useful quantity is @xmath70 defined as @xmath71 \cdot { \hat { \bf z}}.\ ] ] in this equation @xmath72 is the unit vector orthonormal to the plane of the lattice , oriented as @xmath73 .
the set of dual links around the site @xmath35 , @xmath74 , determine a convex region called voronoi cell . the area of this cell is @xmath75 .
as soon as the lattice has been constructed , one can devise tests to check the triangularization .
the first and easiest one is to verify that the number of triangles ( links ) is equal to twice ( three times ) the number of sites @xcite .
other good tests are the integral properties @xcite @xmath76 a third test consists in demanding that the quadratic part of the action has only one zero mode @xcite .
the action for the 2-dimensional @xmath0 non - linear @xmath1-model on a random lattice can be written as @xcite @xmath77 in this equation , @xmath8 is the coupling constant .
@xmath78 denote sites and @xmath79 stands for the value of the field @xmath80 at the site @xmath35 .
this is the action we will make use in our numerical simulations , both on regular and random lattices . for a regular square lattice , @xmath65 is 1 for linked sites and zero otherwise .
hence , eq . ( 2.7 ) becomes the standard action when it is considered on a regular lattice .
one can show that the _ nave _ continuum limit of eq .
( 2.7 ) is the correct action for the model in the continuum , eq .
( 1.1 ) .
we see from figure 4 that random lattices with large ( small ) values of @xmath18 display a higher ( lower ) degree of randomness .
as it was shown in reference @xcite , random lattices with different @xmath18 are different regularizations of the same theory .
in particular , it was shown that the @xmath11 parameter is @xmath18-dependent . in the present paper
, we will check this statement by a numerical simulation and , moreover , we will give hints that the non - universal non - leading coefficients of the @xmath20 function of the model are also @xmath18-dependent .
the level of randomness can be clearly manifested with the distributions of some geometrical properties of the lattice .
the first property we plot is the distance of one site to its nearest site , @xmath81 .
this distance , referred to the site @xmath35 , was written before as @xmath34 . in figure 5 , we show the probability distribution of this distance , @xmath82 for a @xmath58 lattice .
the histogram in the figure is the numerical result , obtained by calculating @xmath34 for a single site on 30000 random lattices of 1000 sites .
the solid line is the theoretical well known poisson distribution , @xmath83 both curves are normalized to 1 and should coincide . in figure 6
, we show the same distribution for a @xmath84 random lattice .
it was obtained as in figure 5 , with 30000 random lattices of 1000 sites .
notice that the distribution displays less dispersion .
the cutoff of the curve is placed at @xmath28 .
given a random lattice constructed with some finite value of @xmath18 , all sites are farther from each other than @xmath28 . if we bring together two of these sites a distance less than @xmath28 , the random lattice will not vary its geometrical properties .
we checked that the results of the simulations performed on it , do not vary either .
indeed , apart from a tiny bump at some position between @xmath85 and @xmath86 , figure 6 will not change .
the more sites the random lattice has , the tinier the bump becomes .
another quantity we can plot is the distribution of link lengths . in figure 7
we show the distribution of link lengths @xmath87 for a random lattice with @xmath58 .
given a site on the lattice , it is always linked with the closest site to it .
but it can also be linked with sites placed quite far away .
this is why figures 5 and 7 differ .
finally , in figure 8 we show the distribution of link lengths for a random lattice with @xmath50 .
again we see that the distribution is sharper as @xmath18 becomes smaller . in table 1
we show the numerical average values of link lengths @xmath88 and distance between closest sites @xmath89 for several values of @xmath18 .
these quantities can be used to characterize the degree of randomness . however
, our method to construct the random lattice needs only the knowledge of @xmath18 as previously explained .
therefore , we will label our random lattices with this @xmath18 parameter .
the column for @xmath88 was obtained averaging the link lengths of random lattices with 10000 sites for @xmath90 and 1000 sites for @xmath91 . the column for @xmath89
was calculated by averaging @xmath81 on a single site for 30000 random lattices of 1000 sites for @xmath58 and @xmath92 ; for @xmath93 we averaged on 14 random lattices of 100 sites .
the numbers shown depend on the lattice size . as this size gets larger ,
the averages tend to stabilize .
for instance , the averages for @xmath94 calculated on random lattices of 100 , 1000 and 10000 sites with @xmath58 are 0.523(2 ) , 0.508(2 ) and 0.504(2 ) respectively . the exact value computed from eq .
( 2.8 ) is @xmath95 .
the errors shown are only statistical and do not include this systematic effect .
the value of @xmath88 for @xmath58 is known theoretically @xcite , it is @xmath96 in good agreement with the value shown in table 1 . the trend shown in figures 5 , 6 and 7 , 8 is consistent with the fact that as @xmath97 the distributions become a dirac delta .
these dirac deltas are equal and centered at @xmath98 .
as the number of sites gets larger , all random lattices with a fixed value of @xmath18 become equivalent . to support this statement
we have generated 100 lattices for several values of @xmath7 and verified that the average link length presents less standard deviation among the 100 lattices as @xmath7 becomes greater . if we define @xmath99 , where @xmath88 is the average link length among the 100 lattices for a fixed value of @xmath7 and @xmath100 is the standard deviation of this average , the results are @xmath101 for @xmath102 respectively .
this analysis was done with @xmath58 random lattices .
the best algorithm for the updating of the @xmath0 non - linear @xmath1-model in a numerical simulation is the non - local wolff algorithm @xcite . on the regular lattice
it does not show any critical slowing down .
let @xmath103 be the time correlation of the markov chain of states generated by the algorithm .
let @xmath6 be the spatial correlation length of the model ( a well defined function of @xmath20 ) .
then , the wolff algorithm has the fundamental property that @xmath103 does not increase as the critical point , corresponding to @xmath104 , is approached .
this nice property is allowed by the intrinsic non locality of the wolff updating and has nothing to do with the details of the lattice .
we argue that it should hold also for the straightforward generalization of the wolff algorithm on the random lattice . indeed
, our numerical data show that the algorithm performs very well at increasing @xmath20 , just like in the regular case .
the above mentioned generalization is obtained as follows : we start by choosing at random one lattice site @xmath35 and an unitary vector @xmath105 .
there is a great freedom in the distribution of @xmath106 . to be definite we chose the uniform distribution on the hypersphere by taking at random a point inside the hypercube @xmath107^n$ ] and rejecting it if @xmath108
then , @xmath109 .
we now mark with a label the site @xmath35 and all of its neighbours according to the probability weights @xmath110.\ ] ] we continue recursively from the new marked sites and proceed further until no new site is added to the list of labelled sites .
when the recursive process ends , we collect in a cluster all the marked sites and transform them with a reflection @xmath111 as wolff has shown @xcite , in the process of construction of the cluster , all the quantities entering the r.h.s . of @xmath112
have already been calculated at each updating step .
the above expression is called an improved estimator for the two point function . in eq .
( 3.3 ) @xmath26 is the total number of sites of the lattice , @xmath113 is the number of sites in the cluster and @xmath114 is the characteristic function of the cluster .
another advantage of this estimator is that it can be measured after each updating step without need of decorrelating updatings .
the physical observables that we have measured are the mass gap for several values of @xmath19 and the topological charge for @xmath15 . for each of these quantities
we have studied the asymptotic scaling behaviour .
the universal 2-loop @xmath20 function of the @xmath0 non - linear @xmath1-model is @xmath115 and therefore the lattice spacing obeys the renormalization group law @xmath116 the first non - universal correction for this scaling function on a regular lattice is known @xcite . on the other hand ,
the value of @xmath117 is known both for regular @xcite and random lattices @xcite . any dimensionful quantity @xmath118 measured on the lattice with the operator @xmath119 must satisfy ( after the subtraction of the relevant mixings ) @xmath120 when @xmath121 .
asymptotic scaling holds when the l.h.s . of
the above expression , with the first terms of the beta function in @xmath122 , depends little on @xmath20 .
the analytical prediction for the mass gap of the @xmath1 model is @xcite @xmath123 which combined with the ratio @xcite @xmath124 gives @xmath125 . in order to evaluate @xmath126
we studied the wall - wall correlation function by integrating the two - point function along the spatial direction and studying it at large temporal separations .
notice that on the random lattice the @xmath43-th temporal slice must be defined as the set of sites with @xmath127 .
moreover , the number of sites that this time slice contains is a function of @xmath43 .
this is a source of systematic error . as @xmath26 , the total number of sites , gets larger
, the width of the time slice @xmath128 can be smaller and reduce the previous systematic error .
numerical studies on the regular lattice indicate that a fit of the wall - wall correlation monte carlo data to the behaviour @xmath129\ ] ] is enough to reproduce the data @xcite .
to avoid correlation among data at different @xmath130 , we used sets of different runs for every @xmath130 . hence , the statistical errors obtained are reliable .
the above mentioned improved estimator , eq .
( 3.3 ) , reduces greatly the statistical noise which affects this measure particularly at large @xmath130 .
we have also measured the topological charge and susceptibility of the @xmath131 model on a random lattice .
the topology of this model is based on the stereographic map from the sphere @xmath132 onto the projective plane @xcite .
the topological charge actually counts the winding or instanton number of this mapping on classical configurations . in the continuum
it is defined as @xmath133 and it rigorously yields integer values on a smooth field configuration . at the quantum level , @xmath134 is a composite operator which requires a renormalization procedure . on a random lattice , we introduce the following definition of topological charge at the site @xmath35 @xmath135 which , by using eqs .
( 2.5 ) and ( 2.6 ) , reproduces the continuum expression , eq .
( 4.7 ) . on a regular lattice , where @xmath136 , @xmath137 is equal either to @xmath138 or to @xmath139 and @xmath65 is 1 for linked sites and 0 otherwise , eq . ( 4.8 ) reproduces the standard definition @xcite . the total topological charge @xmath134 is calculated just by @xmath140 .
an interesting and non - trivial quantity which can be extracted from the topological charge is the topological susceptibility which in the continuum is defined as the two point correlation of the topological charge at zero external momentum @xmath141 on the lattice , this is rewritten in the following way @xmath142 the monte carlo data for the topological susceptibility @xmath143 is @xcite @xmath144 where @xmath145 , eq .
( 4.2 ) , @xmath146 is the non - perturbative vacuum expectation value of the density of action and @xmath147 , @xmath148 and @xmath149 are the renormalizations which can be calculated perturbatively @xcite . on
the regular lattice @xmath146 is negligible @xcite and so is the second term in the r.h.s . of eq .
we have not calculated all of these renormalizations on the random lattice but instead we have used the cooling method @xcite .
the cooling procedure is a relaxation process which after a few cooling steps ( @xmath150 3040 steps ) have almost eliminated all short scale fluctuations leaving the long waves still present . if we assume that these short scale fluctuations , of order @xmath151 , are responsible for the quantum noise which shows up as renormalization effects , then after some cooling steps all renormalizations in eq .
( 4.11 ) will disappear and the physical and monte carlo topological susceptibility will be related by the expression @xmath152 however the situation for the @xmath131 @xmath1 model is not so simple . in this model , instantons tend to be small .
indeed the distribution of instantons with radius @xmath153 in this theory satisfies @xmath154 . as a consequence ,
the previous cooling process can also eliminate small @xmath151 instantons , thus modifying the topological content of the configuration .
this unwanted behaviour of the cooling occurs on regular lattices @xcite . on a random lattice
some sites can group together forming clusters with a typical size less than one lattice spacing .
this fact happens mostly on large @xmath18 lattices ( see figure 4 ) .
this geometry might allow the existence of very small instantons with a size @xmath155 .
this is the main motivation for studying the topological properties of the model on random lattices . in figure 9
we show the evolution of the topological charge for 40 uncorrelated configurations as the cooling process goes on . at zero cooling step the value of the charge on the lattice
is on average @xmath156 .
instead , after 30 or 40 steps , this charge reached an almost integer value which depends only on the underlying instantonic content of the original configuration .
this almost integer value remains stable for a long plateau .
thus , we assumed the value of @xmath157 after 30 or 40 steps of cooling as the correct topological charge of the configuration .
we checked that the value of the topological susceptibility is also stable on the plateau .
indeed , the value obtained is within errors , the same if 30 , 40 or 50 cooling steps are performed .
it can also be seen from figure 9 that the value of @xmath157 after several coolings is not exactly an integer .
this is a general fact and has to do with the fact that instantons are not exact solutions of the theory defined on a lattice .
we also checked that the susceptibility is insensitive to rounding this number to its nearest integer .
we chose for our analysis the values of @xmath157 after 30 cooling steps without rounding it to the nearest integer .
we now turn to a detailed description of the cooling step .
it consists in locally minimizing the action with respect to the field at each site once per step .
we used a controlled cooling @xcite .
this means that for a given positive number @xmath158 , the new field @xmath159 and the old one @xmath160 differ less than @xmath158 , @xmath161 .
first we define the force relative to the site @xmath35 as @xmath162 and then we compute the distance between the force and the field at this site @xmath35 , @xmath163 .
now , if @xmath164 then the new field is going to be @xmath165 . instead , if @xmath166 then the field at the site @xmath35 becomes @xmath167 where @xmath168 is chosen to be @xmath169 in a step of cooling we pass through all sites @xmath35 of the lattice and perform the previous modification on the corresponding field @xmath160 .
in this section we will show the set of monte carlo data obtained for the mass gap and the topological susceptibility and discuss their consequences . in table 2
we show the set of data for the mass gap and the @xmath131 @xmath1 model on lattices of @xmath170 sites . they are also shown in figure 10 .
the data were obtained from 12000 measures of the improved correlator , eq .
( 3.3 ) for each @xmath20 .
we fit these data to the scaling function of eq .
( 4.2 ) @xmath171 where @xmath172 and @xmath173 are the parameters of the fit .
in particular , @xmath173 must be equal to @xmath174 in order for the data to scale as the @xmath20 function of the theory predicts , eq .
( 4.1 ) , and @xmath172 must be @xmath175 .
the fit gives @xmath176 for the regular lattice , @xmath84 and @xmath58 random lattices respectively .
this is in agreement with the result of @xcite concerning the universality of the random lattice : the first coefficient of the beta function is the same in any lattice regularization .
now , imposing @xmath177 and leaving @xmath172 free we obtained @xmath178 for regular lattice , @xmath84 and @xmath58 random lattice respectively , with a value for @xmath179 equal to 6/11 , 8/11 and 9/12 respectively . following @xcite and @xcite , the expected results for @xmath172 are 80.09 , 87.05 and 44.49 respectively .
we excluded any finite size explanation to this discrepancy .
we arrived at this conclusion because @xmath180 the technique of reference @xcite did not improve the results and @xmath181 the fits with the data of table 3 for @xmath182 lattices yielded similar results for @xmath172 .
we think that the disagreement is due to the fact that our range of @xmath20 is narrow enough to collect into @xmath172 all the power - law corrections to the asymptotic scaling function .
unfortunately , another single free parameter is not able to account for the whole non - universal terms correcting eq .
( 5.1 ) and it did not improve dramatically the result for @xmath172 . from the results of the fits for @xmath172 we can get the ratio between @xmath11 parameters .
let us define @xmath183 at a given @xmath18 .
then , @xmath184 and @xmath185 .
these results are in agreement with the average of the ratios obtained from each @xmath20 : @xmath186 and @xmath187 .
these ratios for each @xmath20 are shown in figure 11 .
the theoretical values @xcite are @xmath188 and @xmath189 . for @xmath84 theoretical and monte carlo ratios
are in agreement within errors .
but this is no so for @xmath58 .
we again think that this is due to the lack of asymptotic scaling .
it is well known that asymptotic scaling is rather elusive in the @xmath131 @xmath1 model and here we have realised that this problem does not ameliorate if a random lattice is used .
for this reason , we also performed runs for the @xmath190 and @xmath191 models where it is known that asymptotic scaling is better achieved @xcite . in table 4 the mass gap data for @xmath190 on lattices with @xmath182 sites are shown . in table 5 the same data are shown for @xmath191 and lattices with @xmath192 sites . in both cases 12000 measures of the improved estimator , eq .
( 3.3 ) , were performed for each @xmath20 .
the @xmath190 data of table 4 were fitted to @xmath193 the result for @xmath173 was again close to 1 : 1.02(4 ) , 1.05(7 ) and 1.04(4 ) for regular , @xmath84 and @xmath58 random lattices respectively .
fixing @xmath173 to 1 , we obtained for @xmath172 26.6(5 ) , 28.4(9 ) and 22.3(4 ) for for regular , @xmath84 and @xmath58 random lattices respectively , while the predicted values are @xcite 24.02 , 25.55 and 16.01@xmath194 .
we see a much better agreement between theoretical and monte carlo results .
the ratio of @xmath11 parameters extracted from the monte carlo data is @xmath195 and @xmath196 .
the theoretical ratios are @xcite 0.94(2 ) and 1.5(1 ) respectively .
the agreement for @xmath58 is still not satisfactory .
the data of table 5 for @xmath191 were fitted to @xmath197 again @xmath173 was satisfactorily close to 1 .
the fit for @xmath172 shows a good asymptotic scaling ( @xmath198 for the regular lattice while eqs .
( 4.4 ) and ( 4.5 ) give 9.48 ) .
so we expect that this time the ratios of @xmath11 parameters extracted from monte carlo will be in agreement with the theoretical ones .
the monte carlo ratios are @xmath199 and @xmath200 .
the theoretical ratios are 0.95(2 ) and 1.3(1 ) respectively .
we conclude that the dependence of the @xmath11 parameter on @xmath18 , the degree of randomness of the lattice , is qualitatively correct .
the figures shown in @xcite for these ratios are also correct but the lack of asymptotic scaling prevent us from checking them for the @xmath0 model when @xmath201 68 .
let us now discuss the results for the topological susceptibility obtained for the @xmath131 model .
in table 6 we show @xmath143 after 30 cooling steps which , as described in section 4 , equals @xmath202 .
these data have been obtained on lattices with @xmath170 sites performing the cooling process on 1000 uncorrelated configurations for each @xmath20 . in figure 12
we show this set of data for the three kinds of lattices used and the result of the fits performed on it by imposing the scaling function eq .
( 5.1 ) with @xmath177 .
it is apparent that the data do not scale as eq .
( 5.1 ) imposes .
if @xmath173 is left free , we see again that the data do not scale as they should . indeed ,
for the regular lattice , @xmath84 and @xmath58 random lattices , we obtain @xmath203 respectively .
we repeated the same analysis on larger lattices to check whether this discrepancy is due to finite size effects on the data . in table 7
we show the data for a regular lattice with @xmath192 sites and in table 8 the data for a @xmath58 random lattice with @xmath182 sites .
they are also obtained by cooling 1000 uncorrelated configurations .
the fit on the data of table 7 gives @xmath204 and the fit on table 8 @xmath205 . from these figures
we could hardly conclude that the lack of correct scaling is due to finite size effects .
we think that this problem must be traced back to the elimination of small instantons during the cooling process . probably , the use of large @xmath18 random lattices is not enough to avoid this effect .
the lack of asymptotic scaling prevented us from using the topological susceptibility and the mass gap data to check the physical scaling of the @xmath131 model on random lattices . in figure 13
we show the ratio of @xmath11 parameters for each value of @xmath20 from the data of the topological susceptibility . the average result for this ratio is surprisingly similar to the one obtained from the mass gap data : @xmath206 and @xmath207 .
we have used the random lattice to simulate the 2-dimensional @xmath0 non - linear @xmath1-model .
the sites of the random lattice are considered as the centers of hard spheres of radius @xmath24 where @xmath5 is the lattice spacing .
these hard spheres are located at random on the lattice volume .
the links between neighbouring sites are established by a well known triangularization process @xcite . to compare the performance of different lattices
, we made the simulations on regular lattices and random lattices with both @xmath84 and @xmath58 .
we used the wolff algorithm @xcite for the simulations as well as an improved estimator @xcite for the computation of two - point correlation functions .
we measured the mass gap ( as the inverse of the correlation length measured from the wall - wall two - point correlation function ) and the topological susceptibility .
the topological charge was calculated by using a cooling technique @xcite and we introduced a regularized operator for the topological charge on the lattice ( see eq .
( 4.8 ) ) . the monte carlo results for the mass gap scale as they should , confirming previous claims @xcite about the universality of the random lattice regularization .
they do not present any finite size problems ( for @xmath170 sites ) but for small values of @xmath19 the asymptotic scaling is not fulfilled . for the @xmath191 model ,
where the data display a good asymptotic scaling , we can reproduce the theoretical value for the ratio between lambda parameters @xmath3 for @xmath58 , thus confirming the semi - analytical prediction of reference @xcite . instead , for @xmath84 , the monte carlo value for the previous ratio is in good agreement with the theoretical prediction already in the @xmath131 model .
this could mean that both regularizations ( regular lattice and @xmath84 random lattice ) are quite similar and in particular the non - universal terms in the scaling function , eq .
( 4.2 ) , almost coincide .
this assumption is also supported by the fact that @xmath3 for @xmath84 is close to 1 . in any case , these conclusions are consistent with the scenario where also the non - universal terms in the scaling function , eq . ( 4.2 ) are @xmath18-dependent . the data for the topological susceptibility scale very badly .
we think that the cooling process removes small instantons thus modifying the topological content of the configuration also on random lattices .
the cooling smooths out fluctuations with a length of order @xmath151 . for smaller @xmath20
the lattice spacing is longer in physical units , therefore the number of eliminated instantons when smoothing out @xmath151 fluctuations is also larger .
this explains why the data in figure 12 and tables 6,7 and 8 are shifted downwards for small @xmath20 . in any case
, our results prove that the semi - analytical method used in reference @xcite is reliable to perform analytical calculations on random lattices .
we thank federico farchioni and andrea pelissetto for useful discussions .
we also acknowledge financial support from infn .
99 z. qiu , h. c. ren , x. q. wang , z. x. yang and e. p. zhao , phys . lett .
* b198 * ( 1987 ) 521 ; * b203 * ( 1988 ) 292 .
work in progress .
b. alls , nucl . phys .
* b437 * ( 1995 ) 627 .
a. m. polyakov , phys . lett .
* b59 * ( 1975 ) 79 ; e. brzin and j. zinn - justin , phys . rev .
* b14 * ( 1976 ) 3110 .
a. a. belavin and a. m. polyakov , jetp letters * 22 * ( 1975 ) 245 .
p. hasenfratz , m. maggiore and f. niedermayer , phys .
b245 * ( 1990 ) 522 ; p. hasenfratz and f. niedermayer , phys . lett .
* b245 * ( 1990 ) 529 .
e. m. ilgenfritz , m. l. laursen , m. mller - preussker , g. schierholz and h. shiller , nucl . phys .
* b268 * ( 1986 ) 693 ; m. teper , phys . lett . * b171 * ( 1986 ) , 81 , 86 ; j. hoek , m. teper and j. waterhouse , phys . lett . * b180 * ( 1986 ) 112 ; nucl . phys .
* b288 * ( 1987 ) 589 .
m. campostrini , a. di giacomo , h. panagopoulos and e. vicari , nucl . phys .
* b329 * ( 1990 ) 683 .
u. wolff , phys .
b248 * ( 1990 ) 335 .
u. wolff , phys .
* 62 * ( 1989 ) 361 .
n. h. christ , r. friedberg and t. d. lee , nucl .
b202 * ( 1982 ) 89 .
r. friedberg and h. c. ren , nucl .
phys . * b235 [ fs11 ] * ( 1984 ) 310
. n. h. christ , r. friedberg and t. d. lee , nucl .
b210 * ( 1982 ) 337 .
u. wolff , nucl .
b334 * ( 1990 ) 581 . m. falcioni and a. treves , nucl . phys . * b265 [ fs15 ] * ( 1986 ) 671 .
g. parisi , phys .
b92 * ( 1980 ) 133 .
a. di giacomo , f. farchioni , a. papa and e. vicari , phys . rev .
* d46 * ( 1992 ) 4630 .
a. di giacomo , f. farchioni , a. papa and e. vicari , phys . lett .
* b276 * ( 1992 ) 148 .
f. farchioni and a. papa , nucl . phys .
* b431 * ( 1994 ) 686 .
m. lscher in `` progress in gauge field theory '' ( cargse 1983 ) , g. t hooft et al .
( eds . ) , plenum , new york ( 1984 ) ; i. bender and w. wetzel , nucl . phys .
* b269 * ( 1986 ) 389 . 1
. location of hard discs for a lattice of @xmath208 sites with @xmath37 .
the radius of the discs is equal to @xmath209 . in a )
it corresponds to a regular square lattice . in b )
a column has been shifted : @xmath18 still equals 1 but the lattice is no longer regular .
location of hard discs for a regular hexagonal lattice with 9 sites .
they fit tight and no movement is allowed without breaking the hard nature of the discs .
3 . the ratio between the number of proposed sites @xmath44 and total number of sites @xmath26 in the construction of a random lattice as a function of @xmath18 .
the solid and dashed lines are the results for lattices with @xmath57 and @xmath210 sites respectively .
the result of the triangularization process performed on two random lattices of @xmath211 sites for a ) @xmath58 and b ) @xmath212 . the different level of randomness is apparent .
distribution probability of distances between a site and its closest neighbour on a random lattice with @xmath58 .
the @xmath5 in abscisses stands for one lattice spacing .
the histogram is the numerical result calculated by using a single site on 30000 random lattices of 1000 sites .
this histogram should coincide with the poisson distribution , shown in the figure with a solid line .
both curves are normalized to 1 .
distribution probability of distances between a site and its closest neighbour on a random lattice with @xmath213 .
the @xmath5 in abscisses stands for one lattice spacing .
the histogram is the numerical result calculated by using a single site on 30000 random lattices of 1000 sites .
the curve is normalized to 1 .
distribution probability of link lengths on a random lattice calculated with @xmath58 .
the @xmath5 in abscisses stands for one lattice spacing .
the curve is normalized to 1 .
distribution probability of link lengths on a random lattice calculated with @xmath213 .
the @xmath5 in abscisses stands for one lattice spacing .
the curve is normalized to 1 .
evolution of the measured topological charge along with 50 cooling steps for 40 uncorrelated configurations .
notice the clustering towards integer values after a few coolings .
a random lattice with @xmath214 sites and @xmath58 was used at @xmath215 . 10 .
monte carlo data for the mass gap on a lattice with @xmath170 sites .
the lines are the results of the fits performed on the monte carlo data ( shown with circles , squares and diamonds ) . the solid line ( circles ) , dashed line ( squares ) and dot dashed line ( diamonds ) correspond to the regular lattice , @xmath84 random lattice and @xmath58 random lattice respectively . 11 .
ratio of @xmath11 parameters as obtained from the monte carlo data of the mass gap .
the solid line ( squares ) and dashed line ( circles ) correspond to @xmath84 and @xmath58 respectively .
the average ratios are @xmath216 and @xmath217 .
monte carlo data for the topological susceptibility on a lattice with @xmath170 sites .
the lines are the results of the fits performed on the monte carlo data ( shown with circles , squares and diamonds ) .
the solid line ( circles ) , dashed line ( squares ) and dot dashed line ( diamonds ) correspond to the regular lattice , @xmath84 random lattice and @xmath58 random lattice respectively
ratio of @xmath11 parameters as obtained from the monte carlo data of the topological susceptibility .
the solid line ( squares ) and dashed line ( circles ) correspond to @xmath84 and @xmath58 respectively .
the average ratios are @xmath218 and @xmath219 . 1
. average values for the distance between a site and its closest neighbour , @xmath89 and the link length @xmath88 in units of lattice spacing @xmath5 for several values of @xmath18 .
these numbers have been calculated numerically .
2 . monte carlo data for the mass gap , @xmath220 for regular and random lattices with @xmath170 sites for the @xmath131 @xmath1 model .
monte carlo data for the mass gap , @xmath220 for regular and random lattices with @xmath182 sites for the @xmath131 @xmath1 model .
4 . monte carlo data for the mass gap , @xmath220 for regular and random lattices with @xmath182 sites for the @xmath190 @xmath1 model .
monte carlo data for the mass gap , @xmath220 for regular and random lattices with @xmath192 sites for the @xmath191 @xmath1 model . 6 .
monte carlo data for the topological susceptibility , @xmath221 for regular and random lattices with @xmath170 sites for the @xmath131 @xmath1 model . 7 .
monte carlo data for the topological susceptibility , @xmath221 for a regular lattice with @xmath192 sites for the @xmath131 @xmath1 model
monte carlo data for the topological susceptibility , @xmath221 for a @xmath58 random lattice with @xmath182 sites for the @xmath131 @xmath1 model .
|
the @xmath0 non - linear @xmath1-model is simulated on 2-dimensional regular and random lattices .
we use two different levels of randomness in the construction of the random lattices and give a detailed explanation of the geometry of such lattices . in the simulations , we calculate the mass gap for @xmath2 and 8 , analysing the asymptotic scaling of the data and computing the ratio of lambda parameters @xmath3 .
these ratios are in agreement with previous semi - analytical calculations .
we also numerically calculate the topological susceptibility by using the cooling method .
|
gravitational waves , predicted to exist by einstein s general theory of relativity @xcite , are ripples in space-time propagating at light speed and produced by non - axially symmetric mass accelerations , by analogy with electric charges in any accelerated motion that emit electromagnetic waves ( travelling at light speed ) . however , gravitational waves are quite different from electromagnetic waves .
they are both transverse waves , but gravitational waves are characterized by two polarization states , denoted as `` + '' and `` @xmath0 '' , that differ by a rotation of @xmath1 degrees around the propagation axis , demonstrating the quadrupolar ( spin-2 ) nature of the gravitational radiation . on the contrary , the two polarization states of electromagnetic waves differ by a rotation of @xmath2 degrees , reflecting thus the dipolar ( spin-1 ) characteristics of the electromagnetic radiation .
the emission mechanisms are also quite different : gravitational waves result from the coherent emission from bulk motions of energy , while electromagnetic waves result from an incoherent superposition of waves from molecules , atoms and particles .
the amount of energy radiated as gravitational waves by any mechanical system constructed by man is so small that it will probably never be observed .
for this reason we hope to observe gravitational radiation emitted by sources at astrophysical distances . indeed ,
even though gravitational waves have not yet been directly detected by any detectors , a very strong indirect proof of their existence was given by the observation of the binary pulsar psr b1913 + 16 , discovered in 1974 by the astronomers r. hulse and j. taylor @xcite .
using the giant radio telescope at the arecibo observatory of puerto rico @xcite to search systematically for pulsars , they observed that a particular pulsar ( psr b1913 + 16 ) was changing its motion rapidly and that the variation in pulse rate was caused by the changing doppler effect .
the decrease of the orbital period of such a pulsar around its companion could only be explained if angular momentum and energy were carried away from this system by gravitational waves .
because of the extreme weakness of the interaction of gravitational radiation with matter , gravitational waves travel almost undisturbed from astrophysical sources to earth , without being scattered or absorbed by interstellar dust and debris , carrying thus astronomical information which electromagnetic waves do not carry .
an analysis of gravitational radiation would provide information of great value about the inaccessible and remote locations of the cosmos .
it would tell us something about the behaviour of space - time and matter under the most extreme conditions , and it would also provide a check on einstein s general theory of relativity .
the detection of gravitational waves will help us to understand the dynamics of large - scale events in the universe , like the death of whole stars , the explosion of quasars , the birth and the collisions of black holes ( bhs for short ; for more details see for instance ref .
@xcite and references therein ) .
nowadays laser interferometry is the basis of the most sensitive gravitational wave detectors , such as the ligo ( laser interferometer gravitational wave observatory ) and virgo detectors @xcite , whose current performances are concisely discussed in the next section .
the remainder of the paper is organized as follows . in sec .
[ gwd ] , [ gws ] and [ cws ] , respectively , we briefly overview the current gravitational wave detectors and their sources , with particular attention devoted to a short discussion of the continuous wave signal .
section [ recres ] is devoted to describe some of the main methodologies used in the searches for continuous gravitational waves , highlighting the most recent results obtained analyzing ligo and virgo data .
a summary is finally reported in sec .
[ conclusion ] .
the pioneer of gravitational wave detection was joseph weber in the early 1960s , who developed the first resonant mass detector and later also investigated laser interferometry @xcite .
from that date until today , the experiments that aim at the detection of gravitational radiation , planned in laboratories throughout the world , are in continuous progress @xcite . however , in this paper we consider only the efforts related to ground - based interferometric detectors .
the current world - wide network of gravitational wave detectors consists of the following michelson - type kilometer - scale laser interferometers : 1 . the french - italian virgo detector , at cascina ( pisa , italy ) , with 3 km arm length @xcite ; 2 .
the german - british geo experiment , near hannover ( germany ) , with an arm length of 600 m @xcite ; 3 . the japanese tama , located in tokyo ( japan ) , with 300 m arm length @xcite ; 4 . the american ligo project @xcite , that consists of three detectors working in unison ; one at livingston ( louisiana , usa ) , with an arm length of 4 km ( llo ) and two in the same vacuum container at hanford ( washington , usa ) , with an arm length of 4 km ( lho1 ) and 2 km ( lho2 ) , respectively . in 2007 ligo and virgo achieved their design sensitivities over a wide frequency range and today the performance of such interferometers is improved even more , as can be noticed in fig .
[ fig : allsens ] , where the noise spectra of such detectors is plotted versus the frequency .
ligo reached its design sensitivity with the fifth science run [ s5 in short , started on 2005 november 4 ( 14 ) at lho1 ( llo ) and ended on 2007 october 1 ] , but the sensitivity has continued to improve with time .
in fact , with respect to s5 , the sensitivity curve of a recently completed run , s6 ( started on 2009 july 7 and ended on 2010 october 20 ) , has a factor of 2 improvement above 300 hz , as depicted in fig .
[ fig : allsens ] . moreover ,
virgo 2009 sensitivity measurements show a much better sensitivity than ligo below @xmath3 hz ( see fig .
[ fig : allsens ] ) .
( color online ) . strain sensitivity curves of the present gravitational wave interferometric detectors . in 2007
ligo and virgo reached their goal sensitivities in a wide frequency interval.,width=453 ] the upgrade of ligo and virgo interferometers for their advanced stage is currently underway , with the goal of improving the current strain sensitivity by a factor ten , with a thousandfold increase in the observable volume of space . in 2015
such interferometers will be operational and will gradually improve their sensitivity . at their target sensitivity ,
several gravitational wave events per year should be detected , opening thus the gravitational wave astronomy era .
a future project for an interferometer of comparable sensitivity , lcgt ( large scale cryogenic gravitational wave telescope ) , is going to be built in japan @xcite .
detectors of third generation , such as einstein telescope , are currently in design phase @xcite .
moreover , space - borne detectors , such as lisa ( laser interferometer space antenna ) @xcite and decigo / bbo ( deci - hertz interferometer gravitational wave observatory and big bang observer ) @xcite , are designed to probe the 0.03 mhz to 0.1 hz regime , bringing thus relevant astrophysical information at low frequencies and complementing the ground - based detectors . in order to increase the baseline
, it would be quite convenient to have a detector far away and out of the plane of other detectors in the usa and europe .
this would have a tremendous scientific impact and several advantages , such as to improve the ability to identify exactly where gravitational wave signals come from .
the ligo team is currently investigating some possibilities towards this direction .
different types of gravitational wave sources are expected to be observed by ground - based detectors .
it is well - known that coalescence of compact objects constitutes an interesting source of high - frequency gravitational waves @xcite . in particular , the coalescence of ns - ns ( ns , neutron star ) , ns - bh , bh - bh binary systems are expected to emit gravitational radiation in the khz range @xcite .
such kind of sources are referred to as _ burst sources _ , such as supernovae explosions , whose signals last for a very short amount of time , between a few milli - seconds and a few minutes . a _ stochastic background _ of gravitational waves , either of cosmological or astrophysical origin , is also envisaged to exist .
this consists of a random accumulation of signals from thousands or millions of individual sources .
last , but not least , another class of gravitational wave sources is represented by rapidly rotating non - axisymmetric nss , that are predicted to emit continuously a weak sinusoidal signal . in the next sections we limit ourselves just to the treatment of such _ continuous wave signals_. the main mechanisms by which a ns can radiate gravitational waves consist of non - axisymmetric distortions in the solid part of the star ( i.e. the case treated here , where the signal frequency @xmath4 is twice the star rotation frequency @xmath5 ) @xcite , free precession of the ns ( @xmath6 ) @xcite and fluid _ r_-modes ( @xmath7 ) @xcite .
as already mentioned , continuous gravitational waves are expected to be produced by rapidly rotating nss with non - axisymmetric deformations @xcite .
the general form of a continuous gravitational wave signal is described by the following tensor metric perturbation : @xmath8 where @xmath9 and @xmath10 are the waveforms of the two orthogonal transverse polarizations , `` + '' and `` @xmath0 '' , respectively and are given by @xmath11 with @xmath12 representing the two basis polarization tensors @xcite ; @xmath13 is the time in the detector frame , @xmath14 is the inclination angle of the star s rotation axis with respect to the line of sight ; @xmath15 is the signal phase function and @xmath16 is the amplitude expressed by @xmath17 the constant @xmath18 is the gravitational constant ; @xmath19 represents the light speed ; @xmath20 is the star s principal moment of inertia ( assumed to be aligned with its spin axis ) , @xmath21 is the equatorial ellipticity of the star @xcite , @xmath22 is the distance to the star and @xmath4 represents the signal frequency . as the time - varying components of the mass quadrupole moment tensor are periodic with period half the star rotation period , the gravitational wave frequency @xmath4 is twice the rotation frequency @xmath5 .
the detector response to a metric perturbation is given by the known relation @xmath23 where @xmath24 and @xmath25 are the source right ascension and declination , respectively , @xmath26 is the polarization angle of the wave and @xmath27 are the detector antenna pattern functions for the two orthogonal polarizations @xcite . assuming that all of the frequency s derivative , also denoted with the term of spin - down , is due to emission of gravitational radiation , we can relate @xmath21 to @xmath5 and @xmath28 @xcite : @xmath29 where @xmath30 is the star s moment of inertia in units of the canonical value @xmath31 kg m@xmath32 and @xmath33 is the star s distance from the sun in kiloparsec ( kpc ) .
this is referred to as _ spin - down limit _ on the signal amplitude and represents an absolute upper limit to the amplitude of the gravitational wave signal that could be emitted by the star , where electromagnetic radiation is neglected .
the spin - down limit on strain corresponds to an upper limit on the star s ellipticity , given by @xcite @xmath34
the way to search for continuous wave signals depends on how much about the source is known .
there are different types of searches , briefly described in the text below and whose recent major results are also reported : 1 . _ targeted searches _ ,
where the source parameters ( sky location , frequency , frequency derivatives ) are assumed to be known with great accuracy ( e.g. the crab and vela pulsars ) ; 2 . _
directed searches _ , where sky location is known while frequency and frequency derivatives are unknown ( e.g. cassiopeia a , sn1987a , sco x-1 , galactic center , globular clusters ) ; 3 .
_ all - sky searches _ for unknown pulsars .
this kind of searches is computationally cheap and a fully coherent analysis , based on matched filtering over long observation time , is quite feasible @xcite . the minimum signal amplitude that can be detected over a given observation time @xmath35 , assuming a certain false alarm probability ( typically of 1 % ) and a false dismissal probability ( in general of 10 % ) is given by @xmath36 equation ( [ eq : homints ] ) is obtained by averaging over source and detector parameters and represents the sensitivity of a typical coherent search , with @xmath37 being the detector noise power spectral density .
note that the precise value of the coefficient on the r.h.s of eq .
( [ eq : homints ] ) depends on the analysis method employed ( see for example ref .
@xcite and references therein ) .
a search for continuous wave radiation from the vela pulsar has been quite recently performed using data from the virgo detector second science run ( started on 2009 july 7 and ended on 2010 january 8) @xcite .
the resulting upper limits on continuous gravitational wave emission have been obtained using methods that assume the gravitational wave emission to follow the radio timing .
assuming known orientation of the star s spin axis and value of the wave polarization angle , frequentist upper limits of @xmath38 and @xmath39 , respectively , have been placed on the gravitational wave amplitude with 95 % confidence level .
an independent method , under the same hypothesis , produces a bayesian upper limit of @xmath40 with 95 % degree of belief .
these upper limits are well below the indirect spin - down limit of @xmath41 for the vela pulsar , defined by the energy loss rate inferred from observed decrease in vela s spin frequency , and correspond to a limit on the star ellipticity of the order of @xmath42 . even assuming the star s spin axis inclination and the wave polarization angles unknown , the consequent results exhibit upper limits quite below the spin - down limit @xcite .
these recent results make vela only the second pulsar for which the spin - down limit on gravitational wave emission has been beaten .
the first pulsar for which this important result has been reached is the crab pulsar @xcite .
another search worthy to be mentioned is the directed search for continuous wave signals from the non - pulsating ns in the supernova remnant cassiopeia a over ligo data .
this search has established an upper limit on the signal amplitude over a wide range of frequencies which is below the indirect limit derived from energy conservation @xcite .
it is well - known that all - sky searches for gravitational waves from unknown pulsars over wide - parameter spaces are computationally limited .
the reason is that one needs to search for unknown sources located everywhere in the sky , with signal frequency as high as a few khz and with values of spin - down as large as possible .
long integration times , typically of the order of a few months or years , are needed to build up sufficient signal power .
the data analysis strategy used to extract the faint continuous wave signals from the interferometric noise data was derived in ref .
@xcite and is given by the standard coherent matched filtering method , that is based on the _ maximum likelihood detection_. the resulting optimal coherent search statistic is the so - called @xmath43-statistic .
fully coherent methods based on matched filtering are the approach used in analyses for continuous wave searches over wide parameter space .
however , they become computationally undoable when very long data stretches ( of the order of months or years ) are used and a wide fraction of the parameter space is searched over , because of the increasing number of templates @xcite .
therefore , different incoherent hierarchical methods have been proposed @xcite .
in the hierarchical strategies , the entire data set is split into different shorter fourier transformed data segments , which are then properly combined to account for doppler shifts and spin - down . in other words , at first , every data chunk is analyzed coherently via matched filtering and afterward the information from the different segments is combined incoherently ( that means that the phase information is lost ) .
three different methods have been developed that combine the results from the different segments incoherently , forming sums over power ( `` stack - slide '' @xcite and `` powerflux '' @xcite schemes ) or weighted binary counts ( `` hough transform '' @xcite ) . the sums are then weighted according to the detector noise and antenna - pattern to maximize the signal - to - noise ratio .
the hierarchical methods are computationally faster than the standard coherent methods and have a comparable sensitivity . in general , the whole data set , of duration @xmath35 , is partitioned into @xmath44 smaller segments of duration @xmath45 each .
given such @xmath44 data segments , the typical sensitivity of a continuous wave all - sky search is given by @xmath46 where the exact numerical factor depends on the specific hierarchical method employed ( see for instance ref .
@xcite and references therein ) .
the output of a standard continuous wave hierarchical analysis is given by a set of candidates , i.e. points in the source parameter space with high values of a given statistic and which need a deeper study .
typically coincidences are done among the candidates obtained by the analysis over different data segments in order to reduce the false alarm probability @xcite .
the surviving candidates can be then analyzed coherently over longer time baselines in order to discard them or confirm detection .
early ligo data from s5 have been analyzed using two different methods .
no gravitational waves could be claimed , but interesting upper limits have been placed .
a first search used the first eight months of s5 @xcite , covering the full sky , the frequency band ( 501100 ) hz and a range of spin - down values between @xmath47 hz / s and zero . at the highest frequency
the search would have been sensitive to the gravitational radiation emitted by a ns placed at 500 pc with equatorial ellipticity larger than @xmath48 .
another search was performed over the first two months of s5 using the einstein@home infrastructure @xcite . the analysis consisted of matched filtering over 30 hours - long data segments followed by incoherent combination of results via a concidence strategy . the analyzed parameter space consisted of the whole sky , the frequency interval ( 50 1500 )
hz and spin - down range between @xmath49 hz / s and zero .
this search would have been sensitive to 90 % of signals in the frequency band ( 125225 ) hz with amplitude greater than @xmath50 .
the search sensitivity was estimated through monte carlo methods ( injection of software simulated signals ) .
three new einstein@home searches , covering the full two - years period of the ligo s5 run , will be shortly published @xcite .
as already said , month - long coherent integration is necessary to accumulate a signal - to - noise ratio sufficient for detection .
however , a powerful and effective method that allows us to use the longest possible coherent integration time , and thus improve the search sensitivity , is represented by distributing the computation through the volunteer computing project einstein@home
such a project is built upon the boinc ( berkeley open infrastructure for network computing ) architecture @xcite , namely a system that exploits the idle time on volunteer computers to solve scientific problems that require large amounts of computer power , such as to process data from gravitational wave detectors . at present , it provides roughly 300 tflops of distributed computing resources . a current einstein@home hierarchical search , analyzing data from s6 and using a new technique based on the @xmath43-statistic global correlations @xcite , is expected to bring a relevant increase in terms of sensitivity .
this is due to the longer coherent time baseline ( @xmath45 = 60 hours ) used in such a search .
despite huge efforts on several fronts ( i.e. the improvement of the detector sensitivities and the employment of efficient gravitational wave data analysis algorithms ) , to date no direct gravitational wave detection has been made , but relevant upper limits on gravitational wave signal strength have been derived . moreover , with the advent of advanced ligo and virgo detectors @xcite , an amazing improvement in strain sensitivity , of a factor ten with respect to their initial configuration , is expected to be reached in a few years after 2015 . at this point , the era of gravitational wave astronomy will definitely begin and the possibility of a first direct gravitational wave detection will become much more concrete
. the employment of robust and hard - hitting gravitational wave data analysis techniques will be crucial at that time @xcite .
the authors gratefully acknowledge the support of the united states national science foundation for the construction and operation of the ligo laboratory , the science and technology facilities council of the united kingdom , the max - planck - society , and the state of niedersachsen / germany for support of the construction and operation of the geo600 detector , and the italian istituto nazionale di fisica nucleare and the french centre national de la recherche scientifique for the construction and operation of the virgo detector .
the authors also gratefully acknowledge the support of the research by these agencies and by the australian research council , the international science linkages program of the commonwealth of australia , the council of scientific and industrial research of india , the istituto nazionale di fisica nucleare of italy , the spanish ministerio de educacin y ciencia , the conselleria deconomia hisenda i innovaci of the govern de les illes balears , the foundation for fundamental research on matter supported by the netherlands organisation for scientific research , the polish ministry of science and higher education , the focus programme of foundation for polish science , the royal society , the scottish funding council , the scottish universities physics alliance , the national aeronautics and space administration , the carnegie trust , the leverhulme trust , the david and lucile packard foundation , the research corporation , and the alfred p. sloan foundation .
180 einstein a 1916 _ annalen der physik _ * 49 * 769 hulse r a and taylor j h 1975 _ astrophys . j. _ * 195 * 51 .
leaci p 2008 _ development of wideband acoustic gravitational wave detectors _ ( unpublished doctoral dissertation ) univeristy of trento ( italy ) abramovici a 1992 _ science _ * 256 * 325 barish b and weiss r 1999 _ phys . today _ * 52 * 44 bradaschia c 1990 _ nucl .
methods phys .
res . a _ * 289 * 518 weber j 1960 _ phys .
rev _ * 117 * 306 baggio l. 2005 _ phys .
lett _ * 94 * 241101 astone p 1997 _ astrop .
* 231 astone p 1993 _ phys . rev .
d _ * 47 * 362 mauceli e 1996 _ phys .
d _ * 54 * 1264 bonaldi m 2006 _ phys .
d _ * 74 * 022003 leaci p 2007 _ phys .
* 76 * 062101 leaci p 2008 _ phys .
d _ * 77 * 062001 leaci p 2008 _ class .
quantum grav . _ * 25 * 195018 danzmann k 1995 in _ first edoardo amaldi conference on gravitational wave experiments _ ed .
coccia e , pizzella g , ronga f ( world scientific , singapore ) tsubono k 1995 in _ gravitational wave experiments _ ed .
coccia e , pizzella g , ronga f ( world scientific , singapore ) kuroda k 2006 _ class .
quantum grav . _
* 23 * s215 punturo m 2010 _ class . quantum grav . _
* 27 * 194002 kawamura k 2008 _ j. phys . : conf .
ser . _ * 122 * 012006 thorne k 1987 _ 300 years of gravitation _ hawking s w and israel w , cambridge university press , cambridge , england thorne k 1995 _ gravitational waves in proceedings of the snowmass94 summer study on particle and nuclear astrophysics and cosmology _ eds .
kolb e w and peccei r ( world scientific , singapore ) cutler c 2002 _ phys .
* 66 * 084025 ushomirsky g 2000 _ mon . not .
r. astron .
soc . _ * 319 * 902 stairs i 2000 _ nature _ * 406 * 484 owen b j 1998 _ phys .
d _ * 58 * 084020 andersson n 1999 _ astrophys . j. _ * 516 * 307 bildsten l 1998 _ astrophys .
j. _ * 501 * l89 owen b j 2005 _ phys .
* 95 * 211101 misner c , thorne k , wheeler j a 1973 _ gravitation _ ( san francisco , ca : freeman ) jaranowski p 1998 _ phys . rev .
d _ * 58 * 063001 abadie j ( ligo and virgo scientific collaboration ) 2011 _ astrophys .
j. _ * 737 * 93 krishnan b 2004 _ phys . rev
. d _ * 70 * 082001 abbott b ( ligo scientific collaboration ) 2008 _ astrophys . j. _ * 683 * l45 abbott b ( ligo and virgo scientific collaboration ) 2010 _ astrophys .
j. _ * 713 * 671 abadie j ( ligo scientific collaboration ) 2010 _ astrophys .
j. _ * 722 * 1504 jaranowski p and krlak a 2000 _ phys .
d _ * 61 * 062001 brady p r and creighton t 2000 _ phys .
d _ * 61 * 082001 cutler c 2005 _ phys .
d _ * 72 * 042004 abbott b ( ligo scientific collaboration ) 2008 _ phys .
d _ * 77 * 022001 abbott b ( ligo scientific collaboration ) 2009 _ phys . rev .
* 102 * 111102 abbott b ( ligo scientific collaboration ) 2005 _ phys .
* 72 * 102004 krishnan b ( for the ligo scientific collaboration ) 2005 _ class .
quantum grav _ * 22 * s1265 abbott b ( ligo scientific collaboration ) 2005 _ phys
* 102004 abbott b ( ligo scientific collaboration ) 2009 _ phys . rev .
lett . _ * 80 * 042003 abbott b ( ligo scientific collaboration ) 2009 _ phys .
lett . _ * 102 * 11102 abadie j ( ligo and virgo scientific collaboration ) _ new search for periodic gravitational waves in ligo s5 data through einstein@home grid computing system _ ( in preparation ) pletsch h j 2008 _ phys .
* 78 * 102005 leaci p 2010 _ j. phys .
ser . _ * 228 * 012006 leaci p _ new methods to veto spurious candidates in continuous wave searches from gravitational wave detectors _ ( in preparation )
|
direct and unequivocal detection of gravitational waves represents a great challenge of contemporary physics and astrophysics .
a worldwide effort is currently operating towards this direction , building ever sensitive detectors , improving the modelling of gravitational wave sources and employing ever more sophisticated and powerful data analysis techniques . in this paper
we review the current status of ligo and virgo ground based interferometric detectors and some data analysis tools used in the continuous wave searches to extract the faint gravitational signals from the interferometric noise data .
moreover we discuss also relevant results from recent continuous wave searches .
( some figures in this article are in colour only in the electronic version )
|
rr lyrae , as well as classical cepheids , are considered standard candles for estimating stellar distances in the milky way and to local group galaxies .
they are produced in old stellar populations , and hence they provide important information for an understanding of the age , structure and formation of their parent stellar systems . as typical population ii stars ,
rr lyrae are abundant in globular clusters and have been the subject of a huge number of studies for more than a century .
although the pulsation theory explains quite well the connection between most of the involved physical quantities ( van albada & baker 1971 ; caputo , marconi & santolamazza 1998 ) , the dependence of intrinsic luminosity on metal content has been the subject of debate for nearly three decades ( see smith 1995 for a review ) .
only recently the slope of the luminosity - metallicity relation m@xmath2-[fe / h ] seems to be converging towards a value @xmath30.20 - 0.23 that appears to be `` universal '' as supported by the most accurate studies of field rr lyrae stars in the milky way ( fernley et al .
1998 ; chaboyer 1999 ) and in the large magellanic cloud ( lmc , gratton et al .
2004 ) , and globular clusters in m31 ( rich et al . 2005 ) . near infrared observations of variable stars present several advantages over optical investigations : a smaller dependence on interstellar extinction and metallicity , a smaller pulsational amplitude and more symmetrical light curves , and hence good mean magnitudes .
in the last 20 years the calibration of the rr lyrae period - ir luminosity relation ( pl@xmath0 ) has been the subject of several empirical and theoretical investigations ( see sect .
[ method ] ) , but there still remains some degree of uncertainty on the dependence of @xmath4 on period and metallicity . to solve the problem of the dependence of @xmath4(rr ) on these physical quantities , a large sample of rr lyrae stars is needed , spanning a wide range of [ fe / h ] , for which accurate k and [ fe / h ] measurements are available .
in this paper we present an accurate analysis of the infrared photometric properties of 538 rr lyrae variables ( 376 rrab and 162 rrc ) in 16 globular clusters ( gc ) with @xmath5<-0.9 $ ] .
this more than doubles the number of stars used in the previous largest study of this type ( nemec , linnell nemec & lutz 1994 ) . by means of such a large database
we calibrated the pl@xmath0 relation constraining its dependence on period and metallicity on a strictly observational basis . in 2
we describe the sample of rr lyrae stars and the used photometric ir datasets . 3
is devoted to the description of the adopted method to calibrate the pl@xmath0 relation . in
4 we use the derived pl@xmath0 relation to estimate the distance to the calibrator gcs and to a sample of field rr lyrae stars in the lmc , and compare the results with the previous determinations in literature . finally , we summarize our results in 5 .
the data used in the present analysis consist of k photometry for the central regions of 9 gcs ( m4 , m5 , m15 , m55 , m68 , m92 , m107 and @xmath6 cen ) derived from a set of images secured at the telescopio nazionale galileo ( tng , canary islands ) , using the near - ir cameras arnica , and at the european southern observatory ( eso , la silla ) , using the near - ir camera irac-2 and sofi . a detailed description of the data reduction and calibration procedure can be found in ferraro et al .
( 2000 ) , valenti et al .
( 2004 ) , valenti , ferraro & origlia ( 2004 ) and sollima et al .
all measured instrumental magnitudes were transformed into the two - micron all - sky survey ( 2mass ) photometric system . to extend our analysis to the outer regions of these clusters
, we correlated our catalogs with the database obtained by 2mass that extends to a wide area up to 15 from the cluster centers .
the 2mass k photometry for 5 additional gcs ( m22 , ngc 3201 , ngc 5897 , ngc 6362 and ngc 6584 ) was also considered .
for each cluster we identified a large number of variable stars by cross - correlating the ir catalog with the most comprehensive catalog of gc variable stars available in literature ( clement et al .
. since the adopted k magnitudes are the average of repeated exposures and the k light curves of rr lyrae variables have a fairly sinusoidal low amplitude shape , we assumed our k magnitudes as the mean magnitudes of the identified variables .
note that the individual k values from different datasets generally agree within less than 0.1 mag .
so , we assumed @xmath70.1 mag as a plausible error to attach to the mean k magnitudes used in the following analysis .
in addition , we considered the k photometry of rr lyrae stars taken by longmore et al .
( 1990 , for variables in the clusters m3 , m4 , m5 , m15 , m107 and ngc 3201 ) , storm et al .
( 2004 , ic4499 ) , butler ( 2003 , m3 ) , dallora et al .
( 2004 , reticulum ) and del principe et al .
( 2005 , m92 ) .
all magnitudes were reported to the homogeneous photometric system of 2mass using the transformation equations provided by carpenter ( 2001 ) .
table 1 lists for each calibrator gc the metallicity ( in the carretta & gratton 1997 scale ) and the reddening coefficient e(b - v ) from ferraro et al .
( 1999 ) together with the number of rr lyrae considered in the present analysis .
[ summ ] .the sample of calibrator gcs .
metallicities are in the cg scale . [ cols="<,^,^,^,>",options="header " , ]
from an accurate analysis of 538 rr lyrae variables in 16 gcs using infrared ( k - band ) photometry we derive a pl@xmath0 relation based on purely observational constraints .
the derived dependences of the k magnitude on period and metallicity are in good agreement with those estimated by previous empirical studies .
we confirm that the metallicity coefficient is about 2 - 3 times smaller than that predicted by theoretical models , as it was found in all previous empirical analyses .
the zero point of the calibration has been tied to the trigonometric parallax of rr lyr measured with hst by benedict et al .
this calibration has been used to derive the distances to the 16 calibrator gcs considered in this analysis . as a further check
, this relation has been applied to rr lyrae stars in a few central fields in the lmc yielding a distance modulus @xmath8 , in good agreement with the most recent determinations based on cepheid variables ( persson et al . , 2004 ; gieren et al . , 2005 ) .
this research was supported by the ministero dellistruzione , universit e ricerca .
we warmly thank paolo montegriffo for assistance during catalogs cross - correlation .
we also thank santino cassisi and the anonymous referee for their helpful comments and suggestions .
benedict g. f. et al . , 2002 , apj , 581 , 115 bono g. , caputo f. , castellani v. , marconi m. , 2001 , mnras , 326 , 1183 bono g. , caputo f. , castellani v. , marconi m. , storm j. , deglinnocenti s. , 2003 , mnras , 344 , 1097 borissova j. , minniti d. , rejkuba m. , alves d. , cook k. h. , freeman k. c. , 2004 , a&a , 423 , 97 butler d. j. , 2003 , a&a , 405 , 981 caputo f. , marconi m. , santolamazza p. , 1998 ,
mnras , 293 , 364 carney b. w. , storm j. , jones r. v , 1992 , apj , 386 , 663 carpenter j. m. , 2001 , aj , 121 , 2851 carretta e. , gratton r. g. , 1997 , a&as , 121 , 95 carretta e. , gratton r. g. , clementini g. , fusi pecci f. , 2000 , apj , 533 , 215 catelan m. , pritzl b. j. , smith h. , a. , 2004 , apj suppl .
ser . , 154 , 633 chaboyer b. , 1999 in `` post - hipparcos cosmic candles '' , kluwer ac .
, a. heck , f. caputo eds .
, 237 , 111 clement c. et al . , 2001 ,
aj , 122 , 2587 clementini g. , carretta e. , gratton r. , merighi r. , mould j. r. , mccarthy j. k. , 1995 , aj , 110 , 2319 clementini g. , gratton r. , bragaglia a. , carretta e. , di fabrizio l. , maio m. , 2003 , aj , 125 , 1309 dallora m. et al . , 2004 , apj , 610 , 269 del principe m. , piersimoni a. m. , bono g. , di paola a. , dolci m. , marconi m. , 2005 , aj , 109 , 2714 detre l. , szeidl b. , 1973 , ibvs , 764 fernley j. , skillen i. , burki g. , 1993 , a&as , 97 , 815 fernley j. , skillen i. , carney b. w. , cacciari c. , janes k. , 1998 , mnras , 293 , 61 ferraro f. r. , messineo m. , fusi pecci f. , de palo m. a. , straniero o. , chieffi a. , limongi m. , 1999 aj , 118 , 1738 ferraro f. r. , montegriffo p. , origlia l. , fusi pecci f. , 2000 , aj , 119 , 1282 frolov m. s. , samus n. n. , 1998 , astl , 24 , 171 gieren w. , storm j. b. , thomas g. iii , fouqu p. , pietrzyski g. , kienzle f. , 2005 , apj , 627 , 224 gratton r. g. , bragaglia a. , clementini g. , carretta e. , di fabrizio l. , maio m. , taribello e. , 2004 , a&a , 421 , 937 gratton r. g. , bragaglia a. , carretta e. , clementini g. , desidera s. , grundahl f. , lucatello s. , 2003 , a&a , 408 , 529 jones r. v. , carney b. w. , latham d. w. , 1988 , apj , 332 , 206 jones r. v. , carney b. w. , storm j. , latham d. w. , 1992 , apj , 386 , 646 jones r. v. , carney b. w. , fulbright j. p. , 1996 ,
pasp , 108 , 877 kazarovets e. v. , samus n. n. , durlevich o. v. , 2001 , inf . bull .
variable stars , 5135 , 1 liu t. , janes k. a. , 1990 , apj , 354 , 273 longmore a. j. , dixon r. , skillen i. , jameson r. f. , fernley j. a. , 1990 , mnras , 247 , 684 nemec j. m. , linnell nemec a. f. , lutz t. e. , 1994 , aj , 108 , 222 persson s. e. , madore b. f. , krzemiski w. , freedman w. l. , roth m. , murphy d. c. , 2004 , aj , 128 , 2239 rey s. c. , lee y. w. , joo j. m. , walker a. , baird s. , 2000 , aj , 119 , 1824 rich r. m. , corsi c. e. , cacciari c. , federici l. , fusi pecci f. , djorgovski s. g. , freedman w. l. , 2005 , aj , 129 , 2670 savage b. d. , mathis j. s. , 1979 , ara&a 17 , 73 skillen i. , fernley j. a. , stobie r. s. , jameson r. f. , 1993 , mnras , 265 , 301 skrutskie m. f. et al . , aj , 131 , 1163 smith h. a. , 1990 in `` rr lyrae stars '' , cambridge university press sollima a. , ferraro f. r. , origlia l. , pancino e. , bellazzini m. , 2004 , a&a , 420 , 173 storm j. , 2004 , a&a , 415 , 987 valenti e. , ferraro f. r. , perina s. , origlia l. , 2004 , a&a , 419 , 139 valenti e. , ferraro f. r. , origlia l. , 2004 , mnras , 351 , 1204 van albada t. s. , baker n. , 1971 , apj , 169 , 311 zinn r. , west m. j. , 1984 , apjs , 55 , 45
|
the period - metallicity - k band luminosity ( pl@xmath0 ) relation for rr lyrae stars in 15 galactic globular clusters and in the lmc globular cluster reticulum has been derived .
it is based on accurate near infrared ( k ) photometry combined with 2mass and other literature data . the pl@xmath0 relation has been calibrated and compared with the previous empirical and theoretical determinations in literature .
the zero point of the absolute calibration has been obtained from the k magnitude of rr lyr whose distance modulus has been measured via trigonometric parallax with hst . using this relation
we obtain a distance modulus to the lmc of @xmath1 mag , in good agreement with recent determinations based on the analysis of cepheid variable stars .
[ firstpage ] methods : observational techniques : photometric stars : distances stars : variables : rr lyrae infrared : stars
|
the study of ballistic electron transport in nano - devices has been an interesting field of research.@xcite recently , a si nanowire with a length comparable to the de broglie wavelength of carriers is realized by advanced nanofabrication technique.@xcite the cross - sectional area of si nanowires was designed to show well - separated transverse modes and electrons confined to the wire are expected to suffer from a minimal amount of impurity scattering .
these properties make the si nanowires good candidates for the study of ballistic quantum transport .
in addition , the potential distribution within the wire can be controllable by a metallic gate around the wire .
this provides additional degree - of freedom on currents through the device and one would expect that the basic transistor action is possible for a si nanowire . as a result ,
the gate - all - around si wire may shed the light on one - dimensional structures for future transistor applications .
it is desirable experimentally to make the si wires as intrinsic as possible .
however , to populate the wires with carriers , it is necessary to define source and drain regions where ionized dopants are placed .
these dopants scatter free carriers and the elastic impurity scattering can not be avoided in those regions .
thus , in order to understand transport in the wires , a quantitative treatment of the ionized impurity scattering will be important .
several theoretical works were done to investigate the effects of ionized - impurity scattering on one - dimensional electron gas , and revealed their effects on the electronic structure .
most of these studies were for uniformly doped or remote - impurity systems@xcite and adopted empirical models based on the so - called bttiker probes for simulating the device.@xcite the empirical methods are appealing due to relatively simple implementation but the methods often require parameters that need to be adjusted using more rigorous calculations or values from experiments . in this work ,
we take into account the ionized impurity scattering in simulating the gate - all - around nanowire using non - equilibrium green s function approach . by averaging the green s function over impurity configurations and expanding the arising term perturbatively
, we treat the impurity scattering within a self - consistent born approximation and apply the formula to the si nanowire as realized in ref . @xcite .
since the impurity - scattering strength is a single parameter for the system , the method provides the first - principle approach to understand current - voltage characteristics and compare them with the experimental results .
to see the effects of the impurity scattering clearly , we consider a simple geometry of a quantum wire as in fig .
[ figst ] .
an infinitely - long cylindrical si wire consists of intrinsic channel and heavily doped source and drain regions .
a metallic gate extended over a length of @xmath0 is rolled round the intrinsic region and they are separated from each other by a sio@xmath1 layer with a width @xmath2 . for simplicity , we assume that the si wire is grown along the crystal @xmath3$]-axis(chosen as the @xmath4 direction in the figure ) and the doping profile of @xmath5 in the source and drain regions is symmetric about the @xmath4-axis so that we can utilize the circular symmetry .
we plot a schematic diagram of a cylindrical si wire simulated in this work which is oriented along the [ 001 ] direction . the si wire surrounded with the gate
is assumed to be intrinsic and separate the source and drain regions where ionized dopants are distributed .
, scaledwidth=40.0% ] then , electrons in the si wire are governed by the effective - mass hamiltonian which is given by @xmath6 above hamiltonian describes electrons in six different valleys depending on their effective masses .
for instance , if @xmath7 , transverse mass , and @xmath8 , longitudinal mass of si , the hamiltonian represents electrons in the @xmath9$]-valley , etc . here
, @xmath10 is the macroscopic potential energy resulted from both band discontinuity among the materials , and the coulomb contribution from external charges .
the coulomb part is determined by the poisson s equation , @xmath11 when we know the electron distribution @xmath12 .
@xmath13 describes the impurity potential energy from the ionized dopants . in this work , we assume that the impurity potentials are short - ranged but still vary slowly in the atomic scale . as a result , different valley modes are not coupled by the impurity potential and can be solved independently . since the device has the circular symmetry , it is convenient to express the hamiltonian in terms of the basis diagonalizing the radial motion .
we choose the basis satisfying the following schrdinger equation , @xmath14\mid\ !
\chi_{l}\rangle \!=\!\epsilon_{l}\!\mid\ ! \chi_l\rangle\end{aligned}\ ] ] where @xmath15 is radial coordinates @xmath16 and @xmath17 is a potential energy at @xmath18 , i.e. , in the deep source and drain regions .
then , we expand the field operator @xmath19 as , @xmath20 where we discretize the longitudinal coordinates with a spacing of @xmath21 and @xmath22 is tight - binding basis at the @xmath23th node ( @xmath24 ) . using eq .
( [ basis ] ) and a finite difference approximation , one can express the hamiltonian of eq .
( [ h ] ) as , @xmath25\delta_{mm ' } \big ] \hat{b}_{l'm'}. \label{hp}\end{aligned}\ ] ] here , the first term describes motion along the longitudinal direction for each transverse mode and it s elements are given by , @xmath26\delta_{ll ' } \nonumber\ ] ] with @xmath27 and the hopping energy of @xmath28 ( hereafter , we use bold characters to denote a matrix displayed on the basis @xmath29 ) .
the @xmath30 matrix in eq . ( [ hp ] ) accounts for the deviated potential distribution from that of deep source and drain regions . as a result
, it gives rise to the hybridization among transverse modes as , @xmath31 the last term in eq .
( [ hp ] ) is a contribution from the impurity potential .
now we formulate non - equilibrium green s functions for the hamiltonian of eq .
( [ hp ] ) . in order to take into account the impurity scattering ,
we consider a number of impurity configurations rather than a particular distribution , and average the green s functions over the configurations .
for this we adopt the schwinger - keldysh technique.@xcite according to the scheme , the impurity average gives rise to the quadratic interaction in the action , and we expand it perturbatively to obtain the one - particle irreducible self - energy @xmath32 .
here , we restrict our attention to the first order diagram and treat it self - consistently , which is referred to as the self - consistent born approximation.@xcite the impurity - averaged green s function @xmath33 can be obtained through the dyson s equation , @xmath34 where @xmath35 is the impurity - free green s function ( in fact , the bold characters in this case represent enlarged matrices for taking into account the keldysh space .
however , we keep the notation in the meanwhile because it recovers an original size when we specify it s components explicitly in the keldysh space ) .
the corresponding self - energy from the impurity scattering depends on it s green s functions again through the relation , @xmath36 with @xmath37 here , @xmath38 denotes a configuration average .
we model fluctuating impurity potentials with a @xmath39-correlated function considering the short - ranged form ; @xmath40 here , @xmath41 is a normalized doping profile with respect to the atomic density @xmath42 of si . and
the impurity potential strength is expressed with the impurity potential amplitude of @xmath43 and a screening length @xmath44 , which is approximately equal to the tomas - fermi screening length in the bulk si at carrier density of @xmath45 .
accordingly , the expansion coefficient in eq .
( [ impsigma ] ) becomes , @xmath46 it is noted that the short - ranged potential is diagonal for longitudinal basis @xmath47 but not for transverse modes @xmath48 .
this means that transverse modes are mixed to each other through the impurity scattering . for a given @xmath32 , in order to solve the dyson equation of eq .
( [ dyson ] ) , we should take care of open - boundaries in our problem , i.e. , the infinite number of nodes along the longitudinal direction @xmath49 . for this
, we follow the conventional approach where the device is partitioned into the system being in non - equilibrium and reservoirs.@xcite since the source and drain regions are extended semi - infinitely , we confine our attention to the portion of the system near the gate where physical properties are thought to be deviated from those of deep source and drain regions .
we designate the portion by longitudinal indices @xmath50 .
thus , nodes for @xmath51 ( @xmath52 ) represent the source ( the drain ) being in equilibrium with the chemical potential @xmath53 ( @xmath54 ) . in the source and drain reservoirs ,
we assume that the self - energy @xmath32 is independent of longitudinal coordinates @xmath55 because they are sufficiently far from the gate region where the potential distribution is uneven . within this assumption ,
the schrdinger equation is easily solved and equilibrium green s functions @xmath56 with corresponding self - energies are calculated straightforwardly . in the appendix , we illustrate their simple expressions .
now , we focus on the device region , i.e. , nodes ranging @xmath57 where one expects a non - equilibrium situation for different chemical potentials of @xmath53 and @xmath54 .
the green s functions are obtained by truncating the matrix equation of eq .
( [ dyson ] ) within longitudinal indices of @xmath57 .
instead , the truncation introduces an additional self - energy @xmath58 to the dyson equation owing to the coupling of the source and drains , and a total self energy becomes @xmath59 .
here , the self - energy @xmath60 reads , @xmath61 \label{sigmalead}\end{aligned}\ ] ] where the subscripts of @xmath62 denote that each equilibrium green s function is determined by different chemical potentials of @xmath63 and @xmath64 accounting for applied voltages , @xmath65 and @xmath66 at each reservoir , respectively .
solutions of the dyson equation are obtained by inverting the matrix equation eq .
( [ dyson ] ) .
firstly , it s retarded component is calculated as , @xmath67^{-1}. \label{gr}\ ] ] here , @xmath68^{-1}$ ] is the free - particle green s function and @xmath69 is a retarded component of the self - energy .
detailed form of @xmath70 is given in the appendix .
whereas , the term of @xmath71 depends on diagonal components of it s own green s function , as indicated by eq .
( [ impsigma ] ) .
thus , we should solve the above matrix equation self - consistently . with the obtained @xmath72
and it s hermitian conjugate
@xmath73 , the keldysh components of the green s function and the self - energy become @xmath74 and @xmath75 respectively . according to eq .
( [ sigmalead ] ) , the self - energy contributed from the the source and drain coupling is obtained as , @xmath76 \label{sigmaleadk}\end{aligned}\ ] ] with @xmath77 , the correlated component of the self - energy . however , for the keldysh component of the impurity - induced self - energy @xmath78 the result is not given in a closed form and should be calculated self - consistently as in the case of the retarded one via eqs .
( [ impsigma ] ) and ( [ gk ] ) .
the ensemble average of @xmath79 gives local electron density of the device and , consequently , the electron density distribution in eq .
( [ poisson ] ) becomes @xmath80 .
> from the generating functional technique as in ref .
@xcite , one can express the local electron density with the calculated green s functions .
the result reads , @xmath81\nonumber\\ & = & { \rm tr}\int_{-\infty}^{\infty}de~ \big [ { \bf f}_{fd}(e ) { \bf d}(lm : e ) \big].\end{aligned}\ ] ] here , in the second line we use the functional form of fermi - dirac distribution @xmath82 and the density - of - state @xmath83 for the resemblance with equilibrium results .
since the device is in non - equilibrium condition , two functions are given in a matrix form ; the fermi - dirac distribution matrix is defined by , @xmath84 \label{ffd}\end{aligned}\ ] ] while , using eq .
( [ gk ] ) , the density - of - states matrix at the node @xmath55 and transverse mode @xmath85 , is expressed by , @xmath86 here , @xmath87 is the spin - valley degeneracy , @xmath88 , and @xmath89 is a matrix whose elements are non - zero only at the @xmath90-th diagonal position .
when the impurity scattering is absent , @xmath91 becomes the well - known results as in ref .
@xcite , where non - zero elements are only at @xmath92 and @xmath93 nodes and are equal to the fermi - dirac distribution characterized by @xmath53 and@xmath54 , respectively .
however , due to the impurity scattering of @xmath94 , elements of @xmath91 are deviated from the fermi - dirac distribution function in general .
currents flowing through the device is defined by time - derivatives of total charge at nodes @xmath95 or @xmath96 .
then , through the heisenberg equation of motion , one can find that the currents becomes , @xmath97\nonumber\\ & = & -\frac{e}{2\pi\hbar } { \rm tr } \int_{-\infty}^{\infty } de~ { \bf f}_{fd}(e ) { \bf t}_{m}(e ) \label{ids}\end{aligned}\ ] ] where , by @xmath92 or @xmath98 , the expression means currents at the source or the drain , respectively , and @xmath99 . in the second line of the above equation , we define the transmission matrix @xmath100 by , @xmath101 in the case of free impurities , this form also recovers the previous results.@xcite prior to numerical calculations , let us first look at the approximations used .
firstly , we consider a finite number @xmath102 of transverse modes .
then , the solution of eq .
( [ gr ] ) is obtained by inverting a @xmath103 matrix iteratively .
however , this scheme demands the huge computational cost because the matrix size is large and is deviated from the tridiagonal form due to off - diagonal elements of the self energy @xmath104 and the hamiltonian @xmath30 . as an approximation
, we consider leading terms in green s functions to emphasize mainly the effects of the impurity scattering .
this is equivalent to consider the diagonal components of the green s functions for transverse modes .
namely , the coupling of different transverse modes in the self energy @xmath104 and the hamiltonian matrix @xmath30 are neglected . as indicated in ref .
@xcite , if the potential energy @xmath10 is a slowly - varying function along the radial direction at any node @xmath55 the hamiltonian matrix @xmath30 becomes small and the approximation is well justified . as for the self - energy , leading terms in the green s functions are obtained by writing overlap functions of eq .
( [ overlap ] ) as , @xmath105 and , therefore , the self - energy of eq .
( [ impsigma ] ) becomes diagonal for transverse modes .
however , the approximation of eq .
( [ impapp ] ) still couples transverse modes non - trivially because each diagonal component of the self - energy depends on others .
another approximation is made in the keldysh component of the impurity self - energy @xmath106 .
after various numerical calculations , we find that @xmath106 is well represented by ; @xmath107 where a node @xmath108 is the middle point in the intrinsic si wire .
this indicates that particles at the nodes near the source(drain ) have still the chemical potential @xmath53 ( @xmath54 ) , not an intermediate value between @xmath53 and @xmath54 , even after suffering from scattering .
we attribute this result to a particular potential distribution in the device of a source - to - channel barrier , which prevents particles with different chemical potentials from mixing .
in this section , we numerically illustrate solutions of the non - equilibrium green s functions suffering ionized impurity scattering and related transport properties . we consider a typical case of the device structure which can be realized experimentally . as shown in fig .
[ figst ] , the source and drain regions are doped at @xmath109 and there is no gate - to - source and -drain overlaps to constitute nearly abrupt junctions with the intrinsic channel . the source and drain extensions are @xmath110 nm and the gate length @xmath0 is @xmath111 nm , so that a total device length simulated is @xmath112 nm . by choosing a node spacing of @xmath113 nm , we have the number of @xmath114 nodes along the wire . in order to highlight quantum effects
, we choose a small radius ( @xmath115 nm ) of the wire which exhibits @xmath115 mode occupancies at a zero temperature .
however , to include thermally excited particles as well as the mode coupling from the impurity scattering , @xmath111 transverse states are incorporated .
the gate oxide layer has a thickness of @xmath116 nm and is treated as an infinite potential barrier for electrons . due to this
, wavefunctions at the interface between the si wire and the oxide are assumed to be zero in all of our simulation .
the poisson s equation is solved in the cylindrical coordinates with dirchlet boundary conditions at the gate - oxide interface , otherwise , with neumann conditions . for a rapid convergence of solutions , we use the newton - rhapson method for the gummel form of external charges.@xcite to model a gate material , we choose a work function of @xmath117 , approximately for tin . for a cylindrical si wire with a 20 nm gate length
, we plot local particle density @xmath118 along the wire for impurity scattering strengths of @xmath119 and @xmath120 in @xmath121 and @xmath122 , respectively , at @xmath123v , @xmath124v , @xmath125 , and @xmath126k
. for qualitative comparison , we display higher density with a darker color . dotted lines describes the effective potential energy of each subband before they are renormalized by impurity scattering .
, scaledwidth=40.0% ] in fig .
[ fig2 ] , we show calculated electronic subbands of each level and local particle density along the wire , and compare the results with and without the impurity scattering in @xmath121 and @xmath122 , respectively(@xmath127 and @xmath128 ) . the subband bottoms(dotted lines ) reflect the calculated self - consistent potentials in which electrons at each levels feel at a node @xmath55 .
regardless of the impurity scattering , they exhibit source - channel barriers .
since a high gate voltage lowers the energy barriers , the basic transistor action is achieved by controlling these barriers.@xcite the energy - resolved particle density is plotted in a gray scale ; a darker area in the figure represents higher density . in the impurity - free case of @xmath121 ,
since there is no momentum relaxation , states injected from the drain(source ) end of the device undergo reflections and interfere strongly to the right(left ) of the source - to - channel barrier .
this interference results in coherent oscillations in the particle density as seen in fig .
[ fig2]-(a ) . as a function of energy
, it is found that the local particle density far from the source - channel barrier shows sharp peaks like @xmath129 at every onset of subbands , reminiscence of one - dimensional density of states .
if one turns on impurity scattering , phase information of the electrons within the device is randomized and the energy levels are renormalized .
above all , this makes the interference oscillations washed out in the local particle density as shown in fig . [ fig2]-(b ) .
in addition , electronic states are shifted and broaden , so that the most electrons are found below subband bottoms and it s occupation has no longer @xmath129-dependence , but a monotonically varying function(the abrupt change of darkness along the energy direction comes from a different valley state ) . in both cases of the impurity scattering , one can see that electrons in the source and drain regions are well separated by the source - channel barriers from each other . due to this , the approximation of eq .
( [ sigmakapp ] ) is justified with good accuracy .
we compare calculated @xmath130-@xmath131 results at temperatures of @xmath132k(solid ) , @xmath114k(dotted ) , @xmath133k(dashed ) , and @xmath112k(dot - dashed lines ) , respectively , for impurity scattering strengths of @xmath119 in @xmath121 and @xmath134 in @xmath122 . here
, we assume a small source - drain bias of @xmath135 .
, scaledwidth=40.0% ] in order to examine the electronic transport of the device , we calculate channel currents @xmath130 versus a gate voltage @xmath136 at a small source - drain bias , and plot results in fig .
[ figidvg]-(a ) and ( b ) , respectively , with and without impurity scattering for several temperatures . under this condition , currents exhibit rapidly increasing behavior as a gate voltage becomes larger .
this shows the basic operation of a transistor as indicated in the previous section ; the channel current turns on by lowing the source - channel barrier when a gate voltage is higher than a certain value , called a threshold voltage @xmath137 . by comparing figs . [ figidvg]-(a ) and ( b ) at a given temperature , one can find that the presence of impurities reduces the currents significantly even though electrons in both cases are expected to move ballistically in the intrinsic gate region .
this indicates that transport through the si wire largely depends on the electronic structure of the source and drain regions .
as inspired by flat subbands in the figures , the potential drops across the intrinsic regions are nearly invariant to the impurity scattering strength .
thus , it is reasonable to assume that the suppressed currents do not come from the fermi - dirac matrix of eq .
( [ ids ] ) which crucially depends on the potential drop , but mainly from a reduced transmission coefficient of eq .
( [ tcoeff ] ) .
one of possible explanations for this is that electrons injected from the source are partially reflected from impurities in the source extension in addition to that from the source - channel barriers and , thus electrons tunnel the source - channel barrier at rare intervals .
this type of the reduction for the transmission coefficient is also encountered in problems of tunneling in dissipative environments.@xcite according to the theories , when environments of the device become more dissipative , carriers are harder to tunnel the barriers because more energies should be transferred to the environment . as a function of a temperature ,
curves are shifted with wholly similar shapes and slightly different slopes in both cases of the impurity scattering .
two points are noteworthy .
firstly , the threshold voltage is shifted to a higher value as a temperature is lowered .
this is easily understood because as the temperature decreases , available electrons to overcome source - to - channel barrier thermally are reduced and then more potential energy should be supplied electrostatically to turn on currents .
secondly , we look at the slopes of the @xmath130-@xmath136 curves . in conventional mosfets
, they are related to a channel mobility @xmath138 via a relation of @xmath139 . as seen in the figures ,
our results show linear behavior in some range of gate voltages .
therefore , we may understand the slopes to be proportional to the mobility of electrons in the device . for detailed comparison
, we define the conductance by @xmath140 known as the transconductance in mosfets . in @xmath121 ,
we plot calculated conductances(symbols ) as a function of impurity scattering potential at two different temperatures of @xmath132k and @xmath133k , respectively(@xmath141 . in @xmath122 , calculated conductances are plotted as a function of temperature for given impurity scattering potential of @xmath119(circles ) , @xmath142(crosses ) , and @xmath134(triangles ) , respectively .
solid lines are just guide to the eye . to emphasize their temperature dependence
we normalize them with values at @xmath132k and superimpose the lines of @xmath143(dashed ) and @xmath144(dotted ) .
, scaledwidth=40.0% ] calculated conductance is summarized in fig .
[ figmob ] as functions of impurity scattering strength and temperature . in fig .
[ figmob]-(a ) we compare the conductance with increasing impurity scattering strength for two temperatures .
it is noted that the conductance decreases monotonically when the impurity scattering strength becomes larger at both temperatures and , consequently , suppressed mobilities are expected . in fig .
[ figmob]-(b ) we plot the temperature dependence of the conductance for various impurity scattering strengths . for a bulk material
, it is well known that the mobility resulted from impurity scattering is proportional to @xmath145 to the first order(dotted line in the figure).@xcite in the case of a two - dimensional system , the ionized impurity scattering ( for instance , in a quantum well with a @xmath39-doping ) is enhanced due to the increased overlap of the ionized impurity with electron wavefunctions and the mobility decreases nearly exponentially when a temperature is lowered(dashed line).@xcite in our case of a quasi - one - dimensional system , the conductance shows different temperature dependences from those of higher - dimensional ones ; the conductance of the si wire interpolates from linearly increasing behavior of the impurity - free case to the exponentially decaying dependence of a strong impurity scattering as a function of scattering strength .
curves shown in fig .
[ figmob]-(b ) do not provide a definitive comparison of ionized - impurity scattering among three different dimensional systems because each system has different doping profiles and concentrations . despite of this
, it is interesting to note that the ionized impurity scattering becomes less temperature - dependence when the system has a lower dimension .
in summary , we study transport through a gate - all - around si wire in the ballistic regime by considering the ionized impurity scattering . using the schwinger - keldysh approach
, we include the impurity scattering within the self - consistent born approximation and present expressions for electron density and currents in terms of non - equilibrium green s functions and self - energies . by simulating a typical case of a si wire ,
we compare electron densities and channel currents for zero- and strong - impurity scattering strengths . in the case of the strong impurity scattering , we find that the local particle density profiles are shifted and broaden to result in suppressed currents compared to the zero - impurity scattering case , and the oscillating interference pattern vanishes
. calculated currents and conductances are also presented as functions of temperature and the impurity scattering strength .
it is found that the conductance of a si wire exhibits various behavior by decreasing temperature , which interpolate from a linear increasing function at a zero scattering to an exponentially decreasing function for the strong scattering case .
however , in this work , we do not include other inelastic scattering process such as acoustic and optical phonon scattering which will be occurred in real devices .
therefore , our results show the effects of the ionized impurity scattering alone on electronic transport through a si wire . 25 t. j. thornton , m. pepper , h. ahmed , d. andrews , and g. j. davies , phys .
lett . * 56 * , 1198 ( 1986 ) .
k. k. choi , d. c. tsui , and s. c. palmateer , phys .
b * 32 * , 5635 ( 1985 ) .
b. j. van wees , h. van houten , c. w. j. beenakker , and j. g. williamson , l. p. kouwenhoven , d. van der marel , and c. t. foxon , phys . rev .
lett . * 60 * , 848 ( 1988 ) .
l. dicarlo , y. zhang , d. t. mcclure , d. j. reilly , c. m. marcus , l. n. pfeiffer , and k. w. west , phys .
lett . * 97 * , 036810 ( 2006 ) .
k. nishiguchi and s. oda , appl .
lett . * 76 * , 2922 ( 2000 ) .
k. h. cho , y. c. jung , b. h. hong , s. w. hwang , j. h. oh , d. ahn , s. d. suk , k. h. yeo , d. kim , d. park , and w. s. lee , appl . phys .
lett . * 90 * , 182102 ( 2007 ) ; k. h. cho , s. d. suk , y. y. yeoh , m. li , k. h. yeo , d. kim , s. w. hwang , d. park , and b. ryu , tech .
electron devices meet .
, * 543 * ( 2006 ) . j. masek , z. phys . b * 64 * , 145 ( 1986 ) . b. y. hu and s. das sarma , phys . rev .
b * 48 * , 14388 ( 1993 ) ; h. sakaki , jpn . j. appl .
* 19 * , l735 ( 1980 ) .
r. venugopal , m. paulsson , s. goasguen , s. datta , and m. s. lundstrom , j. appl .
phys . * 93 * , 5613 ( 2003 ) .
j. wang , e. polizzi , and m. lundstrom , j. appl .
phys . * 96 * , 2192 ( 2004 ) .
a. kamenev and a. andreev , phys .
b * 60 * , 2218 ( 1999 ) . h. e. camblong and p. m. levy , phys . rev .
lett . * 69 * , 2835 ( 1992 ) .
r. lake , g. klimeck , r. c. bowen , and d. jovanovic , j. appl .
* 81 * , 7845 ( 1996 ) .
s. datta , in _ electronic transport in mesoscopic systems _
( cambridge university press , cambridge , 1997 ) . j. h. oh , d. ahn , and s. w. hwang , phys .
rev . b * 72 * , 165348 ( 2005 ) .
h. k. gummel , ieee trans . electron dev . * 11 * , 455 ( 1964 ) ; m. shin ( unpublished ) .
e. o. johnson , rca rev .
, * 34 * , 80 ( 1973 ) .
j. h. oh , d. ahn , and s. w. hwang , phys .
b * 68 * , 205403 ( 2003 ) .
g. ingold and y. v. nazarov , in _ single charge tunneling _ , edited by h. grabert and m. h. devoret ( plenum press , new york , 1992 ) .
r. a. smith , in _ semiconductors _ ( cambridge university press , cambridge , 1978 ) .
w. t. masselink , phys .
lett . * 66 * , 1513 ( 1991 ) .
in this appendix , we illustrate green s functions for an infinitely long si wire which is doped uniformly .
eigenstates are plain waves whose wavelength is determined by periodic boundary conditions . since the wire is translational invariant , the self - energy in eq .
( [ impsigma ] ) is independent of a longitudinal position .
then , the retarded component of the green s function can be derived as , @xmath146 where @xmath147 is the number of a longitudinal node and @xmath148 $ ] with an eigenenergy @xmath149 of the @xmath150th transverse mode . in the limit of a large @xmath147 , diagonal components of the green s function reads ; @xmath151 with @xmath152 and , according to eqs .
( [ impsigma ] ) and ( [ impapp ] ) , the self - energy is proportional to the diagonal component of the green s functions like @xmath153 thus , the green s functions are obtained by solving eqs .
( [ agreen ] ) and ( [ aimpapp ] ) self - consistently . on the other hand
, the chemical potential @xmath154 of the uniformly doped wire can be found from the particle density of @xmath155 together with the poisson s equation . the self - energy of eq .
( [ sigmalead ] ) caused by the coupling of the device to the source and drain regions is obtained by solving the uniformly doped wire with vanishing boundary conditions . in the similar way to eq .
( [ agreen ] ) , it is given by , @xmath156 \nonumber \\ & & \left[1-\sqrt{1-\frac{1}{[2y_l(e)-1]^2}}~~ \right].\end{aligned}\ ] ]
|
we investigate transport properties of gate - all - around si nanowires using non - equilibrium green s function technique . by taking into account of the ionized impurity scattering we calculate green s functions self - consistently and
examine the effects of ionized impurity scattering on electron densities and currents .
for nano - scale si wires , it is found that , due to the impurity scattering , the local density of state profiles loose it s interference oscillations as well as is broaden and shifted . in addition , the impurity scattering gives rise to a different transconductance as functions of temperature and impurity scattering strength when compared with the transconductance without impurity scattering .
|
exciton - polaritons are matter - light quasiparticles that arise from the coupling between excitons and photon modes in a semiconductor microcavity and can form bose - einstein condensates ( bec ) at relatively high temperatures @xcite .
polariton condensates are sustained by laser pumping of photons in a two - dimensional quantum well . in a mean - field approximation ,
their wavefunctions produce a rich variety of localised quantum states in the micrometer scale : dark solitons @xcite , bright solitons @xcite , vortices @xcite .
solitons in polaritonic condensates have potential for applications in ultrafast information processing @xcite due to picosecond response times and strong nonlinearities @xcite . in this work ,
we report a frequency band of dark polariton solitons whose exciton wave function develops a discontinuity as the frequency is increased beyond the exciton frequency ( fig .
[ fig : solitons ] ) . at the point of discontinuity ,
the photon field vanishes while the exciton field experiences a half - cycle phase jump .
we investigate a one - dimensional condensate of polaritons in a strongly - coupled exciton and photon system .
our derivation depends crucially on the use of the classic model that retains separate wave functions for the excitons and the photon modes .
exciton interactions are modelled by a nonlinear term , while photons are dispersive . neglecting both pumping and losses ( which are due to radiation and thermalization ) and thus focusing on the synergy of exciton interaction ( nonlinearity ) and photon dispersion allows us to produce analytical formulae for polariton solitons .
conservative solitonic structures of half - light and half - matter have been considered in the literature @xcite .
the solitons we derive apply for a short time after the pumping is removed and the losses have not seriously manifested themselves or the solitons lie outside the pump spot . for example , in refs .
@xcite quasi - one - dimensional structures are observed outside the pump spots . in a different realization ,
polariton condensates can be created at two pump spots @xcite and localised structures can be sustained in the region between the two spots where there is no pumping .
we consider a one - dimensional semiconductor microcavity in which a photon field @xmath0 interacts with an exciton field @xmath1 .
one dimensional or nearly one dimensional polariton structures have been observed in @xcite and @xcite ( radial fields ) .
the pair @xmath2 is a polariton field and is modeled by the system @xcite [ eq : polariton ] @xmath3 the coupling constant is half the rabi frequency @xmath4 ; @xmath5 is the frequency of a free exciton , @xmath6 is the photon frequency at zero wavenumber ; and @xmath7 and @xmath8 are the exciton and photon attenuation rates .
all these are normalized to a reference frequency @xmath9 .
one could set @xmath10 , however , we prefer to keep @xmath11 as an explicit parameter .
the spatial variable @xmath12 is normalized to @xmath13 , where @xmath14 is the effective photon mass .
the wavefunctions @xmath15 are normalised to @xmath16 , where @xmath17 is a reference number of particles .
the nonlinearity parameter @xmath18 is normalised to @xmath19 .
we consider only the case @xmath20 in this paper .
are valid outside the pump spots , at regions that have been the focus of interesting experimental observations @xcite .
we seek stationary harmonic polariton fields @xmath21 for the lossless equations ( @xmath22 ) , with operating frequency @xmath23 .
this assumption is reasonable since experimental results @xcite show that the rate of attenuation is slow enough to allow for the formation of solitons .
letting @xmath24 and inserting ( [ eq : solitonform ] ) into ( [ eq : polariton ] ) yields [ eq : solitonenvelope ] @xmath25 + multiplying eq . by @xmath26 and eq . by @xmath27 and
adding the two integrates the system ( [ eq : solitonenvelope ] ) exactly . the cubic algebraic relation ( [ eq : solitonenvelopeb ] ) allows one to eliminate @xmath28 in favor of @xmath29 to obtain a first - order ode for @xmath30 .
it is then convenient to use the scaled exciton density @xmath31 which eliminates @xmath18 from the equation and results in @xmath32 where @xmath33 $ ] , @xmath34 is an arbitrary real constant of integration , and @xmath35 corresponds to the nonzero equilibrium solution of ( [ eq : solitonenvelope ] ) .
eq . has the structure of an energy equation of a conservative system and admits a rich set of solitons and periodic structures . in this work ,
we focus on continuous and discontinuous dark solitons for @xmath20 .
for a dark soliton @xmath36 to exist , the cubic polynomial @xmath37 must have a double root that serves as the soliton s far - field value .
the value of the constant of integration @xmath34 that provides such a nonzero double root equals @xmath38 where @xmath39 is a convenient dimensionless parameter @xmath40 we calculate the double root to be equal to @xmath41 , given in . the fact that this is also the value of the far - field justifies the notation .
as @xmath12 is varied , @xmath36 varies continuously down to its minimal value ( nadir ) @xmath42 , which is a simple root of the potential in .
we may assume that the nadir occurs at @xmath43 .
the soliton field @xmath44 traces the graph of the cubic relation ( [ eq : solitonenvelopeb ] ) as @xmath12 increases .
[ fig : cubic ] shows the graph of this relation for the two cases @xmath45 and @xmath46 .
the equilibrium points @xmath47 and @xmath48 correspond to the calculated value @xmath41 .
the parameter @xmath39 is convenient for expressing the soliton _ nonlinear dispersion relation _ at zero wavenumber , that relates the soliton amplitude @xmath41 to the operating frequency @xmath23 , which is encapsulated in @xmath39 and @xmath49 , @xmath50 we restrict our attention to @xmath51 , which also implies @xmath52 , given the fact that @xmath53 . under these conditions
, one can show that @xmath54 , a necessary condition for eq . to have real solutions .
a dark soliton appears at @xmath55 ( @xmath56 ) corresponding to a _ threshold frequency _ @xmath57 .
this constitutes the linear limit of the soliton that emerges as the frequency increases ; it is thus no surprise that the frequency @xmath57 coincides with the lower endpoint of the well - known lower band @xmath58 of homogeneous linear @xmath59 polaritons of the form @xmath60 , with @xmath29 and @xmath28 constant @xcite . as the frequency
is increased from its threshold , the value of @xmath39 decreases and the amplitude of the soliton increases until it blows up at the photon frequency @xmath61 ( @xmath62 , @xmath63 ) .
[ fig : bands ] displays the far - field and nadir values of the soliton _
vs._the frequency in the band from threshold to blowup , in the cases of negative detuning and positive detuning . in the case of positive detuning , @xmath64 , ( _ i.e. _ @xmath65 )
, the frequency @xmath5 lies within the soliton frequency band @xmath66 , constituting a _ transition frequency _ above which the soliton field @xmath29 becomes discontinuous .
the obstructing singularity @xmath67 becomes positive , breaking into the soliton range @xmath68 .
the nadir of the soliton is pushed upward from @xmath42 to the value @xmath69 , which is now positive , leading to a jump of the exciton field between the values @xmath70 .
[ fig : cubic ] traces the path of the pair @xmath44 along the graph of the relation ( [ eq : solitonenvelopeb ] ) both for negative detuning and positive detuning .
the system equations remain valid , as the jump in @xmath29 is balanced by a jump in @xmath71 .
physically , the photon field @xmath28 which mediates the coupling between neighboring excitons through the term @xmath72 in ( [ eq : polaritona ] ) , vanishes when @xmath73 takes the special value @xmath74 ( corresponding to @xmath75 in fig .
[ fig : cubic]b ) .
the vanishing of the photon field turns off the coupling between neighboring excitons thus making the jump permissible .
the formulae and for the far - field value @xmath41 remain the same . fig .
[ fig : solitons ] presents four instances of the soliton profile that show the progress towards the discontinuity ( top ) and the progress past the discontinuity of the exciton field ( bottom ) .
the photon field remains continuous .
its second derivative has a discontinuity at @xmath43 , as discussed earlier , but this is not visible in the figure .
notice the monotonic increase of the far - field amplitude as the frequency @xmath23 increases .
it is interesting to visualize the mechanism of the formation of the discontinuity of the exciton field @xmath30 by following the slope of this field at @xmath43 , as one lowers the dimensionless parameter @xmath39 from its value @xmath55 at which the dark soliton is born . in order to calculate this slope
, we express @xmath76 in terms of @xmath73 and @xmath77 from the relation @xmath78 .
we then insert the value for @xmath77 from the differential equation and , finally , set @xmath42 .
we obtain @xmath79 ^ 2 = \frac{\gamma^2 ( \eta-1)^2 } { g\,\eta^2}.\ ] ] for positive detuning and as @xmath80 , the parameter @xmath81 and thus , the slope @xmath82 tends to infinity , while @xmath29 remains finite .
the jump discontinuity of the exciton envelope profile sets on as @xmath39 becomes negative . adopting the slope of the profile at the origin @xmath43 as an indicator of the scale of the slope of the profile
we define the _ healing length _ of a exciton field profile by @xmath83 with a similar equation for the photon field . from the field envelope eq . , and the far - field eq .
, we obtain @xmath84 and @xmath85 . thus , the healing lengths @xmath86 and @xmath87 are related by @xmath88 combining eqs . , and @xmath89 , we obtain for the continuous soliton the healing lengths @xmath90 when @xmath91 , near the blow - up frequency @xmath92 the healing length of the excitons approaches zero , while the photon healing length diverges to infinity . at the same time the far - field value goes to infinity . at the transition frequency @xmath93 (
obtained only for positive detuning ) @xmath87 goes to zero linearly in @xmath39 which one can view as a precursor to the discontinuity .
the photon healing length converges to @xmath94 .
[ fig : solitons ] exemplifies these observations . in the region near the value @xmath55 ,
at which the continuous soliton begins its life , the exciton and the photon fields are nearly proportional to each other and @xmath95 .
the photon field is described well by a gross - pitaevskii ( gp ) model that is derived as a simplification of the two - equation model .
we solve eq . for @xmath96 as a power series in @xmath28 up to the third degree term and we insert this value of @xmath96 into eq . .
there seems to be no analogous way to derive a gp equation for the exciton field .
the gp model derived for the photon field is @xmath97 the notation @xmath98 is a convenient abbreviation of the more descriptive notation @xmath99 .
the parameter @xmath100 measures the deviation from the linear problem and equals @xmath101 while @xmath102 .
multiply by @xmath103 and integrate to obtain @xmath104 where @xmath105 is a constant of integration .
like eq .
, this has the structure of a conservative system .
the left side can be considered as the sum of a kinetic and a potential energy .
it produces the gp approximation of the photon profile of the soliton we are investigating .
the potential has two equal maxima at @xmath106 where @xmath107 these are the far - field values ( @xmath108 ) for soliton solutions obtained from eq . at the peak of the potential @xmath109
we obtain from eq .
@xmath110 taking , as before , the slope @xmath111 as an indicator of the slope of the profile , the healing length for the photons is @xmath112 the photon healing length for the approximate equation ( gp ) underestimates the healing length derived for the full system in by a factor of @xmath39 .
the two agree at the linear limit @xmath55 .
returning to the system involving both the photon and the exciton fields , one can write eqs . ( [ eq : solitonenvelope ] ) as a schrdinger equation for the photon field envelope @xmath28 , @xmath113 in which the effective potential @xmath114 depends on the exciton field : @xmath115 for the dark soliton derived above , @xmath114 exhibits a single symmetric well with far - field value @xmath116 , as shown in fig .
[ fig : potential ] . for the continuous soliton
, @xmath117 has a minimal value of @xmath118 . for the discontinuous soliton ,
the well becomes infinitely deep at the point of discontinuity . in an experimental setup ,
one expects that losses will allow some photons to be trapped by the potential well in the form of bound states at discrete energy levels which lie below @xmath49 . as long as a small enough fraction of the energy of the photon field of the coherent polariton structure
is transferred into lower energy states , the exciton field @xmath30 and therefore also the potential @xmath114 will not be significantly altered and can be considered a fixed potential .
this scenario is consistent with experimental observations @xcite , in which a polariton field is sustained by continuously injecting photons at two pump spots , one on each side of the potential well .
a fraction of the polariton population descends to lower energy states of the well .
we have presented a detailed study of dark solitons in polariton condensates , which result as solutions of a system of equations for strongly coupled excitons and photons .
we have analytically identified soliton solutions for the lossless system .
one type of black soliton studied is of the standard type where the fields vanish at the soliton center .
this corresponds to complete depletion of the condensate at that point .
furthermore , we reported a discontinuous soliton where the exciton field exhibits a jump at the soliton center , so the exciton density does not vanish .
we have shown that the two types of solitons can be unified in one brach , since the discontinuity in the exciton density smoothly increases from zero .
polariton condensates emerge as a fertile ground for solitonic structures .
our results provide an understanding of these structures .
furthermore , they can be used as a basis for a perturbation theory that will include non - conservative features , in particular , sources and losses .
this work was partially supported by the european union s fp7-regpot-2009 - 1 project `` archimedes center for modeling , analysis and computation '' ( grant agreement n. 245749 ) , by the ( us ) national science foundation under grants nsf dms-0707488 and nsf dms-1211638 , and by eu and greek national funds through the operational program `` education and lifelong learning '' thales .
this work has benefited from discussions with p. savvidis , g. christmann , f. marchetti , g. kavoulakis , a. gorbach .
f. m. marchetti and m. h. szymanska , in _ exciton polaritons in microcavities : new frontiers _
springer series in solid - state sciences _ , edited by d. sanvitto and v. timofeev ( springer - verlag , address , 2012 ) .
t. ackemann , w. j. firth , and g .- l .
oppo , in _ advances in atomic , molecular , and optical physics _ , edited by e. arimondo , p. r. berman , and c. c. lin ( academic press , address , 2009 ) , vol .
57 , pp .
6 , 323421 .
|
bose - einstein condensates of exciton - polaritons are described by a schrdinger system of two equations .
nonlinearity due to exciton interactions gives rise to a frequency band of dark soliton solutions , which are found analytically for the lossless zero - velocity case .
the soliton s far - field value varies from zero to infinity as the operating frequency varies across the band . for positive detuning ( photon frequency higher than exciton frequency ) ,
the exciton wavefunction becomes discontinuous when the operating frequency exceeds the exciton frequency .
this phenomenon lies outside the parameter regime of validity of the gross - pitaevskii ( gp ) model . within its regime of validity
, we give a derivation of a single - mode gp model from the initial schrdinger system and compare the continuous polariton solitons and gp solitons using the healing length notion .
|
deep inelastic scattering ( dis ) is a very important tool for investigating the structure of hadrons@xcite .
it involves the interaction between an energetic lepton beam ( electrons , muons , neutrinos ) and the respective hadron .
dis has been described ( i ) in the framework of the quark parton model ( qpm ) using the light - cone formalism and ( ii ) with the aid of covariant field theories .
the qpm is based on the assumption that all quarks ( before and after scattering ) are on - shell @xcite . in this manner gauge invariance
is built - in .
the leading contribution is the pole ( or born ) term and the rescattering term is negligable in the deep inelastic limit . in descriptions based on covariant field theory ,
however , the intermediate quarks ( or nucleons in studies of the deuteron@xcite ) are off - shell , and hence the pole term can not , by itself , be gauge invariant . in this case
it would appear that the pole term can not be the leading term in the deep inelastic limit , and this has been a problem for covariant theories for many years . to avoid this problem ,
de forest @xcite replaced the current by its gauge invariant part .
he presented no proof , and the `` de forest prescription '' has always seemed adhoc .
the problem continues to trouble calculations which rely on the use of off - shell particles and dominance of the pole term . in a recent paper
kelly @xcite outlines three alternatives . assuming that the photon has four momentum @xmath0 , these three alternatives can be summarized as follows : * _ de forest prescription _ :
leave the time component of the current ( the charge ) unchanged , and replace the third component by @xmath1 . * _
weyl prescription _ : leave the third component of the current unchanged , and replace the time component by @xmath2 .
* _ landau prescription _ : replace the four current by @xmath3 , where we use the slac convention that @xmath4 for electron scattering .
note that each of these descriptions gives _ a different result for the charge and @xmath5 components_. in the generalized breit frame , where @xmath6 , the de forest and the landau prescriptions are identical .
one of the major conclusions of this paper is a detailed theoretical argument justifying the landau prescription .
in addition to this , we present a simple toy model for the structure functions of a composite system .
the structure functions we obtain from this model exhibit scaling , are dominated in the deep inelastic limit by the `` gauge invariant part '' of the pole term , and give a simple , qualitative description of the distribution function of valence quarks .
we present the results from two versions . in one ,
the particles are taken to be scalars , and the scalar `` nucleon '' is a bound state of a scalar `` quark '' with unit charge and a scalar diquark with no charge . in a second ,
more realistic model , the composite spin 1/2 nucleon is a bound state of a spin 1/2 quark and a scalar diquark .
stimulated by the success of qcd in 1 + 1 dimensions@xcite , we carry out these model calculations in 1 + 1 dimensions , obtaining finite results without the need for model dependent form factors .
the only parameters in the model are the masses of the quark and diquark , @xmath7 and @xmath8 respectively .
the 1 + 1 dimensional calculations do not necessarily give the right physics , but help in developing the necessary tools for a more complete treatment .
the remainder of this paper is divided into five sections .
first we present our simple bound state model for the nuclon ( which could also be applied to the description of mesons ) .
then we study gauge invariance and show how to uniquely define the gauge invariant part of the pole term . in sec .
iv we show for scalar quarks that the gauge invariant part of the pole term does indeed dominate dis . the model deep inelastic structure functions are calculated and compared with experiment in sec .
v , and conclusions are given in sec .
we begin the discussion by constructing a dynamical model for the nucleon . in the first subsection
we will assume that the scalar `` nucleon '' is a bound state of the two `` fundamental '' scalar particles : a charged `` quark '' @xmath9 of mass @xmath7 and an uncharged `` diquark '' @xmath10 of mass @xmath8 . in the second subsection we will generalize to a charged spin 1/2 quark and a charged spin 1/2 nucleon .
the two `` fundamental '' constituents will interact with each other via the coupling shown in fig . 1 : @xmath11 where @xmath12 is the coupling strength , and the form factor @xmath13 is a function of the square of the diquark four - momentum .
for generality , we keep these form factors for now , but they will be set to unity later when we carry out calculations in 1 + 1 dimensions .
hence the scattering amplitude is : @xmath14 where we have introduced the function @xmath15 defined by @xmath16 with @xmath17 and @xmath18 .
this `` bubble '' integral is represented graphically in fig . [ two ] .
the mass of the bound state must be below threshold , and for simplicity we assume @xmath19 .
= 2.5 in in this simple separable model , the bound state is described by the `` nucleon''@xmath20 vertex function @xmath21 where @xmath22 is a constant to be determined from the normalization of the wave function .
this is shown in fig . [ four ] . denoting the dressed propagator of the bound nucleon by @xmath23 ,
the dressed @xmath20 scattering amplitude is given exactly in this simple model by @xmath24 where @xmath25 eq . ( [ scatamp ] ) is illustrated in fig .
[ three ] .
since the propagator should have a pole at @xmath26 , it is useful to expand the bubble @xmath15 in this vicinity , so the denominator of the rhs of eq .
( [ 1eq6 ] ) is @xmath27 and this must vanish at @xmath26 .
consequently @xmath28 is a necessary condition of the bound state , and near the pole eq .
( [ 1eq6 ] ) becomes @xmath29 knowing that , in the vicinity of @xmath26 , the nucleon propagator should be @xmath30 allows us to find the normalization of the @xmath31 vertex function : @xmath32 note that the existance of the bound state implies that @xmath33 , which will be shown below . in 1 + 1
d the bubble integral will converge without form factors , and we can assume , for simplicity , that @xmath34 .
the bubble integral ( [ 1eq4 ] ) is then model independent , and easily calculated .
if it is parametrized using feynman variables , one can either compute the loop below treshold directly in minkowski space ( doing the momentum integration first and then integrating over the feynman variable ) or compute it in euclidian space and wick rotate back to the minkowski space . introducing the triangle function @xmath35 which is negative
if @xmath36 , the bubble becomes : @xmath37\ , .
\label{1eq9}\ ] ] this is real , and if the mass of the bound state is to be @xmath38 , then @xmath12 must be @xmath39^{-1}\ , .
\label{1eq10}\ ] ] this is real ( so the lagrangian is hermitian , as it should be ) and negative ( as expected from the fact that we have a bound state , implying the interaction is attractive ) .
in fact , the whole 1 + 1 d theory can be derived from the lagrangian @xmath40 where @xmath9 is the `` quark '' and @xmath10 the `` diquark '' fields . from this lagrangian , or from eq .
( [ 1eq1 ] ) , one sees that @xmath12 must be negative in order for a bound state to exist . above threshold
, however , the loop @xmath15 also has an imaginary part @xmath41 which can be calculated in two ways .
the first method is a computation based on cutkosky s rules , which are easily handled in this simple case .
the second follows directly from the feynman parametrization of eq .
( [ 1eq4 ] ) . integrating over the internal momentum
gives @xmath42 introducing the variable @xmath43 transforms ( [ 1eq14 ] ) into : @xmath44 \ , .
\label{1eq15}\ ] ] this expression yields a real value for @xmath15 if @xmath45 is below threshold because @xmath46 is positive above threshold . this relation is a dispersion relation , and comparing it with the general form @xmath47 one concludes that the imaginary part of the bubble is correctly given in eq .
( [ 1eq11 ] ) . as a byproduct
, we have also discovered the correct phase space factor for use with dispersion relations in 1 + 1 d. the real part of @xmath15 above treshold can also be found in two ways : ( i ) using eq .
( [ 1eq16 ] ) , or ( ii ) by analytically continuing eq .
( [ 1eq9 ] ) , a procedure which also provides a third method for computing the bubble s imaginary part . as a result of these calculations ,
the real part above treshold is found to be : @xmath48\ , .
\label{1eq17}\ ] ] at large @xmath45 , @xmath49 , @xmath50 ; so @xmath51 as @xmath52 .
this important result will be used below . before turning to the spin 1/2 case
, we use the representation ( [ 1eq14 ] ) to write the normalization condition ( [ 1eq8 ] ) in the following form : @xmath53 ^ 2 } \ , .
\label{1eqlast}\ ] ] this relation will be used again in the following sections . in this subsection
we will repeat our former calculations for the more realistic case of charged spin 1/2 nucleons and quarks ( retaining , however , the uncharged scalar diquark ) .
the quark - diquark contact interaction is assumed to be a scalar in the dirac space of the quarks , with the momentum space dependence previously given in eq .
( [ 1eq1 ] ) . in this case
the bubble in @xmath10 dimensions becomes : @xmath54 or , after the integration : @xmath55 in 1 + 1 dimensions with the form factors set to unity , the factors @xmath56 and @xmath57 become : @xmath58 where @xmath59 is the feynman parameter used previously in eq .
( [ 1eq14 ] ) . in the presence of spin ,
eq . ( [ 1eq2 ] ) is modified : @xmath60 expanding the denominator near the nucleon pole gives @xmath61 where @xmath62 , @xmath63 , the primes denote derivatives with respect of @xmath64 , and all functions are evaluated at @xmath65 .
the denominator vanishes at @xmath65 if @xmath66 this is the bound state condition . after imposing the bound state condition , in the vicinity of the poles
the scattering amplitude becomes : @xmath67 the normalization factor @xmath22 is then : @xmath68\ , .
\label{1eq25}\ ] ] using the relations ( [ 1eq20 ] ) , and the identity @xmath69 ^ 2}. \label{1eq27a}\ ] ] we may rewrite the renormalization condition ( [ 1eq25 ] ) in the following form : @xmath70 ^ 2}. \label{1eq27}\ ] ] this relation will be needed later to verify baryon number conservation .
we now add electromagnetic interactions to the model , and study the implications of the requirement that the e.m .
interaction be gauge invariant .
we begin by restoring the form factors ; later we will move to 1 + 1 dimensions and set the form factors to unity . as in the previous section
, we will first carry out the discussion for a scalar quark and nucleon , and later extend it to a spin 1/2 quark and nucleon . start with the pole term , shown in fig . [
five ] , which gives @xmath71 where @xmath72 , the propagator of the scalar ( charged ) `` quark '' is @xmath73 and @xmath74 is the one - body current of the quark .
( note that our definition of the current @xmath75 does _ not _ include the charge . )
we assume that the diquark has no electromagnetic interaction , so there are no diagrams with the photon coupling to the diquark .
we require that the quark current satisfy the ward - takahashi ( wt ) identity : @xmath76 the current of a bare , spinless particle @xmath77 satisfies this identity . using the wt identity , and noting that @xmath78 because @xmath79
, we see that the pole term by itself is not gauge invariant @xmath80 \,\delta ( p_1^{\prime } ) \gamma ( p_2 ) \nonumber\\ = & & - \gamma(p_2 ) \ne0\ , .
\label{2eq4}\end{aligned}\ ] ] hence the pole term by itself can not be a good approximation for the scattering amplitude . to obtain a gauge invariant result ,
it is necessary to add the rescattering term shown in fig .
this term is @xmath81 where @xmath82 and @xmath83 is the square of the momentum of the final state after it has absorbed the virtual ( space - like ) photon .
note that @xmath84 but that @xmath85 .
this diagram is also not gauge invariant .
using eq .
( [ vertex ] ) and the wt identity gives @xmath86 = { \cal n}f(p_2 ^ 2 ) = \gamma(p_2)\ , , \label{2eq6}\ ] ] where the bound state condition @xmath28 was used in the second step . this is not gauge invariant , but if we add the rescattering term to the pole term the total amplitude is @xmath87 stated in a different way , the gauge dependent parts of the pole term and the rescattering term cancel .
we now face the question posed in the introduction : how can the pole term dominate the scattering in the deep inelastic limit if it is not gauge invariant ? to answer this question we break the pole term into a gauge invariant part and a gauge dependent part as follows : @xmath88\,\gamma ( p_2 ) \over
m_1 ^ 2-(p_1-q)^2 } \ , , \label{2eq8}\end{aligned}\ ] ] where @xmath89 \ , .
\label{2eq9a } \end{aligned}\ ] ] note that @xmath90 , and that the gauge dependent part of the pole term reduces to @xmath91 now the rescattering term can also be similarly decomposed .
noting that @xmath92 the gauge dependent part of the rescattering term becomes @xmath93 } \nonumber\\ & & \times i\int \frac{d^dk}{(2 \pi)^d } \frac{f^2(k^2)\bigl\{(p - k+q)^2-(p - k)^2\bigr\ } } { ( m_2 ^ 2-k^2)(m_1 ^ 2-(p - k)^2)(m_1 ^ 2-(p - k+q)^2)}\nonumber\\ = & & { q^\mu\over q^2 } \,\gamma(p_2)\;\frac{\lambda } { [ 1- \lambda b(s ' ) ] } \bigl\ { b(m^2)-b(s ' ) \bigr\ } = { q^\mu\over q^2 } \,\gamma(p_2 ) \label{2eq12}\ , , \end{aligned}\ ] ] which exactly cancels the gauge dependent part of the pole term !
we have shown that , if we drop the parts from each terms , the born pole term and the rescattering term are _ redefined _ so that each is _ separately gauge invariant_. as it turns out , the redefined pole term is identical to the landau prescription defined in the introduction .
we have just justified this prescription .
it is clear that the same decomposition works in 1 + 1 dimensions , where the form factor @xmath94 . in this case
the diagrams come from from the lagrangian ( [ 1eq13 ] ) , modified to include electromagnetic interactions .
the new lagrangian is @xmath95 where the fields are the same as in the preceding section , but the gradient of the charged field has been replaced by the covariant derivative @xmath96 we how extend the above results to the spin 1/2 model previously introduced .
the born term for a structureless quark ( with the form factor set equal to unity ) is @xmath97 where now @xmath98 and the quark propagator is @xmath99 the vertex function for the spin 1/2 case is @xmath100 where @xmath101 is the helicity of the nucleon and @xmath102 are the helicity spinor of the nucleon . using the dirac equation for the final quark , the pole term can be written @xmath103 { u}_n(p , \lambda_n ) , \label{5eq1}\ ] ] where @xmath104 the helicity of the outgoing quark and @xmath105 the helicity spinor of the quark .
now the current is spin dependent .
as before , we decompose the pole term into a gauge invariant and gauge noninvariant part : @xmath106 { u}_n(p , \lambda_n ) \nonumber\\ { \cal j}^{\mu}_{ag}=&&-\frac { q^\mu}{q^2}\;{\cal n } { \bar u}(p_1 , \lambda_1 ) { u}_n(p , \lambda_n ) = -\frac { q^\mu}{q^2}\;{\bar u}(p_1 , \lambda_1)\gamma(p,\lambda_n ) \ , .
\label{5eq1b}\end{aligned}\ ] ] note that , except for the addition of the spinors , the gauge dependent part identical to the scalar case , and the gauge independent part is again the landau prescription .
we now look at the spinor rescattering diagram . introducing the decomposition @xmath107 the gauge dependent
part of the rescattering term becomes @xmath108^{-1 } \nonumber\\ & & \times i\int \frac{d^dk}{(2 \pi)^d } \frac{1 } { ( m_2 ^ 2-k^2 ) } s(p - k+q)\bigl\{s^{-1}(p - k+q ) -s^{-1}(p - k)\bigr\}\nonumber\\ & & \qquad\times s(p - k ) \;\gamma(p,\lambda_n ) \nonumber\\ = & & { q^\mu\over q^2}\;{\bar u}(p_1,\lambda_1 ) \left[1- \lambda b(p+q)\right]^{-1 } \bigl\ { \lambda\,b(p)-\lambda\,b(p+q ) \bigr\}\;\gamma(p,\lambda_n ) \nonumber\\ = & & { q^\mu\over q^2 } \;{\bar u}(p_1,\lambda_1)\ ; \gamma(p,\lambda_n ) \label{2xeq12}\ , , \end{aligned}\ ] ] where we used the bound state condition ( [ 1eq23 ] ) and @xmath109 .
note that this term exactly cancels the gauge dependent part of the pole term !
this argument shows that this technique is quite general , and should work in every case .
the pole term is divided into a term which is gauge invariant and one proportional to @xmath110 by adding and subtracting a ( unique ) term proportional to @xmath110 .
then it may be shown that this non gauge invariant term is cancelled when the rescattering term is similarily separated into ( unique ) gauge invariant and non guage invariant parts . knowing that this works means that the non gauge invariant part of the pole term _ can be discarded without checking that it is cancelled by an identical piece from the rescattering term_. our argument can be extended to currents with form factors if we use the technique developed by gross and riska @xcite to define the off - shell currents .
in this section we consider the rescattering term for the scalar model in more detail .
we compute it exactly for the case @xmath111 and confirm that the charge of the bound state is properly normalized
. then we study the behavior of the elastic form factor and the inelastic rescattering term at large @xmath112 .
we will compare the gauge invariant part of the rescattering term with the gauge invariant part of the pole term and show that the pole term dominates at high @xmath112 . in order to achieve convergence in a model independent way
, we return to 1 + 1 dimension .
the key to these calculations is the triangle diagram .
the gauge invariant part of this diagram is @xmath113 feynman parametrizing the integral and doing the momentum integration gives @xmath114 where the denominator is @xmath115 it is convenient to express the integral in terms of @xmath82 and @xmath116 , and to change to the variables @xmath117 which maps the region of integration into @xmath118 the jacobian of this transformation is @xmath119 , and the integral then becomes @xmath120 where @xmath121 we will use the general results ( [ 3eq2 ] ) and ( [ d3eq2 ] ) in the following discussion .
first consider the hadronic part of the elastic electron - nucleon scattering process , shown in fig .
[ seven ] .
this diagram is gauge invariant by itself , and hence the deuteron form factor is given by @xmath122 where , for elastic scattering , the triangle diagram is evaluated with the constraints @xmath123 .
substituting eq .
( [ 3eq2 ] ) into eq .
( [ 3eq8 ] ) gives @xmath124 ^ 2}\ , , \label{3eq5}\ ] ] where @xmath125 if the charge is conserved , the form factor must satisfy the condition @xmath126 ^ 2}\nonumber\\ = & & { \cal n}^2\,{\partial\over \partial m^2}\left\ { \frac{1}{4 \pi } \int^{1}_{0 } dz\;\ ; \frac{1}{\left[m_2 ^ 2 ( 1-z ) + m_1 ^ 2z -m^2 z(1-z)\right ] } \right\ } \label{fat0 } \ , , \end{aligned}\ ] ] comparing this with eq .
( [ 1eq14 ] ) gives the relation @xmath127 which is in agreement with the eq .
( [ 1eq8 ] ) and proves that charge is conserved . in preparation for the discussion of rescattering in the deep inelastic limit , and in order to study the properties of the 1 + 1 d model
, we examine the behavior of the form factor in the high @xmath112 limit .
this is obtained from @xmath128 } \right\}\nonumber\\ = & & { \cal n}^2\,{\partial\over \partial m^2}\left\ { \frac{1}{8\pi q } \int^{1}_{0 } { dz\over z } \;\ ; { 1\over\sqrt{\alpha(z ) + { \textstyle{1\over4}}q^2z^2 } } \log\left({\sqrt{\alpha(z ) + { \textstyle{1\over4}}q^2z^2 } + { \textstyle{1\over2}}qz \over \sqrt{\alpha(z ) + { \textstyle{1\over4}}q^2z^2 } - { \textstyle{1\over2}}qz } \right ) \right\ } \nonumber\\ = & & \frac{{\cal n}^2}{8\pi q } \int^{1}_{0 } ( 1-z ) dz \;\ ; { 1\over \alpha(z ) \,\sqrt{\alpha(z ) + { \textstyle{1\over4}}q^2z^2}}\ , , \end{aligned}\ ] ] where we have introduced @xmath4 used in electron scattering . at large @xmath129 , the last integral
is dominated by values of @xmath130 near zero , and is well approximated by @xmath131 we see that @xmath132 falls off as @xmath129 approaches infinity , as it should for a composite particle .
these results can be applied to realistic cases . denoting the charge of the quark by @xmath133 and the mass of the diquark by @xmath134
the total charge is simply @xmath135 , as required by charge conservation . at high @xmath136 however , we obtain @xmath137 note that the sign of @xmath138 is positive for the proton , but that it is identically zero ( or , more correctly , falls off faster than @xmath139 ) for the neutron , _ unless _ the mass of the @xmath140 diquark differs from the mass of the @xmath141 diquark . in the latter case , @xmath142 ignoring the weak logarithimic dependence on the diquark mass , these are equal ( for example ) if @xmath143 . the behavior of these elastic form factors at high @xmath136 has not yet been determined experimentally , and continues to be a subject of great interest . note that eq .
( [ 3eq188 ] ) also predicts that the electromagnetic form factors of spin 0 @xmath144 mesons which have the same flavor for the quark and antiquark ( such as @xmath145 , @xmath146 ) , is identically zero . in sec .
iii the rescattering term was treated briefly .
we divided the triangle into a gauge dependent and gauge independent part .
the former was calculated exactly , while the latter is proportional to the triangle diagram given in eq .
( [ 3eq2 ] ) : @xmath147 } \;t^\mu(p',p ) \nonumber\\ = & & { \lambda{\cal n}\over \left [ 1-\lambda b(w^2)\right ] } \;\left\ { ( p'+p)^{\mu } -{q^\mu\,(p'^2-p^2)\over q^2}\right\}\ ; t(q^2,w^2 ) \ , , \end{aligned}\ ] ] where @xmath148 and the scalar function @xmath149 is @xmath150 ^ 2 } \ , , \label{3eq2bb}\ ] ] where @xmath151 for electron scattering , @xmath152 and @xmath153 , so @xmath154 .
however , as for the bubble diagram , @xmath155 has a zero if @xmath156 .
this means that the denominator of eq .
( [ 3eq2bb ] ) also has a zero , and that @xmath157 has an imaginary part . from the principles of dispersion theory we know that this is due to the existance of physical scattering in the final state , as illustrated in fig .
[ eight ] . to obtain a tractible form for the integral ,
replace the @xmath158 integration by @xmath159 , which gives @xmath160 ^ 2 } \
, , \label{3eq2aa}\ ] ] where @xmath161 ^ 2-q^2z^2 \beta \ , , \label{3eq2b}\end{aligned}\ ] ] we now replace @xmath162 by @xmath163 where @xmath59 is the bjorken scaling variable @xmath164 and take the deep inelastic limit where @xmath165 with @xmath59 held constant . the integral
is dominated by values of @xmath130 near zero , and hence may be approximated by @xmath166 ^ 2}\nonumber\\ \simeq & & \frac{x}{2 \pi q^2 ( 1-x)}\int^{1}_{0 } dz \left ( \frac{1}{\left[m_2 ^ 2 -w^2z -i\epsilon\right ] } - \frac{1}{m_2 ^ 2 } \right ) \nonumber\\ \simeq & & \frac{x}{2 \pi q^2 ( 1-x)}\left ( { i\pi x\over q^2(1-x ) } - \frac{2}{m_2 ^ 2 } \right ) \ , , \label{tsim}\end{aligned}\ ] ] where we have kept the leading contribution from the imaginary part even though it is negligible compared to the real part . in conclusion , one can state that the gauge invariant part of the rescattering term falls like @xmath167 .
noting that the bubble also falls as @xmath167 , the full rescattering term in the deep inelastic limit approaches @xmath168 this is to be compared with the deep inelastic limit of the born term , which is @xmath169 and does not fall with @xmath136 .
hence , in the deep inelastic limit the gauge invariant part of the rescattering term is negligible in comparison to the gauge invariant part of the pole term . in the next section
we will calculate the cross section for deep inelastic scattering , and find the structure functions predicted by this simple model .
in this section we calculate the cross section and structure functions for the deep inelastic scattering ( dis ) of unpolarized electrons from the composite `` nucleon '' @xmath170 , @xmath171 where @xmath172 is the undetected final hadronic state .
this scattering process is illustrated in fig . [ nine ] .
the nucleon is composite , as described in sec .
ii . for completeness ,
the well known kinematics and cross section formula@xcite are first reviewed in the next two subsections .
the remaining subsections give results for the 1 + 1 d toy models presented in this paper .
the bjorken variable , @xmath59 , was defined in eq .
( [ xdef ] ) and the invariant mass of the final hadronic state , @xmath173 , in eq .
( [ 4eq3 ] ) . in terms of these quantities ,
the energy transferred by the photon to the hadron in the lab system is @xmath174 and the magnitude of the spatial component of @xmath9 , which will be chosen to be in the @xmath175 direction , is : @xmath176 in the cm of the outgoing hadronic pair , the invariant mass of the final state is @xmath177 where the prime denotes that the variable is evaluated in the cm system .
the magnitude of the three - momentum of the `` quark '' ( and `` diquark '' ) in the cm frame , and in the bjorken limit , is @xmath178\ , , \label{4eq1004}\end{aligned}\ ] ] where the non - leading term will be needed below .
the components of @xmath9 in the cm frame can be found by a lorentz boost from the lab : @xmath179 hence the cm components are : @xmath180^{-1/2}\left [ \left(m^2 + \frac{q^2}{x } + \frac{q^4 } { 4m^2x^2}\right)^{1/2}- 2mx-\frac{q^2}{2mx}\right ] \nonumber\\ q_z^{\prime}= & & q\left[m^2+q^2\left(\frac{1}{x}-1\right)\right]^{-1/2 } \sqrt{1+\frac{q^2}{4m^2x^2 } } \nonumber\\ & & \qquad\times\left[\left(m^2 + \frac{q^2}{x } + \frac{q^4}{4m^2x^2}\right)^{1/2}- \frac{q^2}{2mx}\right]\ , . \label{4eq6}\end{aligned}\ ] ] in the deep inelastic limit these become @xmath181 \nonumber\\ q_z^{\prime}=&&{q\over 2 \sqrt{x(1-x ) } } \left [ 1- { m^2x(1 - 2x)^2\over 2q^2(1-x)}\right]\ , , \label{4eq6a}\end{aligned}\ ] ] where the non - leading terms will be needed later . using the same boost , the energy of the initial hadron in the cm of the outgoing hadronic system is @xmath182^{-1/2 } \left(m^2+\frac{q^2}{x } + \frac{q^4 } { 4m^2x^2}\right)^{1/2}\nonumber\\ \to & & { q\over 2 \sqrt{x(1-x)}}\left[1-{m^2x[1 - 4(1-x)]\over 2q^2(1-x)}\right]=q_z ' \left[1+{m^2\over 2q'^2_z}\right]\ , , \label{4eq1006}\end{aligned}\ ] ] as expected .
we now turn to the calculation of the cross section .
the scattering amplitude for the dis process is @xmath183 where @xmath184 is the hadronic current ( kept general for now ) and @xmath185 is the current of the electron @xmath186 the spins of the electron are denoted by @xmath45 and @xmath187 , and the momenta are labeled as in fig . [ nine ] . in this notation
the unpolarized differential cross section in the lab is @xmath188 where @xmath189 contains a sum over the spins of the final particles and an average over the spins of the initial particles , @xmath190 and @xmath191 are the energies of the outgoing quark and diquark , @xmath192 and @xmath193 are the initial and final electron energies , and @xmath194 and @xmath195 are the lepton and hadron current tensors , defined below .
the hadronic tensor contains all of the the physical information about the hadron - photon interaction and includes the integration over the outgoing hadrons and the normalization factors associated with the hadronic wave functions .
if the electron mass can be neglected the outgoing electron differential can be written @xmath196 , which gives the following result for the inelastic cross section : @xmath197 if the electron mass is neglected the lepton current tensor is @xmath198 gauge invariance of the hadronic current implies @xmath199 this in turn implies that the most general form of the hadronic tensor for unpolarized scattering is @xmath200 which _ defines _ the structure functions @xmath201 and @xmath202 .
substituting eq .
( [ 4eq9a ] ) into eq .
( [ 4eq10 ] ) gives @xmath203 \nonumber\\ = & & \sigma_m\,\left[w_2 + 2w_1\,\tan^2 \left(\frac{\theta}{2}\right)\right ] \
, , \label{4eq12}\end{aligned}\ ] ] where @xmath204 is the scattering angle of the electron in the lab system and @xmath205 is the mott cross section .
we now calculate the hadronic tensor .
if the spin of the target is @xmath206 , in 1 + 3 dimensions this tensor is @xmath207 where @xmath206 , @xmath208 , @xmath209 are the spins , and the second line gives the result in the cm system .
the volume integrals @xmath210 and @xmath211 are covariant , insuring that the tensor is also covariant .
however , when we treat the hadronic degrees of freedom in 1 + 1 dimensions , consistency requires that the hadronic tensor also be evaluated in 1 + 1 dimensions . in this case
( [ 4eq7 ] ) must be replaced by its 1 + 1 dimensional equivalent @xmath212 where the second line gives the result in the cm system with @xmath213 corresponding to the cases when @xmath214 is parallel to @xmath215 ( + ) or antiparallel ( @xmath216 ) . except for these two possibilities
the delta functions completely fix the kinematics . to calculate the structure functions for the 1 + 1 dimensional models , we use eqs .
( [ 3eq23 ] ) , ( [ 5eq1b ] ) , and ( [ 4eq7a ] ) . the definitions ( [ 4eq7a ] ) and ( [ 4eq9a ] ) are covariant , so the structure functions can be evaluated in any frame , and it is convenient to evaluate them in the cm frame of the outgoing hadrons . furthermore , since @xmath217 and @xmath218 have components only in the @xmath219 and @xmath175 directions , gauge invariance insures that @xmath220 and that @xmath221 .
hence @xmath201 and @xmath202 can be extracted from @xmath222\ , .
\label{eqcg}\end{aligned}\ ] ] in the next two subsections we calculate the structure functions for the scalar and spin 1/2 models in 1 + 1 dimensions .
using eqs .
( [ 3eq23 ] ) and ( [ 4eq7a ] ) , the structure functions for the scalar theory in the deep inelastic limit are @xmath223 where @xmath224 and @xmath225\nonumber\\ % b=&&q^2 + 2e'_1q^{\prime } _ 0 \rightarrow q^2 + { q^2 ( 1 - 2x)\over 2 { x } } \left[1-{x(m_2 ^ 2-m_1 ^ 2)\over q^2(1-x)}\right ] \nonumber\\ \rightarrow & & { q^2 \over 2 { x } } \left[1- { x(1 - 2x)(m_2 ^ 2-m_1 ^ 2)\over q^2(1-x ) } \right]\ , .
\label{4eq18}\end{aligned}\ ] ] hence @xmath226 , and the @xmath227 term vanishes in the deep inelastic limit .
the ( + ) term is finite however , giving @xmath228 ^ 2 } \ , .
\label{w2}\end{aligned}\ ] ] @xmath201 vanishes in the deep inelastic limit , as expected for scalar particles . but
@xmath229 _ scales _ in the deep inelastic region ; its dependence on @xmath136 and @xmath230 is replaced by dependence on the bjorken scaling variable @xmath59 alone . in the simple quark parton model ,
we expect @xmath231 where @xmath232 is the probability of finding a quark with momentum fraction @xmath59 @xmath233 our simple model therefore predicts @xmath234 ^ 2 } \
, , \end{aligned}\ ] ] which satisfies the normalization condition ( [ 4eq13a ] ) , as can be seen by comparing with eq . ( [ 1eqlast ] ) . in the previous subsection we obtained the familiar result that the callan - gross relation@xcite is not satisfied by scalar quarks .
this relation predicts that @xmath235 where @xmath232 the same function which specified @xmath236 in eq .
( [ 4eq13 ] ) . for scalar quarks
the structure function @xmath201 falls off with @xmath136 . to obtain the relation ( [ 4eq14 ] ) , the quarks must to be treated as spin 1/2 fermions .
[ the diquarks , even when their orbital angular momentum is zero , will have two spin states ( 0 and 1 ) , and our choice of spin zero corresponds to the simplest possible wave function . ] when carrying out the calculation in 1 + 1 d , we must assume that only the momenta are restricted to 1 + 1 d , but that the spin lives in the full 1 + 3 dimensional space .
this assumption is fully consistent with the dynamical calculations of secs .
ii and iii . from the spin zero calculation
we know that it is sufficient to evaluate the ( + ) matrix element only
. in this case the incoming nucleon is traveling in the @xmath237 direction , and the final quark in the @xmath238 direction .
we will calculate this matrix element in the cm system of the outgoing pair using helicity states , which for this case are @xmath239 where @xmath240 with the help of these quantities , the @xmath59-component of the current ( [ 5eq1b ] ) becomes : @xmath241\ , \delta_{\lambda_n , \lambda_1}\nonumber\\ \rightarrow & & -\frac{{\cal n } q}{\sqrt{x}\,[b - a]}\;2\lambda_n \,[mx+m_1]\ , .
\label{5eq4}\end{aligned}\ ] ] this gives @xmath242 ^ 2 } = { \frac{1}{2m}}f(x ) \label{5eq4a}\end{aligned}\ ] ] in a similar manner , it may be shown that @xmath243 in the deep inelastic limit , which is the callan gross relation [ recall eq .
( [ eqcg ] ) ] .
note that the normalization condition for the spin 1/2 model given in eq .
( [ 1eq27 ] ) confirms that this model is also correctly normalized .
however , the @xmath232 obtained from the spin 1/2 model does not go to zero as @xmath244 . in fig .
[ ten ] the structure functions obtained from the scalar and spin 1/2 models are compared with the empirical momentum distribution @xmath245 for @xmath158 quarks in the proton . the empirical distribution was taken from ref .
@xcite and has been renormalized to unity for comparison .
note that the parameters in the momentum distributions predicted by the models can be chosen to give a qualitatively reasonable description of empirical results , but the models fail at both large and small @xmath59 .
the large @xmath59 distributions fall like @xmath246 , while the empirical distribution falls more rapidly ( the fit gives @xmath247 compared to @xmath248 for the naive quark model ) .
similarly , the empirical distribution goes like @xmath249 at small @xmath59 , while the spin 1/2 model goes like @xmath59 and the scalar model goes like @xmath250 . assuming the @xmath158 and @xmath10 quark distributions are identical , the resulting momentum carried by valence quarks is @xmath251 for the empirical distribution , @xmath252 for the spin 1/2 model , and @xmath253 for the scalar model . in every case
some momentum must be carried by glue or sea quarks , and the naive models presented here do not include these contributions .
this will be investigated in a future work .
in this paper we have shown how to extract the gauge invariant part of the pole term so that it gives the _ leading contribution to deep inelastic scattering_. the gauge non - invariant part depends only on @xmath110 and is cancelled exactly by the rescattering term . while we have not shown it explicitly , the argument suggests that this procedure is quite general and works as follows .
first add a ( unique ) term proportional to @xmath110 to the pole term which makes it gauge invariant .
then subtract the same term from all of the remaining terms ( rescattering and interaction currents ) .
it follows that the remaining terms are also gauge invariant , and we _ conjecture _ that the sum of all of these remaining terms will vanish in the deep inelastic limit
. hence the modified pole term will give the _ exact _
result for dis .
more explicitly , imagine that the full current is the sum of a pole term @xmath254 and a remainder @xmath255 , and that it is gauge invariant .
the first step is to write it as @xmath256 where @xmath257 note that @xmath258 is uniquely determined ( unless @xmath259 , which can not happen for dis ) , and that the modified pole term @xmath260 is gauge invariant by construction .
we conjecture that the modified remainder terms ( which are also gauge invariant ) will vanish in the dis limit , leaving ( [ last ] ) to give the exact result for dis .
note that our prescription ( [ last ] ) is identical to the landau prescription , and defined in ref .
@xcite .
the second result of this paper is the construction of toy models for the discription of dis .
these models give a qualitatively reasonable description of the phenomenology , as shown in fig .
[ ten ] , but simplicity is their main virtue .
we plan to use them to study many issues in the theory of dis .
we would like thank richard lebed for calling our attention to ref @xcite , and christian wahlquist for supplying the empirical quark distribution functions used in fig .
[ ten ] . the support of the doe through grant no .
de - fg02 - 97er41032 is gratefully acknowledged .
f. gross and s. liuti , phys .
c * 45 * , 1374 ( 1992 ) ; a. yu . umnikov and f. c. khanna , phys .
c * 49 * , 2311 ( 1994 ) ; w. melnitchouk , a. w. schreiber , and a. w. thomas , phys . lett . * b335 * , 11 ( 1994 ) ; w. melnitchouk , g. piller , and a. w. thomas , phys . lett . * b346 * , 165 ( 1995 ) .
|
in this paper we reconcile two contradictory statements about deep inelastic scattering ( dis ) in manifestly covariant theories : ( i ) the scattering must be gauge invariant , even in the deep inelastic limit , and ( ii ) the pole term ( which is not gauge invariant in a covariant theory ) dominates the scattering amplitude in the deep inelastic limit .
an `` intermediate '' answer is found to be true .
we show that , at all energies , the gauge dependent part of the pole term cancels the gauge dependent part of the rescattering term , so that both the pole and rescattering terms can be separately redefined in a gauge invariant fashion . the resulting , redefined pole term is then shown to dominate the scattering in the deep inelastic limit .
details are worked out for a simple example in 1 + 1 dimensions .
|
recent measurements of anisotropies of the cosmic microwave background ( cmb ) radiation have revealed the detailed distribution of matter in the universe a few hundred thousand years after the big bang @xcite .
observations utilizing large ground - based telescopes and space telescopes have discovered galaxies and black holes that were in place when the age of the universe was less than a billion years . moreover ,
many galaxies have been found at @xmath8 @xcite in the hubble ultra deep field , whereas already a few gamma - ray bursts at @xmath9 have been detected by the _ swift _ satellite @xcite .
these first objects are probably the building blocks of the present day galaxies , thus , solving the puzzle behind their formation will have a profound implication on our understanding of the universe ( see for recent reviews * ? ? ?
* ; * ? ? ?
* and references therein ) .
the formation of the first generation of galaxies in the universe has been studied for many years .
high resolution cosmological simulations can follow complex astrophysical processes , while analytical calculations can provide an over - all understanding , and can be used to decouple different physical effects seen in simulations .
analytic models are also useful for estimating the limitations of numerical simulations such as insufficient resolution and small boxsizes ( yoshida et al .
2003 ; barkana & loeb 2004 ; naoz & barkana 2005 ) .
combining the two approaches may offer many of the advantages of both .
the initial conditions ( hereafter ics ) in a cosmological simulation can have a large effect on the formation of the first galaxies in simulations , i.e. , both on the formation time ( or on the halo abundance at a given time ) and the halo properties at formation time ( such as the average gas fraction ) .
@xcite studied high - redshift structure formation and reionization while testing two different models for power spectra as their ics .
they found that different models have a profound effect on the abundance of primordial star - forming gas clouds and thus on when the reionization was initiated and its progress . in the analytical point of view , @xcite and
@xcite showed that the ics at high redshift have a significant effect on the halo abundance and the gas fraction at virialization . while these effects are largest at the highest redshift , e.g. , @xmath10 for the first star in the universe @xcite , they are still significant for halos forming at @xmath11 .
the first gas rich halos at these redshifts are expected to host the first stars ( @xmath12 * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) and even the first gamma - ray bursts ( e.g. , * ? ? ?
* ; * ? ? ?
thus , investigating the formation properties of these halos is of prime importance .
gas rich halos in the early universe may very well be a nurturing ground for dwarf galaxies , which at high redshift can form stars ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* and references therein ) perhaps even at a high star formation rate @xcite .
their properties are very important as they are responsible for metal pollution and the ionizing radiation at these early times ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
moreover , halos that are too small for efficient cooling via atomic hydrogen , i.e. , minihalos , are most susceptible to the effect of initial conditions .
while they may not normally host astrophysical sources , minihalos may produce a 21-cm signature ( @xcite but see @xcite ) , and they can block ionizing radiation and produce an overall delay in the initial progress of reionization ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
the evolution of the halo gas fraction at various epochs of the universe is of prime importance , particularly in the early universe . in this paper
, we examine the effect of using different initial conditions in simulations on the resulting minimum gas - rich halo mass in the redshift regime @xmath1 .
we perform gadget-2 @xcite simulations using a total of @xmath13 particles .
we compare the initial conditions presented in @xcite , which describe the linear evolution of overdensities in a fully consistent way , to two other alternative ics , often used in the literature .
we also compare to the prediction of the gas - rich mass from linear theory .
we describe our different initial conditions and simulations in sections [ ics ] and [ sim ] , respectively .
our simulation results are presented in section [ res ] where we divide our discussion to the evolution of the non - linear power spectra ( section [ nonlin ] ) and to the minimum gas - rich halo mass resulting from either linear theory or from the simulations ( section [ mc_f ] ) .
finally , we discuss our conclusions ( section [ dis ] ) . throughout this paper , we adopt the following cosmological parameters : ( @xmath14 , @xmath15 , @xmath16 , n , @xmath3 , @xmath17)= ( 0.72 , 0.28 , 0.046 , 1 , 0.82 , 70 km s@xmath18 mpc@xmath18 ) @xcite .
we follow @xcite , who studied the linear evolution of both dark matter and baryon overdensities .
the fluctuations of the temperature of the baryons ( @xmath19 ) can not be described as a simple function of a spatially uniform baryonic sound speed @xmath20 , as was previously assumed ( e.g. , * ? ? ?
furthermore , at high redshifts , the baryon density fluctuations ( @xmath21 ) are not equal to those of dark matter ( @xmath22 ) ( contrary to a common assumption in simulations ; four redshift examples are shown in figure 1 of * ? ? ?
we label the power spectrum model as the `` fid '' ( fiducial ) ics since it follows the evolution of linear overdensities in a complete and consistent way . ) between the fiducial linear initial conditions and the alternative models at @xmath23 .
we consider the relative difference between the fid ics and the mean @xmath4 ics for both the baryons and dark matter ( solid and short - dashed curves , respectively ) , and the relative difference between the fid ics and the e-@xmath2 ics for both the baryons and dark matter ( dotted and long - dashed curves , respectively ) .
note that we have plotted here the absolute value ; the mean @xmath4 model gave a negative value ( i.e. , an underestimate compared to the fid model ) while the e-@xmath2 model gave a positive value ( i.e. , an overestimate).,width=317 ] following @xcite we write the basic equations that describe the evolution of the dark matter , baryon density and temperature fluctuations : @xmath24 where @xmath25 and @xmath26 are the mean cosmic dark matter and baryonic fraction respectively . here
we follow the standard notations for cosmological parameters such as @xmath27 .
the baryons are also subject to a pressure term : @xmath28 where @xmath29 is the mean molecular weight , @xmath30 is the boltzmann constant and @xmath31 is the wavenumber . using the first law of thermodynamics
, @xcite derived the equations for the evolution of the baryon average temperature and temperature fluctuations : @xmath32 where @xmath33/a$ ] is the mean cmb temperature , and the first - order equation for the perturbation : @xmath34 with the second term on the right - hand side accounting for the compton scattering of the cmb photons on the residual electrons from recombination , where @xmath35 is the electron fraction out of the total number density of gas particles at time @xmath36 , and @xmath37 where @xmath38 is the thomson scattering cross section and @xmath39 is the photon energy density . the first term on the right - hand - side of each of these two equations ( [ mean ] ) and ( [ gamma ] ) accounts for adiabatic expansion of the gas , and the remaining terms capture the effect of the thermal exchange with the cmb .
following @xcite we have numerically calculated the evolution of the perturbations by modifying the cmbfast code @xcite according to these equations .
note that similar physics was also explored by @xcite .
we solve the complete set of equations to obtain the power spectrum at different redshifts which can be used as initial conditions for our simulations .
figure [ fig : ics ] shows the ratio between this initial condition to the two alternative models tested in this paper .
in many cosmological ics for n - body simulations and semi - analytical calculations , the fluctuations of the baryons are assumed to be equal to the fluctuations of the dark matter .
we construct a model that includes this incorrect assumption while maintaining the correct overall @xmath40 ( i.e. , conserving @xmath3 at @xmath41 , see appendix [ sig8 ] for more details ) .
thus , in our `` e-@xmath2 '' model we calculate the correct @xmath40 as a combination of @xmath21 and @xmath22 from the fiducial calculation in section ( [ correct ] ) , but then take the baryon perturbation to be the same as for the dark matter , namely : @xmath42 where @xmath43 ( @xmath44 ) is the resulting linear over - density from the fiducial calculation ( e-@xmath2 model ) for the baryons and dark matter , respectively .
we then compare the equal @xmath2 model to our fiducial calculation .
figure [ fig : ics ] shows the ratio between the fid ics and the e-@xmath2 model for both the baryons and dark matter .
we find that the e-@xmath2 model overestimates the baryon fluctuations by @xmath45 on large scales ( @xmath46 kpc ) while the overestimate grows to a much larger factor on small scales . before recombination the baryons
were tightly coupled to the radiation , resulting in suppression of the growth of their overdensity .
however , the dark matter component , which is not affected by the photons , could basically grow once the fluctuation wavelength entered the hubble horizon ( in the linear regime , before equality , the dark mater fluctuations grew logarithmically with the scale factor , where after equality they grew linearly with the scale factor ) .
therefore , this resulted in a suppression of the baryonic overdensity by about three orders of magnitude compare to the dark matter at recombination ( e.g. , fig . 1 in * ? ? ? * ) .
while the baryons subsequently fall into the potential wells of the dark matter , it takes them some time to catch up , and the baryon fluctuations are still suppressed even at lower redshifts .
this point is often overlooked in simulations and analytical calculations .
@xcite showed that the presence of spatial fluctuations in the sound speed modifies the calculation of perturbation growth significantly . nevertheless , for completeness and as a case of comparison with previous results , we compare the simulation results with the results obtained using this approximation .
thus , we proceed by presenting the basic equations of the growth of density fluctuations , in this approximation of a uniform sound speed ( hereafter `` mean @xmath4 '' ) . the evolution of the density fluctuations is described by a different set of coupled second order differential equations : @xmath47 where @xmath48 is assumed to be spatially uniform ( i.e. , independent of @xmath31 ) and is thus calculated from the thermal evolution of a uniform gas undergoing hubble expansion . with this assumption ,
the temperature fluctuations ( as a function of @xmath31 ) are simply proportional at any given time to the gas density fluctuations : @xmath49 @xcite showed that this approximation leads to an underestimation of the baryon density fluctuations by up to 30% at @xmath50 and 10% at @xmath51 for large wavenumbers .
figure [ fig : ics ] shows the ratio between the mean @xmath4 initial conditions and the fiducial ones for both the baryons and dark matter .
it agrees with our previous results , showing that the underestimate by the mean @xmath4 model is greatest at @xmath52 kpc .
the non - linear evolution resulting from these initial conditions will result in shallower potential wells compared to the fiducial calculation , even though it is clear that the precise baryon temperature fluctuations at high redshift are very significant , still many simulations use initial conditions that assume a uniform speed of sound in the universe .
as shown below this leads to significantly different estimates for the gas content of the early halos .
we run a gadget 2 simulation @xcite starting from redshift @xmath53 , for a total of @xmath54 particles ( @xmath55 particles each for the dark matter and baryon components ) and our box size is : @xmath56 mpc . we choose this box size so that a halo mass of @xmath57 m@xmath58 would have @xmath59 particles . this way according to @xcite we are able to estimate the gas fraction in @xmath60 m@xmath58 halos correctly ( see below for the halo definition ) .
our softening length is 0.2 comoving kpc . for all runs ,
glass - like cosmological ics were generated using the zeldovich approximation .
the transfer functions were generated using the various models described above .
we have used a glass file which was randomly displaced thus removing the coupling between nearby dm and gas particles . using this randomization procedure
we achieve essentially the same effect to that shown in @xcite . in generating the ics ,
a convolution between the glass file and the transfer function from the different models was done , thus taking into account the different velocities of the dm and baryons ( for the fiducial and mean cs models ) .
we note that we have used the same phases for the dm and baryons , in all of the simulations .
we set the initial temperature to be @xmath61 k ( as derived from linear theory ) , and thus gadget assumes neutral and monoatomic gas , and converts to thermal energy ( i.e. , adiabatic initial conditions ) .
although this work emphasizes the need for a precise calculation of the baryon overdensities resulting from temperature fluctuations , we actually neglect the temperature fluctuations in the initial conditions .
this may not be a bad approximation since the halos we study are already somewhat non - linear at our initial redshift , and the compton heating is quite small compared to the adiabatic heating during non - linear gravitational collapse ( see appendix [ cheating ] )
. a more complete treatment would be to include in the simulation the precise temperature fluctuations , which we leave for future work .
nevertheless , even with the current treatment our results show consistency with linear theory .
we locate dark matter halos by running a fof group - finder algorithm with a linking parameter of @xmath62 .
we then find the center of mass of each halo and calculate the density profile of the dark matter and baryons , separately . in order to derive the density profile we assume a spherical halo , and divide it to 2000 shells .
combining these density profiles , we find the virial radius @xmath63 at which the overdensity is @xmath64 times the background density , and the gas fraction of each halo .
recently , @xcite performed a resolution analysis in order to study the mass definition of halos in simulations .
their conclusion ( their figure 2 ) is that using the fof algorithm and assuming about 500 particles per spherical halo introduces an error of @xmath65 in the mass definition . in our gas fraction analysis we have chosen only halos with a number of particles larger or equal to @xmath66 , i.e. , we limit our errors in halo mass definition to below @xmath67 . also , according to @xcite , this way we can estimate the gas fraction inside a halo accurately .
one way to probe cosmic structure particularly on small scales is through the non - linear power spectrum .
we begin our simulation at @xmath23 with linear initial conditions overdensities are already slightly non - linear .
the effect of starting the simulation at high redshifts is studied elsewhere ( naoz et .
al , in prep . ) ] .
the main disagreement between the three models lies in the baryonic component ( although the e-@xmath2 calculation also underestimates the dark matter overdensities by @xmath68 ) .
this input difference is then modified by the non - linear evolution . ,
@xmath69 and @xmath70 . ,
width=317 ] following @xcite we compared the linear power spectrum for the fid model , as computed from @xcite , for the dark matter and baryon components , with the non - linear power spectra from the simulation ( see figure [ fig : lin_non ] ) .
the two power spectra agree well as expected in the linear regime .
we note that the other two models approach the fid model at low redshifts ( see appendix [ sig8 ] figure [ fig : psev ] ) .
fig [ fig : nonlinear ] shows the differences among the fid , e-@xmath2 and mean @xmath4 ics , in terms of the non - linear power spectra at the later redshifts at which halos were formed in our simulation .
the mean @xmath4 model maintains over time roughly the same level of discrepancy with the fid model , while in the e-@xmath2 model both the baryonic and dark matter differences decline slightly slower than with the inverse scale factor .
as clearly can be seen from figure [ fig : nonlinear ] , the non - linear evolution of halos is still strongly affected by the choice of initial conditions even at redshift @xmath71 . the fid ics @xcite describe the linear evolution consistently and thus represent the best available prescription for the initial conditions . ) at @xmath72 , 15 , and 12 ( from bottom to top ) ; curves are denoted as in figure [ fig : ics ] .
note that we have plotted here the absolute value ; the mean @xmath4 model underestimates and the e-@xmath2 model overestimates the power spectrum compared to the fid model .
, width=317 ] studying the galaxy evolution and reionization either by using simulations ( both amr and sph ) or by using analytical calculations relies on knowing the amount of gas within the dark matter halos .
the simplest assumption , often used in the literature , is that a dark matter halo has the mean cosmic fraction .
this can lead to incorrect results , especially when one tries to study star formation , galaxy mergers , and related phenomena .
consider the various scales involved in the formation of non - linear objects containing dm and gas . on large scales ( small wavenumbers )
gravity dominates halo formation and gas pressure can be neglected . on small scales , on the other hand
, the pressure dominates gravity and prevents baryon density fluctuations from growing together with the dark matter fluctuations .
the relative force balance at a given time can be characterized by the @xcite scale , which is the minimum scale on which a small gas perturbation will grow due to gravity overcoming the pressure gradient .
as long as the compton scattering of the cmb on the residual free electrons after cosmic recombination kept the gas temperature coupled to that of the cmb , the jeans mass was constant in time . however , at @xmath73 the gas temperature decoupled from the cmb temperature and the jeans mass began to decrease with time as the gas cooled adiabatically .
any overdensity on a scale more massive than the jeans mass at a given time can begin to collapse , due to a lack of sufficient pressure .
however , the jeans mass is related only to the evolution of perturbations at a given time . when the jeans mass itself varies with time , the overall suppression of the growth of perturbations depends on a time - averaged jeans mass .
@xcite defined a `` filtering mass '' that describes the highest mass scale on which the baryonic pressure still manages to suppress the linear baryonic fluctuations significantly .
@xcite suggested , based on a simulation , that the filtering mass also describes the largest halo mass whose gas content is significantly suppressed compared to the cosmic baryon fraction .
the latter mass scale , in general termed the `` characteristic mass '' , is defined as the halo mass for which the enclosed baryon fraction equals half the mean cosmic fraction .
thus , the characteristic mass distinguishes between gas - rich and gas - poor halos .
many semi - analytical models of dwarfs galaxies often use the characteristic mass scale in order to estimate the gas fraction in halos ( e.g. , * ? ? ?
? * ; * ? ? ?
theoretically this sets an approximate minimum value on the mass that can still form stars
. in linear theory the filtering mass , first defined by @xcite , describes the highest mass scale on which the baryon density fluctuations are suppressed significantly compared to the dark matter fluctuations .
@xcite included the fact that the baryons have smoother ics than the dark matter ( see * ? ? ? * ) and found a lower value of the filtering mass ( by a factor of @xmath74 , depending on the redshift ) .
following @xcite , the filtering scale ( specifically , the filtering wavenumber @xmath75 ) is defined by expanding the ratio of baryonic to total density fluctuations to first order in @xmath76 : @xmath77 where @xmath31 is the wavenumber , and @xmath21 and @xmath78 are the baryonic and total ( i.e. , including both baryons and dark matter ) density fluctuations , respectively .
the parameter @xmath79 ( a negative quantity ) describes the relative difference between @xmath21 and @xmath78 on _ large scales _
@xcite , i.e. , @xmath80 where @xmath81 , ( see also * ? ? ?
the ratio @xmath79 is independent of @xmath31 , and its magnitude decreases with time approximately @xmath82 , since @xmath83 is roughly constant and @xmath78 is dominated by the growing mode @xmath84 ( see figure 1 top panel in * ? ? ?
the filtering mass is defined from @xmath75 simply as : @xmath85 where @xmath86 is the mean matter density today .
this relation is one eighth of the definition in @xcite ( who also used a non - standard definition of the jeans mass ) . in figure
[ fig : mc ] ( bottom panel ) we show the filtering mass ( solid curve ) resulting from eq .
( [ mf ] ) , as calculated in @xcite ( see also their figure 3 ) . for each of the models we calculate the filtering mass as described here , assuming the model s initial conditions .
since the simulation is limited in box size , all of the perturbations on large scales are effectively frozen in the simulation .
therefore , we do not extract @xmath79 directly from the simulations , but instead calculate it based on the initial conditions as @xmath87 , where the subscript `` in '' refers to initial .
thus , for example , for the e-@xmath2 case , @xmath88 .
figure [ fig : mc ] ( bottom panel ) shows the analytical results of the filtering mass for the fid calculation , the mean @xmath4 approximation and e-@xmath2 ( solid , dashed and dotted curves , respectively ) .
since the fid calculation is the most consistent calculation , we compare the two other models to it .
the filtering mass represents the competition between gravity and pressure , as it measures the largest scale at which pressure has had a significant overall effect on halo formation .
since it measures an integrated effect over the formation , this mass scale is also very sensitive to the evolution history and the initial conditions ( as shown in * ? ? ?
* ) . in the mean @xmath4 model ,
the temperature fluctuations are greatly overestimated on all relevant scales ( see * ? ? ?
* ) , while in reality the coupling to the cmb ( in the fid model ) keeps the temperature fluctuations highly suppressed for some time after recombination .
moreover , as mentioned in section [ sec : basic ] ( and see also appendix [ cheating ] ) , we do not include explicitly the effect of initial temperature fluctuations in the simulations .
however , the temperature fluctuations from higher redshifts influence the baryon density at the initial redshift ( see figure [ fig : ics ] ) and suppress the baryon density on small scales . as demonstrated in @xcite the system remembers the initial conditions . in other words ,
the initially enhanced filtering mass ( compared to the fid model ) helps maintain a higher filtering mass even at moderately low redshift . in the e-@xmath2 model ,
the baryon perturbations start out much higher than in the other models , so one might expect that the final baryon fraction in halos would tend to be higher as well ; here , however , it is important to separate two issues .
the high initial baryon perturbations in the e-@xmath2 model are present at all scales , so they affect even high - mass halos that are unaffected by pressure
. this can explain why the simulation with the e-@xmath2 ics produced the highest baryon fraction in high - mass halos ( see the top panel of figure [ fig : mc ] ) .
however , when we consider the differences between large and small scales , the high baryon perturbations produce a large pressure term , increasing the effect of pressure relative to gravity and producing a higher filtering mass in the e-@xmath2 model than in the fid model . note
that the filtering mass is particularly sensitive to the importance of pressure at the very highest redshifts ( above 100 ) , since at lower redshifts the gas cools and the jeans mass decreases , reducing the contribution of these redshifts to the final filtering mass .
we note that in @xcite the calculation of the filtering mass in the fiducial model was compared to the time integrated filtering mass in a model that assumes the mean speed of sound model , neglects the @xmath79 factor , and starts out with initial conditions as in the e-@xmath2 model . here , we have separated our discussion into several different cases .
there is no apriori reason to think that the filtering mass can also accurately describe properties of highly non - linear , virialized objects . for halos
, @xcite defined a characteristic mass @xmath89 for which a halo contains half the mean cosmic baryon fraction @xmath90 . in his simulation
he found the mean gas fraction in halos of a given total mass @xmath91 , and fitted the simulation results to the following formula : @xmath92^{-3/\alpha } \ , \ ] ] where @xmath93 is the gas fraction in the high - mass limit . in this function
, a higher @xmath94 causes a sharper transition between the high - mass ( constant @xmath95 ) limit and the low - mass limit ( assumed to be @xmath96 ) .
@xcite found a good fit for @xmath97 , with a characteristic mass that in fact equaled the filtering mass by his definition . by our definition ,
the claim from @xcite is that @xmath98 .
@xcite found that , given their errors , the filtering mass from linear theory is consistent with the characteristic mass fitted from the simulations , for two ( pre - reionization ) scenarios that they tested : the nouv case ( i.e. , no stellar heating ) and the flash case ( i.e. , after a sudden flash of stellar heating ) .
for clarity , we emphasize that this statement ( @xmath99 ) refers to our definition of @xmath100 in equation ( [ mf ] ) .
the characteristic mass is essentially a non - linear version of the filtering mass , and so it also measures the competition between gravity and pressure . at high masses , where pressure is unimportant , @xmath101 , while the low mass tail is determined by the suppression of gas accretion caused by high baryonic pressure . ; different panels show @xmath89 , @xmath94 , and @xmath93 .
we consider the fiducial calculation , mean @xmath4 approximation and the e-@xmath2 model ( boxes , triangles and circles , respectively ) , where we fit equation ( [ f_g - alpha ] ) to all data points from halos with at least 500 particles . in the bottom panel we also show the analytical calculation following @xcite , for all the models , assuming the same ics as in the simulations ( solid , dashed and dotted curves for fid , mean @xmath4 and e-@xmath2 , respectively ) .
we note that at @xmath102 the mean @xmath4 and the e-@xmath2 models have the same value of @xmath89 , and that the fid model and the e-@xmath2 overlap at @xmath103 .
we also note that the data for @xmath104 was unavailable due to a computer failure.,width=317 ] a major conclusion of the simulation results is that different ics result in different gas fractions in the final halos .
specifically , we measure these differences through the characteristic mass at various redshifts . varies for different ics .
we determine for each simulation output the characteristic mass and the parameter @xmath94 using a two - dimensional fit to equation ( [ f_g - alpha ] ) , with @xmath105 separately fixed to equal the average of the highest few mass bins ( see appendix [ app ] for a complete description of the fitting process , together with the @xmath106 errors ) . in figure
[ fig : mc ] we show @xmath105 , @xmath94 and @xmath89 , for all the simulated cases .
the characteristic mass clearly depends on the initial conditions , with the mean @xmath4 model and e-@xmath2 model both yielding gas suppression at systematically higher halo masses then for the fid model .
the parameter @xmath94 shows a less clear pattern with redshift , but it is generally lowest for the fid model .
overall , the most important implication is that the gas fraction in halos is highly sensitive to the assumed initial conditions .
comparing to linear theory allows us to understand some of these results . as noted in section [ sec : mf ] , we calculated the filtering mass from linear theory for each of the ics , and the linear calculation allows us to understand the relative importance of pressure in the various ic models , at least during the linear evolution .
although the simulation results come from non - linear , viralized halos , we find an approximate agreement ( typically to within @xmath107 ) between the filtering mass , as defined here and in our previous work @xcite , and the characteristic mass as measured in the simulation , for all the models .
in particular , the relative sizes of @xmath89 among the various models , and the slow decline of all the characteristic masses with time , are well matched by the corresponding @xmath100 values predicted from linear theory .
this close match can be understood from the fact that while both gravity and pressure increase during the non - linear evolution , their relative strength only changes by a relatively small factor as a halo undergoes non - linear collapse and virialization .
halos in which pressure had a large effect during the early , linear evolution stage , keep sufficient pressure to maintain the suppressed baryon content all through the final collapse . on the other hand , in more massive halos in which gravity overcame pressure early on , the baryons keep up with the collapse of the dark matter and the pressure never has a major role . for the e-@xmath2 alternative model , we find that the resulting characteristic mass is higher than the result in the fid model .
specifically , at @xmath51 we find @xmath108 m@xmath6 and @xmath109 ) .
this can be understood since setting the gas fluctuations to be equal to the dark matter s means that the pressure of the gas is higher compared to the fid model .
as can also be seen from comparison to linear theory , the system retains the memory of the pressure , due to the time integrated nature of the filtering mass .
therefore , the higher pressure translates to a higher filtering / characteristic mass .
the mean @xmath4 approximation starts with effectively smoother ics than in the fid model ( @xmath107 underestimate of the small - scale baryon overdensity ) .
thus , the baryonic components lag behind the dark matter collapse , and the pressure is always overestimated for a given baryon overdensity ( due to the overestimated temperature fluctuations ) , resulting in a lower gas fraction for any given halo mass , i.e. , the characteristic mass is higher than in the fid model .
specifically , at @xmath51 we find @xmath110 m@xmath6 and @xmath109 .
this can be compared with @xmath111 m@xmath6 and @xmath112 for the fid ics .
recently , @xcite and @xcite showed that the characteristic mass scale does not agree with the @xcite filtering mass in the low - redshift , post - reionization regime .
however , it is important to note that at these low redshifts , the heating / cooling and other feedback mechanisms are complex and highly inhomogeneous , so that the `` filtering mass '' calculated from linear theory is not really precisely defined , and the comparison of the linear and non - linear results can not really be considered a direct and precise test . in contrast , @xcite found that the filtering mass gives a good approximation to the characteristic mass , even in the presence of a `` flash '' heating event ( see also * ? ? ? * ) that is physically somewhat contrived but allows for a clear comparison of the linear and non - linear results .
summarizing our results , we find a good agreement between the characteristic mass and the filtering mass in all the models .
figure [ fig : mc ] shows the best fitted parameters at various redshifts for @xmath89 and @xmath94 , and our value for @xmath93 , for all models ( the 1-@xmath113 ( @xmath114 ) confidence regions are listed in table [ table1 ] ) .
it is important to emphasize that in this statement we are referring to our definition in equation ( [ mf ] ) , which is one eighth of the original definition which @xcite claimed was a good fit to the characteristic mass .
while we have been careful to select halos with at least 500 particles , based on the results of @xcite , we do not have the even higher mass resolution needed to perform a resolution convergence test as they did .
our main conclusion is that at least in the redshift range @xmath1 the filtering mass provides a fairly good estimate for the characteristic mass .
this extends the redshift range of the agreement between the filtering mass and the characteristic mass found in @xcite ( @xmath115 ) .
another significant result from this agreement is that previous work ( either analytical , semi - analytical , or using simulations ) that used the filtering or characteristic mass without accounting for the correct initial conditions resulted in inaccurate results .
this is due to the significant ( factor of 23 ) variation among the predictions of the filtering / characteristic mass in the various models .
since this mass scale is of prime importance in early structure formation it is imperative to calculate it accurately .
we have used three - dimensional hydrodynamical simulations to investigate the effect of different initial conditions on the gas fraction in halos in the early universe .
specifically , we studied the minimum `` gas - rich '' mass defined to have half of the mean cosmic baryon fraction .
we tested three different models for the initial conditions ( see text for more details ) 1 . `` fid '' ics ; this model is based on the linear evolution from @xcite , which allows the baryonic speed of sound to spatially vary as a result of the compton scattering with the cmb .
`` e-@xmath2 '' ics ; in this model , the linear evolution from @xcite is modified to match a common assumption in the literature , where the linear initial overdensity of the baryons is taken to be equal to that of the dm , i.e. , @xmath116 , while conserving @xmath3 from the fid model .
3 . `` mean @xmath4 ics '' ; this model assumes that the baryonic speed of sound is spatially uniform .
although @xcite showed that this assumption yields an inaccurate evolution of the baryon density and temperature perturbations , it is still often used in codes that generate initial conditions for simulations .
for all of the tests we used a total of @xmath54 particles of dark matter and baryons with a box size of @xmath56 mpc , starting at @xmath23 .
there are two major findings from the analysis we present here .
the first , shown throughout the paper , is the importance of assuming the correct initial conditions , both for analytical calculations and numerical simulations .
structure formation ( both in the linear and non - linear regime ) and halo gas fractions are very sensitive to the initial conditions even at relatively low redshifts ( @xmath117 ) .
the second major finding is the apparent agreement between the filtering mass and the characteristic mass ( to within @xmath107 ) .
this suggests , as a broader implication , that one can use linear theory in order to predict the overall trend of highly non - linear behavior ( at least in the case of determining the gas fraction of halos ) .
the the fiducial calculation , which was presented in @xcite , follows the time evolution of the linear overdensities correctly .
however , the other ics produce different results for the baryonic structure formation .
for instance , the non - linear power spectrum ( fig . [ fig : nonlinear ] ) shows that the system still remembers its initial condition differences even at redshift 15 .
in particular , the @xmath4 model underestimates the non - linear baryonic fluctuations by about @xmath118 while the e-@xmath2 model overestimates them by @xmath119 on small scales . the mean @xmath4 approximation and the e-@xmath2 model are often used to set the initial conditions in simulations , e.g.
, the cmbfast code @xcite assumes the mean @xmath4 approximation while @xcite is used with the e-@xmath2 assumption .
we have shown that the non - linear evolution is very sensitive to the initial conditions ( figure [ fig : nonlinear ] ) and they affect the gas fraction in small halos down to redshift @xmath117 ( figure [ fig : mc ] ) .
our results emphasize the importance of the differences between the dark matter and baryons and of the spatial sound speed fluctuations , in both the linear calculation and the initial conditions of the simulations .
it is important to emphasize that although compton heating is not included in the gadget code that we used in this analysis ( gadget-2 ) , the fiducial calculation still describes fairly well the non - linear behavior .
actually , the compton heating contribution to the heating of the gas in non - linear objects is negligible compare to the adiabatic heating due to the gravitational collapse ( see appendix [ cheating ] ) .
also , as noted above , much of the contribution to the filtering mass comes from the highest redshifts , above our simulation starting redshift of 99 , since the jeans mass is highest then and so the pressure has the greatest impact at that early time . in each simulation , we calculated the characteristic mass for which a halo keeps most of its baryons ( eq . [ f_g - alpha ] ) .
we found that the fid calculation gives the lowest value , which suggests that with these correct ics , pressure plays only a moderate role in galaxy formation .
in particular , the characteristic mass of @xmath5 m@xmath6 is significantly below the minimum mass for molecular hydrogen cooling , so the gas content is not strongly suppressed even in the smallest star - forming halos . in other words
we find that before significant heating took place the baryon fraction in halos is ( eq . [ f_g - alpha ] with @xmath120 m@xmath58 and @xmath121 ) @xmath122^{-75/16 } \ .\ ] ] the other alternative models give incorrect higher value for the characteristic mass , closer to the minimum mass for forming stars . even with the fid ics , pressure does strongly limit the amount of gas in minihalos below the molecular hydrogen cooling mass .
we note that this value of @xmath123 m@xmath6 assumes adiabatic evolution , in particular with no stellar heating .
this value is consistent with the results of @xcite for a somewhat higher redshift range .
we find that the theoretical linear filtering mass ( as defined in section [ sec : mf ] ) is in fairly good agreement with the characteristic mass .
this finding is true for all the models tested here , throughout a significant redshift range , so this may imply more generally a close relation between linear theory and non - linear halo formation .
in addition , this is consistent with the findings by @xcite from amr simulations , where the filtering mass and the characteristic mass agreed in the `` e-@xmath2 '' model , even when a sudden heating was introduced .
finally , we emphasize that our results are valid only in the pre - reionization era . at the end of the reionization
, @xcite concluded that the characteristic mass is likely to be close to the atomic - cooling threshold of @xmath124 , which is also close to the values found by @xcite and @xcite .
recently @xcite argued that the initial velocity difference between the baryons and dark matter after recombination has not been fully accounted for , because of a higher - order contribution that is not included in the linear theory approach .
they estimated this higher - order effect within the mean @xmath4 approximation and found that it causes an additional suppression of the small - scale power spectrum , in turn affecting the formation of the first structures .
this effect should be further investigated as in our detailed approach here , although this would be more difficult ( analytically , it is a higher - order and anisotropic term , and to simulate it directly would require starting at quite high redshifts ) .
we thank the anonymous referee for useful and helpful comments . we thank ikkoh shimizo for supplying the code for calculating the non - linear power spectrum .
we also thank nick gnedin , andrey kravtsov , matt mcquinn and michele trenti for useful discussions .
sn also expresses special thanks to yoram lithwick for interesting discussions .
the authors acknowledge financial support by the grants - in - aid for young scientists ( s ) 20674003 by the japan society for the promotion of science .
sn acknowledges nasa atp grant nnx07ah22 g and in part the german - israeli project cooperation ( dip ) grant ste1869/1 - 1.ge625/15 - 1 and the generous support of the national post doctoral award program for advancing women in science ( weizmann institute of science ) .
rb is grateful for the support of israel science foundation grant 823/09 .
abel t. , bryan g. l. , norman m. l. , 2002 , sci , 295 , 93 barkana r. , loeb a. , 2002 , apj , 578 , 1 barkana r. , loeb a. , 2004 , apj , 609 , 474 barkana r. , loeb a. , 2005 , mnras , 363 , l36 benson a. j. , frenk c. s. , lacey c. g. , baugh c. m. , cole s. , 2002a , mnras , 333 , 177 benson a. j. , lacey c. g. , baugh c. m. , cole s. , frenk c. s. , 2002b , mnras , 333 , 156 bromm v. , coppi p. s. , larson r. b. , 2002 , apj , 564 , 23 bromm v. , coppi p. s. , larson r. b. , 1999 , apj , 527 , l5 bromm v. , loeb a. , 2006 , apj , 642 , 382 bromm v. , yoshida n. , hernquist l. , mckee c. f. , 2009 , natur , 459 , 49 bouwens r. j. , et al . , 2010 , apj , 709 , l133 bullock j. s. , kravtsov a. v. , weinberg d. h. , 2000 , apj , 539 , 517 clark p. c. , glover s. c. o. , klessen r. s. , bromm v. , 2010 , arxiv , arxiv:1006.1508 ciardi b. , scannapieco e. , stoehr f. , ferrara a. , iliev i. t. , shapiro p. r. , 2006 , mnras , 366 , 689 duffy a. r. , schaye j. , kay s. t. , dalla vecchia c. , battye r. a. , booth c. m. , 2010 , arxiv , arxiv:1001.3447 eisenstein d. j. , hu w. , 1999 , apj , 511 , 5 fan x. , et al . , 2003 , aj , 125 , 1649 furlanetto s. r. , oh s. p. , 2006 ,
apj , 652 , 849 gao l. , yoshida n. , abel t. , frenk c. s. , jenkins a. , springel v. , 2007 , mnras , 378 , 449 gnedin , n. y. & hui , l. 2004 , mnras , 296 .
gnedin n. y. , 2000 , apj , 542 , 535 .
gnedin n. y. , kravtsov a. v. , chen h .- w .
, 2008 , apj , 672 , 765 greif t. h. , glover s. c. o. , bromm v. , klessen r. s. , 2010 , apj , 716 , 510 hoeft m. , yepes g. , gottlber s. , springel v. , 2006 , mnras , 371 , 401 harford a. g. , hamilton a. j. s. , gnedin n. y. , 2008 , mnras , 389 , 880 iliev , i. t. , scannapieco , e. , martel , h. , & shapiro , p. r. 2003 , mnras , 341 , 81 iliev i. t. , scannapieco e. , shapiro p. r. , 2005 , apj , 624 , 491 jeans , j. h. cambridge university press , ( 1928 ) .
komatsu e. , et al . , 2009 ,
apjs , 180 , 330 komatsu e. , et al . , 2010 , arxiv , arxiv:1001.4538 kuhlen m. , madau p. , montgomery r. , 2006 , apj , 637 , l1 iye m. , et al . , 2006 , natur , 443 , 186 lin w. p. , jing y. p. , mao s. , gao l. , mccarthy i. g. , 2006 , apj , 651 , 636 lin l. , liang e. , zhang s. 2010 , scchg , 53 , 64 ma , c. & bertschinger , e. 1995 , apj , 455 , 7 mesinger a. , bryan g. , & haiman z. 2006 , apj , 648 , 835 mesinger a. , dijkstra m. , 2008 , mnras , 390 , 1071 mclure r. j. , dunlop j. s. , cirasuolo m. , koekemoer a. m. , sabbi e. , stark d. p. , targett t. a. , ellis r. s. , 2010 , mnras , 126 mcquinn m. , lidz a. , zahn o. , dutta s. , hernquist l. , zaldarriaga m. , 2007 , mnras , 377 , 1043 morales m. f. , wyithe j. s. b. , 2009 , arxiv , arxiv:0910.3010 naoz s. , barkana r. , 2005 , mnras , 362 , 1047 naoz s. , noter s. , barkana r. , 2006 , mnras , 373 , l98 naoz s. , barkana r. , 2007 , mnras , 377 , 667 naoz s. , barkana r. , 2008 , mnras , 385 , l63 naoz s. , barkana r. , mesinger a. , 2009 , mnras , 377 , 667 naoz s. , bromberg o. , 2007 , mnras , 380 , 757 okamoto t. , gao l. , theuns t. , 2008 , arxiv , 806 , arxiv:0806.0378 ricotti m. , gnedin n. y. , shull j. m. , 2002 , apj , 575 , 49 ricotti m. , gnedin n. y. , shull j. m. , 2008 , apj , 685 , 21 salvaterra r. , et al .
, 2009 , natur , 461 , 1258 u. , seljak , & m. , zaldarriaga 1996 , apj , 469 , 437 ; see http://www.cmbfast.org shapiro p. r. , ahn k. , alvarez m. a. , iliev i. t. , martel h. , ryu d. , 2006 , apj , 646 , 681 shapiro p. r. , iliev i. t. , raga a. c. , 2004 , mnras , 348 , 753 somerville r. s. , 2002 , apj , 572 , l23 springel v. , 2005 , mnras , 364 , 1105 spergel d. n. , et al . , 2003 , apjs , 148 , 175 spergel d. n. , et al . , 2007 , apjs , 170 , 377 springel v. , yoshida n. , white s. d. m. , 2001 , newa , 6 , 79 tanvir n. r. , et al .
, 2009 , natur , 461 , 1254 trenti m. , smith b. d. , hallman e. j. , skillman s. w. , shull j. m. , 2010 , arxiv , arxiv:1001.5037 trenti m. , stiavelli m. , michael shull j. , 2009 , apj , 700 , 1672 trenti m. , stiavelli m. , 2009 , apj , 694 , 879 trenti m. , shull j. m. , 2010 , apj , 712 , 435 tseliakhovich d. , hirata c. , 2010 , arxiv , arxiv:1005.2416 trenti m. , santos m. r. , stiavelli m. , 2008 , apj , 687 , 1 yamamoto , k. , sugiyama , n. , & sato , h. 1997 , prd , 56 , 7566 yamamoto , k. , sugiyama , n. , & sato , h. 1998 , apj , 501 , 442 yoshida n. , 2006 , newar , 50 , 19 yoshida n. , 2009 , arxiv , arxiv:0906.4372 yoshida n. , sokasian a. , hernquist l. , springel v. , 2003 , apj , 598 , 73 yoshida n. , omukai k. , hernquist l. , abel t. , 2006 , apj , 652 , 6 yoshida n. , abel t. , hernquist l. , sugiyama n. , 2003 , apj , 592 , 645 yoshida n. , oh s. p. , kitayama t. , hernquist l. , 2007 , apj , 663 , 687 yoshida n. , omukai k. , hernquist l. , 2008 , sci , 321 , 669 yoshida n. , sugiyama n. , hernquist l. , 2003 , mnras , 344 , 481
we have defined the two different models such that they conserve @xmath125 . from linear theory
we do not expect the evolution of the mean @xmath4 model to be significantly different from that of the fid model ( in terms of halo abundance and total power spectrum ) .
this is indeed the case for the evolution in time of the total fluctuations of the mean @xmath4 model compared to the fid model on large scales ( small @xmath31 ) , as shown in figure [ fig : psev ] ( lower set of thin curves ) .
conservation . the ratio of the _ total _ non - linear fluctuation for different redshifts ( specifically @xmath126 ) .
we show the e-@xmath2 ( upper set of @xmath127 curves ) and the mean @xmath4 model ( lower set of @xmath127 thin curves ) .
we consider redshift @xmath128 and @xmath70 ; dotted , short dashed , long dashed and solid curves respectively ( labeled for the e-@xmath2 model ) . ,
width=317 ] a more delicate treatment is needed for the e-@xmath2 model ( see section [ sec : edelta ] ) . in this case , at high redshift ( such as the initial @xmath129 ) , the baryons are in the process of falling into the dm potential .
this results in a faster growth of the total fluctuations compared to the case in which there is a relative velocity between the dm and the baryons ( such as in the case of the mean cs and fid models , where the relative velocity for the e-@xmath2 model are negligible ) ; see figure [ fig : psev ] dotted thick curve . at later times , the baryon fluctuations approach the dark matter fluctuations , and the large scale behavior ( i.e. , on linear scales ) deviates from the fid model by less then @xmath130 ( see the solid curve in fig .
[ fig : psev ] ) .
we also note that we have checked the overall effect of @xmath131 on the main results .
we have performed two additional simulations for the e-@xmath2 model , where we increased or decreased @xmath3 by @xmath132 .
we found that the calculated @xmath89 is within the fit errors ( see appendix [ app ] and table [ table1 ] ) at @xmath133 . at @xmath134 ,
the difference in the best fitted value is below @xmath135 . "
l c c c redshift & @xmath89 [ @xmath136 m@xmath6 ] & @xmath94 + + & * fiducial * & + & * calculation * & + @xmath69 & @xmath138 & @xmath139 + @xmath140 & @xmath141 & @xmath142 + @xmath143 & @xmath144 & @xmath145 + @xmath146 & @xmath147 & @xmath148 + @xmath149 & @xmath150 & @xmath151 + @xmath152 & @xmath153 & @xmath154 + @xmath155 & @xmath156 & @xmath157 + @xmath158 & @xmath159 & @xmath160 + @xmath71 & @xmath161 & @xmath162 + @xmath70 & @xmath163 & @xmath164 + & * mean * * @xmath4 * & + @xmath69 & @xmath165 & @xmath166 + @xmath140 & @xmath167 & @xmath168 + @xmath143 & @xmath169 & @xmath170 + @xmath146 & @xmath171 & @xmath172 + @xmath149 & @xmath173 & @xmath174 + @xmath152 & @xmath175 & @xmath176 + @xmath155 & @xmath177 & @xmath178 + @xmath158 & @xmath179&@xmath180 + @xmath71 & @xmath181&@xmath182 + @xmath70 & @xmath183&@xmath184 + & * e-@xmath2 * & + @xmath69 & @xmath185 & @xmath186 + @xmath140 & @xmath187 & @xmath188 + @xmath143 & @xmath189 & @xmath190 + @xmath146 & @xmath191 & @xmath192 + @xmath149 & @xmath193 & @xmath194 + @xmath152 & @xmath195 & @xmath196 + @xmath155 & @xmath197 & @xmath198 + @xmath158 & @xmath199&@xmath200 + @xmath71 & @xmath201&@xmath202 + @xmath70 & @xmath203&@xmath204 + and @xmath205 ) .
we consider the fiducial calculation , mean @xmath4 approximation and the e-@xmath2 model ( boxes , triangles and circles , respectively ) , where we fit equation [ f_g - alpha ] to all data points from halos with at least 500 particles .
we also show the fits from table [ table1 ] ( dotted curves ) .
, width=317 ] for each redshift snapshot for each run we find the characteristic mass and @xmath94 using a two dimensional fit . in figure
[ fig : fit ] we consider two example redshifts ( high , @xmath206 and low , @xmath205 ) for which we show the binned data points and the resulting fit . in table
[ table1 ] we show our best fit parameters .
we note that we have checked that the fits give consistent results if we lower the condition on the minimum number of particles per halo to 300 ( instead of 500 ) .
we also note that our determination of @xmath89 relies on an extrapolation ( via the fit ) below our simulations resolution limit the parameter @xmath105 in equation ( [ f_g - alpha ] ) is an average of the gas fraction values in the few highest mass bins . in our simulation
the high - end tail of the masses has large scatter in the estimated gas fraction because of the low number of halos ( each bin among the last 3 or 4 in figure [ fig : fit ] represents just 1 or 2 halos ) , thus we have to average over this scatter to get a reasonable result .
this scatter is in part a result of assuming that the halos are spherical , and thus halos that are undergoing a major merger deviate greatly from a spherical shape and are treated inaccurately in our analysis .
we have tested the resulting @xmath105 when taking a linking parameter of @xmath207 , which indeed resulted in more high - mass halos , but in any case was consistent with the value of @xmath105 we found with the @xmath62 linking parameter .
thus , in this paper , we use the standard value of @xmath62 .
as expected at high redshift , where we have fewer halos , the errors become quite large .
we also tried , following @xcite , to bin the data and to perform the fit for the binned data with the @xmath106 weight for each bin . for the redshifts for which we had more than @xmath208 halos we got that the binned analysis gave results within the non - binned fit errors , and with comparable errors .
we also tried the approach of taking @xmath93 to be a free parameter , but this produced very problematic fits as a free parameter was unproductive . ] .
this is mainly because of the large scatter at the high mass end , so that a three - parameter fit could not strongly constrain the parameter values .
we also note the fact that @xmath105 is lower than the mean cosmic fraction @xmath209 , by about @xmath210 - @xmath211 for the fid and mean @xmath4 models , and @xmath212 for the e-@xmath2 model ( see figure [ fig : mc ] top panel ) .
the result in the fid and mean @xmath4 models may reflect the real suppression of the large - scale baryon fluctuations in these models ; the difference in linear theory is @xmath213 at @xmath51 @xcite , but the non - linear evolution may increase this effect .
the discrepancy in the e-@xmath2 model may reflect a limitation of the simulation ; we note that in @xcite @xmath105 was also lower than @xmath209 and even lower by @xmath214 from our results at the overlapping redshifts ( where we compare the e-@xmath2 model in both cases )
. this might be due to the fact that gas shocks in amr are sharper than in gadget simulations , and thus amr may produce a more realistic gas profile , although the result is still below the universal cosmic baryon fraction @xcite . in our simulation ,
going to a larger radii can result in a more realistic value , but we used @xmath215 for consistency with the common definition .
the fiducial model follows correctly the baryon density and temperature perturbations due to compton scattering on the residual free electrons after recombination .
while this is fully incorporated in our fid ics , our simulation does not take into account compton heating .
below we show that for non - linear objects the heating is actually negligible compared to the adiabatic heating due to the gravitational collapse of baryons into the dark matter potential wells .
therefore , it is sufficient to include compton heating in the linear stage only .
the heating of the gas @xmath216 due to compton heating from the cmb @xcite during the free - fall time @xmath217 of gravitational collapse is @xmath218 where @xmath219 is the thomson scattering cross section , @xmath39 is the photon energy density , @xmath220 and @xmath221 are the cmb and gas temperature and @xmath35 is the electron fraction out of the total number density of gas particles at time @xmath36 .
the virial theorem gives a relation in collapsed objects between the thermal energy @xmath222 and the gravitational energy @xmath223 , i.e. , @xmath224 .
thus , for a halo mass @xmath91 with virial radius @xmath63 the thermal energy can be expressed as : @xmath225 for all relevant redshifts and mass scales we find that @xmath226 .
therefore , neglecting the contribution of the compton heating during the non - linear evolution is justified . however , as we have shown , neglecting the compton heating in the linear evolution and in the initial conditions leads to inaccurate values for the gas fraction in halos .
|
we run very large cosmological @xmath0-body hydrodynamical simulations in order to study statistically the baryon fractions in early dark matter halos .
we critically examine how differences in the initial conditions affect the gas fraction in the redshift range @xmath1 .
we test three different linear power spectra for the initial conditions : ( 1 ) a complete heating model , which is our fiducial model ; this model follows the evolution of overdensities correctly , according to @xcite , in particular including the spatial variation of the speed of sound of the gas due to compton heating from the cmb . ( 2 ) an equal-@xmath2 model , which assumes that the initial baryon fluctuations are equal to those of the dark matter , while conserving @xmath3 of the total matter . ( 3 ) a mean @xmath4 model , which assumes a uniform speed of sound of the gas .
the latter two models are often used in the literature .
we calculate the baryon fractions for a large sample of halos in our simulations .
our fiducial model implies that before reionization and significant stellar heating took place , the minimum mass needed for a minihalo to keep most of its baryons throughout its formation was @xmath5 m@xmath6 .
however , the alternative models yield a wrong ( higher by about @xmath7 ) minimum mass , since the system retains a memory of the initial conditions .
we also demonstrate this using the `` filtering mass '' from linear theory , which accurately describes the evolution of the baryon fraction throughout the simulated redshift range .
|
hybrid inflation @xcite is an attractive mechanism for generating the cosmological density perturbations .
it is naturally realized in the framework of grand unified theories ( guts ) and string theories , especially in the form of d - term inflation @xcite where the gut scale emerges via the fayet - iliopoulos ( fi ) term of an anomalous @xmath2 symmetry .
a further important virtue of d - term inflation is that tree - level supergravity corrections to the inflaton mass of order the hubble parameter are absent .
d - term inflation has been quantitatively analyzed for the canonical khler potential as well as for some non - minimal khler potentials @xcite .
the value of the inflaton field is typically @xmath3 and supergravity corrections are therefore important .
in addition to the primordial fluctuations of the inflaton field , the cosmic microwave background is significantly affected by the production of cosmic strings at the end of inflation @xcite .
generically , it appears difficult to obtain agreement with observational data @xcite . in particular the scalar spectral index
@xmath4 turns out to be rather large and the gauge coupling is constrained to small values , in conflict with the motivation of implementing d - term hybrid inflation in guts . in this paper
we study d - term inflation in the context of superconformal supergravity models @xcite which have recently been considered in connection with higgs inflation @xcite .
these models are motivated by the underlying superconformal symmetry of supergravity , and have several intriguing features .
in particular , there is a jordan frame in which the matter part of the lagrangian takes a particularly simple form , closely resembling global supersymmetry . in the einstein frame ,
supergravity corrections to scalar masses are suppressed by powers of @xmath5 , and , contrary to canonical supergravity , the scalar potential does not contain factors which grow exponentially at large field values .
the superconformal symmetry is broken by fixing the value of the conformal compensator field , which generates the kinetic term of the graviton @xcite .
as we shall see , a fayet - iliopoulos term can be introduced analogously . a further explicit
breaking of superconformal symmetry is a holomorphic contribution to the khler potential @xcite .
this turns d - term inflation into a two - field inflation model . as we have recently shown , the spontaneous breaking of @xmath6@xmath7@xmath8 , the difference of baryon and lepton number , at the gut scale can explain the initial conditions of the hot early universe including baryogenesis and dark matter @xcite .
this analysis was carried out assuming f - term hybrid inflation .
as we shall see , f - term hybrid inflation is inconsistent with superconformal symmetry . on the contrary
, d - term inflation can be implemented with superconformal symmetry and can also incorporate spontaneous @xmath6@xmath7@xmath8 breaking at the gut scale .
the paper is organized as follows . in section [ sec_superconformal_inflation ]
we review the most important features of superconformal models of inflation and in particular discuss the resulting scalar potentials for f- and d - term hybrid inflation . section [ sec_single_field ] deals with an important special case , namely a single - field scenario which arises if inflation lasted sufficiently long before the onset of the final 50 e - folds .
the full two - field inflation model is discussed in section [ sec_two - field ] .
our conclusions are presented in section [ sec_conclusion ] .
an attractive class of supergravity models can be defined by requiring the matter sector and its couplings to supergravity to be invariant under superconformal transformations . the superconformal symmetry is explicitly broken only by the pure supergravity part of the action and the superconformal anomaly .
matter interactions at energies below the planck mass then obey the superconformal symmetry up to corrections suppressed by inverse powers of the planck mass and radiative corrections @xcite .
for these theories there exists a jordan frame in which the lagrangian takes a remarkably simple form which closely resembles globally supersymmetric theories . the bosonic part for metric and scalar fields @xmath9
is given by @xcite @xmath10 here the subscript @xmath11 indicates quantities in the jordan frame , @xmath12 , and @xmath13 , with @xmath14 and @xmath15 acting on the so - called frame function @xmath16 which is the coefficient function of the curvature scalar @xmath17 in eq .
( [ lagj ] ) .
the covariant derivative @xmath18 contains an auxiliary gauge field @xmath19 and the dynamical gauge fields @xmath20 , with @xmath21 being the corresponding generators ; @xmath22 are the auxiliary components of the vector superfields .
the scalar potential is determined by the superpotential @xmath23 and the gauge kinetic function @xmath24 , @xmath25 with @xmath26 ; for the frame function ( [ frame ] ) one has @xmath27 .
local weyl invariance requires a cubic superpotential @xmath23 and a constant gauge kinetic function @xmath24 .
it is remarkable that , up to corrections described by the auxiliary field @xmath19 , the matter part of the lagrangian ( [ lagj ] ) is that of global supersymmetry @xcite .
the lagrangian in the einstein frame with metric @xmath28 is obtained by performing the transformation
@xmath29 eliminating also the auxiliary vector field @xmath19 one obtains , up to a total derivative , @xmath30 with @xmath31 and khler potential @xmath32 note that the covariant derivative @xmath33 does not contain the auxiliary field @xmath19 anymore . for the scalar potential in the einstein frame one
obtains ( @xmath34 ) , @xmath35 where @xmath36 is the inverse khler metric , and @xmath37 . in the jordan frame lagrangian ( [ lagj ] ) superconformal symmetry is explicitly broken by the kinetic term of the gravitational field .
full superconformal symmetry can be achieved by introducing a compensator field @xmath38 and replacing the frame function by the @xmath39 invariant real function @xmath40 the choice @xmath41 , which corresponds to fixing a gauge for the local conformal symmetry , then yields the frame function , @xmath42 as suggested in @xcite , given a gauge singlet @xmath43 with @xmath44 dimensionless , superconformal symmetry can be explicitly broken by using instead of ( [ real1 ] ) the real function @xmath45 after gauge fixing one obtains the modified frame function @xmath46 corresponding to a weyl rescaling between jordan and einstein frame with @xmath47 in the following analysis the symmetry breaking term @xmath48 will play an important role . as we shall see , it will turn the familiar single - field d - term inflation model into a two - field inflation model .
we are particularly interested in adding for a @xmath2 gauge symmetry a fi - term to the lagrangian ( [ lagj ] ) .
naively , this would correspond to the substitution @xmath49 , where @xmath50 is the charge generator .
this , however , would introduce another explicit breaking of superconformal symmetry , since @xmath51 is a constant of mass dimension two . in the lagrangian ( [ lagj ] ) superconformal symmetry breaking only arises from @xmath52 after gauge fixing .
this suggests to add to eq .
( [ lagj ] ) a term with dimensionless constant @xmath53 , @xmath54 where @xmath55 has mass dimension two . note that in the jordan frame the fi - term is field dependent .
using eqs .
( [ einstein ] ) and ( [ potej ] ) and eliminating the auxiliary field @xmath56 , one immediately obtains for the d - term scalar potential in the einstein frame , @xmath57 which is the standard supergravity expression @xcite . note that from now on we work in the einstein frame .
the consistency of a constant fi - term in supergravity is a subtle issue @xcite .
we have in mind a field dependent , effectively constant fi - term at the gut scale , as it can arise in the weakly coupled heterotic string due to the green - schwarz mechanism of anomaly cancellation , where @xmath58 @xcite . here
@xmath59 is the sum over @xmath2 charges and @xmath60 is the string coupling , which depends on the dilaton .
clearly , a gut scale fi - term requires an appropriate stabilization of the dilaton and other moduli fields ( see , for example , refs .
a related problem is the connection between the gut scale and supersymmetry breaking .
a thorough discussion of these important questions goes beyond the scope of the present paper . breaking superconformal symmetry by the holomorphic term @xmath11 in eq .
( [ framechi ] ) significantly modifies the scalar potential . from eq .
( [ kahlerp ] ) one obtains for the frame function @xmath61 given in eq .
the khler metric @xmath62 one easily verifies that the inverse khler metric is given by @xmath63 where @xmath64 inserting eq .
( [ kinv ] ) into the expression ( [ potej ] ) , one obtains for the f - term scalar potential in the einstein frame the compact expression @xmath65 clearly , for superpotentials cubic in the fields and @xmath66 , the second term in the bracket vanishes and one obtains the f - term potential of global supersymmetry up to the rescaling factor @xmath67 between jordan and einstein frame . the expression ( [ ftermp ] ) holds for all superpotentials and it is instructive to apply it to the superpotential of f - term hybrid inflation @xcite , @xmath68 here @xmath69 are ` waterfall ' fields , @xmath70 is a mass parameter and @xmath71 contains the inflaton ; the coupling @xmath72 is chosen to be real . f - term hybrid inflation typically yields a scalar spectral index which is too large compared to observation .
one may hope to improve the situation by a proper choice of the @xmath73-parameter of the frame function @xmath74 where we have used the same symbols for chiral superfields and their scalar components ; the parameter @xmath73 is chosen to be real .
this yields a non - minimal coupling of the inflaton field to gravity . from eq .
( [ ftermp ] ) one then obtains for the scalar potential @xmath75 along the expected inflationary trajectory , i.e. , for @xmath76 , one has @xmath77 unfortunately , this potential has a large tachyonic mass for @xmath71 and is therefore not phenomenologically viable .
let us now consider d - term hybrid inflation .
it has the attractive feature that in string compactifications an fi - term of gut scale size naturally arises , which is welcome for hybrid inflation .
the superpotential reads @xmath78 and the frame function is again given by eq .
( [ frameinf ] ) .
the corresponding f - term scalar potential reads @xmath79 this expression agrees with the potential ( [ fpotfterm ] ) in the case @xmath80 . for field values below the planck mass the potential ( [ fpotdterm ] ) is well behaved .
the potential vanishes identically for @xmath76 , which corresponds to the inflationary trajectory . the potential ( [ fpotdterm ] ) is supplemented by a d - term scalar potential of a @xmath2 gauge interaction under which the chiral superfields @xmath71 and @xmath69 have charge @xmath81 and @xmath82 , respectively .
the corresponding scalar potential with nonvanishing fi - term is given by @xmath83 where @xmath28 is the gauge coupling .
for @xmath76 , @xmath84 provides the vacuum energy @xmath85 which drives inflation .
the slope of the inflaton potential is generated by quantum corrections .
along the inflationary trajectory the weyl rescaling factor reads @xmath86 from eqs .
( [ potej ] ) and ( [ kmetric ] ) one then obtains for the part of the lagrangian quadratic in @xmath69 , @xmath87 from which one reads off the scalar masses @xmath88 for @xmath89 larger than a critical value @xmath90 , both @xmath91 and @xmath92 have positive mass terms and are stabilized at zero , thus allowing inflation to proceed in the @xmath71 direction . at @xmath93 , @xmath94 turns negative , triggering a phase transition which gives an expectation value to @xmath91 and ends inflation . the critical value @xmath95 is determined by the relation @xmath96 supersymmetry is broken along the inflationary trajectory where one has @xmath97 .
hence , quantum corrections to the tree - level potential do not vanish and one obtains the one - loop correction @xmath98 \ .
\label{eq_coleman - weinberg}\end{aligned}\ ] ] here @xmath99 denotes the supertrace running over all fields with @xmath71-dependent masses , i.e. , @xmath69 and their fermionic partners .
@xmath100 is the corresponding mass matrix , and @xmath50 is an appropriate renormalization scale which also determines the argument of the running gauge coupling . according to the mass sum rule ,
the dirac fermion associated with @xmath69 has mass @xmath101 inserting eqs . and into eq . and
choosing the renormalization scale @xmath102 , one obtains for the one - loop potential , @xmath103 where @xmath104 the total potential is given by ( cf .
( [ fpotdterm ] ) , ( [ eq_1-loop ] ) ) @xmath105 note that on the inflationary trajectory one has @xmath106 and @xmath107 .
we are now ready to tackle the slow - roll equations of motion for the field @xmath71 .
note that the inflaton field @xmath71 is not canonically normalized , which leads to a modification of the standard slow - roll equations . expressing the lagrangian for the field @xmath71 in terms of real and imaginary components , @xmath108 , @xmath109 one obtains the slow - roll equations for the homogeneous fields @xmath110 and @xmath111 , @xmath112 where
now we have set @xmath113 for convenience .
these equations can be written as the standard slow roll equations for an effective potential defined by @xmath114 calculating the second derivatives of the potential @xmath115 with respect to @xmath110 and @xmath111 , one finds that for @xmath116 , the trajectory @xmath117 yields a viable inflationary trajectory along which @xmath118 is positive .
hence this trajectory is an attractor for a sufficiently long phase of inflation before the onset of the final 50 e - folds . for @xmath119 ,
the situation is reversed and an equivalent inflationary trajectory corresponds to @xmath120 . for @xmath121 ,
the lagrangian is independent of the phase of @xmath71 and the inflaton can be identified as the absolute value of @xmath71 . in the following we choose @xmath122 . in this section
we will restrict ourselves to the standard case of ` one - field ' inflation described above , postponing the discussion of possible two - field inflation to section [ sec_two - field ] . inserting the khler metric @xmath123 and the one - loop potential ( [ eq_1-loop ] ) into the slow - roll equation ( [ eq_double_sr ] ) , one obtains after integrating from @xmath124 to @xmath125 , @xmath126 here @xmath125 denotes the value of @xmath110 at the end of inflation and @xmath124 is the value of @xmath110 @xmath127 e - folds earlier .
inflation ends when either @xmath128 turns negative ( @xmath129 ) or when the slow - roll conditions are violated ( @xmath130 ) . from eq . and eq . with @xmath131
, one finds @xmath132 for small field values , satisfying @xmath133 , eq .
( [ eq_sigmae ] ) can be solved analytically , leading to @xmath134 however , for most of parameter space this is a bad approximation , and one has to solve eq .
( [ eq_sigmae ] ) numerically . in order to calculate the spectral index and other observables
, we need to evaluate the slow - roll parameters @xmath135 here @xmath136 is the canonically normalized inflaton field which is determined by ( cf .
( [ eq_lagrange ] ) ) @xmath137 on the inflationary trajectory the derivatives of the scalar potential with respect to @xmath138 can be written as @xmath139 from which one obtains the slow - roll parameters @xmath140 note that for @xmath141 , one obtains the results for d - term inflation in global supersymmetry . *
normalization of the scalar power spectrum and cosmic strings * the normalization condition for the amplitude of the primordial power spectrum and the cosmic string bound represent observational constraints which have to be fulfilled by a viable model . allowing for a cosmic string contribution to the power spectrum of the primordial fluctuations implies extending the usual six parameter @xmath142cdm fit to the cmb data by an additional parameter which accounts for the cosmic string contribution .
detailed analyses for nambu - goto strings and abelian higgs ( ah ) cosmic strings have been carried out by several groups @xcite . in the waterfall transition ending d - term hybrid inflation ,
a local @xmath2 symmetry is broken and ah cosmic strings may be formed . in the following discussion
we shall therefore use the results of the recent analysis in ref .
@xcite which is based on the field theoretical simulation of cosmic strings in ref .
@xcite . the analysis in ref .
@xcite yields an upper bound on a cosmic string contribution of about @xmath143 and a best - fit value for the amplitude of the scalar contribution to the primordial fluctuations , @xmath144 where a @xmath145 error has been given . comparing this value with the expression calculated in our model , @xmath146 which , using eq .
, can be simplified to @xmath147 \ , \label{eq_as2}\ ] ] one obtains a relation between @xmath51 and @xmath72 for given values of @xmath73 , @xmath148 and @xmath28 . as an example
, we choose @xmath149 and @xmath150 in the following , which is motivated by identifying the spontaneously broken @xmath2 symmetry with @xmath151 ( c.f .
@xcite ) . for @xmath152 , the implied relation between @xmath51 and @xmath72
is represented by the blue line in fig .
[ fig_cobe_strings ] . a fit to the cmb data assuming scalar perturbations and ah cosmic strings yields an upper bound on @xmath153 , where @xmath154 is newton s constant , @xmath155 denotes the string tension , and @xmath156 gives the deviation from the bogomolnyi bound on the string tension @xcite , @xmath157 inserting the vacuum expectation value of the waterfall field , @xmath158 , as well as the masses for the u(1 ) vector boson and the inflaton in the true vacuum , @xmath159 , one obtains @xmath160 this is to be compared with the @xmath1 upper bound found in the analysis of ref .
@xcite , @xmath161 the solid black lines in fig .
[ fig_cobe_strings ] correspond to the string tensions @xmath162 .
the brighter region on the right of a given line is excluded , whereas the darker region on the left is in agreement with the respective bound .
the upper bounds on the string tension have a considerable theoretical uncertainty . for instance , the upper bounds for nambu - goto strings are more restrictive than the ones for ah cosmic strings by about a factor of three @xcite .
this can be traced back to decay channels into massive radiation for ah cosmic strings @xcite .
note also , that all simulations have been done for a bosonic abelian higgs model , whereas in d - term inflation one is considering a supersymmetric theory .
additional fermionic decay channels may further relax the cosmic string bound by a factor @xmath163 .
last but not least , one has to worry about initial conditions .
clearly , strings can not form until the causal horizon is larger than their characteristic width @xcite , and one should remember that tachyonic preheating proceeds very fast .
in fact , the expectation value @xmath164 of the waterfall field grows with time faster than exponentially @xcite . * spectral index * with the slow - roll parameters from eqs . and the value of @xmath110 @xmath127 e - folds before the end of inflation , cf .
eq . , at hand , we can now easily calculate the spectral index , @xmath165 fig .
[ fig_ns ] shows the resulting @xmath73 dependence for a ( @xmath166 ) pair compatible with the cosmic string bound and the normalization condition at @xmath167 ( cf .
[ fig_cobe_strings ] ) . for reference ,
[ fig_p ] shows the corresponding @xmath73-dependence of the total amplitude .
both curves are shown over the entire range of allowed @xmath73-values for this choice of @xmath51 and @xmath72 , which is bounded from below by the condition that @xmath168 in eq .
is positive .
the dashed lines show the results obtained using the analytical formulas and with @xmath124 determined by eq . , the solid lines show the full numerical results .
the deviation visible in fig .
[ fig_ns ] is due to the approximation of the one - loop potential , which enters in the derivation of eq . and in the expressions for the slow - roll parameters @xmath169 and @xmath170 .
to obtain the numerical result , we do not use this approximation , but proceed with the full expression given in the first line of eq . .
note , however , that these corrections only influence the result for the spectral index at the per mille level , proving that the analytical results obtained above do indeed give a good description of the quantitative results . throughout the parameter region
compatible with the normalization condition and the cosmic string bound , the spectral index is rather high , @xmath171 .
however , taking into account a contribution of cosmic strings close to the current bound significantly modifies the best - fit value of @xmath4 to the cmb data compared to the standard six parameter @xmath142cdm fit . in ref .
@xcite , the spectral index matching the amplitude given in eq . and a cosmic string contribution of about @xmath172
is found to be @xmath173 the obtained values for the spectral index are thus compatible with current observational data at about the @xmath1 level . the qualitative behaviour of the relation between the coupling @xmath72 and the inflationary energy scale @xmath174 , displayed in fig . [ fig_cobe_strings ] , can be easily understood . in the case of small coupling , @xmath175 ,
one has @xmath176 ( cf .
eq.([eq_sigma_f ] ) ) .
the correct fluctuation amplitude is then obtained for small values of @xmath174 and the cosmic string bound can be satisfied .
however , the field value @xmath124 is large , and one therefore obtains a large spectral index , @xmath177 . on the other hand , for large couplings @xmath72
, one has @xmath178 . for large values of @xmath179 , eq .
( [ eq_sigmae ] ) then implies for the field value @xmath124 at @xmath127 e - folds , @xmath180 interestingly , the amplitude of scalar fluctuations is then only determined by the energy density during inflation , @xmath181 ( cf .
( [ eq_as2 ] ) ) , @xmath182 for the spectral index one finds , see ref .
@xcite . ] @xmath183 contrary to the amplitude of scalar fluctuations , the string tension additionally depends on the coupling strength @xmath184 ( cf .
( [ stringbound ] ) ) , @xmath185 hence , for large values of @xmath179 and @xmath72 , it is always possible to satisfy the cosmic string bound by increasing @xmath184 while at the same time keeping @xmath4 small .
this is in contrast to the case where @xmath133 and @xmath186 , with @xmath124 given by eq .
( [ smallfield ] ) . in this case
the amplitude is given by @xmath187 whose value also fixes the string tension .
however , increasing @xmath184 one moves to a regime of strong coupling and the theoretical consistency of the model becomes questionable .
for the other cmb observables , i.e. , the tilt of the spectral index @xmath188 and the tensor to scalar ration @xmath189 , we find small values , well within the experimental bounds @xcite .
for instance , for the parameter point discussed above , @xmath190 gev , @xmath191 , @xmath192 , @xmath193 , @xmath167 and @xmath194 , one obtains @xmath195 in conclusion , fig .
[ fig_cobe_strings ] shows that there is a considerable region in parameter space , which is compatible with the normalization condition as well as cosmic string bounds .
however , for generic gauge coupling strengths @xmath184 , this implies a rather large value for the spectral index .
vice versa , in the region of parameter space which yields a spectral index close to the best fit value @xmath196 , we find a cosmic string tension exceeding the cosmic string bound . in the viable region of parameter space in between these two limiting cases , we thus find a high contribution of cosmic strings close the current bounds as well as a value for the spectral index which is slightly larger than the current best - fit value . clearly , upcoming experiments will provide further stringent tests of superconformal d - term hybrid inflation .
it is worth stressing that the discussed parameter region allows for large values of the gauge coupling constant @xmath28 , compatible with grand unification . in this respect ,
the model presented here differs significantly from d - term inflation with canonical khler potential . in the latter case ,
the masses entering the one - loop potential carry @xmath197 factors , leading to problems for the super - planckian values of @xmath89 typically obtained in d - term inflation . avoiding this
forces the gauge coupling @xmath28 to be small , @xmath198 , as found in ref .
in the previous section , we focused on the situation where one of the two real degrees of freedom of the complex scalar field @xmath71 plays the role of the inflaton , whereas the value of the other degree of freedom is fixed at zero .
this is the case if either the second degree of freedom has a mass of order of the hubble scale or if inflation before the onset of the final @xmath199 e - folds lasted sufficiently long , so that the inflationary trajectory in the direction of the smallest curvature has become an attractor . here , with the mass difference between @xmath110 and @xmath111 governed by the symmetry breaking parameter @xmath73 ,
typically both masses are below the hubble scale , resulting in a two - field inflation model .
this section is hence dedicated to investigating alternative possible trajectories in ( @xmath200 ) field space . in single - field hybrid inflation
, inflation ends at the critical value of the inflaton field , @xmath125 , determined by the zero point of the mass of the waterfall field , @xmath201 .
the starting point @xmath124 of the inflationary trajectory is determined by solving the slow - roll equation . in two - field inflation
, the condition @xmath202 defines a line in ( @xmath203 ) field space . from each point on this line ( @xmath204 )
, a classical inflationary trajectory can be uniquely determined by solving the set of slow - roll equations .
the resulting trajectory ends at ( @xmath205 ) .
the single - field case discussed in section [ sec_single_field ] is reproduced for @xmath206 , where @xmath207 is given by eq . .
hence in two - field inflation , as opposed to single - field inflation , the inflationary predictions are not uniquely determined by the parameters of the lagrangian , but depend on an additional parameter which labels the various possible trajectories . in the notation above , this additional parameter is @xmath125 .
this is illustrated in fig .
[ fig_trajectories ] . a generalization of the usual single - field formulas for the amplitude of the scalar fluctuations and the spectral index to the case of multi - field inflation with a non - trivial metric in field space can be found in ref .
starting from the action @xmath208 \,,\ ] ] with @xmath209 denoting the spacetime metric , @xmath210 the metric on the real scalar field space and @xmath211 the real scalar fields of the theory , the slow - roll conditions read @xmath212 here the usual partial derivatives and covariant derivatives in scalar field space are denoted by @xmath213 and @xmath214 . as usual ,
the metric @xmath210 can be used to raise or lower indices . for inflationary trajectories satisfying these conditions , the authors of ref .
@xcite obtain the following expressions for the amplitude of the primordial power spectrum and the spectral index : @xmath215 ( \partial_a n ) ( \partial^b n)}{(\partial_e n ) ( \partial^e n ) } \ , , \end{split } \label{eq_two - field - pred}\ ] ] with @xmath216 denoting the number of e - folds , @xmath217 the inverse metric , @xmath218 and @xmath219 the scalar field space curvature tensor , @xmath220 with the christoffel symbols @xmath221 .
the number of e - folds @xmath216 as a function of the scalar fields @xmath211 is determined by integrating along all possible classical trajectories .
each point in field space lies on exactly one classical trajectory . integrating along this trajectory yields the value of @xmath216 at this point in field space , which is illustrated by the solid blue contour lines in fig .
[ fig_trajectories ] . fig .
[ fig_ns_p_2field ] shows the spectral index and the amplitude of the scalar power spectrum corresponding to different inflationary trajectories .
the solid lines represent the results for the trajectory along the @xmath110-axis , i.e. for @xmath222 , hence reproducing the single - field results depicted in fig .
[ fig_ns_p ] .
the dotted lines correspond to the other extremal case in which the inflationary trajectory runs along the @xmath111-axis , i.e. , in which @xmath223 .
finally , the dashed lines show the results for an intermediate trajectory with non - trivial evolution in both @xmath110- and @xmath111-direction . as illustrated in fig .
[ fig_p2 ] , the amplitude of the scalar power spectrum becomes smaller the more the inflationary trajectory deviates from the @xmath110-axis .
naively , one might expect a different behaviour , since the gradient of @xmath216 becomes large for inflationary trajectories along the @xmath110- as well as the @xmath111-axis , cf .
[ fig_trajectories ] .
but for negative @xmath73 the entries of the inverse khler metric , @xmath224 , become increasingly smaller the further one moves along the @xmath225 contour away from the @xmath110-axis .
as it turns out , this decrease in @xmath226 dominates over the change in the gradient of the number of e - folds , so that the amplitude ends up going down as soon as one chooses an inflationary trajectory other than the one discussed in section [ sec_single_field ] . in order to understand the decrease in the amplitude more intuitively
, it is useful to consult the single - field expression for @xmath227 in eq . .
interpreting @xmath228 appearing in this expression as the derivative of the scalar potential along the respective inflationary trajectory , the single - field expression for @xmath227 may serve as a lowest - order approximation of the full multi - field expression in eq . . from eq
it is then apparent that a steeper potential , i.e. , a larger @xmath228 , entails a smaller amplitude .
since for negative @xmath73 the scalar potential indeed becomes steeper the further one moves along the @xmath225 contour towards the @xmath111-axis , this explains our observation in fig .
[ fig_p2 ] .
the behaviour of the scalar spectral index @xmath4 is more complicated .
we find that typically the minimal value of @xmath4 as a function of @xmath73 is enhanced when considering trajectories involving a motion in @xmath111-direction . in the limit @xmath229 , the three curves for the scalar spectral index as well as the amplitude in fig .
[ fig_ns_p_2field ] respectively converge to common values .
this reflects the fact that for @xmath121 the phase of the complex inflaton field @xmath71 turns unphysical , rendering all possible trajectories equivalent to each other . for fixed values of the parameters @xmath51 and @xmath72 , the normalization condition , cf .
eq . , can be used to eliminate the parameter @xmath125 , which we introduced in section [ subsec_twovssingle ] to distinguish between the different inflationary trajectories . according to fig .
[ fig_ns_p_2field ] , it is for instance possible to find for @xmath230 and @xmath191 and for each @xmath73 value below @xmath231 one particular @xmath125 , i.e. one inflationary trajectory such that @xmath232 .
it is important to note that it is only these sets of parameter values , which are compatible with the normalization condition , that we are allowed to consider when asking for the range of viable @xmath4 values predicted by our model . in order to determine this range of admissible @xmath4 values
, we perform a numerical scan of the parameter space and record @xmath4 for all values of the parameters @xmath51 , @xmath72 , @xmath73 and @xmath125 that yield an amplitude @xmath227 within the 3-sigma range of the best - fit value @xmath233 .
[ fig_ns_normalized ] presents the results of this analysis for three representative values of the coupling constant , @xmath234 , while keeping @xmath193 and @xmath192 . for each @xmath72 value , we vary @xmath73 between @xmath235 and @xmath81 and @xmath125 between @xmath81 and @xmath207 , where @xmath207 is a function of @xmath73 , cf .
eq . . furthermore , for each @xmath72 value , we vary @xmath51 within a small interval , so that we cover the entire region in parameter space where the amplitude comes out close to the best - fit value @xmath233 . the lower boundaries of these intervals roughly coincide with the respective @xmath51 values one would need in the case of single - field inflation to obtain the correct amplitude , i.e. they lie on the solid blue curve in the equivalent of fig .
[ fig_cobe_strings ] for @xmath236 .
this is due to the decrease in the amplitude with decreasing @xmath237 as well as with decreasing @xmath238 , c.f .
[ fig_p2 ] . in order to compensate for this decrease one has to employ @xmath51 values in the two - field case that are a bit larger than in the single - field case .
the resulting range of @xmath4 values obtained for a given value of @xmath72 is marked by the shaded regions bounded by curves with a given stroke style in fig . [ fig_ns_normalized ] . additionally , the solid - dashed curve marks the cosmic string bound for @xmath239 , c.f .
eq . , with the region to the upper left of this curve in agreement with the bound .
for the two larger values of @xmath72 , the cosmic string bound is violated in the entire @xmath73-range shown . the general trend in fig .
[ fig_ns_normalized ] is the same as in the case of single - field inflation , cf .
[ fig_cobe_strings ] : small @xmath72 values yield a large spectral index , while larger @xmath72 values give smaller @xmath4 values .
for instance , for @xmath240 , we are able to reach @xmath4 values below @xmath241 for nearly the entire range of @xmath73 values .
this illustrates that our model is in principle capable of generating a spectral index of the right magnitude , while simultaneously providing the correct amplitude of the scalar power spectrum .
an obvious problem , however , is that in order to reproduce the observed amplitude @xmath233 , we require quite large @xmath51 values , such that the cosmic string tension becomes unpleasantly large .
considering trajectories different to the @xmath110-axis , i.e. , different to the trajectory studied in section [ sec_single_field ] , sharpens the tension imposed by the cosmic string bound , since the decrease in the amplitude due to the motion in @xmath111-direction forces us to go to even larger values of @xmath51 and hence larger values of @xmath153 .
moreover , we note that among the viable values for @xmath4 for a given value of @xmath72 and @xmath73 , the spectral index comes out smaller the closer to the @xmath110-axis the corresponding inflationary trajectory is . in a universe undergoing a sufficiently long period of inflation
, it may however not require much fine - tuning to end up with an inflationary trajectory running close to the @xmath110-axis during the last @xmath127 e - folds of inflation , cf . the comment below eq . in section [ subsec_slowrolleom ] .
superconformal symmetry is an underlying symmetry of supergravity , broken only by fixing the value of the conformal compensator field , which generates the kinetic term of the gravitational field .
it can also serve as a guideline for coupling matter fields to supergravity .
the resulting supergravity models have several intriguing features .
there is a jordan frame where the lagrangian takes a particularly simple form , closely resembling global supersymmetry .
furthermore , contrary to canonical supergravity , the scalar potential does not contain factors which grow exponentially at large field values , which keeps supergravity corrections to scalar masses under control .
as we have seen , a fayet - iliopoulos term can be introduced analogously to the kinetic term of the graviton by making use of the conformal compensator field . in this paper , we study hybrid models of inflation with superconformal symmetry . as we show , the inflaton acquires a large tachyonic mass in f - term hybrid inflation , which therefore is not viable . on the contrary ,
d - term hybrid inflation is consistent with superconformal symmetry . allowing for an explicit symmetry breaking by a holomorphic contribution to the khler potential involving only dimensionless parameters @xcite , one obtains a two - field inflation model . if inflation lasted sufficiently long before the onset of the last 50 e - folds , the inflationary trajectory along the real part of the complex inflaton field becomes an attractor . for this limiting case
we obtain analytic formulas for the amplitude of scalar fluctuations and the spectral index , which describe the full numerical results very well .
it turns out that the spectral index can become as small as @xmath0 . for generic two - field trajectories , we calculate the resulting amplitude of the primordial power spectrum and the spectral index numerically . comparing the obtained results with current cmb data , we find that for values of the gauge coupling compatible with guts and after fixing the overall normalization of the primordial power spectrum to the observed value , we can identify three different regions of the parameter space . for large values of the superpotential coupling @xmath72 , we obtain a spectral index close to the current best - fit value , @xmath196
. however , in this regime the model is at variance with current bounds on the cosmic string tension . on the other hand , for small values of @xmath72 , the cosmic string bound
can easily be fulfilled , at the price of a 2.4@xmath110 deviation from the best - fit value for the spectral index . in the intermediate regime , the correct spectral index within 2@xmath110 experimental uncertainty can be achieved while simultaneously fulfilling the cosmic string bound .
summarizing , superconformal d - term inflation can successfully account for the primordial power spectrum , with values of the spectral index down to @xmath0 , depending on the inflationary trajectory .
generically , however , there is a tension with the cosmic string bound .
this might be improved by considering a more strongly coupled theory or by considering an embedding of the simple model of d - term inflation described here into a more complete setup containing additional fields , which could yield further contributions to the primordial power spectrum .
it should also be noted that the bound on the cosmic string tension contains considerable theoretical uncertainties ; a better understanding of the related phenomena is necessary before parameter regions in conflict with this bound can be ruled out with certainty . *
acknowledgments * + the authors thank r. kallosh , j. louis , a. westphal and t. yanagida for helpful discussions .
this work has been supported by the german science foundation ( dfg ) within the collaborative research center 676 `` particles , strings and the early universe '' .
a. d. linde , phys .
b * 259 * ( 1991 ) 38 ; phys . rev .
d * 49 * ( 1994 ) 748 , [ astro - ph/9307002 ] .
p. binetruy and g. r. dvali , phys .
b * 388 * ( 1996 ) 241 , [ hep - ph/9606342 ] .
e. halyo , phys .
b * 387 * ( 1996 ) 43 , [ hep - ph/9606423 ] .
j. rocher and m. sakellariadou , jcap * 0503 * ( 2005 ) 004 , [ hep - ph/0406120 ] .
r. a. battye , b. garbrecht and a. moss , jcap * 0609 * ( 2006 ) 007 , [ astro - ph/0607339 ] .
j. rocher and m. sakellariadou , jcap * 0611 * ( 2006 ) 001 , [ hep - th/0607226 ] .
m. b. hindmarsh and t. w. b. kibble , rept .
* 58 * ( 1995 ) 477 , [ hep - ph/9411342 ] .
r. battye , b. garbrecht and a. moss , phys .
d * 81 * ( 2010 ) 123512 , arxiv:1001.0769 [ astro-ph.co ] . for an introduction and references
see , for example , + r. kallosh , l. kofman , a. d. linde and a. van proeyen , class .
* 17 * ( 2000 ) 4269 [ erratum - ibid .
* 21 * ( 2004 ) 5017 ] , [ hep - th/0006179 ] .
f. l. bezrukov and m. shaposhnikov , phys .
b * 659 * ( 2008 ) 703 , arxiv:0710.3755 [ hep - th ] .
m. b. einhorn and d. r. t. jones , jhep * 1003 * ( 2010 ) 026 , arxiv:0912.2718 [ hep - ph ] .
s. ferrara , r. kallosh , a. linde , a. marrani and a. van proeyen , phys .
d * 82 * ( 2010 ) 045003 , arxiv:1004.0712 [ hep - th ] ; phys .
d * 83 * ( 2011 ) 025008 , arxiv:1008.2942 [ hep - th ] .
w. buchmuller , v. domcke and k. schmitz , nucl .
b * 862 * ( 2012 ) 587 , arxiv:1202.6679 [ hep - ph ] .
w. buchmuller , v. domcke and k. schmitz , phys .
b * 713 * ( 2012 ) 63 , arxiv:1203.0285 [ hep - ph ] .
j. wess and j. bagger , princeton , usa : univ .
( 1992 ) 259 p. p. binetruy , g. dvali , r. kallosh and a. van proeyen , class .
* 21 * ( 2004 ) 3137 , [ hep - th/0402046 ] .
z. komargodski and n. seiberg , jhep * 0906 * ( 2009 ) 007 , arxiv:0904.1159 [ hep - th ] ; jhep
* 1007 * ( 2010 ) 017 , arxiv:1002.2228 [ hep - th ] . k. r. dienes and b. thomas , phys . rev .
d * 81 * ( 2010 ) 065023 , arxiv:0911.0677 [ hep - th ] .
j. distler and e. sharpe , phys .
d * 83 * ( 2011 ) 085010 , arxiv:1008.0419 [ hep - th ] .
d. arnold , j. -p .
derendinger and j. hartong , arxiv:1208.1648 [ hep - th ] .
m. dine , n. seiberg and e. witten , nucl .
b * 289 * ( 1987 ) 589 .
w. buchmuller , c. ludeling and j. schmidt , jhep * 0709 * ( 2007 ) 113 , arxiv:0707.1651 [ hep - ph ] .
b. dundee , s. raby and a. westphal , phys .
d * 82 * ( 2010 ) 126002 , arxiv:1002.1081 [ hep - th ] .
a. hebecker , s. c. kraus , m. kuntzler , d. lust and t. weigand , arxiv:1207.2766 [ hep - th ] . m. b. einhorn and d. r. t. jones , arxiv:1207.1710 [ hep - ph ] .
e. j. copeland , a. r. liddle , d. h. lyth , e. d. stewart and d. wands , phys .
d * 49 * ( 1994 ) 6410 [ astro - ph/9401011 ] .
g. r. dvali , q. shafi and r. k. schaefer , phys .
* 73 * ( 1994 ) 1886 , [ hep - ph/9406319 ] .
k. nakayama , f. takahashi and t. t. yanagida , jcap * 1012 * ( 2010 ) 010 , arxiv:1007.5152 [ hep - ph ] .
j. beringer _
[ particle data group collaboration ] , phys .
d * 86 * ( 2012 ) 010001 .
r. battye and a. moss , phys .
d * 82 * ( 2010 ) 023521 , arxiv:1005.0479 [ astro-ph.co ] .
j. dunkley , r. hlozek , j. sievers , v. acquaviva , p. a. r. ade , p. aguirre , m. amiri and j. w. appel _ et al .
_ , astrophys . j. * 739 * ( 2011 ) 52 , arxiv:1009.0866 [ astro-ph.co ] .
j. urrestilla , n. bevis , m. hindmarsh and m. kunz , jcap * 1112 * ( 2011 ) 021 , arxiv:1108.2730 [ astro-ph.co ] . c. dvorkin , m. wyman and w. hu , phys . rev .
d * 84 * ( 2011 ) 123519 , arxiv:1109.4947 [ astro-ph.co ] . n. bevis , m. hindmarsh , m. kunz and j. urrestilla , phys . rev .
d * 75 * ( 2007 ) 065015 , [ astro - ph/0605018 ] .
c. t. hill , h. m. hodges and m. s. turner , phys .
d * 37 * ( 1988 ) 263 .
m. hindmarsh , s. stuckey and n. bevis , phys .
d * 79 * ( 2009 ) 123504 , arxiv:0812.1929 [ hep - th ] .
t. asaka , w. buchmuller and l. covi , phys .
b * 510 * ( 2001 ) 271 , [ hep - ph/0104037 ] .
r. kallosh and a. d. linde , jcap * 0310 * ( 2003 ) 008 , [ hep - th/0306058 ] .
|
we study models of hybrid inflation in the framework of supergravity with superconformal matter .
f - term hybrid inflation is not viable since the inflaton acquires a large tachyonic mass . on the contrary ,
d - term hybrid inflation can successfully account for the amplitude of the primordial power spectrum .
is is a two - field inflation model which , depending on parameters , yields values of the scalar spectral index down to @xmath0 .
generically , there is a tension between a small spectral index and the cosmic string bound albeit , within @xmath1 uncertainty , the current observational bounds can be simultaneously fulfilled .
desy 12 - 149 + october 2012 * superconformal d - term inflation * w. buchmller , v. domcke , k. schmitz + _ deutsches elektronen - synchrotron desy , 22607 hamburg , germany _ .
|
@xcite found that the galactic cepheids follow a spectral type that is independent of their pulsational periods at maximum light and gets later as the periods increase at minimum light .
* hereafter skm ) used radiative hydrodynamical models to explain these observational phenomena as being due to the location of the hydrogen ionization front ( hif ) relative to the photosphere .
their results agreed very well with code s observation .
skm further used the stefan - boltzmann law applied at the maximum and minimum light , together with the fact that radial variation is small in the optical @xcite , to derive : @xmath3 where @xmath4 are the effective temperature at the maximum / minimum light , respectively . if @xmath5 is independent of the pulsation period @xmath6 ( in days ) , then equation ( 1 ) predicts there is a relation between the @xmath7-band amplitude and the temperature ( or the colour ) at minimum light , and vice versa . in other words , if the period - colour ( pc ) relation at maximum ( or minimum ) light is flat , then there is an amplitude - colour ( ac ) relation at minimum ( or maximum ) light .
equation ( 1 ) has shown to be valid theoretically and observationally for the classical cepheids and rr lyrae variables @xcite .
for the rr lyrae variables , @xcite and @xcite used linear and non - linear hydrodynamic models of rrab stars in the galaxy to explain why rrab stars follow a flat pc relation at _ minimum _ light .
later , @xcite used macho rrab stars in the lmc to prove that lmc rrab stars follow a relation such that higher amplitude stars are driven to cooler temperatures at maximum light .
similar studies were also carried out for cepheid variables , as in skm , @xcite , ( * ? ? ? * hereafter paper i ) and ( * ? ? ? * hereafter paper ii ) .
in contrast to the rr lyrae variables , cepheids show a flat pc relation at the _ maximum _ light , and there is a ac relation at the minimum light .
therefore , the pc relation and the ac relation are intimately connected .
all these studies are in accord with the predictions of equation ( 1 ) . in paper
i , the galactic , large magellanic cloud ( lmc ) and small magellanic cloud ( smc ) cepheids were analyzed in terms of the pc and ac relations at the phase of maximum , mean and minimum light .
one of the motivations for this paper originates from recent studies on the non - linear lmc pc relation ( as well as the period - luminosity , pl , relation .
see paper i ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) : the optical data are more consistent with two lines of differing slopes which are continuous or almost continuous at a period close to 10 days .
paper i also applied the the @xmath2-test @xcite to the pc and ac relations at maximum , mean and minimum @xmath7-band light for the galactic , lmc and smc cepheids .
the @xmath2-test results implied that the lmc pc relations are broken or non - linear , in the sense described above , across a period of 10 days , at mean and minimum light , but only marginally so at maximum light .
the results for the galactic and smc cepheids are similar , in a sense that at mean and minimum light the pc relations do not show any non - linearity and the pc(max ) relation exhibited marginal evidence of non - linearity . for the ac relation , cepheids in all three galaxies supported the existence of two ac relations at maximum , mean and minimum light .
in addition , the cepheids in these three galaxies also exhibited evidence of the pc - ac connection , as implied by equation ( 1 ) , which give further evidence of the hif - photosphere interactions as outlined in skm .
to further investigate the connection between equation ( 1 ) and the hif - photosphere interaction , and also to explain code s observations with modern stellar pulsation codes , galactic cepheid models were constructed in paper ii .
in contrast to skm s purely radiative models , the stellar pulsation codes used in paper ii included the treatment of turbulent convection as outlined in @xcite .
one of the results from paper ii was that the general forms of the theoretical pc and ac relation matched the observed relations well .
the properties of the pc and ac relations for the galactic cepheids with @xmath8 can be explained with the hif - photosphere interaction .
this interaction , to a large extent , is independent of the pulsation codes used , the adopted ml relations , and the detailed input physics .
the aim of this paper is to extend the investigation of the connections between pc - ac relations and the hif - photosphere interactions in theoretical pulsation models of lmc cepheids , in addition to the galactic models presented in paper ii . in section 2
, we describe the basic physics of the hif - photosphere interaction .
the updated observational data , after applying various selection criteria , that used in this paper are described in section 3 . in section 4 ,
the new empirical pc and ac relations based on the data used are presented . in section 5
, we outline our methods and model calculations , and the results are presented in section 6 . examples of the hif - photosphere interaction in astrophysical applications are given in section 7 .
our conclusions & discussion are presented in section 8 . throughout the paper , short and long period cepheid
are referred to cepheids with period less and greater than 10 days , respectively .
the partial hydrogen ionization zone ( or the hif ) moves in and out in the mass distribution as the star pulsates .
it is possible that the hif will interact with the photosphere , defined at optical depth ( @xmath9 ) of 2/3 , at certain phases of pulsation .
for example , skm suggested that this happened at maximum light for the galactic cepheids , as the hif is so far out in the mass distribution that the photosphere occurs right at the base of the hif .
the sharp rise of the opacity wall ( where the mean free path goes to zero ) due to the existence of hif prevents the photosphere moving further into the mass distribution and hence erases any `` memory '' of global stellar conditions , including the underlying pc relation .
this lead to a flat relation between period & temperature , period & colour and period & spectral type at maximum light , as seen in skm and paper ii . at other phases , since the hif does not interact with the photosphere , the temperature of the star ( or the colour ) follows the underlying global pc relation .
the hif - photosphere interaction also relies on the properties of the saha ionization equation and the structural properties of the outer envelopes of cepheids .
it is well known that the partition functions in the saha ionization equation are formally divergent unless some atomic physics is used to truncate them . in the pulsation
codes we used , we approximate the partition functions of various atoms by their ground state statistical weights .
the properties of the saha ionization equation in cepheid envelopes are such that hydrogen starts to ionize at a temperature that is almost independent of density , for a certain range of low densities . outside of this range of density , the density dependence increases .
thus , when the photosphere is very close to , or engaged with the hif and the density of these regions is reasonably low , the temperature of the photosphere is less dependent on the surrounding density and hence the global stellar parameters . at higher densities ,
the temperature at which hydrogen ionizes becomes more sensitive to density and hence more sensitive to global stellar parameters .
if the photosphere is far from the hif , or disengaged , then the location of the photosphere and hence the temperature of the photosphere , is again strongly dependent on density and hence on global stellar parameters .
that is why the photosphere needs to be close to , or engaged with the hif for this effect to take place .
moreover , this dependence on density is not sharp so that for `` low '' and `` high '' densities the density dependence of the photospheric temperature is weak and strong respectively .
an examination of figure 15.1 in @xcite demonstrates that this is plausible .
thus as the star pulsates , the photospheric temperature has a density dependence that can be strong or weak depending on phase .
an example where the density dependence is weak are the galactic long period cepheids at maximum light ( skm , paper ii ) : these cepheids display a flat pc relation at maximum light .
these properties of the hif - photosphere interaction can , in turn , affect the temperature of the photosphere and hence the colour of the cepheid .
here we investigate the idea that lmc cepheids with periods below 10 days are such that the hif and photosphere are engaged through most of the pulsation cycle . at periods greater than 10 days
, the photosphere only engages with the hif at maximum light .
the transition is sharp because the photosphere is either at the base of the hif or it is not .
the transition occurs because as the period increases , the @xmath10 ratio increases and this implies the hif is located further inside in the mass distribution , changing the phase at which it can interact with the photosphere @xcite .
the structure of galactic cepheids is such that this interaction only occurs at maximum light , even for cepheids with periods shorter than 10 days .
in paper i , we constructed the light curves of fundamental mode cepheids in the lmc by using the extensive photometric dataset in the ogle ( optical gravitational lensing experiment ) database .
however , the dataset used in paper i was downloaded in 2002 , prior to the updated version of the dataset that was available after april 24 , 2003 ( ogle website , udalski 2004 [ private communication ] ) .
the updated version includes additional @xmath7- and @xmath11-band data for most of the cepheids .
in addition , the periods have been refined by the ogle team using the complete set of photometric data . due to
these reasons , we decided to repeat the light curve construction @xcite with the updated data and periods .
since the cepheids in the ogle database are truncated at @xmath12 , due to the saturation of the ccd detector for the longer period ( hence brighter ) cepheids @xcite , we include some additional lmc cepheid data from @xcite , @xcite and @xcite to extend the period coverage to @xmath13 in our sample .
the requirements that govern our choice of the published photometric data are : ( a ) latest observations that use the modern day ccd cameras ; ( b ) high quality data with large number of data points per light curve , which provide uniform phase coverage and small scatter of the light curve ; and ( c ) as homogeneous as possible ( i.e. , from a minimal number of sources ) to avoid any additional systematic errors .
these requirements are essential to construct accurate light curves to allow the estimation of colours and magnitudes at maximum , mean and minimum light for our pc and ac study .
hence we did not include some of the older photometric data in this study .
the photometric data of all cepheids , comprising 771 from ogle database , 14 from @xcite+@xcite and 39 from @xcite , were mainly fit with @xmath14 to @xmath15 fourier expansions ( @xmath16 is the order of fourier expansion ) using the simulated annealing method described in @xcite to the @xmath7- and @xmath11-band photometric data .
this is in contrast to paper i that only applied @xmath14 fourier fits .
however , for some of the ogle long period cepheids ( @xmath17 days ) , it was found out that the quality of the fitted light curves could be improved by using a higher order fourier expansion , hence we extended the fit to @xmath18 for these long period cepheids .
all the fitted light curves were visually inspected and the best - fit light curves from the different orders of the fourier expansions were selected . to the best of our knowledge , this analysis also represents a major improvement in the fourier analysis of the ogle data .
the extinction is corrected with the standard procedure , i.e. @xmath19 with @xmath20 and @xmath21 @xcite .
the values of @xmath22 for each ogle cepheids are taken from the ogle database @xcite , while for the cepheids in @xcite+@xcite and in @xcite , the values of @xmath22 are adopted from @xcite and/or @xcite .
to guard against some `` bad '' cepheids or other contamination in our sample , and select only the good cepheids in both bands , we removed some cepheids in the sample according to the following criteria ( see also * ? ? ?
* ) : = 7.5 cm 1 .
cepheids without @xmath7- and/or @xmath11-band photometry , or the number of data per light curve ( in either bands or both ) is too low to fit a @xmath14 fourier expansion .
cepheids with poorly fitted or unacceptable @xmath7- and/or @xmath11-band light curves in the sample , such as those with a large scatter of data points or with bad - phase coverage ( large gaps between the phased data points ) .
most of the magnitudes , as well as the colours , at the maximum and/or minimum light from these fitted light curves are very uncertain .
cepheids with possible duplicity in the ogle sample .
some of the possible duplicated cepheids were removed in the ogle database by consulting table 4 of @xcite .
4 . cepheids with unusual colour .
we first plot out ( as in figure [ figcut][a ] ) the extinction corrected pc relation at mean light .
the plot shows that there are number of outliers in the period - colour plane , mostly with @xmath23 .
the presence of these outliers is probably due to : ( a ) their extinction is either over- or under - estimated ; ( b ) they have blue or red companions that can not be resolved due to the problems of blending ; or ( c ) other unknown physical reasons .
a detailed investigation of these outliers is beyond the scope of this paper , but it is clear that they should be removed from the sample . these outliers are removed with the adopted colour - cut of @xmath24 , a compromise between maximizing the number of cepheids in the sample and excluding the cepheids with unusual colour .
cepheids with unusually low ( or high ) amplitude .
some cepheids with unusually low @xmath7- and @xmath11-band amplitudes were found in the sample .
their amplitudes are typically @xmath25 times smaller as compared to the amplitudes of other cepheids at given period .
some examples of the light curves for these low amplitude cepheids are given in @xcite .
in addition , most of the light curves for these low amplitude cepheids can be fitted with @xmath14 fourier expansion , while other cepheids with `` normal '' amplitude may require higher order fits .
@xcite has briefly discussed some possible physical reasons for these cepheids to have such low amplitudes , e.g. they are just entering or leaving the fundamental mode instability strip @xcite or they have different chemical composition ( see , e.g. , * ? ? ?
the detailed investigation of these low amplitudes cepheids is beyond the scope of this paper . here
, we apply a conservative amplitude cut of @xmath26 mag . in the @xmath7-band to remove the low amplitude cepheids . besides that
, we also remove ogle-286532 ( with unusually low amplitude ) and hv-2883 ( with unusually high amplitude ) as they are clear outliers in the @xmath27-amplitude plot ( not shown , but see * ? ? ?
note that @xcite applied a cut of @xmath28 mag . to remove the low amplitude cepheids in ngc 6822 .
other examples of removing the low - amplitude cepheids can also be found in @xcite .
6 . cepheids with @xmath29 and @xmath30 .
in order to guard against possible contamination from the first overtone cepheids @xcite and to be consistent with the previous studies @xcite , we removed cepheids with @xmath29 ( see further justification in @xcite ) . regarding the removal of cepheids with @xmath30 ,
a preliminary analysis of the pc relation reveals that few of the longest cepheids should be removed from the sample , because they are clear outliers in the pc plot at maximum light ( see upper panel of figure [ figpcmax ] ) . without these longest period cepheids , the pc(max ) relation for the long period cepheids
is flat , which is consistent with the results found in paper i. the hypothesis of the hif - photosphere interaction also suggests the flatness of the pc relation at maximum light for long period cepheids .
however , as the period gets longer ( with @xmath31>1.5 $ ] ) , the photosphere disengages from the hif @xcite .
these longest period cepheids have biased the slope of the pc(max ) relation by making the slope becomes steeper .
these selection criteria are guided mainly by the philosophy that it is better to lose some `` bad '' but real cepheids rather than including those spurious and doubtful cepheids in the sample @xcite , or those with bad fitted light curves that will give inaccurate measurements of the maximum and minimum light .
hence , the final sample consists of 641 lmc cepheids that will be considered further .
the locations of the outliers from various selection criteria are shown in figure [ figcut ] for the pc(mean ) relation , @xmath7-band pl relation , @xmath32-@xmath27 relation , where @xmath33 are the fourier amplitudes .
see @xcite and @xcite for details . ] and the colour - magnitude diagram ( cmd ) .
note that some of the outliers are located within the `` good '' cepheids .
however they can be eliminated due to various physical reasons as given above , especially those with poorly fit light curves that will give inaccurate measurements at maximum , mean and minimum light . a simple sigma - clipping algorithm ( e.g. , * ? ? ?
* ) will not be able to remove these outliers @xcite .
to construct the empirical pc & ac relations , we used the following quantities from the fourier fits to the cepheid data as obtained from previous section : * @xmath7-band amplitude : the difference of the numerical maximum and minimum from the fourier expansion , @xmath34 .
* @xmath35 : defined as @xmath36 , where @xmath37 is the @xmath11-band magnitude at the same phase as @xmath38 . *
@xmath39 : defined as @xmath40 , where @xmath41 is the mean value from the fourier expansion ( see * ? ? ?
this is very similar to the conventional definition of the mean colour , @xmath42 , where @xmath43 denotes the intensity mean .
* @xmath44 : defined as @xmath45 , where @xmath46 is the @xmath11-band magnitude at the same phase as @xmath47 .
@xmath47 is the @xmath7-band magnitude closest to @xmath48 , the mean value from fourier expansion .
* @xmath49 : defined as @xmath50 , where @xmath51 is the @xmath11-band magnitude at the same phase as @xmath52 .
these quantities have been corrected for extinction as mentioned in previous section .
the empirical lmc pc and ac relations at maximum , mean and minimum light for all , long and short period cepheids are summarized in table [ c9tabpc ] & [ c9tabac ] , and the corresponding plots are presented in figure [ c9figpc ] & [ c9figac ] , respectively . lccc phase & @xmath53 & @xmath54 & @xmath55 + + maximum & @xmath56 & @xmath57 & 0.099 + mean & @xmath58 & @xmath59 & 0.075 + phmean & @xmath60 & @xmath61 & 0.081 + minimum & @xmath62 & @xmath63 & 0.075 + + maximum & @xmath64 & @xmath65 & 0.098 + mean & @xmath66 & @xmath67 & 0.075 + phmean & @xmath68 & @xmath69 & 0.092 + minimum & @xmath70 & @xmath71 & 0.081 + + maximum & @xmath72 & @xmath73 & 0.097 + mean & @xmath74 & @xmath75 & 0.074 + phmean & @xmath76 & @xmath77 & 0.078 + minimum & @xmath78 & @xmath79 & 0.073 + lccc phase & @xmath53 & @xmath54 & @xmath55 + + maximum & @xmath80 & @xmath81 & 0.093 + mean & @xmath82 & @xmath83 & 0.092 + phmean & @xmath84 & @xmath85 & 0.099 + minimum & @xmath86 & @xmath87 & 0.098 + + maximum & @xmath88 & @xmath89 & 0.086 + mean & @xmath90 & @xmath91 & 0.085 + phmean & @xmath92 & @xmath93 & 0.103 + minimum & @xmath94 & @xmath95 & 0.082 + + maximum & @xmath96 & @xmath97 & 0.076 + mean & @xmath98 & @xmath99 & 0.074 + phmean & @xmath100 & @xmath101 & 0.078 + minimum & @xmath102 & @xmath103 & 0.079 + to test the non - linearity of the pc and ac relations , or the `` break '' at a period of 10 days , we apply the @xmath2-test as given in paper i and in @xcite . the null hypothesis in the @xmath2-test is single line regression is sufficient , while the alternate hypothesis is that two lines regressions with a discontinuity ( a break ) at 10 days is necessary to fit the data .
the probability @xmath104 , under the null hypothesis , can be obtained with the corresponding @xmath2-values and the degrees of freedom . in general
, the large value of @xmath2 ( equivalent to the small value of @xmath105 $ ] ) indicates that the null hypothesis can be rejected . for our sample , @xmath106 when @xmath107 ( the 95% confident level ) , therefore the null hypothesis can be rejected if the @xmath2-value is greater than @xmath108 with more than 95% confident level and the data is more consistent with the two - line regression .
a glance of table [ c9tabpc ] and figure [ c9figpc ] suggests that the lmc pc relations are broken at maximum , mean and minimum light .
these are confirmed with the @xmath2-test results with @xmath109 .
similarly , the @xmath2-test results for the ac relation are : @xmath110 .
hence , the lmc pc and ac relations are non - linear ( hence broken ) at maximum , means and minimum light . note that the flatness of the long period pc(max ) relation as given in table [ c9tabpc ] ( @xmath64 ) is in good agreement with the slope found in paper i ( @xmath111 ) .
recall that equation ( 1 ) predicts that if the pc relation is flat at maximum light , then there is a correlation between the amplitude and the colour at minimum light .
this is seen in table [ c9tabac ] ( and in figure [ c9figac ] ) for the long period ac(min ) relation , with a slope of @xmath112 .
the stellar pulsation codes we used are both linear @xcite and non - linear @xcite . these codes , which include a 1-d turbulent convection recipe @xcite , are the same as in paper ii . briefly speaking ,
the codes take the mass ( @xmath113 ) , luminosity ( @xmath114 ) , effective temperature ( @xmath115 ) and chemical composition ( @xmath116 ) as input parameters .
the chemical composition is set to be @xmath117 to represent the lmc hydrogen and metallicity abundance ( by mass ) .
the mass and luminosity are obtained from the ml relations calculated from evolutionary models .
the @xmath115 are chosen to ensure the models oscillate in the fundamental mode and located inside the cepheid instability strip .
the pulsation periods for the models are obtained from a linear non - adiabatic analysis @xcite .
all other parameters used in the pulsation codes had the same values for the lmc and galactic models ( paper ii ) .
this included the @xmath118 parameters that are part of the turbulent convection recipe , though see section 8 .
of course , one variable parameter was the metallicity .
the only other difference between this study and paper ii , besides the metallicity , is the value set for the artificial viscosity parameter , @xmath119 . in this study , we set @xmath120 for the lmc models to improve the shape of the theoretical light curves , in contrast to the value of @xmath121 used for the galactic models . in paper ii ,
the ml relations are adopted from @xcite and @xcite . in order to be consistent with previous work
, the ml relations used in this paper will also be adopted from these two sources .
however , @xcite only provided two ml relations , one for @xmath122 which are used in paper ii , and another one for @xmath123 .
hence we have to adopt the second ml relation for the lmc models . even though the lmc metallicity is higher than @xmath123 , the lmc is still considered as a low metallicity system in the literature . hence the @xcite ml relation can be approximately applied for the lmc models .
an anonymous referee pointed out that an interpolation of the @xcite ml relations between @xmath122 and @xmath123 should also be used .
we have included the interpolated ml relation in our model calculations . in the context of the hif - photosphere interaction , it is the ml relation which dictates at what period and at what phases this will occur .
stellar evolutionary theory changes the ml relation as a function of metallicity .
hence the coefficients of the ml relation are important in determining the nature of the hif - photosphere interaction ( paper ii ) . in short ,
the ml relations used are : ccccccc @xmath113 & @xmath124 & @xmath125 & @xmath126 & @xmath127 & @xmath128 & @xmath129 + + 11.0 & 4.375 & 5050 & 46.4155 & 0.124 & 28.98 & -0.118 + 10.0 & 4.236 & 5100 & 35.6727 & 0.091 & 22.92 & -0.093 + 9.50 & 4.161 & 5250 & 28.2406 & 0.094 & 18.68 & -0.046 + 9.10 & 4.099 & 5260 & 25.3960 & 0.082 & 16.92 & -0.042 + 8.75 & 4.042 & 5310 & 22.3804 & 0.076 & 15.07 & -0.027 + 8.40 & 3.982 & 5380 & 19.3886 & 0.071 & 13.20 & -0.008 + 7.95 & 3.902 & 5330 & 17.7750 & 0.055 & 12.09 & -0.027 + 7.00 & 3.717 & 5410 & 12.6085 & 0.035 & 8.722 & -0.018 + 6.55 & 3.620 & 5490 & 10.2940 & 0.031 & 7.183 & -0.006 + 6.40 & 3.587 & 5485 & 9.81474 & 0.027 & 6.853 & -0.010 + 6.00 & 3.493 & 5510 & 8.37226 & 0.020 & 5.866 & -0.014 + 5.90 & 3.468 & 5500 & 8.12498 & 0.017 & 5.691 & -0.018 + 5.80 & 3.443 & 5525 & 7.69466 & 0.017 & 5.400 & -0.015 + 5.70 & 3.418 & 5560 & 7.23505 & 0.017 & 5.090 & -0.009 + 5.30 & 3.312 & 5600 & 6.01283 & 0.012 & 4.244 & -0.009 + + 7.20 & 4.272 & 5380 & 40.2561 & 0.275 & 24.51 & -0.162 + 6.80 & 4.192 & 5380 & 35.4374 & 0.264 & 21.91 & -0.122 + 6.20 & 4.063 & 5410 & 28.2629 & 0.225 & 17.94 & -0.076 + 5.95 & 4.005 & 5420 & 25.6378 & 0.211 & 16.43 & -0.060 + 5.40 & 3.869 & 5510 & 19.4314 & 0.170 & 12.82 & -0.015 + 5.15 & 3.803 & 5510 & 17.5523 & 0.160 & 11.66 & -0.007 + 4.65 & 3.660 & 5490 & 14.3143 & 0.131 & 9.611 & -0.010 + 4.20 & 3.518 & 5510 & 11.3659 & 0.101 & 7.729 & -0.011 + 4.00 & 3.450 & 5545 & 10.0011 & 0.089 & 6.854 & -0.005 + 3.95 & 3.432 & 5540 & 9.77393 & 0.085 & 6.701 & -0.008 + 3.80 & 3.378 & 5550 & 8.94637 & 0.075 & 6.157 & -0.009 + 3.70 & 3.341 & 5575 & 8.31297 & 0.070 & 5.745 & -0.005 + 3.65 & 3.322 & 5570 & 8.10751 & 0.066 & 5.605 & -0.008 + 3.60 & 3.302 & 5530 & 8.09994 & 0.058 & 5.583 & -0.022 + 3.60 & 3.302 & 5600 & 7.71463 & 0.065 & 5.352 & -0.001 + + 6.80 & 4.092 & 5280 & 30.9661 & 0.207 & 19.51 & -0.091 + 5.20 & 3.701 & 5340 & 15.9744 & 0.094 & 10.63 & -0.050 + 4.40 & 3.457 & 5460 & 10.0562 & 0.056 & 6.898 & -0.028 + 4.20 & 3.389 & 5550 & 8.51317 & 0.054 & 5.906 & -0.007 + 3.80 & 3.243 & 5630 & 6.46303 & 0.039 & 4.532 & -0.002 + 1 .
ml relation given in @xcite : @xmath130 2 .
ml relation given in @xcite : @xmath131 3 .
ml relation interpolated between two @xcite relations at @xmath132 and @xmath123 to yield a relation at @xmath133 : @xmath134 the units for both @xmath113 and @xmath114 are in solar units .
note that these ml relations cover reasonably broad @xmath10 ratios given in the literature .
the input parameters for the lmc models with these ml relations and the periods calculated from linear non - adiabatic analysis are given in table [ tabinput ] .
after the full amplitude models are constructed from the pulsation codes , the temperature and the opacity profile can be plotted in terms of the internal mass distribution ( @xmath135 $ ] , where @xmath136 is mass within radius @xmath137 and @xmath113 is the total mass ) at a given phase of pulsation . as in paper ii
, the locations of the hif ( sharp rise in the temperature profile ) and photosphere ( at optical depth @xmath138 ) can be identified in the temperature profile . to quantify the hif - photosphere interaction
( if the photosphere is next to the base of the hif or not , see also paper ii ) , we calculate the `` distance '' , @xmath139 , in @xmath140 between the hif and the photosphere from the temperature profile . the definition of @xmath139 can be found in paper ii .
a small @xmath139 means there is a hif - photosphere interaction , and vice versa .
the theoretical quantities from the models can be compared to the observed quantities using the following prescriptions : ccccc @xmath6 & @xmath141 & @xmath5 & @xmath142 & @xmath143 + + 46.4155 & 27481.28 & 5445.00 & 18329.74 & 4826.00 + 35.6727 & 19742.83 & 5434.01 & 14321.20 & 4962.65 + 28.2406 & 16894.15 & 5502.35 & 12093.99 & 4978.72 + 25.3960 & 14460.62 & 5562.28 & 10570.70 & 5010.85 + 22.3804 & 12772.75 & 5615.34 & 9283.257 & 5069.43 + 19.3886 & 11263.84 & 5687.98 & 8143.460 & 5145.20 + 17.7750 & 9034.392 & 5561.39 & 7044.743 & 5147.04 + 12.6085 & 5637.467 & 5528.15 & 4800.840 & 5278.11 + 10.2940 & 4371.170 & 5537.42 & 3839.001 & 5377.80 + 9.81474 & 4004.569 & 5518.39 & 3580.396 & 5383.61 + 8.37226 & 3217.688 & 5596.34 & 2929.354 & 5437.17 + 8.12498 & 3032.639 & 5581.53 & 2793.441 & 5438.96 + 7.69466 & 2864.745 & 5602.37 & 2640.439 & 5467.54 + 7.23505 & 2705.430 & 5639.03 & 2490.271 & 5504.98 + 6.01283 & 2105.589 & 5669.57 & 1986.586 & 5567.91 + + 40.2561 & 23500.98 & 5821.99 & 11853.38 & 4947.11 + 35.4374 & 19604.60 & 5830.18 & 9898.124 & 4962.26 + 28.2629 & 14661.29 & 5875.98 & 7659.896 & 5057.96 + 25.6378 & 12836.04 & 5884.01 & 6916.299 & 5108.02 + 19.4314 & 9538.226 & 5985.21 & 5555.179 & 5303.60 + 17.5523 & 8035.700 & 5921.07 & 4947.967 & 5142.58 + 14.3143 & 5490.345 & 5783.01 & 3639.964 & 5164.11 + 11.3659 & 3850.971 & 5814.49 & 2722.473 & 5242.47 + 10.0011 & 3286.030 & 5817.00 & 2377.950 & 5300.52 + 9.77393 & 3136.139 & 5801.77 & 2301.751 & 5296.73 + 8.94637 & 2738.003 & 5798.30 & 2071.106 & 5331.19 + 8.31297 & 2499.416 & 5794.49 & 1917.696 & 5365.34 + 8.10751 & 2374.713 & 5780.35 & 1850.048 & 5369.40 + 8.09994 & 2235.292 & 5723.18 & 1796.586 & 5355.41 + 7.71463 & 2271.393 & 5805.49 & 1772.270 & 5408.34 + + 30.9661 & 14958.03 & 5611.77 & 8486.049 & 4963.81 + 15.9744 & 5697.042 & 5665.97 & 4209.487 & 5079.81 + 10.0562 & 3197.451 & 5670.64 & 2560.871 & 5283.85 + 8.51317 & 2723.566 & 5716.48 & 2201.628 & 5382.21 + 6.46303 & 1852.537 & 5684.63 & 1593.714 & 5507.38 + = 7.5 cm 1 . as in paper ii
, we use the basel atmosphere database @xcite to construct a fit giving temperature and effective gravity as a function of @xmath144 colour .
the effective gravity is obtained at the appropriate phase from the models ( see paper ii ) .
these prescriptions are used to convert the temperatures to the @xmath144 colours .
the bolometric corrections ( @xmath145 ) are obtained in a similar manner .
the anonymous referee has suggested that @xmath144 may not be a good way to convert between temperature and colour unless both of the micro - turbulence and surface gravity are included . as indicated above this
is the case , and in any case our results and those of paper ii for galactic models , show good agreement between theory and observations .
a number of previous authors have used this method and some authors commented that this colour can be used as an indicator of temperature ( e.g. * ? ? ?
* ; * ? ? ?
the empirical relations we studied in this series were also mainly in the @xmath144 colour .
2 . in addition to the basel atmosphere
, we also use the atmosphere fit from @xcite , referring this as the sbt atmosphere in our paper .
the sbt atmosphere does include both of the effective gravity and the micro - turbulence in their table 6 for the temperature and colour conversion .
these conversions are tabulated for two micro - turbulence velocities of @xmath146 and @xmath147 , as well as for various metallicities . to apply these conversions to our lmc models , we first interpolated the conversions between @xmath148=0.0 $ ] and @xmath148=-0.5 $ ] to @xmath148=-0.3 $ ] , which is appropriate for the lmc metallicity
. the @xmath144 colours at the maximum , mean and minimum light are then obtained from the given effective temperature and the effective gravity for both of the micro - turbulence velocities .
we use the prescriptions given in @xcite to convert the observed colours to the temperatures appropriate for the lmc data as follows : @xmath149 + note that these functions are also obtained from the basel atmosphere database .
we can compare the colours obtained from the basel and sbt atmosphere for our models constructed in this paper .
the results are presented in figure [ figatmos ] . from this figure
it can be seen that the colours obtained from both of the atmosphere fits agree within @xmath150mag .
the difference is even smaller if the micro - turbulence velocity of @xmath147 is used .
this indicates that the @xmath144 colours can be used to indicate the temperature . since the results of our models
are qualitatively compared to the observations ( see next section ) and not used to quantitatively derive any theoretical pc and/or ac relations , an accuracy of @xmath151mag .
, independent of period , from the atmosphere fit is acceptable and would not cause problems for our results .
note that the sbt atmosphere are only defined for @xmath152 and @xmath153 , few of our models either the @xmath115 or @xmath154 or both are beyond these ranges at certain phases , hence no colours can be obtained from the sbt atmosphere ( for example some points are missing at minimum light for few of the long period models , as shown in figure [ figatmos ] ) . due to these reasons ,
we continue adopt the basel atmosphere fits to convert the temperatures and @xmath144 colours , after taking account of the effective gravity in the fits , as a function of phase .
cccccccc @xmath6 & @xmath155 & @xmath156 & @xmath157 & @xmath158 & @xmath159 & @xmath160 ( asc ) & @xmath160 ( des ) + + 46.4155 & 24249.2 & 24423.695 & 5330.06 & 24282.129 & 4882.55 & 5319.42 & 4880.66 + 35.6727 & 17207.7 & 17078.471 & 5293.90 & 17188.051 & 4922.99 & 5305.14 & 4924.47 + 28.2406 & 14484.7 & 14466.869 & 5457.79 & 14529.348 & 5073.65 & 5459.55 & 5069.64 + 25.3960 & 12540.8 & 12461.087 & 5443.66 & 12544.735 & 5096.58 & 5452.77 & 5096.18 + 22.3804 & 10997.1 & 10996.587 & 5493.44 & 11039.123 & 5157.95 & 5493.51 & 5152.87 + 19.3886 & 9587.02 & 9560.1922 & 5545.87 & 9618.0583 & 5232.85 & 5549.58 & 5228.51 + 17.7750 & 7983.35 & 7988.5457 & 5473.94 & 7994.9159 & 5213.83 & 5473.06 & 5211.77 + 12.6085 & 5208.60 & 5208.1564 & 5493.34 & 5202.6440 & 5349.00 & 5493.36 & 5350.84 + 10.2940 & 4170.31 & 4162.1135 & 5560.81 & 4174.0993 & 5450.34 & 5564.43 & 5449.06 + 9.81474 & 3860.04 & 3861.4313 & 5556.01 & 3852.8715 & 5446.51 & 5555.34 & 5449.11 + 8.37226 & 3109.66 & 3106.7337 & 5570.85 & 3110.1806 & 5481.59 & 5572.55 & 5481.37 + 8.12498 & 2939.41 & 2935.0599 & 5553.67 & 2939.6504 & 5476.34 & 5556.38 & 5476.23 + 7.69466 & 2775.73 & 2775.1245 & 5581.66 & 2777.4057 & 5501.67 & 5582.05 & 5500.77 + 7.23505 & 2618.53 & 2620.1965 & 5620.29 & 2618.8674 & 5534.39 & 5619.15 & 5534.20 + 6.01283 & 2051.97 & 2050.8394 & 5646.29 & 2051.3277 & 5583.97 & 5647.27 & 5584.50 + + 40.2561 & 18754.3 & 18580.019 & 5704.60 & 18824.542 & 5144.86 & 5718.84 & 5141.42 + 35.4374 & 15896.6 & 15885.248 & 5749.07 & 15861.179 & 5147.12 & 5750.16 & 5158.63 + 28.2627 & 11157.8 & 11188.594 & 5692.54 & 11143.155 & 5111.26 & 5688.21 & 5112.18 + 25.6378 & 10388.8 & 10366.128 & 5782.33 & 10358.696 & 5184.81 & 5785.59 & 5188.79 + 19.4314 & 7743.89 & 7741.3087 & 5871.14 & 7750.7288 & 5332.20 & 5871.60 & 5330.76 + 17.5523 & 6538.19 & 6537.3819 & 5833.02 & 6532.0823 & 5310.84 & 5833.20 & 5312.28 + 14.3143 & 4564.83 & 4621.4163 & 5750.64 & 4565.7876 & 5280.59 & 5732.80 & 5280.34 + 11.3659 & 3289.17 & 3288.4584 & 5711.02 & 3281.3799 & 5328.36 & 5711.31 & 5331.32 + 10.0011 & 2811.35 & 2823.6619 & 5726.94 & 2800.7972 & 5376.66 & 5721.20 & 5381.50 + 9.77393 & 2702.54 & 2712.0209 & 5715.81 & 2692.4263 & 5378.63 & 5711.29 & 5383.51 + 8.94637 & 2385.34 & 2395.4576 & 5704.28 & 2388.3903 & 5412.32 & 5699.14 & 5410.62 + 8.31297 & 2189.29 & 2193.3351 & 5708.03 & 2188.7842 & 5448.47 & 5705.90 & 5448.80 + 8.10751 & 2094.38 & 2103.6034 & 5698.27 & 2093.0120 & 5449.90 & 5693.36 & 5450.83 + 8.09994 & 2004.90 & 1998.5077 & 5648.26 & 1999.8838 & 5419.02 & 5651.76 & 5422.70 + 7.71463 & 2004.07 & 2009.2157 & 5713.42 & 2004.9007 & 5490.38 & 5710.72 & 5489.80 + + 30.9661 & 12359.6 & 12202.760 & 5566.66 & 12354.904 & 5029.65 & 5585.98 & 5030.07 + 15.9744 & 5011.60 & 5031.4017 & 5550.08 & 5024.6207 & 5173.09 & 5544.24 & 5169.84 + 10.0562 & 2860.25 & 2866.2012 & 5593.28 & 2859.5002 & 5346.96 & 5590.81 & 5347.35 + 8.51317 & 2447.87 & 2441.2040 & 5643.58 & 2453.1647 & 5463.35 & 5646.11 & 5460.12 + 6.46303 & 1749.72 & 1749.6046 & 5719.11 & 1744.6691 & 5577.03 & 5719.24 & 5581.12 + the effective temperatures for the full amplitude models in table [ tabinput ] at the corresponding maximum and minimum light ( or luminosity ) are given in table [ c9tabmaxmin ] .
for the effective temperatures at mean light , the temperatures for the mean light at ascending and descending branch of the light ( or luminosity ) curve are not the same ( e.g. , in paper ii ) , hence table [ c9tabmean ] gives the effective temperature at these phases for our lmc models .
the layout of table [ c9tabmean ] is the same as table 3 from paper ii . following paper ii ,
the locations of the photosphere can be identified in the temperature and opacity profiles .
these are displayed in figure [ c9bono4]-[c9chiosi13 ] with a @xmath161 , a @xmath162 and a @xmath23 model , respectively .
the left and right panels of figure [ c9bono4]-[c9chiosi13 ] are the temperature and opacity profiles respectively .
the photospheres are marked as filled circles in these figures .
finally , the plots of the @xmath139 , the `` distance '' between the photosphere and the hif from the temperature profiles , as a function of pulsating period for the lmc models are presented in figure [ c9deltalmc ] with the three ml relations used . in paper ii
, it is found that the distribution of @xmath139 as a function of period is almost independent of the adopted ml relation .
this is also seen in the lmc models as depicted in figure [ c9deltalmc ] .
figure [ c9bono4]-[c9chiosi13 ] and figure [ c9deltalmc ] bear witness to the fact that at maximum light , the photosphere lies at the base of the hif for all of the models .
although there is a slight deviation for some longer period models , the location of the photosphere is close to the hif within the error bars ( which are defined as the coarseness of the grid points around the location of the hif ) . as in paper ii for the galactic models , the closeness of the photosphere to the base of the hif , for reasonably low densities , results in a flat or almost flat pc relation for the long period lmc cepheids . in the case of minimum light , even though figure [ c9deltalmc ] implies that @xmath163 is nearly constant across the period range and the photosphere is near the base of the hif , as in the case of maximum light , @xmath163 does follow a shallow correlation with period after 10 days .
judging from the error bars of @xmath163 and from figure [ c9bono4]-[c9chiosi13 ] , there is tentative evidence that the photosphere is disengaged from the hif for @xmath161 at minimum light .
hence the temperatures or the colours at minimum light are more dependent on period for @xmath164-@xmath27 relation may not be correlated with the slopes of the pc relation ] and the global properties .
theoretical quantities that can be computed from the models and compared with data include the pulsation periods , the @xmath7-band amplitudes and the fourier parameters , the temperatures and colours at the maximum , mean and minimum light .
these are the pc plots , the ac plots , the period - temperature plots and the fourier parameters plots portrayed in figures [ c9modelpc]-[c9lmcfourier ] . the temperatures in table [ c9tabmaxmin ] & [ c9tabmean ] , after conversion to the @xmath144 colours
as mentioned in previous section , are superimposed along with the observed lmc pc relations as plotted in figure [ c9modelpc ] . similarly ,
figure [ c9modelpt ] graphs the same quantities but on the @xmath165-@xmath27 plane with the observed @xmath144 colours converted to temperatures using the prescriptions given in section 5 .
the theoretical bolometric light curves are converted to the @xmath7-band light curves with the bolometric corrections obtained from the basel database mentioned previously . from the theoretical @xmath7-band light curves ,
the amplitudes can be estimated and these are displayed in figure [ c9modelac ] along with the colours from models to compare with the empirical ac relations .
the fourier parameters of the theoretical @xmath7-band light curves can also be obtained with ( @xmath166 ) fourier expansion .
these fourier parameters are compared with the observational data in figure [ c9lmcfourier ] .
several features are noticed from figure [ c9modelpc]-[c9lmcfourier ] : 1 .
the general trends of the models qualitatively match the observational data .
there are greater discrepancies between the data and short period models , particularly in matching the observed light curve amplitudes .
2 . the models with the ml relation from @xcite , with lower @xmath10 ratio ,
do better in matching the observations
. these models also tend to lie near the envelopes of the pc , ac , @xmath165-@xmath27 and @xmath167-@xmath27 relations defined by the observational data .
3 . the slopes of the period - colour ( or period - temperature ) relations at maximum and minimum light from the models roughly match the observational data , i.e. , the theoretical pc(max ) relation is approximately flat and there is a relation at minimum light .
4 . the temperatures from the models with the @xcite ml relation is cooler ( hence redder ) than the models with the @xcite ml relation and the observed data at maximum light .
in contrast , the temperatures ( or the colours ) at minimum light from the models with these two ml relations are consistent with each other and are located near the blue edge of the observed data .
the means at the descending branches are in better agreement with the observed data than the means at the ascending branches .
this is because the observed means , @xmath168 , are obtained mostly from the descending branches . though previous researchers have noted that temperatures on the ascending and descending branches are not the same at mean light ( as cepheids exhibit loops in cmd ) , what is new here is the way the nature of the hif changes during the pulsation .
the behaviors of the models from the interpolated @xcite ml are closer to the models from @xcite ml relation because their slopes are very similar .
the amplitudes of the theoretical light curves ( in both of the bolometric and @xmath7-band light curves ) are smaller than the observations at given period , especially for the models with the @xcite ml relation .
these can be seen from the ac relations as given in figure [ c9modelac ] and the left panels of figure [ c9lmcfourier ] .
overall , some agreements and disagreements are found between the theoretical quantities and the observational data .
it is also found out that there are some problems associated with the pulsation codes when the lmc models are constructed : these include the smaller amplitude of the model light curves and the cooler temperatures at the maximum light ( especially with @xcite ml relation ) .
note that from equation ( 1 ) , cooler temperatures at maximum light imply that the amplitudes will be lower at given period .
varying other parameters in the pulsation codes , including the @xmath118 parameters , does not improve the situation , though perhaps a more detailed and systematic study of the dependence of lmc cepheid pulsation models on the @xmath169 parameters could resolve this situation .
however , we believe that the qualitative nature of the photosphere - hif interactions as given in figure [ c9deltalmc ] will still hold even in models which fare better in mimicking observed amplitudes .
this is in part because figure [ c9deltalmc ] suggest that the behaviors of @xmath139 as a function of period are nearly , though not completely , independent of amplitudes , as the models with @xcite ml relation have higher amplitudes ( although still smaller than the observations ) than the models with the @xcite ml relation .
however , better codes that fix these problems or the 3-d convection codes are needed in the future studies .
the temperature profiles from the galactic models given in paper ii and the lmc models are compared in figure [ c9pt ] at maximum and minimum light .
the upper panels of figure [ c9pt ] suggest that at maximum light , the photosphere is not far from the base of the hif in both of the galactic and the lmc models .
in contrast , the photosphere is further away from the hif in the galactic models than the lmc models at minimum light .
the hif is located further out in the mass distribution for the galactic models .
the plots of the @xmath139-@xmath27 relation from the galactic and lmc models at maximum and minimum light are also compared in figure [ c9delta ] .
it can be seen from the figure that at maximum light , the behavior of both galactic and lmc models is similar , where the photosphere is near the base of the hif . at minimum light ,
the long period models show that the photosphere is disengaged from the hif , while the behavior of the short period models is different between the galactic and lmc models .
the photosphere of the short period lmc models seems to be located closer to the hif at minimum light , but it is not the case for the short period galactic models .
this could lead to shallower slopes of the pc(min ) relation seen in the lmc cepheids as compared to the galactic counterparts . in terms of the hif - photosphere interaction , there is some tentative evidence from the models that the lmc long period cepheids behave like the galactic cepheids , while the short period lmc cepheids behave like the rr lyrae stars at minimum light .
figure [ c9density ] graphs the density ( defined as @xmath170 , where @xmath7 is the specific volume ) at the photosphere as a function of the period of the model at minimum , maximum and ascending and descending mean light .
galactic models generally tend to have the lowest density and , in particular , have significantly lower densities at minimum light than the lmc models .
we note that the galactic models always have a photospheric density lower than about @xmath171 whereas the photospheric density for the lmc models only falls below this figure after a period of 10 days . at maximum light ,
all long period models have a low photospheric density .
what we get from this figure is that it provides some evidence that there is a difference in photospheric density between the lmc and galactic models .
moreover , this difference appears to be consistent with what is required by our theoretical scenario : short period lmc models have a higher photospheric density than their galactic counterparts .
however , for a discussion of some caveats , see section 8 .
we now discuss two important applications of the photosphere - hif interaction : reddening corrections and the explanation of the observed non - linear lmc pl ( and pc ) relations .
@xcite original interest in the spectral properties of cepheids at maximum light was to estimate reddening .
skm used this to correct a number of reddening for galactic cepheids . @xcite
used equation ( 1 ) and the theoretical explanation provided in skm to derive a relation linking the colour excess to the colour at maximum light , the @xmath7-band amplitude and the period .
such a relation is predicted from equation ( 1 ) .
@xcite estimates the error with this method to be comparable to other multi - colour methods .
a more interesting application of the hif - photosphere interaction is to explain the recent detected non - linear lmc pl relation as presented in @xcite , paper i , @xcite and @xcite .
paper i used the @xmath2-test to provide strong statistical evidence that the optical cepheid pl relation at mean light in the lmc is non - linear around a period close to 10 days .
@xcite used the macho and 2mass datasets together with additional long period cepheids from the literature to further support the existence of non - linear lmc pl relation in the optical and near infra - red wave - bands .
in contrast , current data indicate that the galactic pl relation is linear at mean light @xcite .
non - linearity of the lmc pl relations can be tested using the @xmath2-test with the data given in section 3 .
the empirical results of the fitted lmc pl relations at maximum , mean and minimum light using the updated data are presented in table [ c9tabpl ] .
the plots of the pl relations at maximum / minimum light and at mean light are shown in figure [ c9plmaxmin ] & [ c9plmean ] , respectively .
the @xmath2-test results for these pl relations are : @xmath172 , and @xmath173 .
the large @xmath2-values for both @xmath7- and @xmath11-band pl relations at mean and minimum light strongly indicate that the pl relations at these two phases are not linear , and the data is better described with the broken ( i.e , two regressions ) pl relation .
however , the small @xmath2-values at maximum light , with corresponding @xmath174-values of @xmath175 and @xmath176 for the @xmath7- and @xmath11-band pl(max ) relations respectively , show that the null hypothesis of the @xmath2-test can not be rejected ( a value of @xmath177 and/or @xmath178 is required for doing this ) .
hence there is no observed break seen in the pl(max ) relation and the data is consistent with single line regression .
note that the same slopes of the pl(max ) relations for long period cepheids in both bands are consistent of the finding that the pc(max ) relation is flat for these cepheids . lcccccc phase & @xmath179 & @xmath180 & @xmath181 & @xmath182 & @xmath183 & @xmath184 + + maximum & @xmath185 & @xmath186 & 0.260 & @xmath187 & @xmath188 & 0.170 + mean & @xmath189 & @xmath190 & 0.208 & @xmath191 & @xmath192 & 0.141 + minimum & @xmath193 & @xmath194 & 0.204 & @xmath195 & @xmath196 & 0.140 + + maximum & @xmath197 & @xmath198 & 0.257 & @xmath199 & @xmath200 & 0.174 + mean & @xmath201 & @xmath202 & 0.228 & @xmath203 & @xmath204 & 0.158 + minimum & @xmath205 & @xmath206 & 0.243 & @xmath207 & @xmath208 & 0.174 + + maximum & @xmath209 & @xmath210 & 0.260 & @xmath211 & @xmath212 & 0.169 + mean & @xmath213 & @xmath214 & 0.203 & @xmath215 & @xmath216 & 0.138 + minimum & @xmath217 & @xmath218 & 0.194 & @xmath219 & @xmath220 & 0.131 + our tentative theoretical explanation for the non - linear nature of the lmc pl relations across a period of 10 days replies on the hif - photosphere interaction .
@xcite and @xcite have established the connection between the pc and pl relations : both these relations arise from the more general plc relation .
these relations refer to quantities evaluated at mean light .
the existence of such a connection relies on the period - mean density theorem , the instability strip and the stefan - boltzmann law .
if we assume the stefan - boltzmann law can be applied at every phase , then it is straightforward to show that a plc relation ( though possibly with different coefficients ) exists at every phase point . thus the standard plc relation and
indeed the pc and pl relation expresses at mean light are just the averages of the same relations at different phases points .
consequently one way to understand the behavior of plc / pl / pc relations at mean light is to understand their behavior at different phase points .
what we try to do in this paper is point out some evidence from our models that shows how the changing behavior of the pc relations at different phases can , in principle , arise from a consideration of the photosphere - hif interaction at these phases .
since the mean light pc and pl relation are the average of those at all phases , these properties can affect the pc and , as a consequence , the pl relation ( via the plc relation ) .
in fact , the new data with superb phase resolution from such micro - lensing projects such as ogle and macho demands a multiphase analysis . this approach can potentially lead to a deeper understanding of the pulsation and evolution of cepheid variables .
for example , @xcite looked at pc relations in the galaxy and lmc as a function of phase .
they found that short and long period lmc cepheids have a shallower and steeper slope at most pulsation phases than galactic cepheids respectively .
in this paper , we have confronted updated pc and ac relations at maximum , mean and minimum light for lmc cepheids observed by the ogle team , and additional cepheids from the literature , with theoretical , full amplitude pulsation models of lmc cepheids .
the observed pc and ac relations provide compelling evidence of a non - linearity or break at a period of 10 days .
we also constructed theoretical cepheid pulsation models appropriate for the lmc using the florida pulsation codes @xcite to study the hif - photosphere interaction .
the empirical results presented in this paper , as well as in other papers such as @xcite and @xcite , provide strong empirical evidence that the pc and pl relations for the lmc cepheids are non - linear , in the sense described in previous sections .
issues such as extinction and a lack of long period cepheids that may cause the non - linear lmc pl and pc relations have been addressed and argued against in paper i , @xcite and @xcite , and will not be repeated here .
other arguments against the non - linear lmc pl relation include the results presented in @xcite , as the authors found no evidence for a non - linear pl relation in the lmc at @xmath221-bands .
however , @xcite treated the data of @xcite extensively and found , in a statistically rigorous way , that the reason why @xcite found linear @xmath221 pl relations , is due to the small number of short period cepheids ( @xmath222 ) in their sample .
@xcite also reduce the number of ogle / macho lmc cepheids and show how the @xmath2-test can produce a non - significant result when the number of short / long period cepheids become small . instead , using the 2mass data that are cross - correlated with macho cepheids , @xcite have found that the lmc @xmath223-band pl relations are non - linear - band than in @xmath224-band , as shown in @xcite . ] and the @xmath225-band pl relation starts to become linear . @xcite
also discussed why this is the case .
another argument against the non - linear pl relation is that the pl relation should be universal , as found in @xcite .
we argue that their results are based on a handful of cepheids ( @xmath226 ) and on short periods cepheids in a cluster whose membership to the lmc is in question .
their shallower galactic pl relation based on the revised infra - red surface brightness method also contradicts the steeper galactic pl relation based on independent methods from open cluster main - sequence fitting @xcite .
it is worthwhile to point out that our sample selection does not affect the detection on non - linear lmc pl relation at mean light .
since the mean magnitudes of a cepheid light curve is less affected by our constrains on selecting the cepheids with good light curves , we can use the published ( reddening corrected ) mean magnitudes to test the non - linear lmc pl relation .
the anonymous referee kindly provided a large sample of lmc cepheids that combined the published mean @xmath7-band magnitudes from the ogle , @xcite and @xcite datasets .
there are a total of 115 long period cepheids in this sample and the @xmath2-test still return a significant detection of the non - linear lmc pl relation .
the ogle+@xcite combined data also give very similar results .
similar tests have also been done in @xcite by using the macho data alone and the macho+@xcite combined data .
the non - linear lmc pl relation is still present from the @xmath2-test results on these two datasets
. therefore we believe our sample selection does not affect the detection of the non - linear lmc pl relation .
the detection of non - linear lmc pl relation from totally independent ogle and macho data , using totally independent reddening estimates , suggested that this non - linearity is real and our paper is the first attempt to theoretically explain this non - linearity in terms of the hif - photosphere interaction . due to small number of lmc models , it is impossible to derive the theoretical pc and ac relations with a small error on the slope and compare directly to the empirical relations .
however , these lmc models can be qualitatively compared to the observations by converting some physical quantities to the observable quantities and vice versa , such as the temperature - colour conversion .
hence we compared our model light curves to the observations in terms of theoretical pc and ac relations at the phases of maximum , mean and minimum light and also in terms of the fourier parameters from theoretical light curves with observations .
the theoretical quantities from the models generally agree with the observations , but it was found out that these models tend to have smaller amplitudes and ( hence ) the temperature is cooler at maximum light than the real cepheids . though our models have some drawbacks in this comparison , our main interest is in comparing the interaction of the photosphere and hif as a function of phase with similar results presented in paper ii for galactic cepheid pulsation models .
the aim is _ not _ to compare our models rigorously with observations but rather to study models which match observations reasonably well in the context of the theoretical framework described in previous sections and in paper i & ii . nevertheless we argued that the qualitative nature of the photosphere - hif interaction is not seriously affected by these problems .
our postulate is that at certain phases , this interaction can affect the pc relation due to the properties of the saha ionization equation : specifically for reasonably low densities in cepheid envelopes , hydrogen ionizes at a temperature that is almost independent of period .
consequently , when the photosphere is located at the base of the hif , the photospheric temperature and hence the colour is almost independent of period .
however , when this engagement occurs , but the density is greater , then the temperature at which hydrogen ionizes again becomes sensitive to global surroundings and hence on period .
when the photosphere is not engaged with the hif in this way , its temperature is again dependent on period and global stellar parameters . for galactic cepheids
, this hif - photosphere interaction occurs mainly at maximum light for cepheids with @xmath227 ( paper ii ) . at minimum light ,
there is a strong correlation between the hif - photosphere distance and period leading to a definite ac relation at minimum light for galactic cepheids ( skm , paper i & ii ) . in this paper
, we have found tentative evidence that , for short period lmc models which match observations in the period - color plane , the hif - photosphere interaction occurs at most phases but at densities which are too high to produce a flat pc relation
. why would these short period lmc cepheids be different in this regard to short period galactic cepheids ?
one possibility could be that this is partly because these lmc cepheids are hotter than their galactic counterparts @xcite .
the hif - photosphere are disengaged for most of the pulsation cycle for long period lmc cepheids .
this happens because as the period increases , so does the @xmath10 ratio which pushes the hif further inside the mass distribution . when the hif - photosphere are disengaged in this way , the photospheric temperature is more dependent on density and hence on period .
the change is sudden because the hif - photosphere are either engaged or they are not .
this can lead to a sudden change in the pc relation at 10 days as shown by the observations @xcite .
however , at maximum light the hif - photosphere are engaged at low densities for long period lmc cepheids leading to the observed flat pc relation for these stars . taken together with equation ( 1 ) , this theoretical scenario is consistent with the observed pc - ac behavior described in paper i and in this study .
the anonymous referee has noted that these suggestions about photospheric density can be tested by spectroscopic means .
we now enumerate some caveats to our argument that could be addressed in future papers . 1 . since the smc pc relation at mean light is linear ( e.g. , paper i ) , how do smc ( i.e. , metal - poor ) models fit into the theoretical scenario outlined in this paper and paper ii , if at all ? this is a difficult question and its full answer is beyond the scope of this paper
however , as the metallicity decreases , we do note that the smc has a different ml relation to the lmc and galaxy and so does the temperatures associated with the instability strip .
these will change the relative location of the hif and photosphere @xcite and possibly alter the phase at which they interact .
further the amplitudes for smc cepheids are smaller due to the lower metallicity @xcite .
this will also affect the hif - photosphere interaction .
one difference which can be consistent with this is the fact that the pc relation at maximum light in the smc is not flat ( see paper i ) but it is the case for the galaxy and lmc pc relations .
this indicates that at maximum light , there is less interaction between the hif and photosphere at low densities .
this leads to an observed linear pc relation at mean light for the smc cepheids .
these will be investigated further in a future paper in this series .
2 . could the well - known hertzsprung progression play any part in causing the observed changes in the galactic and lmc pc relations ?
it may also be that higher order overtones becoming unstable or stable , though with the fundamental mode still being dominant , may also have an impact on the pc relation in some as yet unknown way ( paper ii ) .
4 . the behavior of short period lmc cepheids still needs to be understood , for example , what causes the difference between the bottom left panels of figures [ c9deltalmc ] and [ c9delta ] ?
that is , why is it that for short period galactic / lmc cepheids , the hif - photosphere are disengaged / engaged ?
our experience suggests that constructing short period full amplitude fundamental mode cepheids requires more care than the long period case because the first overtone has a non - negligible growth rate .
because of this we feel a thorough study of these short period cepheids merits a separate paper .
5 . would more advanced pulsation codes which , for example , can match the observed amplitudes and which contain a more accurate model of time dependent turbulent convection , yield similar results , especially for figure [ c9delta ] ?
could such codes fare better in modeling short period lmc cepheids ?
smk acknowledges support from hst - ar-10673.04-a .
we thank an anonymous referee for several useful suggestions and providing the data for our testing .
we would also like to thank e. antonello , r. buchler & j. kwan for useful discussions , and r. bell & m. marengo for the discussion regarding the atmosphere fits .
|
period - colour ( pc ) and amplitude - colour ( ac ) relations are studied for the large magellanic cloud ( lmc ) cepheids under the theoretical framework of the hydrogen ionization front ( hif ) - photosphere interaction .
lmc models are constructed with pulsation codes that include turbulent convection , and the properties of these models are studied at maximum , mean and minimum light . as with galactic models , at maximum light the photosphere is located next to the hif for the lmc models . however very different behavior is found at minimum light . the long period ( @xmath0days )
lmc models imply that the photosphere is disengaged from the hif at minimum light , similar to the galactic models , but there are some indications that the photosphere is located near the hif for the short period ( @xmath1 days ) lmc models .
we also use the updated lmc data to derive empirical pc and ac relations at these phases .
our numerical models are broadly consistent with our theory and the observed data , though we discuss some caveats in the paper .
we apply the idea of the hif - photosphere interaction to explain recent suggestions that the lmc period - luminosity ( pl ) and pc relations are non - linear with a break at a period close to 10 days .
our empirical lmc pc and pl relations are also found to be non - linear with the @xmath2-test .
our explanation relies on the properties of the saha ionization equation , the hif - photosphere interaction and the way this interaction changes with the phase of pulsation and metallicity to produce the observed changes in the lmc pc and pl relations .
cepheids stars : fundamental parameters
|
in a previous investigation @xcite , we studied the n - n interaction in the framework of the chromo - dielectric soliton model from a static point of view : we used the born - oppenheimer approximation to derive an adiabatic n - n potential , which showed a soft core repulsion due essentially to the color - electrostatic part of the one - gluon exchange .
previous studies of the n - n interaction in terms of quark degrees of freedom @xcite have pointed out the importance of dynamical methods ( such as generator coordinate or resonating group ) in the calculation of a realistic n - n potential .
for example , in a preceding application of the non - topological soliton model to the n - n problem , schuh et al .
@xcite showed that a significant part of the repulsion was due to dynamics ; the absence of a repulsive core in some previous works was also interpreted as an artifact of the adiabatic approximation @xcite .
the lagrangian of the chromo - dielectric model is defined as in ref .
@xcite : @xmath0 with @xmath1 where @xmath2 is the quark operator and @xmath3 the current quark mass matrix , set here to @xmath4 .
the quark lagrangian @xmath5 is expressed in terms of the covariant derivative @xmath6 , and @xmath7 is the @xmath8-color tensor , where @xmath9 are the @xmath8 structure constants and @xmath10 the @xmath8 generators .
the quantity @xmath11 is the self - interaction of the scalar field , @xmath12 , taken to be of the form : @xmath13 and the dielectric function @xmath14 is : @xmath15 \ , \ ] ] where @xmath16 is the scalar field s vacuum expectation value and @xmath17 the usual step function .
the quark self - energy , due to interactions with confined gluons in the dielectric medium , generates an effective coupling between the quarks and the scalar field : @xmath18 we choose @xmath19 to be of the form : @xmath20 the expression in eq .
( [ cou ] ) is an approximation to what has been calculated in ref .
@xcite , and it is constructed to simulate spatial confinement already at the mean field level .
note that the coupling in eq .
( [ cou ] ) breaks the chiral invariance of the lagrangian of eq .
( [ lag ] ) .
this is an example of dynamical symmetry breaking from which a massless goldstone boson emerges naturally .
the parameters involved in our calculation are @xmath21 and @xmath22 , as discussed in detail in ref .
@xcite . by fitting the nucleon and the @xmath23 masses and the proton s rms charge radius
one remains with two free parameters , for which it is convenient to use the dimensionless quantities c and @xmath24 . in this paper , we have chosen the set @xmath25 and c=10000 taken from table 1 of ref .
contrary to ref .
@xcite , the quarks here are not only coupled to the @xmath12-field but also interact among themselves through one - gluon exchange ( oge ) .
the oge is treated in abelian approximation , and it can be separated into two terms : a self - interaction term ( in addition to @xmath19 of eq .
( [ cou ] ) ) , which is required for color confinement and which contributes to the one - body part of the hamiltonian , and a term of mutual interactions , which gives rise to the two - body part of the hamiltonian . as mentioned earlier , in the adiabatic approximation of ref .
@xcite , it was the color - electrostatic part of the oge , which arises from the time - component of the gluonic quadrivector @xmath26 , and especially the corresponding self - energy diagrams , which were responsible for the soft - core repulsion . in this work ,
we incorporate the dynamics of the n - n interaction by employing the generator coordinate method ( gcm ) ; we derive an approximate differential equation for the n - n wave function describing the relative motion of the two nucleons .
this equation contains a local n - n potential and an effective , coordinate dependent mass . by means of a fujiwara transformation ,
we then define a n - n separation length , x , from the deformation parameter used previously in the adiabatic approximation .
this allows us to introduce a constant mass and to rewrite the effective potential in terms of this coordinate x. one of our objectives is to study the explicit role of the one - gluon exchange effects on the local n - n potential , included for the first time in such type of calculations .
another aim is to establish a connection between our effective deformation parameter and the true internucleon separation .
the latter will enable us to apply our six - quark wave functions to studies of the quark substructure of light nuclei , as has been carried out already , for instance , in ref .
the present numerical results correspond to the ( ts)=(10 ) sector , although the formalism at hand can easily be extended to other isospin - spin channels .
the gcm was introduced in the fifties by hill and wheeler @xcite to describe collective motion in nuclear systems , such as rotation , vibration or center of mass motion @xcite .
starting from a many - body wave function @xmath27 depending on a collective coordinate @xmath28 ( the deformation parameter of the six - quark system in our case ) , a trial wave function is constructed by taking a linear combination of the states @xmath27 with some weight function @xmath29 , @xmath30 where @xmath29 is determined through the variational principle @xmath31 which leads to the hill - wheeler integral equation : @xmath32 this is a homogeneous fredholm - type equation of the first kind , notoriously unstable numerically .
although some methods exist to make it stable ( such as regularization @xcite , removal of the zero normalization eigenmodes @xcite , gaussian transform @xcite , etc . ) , we prefer to solve a differential equation approximately equivalent to the hill - wheeler equation , both for numerical stability and to facilitate comparison with analyses based on the schrdinger equation . in general , @xmath28 is a multidimensional parameter .
it is at least three - dimensional when correspondence is made to @xmath33 .
we here restrict the calculations to the zero - impact parameter case , which reduces the problem to a one - dimensional one , and leave consideration of the angles to a later study .
to derive such a differential equation , it is more convenient to work with mean and relative deformation parameters , @xmath34 and @xmath35 , defined as @xmath36 expanding the weight function in a taylor series around @xmath37 , one has : @xmath38\nonumber\\ & & \langle\,\beta+\frac{\delta}{2}\,|\,h - e\,|\,\beta-\frac{\delta } { 2}\,\rangle\left[\phi(\beta)-\frac{\delta}{2}\phi'(\beta)+ \frac{\delta^2}{8}\phi''(\beta)+\ldots\right ] \ .\end{aligned}\ ] ] it is convenient to introduce the moments : @xmath39 because @xmath40 is an even function of @xmath35 , the odd moments are zero .
supposing , moreover , that @xmath41 is a sharply peaked function of @xmath35 , one can stop the expansion at second order in @xmath35 .
partial integration and variation by @xmath42 leads then to the hill - wheeler differential equation : @xmath43\phi = e \left[n_0+\frac{1}{8}\frac{d^2n_2}{d\beta^2}\right]\phi \ . \label{eqhw1}\ ] ] the introduction of a new function into the hermitian , @xmath44 where @xmath45 allows us to transform eq .
( [ eqhw1 ] ) into hermitian form : @xmath46\tilde{\phi}(\beta ) = e\tilde{\phi}(\beta ) \ , \label{schrod}\ ] ] where @xmath47 is given by : @xmath48 with @xmath49 the term @xmath50 is the effective mass : @xmath51 the total energy e enters the definition of b ; its asymptotic form at threshold is : @xmath52 where @xmath53 is the nucleon mass .
note that because we did nt incorporate center of mass corrections the asymptotic value of the potential in eq .
( [ pot1 ] ) is not equal to the experimental value of @xmath54 .
we have indeed @xmath55 = 2468 mev when gluons are not included and @xmath55 = 2240 mev when they are . in practice , we could obtain a value closer to the experimental value by subtracting recoil corrections from the asymptotic energy : @xmath56 but we prefer to avoid this step . this simplification does not affect our conclusions . following brink and banerjee @xcite ,
we replace @xmath57 in the mass term by : @xmath58 this approximation is consistent with neglecting higher order derivatives of the moments in the hamiltonian . the moments @xmath59 ( n=0 , 2 ) and the corresponding quantities @xmath50 and @xmath47 have been calculated for three distinct cases : @xmath60 where @xmath61 , @xmath62 and @xmath63 are , respectively , the non - gluonic one - body term of the hamiltonian , the color - magnetic and the full one - gluon exchange contribution ; they are given explicitly in ref .
@xcite . in case ( c ) ,
the one - gluon exchange was left out altogether .
this is in the spirit of an earlier investigation where the friedberg - lee soliton model was applied to n - n scattering without considering gluonic degrees of freedom @xcite . in case
( b ) , the color - magnetic hyperfine interaction was accounted for , and in case ( a ) the full color - magnetic and color - electrostatic oge was included .
the reason to distinguish between cases ( a ) and ( b ) is that in the literature it was claimed that the color - magnetic part of the oge itself is responsible for the repulsive core of the n - n interaction @xcite .
we shall return to this point at the end of section v. the plot of @xmath64 as a function of @xmath34 is given in fig . 1 for the three cases ( a ) , ( b ) and ( c ) .
@xmath64 converges towards a constant value @xmath65 , which can be calculated from considering two well - separated non - interacting three - quark bags : @xmath66 we would expect @xmath65 to be equal to the reduced mass , @xmath53/2 .
the discrepancy between the values of @xmath65 and @xmath53/2 which is especially drastic if the oge is included , i.e. , in cases ( a ) and ( b ) is related to the well - known peierls - yoccoz disease @xcite .
the dependence of the effective mass on @xmath34 prevents us from directly interpreting the potential in eq .
( [ pot1 ] ) as an ordinary n - n potential .
moreover , @xmath34 does nt correspond to the true n - n separation distance ( except for large positive @xmath34 when the two nucleons are well separated ) .
therefore , we wish to transform eq .
( [ schrod ] ) into a schrdinger - like equation with a constant , coordinate independent mass term . for this purpose , one can use a fujiwara transformation @xcite , which relates the generator coordinate @xmath34 to an effective n - n separation length : @xmath67 d\beta ' + \beta \ . \label{fuji}\ ] ] if one now redefines the weight function in eq .
( [ schrod ] ) as @xmath68 eq . ( [ schrod ] ) transforms into the familiar form @xmath69\psi(x)=e\psi(x ) \ , \ ] ] with v given by eq .
( [ pot1 ] ) and @xmath70 figure 2 displays the explicit relationship between @xmath71 and @xmath34 , as obtained from eq .
( [ fuji ] ) .
as expected , the deformation parameter @xmath34 converges asymptotically towards the effective internucleon separation @xmath71 .
the correspondence @xmath72 should be very useful in discussions of the quark substructure of nuclei or nuclear matter using schrdinger - based many - nucleon calculations and employing our six - quark wave functions .
we now wish to present detailed results for : @xmath73 where @xmath47 and @xmath74 are given in eqs .
( [ pot1 ] ) and ( [ potf ] ) .
the value of @xmath55 corresponds to the asymptotic value of @xmath75 calculated from two well - separated non - interacting three - quark bags , and it is given in section iii .
this asymptotic value is the same in cases ( a ) and ( b ) because the color - electrostatic mutual and self - energy terms cancel exactly due to color neutrality when the two nucleons are well separated .
the shape of @xmath85 is quite similar to the adiabatic potentials displayed in fig .
10 of ref .
@xcite , both for the full oge " and no - oge " cases .
this tends to confirm our assumption that the matrix elements @xmath86 are rather sharply peaked around @xmath37 .
the term @xmath87 corresponds to the contribution of non - adiabaticity .
it grows important only for @xmath88 fm , and yields in all cases a repulsion due to the dynamics .
this is according to our expectation and in agreement with ref.@xcite .
note that in cases ( b ) and especially ( c ) , we also obtain an intermediate range attraction in @xmath82 .
the fact that our n - n potential extends to negative @xmath71 should not be taken too literally .
it simply reflects inadequacies in the relationship between the deformation parameter @xmath34 and the n - n separation length @xmath71 , which are connected to the peierls - yoccoz disease mentioned earlier .
we recall that one of the main objectives of this and our previous study @xcite was to incorporate explicitly one - gluon exchange effects , in contrast to ref.@xcite where they were neglected . comparing , for instance
, cases ( a ) and ( c ) , one can see that the oge reinforces the repulsive core considerably .
the existence of a repulsive core in all three cases makes us to attribute it to dynamics rather than to the color - magnetic interaction ( case ( b ) ) , as was inferred in ref.@xcite .
in this investigation , we found that the dynamics are manifestly responsible for the hard - core repulsion of the short - range part of the n - n interaction , and we observed that we could obtain both short - range repulsion and some intermediate range attraction if the entire one - gluon exchange or at least its electrostatic part were neglected . in the results containing the full oge effects the lack of attraction is due to the omission of explicit meson exchanges .
then , to reproduce the experimental phase shifts or other two - body data one necessitates to attach a local obe potential beyond a certain internuclear distance @xcite . to obtain this potential in the framework of our model
we could consider extending our calculations by either including quantum surface fluctuations and/or introducing configurations of the form @xmath89 in addition to the @xmath90 states .
this would be a rather cumbersome procedure within the present model .
the most convenient would be to either allow mesonic degrees of freedom and to consider , e.g. , an explicit pion exchange between the individual quarks @xcite or to simply choose a phenomenological potential .
another important result of this work is the evaluation of the relationship between the deformation parameter @xmath34 and the effective n - n separation length @xmath71 through the fujiwara transformation .
this correspondence is very useful for applications of our model to the description of phenomena involving the quark substructure of light nuclei .
it furthermore allows us to relate many - body correlation functions or n - n wave functions given in the literature to the gcm formalism presented here .
an attractive way to confirm our results would be to solve directly the hill - wheeler integral equation in order to obtain phase shifts .
projection on good angular momentum states should also improve our calculations .
|
the present work is an extension of a previous study of the nucleon - nucleon interaction based on the chromo - dielectric soliton model .
the former approach was static , leading to an adiabatic potential .
here we perform a dynamical study in the framework of the generator coordinate method . in practice , we derive an approximate hill - wheeler differential equation and obtain a local nucleon - nucleon potential as a function of a mean generator coordinate .
this coordinate is related to an effective separation distance between the two nucleons by a fujiwara transformation .
this latter relationship is especially useful in studying the quark substructure of light nuclei .
we investigate the explicit contribution of the one - gluon exchange part of the six - quark hamiltonian to the nucleon - nucleon potential , and we find that the dynamics are responsible for a significant part of the short - range n - n repulsion . # 1#2#3 > 0 # 1#2#3#4 > 0 0.10 in > 0
|
with the advent of the science of complexity , numerous complexity measures have been proposed .
these measures can be grouped mainly in two categories : first group follows the route of constructing the shortest computer program corresponding to a given string .
the well - known kolmogorov - chaitin @xcite and logical depth @xcite measures fall into this category .
the second group follows the information - theoretic approaches whose main examples can be given as effective complexity , thermodynamic depth @xcite , shiner - davison - landsberg @xcite and lpez - ruiz - mancini - calbet measures @xcite .
the information - theoretic approaches are founded on the main idea of multiplying a measure of order by that of disorder . in this sense , these approaches rely heavily on the definitions of entropy as a measure of order / disorder . for example , shiner - davison - landsberg @xcite uses boltmann - gibbs - shannon ( bgs ) entropy of the physical system in its definition , while thermodynamic depth as a complexity measure considers the entropy of the ensemble focusing on the entire history of the system under investigation @xcite .
however , to the best of our knowledge , none of these complexity measures have been constructed particularly to deal with the non - equilibrium stationary states resulting from the external influence of a field .
such a complexity measure has been recently introduced by saparin et al . and applied to logistic map @xcite , heart rate variability @xcite and to the analysis of electroencephalograms of epilepsy patients @xcite .
this new complexity measure is an information - theoretic one and is called renormalized entropy , historically originating from klimontovich s s - theorem ( the letter s here stands for the self - organization ) @xcite .
the renormalized entropy is theoretically equivalent to negative relative entropy between a reference distribution and any other distribution obtained either analytically or numerically through a time - series analysis @xcite .
however , it is not only associated with the relative entropy , since an additional procedure of renormalization is also introduced .
the process of renormalization is used in order to compare two states by equating their mean energies so that non - equilibrium stationary states attain the same mean energy , thereby mimicking the ordinary closed system formalism . combining renormalization and the relative entropy , the renormalized entropy decreases as the control parameter increases , indicating the relative degree of order in the system as first suggested by haken @xcite in the context of self - organization . on the other hand ,
another related issue in the context of self - organization is the existence of bifurcations often observed in nonlinear dynamical systems : the systems possessing a stable fixed point become unstable as they recede away from this stable fixed point as a result of increasing nonlinear effects .
these systems eventually pave their way to the new stable branches through bifurcations . for the system away from the stable fixed point
, this process continues until the system sets in a new stationary state , thereby increasing its order as a signature of self - organization due to the non - linearity , dissipation and the non - equilibrium exhibited by the system ( see the detailed discussion by nicolis and prigogine in ref .
physical processes such as rayleigh - benard @xcite , taylor instability @xcite experiments , bacterial @xcite and _ dictyostelium discoideum _
@xcite colonies fall into the aforementioned category .
these systems , being open due to exchange of energy and/or matter with its surroundings , can not be analyzed in terms of usual h - theorem , since it is valid only for isolated systems .
therefore , prigogine proposed a more general form of the second law i.e. @xmath3 where @xmath4 is the entropy produced inside the system and @xmath5 is the transfer of entropy across the boundaries of the system . in this more general
setting , @xmath6 whereas @xmath5 can be positive or negative depending on the flow of energy across the boundaries of the system .
the overall sign of @xmath7 is determined by the interplay between @xmath4 and @xmath5 .
all the aforementioned models of self - organization requires @xmath8 until the system sets in the stationary state corresponding to the most ordered pattern .
the paper is organized as follows . in section 2 , calculation methods of the renormalized entropy is given . in section 3 , this complexity measure is applied to discrete maps possessing different universality classes i.e. to the logistic map which has periodic route to chaos and its self similar windows like period 3 and period 5 to show its robustness and to sine - circle map which has periodic and quasi - periodic route to chaos .
finally , we discuss our results and compare the two different routes to chaos in terms of the renormalized entropy .
let us consider a generic dynamical map @xmath9 where @xmath10 and @xmath11 denote the number of iterations and the corresponding control parameter of the dynamical map , respectively .
a small change in the value of the control parameter , @xmath12 , yields two normalized distributions i.e. @xmath13 and @xmath14 .
the corresponding shannon entropies read @xmath15 let us now assume that the system i.e. dynamical map under scrutiny evolves in such a manner that the state with index @xmath16 is evolved to through increasing order , namely , the system becomes self - organized as the control parameter increases .
then , setting equilibrium temperature @xmath17 , the normalized boltzmann - gibbs distribution reads @xmath18 \label{boltzmann}\ ] ] with the following effective energy @xmath19 s - theorem by klimontovich equates the effective energies of the concomitant states i.e. renormalizes the states in order to apply h - theorem of boltzmann to open systems , implying @xmath20 in the second law formulation of prigogine @xcite and @xmath21 to compensate for the mean energy difference i.e. turning open system into a closed one . denoting the renormalized state by @xmath22 , one can write @xmath23^{\beta_{eff}}= c\,\exp \left [ \frac{-h_{eff}(x)}{t_{eff } } \right ] \label{renormalization}\ ] ] where @xmath24 is the normalization constant . to check whether any heat intake occurs during self - organization , to form spatially more ordered patterns , through dissipation as a result of interaction with the environment , @xmath25
is calculated from the equality of mean energies @xmath26 if heat intake is needed for the process of self - organization such as rayleigh - benard convection resulting in spatially ordered hexagonal patterns at the stationary state @xcite , then one expects , comparing the equilibrium and non - equilibrium states , @xmath27 , since @xmath17 .
otherwise , we deduce that our assumption regarding the more disorderliness of the initial distribution is not correct , indicating the second distribution to be more disordered .
therefore , when this is the case , one renormalizes the second distribution .
the measure of relative degree of order for these compared states can then be given as the difference of entropies @xmath28 which is called renormalized entropy @xcite .
we use the spectral intensities @xmath29 and @xmath30 averaged over @xmath31 periodograms based on the multiplication of the fourier and the inverse fourier transformation of the time series of @xmath32 in the frequency domain @xmath33 , instead of the density distributions @xmath34 and @xmath35 in eqs .
[ [ shannon]-[relatifentropy ] ] , respectively so that @xmath36 where the sequence @xmath37 is a sufficiently long series of values that can be obtained by iterating a mapping by sampling equidistant points and @xmath38 is the length of the samples in every periodogram @xcite , satisfying @xmath39 and @xmath40 .
such a fourier spectra eliminate the zeros in the distributions and detect different regimes of a deterministic dynamical system .
to start with , let us consider a @xmath41-dimensional mapping of the form @xmath42 on some @xmath41-dimensional phase space @xmath43 .
we can numerically generate data from such a mapping equation that describes dynamics of a specific dynamical system .
we particularly focus on two examples .
one is the logistic map given as @xmath44 where @xmath45 $ ] is a control parameter and the map is confined to the interval @xmath46 $ ] , which exhibits periodic route to chaos . for @xmath47 ,
the system is ( semi)conjugated to a bernoulli shift and strongly mixing .
the other is the sine - circle map given as @xmath48 where @xmath49 is a point on a circle and parameter @xmath50 ( with @xmath51 ) is a measure of the strength of the nonlinearity , whichs exhibit periodic route to chaos .
it describes dynamical systems possessing a natural frequency @xmath52 which are driven by an external force of frequency @xmath53 ( @xmath54 is the bare winding number or frequency - ratio parameter ) and belongs to the same universality class of the forced rayleigh - benard convection @xcite .
winding number for this map is defined to be the limit of the ratio @xmath55 where @xmath56 is the angular distance travelled after @xmath10 iterations of the map function . to increase the degree of irrationality of the system ,
one could use frequency ratio parameter @xmath57 corresponding to winding number @xmath58 that approaches the golden mean gradually by following the sequence of ratios of the fibonacci numbers ( @xmath59 ) . the map is monotonic and invertible ( non - monotonic and non - invertible ) for @xmath60 ( @xmath61 ) and develops a cubic inflexion point at @xmath62 for @xmath63 .
considering technical details , it is important to emphasize that we added gaussian white noise into systems in eqs .
( [ logisticmap ] ) and ( [ circlemap ] ) with a small intensity @xmath64 , which is called basic noise , to every state .
the low intensity of the noise is chosen so as not to influence the dynamics of these systems .
this procedure enables a continuous spectral distribution @xmath65 .
after 65536 transients , we generated the discrete sequences with the length of @xmath66 points separately obtained from eq .
( 9 ) and ( 10 ) .
finally , the spectrum of 18 shifted windows of 4096 samples @xmath67 is estimated and averaged .
one of the dynamical systems possessing bifurcation properties can be cited as the logistic map , which moves from a unique stable fixed point to the critical accumulation point possessing infinite periods through the periodic doubling route .
having reached the critical accumulation point , the system enters into the chaotic regime and from this point on it moves under the influence of chaotic band merging ( i.e. , inverse period - doubling ) .
it should also be remarked that periodic windows with different periods but albeit self - similar structures are also found in this chaotic regime .
we now present the results concerning the behavior of the renormalized entropy and the bifurcation properties in the regions of period @xmath0 , @xmath1 and @xmath2 . before proceeding further
, we note that the renormalized entropy has already been applied to the logistic map for period-2 window by saparin _
our aim in this section is to investigate all other self - similar windows to check whether the renormalized entropy behaves consistently as a complexity measure , i.e. , to check its robustness .
1a in particular shows the behavior of the renormalized entropy in the period 2 region where control parameter @xmath11 lies between @xmath1 and @xmath68 . in this region , relative degree of order increases until the period accumulation point i.e. , @xmath69 as the system evolves from the equilibrium state to a new stationary state in accordance with the self - organization process . as a signature of the detection of the self - organization in this region
then , the relative entropy monotonically decreases as expected . after the period @xmath0 accumulation point onward until the most chaotic state with @xmath47 , the relative degree of order decreases , since band - merging ( as opposed to bifurcation in the previous region ) is exhibited by the system in this region .
therefore , the renormalized entropy increases in the aforementioned region in a non - monotonic manner . to sum up , for an open non - equilibrium dynamical system which approaches the stationary state through period - doubling and recedes away from the stationary state by means of band - merging , the behavior of the renormalized entropy conforms to the dictum of prigogine i.e. , order out of chaos . in fig
1b , we zoom in the period @xmath1 window i.e. , the largest self - similar window in the period @xmath0 region .
the control parameter values for this window are confined in the interval between @xmath70 and @xmath71 .
similarly to the period @xmath0 window , the relative degree of order monotonically increases up to the period accumulation point @xmath72 , and thereafter decreases non - monotonically .
accordingly , the relative entropy decreases up to the accumulation point , and begins to increase after the period accumulation point .
it is worth noting the sudden , unexpected changes in the values of the renormalized entropy which signal the existence of the self - similar windows in the chaotic region .
1c shows the behavior of the renormalized entropy in the period @xmath2 window , which is one of the self - similar windows in the logistic map . due to the self - similarity
, a behavior similar to the one in period @xmath1 is exhibited by the renormalized entropy : it decreases almost up to @xmath73 , and then begins to increase in accordance with the decrease in the relative degree of order .
finally , it is interesting to observe the turns in the relative degree of order for each of the three period accumulation points representing the stationary state of a non - equilibrium dynamical system possessing inherent fractal structure . ( a ) , period @xmath1 ( b ) and period @xmath2 ( c ) of the logistic map.,title="fig:",height=245 ] ( a ) , period @xmath1 ( b ) and period @xmath2 ( c ) of the logistic map.,title="fig:",height=245 ] ( a ) , period @xmath1 ( b ) and period @xmath2 ( c ) of the logistic map.,title="fig:",height=245 ] the sine - circle map can exhibit periodic , quasi - periodic or chaotic behaviors depending on the frequency ratio and the nonlinearity parameters i.e. , @xmath57 and @xmath50 , respectively . for @xmath74 ,
the system dynamics is either periodic ( frequency - locked ) or quasi - periodic depending on the value of the frequency ratio parameter @xmath57 being rational or irrational .
as the nonlinearity parameter @xmath50 approaches zero , the system exhibits quasi - periodic behavior for all values of the frequency ratio parameter @xmath57 .
as the nonlinearity parameter @xmath50 approaches one , frequency - locked steps extend and occupy all @xmath57 axes where @xmath50 is equal to one . in this case , there is a special fraction of @xmath57 value called the most irrational @xmath75 , corresponding to the `` golden mean '' winding number @xmath58 if frequency ratio parameter @xmath57 is locked to its critical value @xmath75 . shortly after this critical value on @xmath76 plane , @xmath77 is the edge of quasi - periodic route to chaos since chaotic behavior can occur .
all these characteristic shapes on @xmath76 plane is called `` arnold tongues '' in the literature . for the @xmath61 region where the nonlinearity parameter @xmath50 is dominant on the system dynamics
, there could be periodic regions with different periods , chaotic regions , and so edges of periodic route to chaos .
also , for this region , there could be periodic windows possessing same universality class with the logistic map .
2 shows the behavior of the renormalized entropy and the bifurcations of the sine - circle map for @xmath78 , @xmath79 and @xmath80 obtained from eqs .
( [ circlemap]-[winding ] ) , respectively where the nonlinearity parameter @xmath50 lies between zero and @xmath81 .
the reference state for the renormalized entropy is chosen to be the one with @xmath82 and @xmath83 where the system evolves towards a unique stable point .
note that the degree of irrationality of the system increases as one moves from fig .
2a towards fig .
2c . in each of the aforementioned figures , the oscillatory behavior of the renormalized entropy
is observed when the system is in the quasi - periodic regime .
this oscillatory behavior is exhibited when @xmath84 $ ] , @xmath85 $ ] , @xmath86 $ ] for the cases @xmath57=@xmath87 , @xmath88 and @xmath75 , respectively .
it is interesting to note that the interval of the nonlinearity parameter @xmath50 increases in regard to the oscillatory behavior of the renormalized entropy as the value of the frequency ratio parameter reaches the critical value @xmath75 , at which the winding ratio @xmath58 attains the golden mean . as a result
, the renormalized entropy can detect the quasi - periodic regime as can be seen from fig .
it is worth noting that the lyapunov exponent is zero for all quasi - periodic regions as well as periodic regimes at the bifurcation points @xcite . in this sense , the renormalized entropy is superior to the lyapunov exponent , since the renormalized entropy behaves in a distinct manner in both of the aforementioned regions .
many chaotic and periodic regions with different periods are present in fig .
2 for the nonlinearity parameter @xmath50 values @xmath89 $ ] , @xmath90 $ ] , @xmath91 $ ] corresponding to @xmath57=@xmath87 , @xmath88 and @xmath75 , respectively .
the renormalized entropy always attains values close to zero in these intervals for the chaotic regions , while it decreases with the increasing number of periods in the periodic regions until it reaches the edge of chaos .
this can be considered as the signature of the relative degree of order within the system .
( a ) , @xmath92 ( b ) and @xmath93 ( c ) of the sine - circle map.,title="fig:",height=245 ] ( a ) , @xmath92 ( b ) and @xmath93 ( c ) of the sine - circle map.,title="fig:",height=245 ] ( a ) , @xmath92 ( b ) and @xmath93 ( c ) of the sine - circle map.,title="fig:",height=245 ] it is well - known that the sine - circle map is in the same universality class as the logistic map for @xmath94 $ ] when @xmath95 . fig .
3 shows the bifurcation and the renormalized entropy for this particular window .
the renormalized entropy behaves exactly as it does in fig .
1a for the logistic map , thereby indicating that different dynamical maps exhibit same behavior in the regions falling into the same universality class . of the the sine - circle map.,width=377,height=245
despite the presence of many different complexity measures , the ones enabling a local comparison of the distributions are quite few ( for a recent example , see ref .
one such measure of relative nature is the renormalized entropy introduced by klimontovich , kurths and coworkers @xcite . in this work ,
the renormalized entropy is used to analyze the logistic and sine - circle maps . in the former example of the logistic map , renormalized entropy decreases ( increases ) up to the accumulation point ( after the accumulation point until the most chaotic state ) as a sign of increasing ( decreasing ) relative degree of order in all the self - similar periodic windows , thereby proving the robustness of this complexity measure . by robustness
, we emphasize the similarity of the behavior of the renormalized entropy in all the self - similar windows , therefore removing the doubt concerning a possible accidental feature of the renormalized entropy as a complexity measure . on the other hand ,
the aforementioned observed changes in the renormalized entropy are reasonable , since the bifurcations occur before the accumulation point , after which the band - merging , in opposition to the bifurcations , is exhibited .
on top of the precise detection of the accumulation points in all these windows , we see that the renormalized entropy can detect the self - similar windows in the chaotic regime by exhibiting sudden changes in its values . for the sine - circle map ,
on the other hand , the renormalized entropy detects also the quasi - periodic regimes by signaling oscillatory behavior particularly in these regimes .
moreover , the oscillatory regime of the renormalized entropy corresponds to a larger interval of the nonlinearity parameter of the sine - circle map as the value of the frequency ratio parameter reaches the critical value , at which the winding ratio attains the golden mean .
lastly , we remark that the renormalized entropy is superior to the lyapunov exponent as a complexity measure , since the renormalized entropy can detect the quasi - periodic regimes as well as the periodic regimes at the bifurcation points in a distinct manner whereas the lyapunov exponent is zero for both of these regions , hence detecting no difference at all @xcite .
this work has been supported by tubitak ( turkish agency ) under the research project number 112t083 .
u.t . is a member of the science academy , istanbul , turkey .
|
we apply renormalized entropy as a complexity measure to the logistic and sine - circle maps . in the case of logistic map , renormalized entropy decreases ( increases ) until the accumulation point ( after the accumulation point up to the most chaotic state ) as a sign of increasing ( decreasing ) degree of order in all the investigated periodic windows , namely , period-@xmath0 , @xmath1 , and @xmath2 , thereby proving the robustness of this complexity measure .
this observed change in the renormalized entropy is adequate , since the bifurcations are exhibited before the accumulation point , after which the band - merging , in opposition to the bifurcations , is exhibited .
in addition to the precise detection of the accumulation points in all these windows , it is shown that the renormalized entropy can detect the self - similar windows in the chaotic regime by exhibiting abrupt changes in its values . regarding the sine - circle map , we observe that the renormalized entropy detects also the quasi - periodic regimes by showing oscillatory behavior particularly in these regimes
. moreover , the oscillatory regime of the renormalized entropy corresponds to a larger interval of the nonlinearity parameter of the sine - circle map as the value of the frequency ratio parameter reaches the critical value , at which the winding ratio attains the golden mean .
|
dense matter studies have opened up new dimensions in understanding the nature and behavioral aspects of nuclear matter at extremes .
an ideal laboratory for such studies can be neutron stars , which contains matter around ten times denser than atomic nuclei .
these compact stars are believed to be made in the aftermath of type ii supernova explosions resulting from the gravitational core collapse of massive stars .
all known forces of nature i.e , strong , weak , electromagnetic and gravitational , play key roles in the formation , evolution and the composition of these stars . thus the study of dense matter not only deals with astrophysical problems such as the evolution of neutron stars , the supernovae mechanism but also reviews the implications from heavy - ion collisions .
neutron stars are charge neutral , and the fact that charge neutrality drives the stellar matter away from isospin - symmetric nuclear matter , the study of neutron stars lends important clues in understanding the isospin dependence of nuclear forces . due to @xmath1-stability conditions
, neutron star is much closer to neutron matter than the symmetric nuclear matter @xcite .
however , with increasing densities , the fermi energy of the occupied baryon states reaches eigenenergies of other species such as @xmath2(1116 ) , @xmath3(1193 ) and @xmath4(1318 ) and the possibility of these hyperonic states are speculated in the dense core of neutron stars ( @xcite-@xcite ) .
studies on hypernuclei experiments suggests the presence of hyperons in dense matter such as neutron stars .
theoretically also , it has been found that the inclusion of hyperons in neutron star cores lowers the energy and pressure of the system resulting in the lowering of the maximum mass of neutron stars , in the range of observational limits .
various hadronic models have been applied to describe the structure of neutron stars .
non - relativistic @xcite and relativistic models ( @xcite-@xcite ) predict nearly same maximum mass of neutron star .
relativistic models have been successfully applied to study finite nuclei @xcite and infinite nuclear matter @xcite where they not only satisfy the properties of nuclear matter at saturation but also the extrapolation to high density is automatically causal .
field theories such as the non - linear @xmath5 model @xcite have been phenomenal in this respect .
presently we apply an effective hadronic model to study the equation of state ( eos ) for neutron star matter in the mean - field type approach @xcite . along with non - linear terms , which ensure reasonable saturation properties of nuclear matter ,
the model embodies dynamical generation of the vector meson mass that ensures a reasonable incompressibility .
therefore , one of the motivation for the present study is to check the applicability of the model to the study of high density matter .
secondly , the parameter sets of the model are in accordance with recently obtained heavy - ion data @xcite . with varying incompressibility and effective nucleon mass the study can impart vital information about their dependency and the underlying effect on the resulting eos . also the existing knowledge on the presence of hyperons in the dense core of these compact stars is inadequate , largely because the coupling strength of these hyperons are unknown .
so it would be interesting to see the effect of hyperons in the dense core of neutron stars and the predictive power of the present model in establishing the global properties of the resulting neutron star sequences .
the outline of the paper is as follows : first we give a brief description of the ingredients of the hadronic model that we implement in our calculations . after introducing the tolman - oppenheimer - volkov ( tov ) equations for the static star , we present some general features of the equation of state and then look at the gross properties of the neutron stars in our calculations and compare our results with the observed masses of the neutron stars , and also with predictions from some of the field - theoretical models .
we then discuss a few constraints on the neutron star mass and radius imposed by recent estimates of the gravitational redshift in the m - r plane .
finally we conclude with outlook on the possible extensions of the current approach .
we start with an effective lagrangian generalized to include all the baryonic octets interacting through mesons : @xmath6~ \psi_b \nonumber \\ & & + \frac{1}{2}\big(\partial_\mu\vec \pi\cdot\partial^\mu\vec\pi + \partial_{\mu } \sigma \partial^{\mu } \sigma\big ) - \frac{\lambda}{4}\big(x^2 - x^2_0\big)^2 - \frac{\lambda b}{6}\big(x^2 - x^2_0\big)^3 - \frac{\lambda c}{8}\big(x^2 - x^2_0\big)^4 \nonumber \\ & & - \frac{1}{4 } f_{\mu\nu } f_{\mu\nu } + \frac{1}{2}{g_{\omega b}}^{2}x^2 \omega_{\mu}\omega^{\mu } - \frac { 1}{4}{\vec r}_{\mu\nu}\cdot{\vec r}^{\mu\nu } + \frac{1}{2}m^2_{\rho}{\vec \rho}_{\mu}\cdot{\vec \rho}^{\mu}\ .\end{aligned}\ ] ] here @xmath7 and @xmath8 , @xmath9 is the baryon spinor , @xmath10 is the pseudoscalar - isovector pion field , @xmath11 is the scalar field .
the subscript @xmath12 and @xmath13 , denotes for baryons .
the terms in eqn .
( 1 ) with the subscript @xmath14 should be interpreted as sum over the states of all baryonic octets . in this model for hadronic matter ,
the baryons interact via the exchange of the @xmath11 , @xmath15 and @xmath16-meson .
the lagrangian includes a dynamically generated mass of the isoscalar vector field , @xmath17 , that couples to the conserved baryonic current @xmath18 . in this paper
we shall be concerned only with the normal non - pion condensed state of matter , so we take @xmath19 and also the pion mass @xmath20 .
the interaction of the scalar and the pseudoscalar mesons with the vector boson generate the mass through the spontaneous breaking of the chiral symmetry .
then the masses of the baryons , scalar and vector mesons , which are generated through @xmath21 , are respectively given by @xmath22 in the above , @xmath21 is the vacuum expectation value of the @xmath11 field , @xmath23 , with @xmath24 , the pion mass and @xmath25 the pion decay constant , and @xmath26 and @xmath27 are the coupling constants for the vector and scalar fields , respectively . in the mean - field treatment
we ignore the explicit role of @xmath28 mesons .
the dirac equation for baryons is the euler - lagrange equation of @xmath29 and is obtained as @xmath30\psi_b=0\ .\ ] ] the mass term in the above equation appears in the form @xmath31 , which is referred to as the effective baryon mass , @xmath32 .
we will now proceed to calculate the equation of motion for the scalar field .
the scalar field dependent terms from the lagrangian density are : @xmath33 where in the mean - field limit @xmath15 = @xmath34 . the constant parameters @xmath35 and @xmath36 are included in the higher - order self - interaction of the scalar field to describe the desirable values of nuclear matter properties at saturation point . using equation ( 2 ) and @xmath37 , the above expression divided by @xmath38 becomes @xmath39 differentiating with respect to @xmath40 , we have the equation of motion for the scalar field including all baryons as : @xmath41=0\ , \label{effmass}\ ] ] where the effective mass of the baryonic species is @xmath42 and @xmath43 and @xmath44 are the usual scalar and vector coupling constants respectively .
it should be noted that although the term @xmath45 in the lagrangian does not appear explicitly in eqn .
( 6 ) , however the effect is there through the mass term , following equation ( 2 ) and through @xmath21 . for a baryon species ,
the scalar density ( @xmath46 ) and the baryon density ( @xmath47 ) are , @xmath48 @xmath49 the equation of motion for the @xmath15 field is then calculated as @xmath50 the quantity @xmath51 is the fermi momentum for the baryon and @xmath52 is the spin degeneracy
. similarly , the equation of motion for the @xmath53meson is obtained as : @xmath54 where @xmath55 is the 3rd - component of the isospin of each baryon species ( given in the table ii ) .
traditionally , neutron stars were believed to be composed mostly of neutrons , some of which eventually @xmath1-decay until an equilibrium between neutron , proton and electron is reached .
the respective chemical potentials then satisfy the generic relationship , @xmath56 , among them . along with charge neutrality condition , @xmath57 , the various particle composition
is then determined and the neutron star is believed to be composed of neutrons , protons and electrons .
muons come into picture when @xmath58 , which happens roughly around nuclear matter density , and the charge neutrality condition is altered to @xmath59 .
hyperons can form in neutron star cores when the nucleon chemical potential is large enough to compensate the mass differences between nucleon and hyperons , which happens roughly around two times normal nuclear matter density , when the first species of the hyperon family starts appearing .
the neutron and electron chemical potentials are constrained by the requirements of conservation of total baryon number and the charge neutrality condition given by , @xmath60 with @xmath61 and @xmath62 are the baryon and lepton densities respectively .
these two conditions combine to determine the appearance and concentration of these particles in the dense core of compact objects .
a general expression may be written down for each baryonic chemical potentials ( @xmath63 ) in terms of these two independent chemical potentials , i.e. , @xmath64 and @xmath65 as , @xmath66 where @xmath63 and @xmath67 are the chemical potentials and electric charge of the concerned baryon species .
after achieving the solution to these conditions , one obtains the total energy density @xmath68 and pressure p for a given baryon density as : @xmath69 @xmath70 as explained earlier , the terms in eqns .
( 13 ) and ( 14 ) with the subscript @xmath14 should be interpreted as sum over the states of all baryonic octets .
the meson field equations ( ( 6 ) , ( 9 ) and ( 10 ) ) are then solved self - consistently at a fixed baryon density to obtain the respective fields along with the requirements of conservation of total baryon number and charge neutrality condition given in equation ( 12 ) and the energy and pressure is computed for the neutron star matter . using the computed eos for the neutron star sequences ,
we calculate the properties of neutron stars .
the equations for the structure of a relativistic spherical and static star composed of a perfect fluid were derived from einstein s equations by oppenheimer and volkoff @xcite .
they are @xmath71 \left[m+4\pi r^3 p\right ] } { ( r-2 gm ) } , \label{tov1}\ ] ] @xmath72 with @xmath73 as the gravitational constant and @xmath74 as the enclosed gravitational mass .
we have used @xmath75 . given an eos , these equations can be integrated from the origin as an initial value problem for a given choice of central energy density , @xmath76 . the value of @xmath77 , where the pressure vanishes defines the surface of the star .
we solve the above equations to study the structural properties of the neutron star , using the eos derived for the electrically charge neutral hyperon rich dense matter .
the parameter set for the present model is listed in table-1 , which is in accordance with recently obtained heavy - ion collision data . with varying effective masses
@xmath78 and incompressibility ( @xmath79 mev ) , the study can give us informations on nuclear equation of state and its effect on the properties of neutron stars .
the parameter sets satisfies the nuclear saturation properties , @xmath80 , energy per nucleon , @xmath81 mev at saturation density @xmath82 , effective nucleon landau mass @xmath83 , incompressibility , and asymmetry energy coefficient value ( @xmath84 mev ) , so that our extrapolation to higher density remains meaningful .
we fix the coupling constant @xmath85 by requiring that @xmath86 correspond to the empirical value , 32 @xmath87 6 mev@xcite .
this gives @xmath88 for @xmath86=32 mev .
0.1 in .parameter sets for the model .
[ cols="^,^,^,^,^,^,^,^,^,^,^,^ " , ] the gravitational redshift interpreted by the @xmath89 ratio comes out to be in the range @xmath90=0.12 - 0.23 , which is plotted in figure 14 . for set i , ii and iii ,
the redshift is nearly same because redshift primarily depends on the mass to radius ratio of the star , which in case of first three sets is nearly same .
for all the parameter sets , the redshift obtained at maximum mass lies in the range @xmath91 , which corresponds to @xmath92 km/@xmath93 .
our calculations predicts @xmath94 in the range @xmath95 km/@xmath93 , which is consistent with the observed value .
the predictive power of the model is evident from figure 14 , where we compare the gravitational redshift as a function of the star mass for the five parameter sets
. the overall results of our calculation are presented in table 3 .
we studied the equation of state of high density matter in an effective model and calculated the gross properties for neutron stars like mass , radius , central density and redshift .
we analysed five set of parameters with incompressibility values @xmath96=210 , 300 and 380 mev and effective masses @xmath97 = 0.80 , 0.85 and 0.90 @xmath98 , that satisfies the nuclear matter saturation properties .
the results are then compared with some recent observations and also a few field theoretical models .
it was found that the difference in nuclear incompressibility is not much reflected in either equation of state or neutron star properties , but nucleon effective masses were quite decisive . at maximum mass , the central density of the star for sets i , ii
, iii and iv was found to be @xmath99 ( nuclear matter density ) but for set v , it was found to be @xmath100 , which has the highest effective mass value .
similarly the maximum mass obtained for the the five eos lies in the range 1.21 - 1.96@xmath93 .
set v , which is softest among all parameter sets , predicts lowest maximum mass 1.21@xmath93 , whereas set iv ( stiff ) predicts the maximum mass to be 1.96@xmath93 and also is the star with the largest radius .
the difference in maximum mass and radius of the star in case of set i , ii and iii is negligible , and so the predicted redshift comes out nearly same , whereas set iv and v presents the two extremes in overall properties , which is the reflection of their different effective mass values .
overall , mass predicted by all the parameter sets agree well with most of the theoretical work and observational limits .
the results were also found to be in good agreement with recently imposed constrains on neutron star properties in the m r plane , and the redshift interpreted therein .
further , the precise measurements of mass of both neutron stars in case of psr b1913 + 16 @xcite , psr b1534 + 12 @xcite and psr b2127 + 11c @xcite are available which can put constrains on the nuclear equation of state
. masses of neutron stars in x - ray pulsars are also consistent with these values , although are measured less accurately . in case of radii , the values are still unknown , however some estimates are expected in a few years , which would further constrain the eos of neutron star in the m - r plane . in future
, we intend to study the effect of rotation to neutron star structure and also the phase transition aspects in the model .
it is worth mentioning that the density - dependent meson - nucleon couplings is very much successful in non - linear walecka model@xcite and similar work in this direction would be interesting .
one of us tkj would like to thank facilities and hospitality provided by institute of physics , bhubaneswar where a major part of the work was done .
this work was supported by r / p , under dae - brns , grant no 2003/37/14/brns/669 .
shapiro and s.a .
teukolski , _ black holes , white dwarfs , and neutron stars _ ( wiley , new york , 1983 ) .
n.k . glendening , phys .
b114 * , 392 ( 1982 ) ; n. k. glendening , astrophys . j. * 293 * , 470 ( 1985 ) ; n. k. glendening , z. phys . * a 326 * , 57 ( 1987 ) . m. prakash , i. bombaci , m. prakash , p.j .
ellis , j.m .
lattimer and r. knorren , phys .
rep . * 280 * , 1 ( 1997 ) .
j. schaffner - beilich and i.n .
mishustin , phys . rev . * c 53 * , 1416 ( 1996 ) .
a. akmal , v.r .
pandharipande and d.g .
ravenhall , phys . rev . * c 58 * , 1804 ( 1998 ) .
wiringa , v. fiks and a. fabrocini , phys . rev .
* c 38 * , 1010 ( 1988 ) .
walecka , ann . phys . * 83 * , 491 ( 1974 ) .
a. lang , b. blattel , w. cassing , v. koch , u. mosel and k. weber , z. phys . * a 340 * , 207 ( 1991 ) .
n. k. glendening , f. weber and s.a .
moszkowski , phys . rev .
* c 45 * , 844 ( 1992 ) .
h. heiselberg and m. hjorth - jensen , astro .
* 525 * , l45 ( 1999 ) .
t. sil , s.k .
patra , b.k .
sharma , m. centelles and x. vias , phys .
rev . * c 69 * , 044315 ( 2004 ) ; m. del estal , m. centelles , x. vias , and s.k .
patra , phys . rev .
* c 63 * , 044321 ( 2001 ) ; s. k. patra , m. del estal , m. centelles and x. vias , phys . rev .
* c 63 * , 024311 ( 2001 ) .
p. arumugam , b.k .
sharma , p.k .
sahu , s. k. patra , t. sil , m. centelles and x. vias , phys . lett .
* b 601 * , 51 ( 2004 ) ; p.k .
panda , a. mishra , j.m . eisenberg and w. greiner , phys . rev . *
c56 * , 3134 ( 1997 ) .
j. boguta and a.r .
bodmer , nucl . phys . * a 292 * , 413 ( 1977 ) .
p. danielewicz , r. lacey , w.g .
lynch , science * 298 * , 1592 ( 2002 ) .
oppenheimer and g.m .
volkoff , phys .
rev * 55 * , 374 ( 1939 ) ; r.c .
tolman , phys .
rev * 55 * , 364 ( 1939 ) .
p. mller , w.d .
myers , w.j .
swiatecki and j. treiner , at .
data nucl .
data tables * 39 * , 225 ( 1988 ) .
glendening phys . rev .
* c 64 * , 025801 ( 2001 ) .
moszkowski , phys . rev . * d 9 * , 1613 ( 1974 ) .
n.k . glendening and s.a .
moszkowski , phys .
lett . * 67 * , 2414 ( 1991 ) .
m. rufa , j. schaffner , j. maruhn , h. stocker , w. greiner and p. g. reinhard , phys .
* c 42 * , 2469 ( 1990 ) .
a. mishra , p.k .
panda and w. greiner , j. phys .
* g 28 * , 67 ( 2002 ) .
n.k . glendening , nucl .
phys . * a 493 * , 521 ( 1989 ) .
j. ellis , j.i .
kapusta and k.a .
olive , nucl .
* b 348 * , 345 ( 1991 ) .
pethick , rev .
phys . * 64 * , 1133 ( 1992 ) .
brown , c.h .
lee , m. rho and v. thorsson , nucl phys . * a 567 * , 937 ( 1994 ) .
pandharipande , c.j .
pethick and v. thorsson , rev .
* 75 * , 4567 ( 1995 ) .
d.j . nice , e.m .
spalver , i.h .
stairs , o. loehmer , a. jessner , m. kramer and j.m .
cordes , astrophys .
j. * 634 * 1242 ( 2005 ) .
d. barret , j.f .
olive and m.c .
miller , _ astro - ph/0605486_. o. barziv , l. kaper , m.h .
van kerkwijk , j.h .
telting and j. van paradijs , astron .
& astrophys .
* 377 * 925 ( 2001 ) .
j. casares , p.a .
charles and e. kuulkers , astro .
j. * 493 * l39 ( 1998 ) .
orosz and e. kuulkers , mon . not .
. soc . * 305 * 132 ( 1999 ) .
g. baym , c. pethick and p. sutherland , astrophys .
j. * 170 * , 299 ( 1971 ) .
panda , d.p .
menezes , c. providncia , phys.rev .
* c69 * , 025207 ( 2004 ) ; p.k . panda , d.p .
menezes , c. providncia , phys.rev . *
c69 * , 058801 ( 2004 ) ; d.p .
menezes , p.k . panda and c. providncia , phys
* c 72 * , 035802 ( 2005 ) .
d. sanyal , g.g .
pavlov , v.e .
zavlin and m.a .
teter , astrophys .
j. * 574 * l61 ( 2002 ) .
j.h . taylor and j.m .
weisberg , astrophys .
j. * 345 * , 434 ( 1989 ) .
a. wolsczan , nature * 350 * , 688 ( 1991 ) .
deich , s.r .
kulkarni in _ compact stars in binaries _
, j. van paradijs , e.p.j .
van den heuvel and e. kuulkers eds . , dordrecht , kluwer ( 1996 ) . t. niksic , d. vretenar , p. finelli and p. ring , phys . rev . *
c66 * , 024306 ( 2002 ) .
|
we study the equation of state ( eos ) for dense matter in the core of the compact star with hyperons and calculate the star structure in an effective model in the mean field approach .
with varying incompressibility and effective nucleon mass , we analyse the resulting eos with hyperons in beta equilibrium and its underlying effect on the gross properties of the compact star sequences .
the results obtained in our analysis are compared with predictions of other theoretical models and observations .
the maximum mass of the compact star lies in the range @xmath0 for the different eos obtained , in the model .
|
the accelerated expansion of the universe is one of the most fundamental problems in modern cosmology . the standard cosmological model introducing the cosmological constant is consistent with various observations @xcite .
however , the small value of the cosmological constant raises the problem of fine tuning @xcite . as an alternative to the cosmological constant , the cosmic accelerated expansion
might be explained by modifying gravity theory , e.g. , @xcite . in the present paper ,
we focus on the most general scalar - tensor theory with the second order differential field equations @xcite , which was first discovered by horndeski @xcite .
the horndeski s most general scalar - tensor theory , including 4 arbitrary functions of the scalar field and kinetic term , reduces to various modified gravity models by choosing the specific 4 functions . because the horndeski s theory includes a wide class of modified gravity models , we adopt it as an effective theory of the generalized theories of gravity . in the present paper
, we investigate the aspects of the quasi - nonlinear evolution of the cosmological density perturbations in the horndeski s most general scalar - tensor theory assuming that the vainshtein mechanism is at work @xcite .
the vainshtein mechanism is the screening mechanisms , which is useful to evade the constraints from the gravity tests in the solar system .
we investigate the effects of the nonlinear terms in the matter s fluid equations as well as the nonlinear derivative interaction terms in the scalar field equation . in a previous work @xcite , the second order solution of the cosmological density perturbations
is obtained . in the present paper , we extend the analysis to the third order solution , which enables us to compute the 1-loop order matter power spectrum .
there are many works on the higher order cosmological density perturbations and the quasi - nonlinear matter power spectrum , which have been developed from the standard perturbative approach ( see e.g. , @xcite ) .
improvements to include the non - perturbative effects have been investigated , e.g. , @xcite , but we here adopt the standard perturbative approach of the cosmological density perturbations as a starting place for the analysis of the horndeski s most general scalar - tensor theory . related to the work of the present paper , we refer the recent work by lee , park and biern @xcite , in which a similar solution is obtained for the dark energy model within the general relativity . this paper is organized as follows . in section 2
, we review the basic equations and the second order solution @xcite . in section 3
, we construct the third order solutions of the cosmological density perturbations . here , we carefully investigate independent functions of mode - couplings describing nonlinear interactions . in section 4 , we derive the expression of the 1-loop order power spectra of the matter density contrast and the velocity divergence . in section 5 ,
an expression for the trispectrum for the density contrast is presented . in section 6
, we demonstrate the behavior of the 1-loop order power spectra in the kinetic gravity braiding model .
section 7 is devoted to summary and conclusions . in appendix
a , definitions of the coefficients to characterize the horndeski s theory are summarized . in appendix
b , definitions of the functions to describe the nonlinear mode - coupling for the third order solutions are summarized . in appendix c
, a derivation of the 1-loop power spectra is summarized .
expressions in appendix d are useful for the deviation of the 1-loop power spectra .
appendix e lists the coefficients to characterize the kinetic gravity braiding model .
let us start with reviewing the basic formulas @xcite .
we consider the horndeski s most general scalar - tensor theory , whose action is given by @xmath0 where we define @xmath1 \nonumber\\ & & + g_5(\phi ,
x)g_{\mu\nu}\nabla^\mu\nabla^\nu\phi -\frac{1}{6}g_{5x}\bigl[(\box\phi)^3 -3\box\phi(\nabla_\mu\nabla_\nu\phi)^2 + 2(\nabla_\mu\nabla_\nu\phi)^3\bigr],\label{gg}\end{aligned}\ ] ] where @xmath2 and @xmath3 are arbitrary function of the scalar field @xmath4 and the kinetic term @xmath5 , @xmath6 denotes @xmath7 , @xmath8 is the ricci scalar , @xmath9 is the einstein tensor , and @xmath10 is the lagrangian of the matter field , which is minimally coupled to the gravity .
the basic equations for the cosmological density perturbations are derived in ref .
@xcite . here
, we briefly review the method and the results ( see @xcite for details ) .
this theory is discovered in @xcite as a generalization of the galileon theory ( @xcite , see also @xcite ) , but the equivalence with the horndeski s theory @xcite is shown in @xcite .
we consider a spatially flat expanding universe and the metric perturbations in the newtonian gauge , whose line element is written as @xmath11 we define the scalar field with perturbations by @xmath12 and we introduce @xmath13 .
the basic equations of the gravitational and scalar fields are derived on the basis of the quasi - static approximation of the subhorizon scales @xcite . in the models that the vainshtein mechanism may work , the basic equations can be found by keeping the leading terms schematically written as @xmath14 , with @xmath15 , where @xmath16 denotes a spatial derivative and @xmath17 does any of @xmath18 , @xmath19 or @xmath20 .
such terms make a leading contribution of the order @xmath21 , where @xmath22 is a typical horizon length scale , and we have @xmath23 and @xmath24 where @xmath25 is
the background matter density and @xmath26 is the matter density contrast , and we define @xmath27 the equation of the scalar field perturbation is @xmath28 where we define @xmath29 here the coefficients @xmath30 , @xmath31 , @xmath32 , @xmath33 , etc . , are defined in appendix a. @xmath34 , @xmath35 , and @xmath36 are the coefficients of the linear , quadratic and cubic terms of @xmath19 , @xmath18 , and @xmath20 , respectively . from the continuity equation and the euler equation for the matter fluid , we have the following equations for the density contrast @xmath26 and the velocity field @xmath37 , @xmath38=0 , \label{continue } \\ & & { \partial u^i(t,{\bf x } ) \over \partial t}+{\dot a\over a}u^i(t,{\bf x } ) + { 1\over a}u^j(t,{\bf x})\partial_ju^i(t,{\bf x } ) = -{1\over a}\partial_i\phi(t,{\bf x } ) , \label{euler}\end{aligned}\ ] ] respectively .
the properties of the gravity sector is influenced through @xmath18 in ( [ euler ] ) , where @xmath18 is determined by eqs .
( [ trlseq ] ) , ( [ 00eq ] ) and ( [ seom ] ) . note that @xmath51 does nt have the symmetry with respect to exchange between @xmath52 and @xmath53 .
we find the solution in terms of a perturbative expansion , which can be written in the form @xmath54 where @xmath17 denotes @xmath55 or @xmath20 , and @xmath56 denotes the @xmath57th order solution of the perturbative expansion . neglecting the decaying mode solution , the linear order solution is written as @xcite @xmath58 where @xmath59 is growth factor obeying @xmath60 with @xmath61 and @xmath62 describes the linear density perturbations , which are assumed to obey the gaussian random distribution .
here we adopt the normalization for the growth factor @xmath63 at @xmath64 , and introduced the linear growth rate defined by @xmath65 .
the second order solution is written as ( see @xcite for details ) , @xmath66 where the coefficients @xmath67 , @xmath68 , @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 , and @xmath74 , are determined by the functions in the lagrangian and the hubble parameter , whose definitions are summarized in appendix a. here @xmath75 and @xmath76 are defined as @xmath77 with @xmath78 where @xmath79 is obtained by symmetrizing @xmath51 with respect to @xmath52 and @xmath53 , and @xmath80 is the function to describe the mode - couplings for the nonlinear interaction in the gravitational field equations and the scalar field equation .
@xmath79 , @xmath81 and @xmath80 have the symmetry with respect to exchange between @xmath52 and @xmath53 .
one can easily check that the functions to describe the nonlinear mode - couplings , @xmath82 , and @xmath80 satisfy @xmath83
in this section we consider the third order solutions .
the third order solution of the cosmological density perturbations has been investigated in various models @xcite .
we present the third order solution for the horndeski s theory in the cosmological background .
our results are general and applicable to various modified gravity models . plus
our results are useful for the case of the general relativity because we clarify the independence property of the mode - coupling functions and the relevant parameters to characterize the third order solution .
we start with solving the third order equations for gravity and scalar field @xmath84 + \frac{b_3}{a^2h^2}\bigl(\gamma[t,{\bf p } ; q_1,\phi_2 ] \nonumber\\ & & \hspace{7 cm } + \gamma[t,{\bf p } ; q_2,\phi_1]\bigr ) , \label{thirdgra1 } \\ & & -p^2\left({\cal g}_t\psi_3(t,{\bf p } ) + a_2 q_3(t,{\bf p})\right ) -\frac{a^2}{2}\rho_{\rm m}\delta_3(t,{\bf p } ) = -\frac{b_2}{a^2h^2 } \gamma[t,{\bf p } ; q_1,q_2 ] -\frac{b_3}{a^2h^2}\bigl ( \gamma[t,{\bf p } ; q_1,\psi_2 ] \nonumber\\ & & \hspace{7 cm } + \gamma[t,{\bf p } ; q_2,\psi_1 ] \bigr ) -\frac{c_1}{3a^4h^4 } \xi_1[t,{\bf p};q_1,q_1,q_1 ] , \label{thirdgra2}\\ & & -p^2(a_0q_3(t,{\bf p } ) -a_1\psi_3(t,{\bf p } ) -a_2\phi_3(t,{\bf p } ) ) = - \frac{2 b_0}{a^2h^2 } \gamma[t,{\bf p } ; q_1 , q_2 ] + \frac{b_1}{a^2h^2}\bigl ( \gamma[t,{\bf p } ; q_1 , \psi_2 ] \nonumber\\ & & \hspace{1 cm } + \gamma[t,{\bf p } ; q_2,\psi_1]\bigr ) + \frac{b_2}{a^2h^2}\bigl ( \gamma[t,{\bf p } ; q_1 , \phi_2 ] + \gamma[t,{\bf p } ; q_2 , \phi_1 ] \bigr ) + \frac{b_3}{a^2h^2}\bigr ( \gamma[t,{\bf p } ; \psi_1 , \phi_2 ] \nonumber\\ & & \hspace{1 cm } + \gamma[t,{\bf p } ; \psi_2 , \phi_1 ] \bigr ) + \frac{c_0}{a^4h^4 } \xi_1[t,{\bf p};q_1,q_1,q_1 ] + \frac{c_1}{a^4h^4 } \xi_2[t,{\bf p};q_1,q_1,\phi_1 ] .
\label{thirdsca}\end{aligned}\ ] ] inserting the first and the second order solutions into the above equations , we finally have @xmath85 where we define @xmath86 , @xmath87 and @xmath88 by eqs .
( [ wga ] ) , ( [ wgg ] ) and ( [ wxi ] ) , respectively , in appendix b. then , the gravitational and the curvature potentials , and the scalar field perturbations are written as @xmath89 where the coefficients @xmath90 , etc . , are defined in appendix a. the third order equations for @xmath91 and @xmath92 are , @xmath93 using the first and the second order solutions , these equations are rewritten as @xmath94 where we introduce the functions defined by eqs .
( [ waar ] ) to ( [ war ] ) , for which we find that the following relations hold , @xmath95 then , eqs .
( [ eoc33 ] ) and ( [ eoe33 ] ) reduce to @xmath96 combining these two equations , we have the third order equation for @xmath97 as @xmath98 where we define @xmath99 and @xmath100 where used eqs .
( [ lambdatheta ] ) and ( [ 2ndlambdade ] ) , and @xmath101 which follow from the definition of the growth rate @xmath102 and eq .
( [ lineardp ] ) . we can prove that @xmath103 is equivalent to @xmath104 , using ( [ nagl ] ) and ( [ nalgar ] ) , and @xmath105 , which is demonstrated from eqs .
( [ ngamma ] ) and ( [ sigmaphi ] ) .
then , we write @xmath106 the general solution of eq .
( [ 3rddeq ] ) with ( [ 3rdnh ] ) is @xmath107 where @xmath59 and @xmath108 are the growing mode solution and the decaying mode solution , satisfying equation ( [ lineardp ] ) , @xmath109 and @xmath110 are integral constants , and @xmath111 is the wronskian defined by @xmath112 . since we assume that the initial density perturbations obey the gauss distribution , we set @xmath113 , as is done in deriving the second order solution . then , the solution of the third order density perturbations is given by @xmath114 where we define @xmath115 here note that the parameters in front of @xmath116 and @xmath117 in expression ( [ 3rddel ] ) are the same , which originates from the relation ( [ reql ] ) . in the limit of the einstein de sitter universe in the general relativity , the coefficients , @xmath118 , @xmath119 , @xmath120 , and @xmath121 reduce to @xmath122 .
we can redefine these coefficients using the differential equations . inserting the general form of the solution ( [ 3rddel ] ) into ( [ 3rddeq ] ) , we obtain the following differential equations for the coefficients @xmath123 the homogeneous solution of all these equations is @xmath124 and @xmath125 .
therefore , the differential equations ( [ dea ] ) to ( [ ded ] ) consistently yield the inhomogeneous solutions ( [ 3rdkappa ] ) to ( [ 3rdnu ] ) , respectively .
we next show that @xmath126 identically . using the expression ( [ nalal ] ) , we easily find that @xmath127 is the solution of ( [ dea ] ) .
this means that the inhomogeneous solution ( [ 3rdkappa ] ) reduces to @xmath127 .
we can prove @xmath128 directly from ( [ 3rdkappa ] ) , using partial integral .
furthermore we can show that @xmath129 identically .
we can rewrite eq .
( [ deb ] ) , as follows , @xmath130 where we used ( [ reql ] ) and ( [ nagl ] ) .
we can easily check that @xmath119 and @xmath131 satisfies the same differential equation ( see eq .
( [ 2ndlambdade ] ) ) , which leads to @xmath129 . in summary
, we have the expression equivalent to ( [ 3rddel ] ) , @xmath132 thus the third order solution of density contrast is characterized by @xmath131 , @xmath120 , and @xmath121 .
note that @xmath131 is defined to describe the second order solution , then @xmath120 and @xmath121 are the new coefficients which appear at the third order .
table i summarizes the parameters and the mode - coupling functions necessary to describe the second order solution and the third order solution .
recently , the authors of @xcite investigated the third order solution of the density perturbations , in a similar way , but within a model of the general relativity . in their paper ,
6 parameters are introduced to describe the third order density perturbations .
our results suggest that less number of parameters are only independent . inserting the solution ( [ eoc33f ] ) into eq .
( [ 3rdconeq ] ) , we find the solution for the velocity divergence @xmath133 where we define @xmath134 here note that @xmath135 is the parameter to describe the second order solution , and @xmath136 and @xmath137 are the new parameters which appear at the third order .
+ in summary , we first introduced _ nine _ mode - coupling functions in the third order equations , ( [ eoc33 ] ) and ( [ eoe33 ] ) with ( [ phi3 ] ) .
we find the _ three _ identities ( [ rel1 ] ) , ( [ rel2 ] ) and ( [ rel3 ] ) .
then , only _ six _ mode - coupling functions are independent in the _ nine _ ones .
this conclusion that the number of the linearly independent mode - coupling functions is _ six _ can be proved by using the generalized wronskian .
the coefficients in front of @xmath138 and @xmath139 in equation ( [ eoc33 ] ) are the same , which leads to the final third order solution ( [ eoc33f ] ) and ( [ eoe33f ] ) expressed in terms of the _ five _ mode - coupling functions .
.functions for the mode - couplings and parameters necessary to describe the second order solution and the third order solution . [ cols=">,<,^",options="header " , ] as function of the scale factor @xmath140 . in each panel ,
the blue dash - dotted curve is the @xmath141 model , and the red dotted curve , the yellow dashed curve , and the green thick solid curve are the kgb model with @xmath142 , @xmath143 , and @xmath144 , respectively .
[ fig : six ] , width=680 ] figure [ fig : six ] shows @xmath145 as function of the scale factor @xmath140 . in each panel ,
the blue dash - dotted curve is the @xmath141 model , and the red dotted curve , the yellow dashed curve , and the green thick solid curve are the kgb model with @xmath142 , @xmath143 , and @xmath144 , respectively .
all the curves take the limiting value unity at @xmath146 , but deviate from the unity as @xmath140 evolves .
note that the deviation of @xmath147 from unity is small , of the order of a few percent , but the deviation of @xmath148 is rather large , which could be @xmath149 percent .
this is because the parameters associated with the velocity , @xmath71 and @xmath150 defined by eqs .
( [ lambdatheta ] ) and ( [ nutheta ] ) , respectively , contain the time derivative term , which makes a large contribution .
plus , some part of the difference between the @xmath141 and the kgb model come from the difference of the growth rate @xmath151 .
deviation of @xmath152 in the kgb model from that in the @xmath153cdm model is rather small compared with the deviations of @xmath71 and @xmath150 , which comes from the fact that @xmath154 is not a monotonic increasing function but there exists a maximum value at @xmath155 .
figure [ fig : sixx ] shows the 1-loop power spectra @xmath156 , @xmath157 , @xmath158 , from the top to the bottom , respectively , which are normalized by those of the @xmath141 model .
these are the snapshots at @xmath159 , and we adopted the same normalization begin @xmath160 for each model , which means that all the models have the same linear matter power spectrum . in each panel , the red dotted curve , the yellow dashed curve , and the green thick curve show the kgb model with @xmath142 , @xmath143 and @xmath144 , respectively . in the linear regime @xmath161 $ ] , all the models converge because they have the same linear matter power spectrum due to the same normalization @xmath160 .
the differences between the kgb model and the @xmath153cdm model appear for the quasi - nonlinear regime @xmath162 $ ] due to the nonlinear effect .
because all the model have the same linear matter power spectrum , this figure shows that the enhancement of the power spectrum due to the nonlinear effect is small in the kgb model compared with that in the @xmath141 model .
this is understood as the results of the vainshtein effect .
furthermore , the deviation from the @xmath141 model is more significant in the velocity power spectrum than that in the density power spectrum . in general ,
the amplitude of the 1-loop power spectra @xmath156 , @xmath157 , and @xmath158 are decreased when any of @xmath131 , @xmath120 , @xmath163 , and @xmath136 is increased .
the behavior of @xmath157 and @xmath158 in the quasi - nonlinear regime is dominantly influenced by @xmath135 and @xmath136 .
we found the third order solutions of the cosmological density perturbations in the horndeski s most general scalar - tensor theory assuming that the vainshtein mechanism is at work .
we solved the equations under the quasi - static approximation , and the solutions describe the quasi - nonlinear aspects of the cosmological density contrast and the velocity divergence under the vainshtein mechanism . in this work
, we thoroughly investigate the independence property of the mode - couplings functions describing the non - linear interactions .
we found that the third order solution of the density contrast is characterized by @xmath164 parameters for the nonlinear interactions , one of which is the same as that for the second order solutions .
the third order solution of the velocity divergence is characterized by @xmath165 parameters for the nonlinear interactions , two of which are the same parameters as those of the second order solutions .
the nonlinear features of the perturbative solutions up to the third order are characterized by @xmath166 parameters .
furthermore , the 1-loop order power spectra obtained with the third order solutions are described by 4 parameters . assuming the kgb model
, we demonstrated the effect of the modified gravity in the 1-loop order power spectra at the quantitative level .
we found that the deviation from the @xmath141 model appears in the power spectra of the density contrast and the velocity divergence , which can be understood as the results of the vainshtein mechanism .
the deviation from the @xmath141 model is more significant in the velocity divergence than the density contrast , which is explained by a dominant contribution of the parameters @xmath163 and @xmath136 .
it is interesting to investigate whether this is a general feature of the modified gravity with the vainshtein mechanism or not .
this work is supported by a research support program of hiroshima university .
we summarize the definitions of the coefficients in the gravitational and scalar field equations ( [ trlseq ] ) , ( [ 00eq ] ) , ( [ seom ] ) .
@xmath167g_{5xx } + h\dot\phi x\dot xg_{5xxx } \nonumber\\ & & -2\left(\dot x+2hx\right)g_{5\phi x } -\dot\phi xg_{5\phi\phi x}-x\left(\dot x-2hx\right)g_{5\phi xx}\biggr\ } , \\
b_1&=&2x\left[g_{4x}+\ddot\phi\left(g_{5x}+xg_{5xx}\right ) -g_{5\phi}+xg_{5\phi x}\right ] , \\
b_2&= & -2x\left(g_{4x}+2xg_{4xx}+h\dot\phi g_{5x } + h\dot\phi xg_{5xx}-g_{5\phi}-xg_{5\phi x}\right ) , \\
b_3&=&h\dot\phi xg_{5x } , \\
c_0&=&2x^2g_{4xx}+\frac{2x^2}{3}\left(2\ddot\phi g_{5xx } + \ddot\phi xg_{5xxx}-2g_{5\phi x}+xg_{5\phi xx}\right ) , \\
c_1&=&h\dot\phi x\left(g_{5x}+xg_{5xx}\right),\end{aligned}\ ] ] where we define @xmath168 , \\ { \cal g}_t&=&2\left[g_4 - 2 xg_{4x } -x\left(h\dot\phi g_{5x } -g_{5\phi}\right)\right ] , \\ \theta&=&-\dot\phi xg_{3x}+ 2hg_4 - 8hxg_{4x } -8hx^2g_{4xx}+\dot\phi g_{4\phi}+2x\dot\phi g_{4\phi x } \nonumber\\ & & -h^2\dot\phi\left(5xg_{5x}+2x^2g_{5xx}\right ) + 2hx\left(3g_{5\phi}+2xg_{5\phi x}\right ) , \\
{ \cal e } & = & 2 x k_x - k + 6 x \dot \phi h g_{3x } - 2 x g_{3 \phi } - 6 h^2 g_4 + 24 h^2 x ( g_{4x } + x g_{4xx } ) \nonumber\\ & & - 12 h x \dot \phi g_{4 \phi x}- 6 h \dot \phi g_{4\phi } + 2 h^3 x \dot \phi ( 5 g_{5 x } + 2 x g_{5xx } ) \nonumber\\ & & - 6 h^2 x ( 3 g_{5 \phi } + 2 x g_{5\phi x } ) , \\ { \cal p}&= & k - 2x(g_{3 \phi } + \ddot \phi g_{3 x } ) + 2(3 h^2 + 2 \dot h)g_4 - 12 h^2 x g_{4x } - 4 h \dot x g_{4 x } \nonumber\\ & & - 8 \dot h x g_{4x } - 8 h x \dot x g_{4 x x } + 2 ( \ddot \phi + 2 h \dot \phi)g_{4 \phi } + 4 x g_{4 \phi \phi } + 4 x ( \ddot \phi - 2 h \dot \phi)g_{4\phi x } \nonumber\\ & & - 2 x ( 2 h^3 \dot \phi + 2 h \dot h \dot \phi + 3 h^2 \ddot \phi)g_{5 x } - 4 h^2 x^2 \ddot \phi g_{5xx } + 4 h x ( \dot x - h x)g_{5 \phi x } \nonumber\\&&+ 2\left [ 2 ( h x ) \dot { } + 3 h^2 x \right]g_{5\phi } + 4 h x \dot \phi g_{5 \phi \phi}.\end{aligned}\ ] ] the coefficients in the first and
the second order solutions are defined as follows , @xmath169 some details are described in the previous paper @xcite , but one can show that @xmath131 obeys the differential equation , @xmath170 the coefficients for the third order solutions are defined as @xmath171
in this appendix , we summarize the functions that describe the nonlinear mode - couplings of the third order solutions . in order to derive eqs .
( [ graa ] ) , ( [ grab ] ) and ( [ grac ] ) , we define @xmath172 with @xmath173 in deriving eqs .
( [ eoc33 ] ) and ( [ eoe33 ] ) , we define @xmath174 with @xmath175
the cosmological density contrast @xmath176 and the velocity divergence @xmath177 up to the third order of the perturbative expansion are expressed as @xmath178 were we define @xmath179 and the kernels for the density contrast @xmath180 , and @xmath181 , and those for the velocity divergence @xmath182 , and @xmath183 , as follows , @xmath184 these kernels have the two types of symmetries .
one is the symmetries in replacement of the wave numbers , @xmath185 the second is the symmetries in the conversion of the sign of the wavenumbers , @xmath186 the same relations hold for @xmath187 and @xmath188 .
the above properties are useful in deriving the expressions of the power spectra , @xmath189 , @xmath190 , @xmath191 , defined by eqs .
( [ definedd ] ) , ( [ definedt ] ) and ( [ definett ] ) . using the expressions ( [ delta ] ) and ( [ theta ] ) , we find @xmath192 where @xmath193 is the linear matter power spectrum , @xmath194 and we define @xmath195 and @xmath196 as an example , let us explain the derivation of @xmath197 .
inserting ( [ d2k ] ) into ( [ ps22dd ] ) , we have @xmath198 using the relation that hold for the gaussian variables , we have @xmath199 which yields @xmath200 with eq .
( [ lps ] ) .
then , ( [ p221 ] ) yields @xmath201 using the relation ( [ f2ss ] ) , we have @xmath202 which reduces to ( [ pdd ] ) . in the derivation ,
we define @xmath203 , where @xmath40 is the angle between @xmath52 and @xmath204 .
similarly , the expressions ( [ pdt ] ) and ( [ ptt ] ) are obtained for @xmath205 and @xmath206 . in the limit of the einstein de sitter universe
withtin the general relativity , @xmath207 , which gives the well - known expressions @xmath208 which are constant as functions of time .
next , let us explain the derivation of @xmath209 . inserting ( [ d3k ] ) into ( [ ps13dd ] ) , we have @xmath210 using the relations , @xmath211 and the symmetries , ( [ efu3sym ] ) , we have @xmath212 after performing the angular integration with respect to the spherical coordinate of @xmath213 , we finally have ( [ p13mgst ] ) . note that ( [ p13mgst ] ) does not depend on @xmath121 , which occurs because of the identity @xmath214 .
@xmath215 is characterized by @xmath131 and @xmath120 .
similarly , we have the expressions ( [ p13dt ] ) and ( [ p13tt ] ) for @xmath216 and @xmath217 , respectively .
because of the same reason for @xmath209 , @xmath216 and @xmath217 do not depend on @xmath121 and @xmath137 .
furthermore , @xmath216 and @xmath217 do not depend on @xmath135 .
this is because of the nature of the integration @xmath218 finally , @xmath216 depends on @xmath131 , @xmath120 , and @xmath136 , and @xmath217 depends on @xmath131 and @xmath136 .
we find the following relation holds , in general , @xmath219/2 $ ] , from ( [ 2psldt ] ) .
in the limit of the einstein de sitter universe withtin the general relativity , all the coefficients @xmath131 , @xmath120 , @xmath136 reduce to @xmath122 , which reproduces the well - known expressions @xmath220,\\ 2p_{\delta\theta}^{(13)}(k ) & = & { k^3 \over252(2\pi)^2 } p_{\rm l}(k)\int d r p_{\rm l}(rk ) \nonumber\\ & & \times \left[24 { 1\over r^2 } - 202 + 56 { r^2 } - 30{r^4 } + { 3\over r^3 } \left(r^2 - 1\right)^3 \left(5 r^2 + 4 \right)\ln \left ( { r + 1\over |r - 1|}\right ) \right],\\ 2p_{\theta\theta}^{(13)}(k ) & = & { k^3 \over 84(2\pi)^2}p_{\rm l}(k)\int d r p_{\rm l}(rk ) \nonumber\\ & & \times \left[12 { 1\over r^2 } - 82 + 4 { r^2 } - 6 { r^4 } + { 3\over r^3 } \left(r^2 - 1\right)^3 \left(r^2 + 2 \right)\ln \left ( { r + 1\over |r - 1|}\right ) \right].\end{aligned}\ ] ]
here we summarize the useful expressions , which are useful in deriving the 1-loop order power spectra , @xmath221 and @xmath222,\\ & & \int d^3 q_1 \hspace{0.1 cm } p_{\rm l}(rk ) \alpha\gamma_r({\bf k } , { \bf q}_1 , - { \bf q}_1 ) = 0 , \\ & & \int d^3 q_1 \hspace{0.1 cm } p_{\rm l}(rk ) \alpha\gamma_l({\bf k } , { \bf q}_1 , - { \bf q}_1 ) = { 2 \pi k_1 ^ 3 \over 36 } \int d r p_{\rm l}(rk)\left[6 + 16 r^2 - 6r^4 + { 3\over r^3}(r^2 - 1)^3\ln \left({r + 1\over \left| r - 1\right|}\right)\right],\\ & & \int d^3 q_1 \hspace{0.1 cm } p_{\rm l}(rk ) \gamma\gamma({\bf k}_1 , { \bf q}_1 , - { \bf q}_1 ) = { 2 \pi k_1 ^ 3\over 72 } \int d r p_{\rm l}(rk)\left[- 6 { 1 \over r^2 } + 22 + 22 r^2 - 6 r^4 + { 3\over r^3}(r^2 - 1)^4 \ln \left({r + 1\over \left| r - 1\right|}\right)\right ] , \nonumber\\ \\ & & \int d^3 q_1 \hspace{0.1 cm } p_{\rm l}(rk ) \xi({\bf k}_1 , { \bf q}_1 , - { \bf q}_1 ) = 0.\end{aligned}\ ] ]
in the kgb model , we find the coefficients in basic equations , @xmath223 and the non - trivial expressions , @xmath224 we use the attractor solution which satisfies @xmath225
. then we have @xmath226 where we define @xmath227 .
we also have @xmath228 99 p. j. e. peebles , and b. ratra , rev .
* 75 * ( 2003 ) 559 planck collaboration : p. a. r. ade , et al . ,
arxiv:1502.01589 s. weinberg , rev .
* 61 * ( 1989 ) 1 s. weinberg , _ the cosmological constant problems _ , astro - ph/0005265 j. martin , comptes rendus physique , * 13 * 566 ( 2012 ) , arxiv:1205.3365 w. hu and i. sawicki , phys . rev .
d * 76 * ( 2007 ) 064004 a. a. starobinsky , jetp lett .
* 86 * ( 2007 ) 157 s. tsujikawa , phys .
d * 77 * ( 2008 ) 023507 s. nojiri and s. odintsov , phys .
b * 657 * ( 2007 ) 238 g. r. dvali , g. gabadadze and m. porrati , phys .
b * 485 * ( 2000 ) 208 y - s .
song , i. sawicki and w. hu , phys .
d * 75 * ( 2007 ) 064003 r. maartens and e. majerotto , phys .
d * 74 * ( 2006 ) 023004 r. maartens and k. koyama , living rev .
* 13 * ( 2010 ) 5 c. de rham and g. gabadadze , phys.rev .
d * 82 * ( 2010 ) 044020 c. de rham , g. gabadadze , and a. j. tolley , phys .
* 106 * ( 2011 ) 231101 c. de rham , l. heisenberg , phys .
d * 84*(2011 ) 043503 s. f. hassan and r. a. rosen , phys .
* 108 * ( 2012 ) 041101 a. r. gomes , l. amendola , arxiv1306.3593 c. deffayet , x. gao , d. a. steer , and g. zahariade , phys .
d * 84 * , ( 2011 ) 064039 t. kobayashi , m. yamaguchi , and j. yokoyama , prog .
. phys . * 126 * , ( 2011 ) 511 g. w. horndeski , int .
* 10 * ( 1974 ) 363 a. i. vainshtein , phys . lett .
b * 39 * ( 1972 ) 393 r. kimura , t. kobayashi , and k. yamamoto , phys . rev .
d * 85 * ( 2012 ) 024023 r. kase , s. tsujikawa , jcap * 08*(2013)054 t. narikawa , t.kobayashi , d. yamauchi , and r. saito , phys .
d * 87 * ( 2013 ) 124006 y. takushima , a. terukina , and k. yamamoto , phys .
rev d * 89 * ( 2014 ) 104007
|
we study the third order solutions of the cosmological density perturbations in the horndeski s most general scalar - tensor theory under the condition that the vainshtein mechanism is at work . in this work
, we thoroughly investigate the independence property of the functions describing the nonlinear mode - couplings , which is also useful for models within the general relativity .
then , we find that the solutions of the density contrast and the velocity divergence up to the third order ones are characterized by 6 parameters .
furthermore , the 1-loop order power spectra obtained with the third order solutions are described by 4 parameters .
we exemplify the behavior of the 1-loop order power spectra assuming the kinetic gravity braiding model , which demonstrates that the effect of the modified gravity appears more significantly in the power spectrum of the velocity divergence than the density contrast . * third order solutions of the cosmological density perturbations in the horndeski s most general scalar - tensor theory with the vainshtein mechanism * .45 in yuichiro takushima , ayumu terukina , kazuhiro yamamoto .45 in _ department of physical science , graduate school of science , hiroshima university , + higashi - hiroshima , 739 - 8526 , japan _ .4 in
|
white dwarf ( wd ) cosmochronology provides an independent and accurate age dating method for different galactic populations @xcite . using 43 cool wds in the solar neighborhood , @xcite derived a disk age of 8 @xmath7 1.5 gyr .
@xcite and @xcite significantly improved the field wd sample by using sdss and usno - b astrometry to select high proper motion candidates .
however , their survey suffered from the magnitude limit of the palomar observatory sky survey plates and they were unable to find many thick disk or halo wd candidates .
substantial investment of the @xmath8 time on two globular clusters , m4 and ngc 6397 , revealed clean wd cooling sequences . @xcite and
@xcite use these data to derive cooling ages of @xmath9 gyr for the two clusters .
the coolest wds in these clusters are about 650 @xmath7 230 k cooler than the coolest wds in the disk @xcite .
these studies demonstrate that the galactic halo is older than the disk by @xmath10 gyr @xcite .
even though the wds in globular clusters provide reliable age estimates , these clusters may not represent the full age range of the galactic halo . the required exposure times to reach the wd terminus in globular clusters limit these studies to the nearest few clusters .
in addition , only two - filter ( @xmath11 and @xmath12 ) photometry is used to model the absolute magnitude and color distribution of the oldest wds to derive ages .
the far closer and brighter wds of the local halo field are an enticing alternative as well as complementary targets , with the additional potential to constrain the age range of the galactic halo .
accurate ages for field wds can be obtained through optical and near - infrared photometry and trigonometric parallax measurements .
nearby wds can also be used to understand the model uncertainties and put the globular cluster ages on a more secure footing .
the quest for field halo wds has been hampered by the lack of proper motion surveys that go deep enough to find the cool halo wds .
the initial claims for a significant population of halo wds in the field @xcite and in the hubble deep field @xcite were later rejected by detailed model atmosphere analysis ( see * ? ? ? * and references therein ) and additional proper motion measurements @xcite . to date , the coolest known probable halo wds are wd 0346 + 246 @xcite and sdss j1102 + 4113 , with @xmath13 k @xcite .
there are also about a dozen ultracool wds detected in the sdss @xcite that may be thick disk or halo wds , but current wd atmosphere models have problems in reproducing their intriguing spectral energy distributions ( seds ) .
therefore , their temperatures and ages remain uncertain . here
we report the identification of three old halo wd candidates discovered as part of our bok and usno proper motion survey .
the details of this survey and our follow - up observations are discussed in section 2 , whereas our model fits and analysis are discussed in section 3 .
in january 2006 , we started an @xmath14band second - epoch astrometry survey with the steward observatory bok 90-inch telescope with its 90prime camera @xcite .
the 90prime provides a field of view of 1.0 square degree with 0.45@xmath15 pixel@xmath1 resolution .
since 2009 additional observations have been obtained with the u.s .
naval observatory flagstaff station 1.3 m telescope using the ccd mosaic camera ( 1.4 square degree field of view with 0.6@xmath15 pixel@xmath1 resolution ) .
we limited our program to the sdss data release 3 footprint in order to have a relatively long time - baseline between our program and the sdss observations .
we obtain proper motion errors of roughly 20 mas yr@xmath1 at @xmath16 mag ( @xmath17 mag for cool wds ) .
we select candidates for follow - up spectroscopy based on our proper motion measurements and the photometric colors .
we further limit our sample to objects with high proper motion and relatively red colors in order to find the elusive thick disk and halo wds .
we started the follow - up optical spectroscopy of candidate halo wds at the 6.5 m mmt equipped with the blue channel spectrograph in june 2009 . here
we present low resolution spectroscopy of three halo wd candidates with @xmath18 mag .
these observations were performed on ut 2009 june 19 - 21 .
our targets are sdss j213730.87 + 105041.6 , j214538.16 + 110626.6 , and j214538.60 + 110619.0 ( hereafter j2137 + 1050 , j2145 + 1106n , and j2145 + 1106s , respectively ) .
we used a 1.25@xmath15 slit and the 500 line mm@xmath1 grating in first order to obtain spectra with wavelength coverage @xmath19 and a resolving power of @xmath20 1200 .
the @xmath21band magnitudes of our targets range from 21.0 to 21.8 mag , and the exposure times range from 60 to 100 min .
we obtained all spectra at the parallactic angle and acquired he
ar ne comparison lamp exposures for wavelength calibration .
we use the observations of the spectrophotometric standard star g24 - 9 , which is also a cool wd , for flux calibration .
in addition , we obtained @xmath22 and @xmath23band imaging observations of our targets using the mmt and magellan infrared spectrograph ( mmirs ; * ? ? ? * ) on the mmt on ut 2009 sep 2 and 4 .
the fwhm of the images range from 0.8@xmath15 to 1.3@xmath15 .
we use a 1.0@xmath15 or 1.4@xmath15 aperture for photometry .
the @xmath24 field of view of mmirs enables us to use 20 - 50 nearby 2mass stars to calibrate the photometry .
the optical and near - infrared photometry of our targets , as well as proper motions , are presented in table 1 .
the optical photometry is in the ab system and the @xmath25 photometry is in the 2mass ( vega ) system .
we use the corrections given in @xcite to convert the sdss photometry to the ab system .
two of our targets , j2145 + 1106n and s ( n - for north and s - for south ) , are separated by 10@xmath15 and they have proper motions consistent within the errors .
hence , they are in a common proper motion binary system .
lccc r.a.@xmath26 & 21:37:30.87 & 21:45:38.16 & 21:45:38.60 + dec.@xmath26 & + 10:50:41.6 & + 11:06:26.6 & + 11:06:19.0 + @xmath27 ( mas yr@xmath1 ) & @xmath28228.9 & + 191.9 & + 185.9 + @xmath29 ( mas yr@xmath1 ) & @xmath28473.6 & @xmath28366.9 & @xmath28367.7 + @xmath30 & 23.31 @xmath7 0.69 & 23.74 @xmath7 0.91 & 23.45 @xmath7 0.75 + @xmath31 & 21.77 @xmath7 0.06 & 21.45 @xmath7 0.05 & 21.00 @xmath7 0.03 + @xmath32 & 20.51 @xmath7 0.03 & 20.27 @xmath7 0.03 & 19.93 @xmath7 0.02 + @xmath33 & 20.02 @xmath7 0.03 & 19.75 @xmath7 0.02 & 19.49 @xmath7 0.02 + @xmath34 & 19.73 @xmath7 0.08 & 19.68 @xmath7 0.07 & 19.38 @xmath7 0.06 + @xmath35 & 19.21 @xmath7 0.10 & 18.87 @xmath7 0.07 & 18.54 @xmath7 0.06 + @xmath36 & 19.25 @xmath7 0.18 & 19.00 @xmath7 0.10 & 18.31 @xmath7 0.06 + @xmath37 ( k ) & 3780 & 3730 & 4110 + age@xmath38 ( gyr ) & 9.6 & 9.7 & 8.7 + distance@xmath38 ( pc ) & 78 & 69 & 70 + @xmath39 ( km s@xmath1 ) & 195 & 136 & 136 + @xmath40 ( km s@xmath1 ) & 172 , @xmath2897 , @xmath2835 & 31 , @xmath2875 , @xmath28102 & 31 , @xmath2875 , @xmath28102 out of the three targets , only j2145 + 1106s is detected in the usno - b catalog , and it has a proper motion of @xmath41 181.3 @xmath7 5.2 mas yr@xmath1 and @xmath42 mas yr@xmath1 @xcite .
these proper motion measurements are consistent with our measurements within the errors , and they demonstrate that our proper motion measurements are reliable .
our mmt spectroscopy shows that all three targets have featureless spectra ; they are cool dc wds .
below 5000 k , h@xmath43 disappears in cool wd spectra .
however , hydrogen can still show its presence through the red wing of ly @xmath43 absorption @xcite in the blue and through collision - induced absorption due to molecular hydrogen in the infrared @xcite .
cool helium atmosphere wds do not suffer from these opacities , and they are expected to show seds similar to blackbodies @xcite . therefore , ultraviolet and near - infrared data are crucial for determining the atmospheric composition of cool wds
. we use state of the art white dwarf model atmospheres to fit the optical and near - infrared photometry of our targets .
the model atmospheres include the ly @xmath43 far red wing opacity @xcite as well as non - ideal physics of dense helium that includes refraction @xcite , ionization equilibrium @xcite , and the non - ideal dissociation equilibrium of h@xmath44 @xcite .
since parallax measurements are unavailable , we assume a surface gravity of @xmath45 g = 8 ( @xmath46 ) . we discuss the implications of this mass assumption in section 4.2 . we find that the observed seds of our targets are best matched by pure hydrogen atmosphere models .
figure 1 presents the observed and best - fit seds for our targets assuming a pure hydrogen atmosphere composition .
the temperatures for these models range from 3730 to 4110 k. the seds peak around 1 @xmath47 m . even though the optical portion of the seds may be explained by simple blackbodies ,
our @xmath22 and @xmath23band data show that they differ from blackbodies in the infrared .
the pure hydrogen atmosphere models match the ultraviolet , optical , and near - infrared seds of our targets fairly well .
the fits for j2137 + 1050 and j2145 + 1106n are similar to the fits obtained for the halo wd candidate sdss j1102 + 4113 @xcite .
there are slight differences between the observations and these models .
the synthetic @xmath48band fluxes seem lower than observed and there are related problems with matching the @xmath49 and @xmath22 band fluxes .
systematic problems most likely exist for models below 4000 k. the models for the two coolest stars predict absorption bumps around 1 @xmath47 m .
these bumps have never been observed in the spectra of real wds , indicating that the current collision - induced opacity calculations may be problematic for high - density atmospheres of cool wds ( see e.g. * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) . nevertheless , the overall seds of our targets agree with model predictions over the entire @xmath50 m range .
addition of helium can improve the model fits slightly .
figure 2 shows the best - fit models assuming pure h , pure he , and mixed h / he composition . since mixed h / he atmospheres have higher pressure than pure h atmospheres at the same temperature , the collision - induced absorption is expected to be stronger .
addition of 4 - 16% helium ( relative to hydrogen ) into the atmosphere helps with fitting the infrared portion of the seds .
however , these fits are marginally better than the pure hydrogen atmosphere model fits , and they are not statistically significant .
the best - fit temperature values are also similar to the pure hydrogen atmosphere solutions .
hence , the choice of a pure hydrogen or mixed h / he composition with small amounts of helium does not significantly change our results . in any case , the good match between the optical spectrum and the models including ly @xmath43 absorption indicates that these wds have hydrogen - dominated atmospheres ; helium - dominated or highly helium - enriched atmospheres are ruled out ( see also * ? ? ?
the temperature , wd cooling age , and distance estimates for our targets based on pure hydrogen atmosphere models with @xmath51 ( @xmath52 ) and the cooling models by @xcite are given in table 1 .
this mass assumption implies that our targets are located at 70 - 80 pc away from the sun , and the wd cooling ages are 8.7 gyr or longer .
our model fits to the individual seds give cooling ages of 8.7 - 9.7 gyr and distances of 69 and 70 pc for the members of the j2145 + 1106 common proper motion system .
the difference in cooling ages can be explained by a small mass difference between the two stars .
these results suggest that j2145 + 1106 system is a physical binary and that our model fits are reliable .
based on our proper motion measurements and assuming zero radial velocity , we also estimate the tangential velocity and galactic @xmath6 velocities for our targets .
these wds display tangential velocities of 140 - 200 km s@xmath1 .
@xcite emphasize the importance of determining total stellar ages in order to associate any wd with thick disk or halo .
modelling the optical and near - ir seds of the @xcite wd sample , @xcite find that many of the wds in that sample are fairly warm and too young to be halo wds unless they all have masses near 0.5 @xmath53 .
they find that , with estimated temperatures of 3950 - 4100 k and ages of 8.8 - 9.1 gyr , f351@xmath2850 and wd 0351@xmath28564 are the two most likely halo candidates in the @xcite sample . for an average mass of 0.58 @xmath53 , our temperature estimates corresponds to wd cooling ages of 9.6 - 9.7 gyr for j2137 + 1050 and j2145 + 1106n .
these two stars are the coolest field wds currently known .
although the ultracool wds discovered by @xcite and @xcite are possibly cooler than our targets , current models have problems explaining the observed seds of these wds @xcite . using the initial - final mass relations of @xcite , @xcite , and @xcite , we estimate that a 0.58 @xmath53 wd would be the descendant of a @xmath54 star .
such a progenitor halo star has a main - sequence lifetime of 1.0 - 1.3 gyr @xcite .
therefore , the total ages of our 3 targets range from 9.7 to 11.0 gyr ; they most likely belong to the halo or thick disk .
the theoretical uncertainties due to the unknown core composition , helium layer mass , crystallization , and phase separation are on the order of 1 to 2 gyr for these ages ( * ? ? ?
* ; * ? ? ?
* m. montgomery 2010 , priv .
comm . ) .
figure 3 shows the @xmath6 velocities of our targets , assuming they have 0 km s@xmath1 radial velocities , compared to the 1@xmath55 velocity ellipse of the halo and 2@xmath55 ellipse of the thick disk @xcite .
the velocities for the probable halo object wd 0346 + 246 are also shown for comparison .
the @xmath56 velocity of j2137 + 1050 is more than 3@xmath55 different than the thick disk objects studied by @xcite .
similarly , the @xmath57 velocity of j2145 + 1106 is inconsistent with thick disk objects .
the @xmath6 velocities of the j2145 + 1106 binary are similar to that of wd 0346 + 246 .
the radial velocity assumption does not change these results .
negative radial velocities bring the @xmath6 velocities closer to the 1@xmath55 distribution for the halo , and positive radial velocities move them away from the 2@xmath55 thick disk distribution ( see the dashed lines in fig .
3 ) . hence
, both j2137 + 1050 and j2145 + 1106 systems most likely belong to the halo . , and @xmath57 space velocities for our targets assuming 0 km s@xmath1 radial velocity and @xmath58 7.5 , 8.0 , and 8.5 ( from left to right ) .
the points with error bars correspond to @xmath58 8 .
the probable halo member wd 0346 + 246 is shown for comparison .
the 2@xmath55 velocity ellipse of the thick disk and the 1@xmath55 ellipse of the halo are also shown .
the dashed lines show the effect of changing the radial velocity from @xmath28100 to + 100 km s@xmath1 ( from left to right).,title="fig:",width=326 ] , and @xmath57 space velocities for our targets assuming 0 km s@xmath1 radial velocity and @xmath58 7.5 , 8.0 , and 8.5 ( from left to right ) .
the points with error bars correspond to @xmath58 8 .
the probable halo member wd 0346 + 246 is shown for comparison .
the 2@xmath55 velocity ellipse of the thick disk and the 1@xmath55 ellipse of the halo are also shown .
the dashed lines show the effect of changing the radial velocity from @xmath28100 to + 100 km s@xmath1 ( from left to right).,title="fig:",width=326 ] without a parallax measurement , our age , distance , and velocity estimates are of course uncertain .
a @xmath59 of 8.5 ( @xmath60 ) would imply wd cooling ages of 10.1 - 10.6 gyr and @xmath6 velocities that are still inconsistent with the 2@xmath55 thick disk velocity distribution .
likewise , a @xmath59 of 7.5 ( @xmath61 ) would imply wd cooling ages of 4.1 - 5.0 gyr and @xmath6 velocities that are even more inconsistent with the thick disk sample ( see figure 3 )
. the main - sequence lifetimes would be greater than the age of the universe unless the systems are unresolved double degenerates .
an additional constraint is that j2145 + 1106 is a binary with a separation of 10@xmath15 ( 700 au , assuming @xmath51 ) .
this separation is too large to cause any effect on the evolution of each component and it is small enough that the system can survive the gravitational perturbations from passing stars or galactic tides for billions of years ( @xcite demonstrate that more than 99.9% of the binary stars with initial separations of 0.017 pc ( @xmath62 au ) survive for a hubble time ) .
a scenario involving low - mass wds would require both components of the j2145 + 1106 system to be double degenerates , which seems unlikely .
in addition , our pure hydrogen atmosphere models with @xmath51 fit the seds better than the models with @xmath63 , indicating that our targets are not likely to be low - mass wds .
figure 4 displays a color magnitude diagram of the point sources in the region that encloses the wd population of the globular cluster ngc 6397 and our three halo wd candidates assuming @xmath64 7.5 , 8.0 , and 8.5 .
we use our best - fit wd model spectra to derive synthetic photometry in the @xmath65 and @xmath66 filters . depending on the mass ,
our targets can fall on multiple parts of the wd cooling sequence of ngc 6397 .
if they are _ similar _ to the wds in ngc 6397 , they should have masses ranging from 0.5 @xmath53 to 0.9 @xmath53 ( @xmath67 ) .
@xcite find the masses of the brightest wds in the globular cluster m4 to be 0.53 @xmath53 .
this is a reasonable lower limit for our targets assuming single star evolution . at 0.53 @xmath53
, our targets would have wd cooling ages of 8.0 - 9.1 gyr , distances of 72 - 81 pc , and progenitor masses of 1.25 - 1.48 @xmath53 @xcite .
the main sequence lifetimes would be 1.7 - 2.8 gyr for the progenitor halo stars @xcite , and the total ages would be 9.7 - 11.9 gyr .
the galactic space velocities would be inconsistent with the thick disk velocity distribution .
j2137 + 1050 and j2145 + 1106 are cool wds with hydrogen - dominated atmospheres .
our effective temperature estimates of 3730 - 3780 k make j2137 + 1050 and j2145 + 1106n the coolest wds known in the solar neighborhood .
our best - fit models imply total ages of @xmath68 gyr , distances of 70 - 80 pc , and galactic space velocities that are inconsistent with thick disk population within @xmath69 .
we conclude that these targets most likely belong to the halo .
however , trigonometric parallax observations are required in order to constrain the distances , masses , and ages of our targets accurately .
such observations are currently underway at the mdm 2.4 m telescope .
like wd 0346 + 246 and sdss j1102 + 4113 @xcite , our three halo wd candidates have hydrogen - rich atmospheres .
the oldest wds are likely to accrete from the interstellar medium within their @xmath7010 gyr lifetimes and end up as hydrogen - rich wds even if they start with a pure helium atmosphere . however , the current sample of halo wd candidates is not large enough to conclude that most or all of the oldest wds are hydrogen - rich .
observations of larger samples of field wds will be necessary to check whether all wds turn into hydrogen - rich atmosphere wds or not ( see the discussion in * ? ? ?
the three targets that we present here make up only a small fraction of the halo wd candidates in our proper motion survey .
follow - up observations of these targets will be necessary to confirm many more halo wd candidates that can be used to study the age and age dispersion of the galactic thick disk and halo .
already we can see , however , that these halo ( or possibly thick disk ) wds indicate a gap of 12 gyr between the star formation in the halo and the star formation in the disk at the solar annulus .
our observations further demonstrate that deep , wide - field proper motion surveys ought to find many old halo wds . using the @xcite wd luminosity function for the galactic thin disk and a single burst 12 gyr old population with 10% and 0.4% local normalization for the thick disk and halo
, we estimate that there are 3200 thick disk and 140 halo wds per 1000 square degree ( for a galactic latitude of 45@xmath71 ) down to a limiting magnitude of @xmath72 mag ( our survey limit ) . pushing the limiting magnitude down to @xmath73 mag and assuming 50% sky coverage
, we estimate that future surveys like the pan - starrs and lsst will image @xmath70 1.3 million thick disk and @xmath70 80,000 halo wds .
these surveys will be invaluable resources for halo wd studies .
support for this work was provided by nasa through the spitzer space telescope fellowship program , under an award from caltech .
this material is also based on work supported by the national science foundation under grants ast-0607480 and ast-0602288 .
we thank the mmirs commissioning team for obtaining the near - infrared observations , and j. liebert for extensive help with the bok telescope imaging observations and for many useful discussions .
we also thank e. olszewski for building the 90prime instrument and the steward observatory time allocation committee for supporting our proper motion survey program .
bedin , l. r. , salaris , m. , piotto , g. , anderson , j. , king , i. r. , & cassisi , s. 2009 , , 697 , 965 bergeron , p. 2001
, , 558 , 369 bergeron , p. , & leggett , s. k. 2002 , , 580 , 1070 bergeron , p. , ruiz , m. t. , hamuy , m. , leggett , s. k. , currie , m. j. , lajoie , c .- p .
, & dufour , p. 2005
, , 625 , 838 chiba , m. & beers , t. c. 2000 , , 119 , 2843 eisenstein , d. j. , et al .
2006 , , 167 , 40 fontaine , g. , brassard , p. , & bergeron , p. 2001
, , 113 , 409 gates , e. et al .
2004 , , 612 , 129 hall , p. b. , kowalski , p. m. , harris , h. c. , awal , a. , leggett , s. k. , kilic , m. , anderson , s. f. , & gates , e. 2008 , , 136 , 76 hambly , n. c. , smartt , s. j. , & hodgkin , s. t. 1997 , , 489 , l157 hansen , b. m. s. 1998 , nature , 394 , 860 hansen , b. m. s. , et al .
2002 , , 574 , l155 hansen , b. m. s. , et al .
2004 , , 155 , 551 hansen , b. m. s. , et al .
2007 , , 671 , 380 harris , h. c. , et al .
2006 , , 131 , 571 harris , h. c. , et al . 2008 , , 679 , 697 ibata , r. , irwin , m. , bienaym , o. , scholz , r. , & guibert , j. 2000 , , 532 , l41 jiang , y .- f . ,
& tremaine , s. 2010 , , 401 , 977 kalirai , j. s. , hansen , b. m. s. , kelson , d. d. , reitzel , d. b. , rich , r. m. , & richer , h. b. 2008 , , 676 , 594 kalirai , j. s. , saul davis , d. , richer , h. b. , bergeron , p. , catelan , m. , hansen , b. m. s. , & rich , r. m. 2009 , , 705 , 408 kilic , m. , von hippel , t. , mndez , r. a. , & winget , d. e. 2004 , , 609 , 766 kilic , m. , mndez , r. a. , von hippel , t. , & winget , d. e. 2005 , , 633 , 1126 kilic , m. , et al .
2006 , , 131 , 582 kilic , m. , kowalski , p. m. , reach , w. t. , & von hippel , t. 2009 , , 696 , 2094 kowalski , p. m. & saumon , d. 2004 , , 607 , 970 kowalski , p. m. 2006 , , 641,488 kowalski , p. m. , & saumon , d. 2006 , , 651 , l137 kowalski , p. m. 2007 , , 2007 , 474 , 491 kowalski , p. et al 2007 , phys . rev .
b. 2007 , 76 leggett , s. k. , ruiz , m. t. , & bergeron , p. 1998
, , 497 , 294 liebert , j. , dahn , c. c. , & monet , d. g. 1988 , , 332 , 891 liebert , j. , kilic , m. , williams , k. a. , von hippel , t. , winget , d. m. j. , harris , h. c. , levine , s. , & metcalfe , t. s. 2007 , 15th european workshop on white dwarfs , 372 , 129 marigo , p. , girardi , l. , bressan , a. , groenewegen , m. a. t. , silva , l. , & granato , g. l. 2008 , , 482 , 883 mcleod , b. a. , fabricant , d. , geary , j. , martini , p. , nystrom , g. , elston , r. , eikenberry , s. s. , & epps , h. 2004 , , 5492 , 1306 mndez , r. a. , & minniti , d. 2000 , , 529 , 911 montgomery , m. h. , klumpe , e. w. , winget , d. e. , & wood , m. a. 1999 , , 525 , 482 munn , j. a. , et al .
2004 , , 127 , 3034 oppenheimer , b. r. , hambly , n. c. , digby , a. p. , hodgkin , s. t. , & saumon , d. 2001 , science , 292 , 698 oppenheimer , b. r. , et al . 2001 , , 550 , 448 salaris , m. , serenelli , a. , weiss , a. , & miller bertolami , m. 2009 , , 692 , 1013 saumon , d. & jacobson , s. b. 1999 , , 511 , 107 williams , g. g. , olszewski , e. , lesser , m. , burge , j. , & steward observatory 90-prime team 2001 , bulletin of the american astronomical society , 33 , 791 williams , k. a. , bolte , m. , & koester , d. 2009 , , 693 , 355 winget , d. e. , hansen , c. j. , liebert , j. , van horn , h. m. , fontaine , g. , nather , r. e. , kepler , s. o. , & lamb , d. q. 1987 , , 315 , l77 wood , m. a. 1992 , , 386 , 539
|
we report the discovery of three nearby old halo white dwarf candidates in the sloan digital sky survey ( sdss ) , including two stars in a common proper motion binary system .
these candidates are selected from our 2800 square degree proper motion survey on the bok and u.s . naval observatory flagstaff station 1.3 m telescopes , and they display proper motions of @xmath0 yr@xmath1 .
follow - up mmt spectroscopy and near - infrared photometry demonstrate that all three objects are hydrogen - dominated atmosphere white dwarfs with @xmath2 k. for average mass white dwarfs , these temperature estimates correspond to cooling ages of @xmath3 gyr , distances of @xmath4 pc , and tangential velocities of @xmath5 km s@xmath1 . based on the @xmath6 space velocities , we conclude that they most likely belong to the halo .
furthermore , the combined main - sequence and white dwarf cooling ages are 10 - 11 gyr . along with sdss j1102 + 4113 , they are the oldest field white dwarfs currently known .
these three stars represent only a small fraction of the halo white dwarf candidates in our proper motion survey , and they demonstrate that deep imaging surveys like the pan - starrs and large synoptic survey telescope should find many old thick disk and halo white dwarfs that can be used to constrain the age of the galactic thick disk and halo .
|
few decades ago , artificial atoms emerged timidly in the realm of quantum optics as potential photon sources , eventually alternative to their well established natural counterpart @xcite .
nowadays , due to progress in material fabrication and characterization techniques , they have become not just plausible , but in cases even more suitable for both , basic research and technological applications .
one outstanding example is the initially unambiguous and now impressively diaphanous observation of dressed states in self assembled quantum dots @xcite .
this makes nanostructured qubits ( nq ) most attractive systems for doubly dressing endeavors @xcite . on the other hand , key properties of optoelectronic systems and devices such as emission [ absorption ] frequencies and intensities ,
are directly determined by the local density of optical states ( ldos ) ; which basically describes how efficient a system is to emit [ absorb ] photons at some particular energy . in those devices ,
the ldos is set by the electronic density of energy states , so that reduction in the dimensionality rises sharpness in the ranges of optically accessible energies .
these changes achieve a limit when all the spacial dimensions are comparable to the wavelength associated to the confined charge carriers ( 0d systems ) , case in which the ldos distribution consists of dirac deltas ( actually narrow lorentzians , as in high quality nqs ) @xcite . in this work , a size - independent method to control the ldos of a 0d semiconductor emitter is described ; thus providing an indeed realizable scheme to generate photonic subbands of tunable width from fully discretized states .
the paper is organized as follows ; first a general formulation to describe slightly detuned double driven nqs and simulate their resonance fluorescence spectra , is presented .
then , special attention is paid to the monochromatic double dressing case in which photonic subband generation is demonstrated . in the last part
an analytical expression is derived for the relevant case under study , and conclusions are drawn .
the nq is modeled as a semiconductor quantum dot ( qd ) with well defined vacuum and s - type exciton states , separated by energy @xmath0 , under continuous stimulus by two lasers of frequencies @xmath1 and @xmath2 , respectively [ see figure 1(a ) ] .
the total hamiltonian for the two distinguishable fields interacting with the exciton in the dot , in the jaynes - cummings framework reads @xmath3 where @xmath4 is the rising part of the dot dipole transition operator , @xmath5 is the dot - field coupling , and @xmath6 and @xmath7 ( @xmath8 and @xmath9 ) are the number of photons and photon annihilation operators for the laser a ( b ) , respectively @xcite .
@xmath10 is an offset than allows the energy reference to be chosen at convenience .
currently , highly confined quantum dots exhibit typical neutral exciton excitation energy at the ev scale @xcite , so that detunings @xmath11 and @xmath12 at the order of terahertz or even far infrared , can be considered much smaller than the resonance frequency @xmath13 .
let us assume @xmath14 and take as basis the triple direct product between the energy eigenstates of the dot and each of the fields , i.e. @xmath15 where @xmath16 and @xmath17 it can be noted that if the coupling @xmath5 is artificially turned off , for every fixed integer @xmath18 all states with @xmath19 and @xmath20 , alongside of all states with @xmath21 and @xmath22 ; generate a cluster with @xmath23 eigenenergies ranging from @xmath24 ( corresponding to @xmath25 ) to @xmath26 ( corresponding to @xmath27 ) .
now , if the coupling effects are taken into account , the off diagonal non zero terms are @xmath28 and their corresponding complex conjugates . for a given @xmath29 , we can rewrite the basis elements in the form @xmath30 , which emphasizes the number of photons in the laser b as compared to the total of photons .
we introduce the label @xmath31 for the elements of the basis in the corresponding cluster ; i.e. @xmath32 .
thus we can notate by @xmath33 the basis of a determined subspace with total number of photons @xmath29 ( see inset in fig .
1 ) . the index @xmath31 is related to the number of photons in the laser b and to the qd state through @xmath34 once the hamiltonian in eq . ( 1 ) is written in the basis @xmath33 , it can be numerically diagonalized to obtain @xmath23 eigenvalues with their corresponding eigenvectors .
we represent with @xmath35 the orthonormal basis formed by the eigenstates of the coupled system ; that is @xmath36 the coefficients @xmath37 , are directly obtained from the columns of the unitarian matrix @xmath38 that diagonalizes the hamiltonian , in such a way that @xmath39 ( where @xmath40 is diagonal ) .
they evidently represent the projection of the states of the uncoupled basis on the eigenvectors of the coupled system .
when two lasers with specified powers are applied on the dot , the numbers of photons in lasers a and b interacting with the dot states are set ; let us say @xmath41 and @xmath42 , respectively ( @xmath43 ) .
this means that the optically accessible states of the coupled system are those in which the specific state @xmath44 , is part of the superposition .
hence , under such conditions the ldos of the coupled system associated to the reference state @xmath45 , is @xmath46 we are interested in the strong coupling regime , where the rabi splitting as compared to the qd emission linewidth is large enough to allow steady exciton population . in this case the inelastic part of the light - matter scattering is the dominant one @xcite , and the emission intensity in a particular frequency @xmath47 will be proportional to the transition rate of the system releasing a photon of energy @xmath48 . if the transformation @xmath49 is known [ and consequently the ldos @xmath50 , the fluorescence spectrum can be obtained from the fermi golden rule ( fgr ) @xcite .
namely @xmath51 where @xmath52 is the interaction matrix element in which the lowering operator turns the initial state @xmath53 into the final state @xmath54 , and @xmath55 stands for the availability of the final state in the @xmath56 transition . in order to include the finite exciton lifetime for simulating realistic spectra , the dirac delta distribution in the fgr can be replaced by a lorentzian distribution @xmath57 , where @xmath58 is the @xmath56 transition spectral linewidth ( wigner - weisskopf approximation ) @xcite for an experimentally set total number of photons @xmath29 " , the dominant emitting transitions are those from initial states in the cluster corresponding to the subspace @xmath59 to final states in the cluster corresponding to the subspace @xmath29 .
the relevant matrix element are then those of the form @xmath60 where the @xmath61 " in the coefficients of the final states has been omitted because the elements of the transformation @xmath49 are all real . considering the action of the operator @xmath62 , the correspondences shown in eq .
( [ eq-3 ] ) , and the orthonormality of the basis @xmath33 ; the required dipole transition matrix elements can be expressed in the compact form @xmath63
[ fig1 ] from this point we focus on the resonant monochromatic double dressing case , in which @xmath64 . given this condition , eq .
( [ eq-1 ] ) for each subspace becomes a tridiagonal matrix with constant diagonal term @xmath65 ( a manifold with degree of degeneracy @xmath23 , if the @xmath5 coupling is turned off ) .
this resembles a closest - neighbor tight - binding hamiltonian with position depending hopping @xcite , whose diagonalization straightaway renders formation of energy bands .
therefore , analogously to the pass from completely discretized atomic states to band structures in crystals ; it is natural to expect a drastic change in the ldos , evolving from fully discretized levels to quasi - continuous photonic subbands . figure 1(b ) depicts the manifolds formed due to the presence of the two lasers .
the couplings between levels are presented in a analogous way to the hopping terms in a first - neighbor model for one dimensional atomic chains .
diagonalization of this hamiltonian [ see inset in figure 1(b ) ] produces @xmath23 eigenenergies ( @xmath66 ) which are distributed symmetrically around @xmath67 , along the interval @xmath68 .
figure 2(a ) shows a graphical representation of the matrix transformation @xmath49 which diagonalizes the hamiltonian of eq .
( 1 ) in the resonant monochromatic double dressing case , while figure 2(b ) presents the ldos as function of the energy and number of photons in the laser b ( for a given @xmath69 ) .
they clearly illustrate how the band structure appears as a direct consequence of the presence of laser b , so that the well defined side peaks ( mollow triplet @xcite ) turn into energy sidebands as the number of photons in that laser increases ( upper and lower subbands , respect to the initially degenerated eigenenergy @xmath70 ) .
[ fig2 ] in the purely radiative decay limit , for an exciton lifetime @xmath71 , the spectral linewidths are respectively ; @xmath72 for a transition between the upper - upper and lower - lower subbands ( central peak ) , and @xmath73 for a transition between the upper - lower and lower - upper subbands ( side peaks ) @xcite .
for the sake of physical insight and simplicity in calculations , an analytical approximation to obtain the transformation coefficients from well known functions can be pursued . to do this , for the @xmath29-th manifold we change the basis to symmetric and antisymmetric combinations of the elements of the original one ; i.e. @xmath74 . because the manifold has odd number of states , the last one in the original basis ( @xmath75 ) is intentionally left unpaired for ordering the new basis in reference to it . in this new basis , the dot - field interaction matrix element @xmath76 , can be further evaluated to obtain @xmath77 if the new basis is ordered in the form @xmath78 .
therefore , the hamiltonian takes the explicit form @xmath79 in the above matrix , the effects of the laser a on the qd are contained in the diagonal , where energy splittings of magnitude @xmath80 are observed between the symmetric and antisymmetric combinations @xmath81 and @xmath82 .
taking aside the diagonal part @xmath83 , the diagonalization of the remaining matrix ( @xmath84 ) delivers the energy modifications caused by the laser b on the dressed states of the coupled system qd - laser a. those eigenvalues and eigenvectors define the energy spectrum and ldos of the double driven qd , symmetrically distributed around @xmath85 and @xmath86 ( upper and lower energy subbands ) . matrix @xmath84 can be seen as composed by four equally sized blocks , plus the row and column corresponding to the state @xmath87 .
the off - diagonal terms in the top - right and bottom - left blocks ( off - diagonal blocks , responsible of the mixing between the diagonal ones ) , increase in absolute value as their position are closer to the matrix center , i.e. @xmath88 . hence , for the system in the limit @xmath89 and @xmath90 ( in which laser a is much more intense than laser b and then mixing effects between blocks are negligible ) ; the eigenvectors from diagonalization of @xmath84 are a superposition of the new basis according to @xmath91 where the coefficients @xmath92 are to be associated to a set of orthonormal functions . since the off - diagonal elements in the top - left [ bottom - right ] block , located exclusively right above and below the matrix diagonal , have the same structure as those of the operator @xmath93 [ @xmath94 written in the eigenbasis of the number of photons operator ( @xmath95 ) @xcite ; in this limit the matrix elements of the transformation that diagonalizes @xmath84 can be obtained in good approximation from the harmonic quantum oscillator eigenfunctions @xmath96 @xcite .
this is @xmath97 with @xmath98 ( @xmath99 ) , a dimensionless continuum parameter which multiplied by @xmath100 becomes the eigenenergy corresponding to the eigenvector @xmath101 .
the functions are explicitly @xmath102 where @xmath103 is the @xmath104-th order hermite polynomial @xcite .
for the relevant transitions between each of the two subbands in the @xmath59 and @xmath29 manifolds , there are four possibilities : transitions between the upper ( lower ) subbands , alongside with transitions from the upper @xmath59 ( lower @xmath59 ) subband to the lower @xmath29 ( upper @xmath29 ) one .
the dipole transition matrix elements become @xmath105 and @xmath106 , respectively ; where the transition energy has been also shifted to make @xmath107 . on the other side , the normalized ldos for the reference number of photons @xmath42 as function of the energy of the transition final state ( @xmath108 ) ,
is now given by @xmath109 figure 3 shows the ldos as function of the emission frequency normalized to the coupling @xmath5 , for different numbers of photons in the laser b. it is worth noting how the ldos has absolute maxima at values close to @xmath110 , respectively @xcite . for frequencies more separated from the reference ( @xmath111 ) , it decays rapidly to zero ; setting an approximate width for the subbands of two - times the rabi splitting associated to the laser b. under these considerations , the fgr yields for the emission spectrum @xmath112 \hspace*{1ex } \rho_{m}(e_f ) \hspace*{1ex } l(\hbar\omega + e_f - e_i,\gamma_{i , f } ) \hspace*{1ex } , \end{aligned}\ ] ] which after insertion of eq .
( [ eq-15 ] ) and integration over @xmath113 turns in @xmath114 \hspace*{1ex } \mid \phi_m(\frac{\sqrt{2 } e_f}{\hbar g } ) \mid^2 \hspace*{1ex } .\end{aligned}\ ] ] in figure 4 the resonance fluorescence spectra calculated for different ratios between the numbers of photons in lasers a and b ( @xmath115 ) , are shown for a fixed total number of photons @xmath116 .
they are obtained by using both , the numerical quasi - exact approach of eq .
( [ app-1 ] ) and the analytical approximation of eq .
( [ app-2 ] ) . in the upper frame , for a @xmath117 close to the value of the coupling constant @xmath5 ( i.e. the exciton lifetime is large enough respect to the rabi oscillation periods defined by the laser intensities )
; it is clearly appreciated how the mollow side peaks of the single excitation laser case evolve into sidebands , reflecting the energy band formation under the double dot - field coupling .
markedly the subband width increases with the number of photons in the laser b. in other words , due to the second driving field , well defined peaks spread into optically active regions ; then exhibiting similarities with ldos proper of higher dimensionality systems @xcite .
when @xmath117 is substantially larger than @xmath5 , the effects of the second laser become less noticeable and useful .
this because in the time dominion , the short exciton lifetime inhibits coherent rabi oscillations making almost irrelevant the presence of laser b. this sensitivity of the system to the ratio between @xmath5 and @xmath117 provides a way to estimate the order of magnitude of the dot - field coupling .
figures 4(a ) and 4(b ) , evidence that the analytical approximation behind eq .
( [ app-2 ] ) , as long as the value of @xmath118 is not close to one , works ostensibly well even for a moderate number of photons .
the main discrepancy is found around the subband edges , where the exact calculation predicts slight asymmetry due to the finite nature of the gilbert subspaces .
whereas such an asymmetry is tenuous , actually it has been experimentally observed @xcite .
in conclusion , a theoretical approach to model and simulate doubly driven artificial atoms was implemented . as a main result , tailored
manipulation of the optical density of states in semiconductor quantum dots , has been shown for the case of monochromatic double dressing .
it was described how by coupling a nanostructured qubit simultaneously to two distinguishable lasers whose frequencies match the exciton transition , a discrete eigenstate turns into an energy subband in a process closely analogous to band formation in solid state physics .
the author thanks the chinese academy of sciences for financial support through the fellowship for young international scientist " , grant no . 2011y1jb03 .
valuable discussion with c.y .
lu s group is acknowledged .
|
in this work , a model to study the coupling between a semiconductor qubit and two time - dependent electric fields is developed . by using it in the resonantly monochromatic double dressing regime , control of the local density of optical states
is theoretically and numerically demonstrated for a strongly confined exciton .
drastic changes in the allowed energy transitions yielding tunable broadening of the optically active frequency ranges , are observed in the simulated emission spectra .
the presented results are in excellent qualitative and quantitative agreement with recent experimental observations .
|
the study of cepheids and rr lyrae variables has provided rich insight into countless facets of our universe .
the stars are employed : to establish distances to globular clusters , the galactic center , and to galaxies exhibiting a diverse set of morphologies from dwarf , irregular , giant elliptical , to spiral in nature @xcite ; to clarify properties of the milky way s spiral structure , bulge , and warped disk @xcite ; to constrain cosmological models by aiding to establish @xmath3 @xcite ; to characterize extinction where such variables exist in the galaxy and beyond @xcite ; to deduce the sun s displacement from the galactic plane @xcite ; and to probe the age , chemistry , and dynamics of stellar populations @xcite , etc .
an additional bond beyond the aforementioned successes is shared between rr lyrae variables and type ii cepheids , namely that the stars obey a common distance and wesenheit period - magnitude relation .
a wesenheit period - magnitude diagram demonstrates the continuity from rrc to w vir variables ( fig . [ fig1 ] ) . the wesenheit function describing the ogle lmc data is given by :
@xmath4 the relation is reddening - free and relatively insensitive to the width of the instability strip , hence the reduced scatter in fig .
readers are referred to studies by @xcite , @xcite , @xcite , @xcite , and @xcite for an elaborate discussion on wesenheit functions .
the colour coefficient used here , @xmath5 , is that employed by @xcite .
rr lyrae variables pulsating in the overtone were shifted by @xmath6 so to yield the equivalent fundamental mode period ( e.g. , see * ? ? ?
* ; * ? ? ?
@xcite convincingly demonstrated that the rv tau subclass of type ii cepheids do not follow a simple wesenheit relation that also encompasses the bl her and w vir regimes ( see also * ? ? ? * ) .
rv tau variables were therefore excluded from the derived wesenheit function which characterizes variables with pulsation periods @xmath7 ( eqn .
[ eqn1 ] ) .
a @xmath0-based reddening - free type ii cepheid relation @xcite was used to compute the distance to rr lyrae variables in the galaxies ic 1613 @xcite , m33 @xcite , fornax dsph @xcite , lmc and smc @xcite , and the globular clusters m3 @xcite , m15 @xcite , ngc 6441 @xcite , m54 @xcite , @xmath2 cen @xcite , and m92 @xcite .
the resulting distances are summarized in table [ dgalaxies ] , along with estimates from classical and type ii cepheids , where possible .
the calibrators of the aforementioned relation were ogle lmc type ii cepheids @xcite , with an adopted zero - point to the lmc established from classical cepheids and other means ( @xmath8 , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
the distances to classical cepheids were estimated using a galactic calibration @xcite tied to a subsample of cluster cepheids ( e.g. , * ? ? ? * ) and new hst parallax measures @xcite .
defining the relation strictly as a galactic calibration is somewhat ambiguous given that milky way cepheids appear to follow a galactocentric metallicity gradient @xcite .
the @xcite relation is tied to galactic classical cepheids that exhibit near solar abundances @xcite . applying the @xcite relation to classical cepheids observed in the lmc by @xcite
reaffirms the adopted zero - point ( @xmath9 , fig .
[ fig5 ] ) .
no correction was applied to account for differences in metallicity between lmc and galactic classical cepheids owing to the present results and contested nature of the effect ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
a decrease of @xmath10 magnitudes would ensue if the correction proposed by @xcite or @xcite were adopted .
the distances cited in table [ dgalaxies ] to the globular clusters are consistent with that found in the literature ( e.g. , * ? ? ?
a subsample of period - distance diagrams demonstrate that the inferred distances are nearly constant across the entire period range examined ( fig .
[ fig3 ] ) .
moreover , distances computed to rr lyrae variables in the smc , m33 , and ic 1613 agree with that inferred from classical cepheids ( table [ dgalaxies ] ) . the results reaffirm that the slope and zero - point of @xmath0 reddening - free relations are relatively insensitive to metallicity ( see also * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
no corrections were made to account for differences in abundance .
consider the result for the smc , which is founded on copious numbers of catalogued rr lyrae and classical cepheid variables ( ogle ) .
smc and galactic classical cepheids exhibit a sizeable metallicity difference ( @xmath11\simeq0.75 $ ] , @xcite ) .
by contrast , smc rr lyrae variables are analgous to or slightly more metal poor than their lmc counterparts @xcite . if the metallicity corrections for rr lyrae variables and classical cepheids were equal yet opposite in sign as proposed in the literature ( e.g. , @xmath12 & @xmath13 mag dex@xmath14 ) , then the distances computed for smc rr lyrae variables and classical cepheids should display a considerable offset ( say at least @xmath15 mag ) .
however , that is not supported by the evidence which instead implies a negligible offset separating the variable types ( @xmath16 mag , table [ dgalaxies ] ) .
similar conclusions are reached when analyzing distances to extragalactic rr lyrae variables as inferred from a combined hst / hip parallax for rr lyrae ( see * ? ? ?
in sum , comparing cepheids and rr lyrae variables at a common zero - point offers a unique opportunity to constrain the effects of metallicity .
more work is needed here .
rr lyrae variables in @xmath2 cen provide an additional test for the effects of metallicity on distance since the population exhibits a sizeable spread in metallicity at a common zero - point ( @xmath17 \ge-2.4 $ ] , @xcite ) .
an abundance - distance diagram ( fig .
[ fig2 ] ) compiled for rr lyrae variables in @xmath2 cen using @xmath0 photometry from @xcite , and abundance estimates from @xcite , offers further evidence implying that @xmath0-based reddening - free distance relations are relatively insensitive to metallicity .
a formal fit to data in fig .
[ fig2 ] is in agreement with no dependence and yields a modest slope of @xmath18 mag dex@xmath14 .
if that slope is real metal poor rr lyrae variables are brighter than metal rich ones .
uncertainties linked to the cited slope could be mitigated by acquiring additional abundance estimates and obtaining @xmath0 directly ( see * ? ? ?
a minor note is made that although the variable in @xmath2 cen designated v164 is likely a type ii cepheid ( j2000 13:26:14.86 -47:21:15.17 ) , the variable designated v109 may be anomalous or could belong to another variable class ( j2000 13:26:35.69 -47:32:47.03 , see numbering in * ? ? ?
* ) .
lcccc ic 1613 & @xmath19 & & @xmath20 & ( 6 ) + & & @xmath21 ( @xmath22 ) & @xmath23 & ( 9 ) + smc & @xmath24 & @xmath25 & @xmath26 & ( 7,8 ) + m33 & @xmath27 & & & ( 9 ) + & & @xmath28 ( @xmath29 ) & @xmath30 ( i ) / @xmath31 ( o ) & ( 10 ) + & & @xmath32 & @xmath33 ( i ) & ( 11 ) + fornax dsph & @xmath34 & & & ( 5 ) + & @xmath35 & & & ( 4 ) + m54 & @xmath36 & @xmath37 & & ( 3 ) + m92 & @xmath38 & & & ( 2 ) + ngc 6441 & @xmath39 & @xmath40 & & ( 1 ) + & @xmath41 & & & ( 16 ) + m3 & @xmath42 & & & ( 12 ) + & @xmath43 & & & ( 15 ) + @xmath2 cen & @xmath44 & & & ( 13 ) + m15 & @xmath45 & & & ( 14 ) + equation 2 of @xcite was also employed to compute the distance to the brightest member of the variable class , rr lyrae .
@xmath0 photometry from _ _ the amateur sky survey _ _ was utilized @xcite , although concerns persist regarding the survey s zero - point and the star s modulating amplitude .
nevertheless , the resulting distance of @xmath1 pc is consistent with the star s parallax as obtained using hst ( @xmath46 pc , * ? ? ?
* ) , and within the uncertainties of the hip value @xcite .
that reaffirms the robustness of the aforementioned relation to compute distances to variables of the rr lyrae and type ii cepheid class .
rr lyrae s phased @xmath47 & @xmath48 light - curves are displayed in figure [ fig4 ] .
an ephemeris from the geos rr lyr database was adopted to phase the data @xcite , namely : @xmath49 the slope of the wesenheit function derived from a combined sample of sx phe , rr lyrae , and type ii cepheid variables detected in m3 , @xmath2 cen , and m15 , is consistent with that determined from lmc rr lyrae variables and type ii cepheids ( fig .
[ fig1 ] ) . presently , the distances computed to sx phe variables discovered in m3 and @xmath2 cen via the @xmath0 reddening - free type ii relation of @xcite are systemically offset .
however , the new wesenheit relation performs better ( eqn .
[ eqn1 ] ) .
a reanalysis is anticipated once a sizeable sample of sx phe variables become available .
yet meanwhile eqn .
[ eqn1 ] may be employed to evaluate the distances to sx phe , rr lyrae , bl her , and w vir variables .
uncertainties are expected to be on the order of 5 - 15% .
indeed , the correction factor ( @xmath50 ) established by @xcite could be applied to eqn .
[ eqn1 ] to permit the determination of distances to the rv tau subclass of type ii cepheids .
admittedly , further work is needed but the results are encouraging .
a single @xmath0-based reddening - free relation may be employed to simultaneously provide reliable distances to rr lyrae variables and type ii cepheids . the relation s viability is confirmed by demonstrating that distances to rr lyrae variables in the globular clusters m3 , m15 , m54 , @xmath2 cen , m92 , ngc 6441 , and galaxies ic 1613 , m33 , fornax dsph , lmc , and smc , agree with values in the literature and from other means ( table [ dgalaxies ] , see also * ? ? ?
a distance was computed for the nearby star rr lyrae ( @xmath1 pc ) using mean @xmath0 photometry provided by _
the amateur sky survey_. the estimate is consistent with the hst parallax for the star ( @xmath46 pc , * ? ? ?
the slope and zero - point of the @xmath0-based relation appear relatively unaffected by metallicity to within the uncertainties ( table [ dgalaxies ] & fig .
[ fig1 ] , [ fig3 ] ) .
that assertion is supported by noting that although rr lyrae variables in @xmath2 cen exhibit a sizeable spread in metallicity ( @xmath17 \ge-2.4 $ ] , @xcite ) , no statistically significant effect was observed on the computed distances ( fig .
[ fig2 ] ) .
furthermore , the distances computed to rr lyrae variables and classical cepheids in the smc , m33 , and ic 1613 are consistent to within the uncertainties .
no metallicity correction was applied .
finally , sx phe , rr lyrae , and type ii cepheids essentially follow a common wesenheit period - magnitude relation , although poor statistics for the sx phe variables currently limits an elaborate analysis ( fig .
[ fig1 ] ) .
there remain numerous challenges and concerns to be addressed regarding the use of the distance indicators beyond the contested effects of metallicity ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
for example , achieving a common photometric standardization is difficult and systemic offsets may be introduced , particularly across a range in color ( e.g. , * ? ? ?
* ; * ? ? ?
yet another challenge is to establish a consensus on the effects of photometric contamination ( e.g. , blending , crowding ) on the distances to variable stars in distant galaxies @xcite . increasing the presently small number of galaxies with cepheids observed in both the central and less - crowded outer regions is therefore desirable ( e.g. , * ? ? ?
* ; * ? ? ?
unfortunately , a degeneracy complicates matters since the effects of metallicity and crowding may act in the same sense and be of comparable magnitude .
indeed , @xmath51 ( the ratio of total to selective extinction ) may also vary as a function of radial distance from the centers of galaxies in tandem with the metallicity gradient .
efforts to disentangle the degeneracies are the subject of a study in preparation .
further research is warranted to examine the implications of anomalous values of @xmath51 on the distances obtained from the standard candles ( e.g. , * ? ? ?
* ; * ? ? ?
lastly , the continued discovery of extragalactic sx phoenicis , rr lyrae , and cepheids at a common zero - point shall bolster our understanding and enable firm constraints to be placed on the metallicity effect @xcite .
so too will obtaining mean multiband photometry , particularly @xmath0 , for such variables in the field and globular clusters @xcite .
a forthcoming study shall describe how related efforts are to be pursued from the abbey - ridge observatory ( aro ) @xcite .
aavso members drawn toward similar research may be interested in a fellow member s study entitled : _ using a small telescope to detect variable stars in globular cluster ngc 6779 " _
modest telescopes may serve a pertinent role in variable star research @xcite .
andrievsky s. m. et al . , 2002 ,
a&a , 381 , 32 andrievsky , s. m. , kovtyukh , v. v. , luck , r. e. , lpine , j. r. d. , maciel , w. j. , & beletsky , y. v. 2002 , a&a , 392 , 491 benedict g. f. et al . , 2002 ,
aj , 123 , 473 benedict g. f. et al .
, 2007 , aj , 133 , 1810 benk , j. m. , bakos , g. . , & nuspl , j. 2006 , , 372 , 1657 berdnikov l. n. , efremov y. n. , glushkova e. v. , turner d. g. , 2006 , odessa astronomical publications , 18 , 26 bersier , d. , & wood , p. r. 2002 , , 123 , 840 boninsegna , r. , vandenbroere , j. , le borgne , j. f. , & the geos team 2002 , iau colloq . 185 : radial and nonradial pulsationsn as probes of stellar physics , 259 , 166 bono , g. , caputo , f. , fiorentino , g. , marconi , m. , & musella , i. 2008 , , 684 , 102 caldwell , j. a. r. , & coulson , i. m. 1985 , , 212 , 879 clement , c. m. , et al . 2001 ,
aj , 122 , 2587 corwin , t. m. , borissova , j. , stetson , p. b. , catelan , m. , smith , h. a. , kurtev , r. , & stephens , a. w. 2008 , aj 135 , 1459 demers , s. , & wehlau , a. 1977 , , 82 , 620 dolphin , a. e. , et al .
2001 , , 550 , 554 droege , t. f. , richmond , m. w. , sallman , m. p. , & creager , r. p. 2006
, , 118 , 1666 efremov , y. n. 1997 , astronomy letters , 23 , 579 feast m. , 1999 , pasp , 111 , 775 feast m. , 2001 , arxiv : astro - ph/0110360 feast m. w. , laney c. d. , kinman t. d. , van leeuwen f. , whitelock p. a. , 2008 , mnras , 386 , 2115 feast , m. w. 2008 , arxiv:0806.3019 fernie j. d.,1968 , aj , 73 , 995 fernie , j. d. 1969 , , 81 , 707 fernie d. 1976 , the whisper and the vision - the voyages of the astronomers .
fernie j. d. , 2002 , setting sail for the universe : astronomers and their discoveries , rutgers university press , new brunswick , nj ferrarese l. , mould j. r. , stetson p. b. , tonry j. l. , blakeslee j. p. , ajhar e. a. , 2007 , apj , 654 , 186 fouqu p. et al . , 2007 , a&a , 476 , 73 freedman , w. l. , & madore , b. f. 1996 , clusters , lensing , and the future of the universe , 88 , 9 freedman w. l. et al . , 2001 , apj , 553 , 47 gibson , b. k. 2000 , memorie della societa astronomica italiana , 71 , 693 gieren , w. , pietrzyski , g. , soszyski , i. , bresolin , f. , kudritzki , r .-
p . , storm , j. , & minniti , d. 2008 , apj , 672 , 266 groenewegen m. a. t. , udalski a. , bono , g. , 2008 , a&a , 481 , 441 gruberbauer , m. , et al .
2007 , , 379 , 1498 harris , w. e. 1996 , aj , 112 , 1487 hartman , j. d. , kaluzny , j. , szentgyorgyi , a. , & stanek , k. z. 2005 , , 129 , 1596 hoffleit , d. 2002 , misfortunes as blessings in disguise : the story of my life , by dorrit hoffleit cambridge , ma : american association of variable star observers ( aavso ) , 2002 .
horne , j. d. 2005 , journal of the american association of variable star observers ( jaavso ) , 34 , 61 kopacki , g. 2001 , , 369 , 862 kovtyukh v. v. , soubiran c. , luck r. e. , turner d. g. , belik s. i. , andrievsky s. m. , chekhonadskikh f. a. , 2008 , mnras , 389 , 1336 kubiak m. , udalski a. , 2003 , acta astr . , 53 ,
117 laney c. d. , caldwell j. a. r. , 2007 , mnras , 377 , 147 lane d. j. , 2007 , 96th spring meeting of the aavso , http://www.aavso.org/aavso/meetings/spring07present/lane.ppt layden , a. c. , ritter , l. a. , welch , d. l. , & webb , t. m. a. 1999 , , 117 , 1313 layden , a. c. , & sarajedini , a. 2000 , , 119 , 1760 le borgne , j. f. , klotz , a. , & boer , m. 2004 , ibvs , 5568 , 1 le borgne , j. f. , et al .
2007 , , 476 , 307 luck r. e. , moffett t. j. , barnes t. g. , gieren w. p. , 1998
, aj , 115 , 605 madore b. f. , 1982 , apj , 253 , 575 madore , b. f. , & freedman , w. l. 2009 , , 696 , 1498 mackey , a. d. , & gilmore , g. f. 2003 , , 345 , 747 macri , l. m. , 2001 , ph.d .
thesis macri , l. m. , stanek , k. z. , sasselov , d. d. , krockenberger , m. , & kaluzny , j. 2001 , aj , 121 , 870 macri , l. m. , et al . 2001 ( b ) , apj , 549 , 721 macri , l. m. , stanek , k. z. , bersier , d. , greenhill , l. j. , & reid , m. j. 2006 , apj , 652 , 1133 mochejska , b. j. , 2002 , ph.d .
thesis majaess d. j. , turner d. g. , lane d. j. , 2008 , mnras , 390 , 1539 majaess d. j. , turner d. g. , lane d. j. , moncrieff k. e. , 2008 ( b ) , jaavso , 36 , 90 majaess , d. j. , turner , d. g. , & lane , d. j. 2009 , mnras , 398 , 263 majaess , d. j. , turner , d. g. , & lane , d. j. 2009 ( b ) , jaavso , 37 , 179 majaess , d. j. , turner , d. g. , & lane , d. j. 2009 ( c ) , arxiv:0909.0181 marconi , m. 2009 , arxiv:0909.0900 matsunaga , n. , et al .
2006 , mnras , 370 , 1979 matsunaga , n. , feast , m. w. , & menzies , j. w. 2009 , mnras , 397 , 933 mochejska , b. j. , macri , l. m. , sasselov , d. d. , & stanek , k. z. 2000 , aj , 120 , 810 mochejska , b. j. , macri , l. m. , sasselov , d. d. , & stanek , k. z. 2001 , arxiv : astro - ph/0103440 mochejska , b. j. , 2002 , , ph.d . thesis mottini m. , 2006 , ph.d .
thesis ngeow , c. , & kanbur , s. m. 2006 , , 650 , 180 opolski a. , 1983 , ibvs , 2425 , 1 opolski a. , 1988 , acta astr .
, 38 , 375 paczyski , b. 2006 , , 118 , 1621 percy j. r. , 1980 , jrasc , 74 , 334 percy , j. r. 1986 , study of variable stars using small telescopes pietrzyski , g. , gieren , w. , udalski , a. , bresolin , f. , kudritzki , r .-
p . , soszyski , i. , szymaski , m. , & kubiak , m. 2004 , aj , 128 , 2815 pietrzyski , g. , et al .
2006 , aj , 132 , 2556 pritzl b. j. , smith h. a. , stetson p. b. , catelan m. , sweigart a. v. , layden a. c. , rich r. m. , 2003 , aj , 126 , 1381 poretti , e. , et al .
2006 , memorie della societa astronomica italiana , 77 , 219 rabidoux , k. , et al . 2007 , bulletin of the american astronomical society , 38 , 845 randall , j. m. , rabidoux , k. , smith , h. a. , de lee , n. , pritzl , b. , & osborn , w. 2007 , bulletin of the american astronomical society , 38 , 276 rey , s .- c . , lee , y .- w . ,
m . , walker , a. , & baird , s. 2000 , , 119 , 1824 saha , a. , thim , f. , tammann , g. a. , reindl , b. , & sandage , a. 2006 , , 165 , 108 sakai , s. , ferrarese , l. , kennicutt , r. c. , jr . , & saha , a. 2004 , , 608 , 42 sarajedini , a. , barker , m. k. , geisler , d. , harding , p. , & schommer , r. 2006 , , 132 , 1361 sawyer , h. b. 1939 , publications of the david dunlap observatory , 1 , 125 schmidt , e. g. , johnston , d. , langan , s. , & lee , k. m. 2004 , aj , 128 , 1748 schmidt , e. g. , johnston , d. , langan , s. , & lee , k. m. 2005 ( c ) , aj , 130 , 832 schmidt , e. g. , johnston , d. , langan , s. , & lee , k. m. 2005 ( b ) , aj , 129 , 2007 schmidt , e. g. , langan , s. , rogalla , d. , & thacker - lynn , l. 2005 ( d ) , bulletin of the american astronomical society , 37 , 1334 schmidt , e. g. , hemen , b. , rogalla , d. , & thacker - lynn , l. 2009 , aj , 137 , 4598 scowcroft , v. , bersier , d. , mould , j. r. , & wood , p. r. 2009 , mnras , 396 , 1287 sebo , k. m. , et al .
2002 , , 142 , 71 shapley , h. 1918 , , 48 , 279 smith , h. a. 2004 , rr lyrae stars , by horace a. smith , pp . 166 .
isbn 0521548179 .
cambridge , uk : cambridge university press , september 2004 soszynski , i. , et al . 2002 ,
acta astronomica , 52 , 369 soszynski , i. , et al .
2003 , acta astronomica , 53 , 93 soszynski , i. , et al .
2008 , acta astronomica , 58 , 293 soszynski , i. , et al .
2009 , acta astronomica , 59 , 1 stanek , k. z. , zaritsky , d. , & harris , j. 1998 , apjl , 500 , l141 stanek , k. z. , & udalski , a. 1999 , arxiv : astro - ph/9909346 szabados , l. 2003 , astrophysics and space science library , 289 , 207 szabados , l. 2006 , commmunications of the konkoly observatory hungary , 104 , 105 szabados , l. 2006 ( b ) , odessa astronomical publications , 18 , 111 tammann , g. a. 1970 , the spiral structure of our galaxy , 38 , 236 tammann , g. a. , reindl , b. , thim , f. , saha , a. , & sandage , a. 2002 , a new era in cosmology , 283 , 258 thim , f. , tammann , g. a. , saha , a. , dolphin , a. , sandage , a. , tolstoy , e. , & labhardt , l. 2003 , apj , 590 , 256 turner , d. g. 1990 , , 102 , 1331 turner d. g. , 1996 , jrasc , 90 , 82 turner d. g. , 2001 , odessa astr .
, 14 , 166 turner d. g. , burke j. f. , 2002 , aj , 124 , 2931 turner , d. g. , savoy , j. , derrah , j. , abdel - sabour abdel - latif , m. , & berdnikov , l. n. 2005 , , 117 , 207 turner , d. g. 2010 , , 5 turner , d. g. , majaess , d. j. , lane , d. j. , szabados , l. , kovtyukh , v. v. , usenko , i. a. , & berdnikov , l. n. 2009 , american institute of physics conference series , 1170 , 108 udalski , a. , szymanski , m. , kubiak , m. , pietrzynski , g. , wozniak , p. , & zebrun , k. 1998 , acta astronomica , 48 , 1 udalski , a. 1998 ( b ) , acta astronomica , 48 , 113 udalski a. et al . , 1999 , acta astr .
, 49 , 223 udalski , a. 2000 , acta astronomica , 50 , 279 udalski , a. , wyrzykowski , l. , pietrzynski , g. , szewczyk , o. , szymanski , m. , kubiak , m. , soszynski , i. , & zebrun , k. 2001 , acta astronomica , 51 , 221 udalski , a. 2003 , apj , 590 , 284 van den bergh s. , 1968 , jrasc , 62 , 145 van leeuwen , f. , feast , m. w. , whitelock , p. a. , & laney , c. d. 2007 , mnras , 379 , 723 wallerstein , g. , & cox , a. n. 1984 , , 96 , 677 wallerstein , g. 2002 , pasp , 114 , 689 weldrake , d. t. f. , sackett , p. d. , & bridges , t. j. 2007 , , 133 , 1447
|
preliminary evidence is presented reaffirming that sx phe , rr lyrae , and type ii cepheid variables may be characterized by a common wesenheit period - magnitude relation , to first order .
reliable distance estimates to rr lyrae variables and type ii cepheids are ascertained from a single @xmath0-based reddening - free relation derived recently from ogle photometry of lmc type ii cepheids .
distances are computed to rr lyrae ( @xmath1 pc ) , and variables of its class in the galaxies ic 1613 , m33 , fornax dsph , lmc , smc , and the globular clusters m3 , m15 , m54 , @xmath2 cen , ngc 6441 , and m92 .
the results are consistent with literature estimates , and in the particular cases of the smc , m33 , and ic 1613 , the distances agree with that inferred from classical cepheids to within the uncertainties : no corrections were applied to account for differences in metallicity .
moreover , no significant correlation was observed between the distances computed to rr lyrae variables in @xmath2 cen and their metallicity , despite a considerable spread in abundance across the sample . in sum ,
concerns regarding a sizeable metallicity effect are allayed when employing @xmath0-based reddening - free cepheid and rr lyrae relations .
|
type ia supernovae ( sne ia ) are thought to be thermonuclear disruptions of white dwarf ( wd ) stars @xcite , but the details remain uncertain .
one possibility for the progenitors is the single degenerate model in which main sequence stars or post main sequence red giants transfer mass to a wd through roche lobe overflow or a common envelope and the wd grows close to the chandrasekhar mass @xmath6 . due to compression , the thermonuclear runaway starts near the center leading to the explosion of the wd and a rapidly expanding envelope with a mass close to @xmath6 .
a second possibility is the double degenerate model in which a pair of wds merge and lead to an explosion . in most such cases ,
the resulting mass of the rapidly expanding envelope will be different from @xmath6 .
residual material from these mergers surrounding the explosions will get swept up by the ejecta forming dense , shell - like structures .
although the explosion of single wds seems to be favored for the majority of objects , we may expect mergers to contribute to the sne ia population ( see [ obs ] ) .
the possibility of different progenitor channels , the population of which may vary with redshift , may pose a challenge for the use of sne ia in cosmological studies that rely on a single parameterization , such as a light curve width to peak luminosity relation , lwr , to reduce the intrinsic scatter in the peak magnitudes and render them standard candles @xcite .
to first order , the lwr relation can be understood as a result of different amounts of @xmath7ni produced during the explosion @xcite .
there may be some spread and an offset in lwr introduced by one of the channels if the masses of the envelope differ from @xmath6 , and/or the density structures differ .
this can lead to a systematic shift of lwr with redshift if the evolutionary time scales of the progenitor systems differ . even if the different progenitor scenarios obey the same lwr
, differences in the color could introduce systematic errors in cosmological studies because sne ia are known to suffer to some degree from reddening in their respective host galaxies which has to be taken into account . to correct for this , the maximum light color excess ( usually @xmath8 ) and
an average reddening law are used to determine the amount of absorption .
sne ia that are intrinsically redder as compared to the average local sample will thus be over - corrected in this fashion to a higher luminosity .
similarly to the two distinct progenitor channels , qualitative variations in the explosion physics may lead to various classes of sne ia even within the single degenerate scenarios .
standard explosion models include delayed detonations ( dd ) and deflagrations . in these scenarios , burning during the deflagration phase
leads to an unbound wd . in dd models ,
the deflagration turns into a detonation in an expanding envelope . because the density structure of the wd declines monotonically with radius , the resulting density structure in the expanding envelope also smoothly declines with mass and radius .
a variation of dd models are the pulsating delayed detonation models ( pdd ; @xcite ) . in these models ,
the total energy production during the deflagration phase is , by construction , lower and insufficient to unbind the wd .
this results in large amplitude pulsations . because the fall - back time increases with distance , the inner regions contract and
leave a shell behind at larger distances . due to infall driven compression , a detonation
is triggered , the material behind the burning front is accelerated , and this expanding material runs into the low - velocity material left at larger distances .
similar to the merger scenario , a shell - like structure is formed with very similar light curve and spectroscopic properties , but with a total mass close to @xmath6 @xcite .
these two groups , consisting of dd and deflagration models such as w7 @xcite , which lack shells , and the models with shells ( mergers and pdds ) , can be differentiated by their predictions for the photospheric evolution and maximum light colors @xcite . for the former group ,
the photospheric velocities , @xmath9 , smoothly decline with time and the models show a blue color at maximum light , @xmath10 ; in the latter group , @xmath9 shows a plateau in the evolution as the photosphere recedes through the shell .
these models are intrinsically redder and slightly over - luminous because of the lower expansion rate in the inner region . as shown in @xcite the color , length and velocity of the plateau
are correlated with the mass of the shell , and this potentially allows the two groups to be distinguished even for similar brightnesses .
indeed , there is a growing sample of sne ia showing photospheric velocity plateaus ( e.g. 1990n ; @xcite ; 1991 t , 1999aa ; @xcite ; 1999ee ; @xcite ; 2000cx ; @xcite ; see also @xcite ) .
many of these sne ia have been reported as having a red color @xmath11 at maximum , but this is typically attributed to reddening along the line of sight .
alternatively , this sample may suggest the contribution of events with shell - like density structures in the observed population .
these events may be understood in terms of mergers or pdds ; however , the inhomogeneities and incompleteness of individual data sets in the literature preclude definite conclusions . to address this problem and others , we started the texas supernovae search ( tss ; quimby et al . in prep . ) with the goal of providing a homogeneous set of quality data for several supernovae beginning well before maximum light .
in this paper , we present our observations of sn 2005hj and analysis of the data . in [ obs ]
we describe the discovery and give the details for both the photometric and spectroscopic follow - up . in [ models ] we discuss generic properties of explosion models and suggest a secondary parameter to separate models with and without shells , and analyze the peculiarities of sn 2005hj .
conclusions and discussion are presented in [ conclusions ] .
sn 2005hj was discovered on october 26.13 ut in the field of abell 194 as part of the tss .
the tss uses the wide field ( @xmath12 ) 0.45 m rotse - iiib telescope @xcite at the mcdonald observatory in texas to scan nearby galaxy clusters nightly for transients with a modified version of the psf - matched image subtraction code from the supernova cosmology project .
sn 2005hj was found at an unfiltered magnitude ( calibrated against the usno - b1.0 r2 ) of @xmath13 and is located at @xmath14 , @xmath15 .
the foreground reddening at this location is @xmath16 @xcite .
examination of rotse - iiib images from oct .
20 and oct .
22 shows the sn was detected prior to discovery , but not significantly well to pass the search pipeline s automatic cuts .
figure [ lc ] shows the rotse - iiib light curve for sn 2005hj through 40 days after maximum light . to construct the light curve
, we co - added images taken on a given night ( usually 6 ) excluding any frames of significantly lower quality due to passing clouds or wind sheer , and then subtracted the reference image convolved to the same psf .
magnitudes were determined by fitting the local psf ( derived from the co - added nightly images ) to the location of the sn on the subtracted frame using custom software and the daophot psf - fitting routines ( @xcite ported to idl by @xcite ) . the unfiltered ccd response of rotse - iiib has an approximate full width of @xmath17 centered in the @xmath18-band around 6000 .
because we do have some sensitivity in the blue and since the @xmath11 colors of sne ia typically grow @xmath19 mag redder in the 30 days after maximum @xcite , there is a blue deficit at later times that causes our unfiltered magnitudes to decline more rapidly than the true @xmath18-band fading .
note that @xmath20 colors of sne ia are close to zero at maximum light .
we therefore limit the light curve fitting to data taken before 10 days after maximum ( determined through several iterations of the fit ) , during which the color evolution is minimal . the best fit @xmath18-band template from @xcite is also shown in figure [ lc ] .
the date of maximum light determined from the fit is nov .
1.6 with a formal error of 0.7 days ( note the template phases are relative to the @xmath21-band maximum ) .
the best fit stretch factor @xcite for the light curve width is @xmath22 .
the preliminary measurement of the observed @xmath11 color at @xmath23 maximum from the carnegie supernova project is @xmath24 after removal of the host light but before any extinction or @xmath25-corrections are applied ( m. m. phillips , private communication ) . near real - time photometric analysis combined with target of opportunity ( too ) time on the neighboring 9.2 m hobby - eberly telescope ( het ) allowed us to obtain optical spectra just 4 hours after the discovery images were taken and every few days over the next 6 weeks .
these observations are detailed in table [ spec ] .
the instrumental response is such that very little second order light is expected blue of 8900 even with the gg385 blocking filter .
the data were reduced in the optimal manner using iraf and custom idl scripts .
the wavelength scale was calibrated against cd and ne lamps and its accuracy was verified by comparing night sky lines to the spectral atlas of @xcite . because the het pupil size varies for different tracks , absolute flux calibration can not reliably be achieved ; however , we used the standard stars of @xcite and @xcite , which were observed using the same setups , to achieve _ relative _ spectrophotometric calibration and to remove strong telluric features .
the redshift of the host galaxy was derived from narrow emission lines around 7000 ( observed ) , which we attribute to h-@xmath26 , [ ] , and [ ] in the host galaxy .
we combined all the spectra and simultaneously fit these lines with gaussians to determine the line centers .
the line redshifts are best fit by @xmath27 , and we adopt this value for the sn .
this gives sn 2005hj an absolute peak magnitude of @xmath1 in our unfiltered band pass ( assuming h@xmath28=71 km s@xmath3 mpc@xmath3 , @xmath29 , and @xmath30 ) , and places the host well behind abell 194 ( @xmath31 ; @xcite ) .
the brightness and broad light curve shape suggest that sn 2005hj is a slightly over - luminous sn ia .
the unfiltered rotse - iiib reference image shows that the host for sn 2005hj is relatively bright ( @xmath32 ) and compact , and is therefore likely a significant contaminant to our spectra .
thus , we have to subtract the galaxy contribution ( see fig . [ spec0 ] ) . lacking an observed spectrum for the host galaxy excluding the sn light , we constrained the galaxy sed using archival sloan digital sky survey ( sdss ) @xmath33 observations and obtained a template galaxy spectrum ( n. drory 2005 , private communication ) .
the relative amounts of sn and galaxy light in the spectral apertures will vary not only with the changing sn brightness , but also with the seeing , slit width and positioning .
also plotted in figure [ spec0 ] is a spectrum of sn 1999aa ( blue curve ) constructed via a linear interpolation of the @xmath34 day and @xmath35 day spectra presented by @xcite . noting the similarity of the spectral features of sn 1999aa and sn 2005hj
, we assume that we can model our observed spectra as a linear combination of our galaxy template and the sn 1999aa spectra interpolated to the same phase as the sn 2005hj observations .
we perform a least squares fit to determine the relative contributions of each component .
the red line in figure [ spec0 ] shows the derived contribution of galaxy light in the @xmath36 day spectrum . aside from a few small differences ( most noticeably in the @xmath374481 triplet ) ,
some of which may be explained by calibration errors , the combined sn 1999aa + host spectrum ( purple curve ) is a good fit .
the over all fit is improved if we interpolate the sn 1999aa spectra to @xmath38 days instead of @xmath36 , especially in the 5400 to 6500 range , which could imply a @xmath391 day error in the date of maximum light or different time scales for the spectral evolutions of the two sne .
we repeated this process for all the sn 2005hj spectra , each time using the same galaxy template and the sn1999aa spectra ( interpolated to the appropriate phase ) as reference to determine the relative amount of galaxy light .
in general , the galaxy template added to the sn 1999aa spectra does an excellent job of reproducing the observed sn 2005hj spectra .
the galaxy light typically dominates the flux red of 7000 .
figure [ spec ] shows the spectral evolution of sn 2005hj recorded by the het between days @xmath36 and @xmath40 with the derived galaxy contribution subtracted .
overall , sn 2005hj shows spectra with lines dominated by intermediate mass and iron group elements as is typical for sne ia .
while the lines show normal expansion velocities , the absorption components are more narrow and , for the early phases , weaker than typically observed , as exemplified by the @xmath376355 line ( see fig .
[ spec1 ] ) .
sn 2005hj also shows an atypical velocity evolution of these features over time .
line minima are useful diagnostic indicators of the ejecta structure as they give the abundances and velocities of the material .
the actual measurement of the velocity at the minimum of the line profile is complicated by the presence of the continuum , other blended lines , and some uncertainty in the true line profile shape .
detailed modeling is required to accurately sort out all the components and how they relate to the photospheric layer to reveal the velocity distribution of the ejecta larger than those measured from weak lines . however , for shell models the steep density gradients cause even strong lines to from very close ( in radius ) to the actual photosphere . ] .
such models have shown that the absorption minima approximate the photospheric expansion velocities to within about 1000 km s@xmath3 at maximum light @xcite .
thus , simple line fitting can lead to a rough description of the ejecta velocities , and allows a useful comparison to discriminate between different models . at late times
the photosphere will recede below the si rich layer and so the velocities derived from the @xmath376355 line will become increasingly discrepant with the photospheric velocity . for deflagration and classical detonation models , this departure will begin to set in 1 - 2 weeks after maximum light @xcite .
the strength of the @xmath376355 line and its persistence from at least 2 weeks before to 4 weeks after maximum light make it a valuable tool for probing the ejecta .
its evolution with time is shown in figure [ siii ] for the case of sn 2005hj . to determine the velocity , we smooth the spectra by fourier transform filtering , divide by the estimated continuum , and then select the lowest point using spline interpolation over a selected range . the continuum is represented by a medium order ( 6th - 7th ) polynomial fit to regions of the spectra that are not strongly affected by lines . to smooth the spectra
, we use a fourier transform to convert the data into a power spectrum , and then multiply this by a filter to remove high frequency variations .
we then apply a reverse ft to the filtered power spectrum to recover the smoothed spectrum .
the filter has the functional form @xmath41 & \rm{otherwise } \end{array } \right.\ ] ] the filter cutoff frequency , @xmath42 , and attenuation scale , @xmath43 , were determined as follows : 1 ) the spectra were converted into a power spectrum , @xmath44 , via fourier transform ; 2 ) the slope of @xmath45 is fit over the noise dominated high frequencies and interpolated through the low frequencies to determine the noise spectrum ; 3 ) @xmath42 is taken as the frequency at which @xmath45 drops to within three times the dispersion about the noise spectrum ; 4 ) @xmath43 is chosen such that the slope of @xmath46 $ ] is twice the noise spectrum slope ( i.e .. @xmath47 is the frequency above which noise is clearly the dominate component ) . for this analysis ,
only the spectral bins with signal to noise above 25 were considered ( note the peak throughput for het / lrs is near the @xmath376355 line ) . for consistency , we adopt a single filter for all our analysis , choosing the results from our nosiest data , @xmath48 @xmath3 and @xmath49 @xmath3 , which removes noise in the data but also some real information related to `` sharp '' features in the spectra such as the narrow core to the @xmath376355 absorption in the day @xmath50 spectrum . using the relativistic doppler formula and the @xmath51-weighted @xmath376355 rest velocity in the host galaxy frame , we convert the wavelengths of the line profile minima into expansion velocities . for each spectrum
we conducted 250,000 monte carlo simulations in which normally distributed noise based on the statistical flux errors was added to the data and the ft smoothed minimum was found .
the peak of the distribution and the interval containing 68% of the simulation results were used to calculate the velocity of the minimum and its error , respectively .
we also measured the relative shift in the region lines over all epochs and found the scatter to be 80 km s@xmath3 , which we add in quadrature to the individual errors .
the results are given in table [ linedata ] and plotted in figure [ linevel ] .
we find that the data points are at 10,600 @xmath52 150 km s@xmath3 between maximum light and @xmath53 days , somewhat faster prior to maximum , and significantly slower on day @xmath54 . by day @xmath40
, the @xmath376355 absorption has all but completely disappeared . from maximum light through day @xmath50
, the @xmath376355 line profile shows little change in both depth and width in addition to maintaining a constant absorption minimum velocity .
of specific relevance is the blue wing of the absorption profile ; this section of the line is formed by the material at the greatest distance from the photosphere and at the highest velocities , and as such it should be the first to vanish as the photosphere recedes . the consistency of this blue wing from maximum light through day @xmath50 suggests the photosphere falls within the si enriched layers for at least this period . by day @xmath53
the blue wing has shifted significantly to the red , while the red wing remains constant except for the effects of an blend around 6250 .
other features begin to appear or strengthen at this phase as well .
this behavior could be a signal that the si layers are becoming detached from the photosphere by day @xmath53 .
the day @xmath54 spectra show a double minimum at the location of the @xmath376355 feature ( see figure [ day25 ] ) .
telluric absorption is weak in this wavelength range , and the line profile is clearly seen in each of the three individual exposures , which support the reality of this feature .
a possible explanation for this feature is contamination from the host that is not removed by the template subtraction ; however , galaxy spectra do not typically exhibit features in this range that could cause such interference , and even if such were the case , we would expect to see similar behavior in the @xmath40 day spectra .
a second possibility is contamination from lines . using the spectral analysis tool synow @xcite , and the example of sn 1994d as a starting place @xcite
, we find that while likely produces the absorption dips @xmath55 away on either side of the @xmath376355 line , it is unlikely responsible for the double minimum .
the third possibility , which we favor , is that this double minimum simply appears because we are resolving the @xmath376355 doublet .
this result implies that the seen in the @xmath54 day spectra is confined to a very narrow region of velocity space ( @xmath56 km s@xmath3 ) .
if accurate , the true minimum of the @xmath376355 doublet would be about 100 - 200 km s@xmath3 faster than indicated in figure [ linevel ] and table [ linedata ] , but still significantly below the plateau velocity
. the emergence of this thin layer may also be responsible for the appearance of the narrow core in the @xmath50 day spectrum as well as the apparent double minimum to the @xmath53 day data .
some remnant of the blue component to the doublet may persist to the @xmath40 day spectrum .
figure [ day25 ] also shows the spectra of several other sne ia taken around 25 days after maximum light . while the distinctly double minimum appears unique to sn 2005hj
, the width and depth of the feature is roughly consistent with the others .
sn 2005hj clearly belongs to the low velocity gradient ( lvg ) group in the classification scheme of @xcite , but moreover the velocity derivative from maximum light through day @xmath53 , @xmath57 km s@xmath3 day@xmath3 , is consistent with no change is the average daily rate of decrease in the expansion velocity from maximum light through the last available spectrum before the @xmath376355 line disappears ; therefore , including the day @xmath54 spectrum , sn 2005hj formally has @xmath58 km s@xmath3 day@xmath3 , but with a @xmath59 per degree of freedom of 3.2 ] . from the line profile evolution ( table [ linedata ] , figures [ siii ] and [ linevel ] ) we can deduce a plateau phase starting at @xmath60 days which lasts no more than 30 days . noting the change in the @xmath376355 line profile in the @xmath53 day spectrum , we conservatively mark the end of the plateau phase as day @xmath61 days , which gives the plateau phase a total duration of @xmath62 days .
the @xmath376355 velocity evolution derived from the minima of ft smoothed spectra of several selected sne ia is plotted in figure [ linevels ] .
the velocity plateau of sn 2005hj is similar to that of other over - luminous sne ia such as sn 1999aa @xcite and sn 2000cx @xcite , but it is distinct from normal sne ia such as sn 1994d @xcite and sn 1992a @xcite that do not show a plateau phase from the sn 1999aa curve because all spectral features in these data seem to be frequency shifted including the telluric features . ] .
there is general agreement that sne ia result from some process involving the combustion of a degenerate c / o white dwarf @xcite . within this general picture ,
two classes of models are most likely .
the first is an explosion of a c / o - wd with a mass close to the chandrasekhar limit ( @xmath6 ) that accretes matter through roche - lobe overflow from an evolved companion star @xcite . in this case , the explosion is triggered by compressional heating near the wd center .
alternatively , the sn could be an explosion of a rotating configuration formed from the merging of two low - mass wds , after the loss of angular momentum @xcite .
candidate progenitor systems have been observed for both scenarios : wd binary systems with the correct period to merge in an appropriate time scale with an appropriate total mass @xcite ; and supersoft x - ray sources @xcite showing accretion onto the wd from an evolved companion .
there are still open questions about the details of both the merging and accretion processes ( e.g. * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
. from the observed spectral and light curve properties , the first scenario appears to be the most likely candidate for the majority of normal sne ia . in particular , delayed detonation ( dd ) models @xcite have been found to reproduce the majority of the observed optical / infrared light curves ( lc ) and spectra of sne ia reasonably well @xcite . in the dd scenario
, a slow deflagration front turns into a detonation .
the wd pre - expands during the deflagration and undergoes complete burning during the detonation phase .
similarly , the classical deflagration models w7 @xcite show similar behavior to dds but only by neglecting instabilities due to the deflagration fronts @xcite . for recent reviews
see @xcite .
despite the success of classical dd and w7 models , both lack the basic features seen in sn 2005hj .
neither predicts a long plateau in velocity ; they instead show a smooth decline of the photospheric velocity6355 line is an imperfect tracer of the photospheric velocity as mentioned in
[ spec_char ] , the observed sne ia population typically exhibits a 1000 - 3000 km s@xmath3 decrease in the measured line velocities between 1 week before maximum light to 2 weeks after@xcite , and the deflagration and classical delayed detonation models employed to explain these events have shown a correspondingly large decrease in photospheric velocities over the same period @xcite .
these models are inconsistent with the corresponding @xmath63 km s@xmath3 shift measured for sn 2005hj . ] as a function of time ( fig .
[ modelvel ] ) .
this happens because in expanding envelopes the photosphere recedes in mass and , because of the homologous expansion , in velocity as well .
this behavior results from the smoothly declining density structure of the wd and the fact that variations in the specific energy production are small .
in contrast , shell - like density structures will produce velocity plateaus in a natural way because the photosphere remains in the shell for some time as shown by @xcite . to form a shell - like structure
requires interaction of rapidly expanding material with a surrounding envelope .
various mechanisms have been suggested to supply this surrounding matter : the pulsating delayed detonation scenario @xcite , mergers or tamped detonation models .
shells may also form by the interaction of an exploding single wd within the progenitor system @xcite .
we analyzed the observations of sn 2005hj based on detailed , spherical models for supernovae published in the literature .
the models are based on detailed calculations for the explosion , light curve and spectra .
the models considered include delayed detonations , deflagrations , pulsating delayed detonations and tamped detonation / merger scenarios . in figure
[ modelvel ] , we show the photospheric velocities as a function of time for these models along with the branch - normal sne ia to illustrate the formation of a plateau in the models that naturally form a shell . note for lower shell masses , this `` plateau '' is more accurately described as a period of slowly declining velocities . in classical delayed detonation models and for normal - bright sne
ia , si is present over a wide range in mass , spanning about @xmath64 to @xmath65 , which corresponds to velocities from about 8,000 to 9,000 km s@xmath3 to more than 20,000 km s@xmath3 .
the si layer is thick ( in the mass frame ) because explosive oxygen burning occurs over a wide range of temperatures .
the density gradient is smooth and si is mostly in , so initially the velocity derived from the minimum of the @xmath376355 line smoothly declines with the receding photosphere governed by the geometrical dilution of the expanding envelope .
eventually , the photosphere begins to recede below the si layer at which point the evolution of the @xmath376355 line profile will show the following behavior : 1 ) the optical depth of the highest velocity material at the largest radii will begin to decline below 1 and as such the blue wing of the line profile will start to drift toward the red ; 2 ) as the optical depth decreases , the strength of the line as measured from the line depth will decrease ; 3 ) the line minimum may continue to slow , but it will grow increasingly discrepant with the photospheric velocity .
this phase typically begins 1 - 2 weeks after maximum light for normal sne ia and is heralded by the appearance of lines .
while this behavior is commensurate with observations of normal - bright sne ia such as sn 1994d , this behavior is not consistent with the observations of sn 2005hj .
the @xmath376355 line seen in sn 2005hj is narrow , and during the plateau phase the wings do not change , the depth does not change , and the velocity of the minimum does not change to within the errors .
the data require a narrow distribution of in velocity space , and we suggest this may be explained by an interaction that compresses the si rich layers as predicted by merger and pulsating delayed detonation models .
the shell models are also consistent with the velocity drop seen after the plateau because a significant amount of si is located below the shell @xcite . in fig .
[ modelprop ] , we show general properties of these models .
as discussed in the papers above , to first order , the observational signatures of the shell depend on the mass of the shell @xmath4 .
almost the entire wd is burned , and momentum conservation determines the amount of high velocity matter that can pile up in the expanding shell . with increasing shell mass , more material of the sn envelope
is slowed down . as a consequence ,
the velocity of the shell @xmath66 decreases with @xmath4 . because it will take longer for the photosphere to recede into the lower velocity matter , the time until the beginning of the plateau phase , @xmath67 , increases with @xmath4 .
the optical depth increases with @xmath4 , duration of the plateau , @xmath68 , also increases , the temperature gradient becomes steeper , and the photosphere becomes cooler ( i.e. @xmath11 increases ) with increasing @xmath4 @xcite .
the duration of the plateau , @xmath68 , is defined by the velocity spread @xmath69 around @xmath70 with @xmath71 km s@xmath3 , which puts the end of the plateau phase safely into the parts of a rapidly declining @xmath9 .
we choose a larger value than in the observations to avoid ambivalences due to discreetness , which , in some of the models , is of the order of @xmath72 km s@xmath3 . by increasing @xmath73 from 200 to 500 km
s@xmath3 the nominal duration is increased by @xmath74 day
. however , we also note that the actual width depends on the velocity spread in the shell ( see [ conclusions ] )
. given the model predictions , we can use different observational indicators to test which @xmath4 is consistent with sn 2005hj ( fig .
[ modelprop ] ) .
all three parameters , @xmath66 , @xmath75 , and @xmath67 suggest @xmath76 , with the allowed ranges specifically bracketed by 0.15 - 0.6 , 0.1 - 0.25 , and 0.1 - 0.25 @xmath77 for the plateau length , shell velocity , and plateau onset , respectively , taking the observed errors into account .
the comparison between the @xmath11 color as a function of @xmath66 , @xmath75 , or @xmath67 , however , shows only marginal consistency between the observations and the models if we assume only foreground redding by the galaxy .
we note , that the intrinsic @xmath11 color of the models is uncertain by about 0.05 to @xmath78 at maximum light .
the two best - fitting models , pdd3 and det2env2 , show a peak brightness , @xmath79 , of @xmath80 and @xmath81 , respectively , with an uncertainty of @xmath82 @xcite vs. a typical dd model with @xmath83 @xcite ,
i.e. they are brighter by about 20 % mostly due lower escape probablity of @xmath84-rays that results when the @xmath7ni layers are slowed down because of the interaction @xcite .
we have presented photometric and spectroscopic data for sn 2005hj , a slightly over - luminous type ia .
the most striking feature is an apparent plateau in the expansion velocity evolution , which we derive from the @xmath376355 line .
the velocities remain at about @xmath85 km s@xmath3 for about 3 weeks starting slightly before maximum light , and this plateau is bracketed by preceding and succeeding decelerations .
we find that si is confined to a relatively narrow velocity region .
analysis of the detailed observations in concert with published models suggest there may be some physical distinction between sn 2005hj and other normal - bright sne ia that may systematically affect their use as distance indicators if not properly taken into account .
the models considered include delayed detonations , deflagrations , pulsating delayed detonations and tamped detonation / merger scenarios . in order to explain the narrow @xmath376355 line and its plateau in velocity ,
we suggest an early interaction that forms a dense shell as predicted by merger and pdd models . the spectral and photometric peculiarities are consistent with respect to the velocity , duration , and onset of the plateau , and marginally consistent with the maximum light color , for models that have shells of about @xmath86 .
as indicated by earlier works @xcite , the mass of the interacting shell has been found to be the parameter that dominates the details of these observational signatures independent of how this shell may form .
the tight predicted relation between each of @xmath66 , @xmath75 , and @xmath67 may provide a stable means to separate sn 2005hj like events from regular branch - normal sne ia .
although the agreement between the shell models and the observations is good , the predictions are not necessarily unique and other possibilities may exist . for example , we have not considered 3-d models such as the detonation from a failed deflagration scenario recently examined by @xcite . for sn 2005hj
then the agreement of the plateau velocity and its duration to that predicted by shell models may simply be a fluke , and in such case this concordance should then not hold for other sne with similar @xmath376355 evolution .
given the data and models considered , we suggest either pdds or merger events are responsible for sn 2005hj , and this implies the existence of two different progenitor channels . it is important to understand how these two progenitor channels , which may occur in relatively varying fractions as a function of redshift , will impact studies using sne ia as distance indicators .
@xcite estimate that 20% of sne ia in their sample are either 1991t - like or 1999aa - like .
these sne show spectral features and a velocity plateau similar to sn 2005hj .
@xcite found 5 1999aa - like events in the @xcite sample out of 20 total sne ia that were observed early enough to show 1999aa - like spectral features , and one that was 1991t - like ; however , in the pre - loss sample they do not classify any of the 7 sne ia with early spectra as 1999aa - like .
these nearby samples are constructed from targeted galaxy searches that have different selection biases than the deep cosmological surveys , but we will assume a uniform 2005hj - like rate of 25% for all sne ia .
sne ia that appear spectroscopically similar to sn 2005hj in a single epoch could none the less arise from different progenitors , and the mass of the low - density envelope around pdds or mergers may effect their peak magnitudes and/or light curve shapes , but we will further assume that all such events deviate uniformly from the lwr of branch normal sne ia .
@xcite calculated the relation between peak @xmath23 band magnitudes , @xmath87 , and the fading between maximum light and + 20 days , @xmath88 , for a variety of theoretical models and found that shell models produced @xmath88 that were 0.2 to 0.3 mag smaller than for ( in vacuum ) delayed detonations reaching the same peak magnitude . therefore using the same lwr for shell models will result
in corrected peak magnitudes systematically offset by 0.1 to 0.2 mag .
also , the observed peak magnitudes of sne ia are usually corrected for absorption along the line of sight using the observed @xmath11 color at maximum light and a reddening law . for events that are intrinsically red
, this will increase the estimated peak magnitude above its already over - luminous intrinsic value .
cosmological studies may therefore need to remove or at least separately handle sn 2005hj - like events to avoid systematic errors in distance . as a case for
the importance of separating different progenitors , let us consider sn 1999ee . very similar to sn 2005hj
, sn 1999ee shows a plateau with @xmath89 km sec@xmath3 , a duration of @xmath90 days , and an onset at day @xmath91 relative to maximum ( @xcite ; see fig .
[ linevels ] ) .
the @xmath11 color of sn 1999ee was also quite red at maximum light ; @xmath92 after correction for galactic extinction @xcite . based on the standard brightness decline relation and the corresponding colors ,
@xcite derived reddening in the host galaxy of @xmath93 which implies an absolute brightness of @xmath94 similar to snls-03d3bb which @xcite attributed to a super - chandrasekhar mass wd . taking into account the spectroscopic information about the velocity plateau , its length and onset
, we attribute a portion of the red color to the intrinsic properties of the supernova .
we find that the duration of the velocity plateau , its onset and size are consistent with a shell mass of @xmath95 which suggests an intrinsic color @xmath11 of @xmath96 ( see fig .
[ modelprop ] ) .
this reduces the reddening in the host galaxy to @xmath97 and the absolute brightness @xmath87 to @xmath98 , which compares favorably to the model predictions of @xmath99 and @xmath81 for pdd3 and detenv2 , respectively , within the model uncertainties .
note that there is an interstellar sodium line in the spectra that implies some degree of reddening within the host .
there are some apparent spectral differences when compared to sn 2005hj , namely sn 1999ee has a slightly broader blue wing in and stronger absorption around 4900 .
this may either hint toward different explosions scenarios ( i.e. pulsations versus mergers ) , or different viewing angles of asymmetric envelopes .
this brings us to the limitation of our studies . except for the color ,
sn 2005hj fits remarkably well with the merger and pdd model predictions but , still , it is a single event and the good agreement may be coincidental .
we need a large , well - observed sample of similar objects to test and verify or falsify the models and to determine the shell mass distribution .
moreover , 3d effects have been neglected . in reality
, we must expect some dispersion . though pulsating delayed detonation models may be expected to be rather spherical
, mergers may be asymmetric with directionally dependent luminosities and colors .
in fact , both classes may be realized in nature .
as mentioned above , the duration of the plateau , @xmath68 , is defined by the velocity spread around @xmath70 .
the physical width of the shell depends , to first order , on the distance at which the interaction occurs and the density distribution of the interacting expanding media and shell during the hydrodynamical phase of the interaction @xcite .
for obvious reasons , asymmetries of the shell will increase the velocity gradient seen over the shell .
the observations of sn 2005hj indicate a very flat plateau that , in principle , may further constrain the properties of the shell . for sn 2005hj , this may already indicate a rather spherical shell and hint toward the pdd scenario or mergers with an intermediate disk of very large scale heights .
however , additional information needs to be taken into account such as detailed line profiles and statistical properties to break this degeneracy between mergers and pdds . as a next step ,
detailed models for the explosion , light curves and spectra tailored toward individual events need to be constructed .
whereas the mean velocity of the shell for a given mass is dictated by momentum conservation , the thickness of the shell is limited by the distance of the shell material , the distance sound can travel during the interaction , and the specific density profile within the shell . with increasing distance of the shell , the relative size ( and corresponding velocity spread ) becomes smaller because the sound speed remains about the same .
the intrinsic color will be sensitive to the optical depth of the shell , which is governed by the magnitude of the density jump and thus depends on the distance of the interacting shell from the wd @xcite . the blue @xmath11 color for sn 2005hj may hint of a need to modify the distance and structure of the shell .
precise analysis of such `` non - stable '' features requires detailed model fitting beyond the scope of this paper . in the recent past ,
both the scenarios leading to shell - like structures have been discounted .
pdd models have been dismissed because 3d deflagration models showed that the wd becomes unbound and thus pulsations would not occur @xcite .
however , it has recently been shown that this solution depends mainly on the ignition conditions , namely the number and locations of ignition points leading to single or multiple bubble solutions , and mixture of bubble solutions leading to raleigh - taylor instabilities . as a result ,
solutions with fewer bubbles are likely to result in a reduced amount of burning , thus only slightly unbinding the wd and increasing the possibility of pdds @xcite .
similarly , the merging scenario has been dismissed because the wd may undergo burning during the merger and result in an accretion induced collapse @xcite , and also on the basis of the long merging time scale .
however both of these results depend sensitively on the initial conditions , and new pathways to the actual merging may effect the results @xcite . in light of our results ,
the predicted death of both of these scenarios may be premature , and further studies are needed .
we would like to thank the staff of the hobby - eberly telescope and mcdonald observatory for their support and the rotse collaboration .
we give specific thanks to j. caldwell , s. odewahn , v. riley , b. roman , s. rostopchin , m. shetrone , e. terrazas , and m. villarreal for their skilled observations with the het , and to f. castro , p. mondol , and m. sellers for their efforts in screening potential sn candidates .
this work made use of the suspect on - line database of sne spectra ( http://bruford.nhn.ou.edu/$\sim$suspect/index1.html ) .
this research is supported , in part , by nasa grant nag 5 - 7937 ( ph ) and nsf grants ast0307312 ( ph ) and ast0406740 ( rq & jcw ) .
cccccc oct 26.30 & 53669.30 & -6 & 2x600 & gg385 & 2.0 + oct 27.20 & 53670.20 & -5 & 4x600 & og590 & 1.5 + oct 27.28 & 53670.28 & -5 & 2x600 & gg385 & 1.5 + nov 1.27 & 53675.27 & 0 & 2x600 & gg385 & 1.5 + nov 1.29 & 53675.29 & 0 & 2x550 & og515 & 1.5 + nov 3.27 & 53677.27 & 2 & 2x480 & gg385 & 1.5 + nov 4.26 & 53678.26 & 3 & 2x600 & og515 & 1.5 + nov 6.17 & 53680.17 & 5 & 2x600 & gg385 & 1.5 + nov 6.18 & 53680.18 & 5 & 2x600 & og515 & 1.5 + nov 11.25 & 53685.25 & 10 & 3x600 & og515 & 1.5 + nov 19.22 & 53693.22 & 18 & 4x600 & og515 & 1.5 + nov 26.20 & 53700.20 & 25 & 2x600 & og515 & 1.5 + dec 5.18 & 53709.18 & 34 & 3x600 & og515 & 2.0 + rrrrr @xmath36 & 10820 & 140 & 0.38 & 140 + @xmath38 & 10800 & 110 & 0.32 & 120 + 0 & 10640 & 90 & 0.52 & 110 + 2 & 10440 & 100 & 0.60 & 110 + 3 & 10640 & 90 & 0.57 & 110 + 5 & 10680 & 80 & 0.57 & 110 + 10 & 10530 & 100 & 0.60 & 100 + 18 & 10550 & 120 & 0.48 & 90 + 25 & 9850 & 90 & 0.25 & 60 + 34 & & & & +
|
het optical spectra covering the evolution from about 6 days before to about 5 weeks after maximum light and the rotse - iiib unfiltered light curve of the `` branch - normal '' type ia supernova sn 2005hj are presented .
the host galaxy shows region lines at redshift of @xmath0 , which puts the peak unfiltered absolute magnitude at a somewhat over - luminous @xmath1 .
the spectra show weak and narrow lines , and for a period of at least 10 days beginning around maximum light these profiles do not change in width or depth and they indicate a constant expansion velocity of @xmath2 km s@xmath3 .
our observations indicate that si is confined to a relatively narrow velocity region .
we analyzed the observations based on detailed radiation dynamical models in the literature .
the models considered include delayed detonations , deflagrations , pulsating delayed detonations , and tamped detonation / merger scenarios .
whereas the first two classes of models have been used to explain the majority of sne ia , they do not predict a long velocity plateau in the minimum with an unvarying line profile . pulsating
delayed detonations and merger scenarios form shell - like density structures with properties mostly related to the mass of the shell , @xmath4 , and we discuss how these models may explain the observed line evolution ; however , these models are based on spherical calculations and other possibilities may exist .
sn 2005hj is consistent with respect to the onset , duration , and velocity of the plateau , the peak luminosity and , within the uncertainties , with the intrinsic colors for models with @xmath5 .
our analysis suggests a distinct class of events hidden within the branch - normal sne ia . if the predicted relations between observables are confirmed , they may provide a way to separate these two groups .
we discuss the implications of two distinct progenitor classes on cosmological studies employing sne ia , including possible differences in the peak luminosity to light curve width relation .
|
since the terrestrial particles accelerators like large hadron collider probe particle physics at the energy scales that are almost @xmath4 orders of magnitude smaller than the planck scale , it would interesting to investigate whether or not various naturally occurring high energy astrophysical phenomenon could shed light on the new physics at higher energy scales that remain unexplored . stepping ahead towards this exciting possibility , an interesting proposal was made recently which suggests that the kerr black holes could act as particle accelerators@xcite .
it was shown that the two particles dropped in from infinity at rest , traveling along the timelike geodesics can collide and interact near the event horizon of a kerr black hole with divergent center of mass energy , provided the black hole is close to being extremal and angular momentum of one of the particles takes a specific value of the orbital angular momentum .
the possible astrophysical implications of this process around the event horizon of the central supermassive black hole in the context of annihilations of the dark matter particles accreted from the galactic halo were also investigated @xcite .
this process of particle acceleration suffers from several drawbacks and limitations pointed out in@xcite .
the angular momentum of one of the colliding particle must take a single fine tuned value . the proper time required for the particle with fine tuned angular momentum to reach
the horizon and thus the time required for the collision to take place is infinite .
the gravity produced by the colliding particles themselves was neglected .
there were many investigations of this acceleration mechanism in the background of kerr as well as many other black holes@xcite .
two of present authors , pm and psj , studied and extended the particle acceleration mechanism to the kerr naked singular geometries transcending kerr bound by arbitrarily small amount @xmath5 @xcite .
we considered two different scenarios where the colliding particles follow a geodesic motion along the equatorial plane as well as along the axis of symmetry of the kerr geometry . in the first case ,
the particles are released from infinity at rest in the equatorial plane .
one of the initially infalling particle turns back as an outgoing particle due to its angular momentum .
it then collides with an another infalling particle around @xmath6 .
we showed that the center of mass energy of collision between these two particles is arbitrarily large .
the angular momentum of the colliding particles is required to be in a finite range as opposed to the single fine tuned value in case of kerr black holes .
thus the extreme fine tuning of the angular momentum is avoided in such a collision .
the proper time required for such a collision to take place is also shown to be finite . in the second case , the particles are released from rest along the axis of symmetry , from large but finite distance .
these particles have zero angular momentum .
one of the particles initially falls in and then turns back due to the repulsive effect of gravity in the vicinity of a kerr naked singularity .
this particle then collides with an ingoing particle at @xmath7 .
the center of mass energy of collision is arbitrarily large and the proper time required for the process to take place is finite .
thus two issues related to acceleration mechanism in kerr black hole case , namely the fine tuning of the angular momentum and the infinite time required for the collision , are avoided in case of kerr naked singularities .
the issue of the self - gravity of the point particles is difficult to deal in general .
the accretion of the particles onto an astrophysical object can be expected to be more or less isotropic in many cases .
thus it would be interesting and more physical to study the motion and collisions of the shells of particles instead .
the rigorous mathematical analysis of the shells would be very extremely difficult in the kerr spacetime due to the lack of sufficient symmetry .
by contrast , the motion and collision of the spherical shells would be exactly tractable in the spherically symmetric spacetimes following the israels thin shell formalism@xcite .
we first note that while no gravitational radiation is emitted by a perfectly spherical shell , the gravitational radiation per particle emitted by a quasispherical shell of particles will be significantly lower than the radiation emitted by a single particle@xcite
. thus it might be reasonable to ignore the gravitational radiation effects and focus entirely on the backreaction while dealing with the shells .
the acceleration of the particles around the extremal reissner - nordstrm black hole was studied in @xcite,@xcite .
this process is mathematically similar to the acceleration process in kerr geometry .
the center of mass energy of collision near the horizon of the extremal reissner - nordstrm black hole , of the charged and uncharged particles is shown to be divergent .
the collision of the charged and uncharged spherical shells was investigated in@xcite .
the dynamics of the shells when their gravity is ignored is same as that of the test particles .
whereas when the exact calculation is done taking into account the self - gravity effects , the center of mass energy turns out to be finite .
thus it was speculated that the center of mass energy of collision of particles around kerr black hole might also turn out to be finite when the gravity due to the colliding particles is taken into account . in this paper
we first describe the particle acceleration process in the background of reissner - nordstrm naked singularities .
we show that the center of mass energy of collision between two uncharged particles , one of them initially ingoing and other one initially ingoing , but turning back due to the repulsive effect of gravity in the vicinity of naked singularity is arbitrarily large , when the collision happens around @xmath8 , provided that the deviation of the reissner - nordstrm charge from the mass is extremely small .
we calculate the coordinate time as seen by the distant observer , associated with the ultra - high energy collisions for extremal black hole as well as for naked singularity .
we show that the time scale associated with the trans - plankian collisions around naked singularity with one solar mass is of the order of million years which is significantly smaller than the hubble scale , whereas the timescale for the extremal black hole with the same mass as that of the naked singularity is fifteen orders of magnitude larger than the age of the universe .
thus collision process around black hole suffers from the inflating timescale problem while such issue is absent in case of the naked singularity .
we then investigate the collision between two uncharged shells made up of dust particles , in a situation analogous to the particle collision , taking into account their gravity .
we find that the center of mass energy of a collision between the shells is bounded above .
however , the center of mass energy of a collision between two of constituent particles of the shells can exceed the planck energy which might be a threshold value of the quantum gravity . in this paper
, we adopt the geometrized unit in which the speed of light and newton s gravitational constant are unity .
the reissner - nordstrm spacetime is a unique solution of einstein equations under the assumptions of spherical symmetry , asymptotic flatness with the @xmath9(1 ) gauge field as a source of spacetime curvature .
the line element of the reissner - nordstrm geometry in the spherical polar coordinates is given by @xmath10 where @xmath11 the gauge field is given by @xmath12 this solution contains two parameters @xmath1 and @xmath0 , namely the mass and @xmath9(1 ) charge . in this paper
, we assume that @xmath1 and @xmath0 are positive , @xmath13 in the reissner - nordstrm spacetime , there is a spacetime singularity at @xmath14 .
this singularity is timelike and thus is necessarily locally naked .
the location of the horizon in the reissner - nordstrm spacetime is given by a solution to the equation @xmath15 .
there are two roots to this quadratic equation given by @xmath16 there are two real positive roots to the equation if @xmath17 .
the larger root @xmath18 is the location of the event horizon and this spacetime corresponds to a spherically symmetric charged black hole .
the smaller root @xmath19 corresponds to the cauchy horizon associated with the timelike singularity at @xmath14 .
if @xmath20 , there is only one positive root . in this case
the black hole is known as the extremal black hole with a degenerate event horizon at @xmath21 . in the case of @xmath22
, there is no real root to the equation @xmath15 .
thus , the event horizon is absent and the timelike singularity at @xmath14 is exposed to the asymptotic observer at infinity .
this configuration thus contains a globally visible naked singularity .
we will investigate the last case in this paper from the perspective of particle acceleration . before proceeding further
, it is worthwhile to mention that , the naked singularities are associated with pathological features like the breakdown of predictability and so on . that was precisely the reason why penrose came up with the cosmic censorship conjecture abandoning the existence of naked singularities in our universe@xcite .
however there were recent developments in the framework in string theory , which suggests by means of the specific worked out examples , that the naked singularities might be resolved by high energy stringy modification to the classical general relativity @xcite and various pathological features disappear .
this renders the classical naked singular solutions legal as long as one stays sufficiently away from high curvature region where quantum gravity would prevail .
we now study the motion of a point test particle following a timelike geodesic in the reissner - nordstrm spacetime .
let @xmath23 be the 4-velocity of the particle .
without loss of generality , we assume that the motion of the particle is confined to the equatorial plane @xmath24 .
all of the metric components ( [ rn ] ) are manifestly independent of time coordinate and azimuthal angular coordinate .
this means that both of the time coordinate basis @xmath25 and azimuthal angular coordinate basis @xmath26 are killing vectors the following quantities are conserved along the geodesic of the particle @xmath27 @xmath28 can be interpreted as the conserved energy of the particle per unit mass and @xmath29 can be interpreted as the conserved angular momentum of the particle per unit mass . using these constants of motion and the normalization condition for 4-velocity of the particle , the components of the 4-velocity @xmath23
are written as @xmath30 @xmath31 stands for the radially outgoing and infalling particles respectively .
the second one in the above equations can also be written in the following form @xmath32 where @xmath33 is the proper time of the particle , and @xmath34 @xmath35 can be thought of as a effective potential . for simplicity and from the perspective of the comparison to shell collision that would be discussed in the next section
, we assume that the angular momentum of the particle is zero @xmath36 .
this implies that the motion of the particle is purely radial .
the effective potential now can be written as @xmath37 the effective potential is plotted as a function of @xmath38 for a test particle following a radial geodesic in reissner - nordstrm naked singular geometry with @xmath39 . an allowed domain for the motion of a particle
is depicted by a dashed line for each case of specific energy .
it admits a minimum at the classical radius @xmath40 , depicted by min , where gravity changes it s character from being attractive to repulsive in the close neighborhood of singularity .
the ingoing particle thus gets reflected back as an outgoing particle close to singularity .
the motion of a particle having energy @xmath41 is bound and oscillates .
the motion of a particle with energy @xmath42 is unbound , has only one turning point .
the motion of a particle with @xmath43 is marginally bound , also has only one turning point .
the potential energy curve asymptotes to the @xmath43 as @xmath44 .
, scaledwidth=50.0% ] the effective potential is plotted as a function of radius @xmath45 in fig .
[ veff_particle ] . for large values of radial coordinate @xmath44
, we have @xmath46 .
as one approaches the naked singularity @xmath47 , effective potential blows up positively , i.e. , @xmath48 . it always remains greater than zero and admits a minimum at @xmath49 which is given by @xmath50 and we have @xmath51 note that @xmath52 coincides with the classical radius associated with an object of charge @xmath0 and mass @xmath1 .
it is clear from the shape and slope of the effective potential curve that the gravity of the reissner - nordstrm naked singularity is attractive in the domain @xmath53 , from the classical radius all the way upto infinity . whereas the gravity is repulsive in the region extending from the singularity to the classical radius @xmath54 .
similar behavior is also observed in case of other known examples of the stationary naked singularities@xcite .
an ingoing particle at initially speeds up upto the classical radius .
it then slows down due to the repulsive gravity and gets reflected back eventually .
it them emerges as an outgoing particle .
if the conserved energy of the particle is less than unity @xmath55 then the particle is bound , i.e. , it oscillates back and forth in the radial domain @xmath56 , where @xmath57 in the case of @xmath58 , @xmath59 is equal to @xmath60 .
this means that the particle stays stably at rest at the classical radius @xmath40 .
if the conserved energy is identical to unity @xmath43 , then there is only one turning point given by @xmath61 . in this case , the particle is at rest at infinity , and the motion of the particle is said to be marginally bound . in the case
when energy is larger than unity @xmath62 , again there is only one turning point given by @xmath63 since @xmath59 is negative in this case .
the asymptotic velocity of the particle as it reaches infinity is positive @xmath64 .
such a particle trajectory is called the unbound one .
we should note that there is an important difference between the black hole case @xmath65 and the naked singular case @xmath22 . in the case of the black hole @xmath65
, the radial motion can not be restricted to only one asymptotically flat region . since the inner turning point , @xmath63 is less than or equal to the radius of the cauchy horizon @xmath19 , the particle can not return to the asymptotically flat region where it comes from .
by contrast , in the case of the naked singularity @xmath17 , there is only one asymptotically flat region .
hereafter , we focus on the naked singular case @xmath17 .
we now consider a collision between two particles moving along a radial geodesics i.e. , @xmath36 , each with mass @xmath66 and conserved energy @xmath43 : particles are assumed to be marginally bound , or in other words , they are released from rest from infinity .
one could replace marginally bound particles by either unbound or bound particles .
it does not change the conclusions .
let @xmath67 and @xmath68 be components of their 4-velocities with respect to the coordinate basis .
we assume that one of the particles is initially ingoing particle which gradually slows down and eventually turns back as an outgoing particle due to the repulsive gravity in the vicinity of the naked singularity .
such a particle then collides with another ingoing particle at the radial coordinate @xmath45 . by the assumption , @xmath67 and @xmath68
are given by @xmath69 the energy of a collision between two particles at the center of mass frame is then given by @xcite @xmath70 where @xmath71 is the metric tensor given in eq .
( [ rn ] ) .
it is seen from the above equation that the center of mass energy @xmath72 of collision depends on the location for the collision , for given values of charge @xmath0 and mass @xmath1 .
@xmath72 takes maximum when the effective potential @xmath35 takes minimum .
the minimum of @xmath35 is realized at the classical radius @xmath73 .
if the collision takes place at @xmath49 , @xmath72 is given by @xmath74 @xmath75 depends on the ratio of mass to the charge of reissner - nordstrm spacetime .
@xmath75 is very large if the charge transcends the mass by infinitesimally small amount .
here , we introduce a parameter defined by @xmath76 in the limit @xmath77 , t@xmath75 becomes infinite , @xmath78 the above equation implies that the energy of collision measured at the center of mass frame would be arbitrarily large . in case of the black hole
, the divergence of center of energy in the collision has been demonstrated in near extremal or extremal geometries when the mass transcends the charge by arbitrarily small amount @xmath79 . in this paper , we have shown the possibility of the indefinitely large center of mass energy in the naked singular geometry , which can be thought to be near extremal , with the charge transcending the mass by arbitrarily small amount @xmath80 .
we now estimate the time scale associated with the ultra - high energy particle collisions in the reissner - nordstrm naked singular geometry as well as in the extremal black hole geometry and make a critical comparison .
we compute the proper time in the reference frame attached to the colliding neutral particle , as well as the coordinate time measured by a distant static observer , required for the particle to reach the collision point @xmath81 in the case of naked singularity , and the horizon @xmath82 in the extremal black hole case .
the particle starts from a distant location with @xmath83 and participates in the high energy collision . in the extremal reissner - nordstrm black hole geometry with @xmath84 ,
the high energy collision between the particles takes place at a location extremely close to the event horizon .
one of the colliding particles is charged and the other one is charge neutral .
the charged particle experiences an outward repulsive electromagnetic force during its inward motion .
for such a particle it turns out that @xmath85 as it approaches the event horizon , as a consequence of which the proper time required for it to reach the horizon and participate in the high energy collision turns out to be infinite .
the neutral particle , however , falls freely following a geodesic motion and reaches the event horizon in a finite proper time as we show later in this section .
we also estimate the coordinate time as seen by the static observer at infinity , required for the neutral particle to participate in the high energy collision . we show that it diverges in the limit of approach to the horizon
as it is an infinite blueshift / redshift surface and the timescale associated with the trans - plankian collision is much larger than the age of the universe . in the reissner - nordstrm naked singular geometry
, the collision is between two charge neutral particles following a geodesic motion as they fall freely under the gravity .
both the conditions mentioned above in the last paragraph namely @xmath85 are not satisfied simultaneously anywhere along the trajectory of either of the two particles . thus the proper time required for the collision to take place for both
the particles in their own frame is finite as we demonstrate later in this section .
however , since it is necessary to have @xmath86 , for high energy collision to occur , which is precisely the condition for extremely large blueshift / redshift , one would expect that coordinate time as measured by the static observer at infinity would diverge .
we show that for trans - plankian collisions the coordiante time required is of the order of million years which is much smaller than the hubble time . for a particle moving along a radial geodesic with @xmath43 , from ( [ p - eom ] )
, we have @xmath87 where @xmath33 is a proper time and @xmath31 corresponds to radially outgoing and ingoing particles respectively . the proper time as measured in the reference frame attached to the particle when it travels from @xmath88 to @xmath89 can be obtained by integrating the ( [ tm1 ] ) and is given by @xmath90^{r_{\rm f}}_{r_{\rm i } } , \label{ptime } \end{split}\ ] ] where @xmath31 corresponds to the case where @xmath91 and @xmath92 , i.e. , when particle moves radially onwards and radially inwards respectively .
* extremal black hole * the proper time required for the neutral particle to reach horizon from the initial location @xmath93 using ( [ ptime ] ) is given by @xmath94_m^{r_{\rm i } } -\frac{2}{3}m\ ] ] which is clearly finite .
the proper time required for the charged particle to reach the horizon however diverges as discussed earlier since its effective potential for the radial motion as well as its derivative goes to zero at the horizon .
* naked singularity * in the naked singularity case , one of the particles starts out as an ingoing particle at @xmath88 , gets reflected back at @xmath95 due to the repulsive effect of the naked singularity and arrives at the collision point @xmath81 as an outgoing particle .
the proper time required in its rest frame from ( [ ptime ] ) is given by @xmath96 \label{pt1 } \end{split}\ ] ] the second particle starts out at @xmath88 and reaches @xmath97 as an ingoing particle where it collides with the first particle .
the proper time required in its rest frame is given by @xmath98 \label{pt2 } \end{split}\ ] ] it is evident from ( [ pt1]),([pt2 ] ) that the proper time required for the collision is finite in the rest frame of both the particles .
we now compute the coordinate time required for the collision as measured by the static distant observer in the extremal black hole and naked singularity cases . from ( [ tm1 ] )
we get @xmath99 the time observed by the distant observer as the particle moves from @xmath93 to @xmath100 can be obtained by integrating the equation above and is given by @xmath101 \label{tb } \end{split}\ ] ] where @xmath31 stands for the radially outgoing and radially ingoing particles with @xmath91 and @xmath92 respectively as stated earliar , and @xmath102 is the indefinite integral @xmath103 * extremal black hole * we now compute the coordiate time required for the neutral particle to reach the event horizon of the extremal reissner - nordstrm black hole . in this case
the function @xmath102 is given by the expression @xmath104 thus it follows from eqs .
( [ tb ] ) and ( [ tbh ] ) that the time required for the ingoing neutral particle to reach @xmath105 diverges in the limit @xmath106 as @xmath107 the center of mass energy of collision @xmath72 between the charged and uncharged particles as a function of the collision location @xmath105 varies as @xcite @xmath108 where @xmath66 is the mass of each of the colliding particles .
it follows from eqs .
( [ tbh1 ] ) and ( [ ecmbh ] ) that the time required for the neutral particle to participate in the collision at the radial location @xmath109 is thus given by @xmath110 where @xmath111 is the solar mass , @xmath112 is planck energy and @xmath113 is mass of the proton .
the time required for the neutral particle with mass @xmath114 such as neutron , to participate in a planck scale collision around a solar mass extremal black hole is approximately @xmath115 times larger than the age of the universe .
the time required for the charged particle to reach the collision point will be even larger .
therefore the phenomenon of ultra - high energy collisions around charged black holes does not occur within the hubble time scale and thus has no observable consequences whatsoever .
* naked singularity * we now discuss the timescale associated with the ultra - high energy collision around the reissner - nordstrm naked singularity .
the function @xmath102 in this case is given by the following expression .
@xmath116 the time required for the ingoing neutral particle starting at @xmath88 to get reflected at @xmath95 as an outgoing particle and to reach the collision point @xmath81 in the limit @xmath117 , from eqs .
( [ tb ] ) and ( [ nsb ] ) is given by @xmath118 whereas the time required for the second ingoing neutral particle to reach @xmath49 starting from @xmath88 , from eqs .
( [ tb ] ) and ( [ nsb ] ) is given by @xmath119 it is clear from ( [ t1]),([t2 ] ) that @xmath120 and @xmath121 diverge as @xmath122 in the limit @xmath123 .
the center of mass energy of collision between two particles at @xmath81 in the reissner - nordstrm naked singularity case in the limit @xmath123 is given by eq .
( [ ens ] ) @xmath124 where @xmath66 is the mass of each of the colliding particles . from eqs .
( [ t1 ] ) , ( [ t2 ] ) and ( [ ecmns ] ) , the time scale associated with the collision is given by @xmath125 where as before @xmath111 is mass of the sun , @xmath112 is the planck energy and @xmath113 is mass of the proton .
we see that the time scale associated with the planck scale collision of two neutrons around a solar mass naked singularity is merely of the order of million years which is @xmath126 times smaller than the age of the universe .
this implies that the trans - plankian collisions around naked singularities are conceivable and might be observable either in our galaxy or at very high cosmological redshifts .
furthermore if the particles continuously accrete from a distant location @xmath127 , in a steady state , the rate of occurrence of the collisions will be same as the accretion rate .
thus one could say that there is no inflatiing time - scale problem in the naked singular reissner - nordstrm spacetime while it does exists in the extremal black hole geometry .
in this section , we discuss the validity of test particle approximation on the particle collision around reissner - nordstrm naked singular geometry .
we should consider two type of `` back reaction '' , i.e. , the effects of the gravitational radiation and the conservative self - force .
as long as we consider the radially moving particles , the effect of gravitational radiation does not change it to non - radial one . however
, if the energy of the particle is released by the gravitational radiation , initially marginally bound particle will be bound .
we denote the energies of the particles by @xmath128 and @xmath129 . if the gravitational emission is negligible , @xmath128 and @xmath129 are constants of motion , but it might vary with time if the gravitational emission is not negligible .
the 4-velocities of the particles are written in the form @xmath130 as before , we assume that the collision occurs at the minimum of @xmath131 , i.e. , @xmath132 , where @xmath133 has been defined by eq .
( [ e - def ] ) , and then the collision energy at the center of mass frame is given by @xmath134\ ] ] since the @xmath45-components of @xmath67 and @xmath68 should be real , @xmath128 and @xmath129 should be larger than or equal to @xmath135 .
if @xmath128 and @xmath129 become several times @xmath135 by the emission of the gravitational radiation , the collision energy @xmath72 takes small value which is several times @xmath66 .
however note that in this case the emitted gravitational radiation would be so large that the conserved energies , which were assumed to have unit value to begin with , drastially reduce to a value that is nearly equal to zero .
it is beyond the scope of this paper to estimate how large is the amount of energies of the particles are released by the gravitational radiation , and hence , we can not make any quantitative statement .
if the colliding particles do not drastically loose the energies to a value close to zero and carry a descent fraction of the initial energies , the ultra - high energy collision can occur . if so , it is sufficient to consider the effect of conservative self - force .
consideration of conservative self - force is important since it can turn a near extremal naked singular configuration into a black hole and thus hiding the ultra - high energy collisions below the event horizon . in this section ,
since to treat conservative self - force for the point particle is difficult , we study analogous system , i.e. , the collision of spherical shells in the reissner - nordstrm naked singular geometry .
it is also well justified on the physical grounds , since in a realistic situation the accretion of the matter onto a massive compact object would be more or less isotropic .
therefore the amount of gravitational radiation emitted per particle will be significantly reduced @xcite and its effect on the process of ultra - high energy collisions can be ignored to a very good approximation .
thus would suffice to consider only the conservative self - force .
the dynamics of the spherical thin shells is tractable exactly owing to the spherical symmetry of the system . due to the gravity generated by the shells themselves , the equations describing the motion of shells are no longer the geodesic equations in the reissner - nordstrm spacetime .
we deal with the situation that is analogous to the scenario described in the previous section , in order to draw a parallel to and compare with the test particle case .
we assume that the deviation of the charge from the mass associated with the naked singularity is vanishingly small .
we first describe the procedure to deal with the thin shells with taking into account their gravity @xcite .
we basically follow notation and convention of ref .
a trajectory of a shell that is being considered here is a timelike hypersurface with a thin surface layer of matter in the four dimensional ambient spacetime manifold : we denote it by @xmath136 .
then , a shell @xmath137 means an intersection between @xmath136 and a spacelike hypersurface with constant time coordinate chosen appropriately . since the finite amount of energy exists within the infinitesimally thin layer , the energy - momentum tensor is infinite , but it is possible to define it as a distribution .
the geometry of a hypersurface can be described by specifying a three dimensional metric @xmath138 within it ( also known as the induced metric ) and an extrinsic curvature @xmath139 , which is a three dimensional tensor describing how the hypersurface is embedded in the ambient spacetime . even if the trajectory of the shell is a singular hypersurface
, we assume that the metric of four dimensional spacetime is everywhere continuous .
thus , the induced metric @xmath138 of the shell is assumed to be continuous .
by contrast , the extrinsic curvature @xmath139 of the shell may be discontinuous across the shell due to the distributional energy - momentum tensor on the shell .
@xmath136 separates the spacetime in two regions @xmath140 and @xmath141 .
the coordinates defined in these two regions is denoted by @xmath142 ( @xmath143 ) , whereas the coordinates within @xmath136 is denoted by @xmath144 ( @xmath145 .
although the metric is continuous , the components of it may not be continuous , since the coordinate systems may be discontinuous at @xmath136 .
the projection operator from the four dimensional ambient spacetime to @xmath136 is given by @xmath146 here , the index @xmath147 of the projection operator indicates side of @xmath136 on which the quantity is defined , @xmath140 or @xmath141 . denoting the components of the metric by @xmath148 , the induced metric on the hypersurface
is given by @xmath149 the extrinsic curvature of the shell is given by @xmath150 where @xmath151 denotes a component of the covariant derivative of the unit normal vector @xmath152 to @xmath136 , which is directed from @xmath153 to @xmath154 . here , note that the unit normal vector to @xmath136 is unique , since the metric tensor is everywhere continuous . by denoting the components of the energy momentum tensor of the shell by @xmath155 , it is given by the following form @xmath156 where @xmath157 is a three dimensional tensor defined over @xmath136 shell , which is called the surface energy - momentum tensor
, @xmath158 is dirac delta function , and @xmath159 is the gaussian normal coordinate which is equal to zero on @xmath136 .
the junction condition is given in the form of the condition on the discontinuity of the extrinsic curvature of @xmath136 as follows ; @xmath160 we now consider a case where the both regions @xmath153 and @xmath154 are the reissner - nordstrm and the shell @xmath137 is spherically symmetric and made up of charge neutral dust . due to the charge neutrality of the shell ,
the charge parameters in the both regions are identical , and we denote it by @xmath0 . by contrast , the value of the mass parameter would be different in two regions ( see fig . [ 1-shell ] ) .
we use the coordinate systems @xmath161 and @xmath162 in the region @xmath153 and @xmath154 , respectively . note that the time coordinate is not continuous , whereas @xmath45 , @xmath163 and @xmath164 are everywhere continuous .
the metric in these two regions can be written as @xmath165 where @xmath166 the misner - sharp mass of the shell is defined by @xmath167 and we assume @xmath168 .
the positivity of @xmath169 naturally introduce a picture that @xmath153 is the inside of the shell @xmath137 , whereas @xmath154 is the outside .
we use the coordinates @xmath170 on @xmath136 , where @xmath33 is taken to be the proper time for an observer comoving with the shell .
the intrinsic metric of the shell is then written as @xmath171 the proper time of the shell can parametrize the trajectory of the shell , i.e. , @xmath172 the projection operator is then given by @xmath173 the induced metric is given by @xmath174d\tau^2 \cr & & \cr & + & r(\tau)^2 ( d\theta^2+\sin^2\theta d\phi^2 ) , \label{im2}\end{aligned}\ ] ] where a dot represents a derivative with respect to @xmath33 .
equations ( [ im1 ] ) and ( [ im2 ] ) imply @xmath175 equation ( [ tnsame ] ) implies that the time coordinate in the regions @xmath153 and @xmath154 necessarily have to be different .
this is schematic diagram of the spherically symmetric spacetime divided into two domains @xmath153 , @xmath154 by the trajectory of a thin shell @xmath136 depicted by a dashed curve .
the spacetime metric in the two domains @xmath153 , @xmath154 is reissner - nordstrm with different values of mass parameters , namely @xmath176 and @xmath177 , but with the same charge @xmath0 .
, scaledwidth=30.0% ] the unit normal vector @xmath152 to @xmath136 is given by @xmath178 substituting the above expression into eq .
( [ k - def ] ) , we find that the non - vanishing components of the extrinsic curvature are given by @xmath179 the energy - momentum tensor of the dust within the thin shell is given by @xmath180 where @xmath181 is the surface density and @xmath182 is a component of the 4-velocity of the dust , which is equivalent to @xmath183 .
we assume that @xmath181 is non - negative . comparing the above equation with eq .
( [ t1 ] ) , we get @xmath184 where @xmath185 is the 3-velocity of the dust within the shell , whose components are given by @xmath186 by using eqs .
( [ kab ] ) and ( [ sab ] ) , we now write down eq .
( [ eom ] ) and obtain @xmath187 equations ( [ meq ] ) and ( [ vel ] ) taken together give @xmath188 the constant @xmath66 is interpreted as the proper mass of the shell , and we get an energy equation for the shell as follows @xmath189 as in the case of the test particle , let us introduce the following effective potential @xmath190 where @xmath191 then , eq . ( [ eom2 ] )
is written in the very similar form to eq .
( [ p - eom ] ) for the test particle as @xmath192 where @xmath193 is the energy of the shell per unit proper mass . minus
the square of specific energy @xmath194 of the shell is depicted for the case of @xmath195 and @xmath55 . the allowed domain for the motion of the shell
is specified by the dashed line . ,
scaledwidth=43.0% ] , but @xmath62 .
, scaledwidth=43.0% ] , but @xmath196 .
, scaledwidth=40.0% ] , but @xmath196 and @xmath62 .
, scaledwidth=40.0% ] we depict the effective potential @xmath35 for the shell as a function of @xmath197 in figs . 3 - 6 .
first , we consider the case of @xmath195 . for @xmath55
( see fig .
[ b_ml2q ] ) , the motion of the shell is restricted within the domain @xmath198 , where @xmath199 in the case of @xmath200 , @xmath201 is equal to @xmath202 , and the particle stays stably at rest at the radius @xmath203 . for @xmath204 ( see fig . [ unb_ml2q ] ) , the allowed domain for the motion is @xmath205 . initially outgoing shell monotonically approaches to infinity , @xmath206 , whereas initially ingoing shell turns to be outgoing .
these behaviors are basically the same as that of the test particle .
the repulsive nature of the charged singularity halts the collapse of the shell .
by contrast , in the case of @xmath207 , the allowed domain for the motion is @xmath208 for @xmath55 ( see fig .
[ b_mg2q ] ) and thus the shell with @xmath55 necessarily collapses to the singularity at @xmath14 . in the case of @xmath204
( see fig .
[ unb_mg2q ] ) , whole domain is allowed for the motion of the shell ; the initially outgoing shell goes to infinity , whereas the initially ingoing shell collapses to the singularity at @xmath14 .
the self - gravity of the shell overcomes the repulsive gravity of the charged singularity .
we rewrite the effective potential in the form @xmath209 from the above equation , we have @xmath210 where @xmath211 is a larger root of @xmath212 .
thus , in the case of @xmath213 , an ingoing shell necessarily enters into the black hole and goes to the another asymptotically flat region .
by contrast , in the case of @xmath214 , there is only one asymptotically flat region , and hence even in the case of the ingoing shell , the shell remains in this asymptotically flat reg , ion as long as it does not hit the spacetime singularity at @xmath14 .
the situation is similar to the case of a radially moving test particle .
now we describe the process of acceleration and collision of charge neutral shells whose motion has been considered in the preceding section .
we consider two concentric spherical thin shells @xmath215 and @xmath216 .
these shells divide the spacetime into three regions each of which is denoted by @xmath217 @xmath218 : @xmath215 faces @xmath153 and @xmath154 , whereas @xmath216 faces @xmath154 and @xmath219 ( see fig . [ 2-shell ] ) .
the metric in the three regions would be given by reissner - nordstrm geometry with the different values of parameters in three regions .
the shells are assumed to be electrically neutral and thus the charge parameters in the three regions take an identical value @xmath0 . by contrast ,
mass parameters take different values in three regions .
we denote them by @xmath220 . for simplicity , we assume that these shells have identical misner - sharp mass @xmath169 , and they are given by @xmath221 , @xmath222 and @xmath223 with @xmath168 .
further , we assume that the charge @xmath0 is somewhat larger than @xmath224 . here
, we replace the define a small parameter @xmath133 by @xmath225 by the assumption , we have @xmath226 , and by another assumption @xmath227 , we have @xmath228 .
thus , we may write @xmath169 in the form @xmath229 the above condition ensures that the naked singularity does not turn into a black hole by the shells @xmath215 and @xmath216 .
this is a schematic diagram showing the motion and collision of two shells @xmath215 and @xmath216 .
there is a reissner - nordstrm naked singularity at the center whose charge is slightly larger than the mass .
the shell @xmath215 which is initially ingoing turns back as an outgoing shell and then collides with the ingoing shell @xmath216 at @xmath230 .
the similar picture also can be drawn in case of particle collision replacing shells by particles .
, scaledwidth=40.0% ] hereafter , for simplicity , we assume that the shells are marginally bound , i.e. , @xmath231 . following exactly the same procedure as in the one - shell case ,
the radial components of 4-velocities of @xmath232 ( @xmath143 ) whose radii are denoted by @xmath233 can now be written as @xmath234 where @xmath235 are identical to eq .
( [ f1-def ] ) , and @xmath31 stands for outgoing and ingoing shell , respectively . using the normalization condition for 4-velocity @xmath236 , we obtain the time components of the 4-velocity with respect to the coordinate basis in the domain @xmath154 , @xmath237 the radial components of 4-velocities of both @xmath215 and @xmath216 would go to zero at infinity by the assumption of @xmath231 . by carefully taking a limit @xmath238 for @xmath202 in eq .
( [ b - def ] ) , we have the turning points @xmath239 for @xmath215 and @xmath240 for @xmath216 , where @xmath241 both the turning points are the same order , but @xmath242 .
we consider a situation where the inner shell @xmath215 starts off at infinity as an ingoing shell ; @xmath215 then turns back at @xmath243 and emerges as an outgoing shell and collides with the outer ingoing shell @xmath216 at @xmath244 .
this situation is exactly analogous to the situation encountered in the case of test particles in sec .
the energy of two shells at the center of mass frame " was defined in @xcite in a following way by generalizing the definition of the center of mass energy of the particles . in case of the particle collisions , in order to compute the center of mass energy , one goes to the orthonormal frame in which the spatial components of the total momentum of the two particles is zero .
the time component yields the center of mass energy . while dealing with the collision event of the shells , the center of mass frame
was defined to be an orthonormal frame in which the energy flux along the spatial direction is zero and the center of mass energy is defined analogously .
we obtain for the shells @xmath245 we can compute @xmath246 by using their components with respect to the coordinate basis in the region @xmath154 , and we have the center of mass energy of collision at any given value of @xmath197 as @xmath247 \label{ecm}\end{aligned}\ ] ] the circumferential radius at the the minimum of @xmath248 is @xmath249 , and let us consider the collision there . from eqs .
( [ e - def-2 ] ) and ( [ mu - def ] ) , we have @xmath250 then , if the signs of @xmath251 and @xmath252 are different from each other , we have , for @xmath253 , @xmath254 we can see from the above equation that as in the case of the test particles , the energy of two spherical shells at the center of mass frame can be arbitrarily large . here , we assume that a shell is composed of @xmath255 particles each of which has a mass @xmath256 .
the center of mass energy @xmath257 of a collision between two of constituent particles is given by @xmath258 using the above equation and eq .
( [ ecm - max ] ) , the collision energy at @xmath249 with @xmath259 is given by @xmath260 the above equation seems to imply that the center of mass energy can be indefinitely large .
however , in order that the description by a spherical shell is valid , the number of particles @xmath255 should be much larger than unity , i. e. , @xmath261 or , by assuming @xmath262 , @xmath263 due to this constraint , we have @xmath264 the above equation implies that if @xmath1 is order of the solar mass @xmath265 kg , the collision energy @xmath257 between particles at the center of mass frame can exceed planck scale @xmath266 even if @xmath267 is the order of the proton mass @xmath268gev .
in this paper , we studied the particle and shell acceleration by reissner - nordstrm naked singularities .
the phenomenon of particle acceleration and collision with extremely large energy at the center of mass frame was previously studied and explored in the background of extremal and near extremal black holes .
we extended this result to the near extremal naked singularities .
we showed that there are significant qualitative differences in the particle acceleration mechanism between black holes and naked singularities . in case of black ,
the particle collision between ingoing particles should be considered , and in order to achieve large collision energy at the center of mass frame , fine tuning of parameters is necessary , and further the proper time of one of two particles required for such a collision is very long . on the contrary , in case of naked singularity , it is possible to consider a collision between ingoing and outgoing particles , since due to the absence of the event horizon and the repulsive gravity effects near singularity , initially ingoing particle turns back as an outgoing particle .
this fact eliminates the necessity of the fine tuning of some parameters and also the required proper time required for such a collision need not be so long .
we also calculate the coordinate time as seen by the observer at infinity required for the ultra - high energy collisions to occur for extremal black hole as well as naked singular geometry .
we show that the time required for the planck - scale collisions around naked singularity is of the order of million years which is much smaller than the age of the universe . whereas the time scale in extremal black hole case in the analogous process is many orders of magnitude larger than the hubble time .
therefore the high energy collisions occuring around the naked singularities , subject to their existence will be observable .
rate of occurence of the collisions will be same as the rate of the accretion of the matter in a steady state . on the contrary , in the black hole case high energy collisions would not occur within the hubble time and thus would have no observational consequences .
particles participating in the collision are assumed to be test particles following the geodesics on the background geometry .
the effects of gravity generated by the particles are ignored .
thus , to study whether or not the phenomenon of divergence of center of mass energy survives , we studied the collision between the concentric spherical shells .
the gravity of the shells is taken into account in an exact calculation , and the energy of collision between shells at the center of mass frame " is computed in a situation analogous to the test particle case .
it is shown that , in this case , due to the condition that the outermost region is described by the over - charged rn spacetime , the center of mass energy of a collision between two of the constituent particles of the shells is bounded above .
however , if the mass of the central naked singularity is order of the solar mass , and if the mass of a constituent particle of the shells is order of the proton mass , the upper bound exceeds @xmath269gev which is much larger than the planck scale .
mk is supported by the jsps grant - in - aid for scientific research no.23@xmath2702182 .
we would like to thank the anonymous referee for valuable comments that led to an improvement of the manuscript .
m. banados , j. silk , s.m .
west , phys .
lett . * 103 * , 111102 ( 2009 ) .
m. banados , b. hassanain , j. silk , s. m. west , phys .
d * 83 * , 023004 ( 2011 ) ; a. williams , arxiv / astro.ph/1101.4819 ( 2011 ) .
e. berti , v. cardoso , l. gualtieri , f. pretorius , u. sperhake , phys .
103 * , 239001 ( 2009 ) ; t. jacobson , t. p. sotiriou , phys .
lett . * 104 * , 021101 ( 2010 ) .
t. harada , m. kimura , phys .
d * 83 * , 084041 ( 2011 ) ; phys .
d * 83 * , 024002 ( 2011 ) .
grib , y.v .
pavlov , astropart . phys . * 34 * , 581 ( 2011 ) ; arxiv:1004.0913 ; arxiv:1007.3222 ; k. lake , phys .
lett . * 104 * , 211102 ( 2010 ) ; grav . cosmol . * 17 * , 42 ( 2011 ) ; o.b .
zaslavskii , phys .
d * 84*,024007 ( 2011 ) ; class .
quant . grav .
* 28 * , 105010 ( 2011 ) ; jetp lett . *
92 * , 571 ( 2010 ) ; phys .
d * 82 * , 083004 ( 2010 ) ; s. gao , c. zhong , arxiv:1106.2852 ; j. said , k.z .
adami , phys .
d * 83 * , 104047 ( 2011 ) ; c. liu , s. chen , j. jing , arxiv:1104.3225 ; y. zhu , s. wu , y. liu , y. jiang , arxiv:1103.3848 ; c. liu , s. chen , c. ding , j. jing , arxiv:1012.5126 ; y. li , j. yang , y - l .
li , s. wei , y. liu , arxiv:1012.0748 ; p. mao , r. li , l. jia , j. ren , arxiv:1008.2660 ; s. wei , y. liu , h. li , f. chen , jhep * 1012 * , 066 ( 2010 ) ; s. wei , y. liu , h. guo , c. fu , phys .
d * 82 * , 103005 ( 2010 ) ; w. yao , s. chen , c. liu , j. jing , arxiv:1105.6156 ; m. patil , p. joshi , arxiv/1103.1082(2011 ) ; arxiv/1103.1083(2011 ) . w. israel , nuovo cimento b * 44 * , 1 ( 1966 ) ; 463(e ) ( 1967 ) ; nuovo cimento b * 44 * , 1(1966 ) . t. nakamura , k. oohara , phys . lett .
a * 98 * , 403 ( 1983 ) e. poisson , relativists toolkit , the mathematics of black hole mechanics , cambridge university press , cambridge(2004 ) . o. zaslavskii , jetp lett .
* 92 * , 571(2010 ) .
m. kimura , k. nakao , h. tagoshi , phys.rev .
d * 83 * , 044013 ( 2011 ) .
r. penrose , riv .
nuovo cim * 1 * , 252 ( 1969 ) [ gen .
. grav . * 34 * , 1141 ( 2002 ) ] e. gimon , p .
horava , arxiv / hep - th/0706.2873 ; arxiv : hep - th/0405019 ; e. boyda , s. ganguli , p. horava , u.vardarajan , phys . rev .
d * 67*,106003(2003 ) ; g. horowitz , a. sen , phys .
d * 53*,808 ( 1996 ) .
g. preti , f. de .
felice , am .
* 76 * , 7 ( 2008 ) ; o. luongo , h. quevedo , arxiv / gr - qc/1005.4532 .
|
we explore the reissner - nordstrm naked singularities with a charge @xmath0 larger than its mass @xmath1 from the perspective of the particle acceleration .
we first consider a collision between two test particles following the radial geodesics in the reissner - nordstrm naked singular geometry .
an initially radially ingoing particle turns back due to the repulsive effect of gravity in the vicinity of naked singularity .
such a particle then collides with an another radially ingoing particle .
we show that the center of mass energy of collision taking place at @xmath2 is unbound , in the limit where the charge transcends the mass by arbitrarily small amount @xmath3 .
the acceleration process we described avoids fine tuning of the parameters of the particle geodesics for the unbound center of mass energy of collisions and the proper time required for the process is also finite . we show that the coordinate time as observed by the distant observer required for the trans - plankian collisions to occur around the naked singularity with one solar mass is merely of the order of million years which is much smaller than the hubble time . on the contrary ,
the time scale for collisions associated with extremal black hole in an analogous situation is many orders of magnitude larger than the age of the universe .
we then study the collision of the neutral spherically symmetric shells made up of dust particles . in this case
, it is possible to treat the situation by exactly taking into account the gravity due to the shells using israels thin shell formalism , and thus this treatment allows us to go beyond the test particle approximation . the center of mass energy of collision of the shells
is then calculated in a situation analogous to the test particle case and is shown to be bounded above .
however , we find that the energy of a collision between two of constituent particles of the shells at the center of mass frame can exceed the planck energy .
|
the berry phase theory @xcite allowed to generalize the idea of aharonov - bohm effect @xcite on electrons in the electromagnetic potential , to an analogous effect related to a gauge potential , which arises during the adiabatic motion of a quantum system in a parametric space . up to now
a lot of efforts has been directed to understand better and to find an experimental confirmation for the berry phase of electrons like , for example , in the case of electrons moving in a varying magnetization field of the inhomogeneous ferromagnet .
@xcite one of the most intriguing consequences of the berry phase theory is a possibility of the aharonov - bohm - like effect on electrically neutral particles or boson fields.@xcite an example of the adiabatic phase for the polarized light has been investigated by pancharatnam @xcite and berry .
@xcite the other example is the aharonov - bohm effect for the exciton,@xcite which is a bound state of an electron and a hole in semiconductors . here
we consider the effect of the gauge potential and berry phase on the propagation of magnons in textured ferromagnets .
such quasiparticles are usually viewed as the elementary excitations of the ordered homogeneous state of a ferromagnet but they can be also used to classify the excited states near a metastable inhomogeneous magnetic configuration .
these magnons describe the dynamics of weakly excited inhomogeneous ferromagnet .
the dynamics of magnetization in nanomagnets is in the focus of recent activity@xcite because of the importance of this problem for magnetoelectronic applications.@xcite it includes the switching of magnetization by electric current , spin pumping , magnetization reversal in microscopic spin valves , etc .
usually , the magnons play a negative role in the magnetization dynamics limiting the frequency of magnetic reversal , and also leading to the energy dissipation .
however , they can be probably used in the spin transport phenomena like the spin currents of magnetically polarized electrons . here
we study the energy spectrum of spin waves in ferromagnets with a static inhomogeneous magnetization profile , and we demonstrate a possibility of observation of the berry phase in the interference experiments on spin waves in magnetic nanostructures .
recent results of the micromagnetic computer simulation@xcite of such systems demonstrate that the interference of spin waves can be really observed in magnetic nanorings with domain walls .
the equation for spin wave excitations in a general case of arbitrary local frame depending on both coordinate and time , has been found long ago by korenman _
et al_@xcite in the context of local - band theory of itinerant magnetism.@xcite here we use an idea of this method to relate the adiabatic space transformation to the berry phase and to find corresponding properties of the spin waves in a topologically nontrivial inhomogeneous magnetic profile , which is a metastable state of the ferromagnet .
we show that the magnetic anisotropy is a crucial element determining the possibility of observation of the berry phase in real experiments .
we consider the model of a ferromagnet described by the hamiltonian , which includes the exchange interaction , anisotropy , and the interaction with external magnetic field .
it has the following general form @xmath0 , \end{aligned}\ ] ] where @xmath1 is the unit vector oriented along the magnetization @xmath2 at the point @xmath3 , @xmath4 is the constant of exchange interaction , @xmath5 is a function determining the magnetic anisotropy [ correspondingly , it includes a certain number of tensors relating the components of vector @xmath1 ] and the dependence on external field , and @xmath7 is the magnitude of magnetization . due to the condition @xmath8 ,
the model is constrained and belongs to the class of nonlinear @xmath9 models.@xcite the stationary ( saddle point ) solutions for the magnetization vector @xmath10 describing metastable states of the ferromagnet , can be found by minimizing hamiltonian ( 1 ) with the constraint @xmath8 .
it was shown ( see , e.g. , refs . [ ] ) that such metastable states with inhomogeneous magnetization profile are related to the topology of ferromagnetic ordering , and they can include skyrmions , magnetic vortices , and other topological objects
. we will be interested in describing the dynamics of small deviations @xmath11 from a certain metastable profile @xmath10 with a nonuniform magnetization , @xmath12 , @xmath13 .
correspondingly , we assume that the solution of a saddle - point equation describing the state @xmath10 is already known .
we perform a local transformation @xmath14 using the orthogonal transformation matrix @xmath15 . by definition , it determines the rotation of local frame in each point of the space , so that the magnetization in the local frame is oriented along the @xmath16 axis , @xmath17 . then we consider small deviations of magnetization @xmath18 from @xmath19 .
since @xmath18 is small and vectors @xmath19 are oriented along @xmath16 , the vectors @xmath18 lie in the @xmath20-@xmath21 plane . the transformation matrix in eq .
( 2 ) is taken in a general form of orthogonal transformation @xmath22 where @xmath23 , @xmath24 , @xmath25 are the euler angles determining an arbitrary rotation of the coordinate frame , and @xmath26 , @xmath27 , and @xmath28 are the generators of 3d rotations around @xmath20 , @xmath21 and @xmath16 axes .
two rotation parameters ( for definiteness , the angles @xmath24 and @xmath25 ) can be used to define the frame with the @xmath16 axis along the vector @xmath10 . in the absence of anisotropy , the additional rotation to the angle @xmath23 is purely gauge transformation .
however , in a general case of anisotropic system , this rotation allows to choose the local frame in correspondence with the orientation of anisotropy axes .
the hamiltonian of exchange interaction ( the first term in eq .
( 1 ) ) in the rotated frame has the following form @xmath29 where the gauge field @xmath30 is defined by @xmath31 transformation ( 3 ) and gauge potential ( 5 ) are @xmath32 matrices acting on the magnetization vectors .
the matrix @xmath30 can be also presented as @xmath33 where @xmath34 belongs to the adjoint representation of the rotation group .
using ( 3 ) and ( 6 ) , we find the explicit dependence of the gauge potential on the euler angles @xmath35 the magnetic anisotropy described by the second term in the right hand part of ( 1 ) gives after transformation to the local frame a function @xmath36 with correspondingly transformed tensor fields .
here we do not restrict the general consideration of the problem by any specific form of the anisotropy but in the following we consider the most important examples of easy plane and easy axis anisotropy .
the landau - lifshitz equations for the magnetization in the locally transformed frame are @xmath37 where @xmath38 is the unit antisymmetric tensor , and @xmath39 is the covariant derivative . the right - hand part of eq .
( 8) vanishes for the magnetization profile corresponding to a metastable state .
this is seen from the landau - lifshitz equation in the unrotated original frame . in the following
, we will use eq .
( 8) for the small deviations of magnetization from the metastable state .
hence , we will consider in the right part of ( 8) only the terms linear in deviations . using ( 1 ) , ( 4 ) and ( 8) we find the equations for weak magnetic excitations near the metastable state ( spin waves ) @xmath40 \nonumber \\ + \frac{\gamma } { m_0}\ ; \frac{\partial ^2\tilde{\mathcal{f } } } { \partial \tilde{n}_x\ , \partial \tilde{n}_y}\ ; s_x + \ , \frac{\gamma } { m_0}\ ; \frac{\partial ^2\tilde{\mathcal{f } } } { \partial \tilde{n}_y^2}\ ;
s_y % + \delta _ 0\ , s_y \ ; , \end{aligned}\ ] ] @xmath41 \nonumber \\ -\frac{\gamma } { m_0}\ ; \frac{\partial ^2\tilde{\mathcal{f } } } { \partial \tilde{n}_x\ , \partial \tilde{n}_y}\ ; s_y -\ , \frac{\gamma } { m_0}\ ; \frac{\partial ^2\tilde{\mathcal{f } } } { \partial \tilde{n}_x^2}\ ; s_x % -\delta _ 0\ , s_x \ ; , \end{aligned}\ ] ] where @xmath42 is the stiffness . using ( 10 ) and
( 11 ) we can also present the equations for circular components of the spin wave , @xmath43 , @xmath44 s_\pm + \left [ -w({\bf r } ) -ic_s\ , \mathcal{a}^x_i\mathcal{a}^y_i \right . \nonumber \\ \left . + \
, \frac{i\gamma}{m_0}\ ; \frac{\partial ^2\tilde{\mathcal{f } } } { \partial \tilde{n}_x\ , \partial \tilde{n}_y } + \frac{\gamma } { 2m_0}\ ; \frac{\partial ^2\tilde{\mathcal{f } } } { \partial \tilde{n}_x^2 } -\frac{\gamma } { 2m_0}\ ; \frac{\partial ^2\tilde{\mathcal{f } } } { \partial \tilde{n}_y^2 } \right ] s_\mp \ , , \hskip0.3cm\end{aligned}\ ] ] where @xmath45 and @xmath46 are , respectively , the effective potential and a mixing field acting on the spin wave : @xmath47 , \ ] ] @xmath48 .\ ] ] equations ( 12 ) for @xmath49 and @xmath50 are complex conjugate to each other since they both describe the same spin wave with real components @xmath51 and @xmath52 .
we can see that @xmath45 is an effective potential profile for the propagation of spin wave .
due to the terms @xmath53 and @xmath54 in ( 12 ) , the equations for circular components @xmath55 and @xmath56 are coupled even in the absence of anisotropy .
all these terms are of the second order in derivative of the rotation angle , and they are small in the adiabatic limit corresponding to a smooth variation of the magnetization vector @xmath1 .
equations ( 10 ) and ( 11 ) can be solved in the semiclassical approximation .
the condition of its applicability is a smooth variation of gauge potential @xmath34 and fields related to the anisotropy , as well as the external magnetic field , at the wavelength of the spin wave , @xmath57 , where @xmath58 is the wavevector of the spin wave and @xmath59 is the characteristic length of the variation of @xmath34 and @xmath60 ( more exactly , the minimum of the corresponding characteristic lengths ) .
note that the condition of applicability of the semiclassical approximation to solve the spin - wave equations , does not require any smallness of the gauge potential itself .
starting from eqs .
( 10 ) and ( 11 ) , we look for a general semiclassical solution in the form @xmath61 + b\ , \sin \left [ \xi ( { \bf r})-\omega t\right ] , \ ] ] @xmath62 + f\ , \cos \left [ \xi ( { \bf r})-\omega t\right ] , \ ] ] with arbitrary coefficients @xmath4 ,
@xmath63 , @xmath64 , @xmath65 , and a smooth function @xmath66 , so that we can neglect the second derivative of @xmath66 over coordinate @xmath3 . substituting ( 15 ) and ( 16 ) in ( 10 ) and ( 11 ) , we can find four equation for the @xmath4 , @xmath63 , @xmath64 , @xmath65 coefficients .
the solution ( 15 ) , ( 16 ) describes the elliptic spin wave with an arbitrary choice of the axes @xmath20 and @xmath21 , and , generally , with a varying in space orientation of the principal axes of the ellipse .
we can simplify our consideration by choosing the angle @xmath67 at each point of the space in accordance with the orientation of the principal axes . the corresponding equation for @xmath67
can be found from the condition of @xmath68 in eqs .
( 15 ) and ( 16 ) @xmath69 using ( 17 ) and neglecting the terms with derivative of @xmath70 , which are small in the semiclassical approximation , we write the spin - wave equations ( 10 ) and ( 11 ) as @xmath71 \nonumber \\ + \ , \frac{\gamma } { m_0}\ ; \frac{\partial ^2\tilde{\mathcal{f } } } { \partial \tilde{n}_y^2}\ ; s_y\ ; , \hskip0.5cm\end{aligned}\ ] ] @xmath72 \nonumber \\ -\ , \frac{\gamma } { m_0}\ ; \frac{\partial ^2\tilde{\mathcal{f } } } { \partial \tilde{n}_x^2}\ ; s_x\ ; .\hskip0.5cm\end{aligned}\ ] ] note that by fixing the angle @xmath23 in eq .
( 17 ) , we are choosing the gauge , which defines completely the potential @xmath73 .
we do it in spirit of the usual fixing gauge in the wkb approximation .
after substitution of ( 15 ) and ( 16 ) with @xmath68 into ( 18 ) and ( 19 ) , we come to the following equation for the momentum @xmath74 @xmath75 \right .
+ 2p_x\right\ } \left\ { c_s\left [ k_i^2+(\mathcal{a}^z_i)^2-(\mathcal{a}^x_i)^2\right ] + 2p_y\right\ } = 0,\hskip0.1cm\end{aligned}\ ] ] where @xmath76 are the anisotropy parameters .
equation ( 20 ) should be solved for @xmath77 as a function of smooth inhomogeneous field @xmath78 .
this equation does not constraint the orientation of @xmath79 but determines the magnitude of vector @xmath79 for each direction in the momentum space .
let us take vector @xmath79 along an arbitrary direction , defined by a unity vector @xmath80 .
then we can rewrite ( 20 ) as @xmath81 ^2 -\left\ { c_s\left [ k_i^2+(\mathcal{a}^z_i)^2-(\mathcal{a}^x_i)^2\right ] % \right .
\nonumber \\ \left .
+ 2p_x\right\ } \nonumber \\ \times \left\ { c_s\left [ k_i^2+(\mathcal{a}^z_i)^2-(\mathcal{a}^y_i)^2\right ] + 2p_y\right\ } = 0,\hskip0.5cm\end{aligned}\ ] ] and we come to the fourth - order algebraic equation for @xmath82 .
it can be solved numerically , and a resulting dependence of @xmath77 on the gauge field in the integral @xmath83 leads to the berry phase acquired by the spin wave propagating along the contour @xmath84 .
we can find the solution of eq .
( 22 ) analytically in the limit of weak gauge potential @xmath85 , which corresponds to the adiabatic variation of the magnetization direction @xmath1 and also the adiabatic rotation in space of the elliptic trajectory , @xmath86 .
then in the first order of @xmath87 we find @xmath88 ^{1/2 } \nonumber \\ + \ , g_i\ , g_j\ , \mathcal{a}^z_j({\bf r})\ , \left ( 1+p^2/\omega ^2 \right ) ^{-1/2},\end{aligned}\ ] ] where @xmath89 .
using eq .
( 23 ) and taking the vector @xmath80 along the tangent at each point of a closed contour @xmath84 , we find the berry phase @xmath90 as follows from ( 24 ) , the berry phase @xmath91 in the anisotropic system acquires an additional factor @xmath92 depending on the magnetic anisotropy parameter @xmath93 .
the denominator in ( 24 ) has a simple geometrical interpretation .
indeed , the coefficients @xmath4 and @xmath64 in the semiclassical solution ( 15 ) , ( 16 ) are the ellipse parameters , which are related to the anisotropy factor @xmath93 @xmath94 correspondingly , we can relate the parameter @xmath95 in eq . (
24 ) to the geometry parameters of the ellipse @xmath96 where @xmath97 .
using definition ( 7 ) , the berry phase can be finally presented as @xmath98 dr_i.\ ] ] in this expression we extracted a term proportional to @xmath99 , @xmath100 .
this allows to avoid the multivaluedness of berry phase in the absence of anisotropy when @xmath101.@xcite the first term in ( 27 ) is proportional to the total winding number of rotations associated with the angles @xmath23 and @xmath25 , whereas the second term is a spherical angle on @xmath102 , which is the mapping space of the vector field @xmath1 .
the second term in ( 27 ) has a standard interpretation of the berry phase as the magnetic flux penetrating the contour on @xmath102 , when the field is created by monopole at the center of berry sphere .
following this idea , one can interpret the first term in ( 27 ) as the flux created by the magnetic string along the @xmath16 axis , penetrating through the mapping contour on the unit circle.@xcite in accordance with eq .
( 27 ) , this contribution to the berry phase vanishes for isotropic magnetic systems , @xmath101 .
the first term in ( 27 ) is the _ topological _ berry phase ( it depends only on the winding number ) in contrast to the _ geometric _ berry phase of the second term in ( 27).@xcite as follows from ( 24 ) , the effective gauge field for spin waves in the anisotropic system is @xmath103 , and the corresponding topological field acting on the magnons can be calculated as the curvature of connection @xmath104 @xmath105 note that there is a contribution related to the variation in space of the anisotropy parameters [ second term in eq .
( 28 ) ] .
we consider now in more details the motion of elliptic spin wave in the adiabatic regime .
the anisotropy suppresses one of the components @xmath106 or @xmath107 breaking the symmetry with respect to rotations around @xmath16 axis .
correspondingly , there is no gauge invariance @xmath108 and @xmath109 for the circular components , and the motion of magnetization in the spin wave is elliptical . in the adiabatic limit of @xmath85 , the solutions for @xmath106 and @xmath107 are given by eqs .
( 15 ) and ( 16 ) with @xmath68 and the ratio of amplitudes @xmath110 .
thus , we could expect the local invariance to transformations preserving the value of @xmath111 instead of simple rotations in @xmath112 plane . using the fourier transformation of eqs .
( 18 ) and ( 19 ) for @xmath113 we find the following equation for the elliptic components of spin wave , @xmath114 @xmath115 \tilde{s}_+ \nonumber \\ -\frac{a}{2d } \left [ c_s\left\ { ( k_i^2+(\mathcal{a}^z_i)^2\right\}\left ( 1-\frac{d^2}{a^2}\right ) -\frac{pd^2}{a^2}\right ] \tilde{s}_-=0,\hskip0.3cm\end{aligned}\ ] ] and the complex conjugate to ( 29 ) , where @xmath116 , and we determine the @xmath117 from the condition of vanishing of the second bracket in eq . ( 29 )
. this condition determines the ellipticity factor , and we find that it coincides with eq .
( 25 ) in the limit of @xmath85 .
thus , we come to the following equation for the elliptic wave in the gauge field @xmath118 % \right .
\\ \left . -\frac{pd}{2a}\right\ } \tilde{s}_+=0.\hskip0.3cm\end{aligned}\ ] ] this equation is not gauge invariant but in the adiabatic regime , neglecting the difference in small terms of the order of @xmath119 , we can present it as @xmath120 \tilde{s}_+=0.\end{aligned}\ ] ] equation ( 31 ) contains a factor @xmath121 before @xmath122 and formally looks like the equation of motion of a particle moving in the reduced gauge field , which in turn leads to an effective suppression of the berry phase .
the calculation of berry phase using eq .
( 31 ) with the gauge field suppressed by factor @xmath123 leads us again to eq . ( 24 ) . in the absence of anisotropy and in the adiabatic approximation
, the solution of spin wave equations has a simple form .
the equations for circular components ( 12 ) are separated @xmath124 and the corresponding solution is @xmath125 , \end{aligned}\ ] ] with @xmath126 .
the spin wave propagating along a closed contour @xmath84 acquires the berry phase of eq .
( 24 ) with @xmath127 . using eqs .
( 7 ) we can present the topological field ( 28 ) in the absence of anisotropy as @xmath128 it does not depend on the angle @xmath23 , related to the choice of gauge like in the case of electromagnetism . by creating a certain metastable configuration of the magnetization @xmath10 in the ferromagnet , we simulate an effective gauge potential @xmath129 , acting on the spin waves similar to the magnetic field in case of electrons .
in particular , when the averaged in space topological field ( 28 ) is not zero , there arises the landau quantization of the energy spectrum of magnons . in the absence of anisotropy
, we find the quantized spectrum @xmath130 , where @xmath131 means the average in space .
let us consider now the ring geometry of a ferromagnet with a topologically nontrivial metastable magnetization @xmath132 .
it can be , for example , a magnetization vortex ( fig .
1a ) or an even number of domain walls in one branch of the ring like presented in fig . 1b .
such a magnetization profile presents a metastable magnetic state .
let us consider first the case when there is no anisotropy .
if @xmath133 ( adiabatic regime ) , the low - energy magnetic excitations of the metastable state are described by eq .
( 32 ) . due to the presence of gauge potential @xmath134
, there is a phase shift of waves propagating from the point @xmath135 , where the waves are excited , to the observation point @xmath136 ( see fig . 1 ) . the phase shift ( berry phase )
equals to the integral @xmath137 along the ring , and by using stokes theorem can be calculated as the flux @xmath138 of topological field @xmath136 defined in eq .
it can be also presented as the spherical angle enclosed by the mapping of the ring to the circle at the unit sphere @xmath102 . this way we can find the phase shift of @xmath139 and @xmath140 for figs .
1a and 1b , respectively , where an even @xmath141 is the number of domain walls in the right arm of the ring .
for example , in the case of fig .
1b with two domain walls in the right arm , there is no interference of spin waves excited in @xmath135 and coming to the point @xmath136 because the corresponding phase shift is @xmath139 . in the absence of anisotropy
, the interference in the ring can be induced by rotating all magnetic moments from the plane to a certain angle @xmath142 ( the corresponding mapping is presented in fig .
the berry phase associated with the path along the ring will be smaller than @xmath139 . for @xmath143
the berry phase turns out to be @xmath144 .
it means that the experiment with interference of spin waves propagating from @xmath135 to @xmath136 through two different arms of the ring , would result in a complete suppression of the outgoing from @xmath136 spin wave .
physically , it can be realized using the ring with very small easy - plane anisotropy , @xmath145 in a weak external magnetic field along @xmath16 axis .
a similar idea was recently proposed by schtz _
@xcite for the radial orientation of magnetic moments under inhomogeneous magnetic field directed from some point at the axis of the ring .
this magnetic field creates a `` crown '' of magnetic moments , and the corresponding mapping is similar to that presented in fig .
-space @xmath102 ( red contour ) in the case of in - plane vortex magnetization shown in fig.1a ( a ) and for the same geometry with magnetization vector deviating from the plane to the angle @xmath142 ( b ) .
the berry phase is @xmath139 in case ( a ) and @xmath144 for @xmath143 in case ( b ) . ]
however , in the case of nonvanishing easy - plane anisotropy , there is no need to apply magnetic field to provide the interference of spin waves propagating in the geometry of fig .
in this case the berry phase is given by eq .
( 27 ) with @xmath146 , and we obtain the difference in phases for two waves @xmath147 .
thus , the interference of spin waves should be clearly seen for the two - arm geometry with domain walls .
the computer simulation experiments @xciteconfirm this expectation . .
] another possibility to observe the interference of spin waves can be presented by the geometry of a wide ring ( thin - wall cylinder ) like presented in fig .
3 . assuming the easy - plane anisotropy of the ribbon , we obtain the ground state with a homogeneous magnetization along the axis of cylinder .
the anisotropy axis is oriented radially in each point of the cylinder , and the corresponding local frame is shown in the figure as @xmath148 . due to the homogeneous magnetization , we get @xmath149 , and from ( 7 ) we obtain @xmath150 . the components of anisotropy vector are @xmath151 and @xmath152 . using eq .
( 34 ) we find that the condition @xmath153 reduces to @xmath154 , and from ( 34 ) we obtain @xmath155 . correspondingly , the anisotropy parameter @xmath156 , and we obtain the berry phase for the closed contour on the ring @xmath157 , were @xmath158 is the winding number of the contour @xmath84 .
the berry phase of the spin wave propagating in magnetic ring , plays the similar role as the phase of electron wavefunction in the aharonov - bohm effect with magnetic flux penetrating through the ring .
the role of magnetic flux plays a string through the ring.@xcite however , the flux created by the string does not depend on the size or shape of the magnetic ring .
correspondingly , the berry phase associated with the string has the topological origin , which makes it different from the aharonov - bohm effect induced by the magnetic - field flux thorough the conductive ring .
the other example is a ring with uniaxial anisotropy in a homogeneous magnetic field @xmath159 along the axis @xmath16 like presented in fig .
4 . due to the anisotropy and exchange interaction , the magnetization along the ring is oriented like in fig .
4 , creating a certain angle @xmath24 out of the ring plane .
we take the anisotropy function @xmath160 in the form @xmath161 which corresponds to the uniaxial anisotropy along the ring .
the local frame is chosen with the axis @xmath16 along the magnetization at each point , and with the @xmath16-@xmath20 plane tangential to the ring ( parallel to the axis @xmath16 ) . in this case
we find @xmath162 .
the angles determining the orientation of local frame are @xmath163 and @xmath164 .
we can calculate the angle @xmath24 using eqs .
( 1 ) and ( 32 ) with vectors @xmath165 and @xmath159 along the axis @xmath16 . then , using the polar coordinates and the condition that @xmath166 and @xmath167 do not depend on the point along the ring , we find the energy @xmath168 substituting @xmath169 , @xmath170 , we calculate the angle @xmath24 minimizing the energy ( 36 ) @xmath171 for @xmath172 , and @xmath149 for @xmath173 . axis . ] , where @xmath174 . ]
the gauge potential in magnetic ring induces the energy splitting of magnons propagating in the opposite directions.@xcite using eqs .
( 7 ) we find @xmath175 , @xmath176 , and @xmath177 .
it should be noted that the gauge potential is constant along the ring , so that there is no need to use the adiabatic approximation to determine the energy spectrum of magnons . using eqs .
( 10 ) and ( 11 ) after fourier transformation over time @xmath178 and coordinate @xmath179 along the ring , we find @xmath180 \nonumber \\ + \
, i\delta _ 0s_y\ ; , \hskip0.5 cm \nonumber
\\ \omega s_y =- ic_s\left [ k_n^2s_x-2ik_n\mathcal{a}^zs_y+(\mathcal{a}^z)^2s_x\right ] + ips_x \nonumber \\ -i\ , \delta _ 0s_x\ ; , \hskip0.5cm\end{aligned}\ ] ] where @xmath181 .
here the momentum takes discrete values @xmath182 ( @xmath183 ) to provide the periodicity of solution for the spin wave along the ring .
we can use ( 38 ) to find the equation for elliptic components of the spin wave @xmath184 \frac{\eta ^2 + 1}{\eta } \right .
\nonumber \\ \left .
+ \frac{c_s(\mathcal{a}^x)^2}{\eta } -\eta p\right\ } \tilde{s}_+ + \left\ { \left [ c_sk_n^2+c_s(\mathcal{a}^z)^2+\delta _ 0\right ] \frac{\eta ^2 - 1}{\eta } \right
-\ , \frac{c_s(\mathcal{a}^x)^2}{\eta } -\eta p\right\ } \tilde{s}_-\ ; , \hskip0.5cm\end{aligned}\ ] ] where @xmath185 is the ellipticity .
the expression in second curved brackets vanishes for @xmath186 then , using ( 43 ) we find the energy spectrum of magnons in the ring @xmath187 the spectrum is shown in fig . 5 for several first values of @xmath188 as a function of the magnitude of field @xmath136 .
we take the parameters : @xmath189 ( it corresponds to the dipolar shape anisotropy of a magnetic cylinder ) , @xmath190 nm , and @xmath191 . all the curves have a critical point @xmath192 corresponding to the magnetic field , for which the magnetization @xmath193 starts to deviate from the direction with @xmath149 . in view of eq .
( 37 ) , @xmath194 . for @xmath195
the energy of magnons at this point is the soft mode with @xmath196 .
this mode corresponds to a uniform rotation of spins at each point of the ring toward a tangential to the ring direction . in the local frame
it is the uniform rotation corresponding to the state with @xmath195 . for @xmath197 and @xmath198 we can find the spectrum in linear in @xmath136 approximation @xmath199 where @xmath200 .
it shows that the spectrum is degenerate at @xmath201 with respect to the sign of @xmath188 .
the splitting for @xmath202 in linear approximation gives the curve for positive @xmath188 below the one for the same negative in accordance with fig .
5 . at large magnetic field @xmath136 ,
the systematics of levels should be changed .
namely , the lowest energy mode corresponds to the uniform _ global _ deviation of orientation of all spins from the direction along the @xmath16 axis . in the local frame
, it corresponds to the mode with @xmath203 .
thus , it would be natural to label the modes with index @xmath204 , so that the lowest in energy is the spin wave with @xmath205 . in the limit of @xmath206 , each pair of modes @xmath207 is degenerate .
it is clearly seen from eq .
( 41 ) with @xmath208 .
the splitting of these modes for @xmath209 demonstrates the existence of _ topological _ berry phase for the magnons on the ring@xcite because the equilibrium state is the homogeneous magnetization , which leads to the vanishing of _ geometric _ berry phase .
we calculated the berry phase associated with the propagation of magnons in inhomogeneous ferromagnets and mesoscopic structures with topologically nontrivial magnetization profile .
we found that the most important effect is related to the magnetic anisotropy . due to the anisotropy
, the berry phase for magnons is lower than a standard value of the spherical angle on the berry sphere with a monopole .
besides , an additional contribution to the berry phase arises in anisotropic systems , which can be viewed as an effect of the gauge string penetrating through the mapping contour on the unit circle .
thanks the university joseph fourier and laboratory louis nel ( cnrs ) in grenoble for kind hospitality .
this work was supported by fct grant no .
pocti / fis/58746/2004 in portugal , polish state committee for scientific research under grants nos .
pbz / kbn/044/p03/2001 and 2 p03b 053 25 , and also by calouste gulbenkian foundation .
gerrits , h. a. m. van der berg , j. hohfeld , l. br , and th .
rasing , nature * 418 * , 509 ( 2002 ) ; b. heinrich , y. tserkovnyak , g. wolrersdorf , a. brataas , r. urban , and g. e. w. bauer , phys .
lett * 90 * , 187601 ( 2003 ) ; y. tserkovnyak , a. brataas , and g. e. w. bauer , phys . rev .
b * 67 * , 140404 ( 2003 ) ; j. grollier , v. cros , h. jaffrs , a. hamzic , j. m. george , g. faini , j. b. youssef , h. le gall , and a. fert , phys . rev .
b * 67 * , 174402 ( 2003 ) .
k. nielsch , r. b. wehrspohn , j. barthel , j. kirschner , u. gsele , s. f. fischer , and h. kronmller , appl .
. lett . * 79 * , 1360 ( 2001 ) ; k. nielsch , r. b. wehrspohn , j. barthel , j. kirschner , s. f. fischer , h. kronmller , t. schweinbck , d. weiss , and u. gsele , j. magn .
magn . mater . *
249 * , 234 ( 2002 ) .
|
we study the energy spectrum of magnons in a ferromagnet with topologically nontrivial magnetization profile . in the case of inhomogeneous
magnetization corresponding to a metastable state of ferromagnet , the spin - wave equation of motion acquires a gauge potential leading to a berry phase for the magnons propagating along a closed contour .
the effect of magnetic anisotropy is crucial for the berry phase : we show that the anisotropy suppresses its magnitude , which makes the berry phase observable in some cases , similar to the aharonov - bohm effect for electrons .
for example , it can be observed in the interference of spin waves propagating in mesoscopic rings .
we discuss the effect of domain walls on the interference in ferromagnetic rings , and propose some experiments with a certain geometry of magnetization .
we also show that the nonvanishing average topological field acts on the magnons like a uniform magnetic field on electrons .
it leads to the quantization of the magnon spectrum in the topological field
. 0.5 cm
|
a long standing question in the context of magnetic fusion is the impact of 3d shaping on the fundamental properties of the plasma . the two most studied toroidal configurations , stellarators and tokamaks , manifest in general different magnetohydrodynamic ( mhd ) stability and neoclassical confinement .
comparisons of this sort have already been addressed @xcite , and the underlying reasons seem to be well understood .
conversely , the difference in the behavior of turbulent transport is a topic which remains to date largely unanswered : while the tokamak line has been extensively investigated in this regard during the last decades , it is only recently that the stellarator community has started taking sophisticated steps in this direction , due to the difficulty in dealing with helical fields @xcite . in this paper
we exploit the unique flexibility offered by the reversed field pinch ( rfp ) device , thanks to its ability to produce both helical and axisymmetric plasmas in the course of a single discharge .
the rfp is nominally an axisymmetric configuration , which at low plasma current is characterized by a wide spectrum of resonant mhd modes maintaining an overall ( quasi)axisymmetry of the plasma .
interestingly though , the rfp plasma at high currents experiences a transition to a helical state , named single helicity @xcite . depending on the current intensity , the duration of the helical state can be long enough to reach an equilibrium state , which can be described numerically by special codes like vmec @xcite . despite the fact that the stability of the ion temperature gradient ( itg ) mode @xcite has already been studied in the rfp , so far only axisymmetric equilibrium models
have been taken into account @xcite .
the main conclusion from these geometrically simplified investigations is that the itg stability threshold in the rfp is larger than in the tokamak , typically by a factor @xmath0 ( with @xmath1 the major radius and @xmath2 the minor radius of the torus ) .
an explanation comes from a detailed analysis of the parallel dynamics , showing a relevant landau damping of the mode due to the short field connection length @xcite .
nonlinear simulations of itg turbulence have been performed with and without impurities @xcite , showing a relatively low ion heat transport and a significant dimits shift . in this work
we aim at revisiting these findings , considering what the introduction of a helical deformation may cause to itg turbulence .
to do this , we rely on the massively - parallel eulerian gyrokinetic code gene @xcite , applied to the vmec helical / axisymmetric rfp equilibria with the aid of the code gist @xcite . after introducing the mhd equilibria of the rfp in sec .
[ sec : equilibria ] , we present the comparison of the respective itg modes in sec . [
sec : linear ] , to proceed with the itg turbulence and the behavior of zonal flows in axisymmetric and helical systems in sec .
[ sec : nonlinear ] .
we conclude with a short discussion in sec .
[ sec : concl ] .
( bottom ) for the helical ( left ) and axisymmetric ( right ) vmec equilibrium reconstructions .
the grey - shaded areas depict the selected radial domain for the gyrokinetic investigation . ] ( blue ) and the @xmath3 ( red ) flux - tubes , where @xmath4 is the field - label coordinate .
the dot on each tube marks the origin in the parallel direction , @xmath5 . ] on the surface @xmath6 , for the axisymmetric case and for the two stellarator - symmetric tubes , @xmath7 and @xmath3 , for the helical case . ] tube ( left frames ) and axisymmetric ( right frames ) geometry , on the respective @xmath6 surface . here
, @xmath8 is the ion sound speed and @xmath2 is the minor radius . ] as a function of the ballooning angle @xmath9 for the @xmath7 and @xmath3 helical flux tubes on the @xmath6 surface , for binormal wavenumber @xmath10 and ion - temperature gradient @xmath11 ( a ) .
the maxima of the @xmath12 component of the metric tensor ( b ) correspond to the minima of @xmath13 ( with @xmath9 such that @xmath14 is maximum , see text for definitions ) ( c ) , with the structure of the corresponding normalized eigenfunctions in ( d ) . ]
$ ] for the helical @xmath7 tube ( left frames ) and axisymmetric ( right frames ) geometry . ] surface , for different ion temperature gradients , for the @xmath7 and @xmath3 tubes . ] on the @xmath6 surface . ]
surface , for the helical and the axisymmetric tubes .
the linear growth rates are also plotted ( red ) , suggesting the usual nonlinear shift , which is particularly pronounced in the axisymmetric case . ]
space , on the @xmath6 surface , for the helical @xmath7 and the axisymmetric tubes . ]
tube and the axisymmetric one for @xmath15 , on the @xmath6 surface . ]
the rfp configuration is characterized by a low safety factor profile , @xmath16 in the core , which further decreases in the outer region , reaching slightly negative values at the very edge . in case
the plasma can be assumed axisymmetric , the connection length of the field turns out to be @xmath17 , with @xmath18 , which is much lower than in a tokamak ; in particular @xmath19 for @xmath20 , where the field is purely poloidal . in rfx - mod , pushing large plasma currents ( @xmath21 1 ma ) makes the plasma undergo a transition to a helical state , named single helicity , with a single tearing mode saturating well above the others .
axisymmetric and helical rfp plasmas have rather different features in terms of overall transport properties , mhd dynamics , impurity behaviour @xcite .
we mention here only the sharp reduction of magnetic field line stochasticity in the helical states , with the occurrence of transport barriers , at least in the electron heat channel . for this reason , we expect turbulence ( at least temperature - gradient driven ) to play a potential role in such states . from the point of view of mhd , the most relevant parameter differentiating the axisymmetric from the single - helicity state is the @xmath22 profile .
axisymmetric states have a monotonically decreasing @xmath22 as a function of the radius ; for the helical equilibria the @xmath22 profile has a maximum , which statistically corresponds to the maximum electron temperature gradient @xcite .
there has been a considerable effort of the rfx - mod team to describe such equilibria by means of the equilibrium code vmec , and more recently by means of v3fit @xcite , which uses vmec as a solver to determine the equilibrium that best matches the experimental data . in particular
, vmec has been modified to work in the rfp : in a reversed field configuration the toroidal flux @xmath23 can not be used as a radial coordinate in this configuration , as it is not monotonic . for this reason
a new version of the code has been released , which uses the poloidal flux @xmath24 as the radial coordinate . however , since the magnetic field representation in gist makes use of the toroidal flux as a radial coordinate , we prefer to slightly modify the @xmath22 profile in the very edge , keeping it slightly positive .
this approximation does not influence the @xmath22 profile and other equilibrium - related quantities in the radial region where we are going to perform our study , i.e. , the mid - radius region where transport barriers emerge .
thus , in the following , the toroidal flux normalized at the last closed magnetic surface , @xmath25 , @xmath26 $ ] , will play the role of the radial label . in fig .
[ fig : vmec ] we show two equilibrium reconstructions . on the left , the vmec reconstruction is made using v3fit , therefore a minimization procedure is performed on the experimental data .
on the right , the reconstruction is based on an axisymmetric @xmath22 profile @xcite , i.e. , it does not make use of the @xmath27 tearing mode which drives the helical deformation .
we prefer to use two different reconstructions at the same time instead of considering different time instants of a discharge , so as to keep most of the plasma parameters equal .
the radial region where we are performing gyrokinetic simulations is @xmath28 $ ] .
the majority of them are on the @xmath6 surface , where the ( helical ) @xmath22 profile peaks and the magnetic shear is almost vanishing .
the gist code prepares a flux tube domain with the full geometric description in terms of metric coefficients , curvature operators , jacobian , magnetic field strength , and parallel gradient which the gene simulation is based on . in principle , it is possible to obtain either a boozer or a pest coordinate representation of the flux tube .
we will use the latter in the following , as that representation is more directly linked to the vmec @xmath29 coordinates , @xmath30 and @xmath31 being the vmec poloidal angle and the cylindrical toroidal angle , respectively . in particular , gist makes use of the straight - field - line poloidal angle @xmath32 , with @xmath33 the stream - function of vmec .
each point in the flux - tube domain at @xmath34 is described by a triplet @xmath35 , with @xmath36 ( radial coordinate ) , @xmath37 ( binormal ) , @xmath38 ( parallel ) . due to the peculiar helicity of the helical states , @xmath27 , the overall magnetic geometry has a seven - fold symmetry .
two stellarator - symmetric @xcite flux tubes are used for gyrokinetic studies , one centered in the outboard midplane of the @xmath39 section , the other one centered in the outboard midplane at @xmath40 , see fig .
[ fig:3d ] .
the two tubes have @xmath7 and @xmath3 respectively . in fig .
[ fig : gist ] we show some relevant ( normalized ) parameters as a function of the parallel flux - tube coordinate @xmath41 @xcite : the @xmath42 and @xmath43 components of the metric tensor , the normalized magnetic field @xmath44 and its curvature @xmath45 ( with @xmath46 and @xmath47 the normal and geodesic components of the field line curvature , respectively ; negative @xmath48 corresponds to unfavorable curvature causing itg instability ) , the jacobian @xmath49 , and the local magnetic shear @xmath50 , with the dashed lines representing the values of the ( global ) magnetic shear , given by @xmath51 .
the even - parity with respect to @xmath5 reflects in the stellarator symmetry of the tubes .
the other case shown in the figure is the axisymmetric configuration , where , of course , @xmath52 does not play any role .
in rfx - mod , a local estimate of the ion temperature gradient is not available in the plasma core .
ion temperature profiles have been obtained recently from neutral - particle - analyzer data , these profiles being reliable in the edge only , because of the increased neutral absorption probability experienced towards the plasma center @xcite .
the doppler broadening of spectral lines from impurities provides , too , good local estimates of @xmath53 in the outer plasma only , where a ratio @xmath54 is commonly evaluated@xcite .
therefore , we consider the electron temperature @xmath55 as a reference profile in the core . a detailed study of the dynamics of the electron temperature barriers arising during the helical states can be found in ref . .
in particular , the @xmath56 barriers turn out to have logarithmic gradients @xmath57 . our studies on the ion temperature gradient must cover a larger interval . in the previous section
we have seen how the flux - tube domain is properly defined .
the code gene can now solve the system of gyrokinetic equations to investigate the electrostatic itg mode .
since we are emphasizing the geometrical aspects of the instability , we consider for simplicity adiabatic electrons .
in addition , the density gradient @xmath58 is assumed to vanish everywhere .
this is a good approximation in the region @xmath59 , as the experimental density profiles turn out to be essentially flat both in the axisymmetric and in the helical state .
high density gradients do exist in the edge , but that domain is outside the scope of this work .
other assumptions are zero collisionality and plasma @xmath60 . while the ion collisionality @xmath61 does not introduce relevant differences in the linear results ( in rfx - mod experimental conditions , @xmath62 , with @xmath63 ) , assuming vanishing @xmath60
can be more questionable ( in rfx - mod , @xmath64 ) .
as is known , a finite @xmath60 stabilizes the itg mode both in the tokamak and the stellarator , see , e.g. , ref . for a recent study on this topic . in axisymmetric rfps
, the @xmath60 suppression turns out to be moderate with respect to the tokamak @xcite , with itg modes being unstable until @xmath65 . whether this slow quenching is a feature also of a helical rfp is an open issue , left for a future work .
let us consider two cases , the helical equilibrium with flux tube @xmath7 and the axisymmetric one , both on the surface @xmath6 .
the other tube for the helical case ( @xmath3 ) will be considered in a while .
the parallel domain is set to cover @xmath66 $ ] . in the velocity space , the box size in the @xmath67 direction is @xmath68 with 32 grid points , while in the @xmath69 direction @xmath70 with 8 grid points .
these conditions are chosen both for the helical and the axisymmetric equilibria , even though shorter domains in the parallel direction are typically required for the axisymmetric equilibria .
increasing the ion temperature gradient @xmath71 , all the other parameters fixed , the linear spectra in fig .
[ fig : linear_spec ] are obtained ; the introduction of a helical deformation strongly destabilizes itg modes . to understand this feature ,
let us recall that in the ballooning representation @xcite perturbed quantities vary as @xmath72 , where @xmath73 is slowly varying in the parallel direction and @xmath74 describes the rapid variation across the field line .
the perpendicular wave vector is @xmath75 , the components @xmath76 and @xmath77 being constants .
given @xmath78 , with @xmath9 the ballooning angle , it follows that @xmath79 ( or , in normalized units , @xmath80 , with @xmath81 magnetic shear ) .
thus @xmath82 .
for @xmath83 , which is the case of fig .
[ fig : linear_spec ] ( this choice will be justified later ) , the wave vector is reduced to @xmath84 . as can be seen in fig .
[ fig : gist ] , the metric component @xmath85 is much smaller in the helical case than in the axisymmetric case .
this implies a reduced finite larmor radius ( flr ) suppression of the itg mode via the bessel function @xmath86 in the linear gyrokinetic equation . even if the helical curvature is not so unfavorable as the axisymmetric one about one half where the mode peaks the reduced flr suppression justifies the higher growth rate instabilities in the helical case .
it is useful to remark that the @xmath87 metric component , which does not enter the linear gyrokinetic equation if @xmath83 , determines the magnitude of @xmath85 via the relation @xmath88 ( which is a direct consequence of the clebsch representation of the magnetic field @xmath89 ) . where the itg mode peaks , @xmath90 $ ] , we have @xmath91 , so that @xmath92 .
the different weight of the flr suppression is not the only mechanism acting differently on the two geometries under consideration .
as already mentioned in the previous section , the connection length for an axisymmetric rfp is @xmath93 .
such a low connection length is responsible for the mode suppression via landau damping at low wavenumbers , and for the high itg stability threshold in axisymmetric geometry @xcite . in a helical rfp
, @xmath94 is not the length of a field line corresponding to a poloidal turn .
contrary to the stellarator case @xcite , where @xmath94 is usually considered as the distance between the ( stabilizing ) spikes of @xmath95 , in the rfp case the local shear has a more oscillatory behaviour ( see fig . [
fig : gist ] ) , so it makes sense to consider for @xmath94 the distance along the field line between two consecutive ( destabilizing ) peaks of @xmath87 .
as such , we typically have @xmath96 , thus @xmath97 .
this local property makes the mode suppression via landau damping less effective in the helical case .
it is known that the ( global ) magnetic shear @xmath81 influences itg mode stability . since the @xmath22 profile is essentially flat in the helical case for @xmath98 , this provides mode destabilization in the inner region . however , this effect is only marginal with respect to the others mentioned above : forcing axisymmetry with a helical @xmath22 profile yields a slightly higher growth rate with respect to the axisymmetric @xmath22 , without modifying the wavenumber range where destabilization occurs , @xmath99 , see fig .
[ fig : linear_spec](b ) .
we can now compare the two helical tubes for linear itg stability .
starting from the @xmath100 spectrum of the @xmath7 tube , we can fix the binormal wavenumber such that the growth rate is maximum , @xmath101 in our case .
varying @xmath9 ( or , equivalently , @xmath102 ) , it is confirmed that the maximum @xmath103 corresponds to @xmath83 ( i.e. , @xmath104 ) , see fig . [
fig : ballooning]-a . for the @xmath3 tube , @xmath103 is maximum for a ballooning angle @xmath105 , which corresponds to the maximum of @xmath87 along the field line , fig .
[ fig : ballooning]-a / b
. interestingly , the highest growth rate is approximately the same in the two flux tubes .
the dependence of @xmath106 on @xmath41 in the two flux tubes is shown in fig .
[ fig : ballooning]-c , where the value of @xmath9 ( or , equivalently , of @xmath102 ) is set such that @xmath14 is maximum . for the two tubes the lowest
@xmath106 is approximately the same . for both tubes of the helical configuration a smaller flr suppression
is observed compared to the axisymmetric case . for these parameters ,
the structure of the normalized electrostatic potential as a function of the parallel coordinate @xmath41 is shown in frame ( d ) .
the itg mode growth rate as a function of the ballooning angle @xmath9 is shown for several neighboring flux tubes in fig .
[ fig : ballooning3d ] .
it is evident that the modes have a `` helical ballooning '' structure , being large in correspondence to the peaks of @xmath107 , i.e. , along the external ridge of the helical structure .
heuristically , the compression of the magnetic surfaces enhances the local temperature gradients , with a consequent growing instability .
this is a common feature to any helical plasma configuration , not only the rfp .
the last piece of the linear analysis on itg stability refers to the radial dependence of the growth rates , fig .
[ fig : linear_s ] .
we perform this study for @xmath28 $ ] , this interval fully including the region where the transport barrier usually arises .
both geometries show a larger growth rate in the core than in the edge .
while in the axisymmetric case the growth rate monotonically increases as @xmath108 , in the helical case it is larger for @xmath109 , i.e. , in the proximity of the maximum of @xmath22 .
the radial domain of largest growth rate is characterized , in both geometries , by an overall increased @xmath87 , thus by a decreased @xmath85 and @xmath13 , and by a decreased global magnetic shear @xmath110 ( see fig . 4 of ref . for an analogous study with gs2 ) . again , with respect to the axisymmetric case , the helical one clearly yields higher growth rates in the core , where the two sets of geometric coefficients largely deviate .
conversely , at @xmath111 the two configurations become equivalent in terms of itg stability . the footprint of the helical core is rapidly vanishing in the outer region : the peaks in @xmath87 are no longer present and the curvature starts to have a tokamak - like oscillating behaviour .
itg modes are strongly stabilized by the ( negative ) magnetic shear , which is rapidly increasing in absolute value towards the edge .
having completed the analysis of the itg mode stability for the helical and axisymmetric rfps , we turn to the effect of the geometry on the turbulence .
it is well known that , apart from the strength of the instability itself , itg turbulence is largely regulated by the zonal flows .
therefore , again focusing on the different geometric features , in this section we will also provide results on the linear zonal flow response for each configuration under study .
we provide some details concerning the setup of the nonlinear simulations . for the @xmath7 tube on the @xmath6 surface ( where @xmath112 and @xmath113 ) , the simulations are made with the following discretization in the 5-dimensional space : @xmath114 , with @xmath115 , @xmath116 , and @xmath66 $ ] . for the @xmath3 tube ,
the @xmath41 domain must be larger , so as to fully include the peaks in @xmath12 , @xmath117 $ ] , see fig .
[ fig : ballooning ] . accordingly , in order to allow for a full @xmath102 spectrum resolution ( not centered at @xmath104 ) a larger number of @xmath118 grid points is used , namely @xmath119 , with @xmath120 .
convergence tests have shown that this phase space resolution is adequate .
as one can see , the most evident difference with a `` standard '' ( axisymmetric ) discretization for itg turbulence simulations is the large box size in the parallel direction : this choice is of central importance , so as to capture all the geometric details , especially for the @xmath3 tube . due to the length of the @xmath41 domain and to the vanishing collisionality , numerical dissipations providing damping in the parallel direction and mimicking diffusion in velocity space have been included by means of hyper - diffusion terms in the @xmath121 subspace . in the axisymmetric case
the numerical setup is less demanding .
for the box size on the @xmath6 surface ( where @xmath122 and @xmath123 ) we set @xmath124 , with @xmath125 , and @xmath90 $ ] .
of course the box size , both in the helical and in the axisymmetric case , may differ from surface to surface , depending on the value of the magnetic shear @xmath126 . with this simulation setup the time history of the ion heat flux @xmath127 for the helical geometry
is shown in fig .
[ fig : qi_t ] for several ion temperature gradients . for two of them ( @xmath128 and 5 )
we show the time trace for both the @xmath7 and the @xmath3 tube .
for the two tubes the same level of saturated turbulence is observed . as a consequence , the two tubes , as in the linear case ,
can be considered equivalent in terms of physical results , the ion heat flux in this case .
however we remark that all the physical fluctuating quantities , e.g. the electrostatic potential @xmath31 , the density @xmath129 , the parallel and perpendicular temperature @xmath130 and @xmath131 , exhibit a clear modulation in the @xmath41 direction .
for instance , the electrostatic potential is peaked where @xmath12 peaks , see fig .
[ fig : phi_z ] . in this light , from now on we can restrict our study to the @xmath7 flux tube .
performing simulations for several ion temperature gradients allows us to compare linear and nonlinear trends , and in particular the temperature gradient threshold in the two cases . in fig .
[ fig : qi_lti ] we focus again on the half - flux surface @xmath6 .
while the linear growth rate ( dotted lines ) already suggests a sharp distinction between the two geometries , in a nonlinear environment their difference is even more pronounced ( solid lines ) . in both cases
a dimits shift@xcite exists , and the axisymmetric configuration requires ion temperature gradients which must be @xmath132 times larger than the linear threshold for an effective itg turbulence to take place . a less pronounced upshift occurs in the helical case . as already mentioned , these gradients in rfx - mod can be compared , as a reference , to the electron temperature profiles only , which are characterized by @xmath133 in the core of the helical geometry , and lower in the axisymmetric one .
our conclusion so far is that , for an axisymmetric rfp , itg turbulence is not likely to be a concern from the point of view of particle / heat transport , at least away from the edge .
high current helical states , on the contrary , are more prone to itg instabilities and present a high level of ion heat flux . besides the difference in the linear spectrum , discussed in the previous section , in fig .
[ fig : nl_spec ] we show the wavenumber spectra of the ion heat flux in the @xmath134 space , averaged over the remaining coordinates and time .
the helical and axisymmetric @xmath135 values are chosen to provide approximately the same ion heat flux , @xmath136 , see fig .
[ fig : qi_lti ] . in analogy with the linear result , the helical @xmath137 spectrum peaks at smaller scales than the axisymmetric one , a feature which , based on quasi - linear estimates , is however not enough to compensate for the much larger growth rates .
we turn our attention to the effect of zonal flows ( zf ) @xcite , whose role in controlling itg turbulence in tokamaks is already established . in addition , in the case of non - axisymmetric configurations , it has been recently shown @xcite , that although the nature of zf can be entirely different compared to tokamaks , as well as among different stellarator designs , their impact is still measurable and beneficial . in fig .
[ fig : zf ] , we calculate the linear zf response for the rfx configuration having selected the normalized radial wavenumber equal to @xmath138 .
interestingly , oscillations similar to the ones seen in the other stellarators are also here observed , which are attributed to the radial drift of locally trapped particles , and the residual level takes relatively small values . in this situation
, it is suggested that the regulation of itg turbulence takes place during the dynamical state of zf evolution and is mildly , if at all , affected by the residual level . in the same figure ,
we show for comparison the linear zf response for the axisymmetric rfp surface . due to the small safety factor and to the toroidal symmetry of the system ,
the residual turns out to be unusually large .
its value is in agreement with the rosenbluth - hinton estimate @xmath139 for the undamped zonal flow @xcite , which for the case shown in the figure yields @xmath140 .
coming back to the nonlinear simulations , we conclude with the snapshots of the electrostatic potential and of the density fluctuations in the @xmath141 space in fig .
[ fig : snap ] , for the two geometries under consideration ( the respective @xmath135 gradients are those of fig . [
fig : nl_spec ] ) .
the colour contours confirm the enhanced generation of the zonal flows in the axisymmetric configuration .
while a helical core has generally beneficial consequences on the rfp plasma performance , we have shown that it imposes an unfavorable effect in terms of itg stability and turbulent transport .
focusing on the role of geometry , thus simplifying the physical problem ( we have assumed adiabatic electron response , vanishing plasma @xmath60 and collisionality , flat density profile ) , it turns out that itg modes are localized in the proximity of the peaks of @xmath107 , i.e. where the magnetic surface compression is higher . here
the local temperature gradients become larger , with a consequent growing instability . from the point of view of turbulence , the maxima of @xmath107 reflect in local maxima of turbulent fluctuations , though turbulence turns out to be quite uniformly spread along the flux tubes .
with respect to the linear threshold , a moderate nonlinear upshift occurs , with the ion heat flux rapidly increasing with the ion temperature gradient and a consequent high stiffness of the temperature profile . on the contrary , for an axisymmetric rfp ,
already more linearly stable to itg modes , the nonlinear shift is much larger .
the different behaviour is confirmed by the very high zonal flow residual level in the axisymmetric case , while helical states exhibit a lower zonal flow residual and a stellarator - like oscillating behaviour of the zonal potential .
itg turbulence is consequently expected to be a scarce contributor to particle transport in an axisymmetric rfp , at least without impurities .
the inclusion of more realistic parameters of the plasma and a comparison with experiment are the steps to face henceforth .
40 p. helander , c. d. beidler , t. m. bird , m. drevlak , y. feng , r. hatzky , f. jenko , r. kleiber , j. h. e. proll , y. turkin , p. xanthopoulos , plasma phys .
fusion * 54 * , 124009 ( 2012 ) .
m. nunami , t .- h .
watanabe , h. sugama , k. tanaka , phys .
plasmas * 19 * , 042504 ( 2012 ) .
p. xanthopoulos , h. e. mynick , p. helander , y. turkin , g. g. plunk , f. jenko , t. grler , d. told , t. bird , and j. h. e. proll , phys .
lett . * 113 * , 155001 ( 2014 ) .
l. carraro , p. innocente , f. auriemma , r. cavazzana , a. fassina , p. franz , m. gobbin , i. predebon , a. ruzzon , g. spizzo , d. terranova , t. bolzonella , a. canton , s. dal bello , l. grando , r. lorenzini , l. marrelli , e. martines , m. e. puiatti , p. scarin , a. soppelsa , m. valisa , l. zanotto , m. zuin , nucl .
fusion * 53 * , 073048 ( 2013 ) .
s. p. hirshman and j. c. whitson phys .
fluids * 26 * , 3554 ( 1983 ) .
f. romanelli , phys .
fluids b * 1 * , 1018 ( 1989 ) . w. horton , rev .
phys . * 71 * , 735 ( 1999 ) .
s. c. guo , phys .
plasmas * 15 * , 122510 ( 2008 ) .
i. predebon , c. angioni , and s.c .
guo , phys .
plasmas * 17 * , 012304 ( 2010 ) .
f. sattin , x. garbet and s. c. guo , plasma phys .
fusion * 52 * , 105002 ( 2010 ) .
v. tangri , p. w. terry and r. e. waltz , phys .
plasmas * 18 * , 052310 ( 2011 ) . d. carmody , m. j. pueschel and p. w. terry , phys .
plasmas * 20 * , 052110 ( 2013 ) .
i. predebon , l. carraro and c. angioni , plasma phys .
fusion * 53 * , 125009 ( 2011 ) .
f. jenko , w. dorland , m. kotschenreuther , and b. n. rogers , phys .
plasmas * 7 * , 1904 ( 2000 ) .
p. xanthopoulos , w. a. cooper , f. jenko , y. turkin , a. runov , j. geiger , phys .
plasmas * 16 * , 082303 ( 2009 ) . m. gobbin , d. bonfiglio , d. f. escande , a. fassina , l. marrelli , a. alfier , e. martines , b. momo , d. terranova , phys . rev
. lett . * 106 * , 025001 ( 2011 ) .
d. terranova , d. bonfiglio , a. h. boozer , a. w. cooper , m. gobbin , s. p.
hirshman , r. lorenzini , l. marrelli , e. martines , b. momo , n. pomphrey , i. predebon , r. sanchez , g. spizzo , m. agostini , a. alfier , l. apolloni , f. auriemma , m. baruzzo , t. bolzonella , f. bonomo , m. brombin , a. canton , s. cappello , l. carraro , r. cavazzana , s. dal bello , r. delogu , g. de masi , m. drevlak , a. fassina , a. ferro , p. franz , e. gaio , e. gazza , l. giudicotti , l. grando , s. c. guo , p. innocente , d. lopez - bruna , g. manduchi , g. marchiori , p. martin , s. martini , s. menmuir , s. munaretto , l. novello , r. paccagnella , r. pasqualotto , g. v. pereverzev , r. piovan , p. piovesan , l. piron , m. e. puiatti , m. recchia , f. sattin , p. scarin , g. serianni , a. soppelsa , s. spagnolo , m. spolaore , c. taliercio , m. valisa , n. vianello , z. wang , a. zamengo , b. zaniol , l. zanotto , p. zanca , m. zuin , plasma phys .
fusion * 52 * , 124023 ( 2010 ) .
d. terranova , l. marrelli , j. d. hanson , s. p. hirshman , m. cianciosa , p. franz , nucl .
fusion * 53 * , 113014 ( 2013 ) .
p. zanca and d. terranova , plasma phys .
fusion * 46 * , 1115 ( 2004 ) .
r. l. dewar , s. r. hudson , physica d * 112 * , 275 ( 1998 ) .
f. auriemma , r. lorenzini , m. agostini , l. carraro , g. de masi , a. fassina , m. gobbin , e. martines , p. innocente , p. scarin , w. schneider , m. zuin , nucl .
fusion * 55 * , 043010 ( 2015 ) .
l. carraro , a. alfier , f. bonomo , a. fassina , m. gobbin , r. lorenzini , p. piovesan , m. e. puiatti , g. spizzo , d. terranova , m. valisa , m. zuin , a. canton , p. franz , p. innocente , r. pasqualotto , f. auriemma , s. cappello , s. c. guo , l. marrelli , e. martines , m. spolaore , l. zanotto , nucl .
fusion * 49 * , 055009 ( 2009 ) .
m. gobbin , p. franz , r. lorenzini , i. predebon , a. ruzzon , a. fassina , l. marrelli , b. momo , d terranova , plasma phys .
fusion * 55 * , 105010 ( 2013 ) .
a. ishizawa , s. maeyama , t .- h .
watanabe , h. sugama , n. nakajima , nucl .
fusion * 53 * , 053007 ( 2013 ) .
r. l. dewar and a. h. glasser , phys .
fluids * 26 * , 3038 ( 1983 ) .
g. g. plunk , p. helander , p. xanthopoulos , j. w. connor , phys .
plasmas * 21 * , 032112 ( 2014 ) .
a. m. dimits , g. bateman , m. a. beer , b. i. cohen , w. dorland , g. w. hammett , c. kim , j. e. kinsey , m. kotschenreuther , a. h. kritz , l. l. lao , j. mandrekas , w. m. nevins , s. e. parker , a. j. redd , d. e. shumaker , r. sydora , j. weiland , phys .
plasmas * 7 * , 969 ( 2000 ) .
h. diamond , s - i .
itoh , k. itoh , t. s. hahm , plasma phys .
controlled fusion * 47 * , r35-r161 ( 2005 ) .
p. xanthopoulos , a. mischchenko , p. helander , h. sugama and t .- h .
watanabe , phys .
. lett . * 107 * , 245002 ( 2011 ) .
m. n. rosenbluth and f. l. hinton , phys .
80 * , 724 ( 1998 ) . h. sugama , t .- h .
watanabe , phys .
plasmas * 13 * , 012501 ( 2006 ) .
|
turbulence induced by the ion temperature gradient ( itg ) is investigated in the helical and axisymmetric plasma states of a reversed field pinch device by means of gyrokinetic calculations .
the two magnetic configurations are systematically compared , both linearly and nonlinearly , in order to evaluate the impact of the geometry on the instability and its ensuing transport , as well as on the production of zonal flows . despite its enhanced confinement ,
the high - current helical state demonstrates a lower itg stability threshold compared to the axisymmetric state , and itg turbulence is expected to become an important contributor to the total heat transport .
|
the motion of test bodies in the kerr metric of a rotating black hole has been studied for almost half a century ( see , e.g. , @xcite ) .
much of the more recent work is motivated by the need to understand radiative inspirals into a kerr black hole as sources of gravitational waves for future detector experiments .
examples of recent work include an action - angle formalism @xcite , a frequency - domain method for computing functionals of the orbit ( such as the gravitational perturbation from an orbiting test particle ) @xcite , a system for classifying kerr orbits @xcite , and an analytic method for solving the geodesic equations of motion @xcite .
timelike geodesics of the kerr geometry are completely integrable .
they admit three nontrivial constants of motion ( `` first integrals '' ) , each associated with a killing field of the kerr background : the time - translation and rotational killing vectors give rise to conserved ( specific ) orbital energy @xmath0 and azimuthal angular momentum @xmath1 , and the second - rank killing tensor discovered by carter @xcite gives rise to what is known as the carter constant , @xmath2 .
up to initial conditions , these three constants of motion uniquely label all timelike geodesics of the kerr geometry . this paper is concerned with the family of _ bound _ geodesic orbits .
each bound orbit is confined to the interior of a compact spatial torus given by @xmath3 and @xmath4 , where hereafter @xmath5 are boyer - lindquist ( bl ) coordinates , @xmath6 are two radial turning points ( `` periastron '' and `` apastron '' , respectively ) , and @xmath7 are two longitudinal turning points .
generically , the motion is ergodic , in the sense that a generic orbit will pass arbitrarily close to any point on the torus within a finite time @xmath8 ( exceptional are `` resonant '' orbits , mentioned briefly below ) .
the triplet @xmath9 provides an alternative parametrization of bound geodesics , which is in a one - to - one correspondence with that of @xmath10 .
is one - to - one can be establishing in the following way .
we first note that schmidt @xcite provides formula for @xmath10 in terms of @xmath11 , and that there is a bijection between @xmath12 ( straightforward to see from eqs . and their inverse ) .
furthermore eqs . and
imply that @xmath13 , @xmath14 and @xmath15 ( and hence @xmath16 ) are given uniquely in terms of @xmath17 .
the existence of these relations asserts that the original mapping is one - to - one . ] generically , bound orbits are triperiodic , with three frequencies @xmath18 , @xmath19 and @xmath20 associated with the motions in the radial , longitudinal and azimuthal directions , respectively .
of these , @xmath18 and @xmath19 are `` libration''-type frequencies , defined from the ( average ) radial and longitudinal periods , while @xmath20 is a `` rotation''-type frequency , describing the average rate at which the bl azimuthal phase @xmath21 accumulates in time .
we define the above frequencies with respect to bl time @xmath8 ; this is useful for many purposes , because @xmath8 is also the proper time of an asymptotically far static observer ( e.g. , a gravitational - wave detector ) .
it is important to note that , in general , the orbital radius @xmath22 and polar angle @xmath23 of a given orbit are _ not _ ( separately ) periodic functions of @xmath8 : the @xmath8-interval between successive periastron passages is not constant , and the @xmath8-interval between successive @xmath24 passages is not constant either .
there is a choice of a time variable ( the so - called `` mino time''see sec .
[ sec : kerr_freqs ] below ) in terms of which the radial and longitudinal motions completely separate and become precisely periodic .
however , in terms of bl time @xmath8 , the orbital periodicity can generally only be defined through an infinite time average ( or , equivalently , through an average over the orbital torus @xcite ) .
we shall define the bl - time frequencies more precisely below , following schmidt @xcite and drasco and hughes @xcite the above general description simplifies in several special cases . if the ratio @xmath25 is a rational number ( `` resonant orbits '' ) , then the trajectory traced by the orbit in the @xmath22@xmath23 plane is closed ( with a finite @xmath8-period ) , and the ergodicity property
if the orbit is equatorial ( @xmath26 ) , then @xmath27 loses its meaning , and the orbit becomes biperiodic with frequencies @xmath18 and @xmath28 ; in this case the radial motion is strictly periodic , with a radial period @xmath29 .
similarly , if the orbit is circular ( @xmath30 ) , then @xmath31 loses its meaning , the orbit becomes biperiodic with frequencies @xmath27 and @xmath28 , and the longitudinal motion is strictly periodic with period @xmath32 .
( orbits that are both equatorial and circular are singly periodic with frequency @xmath28 . ) finally , in the special case of a schwarzschild black hole , one can always set up the bl system so that the orbit is equatorial and biperiodic with frequencies @xmath18 and @xmath28 .
the purpose of this article is to challenge the commonly held notion ( see , e.g. , @xcite ) that the trio of frequencies @xmath33 provides a good parametrization of generic bound geodesics in kerr , i.e. , one which is in a one - to - one correspondence with @xmath10 or @xmath9 .
we show that this is not the case : there are infinitely many pairs of `` isofrequency '' orbits , which are physically distinct ( i.e. , have different @xmath10 values ) and yet they share the same values of @xmath33 .
this point was already made briefly by two of us in appendix a of @xcite in reference to a schwarzschild black hole ( where orbits are biperiodic , and two isofrequency orbits share the same values of @xmath28 and @xmath18 ) .
here we first revisit the schwarzschild problem to provide a further illumination of this phenomenon , and then extend the analysis to the kerr case , showing that isofrequency pairing occurs even among triperiodic orbits .
we shall on occasion refer to a pair of isofrequency orbits as `` synchronous '' , because the phases of such orbits remain synchronized in an average sense .
for example , two equatorial isofrequency orbits that pass through their periastra simultaneously at @xmath34 will reach their next periastra at the same time and with the same azimuthal phase ; they will have experienced an identical amount of periastron advance .
although such orbits go `` in and out of phase '' between periastron passages , their phase remains synchronized `` on average '' .
we will present some graphics to illustrate this behavior . throughout this article
we use geometric units such that the gravitational constant and the speed of light are both equal to unity .
we denote the black hole s mass and spin by @xmath35 and @xmath36 , respectively .
we use an over - tilde to denote adimensionalization using @xmath35 ; for example , @xmath37 and @xmath38 .
we adopt a convention whereby @xmath39 and @xmath40 correspond to prograde and retrograde orbits , respectively , with @xmath1 always positive .
we use the term `` orbit '' synonymously with `` timelike geodesic orbit '' . in sec .
[ sec : schwarzschild ] we consider ( biperiodic ) synchronous orbits in schwarzschild geometry ( @xmath41 ) .
we delineate the region in the parameter space where such orbits occur , and also provide an intuitive explanation as to why isofrequency pairing must occur . in sec .
[ sec : kerr ] we generalize our discussion to the kerr case , where we consider first equatorial orbits and then generic , triperiodic orbits .
the radial motion of geodesic test particles in the equatorial plane of a schwarzschild black hole satisfies @xmath42 where a dot denotes differentiation with respect to proper - time , and @xmath43 is an effective potential for the radial motion .
bound orbits exist for @xmath44 with @xmath45 . for each @xmath46 in this range , @xmath47 has three real roots , and motion is allowed between the second largest and largest of these , which we label @xmath48 and @xmath49 , respectively .
a convenient alternative parametrization of bound orbits is provided by the pair of values @xmath50 defined through @xmath51 which are relativistic generalizations of semi - latus rectum and eccentricity , respectively @xcite .
this parametrization is in a one - to - one correspondence with that of @xmath46 .
explicitly , @xmath52 which can be inverted ( for real @xmath53 ) to give unique expressions for @xmath54 and @xmath55 . in the @xmath56 space ,
bound orbits span the range @xmath57 with @xmath58 .
the boundary @xmath59 ( `` separatrix '' ) separates between stable and unstable orbits in the @xmath56 space @xcite .
the @xmath60 terminus of the separatrix curve is known as the innermost stable circular orbit ( isco ) .
the existence of a separatrix is one of the salient features of motion in black hole spacetimes , and it marks a major qualitative departure from newtonian dynamics .
as we shall see , the occurrence of isofrequency pairing of orbits is intimately related to the existence of a separatrix .
the function @xmath61 is periodic with ( @xmath8-)period @xmath62 .
following darwin @xcite , it is convenient to introduce the `` relativistic anomaly '' parameter @xmath63 , which is related to @xmath8 via @xmath64^{1/2}(p-6 - 2e\cos\chi)^{-1/2}}{(p-2 - 2e\cos\chi)(1+e\cos\chi)^2}\ , , \hskip3 mm \label{eq : dt_dchi}\ ] ] and in terms of which the radial motion is given simply by @xmath65 ( taking @xmath66 at a periastron passage ) .
the radial period can then be computed via @xmath67 with associated radial frequency @xmath68 the _ azimuthal _ frequency of the orbit is defined as the average of @xmath69 ( with respect to @xmath8 ) over a complete radial period : @xmath70 where @xmath71 is the azimuthal phase accumulated over time interval @xmath62 .
the latter can be computed via @xmath72 where @xmath73 and @xmath74 is the complete elliptic integral of the first kind . at the separatrix limit ,
@xmath75 , both @xmath71 and @xmath62 diverge at a similar rate [ see eqs .
( [ deltaphiasy ] ) and ( [ trasy ] ) below ] , so that @xmath76 while @xmath20 attains a finite value [ @xmath77 , corresponding to the frequency of the unstable circular orbit of radius @xmath78 .
this gives rise to the well known `` zoom - whirl '' behavior @xcite : orbits with @xmath79 can `` whirl '' around the black hole many times near the periastron before `` zooming '' back out towards the apastron .
as pointed out in ref .
@xcite , the jacobian matrix of the transformation @xmath80 turns out to be singular along a certain curve in the parameter space , well _ outside _ the separatrix .
this indicates that the transformation is not bijective . to see this most
clearly it is instructive to move to a new orbital parametrization given by the pair @xmath81 .
this reparametrization is admissible because ( i ) as argued above , the original parametrization @xmath56 is a good one , and ( ii ) as can be easily checked , @xmath28 is a monotonically decreasing function of @xmath82 for any fixed @xmath83 .
our argument now follows from examining the structure of the @xmath84 contour lines in the @xmath81 plane , as shown in figure [ fig : schwarzschild_omega_phi_e ] .
the key feature here is that some @xmath84 contours have vertical tangents ( the locus of which is shown by the dashed black line in the figure ) .
each of these contour lines is intersected _ twice _ by vertical lines just right of the vertical tangent .
but vertical lines are also @xmath85 contours , and so the two intersections mark a pair of isofrequency orbits .
( any two such isofrequency orbits are clearly physically distinct : they have different eccentricities . ) in fig .
[ fig : schwarzschild_sample_pair ] we show , superimposed , the orbital trajectories of a sample pair of isofrequency orbits of rather different eccentricities .
the radial and azimuthal motions of these two orbits are plotted in fig .
[ fig : schwarzschild_example_r_phi ] .
since the rate of relativistic periastron advance depends only on the frequency ratio @xmath86 , two isofrequency orbits will exhibit the same rate of advance .
this means that their phase remains `` synchronized '' on average , a behavior illustrated in the figures .
parameter space for bound geodesic orbits in schwarzschild geometry .
bound orbits are confined to the region right of the curve marked _
separatrix_. thin ( blue ) curves are contour lines of constant @xmath18 . the marginal contour line @xmath87 is shown as a thick ( red ) line .
@xmath18 takes its greatest value at the point marked @xmath88 , representing a ( slightly perturbed ) circular orbit of radius @xmath89 .
the dotted ( black ) line shows the curve along which the jacobian matrix of the transformation @xmath90 becomes singular .
the singular curve intersects the @xmath91 axis at @xmath92 , corresponding to a circular orbit of radius @xmath93 .
any vertical ( @xmath94 ) line left of @xmath92 intersects some @xmath95 contours _ twice_. each pair of intersections identifies a pair of isofrequency orbits ; a sample pair is marked in the plot .
each and every orbit between the separatrix and the singular curve has an isofrequency dual between the singular curve and the dashed ( green ) curve marked cod ( for _ circular - orbit duals _ ) . the cod is the locus of all orbits dual to circular orbits of radius @xmath22 with @xmath96 . ] before giving a more detailed analysis , let us remark on the practicalities of producing the contour map of fig .
[ fig : schwarzschild_omega_phi_e ] .
the relation @xmath97 is not known analytically , so we resort to a numerical calculation : first , for a given @xmath83 , we numerically invert the relation @xmath98 [ eq .
( [ eq : schwarzschild_phi_frequency ] ) ] to find @xmath99
. then we use eq . to obtain @xmath100 .
much of the interesting portion of the parameter space for our purpose lies very near the separatrix , where it becomes numerically challenging to evaluate the divergent quantities @xmath101 and their ratio in eq .
( [ eq : schwarzschild_phi_frequency ] ) . in this problematic domain
we instead use the near - separatrix analytic expansions @xcite @xmath102 + \mathcal{o}(\epsilon\log\epsilon)\ , .
\label{trasy}\end{aligned}\ ] ] here , the integral @xmath103 , with @xmath104^{1/2 } \nonumber\\ & - & 3+e-\frac{1}{4}(7e-3)(1+\cos\chi)\ , , \end{aligned}\ ] ] is easily evaluated numerically . , and orbit 2 ( blue , square markers ) has parameters @xmath105 @xmath106 .
both share the same orbital frequencies , @xmath107 .
the orbital period of both orbits is @xmath108 and each accumulates @xmath109 radians during that period .
both orbits start at their periastron marker ` 0 ' along the radial line @xmath110 .
each successive marker shows the orbital phase after a time period of @xmath111 , where @xmath112 is the marker number . at @xmath113 ( marker 4 )
both orbits are synchronized again at their apastra along the line @xmath114 . when each test body has completed one orbit ( marker 8) they are again synchronized at their periastra along the line @xmath115 .
both orbits have precessed by the same amount over their common radial period.,width=321 ] and @xmath116 for the isofrequency pair shown in fig .
[ fig : schwarzschild_sample_pair ] . both radial and azimuthal motions are `` phase - synchronized '' on average .
, width=321 ] it is in fact not hard to demonstrate the existence of isofrequency orbits without resorting to a numerical calculation as above .
the argument follows from a few simple observations , which we now describe .
first , it is easily established that , in the @xmath117 plane , the separatrix @xmath118 is a curve of a _ positive _ slope as shown in fig .
[ fig : schwarzschild_omega_phi_e ] ( noting that in the figure we have chosen the horizontal axis with @xmath28 increasing to the _ left _ , so that , e.g , the radius of circular orbits increases to the right ) . to see this , use eq .
( [ eq : schwarzschild_phi_frequency ] ) with ( [ deltaphiasy ] ) and ( [ trasy ] ) to derive the relation @xmath119 along the separatrix , and invert to obtain @xmath120 where , recall , @xmath37 .
this gives @xmath121 in the relevant range @xmath122 .
next , examine the curve @xmath123 in the @xmath117 plane : it runs up along the separatrix , then proceeds horizontally along the line @xmath124 ( which represents orbits with @xmath125 and hence @xmath126 ) , and finally descends along the line @xmath127 ( which represents weak - field orbits with @xmath128 , for which both frequencies vanish ) . hence , the @xmath123 contour is represented by the thick red line in fig .
[ fig : schwarzschild_omega_phi_e ] , circumscribing the parameter space of bound orbits on 3 sides . from continuity
, it is now clear that a contour line of sufficiently small @xmath18 must `` bend backward '' inside the wedge formed by the separatrix and the @xmath124 line , so that it becomes vertical at a point .
the existence of isofrequency pairs follows immediately , as discussed above .
let us now delineate the region in the parameter space where isofrequency pairing occurs . in fig .
[ fig : schwarzschild_omega_phi_e ] we have indicated in a dotted black line the curve along which the transformation @xmath129 becomes singular .
each and every orbit left of this singular curve has an isofrequency dual right of the curve . in particular , each and every circular ( @xmath91 ) orbit on the open segment @xmath130 has an isofrequency dual on the dashed green line marked as _ circular - orbit duals _ ( cod ) .
( here we define the radial frequency of a circular orbit to be that of a slightly eccentric orbit , at the limit @xmath131 . ) hence , each and every orbit between the separatrix and the singular curve has an isofrequency dual between the singular curve and the cod , and vice versa .
we conclude that ( i ) all isofrequency pairs are confined to the region left of the cod , and ( ii ) every orbit left of the cod has an isofrequency dual .
how `` strong field '' is the region left of the cod , where isofrequency pairing occurs ?
the isofrequency pair of lowest azimuthal frequency sits where the singular curve intersects the @xmath91 axis , at point @xmath92 ( refer again to fig .
[ fig : schwarzschild_omega_phi_e ] ) . to calculate the value of @xmath20 at @xmath92 ,
we analytically taylor - expand the jacobian determinant @xmath132 in @xmath83 about @xmath91 ( for fixed @xmath82 ) .
we find , to leading order , @xmath133 of which the relevant root is @xmath134 this corresponds to a circular orbit of radius @xmath135 and frequency @xmath136 .
recall this is the _
lowest _ frequency of any isofrequency pair .
the isofrequency pair of _ highest _ frequency sits at the upper - left corner of the diagram in fig .
[ fig : schwarzschild_omega_phi_e ] ; it has @xmath137 . hence , for a schwarzschild black hole
, the range of isofrequency pairing is given by @xmath138 ( for comparison , the isco frequency is @xmath139 . ) evidently , the phenomenon is confined to the very strong - field regime of the schwarzschild black hole .
finally , we note that all orbits in isofrequency pairs are strongly zoom - whirling .
for example , the lowest - frequency isofrequency pair mentioned above ( slightly perturbed circular orbits of radii @xmath140 ) have @xmath141 , i.e. , they each complete more than 4 full revolutions in @xmath21 over a single radial period .
this behavior is also manifest in the example shown in fig .
[ fig : schwarzschild_sample_pair ] .
we consider first the case of equatorial orbits , in which the treatment is entirely analogous to that of orbits in schwarzschild spacetime .
equatorial orbits have @xmath142 , and are therefore parametrized by the pair @xmath46 alone . as in the schwarzschild case ,
bound equatorial orbits may instead be parametrized by the ( bl coordinate values of the ) turning points @xmath143 , or by a pair @xmath50 defined from them as in eq .
( [ pe ] ) .
one can then write integral expressions analogous to eqs .
( [ eq : schwarzschild_r_frequency ] ) [ with ( [ eq : t_r ] ) and ( [ eq : dt_dchi ] ) ] and ( [ eq : schwarzschild_phi_frequency ] ) [ with ( [ eq : deltaphi ] ) ] for the radial and azimuthal frequencies of the motion ; the dependence upon the black hole s
spin @xmath144 only enters via the explicit form of the functions @xmath145 and @xmath146 , which are significantly more complicated than their schwarzschild ( @xmath41 ) reductions .
the integral formulas for @xmath18 and @xmath20 , for arbitrary spin , can be found in sec .
ii.a of ref .
@xcite , and an analytic formula for the separatrix curve , @xmath59 , again for arbitrary spin , is given in ref .
we will not reproduce these expressions here given their complexity , and since we will be giving explicit formulas for generic orbits in the next subsection .
one finds that our intuitive argument for the existence of isofrequency orbits carries over directly from the schwarzschild case to equatorial orbits in kerr . along the separatrix of the kerr black hole ,
the function @xmath147 is most neatly expressed in terms of the periastron radius @xmath148 ( which , on the separatrix , corresponds to the radius of an unstable circular orbit of frequency @xmath28 ) @xcite : @xmath149 it can be easily checked that @xmath150 and @xmath151 for all @xmath144 and all @xmath20 in the relevant range @xmath152 , leading , again , to @xmath153 . [ here
@xmath154 is the whirl frequency of the marginally bound and marginally stable orbit with @xmath155 ( and @xmath124 ) , an expression for which will be given in eq .
( [ omegamax ] ) below . ]
the pattern of the @xmath156 contour lines in the @xmath157 plane should therefore be qualitatively as in fig .
[ fig : schwarzschild_omega_phi_e ] , including the crucial feature that contour lines `` curve back '' inside the wedge formed by the separatrix and the @xmath124 line .
it follows that isofrequency pairing should be a feature of equatorial orbits for any black hole spin @xmath144 ( and , in particular , we expect to see it in both prograde and retrograde orbits ) .
figure [ fig : kerreq_omega_phi_e ] shows an actual contour - line map , similar to that in fig .
[ fig : schwarzschild_omega_phi_e ] , for the sample case @xmath158 .
the @xmath156 contours were computed numerically as in the schwarzschild case , this time using the integral expressions from ref .
@xcite . near the separatrix
we have used the asymptotic expressions also given in @xcite . evidently , the essential features are as in the schwarzschild case .
one again identifies a singular curve and a cod curve in the ( @xmath159 ) plane , so that for any orbit between the separatrix and the singular curve there exists a dual isofrequency orbit between the singular curve and the cod , and vice versa .
the situation is qualitatively the same for other values of the spin and for retrograde orbits .
parameter space for bound equatorial geodesic orbits in kerr geometry with @xmath158 .
compare with fig .
[ fig : schwarzschild_omega_phi_e ] .
the relevant features are as in the schwarzschild case , and the existence of isofrequency pairing below the cod is similarly evident .
we indicate a sample pair with @xmath160 and @xmath161 , both having frequencies @xmath162 .
labelled points on the horizontal axis correspond to circular orbits of radii ( left to right ) @xmath163 ( whirl radius of marginally bound marginally stable orbit ; orbit of highest azimuthal frequency ) , @xmath164 ( isco ) , @xmath165 ( outermost orbit in an isofrequency pair ) , and @xmath166 ( orbit of highest radial frequency , @xmath167 ) . ]
let us identify the frequency range @xmath168 where isofrequency pairing occurs . the @xmath169 version of eq .
( [ j ] ) is too complicated to be solved analytically for @xmath170 ( the radius of the outermost circular orbit belonging to an isofrequency pair ) as we have done in the schwarzschild case , so we resort to numerical solutions .
table [ table ] lists @xmath171 values for a sample of black hole spins .
once a numerical value for @xmath171 is at hand ( for a given @xmath144 ) , @xmath172 is obtained via @xmath173 where we have used the general relation between the frequency of a circular equatorial orbit and its bl radius @xcite .
the _ maximal _
value @xmath154 corresponds to the whirl frequency of the marginally bound marginally stable orbit with @xmath124 ( top left corner in fig .
[ fig : kerreq_omega_phi_e ] ) .
it is given by @xmath174 the range @xmath168 is illustared in fig .
[ fig : omegaminmax ] . .
numerical values for @xmath171 , the bl radius of the outermost circular orbit belonging to an isofrequency pair ( cf .
[ fig : kerreq_omega_phi_e ] ) .
the frequency @xmath172 of this orbit [ given in eq .
( [ omegamin ] ) ] marks the lower end of the frequency range where synchronous pairing occurs . for comparison
, the second column displays the isco radius @xmath175 ( elsewhere in this paper denoted @xmath176 ) ; it is given by @xcite @xmath177^{1/2}$ ] , where @xmath178 $ ] and @xmath179 .
numerical values are truncated at the 5th decimal place , rounding up .
[ cols="^,^,^",options="header " , ] .
the orbits depicted correspond to ` sample pair 3 ' from table [ table : kerr_sample_pairs ] ( also leftmost pair in fig .
[ fig : kerr_case ] ) , with ` orbit 1 ' shown on the left and ` orbit 2 ' shown on the right .
the top row shows the motion in the @xmath180-plane and the bottom row shows the motion in the @xmath181-plane , where @xmath182 , @xmath183 and @xmath184 .
the black hole is shown to scale . in both orbits
the motion begins at @xmath185 at periastron , with @xmath34 and @xmath186 . in integrating the geodesic equations we used the method of drasco and hughes @xcite , which avoids numerical difficulties near the orbital turning points .
we show the portion of the orbits between @xmath187 and @xmath188 .
, width=321 ] , @xmath116 and @xmath189-\cos[\theta_2(t)]$ ] for ` sample pair 3 ' of table [ table : kerr_sample_pairs ] and fig .
[ fig : kerr_sample_pair ] .
both orbits begin at @xmath190 at periastron with @xmath34 and @xmath186 .
triperiodic isofrequency orbits are `` synchronized '' only in a long - time average sense .
periastra are reached only approximately at the same time ( as a closer inspection of the upper panel would reveal ) but the time differences should average to zero over a long time .
the same applies to the average azimuthal motion ( middle panel , where a close inspection reveals that the azimuthal phases of the two orbits are not in precise agreement at the periastra ) , and to the motion in @xmath23 ( lower panel ) . in the latter case
we show the _ difference _ between the two longitudinal phases , which remains quasi - periodic . it would have not remained quasi - periodic had the two orbits not been in an isofrequency pair.,width=321 ] before concluding ,
let us comment on the validity of our numerical algorithm , which , as already mentioned , involves delicate high precision computation of the orbital frequencies . to establish confidence in our results we tested our code in a number of ways .
first , we checked that our code reproduces all the double - precision - accurate results for @xmath191 ( given @xmath192 ) tabulated in ref .
we also verified , to over one hundred significant figures , that the results of fujita and hikida s orbital frequency formulas ( in the form given above ) agree with the results of schmidt s less explicit formulas @xcite .
we further validated our equations using a direct numerical integration of the @xmath193-time geodesic equations in a few test cases . we were able to reproduce the analytically calculated @xmath193-frequencies @xmath194 and @xmath195 to within 25 significant figures .
( the quantities @xmath196 and @xmath197 involve infinite time averages and are therefore less easily tested in this manner . )
in this article we have shown that the three fundamental frequencies of bound geodesics in kerr geometry do not constitute a good parametrization of the orbits in the strong - field regime .
we identified a mapping between pairs of physically distinct orbits that possess the same set of orbital frequencies .
a pair of isofrequency orbits are `` synchronous '' in that they exhibit the same periastron and lense - thirring precession rates .
all orbits in isofrequency pairs are confined to the very strong - field regime near the innermost stable orbit cf .
table [ table ] and fig .
[ fig : omegaminmax ] .
( some orbits in isofrequency pairs have very large eccentricities and apastra at arbitrarily large radii , but their periastra are in the very strong field . )
our numerical experiments suggest that all members of isofrequency pairs are of `` zoom - whirl '' type , but this is yet to be checked more thoroughly in the case of triperiodic orbits and across all spin values .
the first practical lesson from our analysis is a cautionary note for colleagues studying the data - analysis problem for gravitational - wave detectors , in particular the problem of parameter extraction for systems of extreme - mass - ratio inspirals ( emris ) .
the fundamental frequencies extracted from a `` snapshot '' of an emri waveform , on their own , as a matter of principle , do not necessarily provide enough information from which to extract the system s intrinsic physical parameters @xmath198 ( or @xmath199 ) .
if the system is sufficiently close to the innermost stable orbit , a measurement of the instantaneous frequencies could at most narrow down on two possible sets of system parameters .
this `` degeneracy '' , however , can be removed in any one of the following ways : ( i ) by examining the power spectrum of the waveform ( the power distribution among the various harmonics of the fundamental frequencies will be different for the two orbits ) ; ( ii ) by inspecting the waveform snippet in the time domain ( the shape of the waveform is strongly dependent upon the eccentricity , for instance ) ; or ( iii ) by accounting for radiation - reaction evolution effects ( two orbits which are instantaneously isofrequency will evolve radiatively in different ways ) . at a more fundamental level
, our analysis identifies a new feature in the strong - field dynamics of compact - object binaries in general relativity .
the fundamental frequencies in a bound binary ( of any mass ratio ) are important invariant characteristics of the `` conservative '' sector of the dynamics . as such
they have long been studied in the context of post - newtonian ( pn ) theory .
the instantaneous frequencies in a binary of inspiralling black holes can even , nowadays , be extracted from high - precision fully nonlinear simulations in numerical relativity ( nr)see , for example , ref . @xcite .
our analysis here revealed the occurrence of isofrequency pairing in the test - particle limit ( i.e. , the limit of vanishing mass ratio ) , but it is not unreasonable to speculate that the phenomenon is a general feature of the dynamics in strongly gravitating binaries , and would reveal itself also when the mass ratio is finite .
it is not clear if available pn theory can predict isofrequency pairing this would be interesting to check .
when new , higher - order pn terms are calculated in the future , it would again be interesting to check if they reveal the phenomenon , as a way of assessing the faithfulness of the pn expressions in the strong - field regime .
it would also be interesting to examine whether the phenomenon manifests itself in nr simulations of inspiralling black holes of comparable masses near the innermost stable orbit . because the fundamental frequencies are _ invariant _ characteristics of the conservative dynamics , they are useful as reference points for comparing the predictions of different approaches to the relativistic two - body problem . recent examples of such `` cross - cultural '' comparisons include ( i ) calculations of the isco frequency in the self - force ( sf )
, pn and effective - one - body ( eob ) approaches @xcite ; and ( ii ) calculations of the periastron advance in slightly eccentric orbits in sf , pn , eob and nr @xcite . in both examples ( which involve two nonrotating black holes ) relations between the two invariant frequencies associated with infinitesimally perturbed circular orbits were utilized as benchmarks for comparison .
the singular curve / surface identified in our current work is an _ invariant structure _ in the parameter space , . ] which provides yet another , independent , comparison point in the strong field , this time utilizing eccentric orbits . as a first example
, one could consider the function @xmath200 along the singular curve in the parameter space of nonrotating binaries ( fig .
[ fig : schwarzschild_omega_phi_e ] shows this curve in the test - particle limit ) . in principle
, one could compute this function in the sf approximation ( i.e. , order by order in the mass ratio ) , and perhaps also in fully nonlinear nr , making for an interesting comparison
. there may be a way of using the results of such a calculation to calibrate the potentials of eob theory in the strong field , although how this could be done in practice is yet unclear @xcite .
comparison with existing pn expressions could test the performance of the pn expansion in the strong field .
a more constructive synergy could be achieved within the recent `` phenomenological '' approach to pn calculations , whereby high - order terms in the pn expansion are determined by fitting to numerical data from sf or nr calculations @xcite .
a faithful phenomenological pn model would need to be able to recover the singular curve in the strong field , perhaps through the inclusion of suitable `` poles '' in pn expressions . finally , let us mention the intriguing possibility that isofrequency pairing in astrophysical black holes ( e.g. , between clumps of accreting matter ) could have observational implications .
the question is worth asking because we are at an era where astronomical observations in a range of electromagnetic wavelengths routinely peer into processes deep in the strong - field potentials of accreting black holes .
quasi - periodic oscillations ( qpos ) in x - rays from accreting black - hole systems probe the innermost regions of accretion disks @xcite , and ( to a lesser extent ) so do x - ray flares from the galactic center @xcite .
could the peculiar strong - gravity phenomenon of isofrequency pairing have a dynamical effect on matter orbiting the black hole , perhaps through resonant interaction ?
although admittedly far - fetched , this possibility deserves exploration .
we thank sam dolan , steve drasco , carsten gundlach , scott hughes , amos ori and eric poisson for helpful discussions .
we are also grateful to maarten van de meent for feedback on the first version of this paper .
nws work was supported by stfc through a studentship grant and by the irish research council , which is funded under the national development plan for ireland .
lb acknowledges support from the european research council under grant no .
304978 , and from stfc through grant number pp / e001025/1 .
31ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ]
+ 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1103/physrev.174.1559 [ * * , ( ) ] link:\doibase 10.1086/151796 [ * * , ( ) ] link:\doibase 10.1103/physrevd.5.814 [ * * , ( ) ] @noop _ _ ( , ) @noop * * , ( ) link:\doibase 10.1088/0264 - 9381/19/10/314 [ * * , ( ) ] link:\doibase 10.1103/physrevd.69.044015 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.77.103005 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.85.023012 [ * * , ( ) ] , link:\doibase 10.1088/0264 - 9381/26/13/135002 [ * * , ( ) ] link:\doibase 10.1103/physrevd.79.104016 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.83.084023 [ * * , ( ) ] link:\doibase 10.1098/rspa.1961.0142 [ * * , ( ) ] link:\doibase 10.1103/physrevd.50.3816 [ * * , ( ) ] link:\doibase 10.1103/physrevd.66.044002 [ * * , ( ) ] link:\doibase 10.1103/physrevd.79.124013 [ * * , ( ) ] link:\doibase 10.1103/physrevd.67.084027 [ * * , ( ) ] link:\doibase 10.1103/physrevd.73.024027 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.77.124050 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.107.141101 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.102.191101 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.81.024017 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.83.024028 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.82.084036 [ * * , ( ) ] , @noop link:\doibase 10.1103/physrevd.81.084033 [ * * , ( ) ] , `` , '' in link:\doibase 10.1007/978 - 90 - 481 - 3015 - 3_15 [ _ _ ] , ( ) pp .
link:\doibase 10.1146/annurev.astro.44.051905.092532 [ * * , ( ) ] , link:\doibase 10.1088/1742 - 6596/372/1/012022 [ * * , ( ) ] ,
|
bound geodesic orbits around a kerr black hole can be parametrized by three constants of the motion : the ( specific ) orbital energy , angular momentum and carter constant .
generically , each orbit also has associated with it three frequencies , related to the radial , longitudinal and ( mean ) azimuthal motions . here
we note the curious fact that these two ways of characterizing bound geodesics are not in a one - to - one correspondence .
while the former uniquely specifies an orbit up to initial conditions , the latter does not : there is a ( strong - field ) region of the parameter space in which pairs of physically distinct orbits can have the same three frequencies . in each such isofrequency
pair the two orbits exhibit the same rate of periastron precession and the same rate of lense - thirring precession of the orbital plane , and ( in a certain sense ) they remain `` synchronized '' in phase .
|
the enhanced mixing of a passive scalar is one of the most direct consequences of chaos : a flow whose trajectories exhibit sensitivity to initial conditions will lead to rapid mixing .
there are powerful theories based on the distribution of lyapunov exponents @xcite that link the mixing rate of the passive scalar with the chaotic properties of the flow . it has been recently suggested , following earlier work of pierrehumbert @xcite , that the mixing properties of the flow can often be elucidated only by solving a full eigenvalue problem for the advection diffusion operator , in an analogous manner to what is done for the kinematic dynamo @xcite .
the resulting eigenfunctions have been dubbed _
strange eigenmodes _ by pierrehumbert , and are closely related to pollicott
ruelle resonances in ergodic theory @xcite , which describe the long - time decay of correlations in mixing hyperbolic dynamical systems .
strange eigenmodes have also been observed experimentally @xcite .
they are often called _ persistent patterns _ or _
large - scale eigenfunctions_. the strange eigenmodes reflect a balance between advection and diffusion . on its own
, advection is incapable of achieving mixing : it shuffles the concentration field of the passive scalar but does not decrease its fluctuations .
the role of advection is to stir the concentration field , thereby creating sharp gradients in concentration .
physically , these gradients are reflected in the filamentation experienced by a blob of dye when it is stirred .
the sharp gradients enhance the role of diffusion tremendously , and this allows mixing to proceed . as the diffusivity of the scalar is made smaller , the scale at which this mixing occurs decreases , so the concentration field appears very rough . in the limit of arbitrarily small diffusivity ,
the concentration field is not smooth : it consists of a superposition of strange eigenmodes .
the dominant one among these eigenfunctions is called _ the _ strange eigenmode , although there may be several of comparable importance . in the present work we tie the two types of theories together ( _ i.e. _ , lyapunov exponent - based and strange eigenmode ) for a specific system , which was already studied in @xcite from the strange eigenmode viewpoint .
the strange eigenmode represents a fundamentally eulerian ( spatial ) view of mixing , whereas the lyapunov exponent view is lagrangian ( material ) , following as it does the stretching history of fluid elements . in the strongly chaotic systems we deal with in the present context ,
lagrangian and eulerian coordinates are related by a convoluted transformation .
this transformation is so complex that its specific form is inaccessible ( even numerically ) after some time , a reflection of sensitivity to initial conditions . for long times
, the two frames must be regarded as essentially independent : we can not simply solve a problem in eulerian coordinates and transform to lagrangian coordinates ( or vice - versa ) .
thus we believe it is worthwhile to take a chaotic system whose solution has already been obtained in eulerian coordinates and solve it in lagrangian coordinates . as indicated above , this links two views of mixing together , and in particular illustrates the nature of strange eigenmodes in lagrangian coordinates .
this also indicates the source of the breakdown of local theories . we will show that there exists a kind of _ lagrangian strange eigenmode _ , which is not quite an eigenmode but which exhibits similar features :
specifically , it is an eigenmode if an appropriate time - dependent rescaling of coordinates is performed ( exponential in time ) .
this rescaling is closely related to the `` cone '' involved in diffusive problems in the presence of an exponentially stretching flow @xcite .
we call it the _ cone of safety _ , because modes inside it are sheltered from diffusion at a given time .
we introduce the system to be studied , the perturbed cat map , in section [ sec : torusmap ] .
we find its finite - time lyapunov exponents and eigenvectors using first - order perturbation theory . in section
[ sec : ad ] we partially solve the advection diffusion equation for our map , again using perturbation theory . some numerical work is needed to complete the solution , and this is described in section [ sec : numres ] . finally , a few concluding remarks are offered in section [ sec : disc ] .
the strange eigenmode has for the most part been studied in maps , because these present great advantages for analytical work .
this is also reasonable since experimental work has so far focused on time - periodic flows @xcite . as in @xcite
, we consider the map @xmath0 defined on the unit two - torus , . here
@xmath1 is a matrix of integers with unit determinant , and @xmath2 is a doubly - periodic function , so that @xmath3 is a diffeomorphism ; specifically , we take @xmath4 where @xmath2 is chosen such that @xmath3 is area - preserving . for @xmath5 ,
is the usual cat map of arnold @xcite .
the map inherits much of the simplicity of the cat map , but the perturbation allows for more complex and less singular behavior .
the action of the map is depicted in fig .
[ fig : catmap ] . for small @xmath6 ,
the map is very close to the cat map , but the implications of the perturbation for mixing are profound , as we will now discuss . .
( a ) initial pattern ; ( b ) first iterate ; ( c ) second iterate ; ( d ) third iterate .
the gray shading shows the action of the unperturbed cat map . ]
we are interested in the mixing properties of the map @xmath3 . in ref .
@xcite mixing in this map was investigated from an eulerian perspective : advection alternated with diffusion and the central object was the distribution of the concentration field in eulerian coordinates . for @xmath5
, mixing occurs superexponentially in the map , due to the lack of dispersion in fourier space .
the concentration in a given fourier mode is mapped entirely to one mode of higher wavenumber , and so on to ever higher wavenumbers .
this sequence of wavenumbers have exponentially - growing magnitudes . because diffusion is exponential in the wavenumber , the net result is superexponential decay ( _ i.e. _ , the exponential of minus an exponential in time ) . for @xmath7 ,
the situation is radically different .
the map now disperses concentration among many fourier modes at each iteration .
in particular , some concentration is always mapped back to the lowest allowable wavenumber ( the grave mode ) , on which the weak diffusion is ineffective .
the decay of the scalar is then limited by how much concentration is mapped to this grave mode at each iteration .
the grave mode thus forms the seed of a strange eigenmode , since an eigenmode is by definition a _ recurring _ feature .
the concentration field thus settles into the slowest - decaying eigenfunction of the advection
diffusion operator the strange eigenmode
analogously to earlier work @xcite . here
we wish to solve the same problem as in ref .
@xcite , but in lagrangian coordinates .
this means that we focus on following fluid elements and describing how they deform under the action of the map . to solve the advection
diffusion problem in lagrangian coordinates , it is necessary to have expressions for the finite - time lyapunov exponents of the map @xcite ( or equivalently the coefficients of expansion ) and their associated characteristic directions , as a function of lagrangian coordinates , and not just their distribution ( we will see why this is so in section [ sec : ad ] ) . because the finite - time lyapunov exponents are easily derived for the cat map ( @xmath5 ) , we shall proceed perturbatively , assuming @xmath6 is small .
first let us give the lyapunov exponents and associated characteristic directions for the unperturbed cat map .
the lyapunov exponents are the logarithms of the eigenvalues of @xmath1 , and the characteristic directions are the corresponding eigenvectors .
it is convenient to introduce an angle @xmath8 in terms of which the eigenvectors of @xmath1 are @xmath9 with @xmath10 and @xmath11 .
then the corresponding eigenvalues of @xmath1 are @xmath12 [ eq : nugr ] these equalities are specific to this particular angle , as is the relation @xmath13 .
the @xmath14 direction is associated with stretching , and @xmath15 with contraction .
the coefficients of expansion ( given by @xmath16 and @xmath17 after @xmath18 iterations of the map ) and characteristic directions for the linear cat map are uniform in space .
now we derive their value for @xmath6 nonzero but small , using perturbation theory .
the problem is to find the eigenvalues and eigenvectors of the matrix @xmath19 , with components @xmath20 often called the metric tensor ( or cauchy green strain tensor in fluid mechanics ) . here
@xmath21 is the lagrangian label ( coordinate ) and @xmath22 is the @xmath18th iterate of the point @xmath21 under the action of the map ( so that ) .
the metric tensor describes the stretching experienced at the @xmath18th iteration by a fluid element initially at @xmath21 .
its eigenvalues and eigenvectors give the shape and orientation at the @xmath18th iteration of an ellipsoid representing an initially spherical infinitesimal element of fluid .
before we can apply perturbation theory to the metric tensor , we must find the form of the perturbation itself . to first order in @xmath6 ,
the @xmath18th iterate of the map is @xmath23 the jacobian matrix of this transformation is @xmath24 where the numerator corresponds to rows and the denominator to columns of a matrix .
we must now construct the metric tensor , which to leading order in @xmath6 is @xmath25 with @xmath26 and the tilde denotes the transpose of a matrix .
the unperturbed metric is , with eigenvalues @xmath27 and @xmath28 and eigenvectors given by . the perturbation is the bracketed term in . finding the eigenvalues and eigenvectors of the symmetric matrix @xmath19 to leading order in @xmath6 is a straightforward application of perturbation theory for symmetric matrices , familiar from quantum mechanics . for more details
we refer the reader to standard texts on the subject @xcite . to leading order in @xmath6 ,
the coefficient of stretching is written as @xmath29 where @xmath16 is the coefficient of stretching of the unperturbed cat map , and the correction is obtained from the perturbation by contraction with the unperturbed eigenvectors @xcite , @xmath30 observe that because the coefficient of stretching is the square root of the largest eigenvalue of @xmath31 , there is an extra factor of @xmath32 to leading order in @xmath6 . using the fact that , for @xmath1 symmetric , @xmath33 , we find eqs . and give @xmath34 where we have substituted the specific form of the map , given by .
the subscript ` 1 ' in indicates the @xmath35 component of a vector .
the perturbed eigenvectors can be written as @xmath36 where @xmath37 may be regarded as a small angle of rotation .
again we follow standard matrix perturbation theory @xcite , so that the angle of rotation is given by @xmath38 which after some reduction and the use of yields @xmath39 with @xmath40 note that the asymptotic direction ( @xmath41 ) is dominated by @xmath42 , so that @xmath43 in this form it is easy to check that @xmath44 for @xmath41 , as required by the differential constraint @xcite .
( the derivatives are taken with respect to the lagrangian coordinates @xmath21 . ) the first - order perturbative solution for the coefficient of stretching @xmath45 is compared to numerical results in fig .
[ fig : torusmap_lyap ] , and similarly for the eigenvector @xmath46 in fig .
[ fig : torusmap_s ] .
( dashed ) from eq . for .
the solutions diverge after several iterations because we are perturbing off a chaotic trajectory . ] with @xmath47 , using the asymptotic result in .
the error is of order @xmath48 . ] unlike the perturbed coefficients of stretching , which eventually diverge from the numerical solution because of the sensitivity to initial conditions , the perturbed eigenvectors converge very rapidly and are always close to the numerical result
. it will be more convenient to express the metric tensor in terms of its eigenvalues and eigenvectors .
the metric tensor can be written in terms of the coefficients of expansion and characteristic directions as @xmath49 ^ 2{\ensuremath{\hat{{\ensuremath{\mathbf{u}}}}}}^{({\ensuremath{n}})}{\ensuremath{\hat{{\ensuremath{\mathbf{u}}}}}}^{({\ensuremath{n } } ) } + [ { \ensuremath{\lambda}}^{({\ensuremath{n}})}]^{-2}{\ensuremath{\hat{{\ensuremath{\mathbf{s}}}}}}^{({\ensuremath{n}})}{\ensuremath{\hat{{\ensuremath{\mathbf{s}}}}}}^{({\ensuremath{n}})}\,.\ ] ] to leading order in @xmath6 , we have @xmath50 the only dependence on @xmath21 in is contained in @xmath45 and @xmath51 .
having derived the coefficients of expansion and characteristic directions of stretching ( to leading order in @xmath6 ) , we can now solve the advection diffusion equation in lagrangian coordinates .
we will first discuss the case for an incompressible flow , and then make the transition to a volume - preserving map .
the advection diffusion equation for the concentration of a scalar , @xmath52 , advected by an incompressible velocity field @xmath53 is @xmath54 where @xmath55 is the diffusion coefficient .
we define the transformation @xmath56 from lagrangian coordinates @xmath21 to eulerian coordinates @xmath57 by @xmath58 where the overdot denotes a time derivative at fixed @xmath21 .
we can then transform to lagrangian coordinates @xmath21 @xcite , @xmath59 where we reused the same symbol for @xmath60 .
the anisotropic , nonhomogeneous , time - dependent diffusion tensor @xmath61 is @xmath62 where @xmath63 is the metric tensor encountered in section [ sec : torusmap ] . by construction
, the advection term has disappeared from .
the flow enters indirectly through the metric tensor in , reflecting the enhancement to diffusion due to the deformation of fluid elements @xcite .
we now make the leap from a flow to a map : because the velocity field does not enter directly , we may regard the time dependence in @xmath61 as given by a map rather than a flow , and use the metric in the diffusion tensor @xmath61 .
we also write @xmath64 for @xmath60 , where @xmath18 denotes the @xmath18th iterate of the map . since our map is defined on the torus , we can expand @xmath64 in fourier components @xmath65 ; the resulting map , obtained by first fourier transforming and then solving , is @xmath66 where @xmath67 with @xmath68 the period of the map .
this is an exact result , but the great difficulty lies in calculating the exponential of @xmath69 .
again , we shall accomplish this perturbatively .
for the torus map introduced in section [ sec : torusmap ] , from eq .
we obtain @xmath70^{-1 } = { \ensuremath{\lambda}}^{2{\ensuremath{n}}}\,{\ensuremath{\hat{{\ensuremath{\mathbf{s}}}}}}\,{\ensuremath{\hat{{\ensuremath{\mathbf{s}}}}}}+ { \ensuremath{\lambda}}^{-2{\ensuremath{n}}}\,{\ensuremath{\hat{{\ensuremath{\mathbf{u}}}}}}\,{\ensuremath{\hat{{\ensuremath{\mathbf{u}}}}}}+ 2{\ensuremath{k}}\,\eta^{({\ensuremath{n}})}({\ensuremath{\lambda}}^{2{\ensuremath{n}}}\,{\ensuremath{\hat{{\ensuremath{\mathbf{s}}}}}}\,{\ensuremath{\hat{{\ensuremath{\mathbf{s}}}}}}- { \ensuremath{\lambda}}^{-2{\ensuremath{n}}}\,{\ensuremath{\hat{{\ensuremath{\mathbf{u}}}}}}\,{\ensuremath{\hat{{\ensuremath{\mathbf{u } } } } } } ) - { \ensuremath{k}}\,\zeta^{({\ensuremath{n}})}{\left}({\ensuremath{\lambda}}^{2{\ensuremath{n } } } - { \ensuremath{\lambda}}^{-2{\ensuremath{n}}}{\right } ) ( { \ensuremath{\hat{{\ensuremath{\mathbf{u}}}}}}\,{\ensuremath{\hat{{\ensuremath{\mathbf{s}}}}}}+ { \ensuremath{\hat{{\ensuremath{\mathbf{s}}}}}}\,{\ensuremath{\hat{{\ensuremath{\mathbf{u } } } } } } ) , \label{eq : ginveps}\ ] ] to leading order in @xmath6 , where the only functions of @xmath21 are @xmath45 and @xmath51 . inserting into , we find @xmath71 where @xmath72 and @xmath73 here we have defined @xmath74 to agree with the notation in ref .
@xcite , as well as @xmath75 and similarly for @xmath76 and @xmath77 . upon making use of
the fourier - transformed , , and in , we find @xmath78 where @xmath79 is a unit vector in the @xmath35 direction , and @xmath80 to obtain the full solution , we must now exponentiate to give the transfer matrix in . fortunately ,
for @xmath81 diagonal there is a simple expansion , @xmath82_{{\ensuremath{\bm{k}}}{\ensuremath{\bm{\ell } } } } = { \ensuremath{{\mathrm e}}}^{{\ensuremath{a}}^{({\ensuremath{n}})}_{{\ensuremath{\bm{k}}}{\ensuremath{\bm{k}}}}}\,\delta_{{\ensuremath{\bm{k}}}{\ensuremath{\bm{\ell } } } } + { \ensuremath{k}}{\ensuremath{e}}^{({\ensuremath{n}})}_{{\ensuremath{\bm{k}}}{\ensuremath{\bm{\ell } } } } ; \qquad { \ensuremath{e}}^{({\ensuremath{n}})}_{{\ensuremath{\bm{k}}}{\ensuremath{\bm{\ell } } } } { \ensuremath{\mathrel{\raisebox{.069ex}{:}\!\!=}}}{\ensuremath{b}}^{({\ensuremath{n}})}_{{\ensuremath{\bm{k}}}{\ensuremath{\bm{\ell}}}}\,\frac{{\ensuremath{{\mathrm e}}}^{{\ensuremath{a}}^{({\ensuremath{n}})}_{{\ensuremath{\bm{k}}}{\ensuremath{\bm{k } } } } } - { \ensuremath{{\mathrm e}}}^{{\ensuremath{a}}^{({\ensuremath{n}})}_{{\ensuremath{\bm{\ell}}}{\ensuremath{\bm{\ell } } } } } } { { \ensuremath{a}}^{({\ensuremath{n}})}_{{\ensuremath{\bm{k}}}{\ensuremath{\bm{k } } } } - { \ensuremath{a}}^{({\ensuremath{n}})}_{{\ensuremath{\bm{\ell}}}{\ensuremath{\bm{\ell}}}}}\ , , \label{eq : edef}\ ] ] valid to first order in @xmath6 .
we say fortunately because without such a formula it is very difficult to compute this matrix exponential even numerically due to the large size of the matrices ( _ i.e. _ , infinite ) and their magnitude ( _ i.e. _ , growing exponentially in time ) .
the @xmath27 term in @xmath83 seems to imply that @xmath65 decays _ superexponentially _ fast as @xmath84 . from eulerian considerations @xcite , we know that for the decay is actually exponential after a short superexponential transient .
this is because the @xmath85 term must be taken into account : it breaks the diagonality of @xmath69 , so that given some initial set of wavevectors , the concentration contained in those modes can be transferred elsewhere .
in particular , it can transfer concentration to modes aligned with the unstable direction .
we will see how this avoids superexponential decay in section [ sec : numres ] .
at this point , solving and numerically seems like the only way forward .
clearly , attempting the solve this on a grid in fourier space is hopeless : very high wavenumber modes are quickly populated so the resolution is exhausted very rapidly .
instead , the procedure we use involves keeping track of a list of excited fourier modes ( _ i.e. _ , those that are nonzero to machine precision ) .
we now describe this scheme .
first , an initial wavenumber is seeded with some initial concentration .
this mode will be damped by the diagonal part of the matrix in , and will also excite two new modes as seen in . repeating this , starting now from three modes , we see that the number of excited modes grows exponentially .
thus it would seem that this procedure is not very advantageous ; however , after a few iteration the diffusivity ( the diagonal part in ) will damp most modes because @xmath86 is growing exponentially . thus the modes that have been damped beyond redemption can be removed from the list . in this manner
the number of excited modes eventually reaches a constant , though they consist of ever higher wavenumbers .
thus one can think of a `` packet '' of modes cascading through fourier space towards larger wavenumbers .
it is this packet that is the lagrangian analogue of the strange eigenmode in eulerian space , as we will discuss in section [ sec : lagrstr ] .
let us first present some results . and
different values of @xmath87 , compared to the result from eulerian coordinates ( dotted lines ) .
the dashed line shows the exact result for superexponential decay ( @xmath5 ) for @xmath88 . ]
figure [ fig : var_eps=0.001 ] shows the decay of the scalar variance for and four values of the diffusivity .
the agreement with the eulerian results is excellent for early times , but inevitably breaks down later .
( in fact the variance eventually begins to _ increase _ , which is forbidden . )
the agreement is also worse for smaller diffusivity .
both of these disagreements are a manifestation of the wavenumber dependence of the perturbation in : for @xmath89 too large the perturbation becomes large , invalidating the approach .
nevertheless , fig .
[ fig : var_eps=0.001 ] clearly validates the calculation for times that are not too long .
, at the fourth iteration .
the dashed line indicates an @xmath48 dependence , showing that the two agree to first order in @xmath6 . ]
another validation is shown in fig .
[ fig : converg ] , where we plot the difference between the eulerian and lagrangian results as a function of @xmath6 .
the difference clearly scales as @xmath90 , showing that the two agree at leading order , as required for a first - order asymptotic result .
we now interpret our results in greater detail , and look for a manifestation of the strange eigenmode in lagrangian coordinates .
the mechanism described in section [ sec : nummet ] is similar to that originally introduced ( in the context of the kinematic dynamo problem ) by zeldovich _ et al . _
@xcite : they basically solved the advection
diffusion equation in lagrangian coordinates for a linear velocity field , and found that in order to avoid rapid superexponential decay one had to restrict attention to a `` cone '' of wavenumbers that are closely aligned with the unstable manifold of the flow ( a similar approach was used later in refs .
the exponential shrinking in time of this `` cone of safety '' leads to an exponential decay of scalar variance at a rate given by the lyapunov exponents .
the problem with that approach is that a linear velocity field offers no possibility of _ dispersion _ in fourier space .
the wavenumbers in the cone of safety must have some concentration associated with them initially .
what our numerical results show is that if one considers dispersion in fourier space ( of the type allowed by the @xmath85 term in ) then it is possible for concentration to be moved inside the cone from elsewhere .
the lagrangian equivalent of the strange eigenmode must live within the cone of safety , otherwise it would decay away superexponentially .
but unlike ref . @xcite the decay rate in the present case is not determined by the shrinking of the cone : it is set by how much variance gets transferred into the cone at each iteration .
figure [ fig : spectrum ] shows a plot of the power spectrum of concentration .
( circles ) and @xmath91 ( black dots ) .
the large black dots are points that are the same for all iterations ( after rescaling ) : this is the dominant strange eigenmode in lagrangian coordinates . both axes
have been rescaled such that the dominant peak has unit amplitude and wavenumber ( @xmath92 is defined in eq . ) .
the hollow circles are due to an admixture of another , faster - decaying eigenfunction . ]
the magnitude of the concentration ( @xmath93 ) is plotted vs the magnitude of the wavenumber normalized by @xmath94 ( its maximum value ) , which is proportional to @xmath16 .
the concentration is normalized at each iteration such that the mode with largest concentration has unit magnitude .
the iterations plotted are @xmath95 ( circles ) and @xmath91 ( black dots ) .
most of the circles appear as large black dots , because all these points lie on top of each other .
hence , the concentration is in an eigenstate , given that the wavenumber has been rescaled by a factor proportional to @xmath96 ( _ i.e. _ , such that the dominant peak is at unit rescaled wavenumber ) .
this is what we interpret as the lagrangian equivalent of the strange eigenmode ( a good name might be `` stretched eigenmode '' in fourier space ) . some points in fig .
[ fig : spectrum ] exhibit a decay with iteration number ( appearing as columns of circles , with higher iteration numbers lower on the graph ) : they belong to a more rapidly - decaying eigenfunction .
note that the peaks do not sharpen with iteration number , but more points are added to some of the tails .
the eigenfunction appears extremely rough and discontinuous , though the peaks are indicative of some underlying continuum behavior .
the seemingly isolated points actually tend to line up with a peak far below .
finally , note that the relative height ( but not position or shape ) of the peaks depends on @xmath6 : the whole shape is stretched vertically as @xmath6 is made smaller .
this is because the term proportional to @xmath6 controls the transfer of concentration `` vertically '' ( with respect to fig .
[ fig : spectrum ] ) in the eigenmode at each iteration .
the lagrangian strange eigenmode has some intriguing features : ( i ) it is rescaled exponentially in time , in order to remain within the cone of safety ( so it is not a true eigenmode ) ; ( ii ) its power spectrum is very discontinuous , in sharp contrast to its eulerian counterpart @xcite ; ( iii ) its decay rate is set by how much concentration is moved into the `` new '' cone of safety at each iteration ( since the cone is shrinking ) . in the appendix we present an analytic result for a two - mode system which gives a simplified representation of the cone of safety . in the map analyzed here
the exponential time - rescaling gives a proper eigenmode , since only the constant stretching is important at leading order . in a generic map
the stretching is a strong function of space , so the necessary time - rescaling would be position - dependent .
it is hard to see how the lagrangian approach presented here could be used in more realistic problems : perturbation theory was used extensively ( which would not be applicable in most real situations ) , both for computing the finite - time lyapunov exponents and the matrix exponential .
we believe the approach is instructive nonetheless , giving as it does a picture of the strange eigenmode in lagrangian coordinates .
our approach does not yield much information about the long - time behavior of the decay .
there is currently a debate as to whether the mechanism presented in refs .
@xcite gives a lower bound on the decay rate @xcite .
our perturbation expansion breaks down before this question can be answered .
the author wishes to thank steve childress for stimulating discussions .
10 t. m. antonsen , jr . , z. fan , e. ott , and e. garcia - lopez , `` the role of chaotic orbits in the determination of power spectra , '' phys .
fluids * 8 * , 3094 ( 1996 ) .
d. t. son , `` turbulent decay of a passive scalar in the batchelor limit : exact results from a quantum - mechanical approach , '' phys .
e * 59 * , r3811 ( 1999 ) .
e. balkovsky and a. fouxon , `` universal long - time properties of lagrangian statistics in the batchelor regime and their application to the passive scalar problem , '' phys .
e * 60 * , 4164 ( 1999 ) .
d. r. fereday , p. h. haynes , a. wonhas , and j. c. vassilicos , `` scalar variance decay in chaotic advection and batchelor - regime turbulence , '' phys .
e * 65 * , 035301(r ) ( 2002 ) .
a. wonhas and j. c. vassilicos , `` mixing in fully chaotic flows , '' phys .
e * 66 * , 051205 ( 2002 ) .
a. pikovsky and o. popovych , `` persistent patterns in deterministic mixing flows , '' europhys . lett . *
61 * , 625 ( 2003 ) . r. t. pierrehumbert , `` tracer microstructure in the large - eddy dominated regime , '' chaos solitons fractals * 4 * , 1091 ( 1994 ) .
thiffeault and s. childress , `` chaotic mixing in a torus map , '' chaos * 13 * , 502 ( 2003 ) .
w. liu and g. haller , `` strange eigenmodes and decay of variance in the mixing of diffusive tracers , '' physica d * 188 * , 1 ( 2004 ) .
j. sukhatme and r. t. pierrehumbert , `` decay of passive scalars under the action of single scale smooth velocity fields in bounded two - dimensional domains : from non - self - similar probability distribution functions to self - similar eigenmodes , '' phys .
e * 66 * , 056032 ( 2002 ) .
s. childress and a. d. gilbert , _ stretch , twist , fold : the fast dynamo _ ( springer - verlag , berlin , 1995 ) .
m. pollicott , `` on the rate of mixing of axiom a flows , '' invent .
math . * 81 * , 413 ( 1981 ) .
m. pollicott , `` meromorphic extensions of generalised zeta functions , '' invent . math . *
85 * , 147 ( 1986 ) .
d. ruelle , `` resonances of chaotic dynamical systems , '' phys .
* 56 * , 405 ( 1986 ) .
d. rothstein , e. henry , and j. p. gollub , `` persistent patterns in transient chaotic fluid mixing , '' nature * 401 * , 770 ( 1999 ) .
g. a. voth , t. c. saint , g. dobler , and j. p. gollub , `` mixing rates and symmetry breaking in two - dimensional chaotic flow , '' phys .
fluids * 15 * , 2560 ( 2003 ) .
y. b. zeldovich , a. a. ruzmaikin , s. a. molchanov , and d. d. sokoloff , `` kinematic dynamo problem in a linear velocity field , '' j. fluid mech .
* 144 * , 1 ( 1984 ) .
v. i. arnold , _ mathematical methods of classical mechanics _ , 2nd ed .
( springer - verlag , new york , 1989 ) .
thiffeault , `` advection diffusion in lagrangian coordinates , '' phys .
a * 309 * , 415 ( 2003 ) .
t. kato , _ perturbation theory for linear operators _ ( springer - verlag , berlin , 1980 ) .
e. merzbacher , _ quantum mechanics _ ( john wiley & sons , new york , 1970 ) . x. z. tang and a. h. boozer , `` finite time lyapunov exponent and advection - diffusion equation , '' physica d * 95 * , 283 ( 1996 ) .
thiffeault and a. h. boozer , `` geometrical constraints on finite - time lyapunov exponents in two and three dimensions , '' chaos * 11 * , 16 ( 2001 ) .
thiffeault , `` derivatives and constraints in chaotic flows : asymptotic behaviour and a numerical method , '' physica d * 172 * , 139 ( 2002 ) . e. ott , private communication . d. r. fereday and p. h. haynes , `` scalar decay in two - dimensional chaotic advection and batchelor - regime turbulence , '' preprint ( 2003 ) .
though we have not found a general method of solution of with the exponential given by , there is at least an approximate solution available that illustrates the broad features of a full solution . it also shows how the decay rate of the variance can become independent of the diffusivity in the lagrangian viewpoint , as in ref .
@xcite .
the method is based on defining a class of `` aligned '' wavenumbers ( _ i.e. _ , that live inside the cone of safety ) , and retaining only two of these modes .
these wavenumbers @xmath97 are defined by @xmath98 that is , @xmath99 , where @xmath100 is any initial wavenumber for large enough @xmath101 .
then @xmath102 satisfies @xmath103 so that with the choice @xmath104 , @xmath105 , the first kronecker delta in is unity for @xmath106 .
define @xmath110 which is independent of @xmath18 , since we have defined @xmath101 relative to the current iteration of the map .
it can be shown that then the coupling from @xmath108 to @xmath107 takes the simple form @xmath111 to recapitulate : at the @xmath18th iteration , the mode @xmath107 is mapped to itself with coupling amplitude @xmath112 , and @xmath108 is mapped to @xmath107 with amplitude @xmath113 .
it is easy to show that in this two - mode situation the decay rate is determined by the magnitude of @xmath113 .
all that remains is to find @xmath109 .
the `` lag '' @xmath109 is obtained by maximizing @xmath113 over @xmath109 ; @xmath109 has to be large enough that @xmath114 overcomes the tiny diffusivity in so the two @xmath115 terms do nt cancel in but not so large that @xmath112 is damped .
we are thus justified in approximating ( the other term in is smaller by a factor @xmath116 , which is small even for @xmath117 ) .
we then have @xmath118 since .
this is easily extremized over @xmath119 : the maximum @xmath120 is achieved for , for which @xmath121 .
the lag is then given by solving for @xmath109 in terms of the extremizing @xmath119 , @xmath122 which scales logarithmically with the diffusivity .
note that the decay rate is now completely independent of the actual value of the diffusivity : the lag adjusts itself to compensate , introducing a separation of scale between the dominant wavenumber @xmath107 and the largest wavenumber in the system , @xmath123 .
the actual decay rate as @xmath124 ( from the eulerian solution in @xcite ) is @xmath125 for small @xmath6 , compared to the two - mode lagrangian solution @xmath126 .
thus , most of the important behavior is captured by the two - mode solution .
the two modes can be seen in the spectrum of the strange eigenmode in fig .
[ fig : spectrum ] : the dominant peak at is @xmath107 ( @xmath127 in this case ) , and the peak at is @xmath108 .
the other peaks are modes that could be included to get a more accurate expression for the decay rate .
the two - mode solution also nicely illustrates the idea of the cone of safety : both modes are always inside it , and because the cone is shrinking by a factor @xmath128 at each iteration then the modes have to follow suit . the key difference with @xcite
is that here the concentration in the modes is _ mapped _ from one cone to another at each iteration , and is not part of the initial condition .
|
for a distribution advected by a simple chaotic map with diffusion , the `` strange eigenmode '' is investigated from the lagrangian ( material ) viewpoint and compared to its eulerian ( spatial ) counterpart .
the eigenmode embodies the balance between diffusion and exponential stretching by a chaotic flow .
it is not strictly an eigenmode in lagrangian coordinates , because its spectrum is rescaled exponentially rapidly .
* there are two main types of coordinates used to represent fluid flow and dynamical systems .
eulerian ( or spatial ) coordinates are fixed in space , while lagrangian ( or material ) coordinates follow parcels of fluid .
strange eigenmodes are persistent patterns in mixing they can decay slowly , and hence remain visible in the concentration field for a long time .
so far , these have been studied from the eulerian viewpoint . here
we describe the nature of the strange eigenmode in lagrangian coordinates for a simple map .
it is not a true eigenmode because its wavelength is continuously rescaled in time . *
|
data from tidal stream debris is a valuable resource for constraining galactic structure . in the last decade ,
several streams , with both globular cluster and dwarf galaxy progenitors , have been discovered in the milky way @xcite , as well as in nearby galaxies @xcite . by examining the density , kinematics , distribution , and structure of various tidal streams surrounding the milky way , a clearer picture of how our halo was built
can be developed @xcite .
in addition , stellar streams can be used as probes of the galactic gravitational potential , and thus constrains the shape of the dark matter halo @xcite .
@xcite , hereafter gd , announced the detection of a @xmath0 cold stellar stream in the galactic halo ( the stream itself we refer to henceforth as gd-1 , following gd ) , using stellar density counts extracted from the sloan digital sky survey ( sdss ; york et al .
this stream is extremely narrow , less than @xmath5 degrees in width , which is less than 50 pc at their measured distances of 7.3 to 9.1 kpc from the sun .
gd therefore concluded that the progenitor was a globular cluster , but the progenitor remains unidentified and could be completely disrupted . in this work
, we utilize newly available sloan extension for galactic understanding and exploration ( segue ; see yanny et al .
2009 ) spectroscopy of stars identified along the stream , that are available in the sloan digital sky survey ( sdss ) data release 7 ( dr7 ) , to constrain the orbit of the stream and search for possible progenitors .
the sdss imaging survey @xcite has made it possible to detect faint milky way halo streams from the spatial distribution of stars because it provides very accurate multicolor photometry for millions of galactic stars .
even with the well - calibrated sdss photometry @xcite , gd separated the faint structure from the background of milky way halo and disk stars only with careful filtering and smoothing techniques @xcite . to characterize the stream in more detail and to compute an orbit , we reanalyze the imaging data , supplemented through dr7 , and then add to it all newly available spectroscopic observations of gd-1 stream stars available in sdss dr7 . using the analysis of gd as a guide
, stars were selected from the sdss dr7 @xcite footprint from a rough color - magnitude box restricted to blue f turnoff and upper main sequence stars at distances of 7 to 20 kpc from the sun ( taking into account that the turnoff spans more than a magnitude of absolute magnitudes ) : @xmath6 .
magnitudes with @xmath7 subscript indicate those which have been corrected for reddening using the @xcite maps as implemented in sdss ( dr7 ) .
all sdss stars in the dr7 north galactic cap footprint which are in this color - magnitude box were selected and plotted in an ( @xmath8 ) stellar density diagram with pixels @xmath9 on a side .
examination of this diagram by eye showed that gd-1 stream stood out weakly from the background , with enough contrast so that the location of several fiducial points along the stream in ( @xmath8 ) could be mapped .
low - order polynomials were then fit to the positions of these points .
the lowest order best fit was of third order : @xmath10 by comparison with likely identified spectroscopic stream members we later verified that this polynomial matches the stream position within about @xmath11 for @xmath12 , and within about @xmath13 for @xmath14 and @xmath15 .
next , 147,537 stars within @xmath16 of this polynomial fit ( a conservative width is used to increase the signal to noise , it is not necessary to identify every star ) were selected from @xmath17 .
they represent the ` on - stream ' data set . for a control sample ( ` off - stream ' ) , 158,147 stars were selected within @xmath18 degrees of a curve offset by 5 degrees in @xmath19 from the ` on - stream ' data over a similar @xmath20 range . a hess diagram in ( @xmath21 ) of the difference between the on and off stream data @xcite
was generated and the results are shown in figure 1 .
figure 1 shows a clear faint turnoff around @xmath22 and upper main sequence descending to @xmath23 .
other features in figure 1 include an imperfectly subtracted thick disk residual at @xmath24 from @xmath25 and a residual from nearby m dwarfs at @xmath26 and @xmath27 .
the stars below the turnoff are concentrated in a relatively narrow band of color and magnitude , as expected for a stream localized in distance from the sun .
since the stream varies in distance from 7 to about 10 kpc for the stars in this figure , the actual width of the main sequence is substantially narrower than demonstrated here .
the blue turnoff of @xmath28 suggests a lower metallicity or younger age than that of the spheroid , which has a turnoff of @xmath29 and @xmath30 \sim -1.6 $ ] .
we superimpose with black dots in figure 1 a fiducial sequence from the cluster m92 .
the m92 sequence was calculated by starting with the fiducial sequence in undereddened @xmath31 from @xcite , converting it to @xmath32 using e(b - v ) of 0.02 and a distance to m92 of 8.2 kpc , as tabulated by @xcite , and then shifting it to the approximate distance to the gd-1 stream , which is approximately 9 kpc over this range of ra ( distance modulus @xmath33 ) .
this low - metallicity cluster , with @xmath30 = -2.3 $ ] @xcite , fits the main sequence and the turnoff reasonably well .
we do not see the giant branch or the horizontal branch ( which should be at @xmath34 ) in this figure , but given the faintness of the stream that is not unexpected .
we now refine the rough color - magnitude box used to detect the stream in equatorial coordinates .
the refined box is selected to allow for a stream which changes distance by @xmath35 over @xmath36 .
the box is defined as the union of three selection regions heavily outlined in figure 1 : a ) @xmath37 ( turn off ) b ) @xmath38 ( lower turn off ) and c ) @xmath39 ( upper main sequence ) .
all stars with images in the sdss dr7 northern galactic cap region which meet these criteria ( and have @xmath40 to exclude quasars ) , are selected and plotted in @xmath41 in the upper panel of figure 2 .
the gd-1 stream is faintly visible , running in an arc from about @xmath42 through @xmath43 , then down to @xmath44 where it crosses in front of the sagittarius stream , and then it is not clearly visible as it is lost in monoceros and other galactic halo stars near @xmath45
. the lower panel of figure 2 presents the same data in a galactic polar projection .
the gd-1 stream is clearly visible .
there are several features in the density of stars along the stream of unknown origin
. they could be the remains of a nearly dissolved progenitor , the result of interactions between the stream and the potential of the milky way s disk and halo , or unassociated spheroid substructure .
the kinematics of the stream , detailed below , reveal that it is on a retrograde orbit moving through the sequence of points in the order just described for the upper panel of figure 2 .
numerous other dwarf galaxies , streams and halo overdensities are present in figure 2 ; these will not be discussed further here . the segue survey @xcite , which is one of three surveys carried out as part of sdss - ii , obtained spectra of approximately 240,000 milky way stars toward @xmath46 sightlines that each covered seven square degrees of the sky , with an emphasis on obtaining spectra of fainter halo stars . while most of segue s 200 observing tiles were randomly distributed across the sdss imaging footprint , a few were placed on streams of known interest , including the gd-1 stream .
all segue spectra were processed through the standard sdss spectroscopic reduction pipelines @xcite , from which radial velocities accurate to about @xmath47 for objects at @xmath48 were determined .
in addition , the stellar spectra were processed through the segue stellar parameter pipeline ( sspp ) @xcite in order to obtain abundance ( [ fe / h ] ) , surface gravity ( log g ) , and other atmospheric parameter estimates .
we select from the sdss - ii / segue dr7 database all measured parameters of the @xmath49 spectra of stars within @xmath50 of the gd-1 stream described by eq .
1 . we further refined the selection to exclude objects far away from the fiducial m92 curve of figure 1 by requiring that they meet these color magnitude cuts : @xmath51 or @xmath52 or @xmath53 or @xmath54 or @xmath55 or @xmath56 .
the region bounded by these cuts is outlined with a light line in figure 1 .
most of the 4568 remaining spectra are concentrated in @xmath57 diameter patches centered on segue tiles , but some are part of the sdss - i and sdss - ii legacy surveys .
these latter surveys targeted nearly the entire sdss footprint spectroscopically , but with few and limited signal - to - noise on stellar targets ( since the sdss legacy survey primarily targets galaxy and quasar candidates ) .
we show in figure 3 the line - of - sight , galactic standard of rest velocities , @xmath58 , for each star in the sample , as a function of galactic longitude .
we calculate @xmath58 using : @xmath59 , where rv is the heliocentric radial velocity in @xmath60 and @xmath61 are the standard , sun - centered galactic coordinates of each star . a sine curve with amplitude 110 @xmath60 traces an approximate locus of nearby disk stars co - rotating with the sun .
spheroid stars occupy a broad range of @xmath62 centered at @xmath63 .
seven regions of interest are marked along the bottom of figure 3 , indicating areas with segue plates , where stars identified with the gd-1 stream will be selected .
the positions of these seven regions on the sky in equatorial coordinates are also indicated with circles and numbered in the upper panel of figure 2 .
regions 5 and 6 were specially targeted by segue with a tile directly on locations along the gd-1 stream . from examination of figure 3
, it appears that there is an excess of stars off the rotating disk locus at @xmath64 in regions 5 and 6 . to confirm that these are in fact gd-1 stream stars
, we isolate the stars in regions 5 and 6 and plot their velocity histogram in figure 4 .
the distribution in figure 4 is overlayed with gaussians for the thick disk ( dispersion of @xmath65 and an offset of @xmath66 ) , and inner halo ( dispersion of @xmath67 ) .
a significant peak is detected at @xmath68 which we associate with the gd-1 stream member stars .
we next examine the metallicity distribution of stars in this velocity peak in order to estimate the elemental abundance of the gd-1 stream . later in the paper
we will show that the individual @xmath62 velocities in regions 5 and 6 are @xmath69 and @xmath70 km s@xmath71 , respectively , so we chose a peak " velocity range of @xmath72 km s@xmath71 .
figure 5 shows the sspp abundance estimates for all stars with good metallicity estimates ( for a good estimate a turnoff star generally needs to be brighter than about @xmath73 ) .
errors on individual stars @xmath30 $ ] are approximately 0.3 dex for spectral type f objects with @xmath74 .
the histogram for all abundances of stars in regions 5 and 6 are plotted with a light line ( 1311 stars ) , those for stars in the velocity peak of figure 4 are indicated with a heavy line ( 115 stars ) .
the stars with velocities of the gd-1 stream are heavily biased towards lower metallicity stars , compared with those of the thick disk ( @xmath30 \sim -0.7 $ ] ) , or inner halo ( @xmath30 \sim -1.6 $ ] ) .
we estimate from figure 4 that about 30 stars in the spectroscopic dataset are from the gd-1 stream . to see the metallicitydistribution of the stars in the gd-1 stream
, we subtract a scaled version of the histogram with the light line from the histogram with the heavy line .
the scaling factor is ( 115 - 30)/(1311 - 30 ) . since the stars in the velocity selected region contain a smaller fraction of thick disk stars , the subtracted histogram is oversubtracted at high metallcities , and likely undersubtracted at spheroid metallcitites .
the mean of the stars in the shaded region is [ fe / h]=-1.9 , but the real metallicity of the stream is probably somewhat lower than this .
bins with negative counts do not appear in figure 5 .
we now return to the sample of stars in figure 3 , and select only those of very low metallicity ( @xmath75 < -1.65 $ ] ) in order to isolate stream members from the thick disk and halo field star populations . the low metallicity spectra with positions , colors , and magnitudes that make them candidates for gd-1 stream members
are shown in figure 6 .
several velocity peaks are now clearly separated from the disk and spheroid .
we now examine stars in each of the seven regions numbered in figure 6 and determine their observational properties .
we estimated the distance to the gd-1 stream at the positions of each of the seven regions that it overlaps using a matched filter algorithm .
we first generated the hess diagram from sdss dr7 data from a region about @xmath76 wide in ra and 1@xmath77 wide in dec in the vicinity of each plate , centered on the polynomial fit to the gd-1 stream .
then the hess diagram of the background was generated from two regions of sky with the same angular extent on the sky , but offset 1.5@xmath77 higher and 1.5@xmath77 lower in declination . the background hess diagram ( divided by two to correct for the difference in sky area )
was subtracted from the corresponding hess diagram on the gd-1 stream .
the subtracted hess diagrams are shown in figure 7 .
since the gd-1 stream is of quite low metallicity , we selected the globular cluster m92 ( [ fe / h]@xmath78 ) to compare with the observations .
we then constructed a m92 filter hess diagram with the similar method to @xcite .
we first broaden the m92 fiducial sequence from @xcite with the sdss photometric errors .
because we do not have a luminosity function for m92 stars , we used the luminosity function of m13 , estimated from sdss survey counts vs. magnitude for stars away from the core of m13 , to create the hess diagram .
as before , we assume the distance to m92 is 8.2 kpc .
the m92 filter was shifted from -0.5 to 1.5 magnitudes in r in steps of 0.05 mag . for each shift ,
we cross - correlated the m92 filter with the subtracted hess diagram : @xmath79\cdot f(g - r , r+{\delta}r)\ ] ] and estimated the error of cross - correlation function as : @xmath80\cdot f^2(g - r , r+{\delta}r)\}^{0.5}.\ ] ] in the above equations , o and b represent the hess diagrams of orbit and background segments , respectively .
f is the value of m92 filter .
then we define the maximum position , @xmath81 , of the cross - correlation function to be the actual distance modulus to the stream . near the maximum
, we can estimate the cross - correlation function by the taylor expansion : @xmath82 the second term in the right equals zero , since the first derivative is zero at maximum .
we use the above taylor expansion , using only the lowest order non - zero terms , to estimate the error in the cross - correlation function : @xmath83 since @xmath84 , @xmath85 the distance and corresponding error of each of the seven points along the stream with segue spectra are listed in table 1 .
figure 8 shows the same data as figure 7 , but overplotted with the m92 fiducial sequence shifted to the best estimated distance for each sky position .
figure 9 shows velocity histograms of low metallicity stars from figure 6 within @xmath86 of each selected region .
a gaussian velocity histogram , representing a halo distribution with @xmath87 and are normalized so that the area under the curve equals the number of stars in the histogram , is also shown .
the region number is indicated in the upper right corner of each panel , and the velocity of the peak most likely associated with the gd-1 stream is also indicated .
we list below some details of each region s selection , including the sdss / segue plates on which most of the objects were detected . in each region with a clear stream detection ,
the velocity and velocity dispersion for the gd-1 stream were computed using an iterative method that used only stars within one standard deviation of the mean velocity .
we computed the mean and standard deviation of the stars near the velocity peak .
then , we selected stars that were within one standard deviation of the mean and re - computed the mean and standard deviation
. the standard deviation calculated this way is an underestimate , since we have removed the tails of the distribution .
we corrected the standard deviation assuming gaussian tails .
this process was repeated until the computed mean and standard deviation matched the mean and standard deviation used to select the stars in the stream .
table 2 lists 48 high confidence gd-1 stream members .
the sample includes the stars in figure 6 ( which are selected based on angular distance from the gd-1 stream , photometric color and magnitude , and metallicity ) which have galactic longitude within two degrees of the seven plate centers , have velocities near the measured or expected velocities for the gd-1 stream , and which have proper motions that are consistent with our gd-1 stream model ( presented in 7 ) .
the velocity cuts for each of the seven regions are : 1 ) @xmath88 , 2 ) @xmath89 , 3 ) @xmath90 , 4 ) @xmath91 , 5 ) @xmath92 , 6 ) @xmath93 , and 7 ) @xmath94 .
the proper motions expected for each of the seven regions are listed in table 1 .
the stars in table 2 are within two sigma of the expected proper motions , where one sigma is 3 mas / yr in each of @xmath95 and @xmath96 .
table 2 includes each objects sdss - id number , coordinates , magnitude , colors , velocity , estimated metallicity , surface gravity and proper motion .
the proper motions ( @xmath97 ) listed here are from the usno - b / sdss catalog of @xcite , as extracted from the dr7 database s ` propermotions ' table .
errors on an individual @xmath98 star s measurement are about @xmath99 on each coordinate .
regions 4 , 5 , and 6 were specifically targeted by segue with plates along the gd-1 stream .
these regions , along with region 1 , targeted by the sdss legacy survey , constitute the 4 regions along the stream with spectroscopy used to fit a model orbit for gd-1 .
_ region 4 , plates 2889 and 2914 : _ the gd-1 stars are well sampled in this plate pair ; there is a clearly detected peak of ( primarily ) f turnoff stars at @xmath100 km s@xmath71 with @xmath101 km s@xmath71 , in the magnitude range @xmath102 .
the metallicity of stars in the velocity peak is about @xmath30 \sim -2.2 $ ] .
there are two bhbs in this sample near @xmath103 , corresponding to a distance of 7.2 kpc @xcite .
the distance to this region derived in section 3 from the turnoff photometry is @xmath104kpc , in good agreement with the bhb magnitudes .
the velocities of the two bhbs are -6 and -3 @xmath60 , in excellent agreement with the average of the turnoff star velocities , as are the metallicities ( [ fe / h ] = -2.0 and -2.1 , respectively ) .
_ region 5 , plates 2557 and 2567 : _ the plate spans the full width of the gd-1 stream .
there is a strong peak in the velocity distribution .
this peak has @xmath105 with @xmath106 km s@xmath71 .
the average @xmath107 magnitude in this range is @xmath73 , which is consistent with the distance estimation from color magnitude turnoff fitting .
the metallicity distribution of f turn - off stars in the peak shows that [ fe / h ] of this stream is about -2.05 .
the distance to this piece of the stream is @xmath108 .
_ region 6 , plates 2390 and 2410 : _ a strong and narrow peak is detected in figure 9 , at @xmath109 , @xmath110 km s@xmath71 and @xmath111 .
[ fe / h ] distributions show the metallicity peaks at about -2.05 , which is consistent with the [ fe / h ] peak find in plate 2567 and slightly higher than plate 2914 , this indicates that they are from the same stream .
a distance estimate puts stars in this region of the stream at @xmath112 from the sun .
_ region 1 , plate 1154 : _ this is a special sdss legacy plate , rather than a segue plate , but since it is at low galactic latitude it does have a large number of stellar candidates .
its important gd-1 stream candidate stars cluster at velocity @xmath113 , with @xmath114 km s@xmath71 , and ( lower s / n ) metallicity [ fe / h ] @xmath115 , which anchors the stream orbit away from regions 4,5 and 6 . with larger errors ,
the distance to this stream here is @xmath116 .
there is a second peak ( 3 stars ) at @xmath117 .
we discount this second peak as being associated with gd-1 , since two of its three members have [ fe / h ] = -1.8 , significantly higher than the average for other stream peak members .
this secondary peak could be related to the anti - center stream noted by @xcite , as the radial velocity of these stars is @xmath118 , is close to their value for acs - c ( see lower panel of figure 1 and figure 2 of that work ) .
the following 3 regions were not involved in the model fit ( see below ) , but as there are segue spectra available here along the orbit defined by the imaging arc defined in equation 1 , we examine these segue plates for possible stream members . _
region 2 , plates 1760 , 2433 , 2667 and 2671 : _ this plate pair was targeted by segue as it contains the open solar - metallicity cluster m67 .
the sagittarius stream also passes near by , along with the anti - center stream . by chance
, the gd-1 stream arc appears to pass within 1.3@xmath77 of the plate center , and several very low metallicity turnoff stars are detected with average @xmath119 .
the distance to the stream is @xmath120 .
_ region 3 , plates 2304 and 2319 : _ several stars with metallicity have velocities in a broad excess near @xmath121 , however there is not a significant candidate peak here .
a peak at @xmath122 has stars with @xmath123 >
\sim -1.8 $ ] , and is not a viable stream maximum . the distance from section 3 is estimated at @xmath124 .
_ region 7 , plate 2539 and 2547 : _ this plate pair is on the fitted arc of equation 1 , but the stream becomes too tenuous to be defined . we do not see a velocity peak here , but following the trend in the peaks of points 1 - 6 there may be two or three stars ( above background ) at @xmath125 , with the correct metallicity and proper motion to be associated with gd-1 .
we do not place as strong a confidence in the gd-1 membership of stars in this peak as the other 6 regions . a very uncertain distance estimate to the stream here ,
is about @xmath126 from the sun .
orbit fits , which were not fit to data in region 7 , confirm that this is the correct velocity to be looking for stream stars , but that the distance is somewhat underestimated .
we note that the estimated distances to individual regions are in very good agreement with those quoted by @xcite , indicating that the results are somewhat robust to details of the method and potential parameters .
the observed properties of the stars in the seven gd-1 stream candidate regions are summarized in table 1 , where we list region number ( @xmath127 ) ; equatorial coordinates , @xmath41 ; the corresponding galactic coordinates ( @xmath128 ) with errors ( we use @xmath19 to denote a measured error , to distinguish it from the intrinsic dispersion , which we denote with the symbol @xmath129 ) ; the average galactocentric standard of rest velocity and heliocentric radial velocity with an error , and the velocity dispersion . the velocity mean and dispersion were calculated as described in 3 .
the tabulated intrinsic dispersions are upper limits to the actual velocity dispersion of the stream ; since they are similar in size to the velocity errors for each individual spectrum , the measurement is consistent with an intrinsic velocity distribution of zero .
the error in the mean is the velocity dispersion divided by the square root of the number of stars used to compute it .
regions 2 , 3 , and 7 do not have clear , narrow peaks in the velocity distributions and therefore were not used to fit the orbit , though figure 6 shows there are excess stars at about the right velocities .
table 1 lists the computed galactic @xmath130 positions ( with respect to a right handed coordinate system with the sun at ( -8,0,0 ) kpc ) for each stream region and a distance to the stream at that region , again with errors . for the four regions numbered 1,4,5 and 6 , the velocity and position accuracies are highly significant , and there is high confidence of the gd-1 stream membership of stars highlighted in the corresponding boxes of figure 6 .
figure 10 highlights this confidence by showing a color - magnitude diagram of all high - confidence spectroscopic gd-1 stream candidates in regions 1 - 7 , shifted to a standard distance of 9 kpc based on the photometric hess diagram fitting .
these objects are also proper motion selected , in that only objects within @xmath131 of the proper motion of the best fit model are kept . superimposed over the spectral candidates
is the m92 fiducial sequence of @xcite shifted to a distance modulus of @xmath33 , identical to that used in figure 1 .
distances to the other regions were estimated as described in section 3 .
we now calculate our best metallicity estimate for gd-1 by selecting only the 48 stars with spectra in figure 10 .
we note that these stars were pre - cut on metallicity at an earlier stage ( figure 6 ) to have @xmath132 < -1.65 $ ] . a histogram with bins similar to the measurement error
yields a gd-1 stream metallicity of [ fe / h]@xmath133 dex with a dispersion of @xmath134 dex ( essentially the measurement error ) .
in addition to these statistical errors , there may be systematic errors in the metallicity determinations from sdss dr7 of @xmath135 dex @xcite .
we now use the data listed in table 1 to fit an orbit for the stream , assuming a fixed galactic potential .
@xcite postulated the progenitor of this stream is a globular cluster because it has a narrow width in the sky .
as a cluster orbits the galaxy , stars farther from the progenitor will depart from the orbit due to dynamical friction and scattering of the stream stars . because the progenitor is presumed to be a compact object with a few km s@xmath71 velocity dispersion , it is reasonable to assume that the stars in the tidal stream lie approximately on the orbit of the globular cluster @xcite .
dwarf galaxies , on the other hand , will experience larger spatial dispersions because they have larger dispersions in their energies . therefore , in this paper , we fit the orbit to the positions and velocities of the stars in the tidal stream .
the galactic potential model used in this work follows directly from @xcite .
we use a 3 component potential - disk , bulge , and halo , of the form given in equations ( [ disk ] ) , ( [ bulge ] ) , and ( [ halo ] ) .
@xmath136 @xmath137 @xmath138 in these potentials , @xmath139 and @xmath140 , where @xmath141 are galactocentric cartesian coordinates .
we adopt the sun - galactic center distance so that @xmath142 .
the symbol @xmath143 is the cylindrical radius from the center of the galaxy , whereas @xmath144 refers to the sun - centered distance to an arbitrary point along the stream . in these potentials ,
@xmath145 and @xmath146 are the masses of the disk and bulge , respectively .
the spatial extent of the potentials are determined by @xmath147 , the disk scale length , @xmath148 the disk scale height , @xmath149 the bulge scale radius , and @xmath150 the dark matter halo scale length .
additionally , @xmath151 describes the dark matter halo dispersion speed ( which is related to the total dark matter halo mass ) , and @xmath152 represents the dark matter halo flattening in the @xmath153 direction .
we found that the parameters of the galactic potential were not well constrained by the path of the tidal stream , so these parameters ( table [ parameters ] ) were held constant , with the same values used by @xcite . in this work
, we will fit four orbital parameters , given in table [ fitpars ] .
@xmath154 are the distance ( from the sun ) to , and velocities of , region 5 . given a galactic potential , these parameters fully specify the gc orbit .
we then evolve the test particle forward and backward from the starting location , using the @xmath155 and @xmath156 tools in the nemo stellar dynamics toolbox @xcite .
we convert the resulting orbit into @xmath61 and perform the goodness of fit calculation .
to find reasonable initial values for these parameters , we imagine placing a test particle in region 5 at @xmath157 kpc ) . we then construct a vector between the @xmath158 and @xmath159 points .
this gives us the direction of the total velocity , because we are assuming the orbit passes through both of these points .
the principle initial values for the parameters in region 5 are @xmath154 = ( 8.0 kpc , -94 km / s , -285 km / s , -104 km / s ) . in practice
we start searching for the best parameters in a range of values near these approximate values for the orbital parameters .
the parameter selection ranges are given in table [ fitpars ] . in order to find the best - fit orbit to the data given in table 1 ,
it is necessary to define a goodness - of - fit metric , and search the relevant parameters for the minimum value of this metric .
the metric for an orbit of this type involves three important factors : the orbit passing through the plate locations in the sky , having the proper velocities at these locations , and having the correct distances . in order to consider all of these factors , we define three chi - squared values , one for each of the relevant variables , and simply sum them to create the total goodness of fit metric . specifically , @xmath160 @xmath161 @xmath162 @xmath163 where @xmath164 , @xmath127 being the number of data points , and @xmath165 being the number of parameters . to calculate these @xmath166 values
, we calculate a model orbit using the selected parameters .
we search the orbit for the @xmath167 values from the plates , and use the associated values of @xmath148 , @xmath62 , and @xmath144 to compute @xmath166 .
we now optimize the orbital parameters so that @xmath166 is minimized . to do this
, we choose an initial set of parameters , calculate an orbit using the nemo stellar dynamics toolbox @xcite , and calculate @xmath166 .
we then use a gradient descent method to adjust the parameters to new values , generate a new orbit , and recalculate @xmath166 .
this process is continued until the true minimum value of @xmath166 is found , and the associated parameters describe the best fit orbit .
let the vector @xmath168 describe the parameters .
for each @xmath169 there is an associated @xmath170 .
we choose an initial set of parameters @xmath171 , and find @xmath172 .
we then iterate the parameters by equation ( [ gradientdescent ] ) .
@xmath173 we calculate the gradient using a finite difference method , shown in equation ( [ finitedifference ] ) . @xmath174 different values of @xmath175 are used because the parameters are on different scales , it would not be appropriate to use the same step size for them all .
the step sizes for the parameters are given in table [ fitpars ] .
@xmath176 is a variable - learning parameter .
it initially begins at @xmath177 , and if the new value of @xmath178 is smaller than the old , then @xmath176 is multiplied by @xmath179 , if not , it is multiplied by @xmath180 .
the purpose of this is to ensure if a minimum is being found , then it is found faster than with a constant - learning parameter .
we also multiply it by the associated @xmath175 value to make the step size appropriate for the parameter being considered .
the uncertainties of the best - fit parameters can be estimated from the shape of the @xmath166 surface at its minimum . to do this ,
we follow the method outlined by @xcite .
we construct a matrix @xmath181 of second partial derivatives of the @xmath166 surface , evaluated at the minimum found by the gradient descent . the error estimate for the @xmath182 parameter
is @xmath183 .
the @xmath181 matrix is defined as @xmath184 the matrix @xmath185 is the hessian matrix , whose elements are given by @xmath186
we select five initial sets of parameters by randomly choosing values within the ranges given in table [ fitpars ] .
we perform the gradient descent to reach the best fit parameters .
we then estimate the parameter errors using the hessian method outlined above .
the best - fit parameters and their errors are @xmath187 .
the chi - squared of this fit is 2.07 .
the negative velocities indicate a retrograde orbit .
the perigalacticon for this orbit is located @xmath188 kpc from the galactic center at @xmath189 , near region 6 .
the space velocity of stars in the model at this position is 276 @xmath60 .
the apogalacticon is at @xmath190 kpc from the galactic center , toward @xmath191 , though we do not observe this direction on the sky .
all errors are @xmath192 .
these orbital parameters are fairly insensitive to the choices of parameters in the galactic potential . in particular , they are good estimates of the orbital parameters for a wide range of @xmath152 and @xmath150 .
to investigate whether we could determine anything about the shape of the dark matter halo from this tidal tail , we performed parameter sweeps on @xmath152 and @xmath150 while keeping the kinematic parameters constant ( figure 11 ) .
we see that very low flattenings are inconsistent with the data , but a wide range of @xmath152 and @xmath150 are allowed .
the potential assumed in fitting the orbit is a standard logarithmic flat - rotation curve dark matter halo plus a stellar disk .
since the gd-1 stream approaches within 15 kpc of the galactic center , the effects of the massive disk are felt by the orbit , and increasing the relative mass of the disk vs. the halo can mimic the effects of a flattened halo . at perigalacticon ,
the disk exerts twice as much gravitational force as the halo . given this , it is not surprising that the stream orbit depends minimally on the halo parameters .
more models and constraints , from this and other streams , are clearly needed to constrain the shape of the dark matter halo .
the upper panel of figure 12 shows the orbit in @xmath61 with the stream locations shown .
the model prediction is in very good agreement with the experimental observations .
the middle and lower panels of figure 11 show the orbit in @xmath167 versus @xmath58 and @xmath167 versus distance from the sun .
we also see good agreement with the experimental observation .
figure 13 shows the orbit projected into the three planes of galactic coordinates @xmath141 .
we deduce an orbital eccentricity @xmath193 ( one sigma error ) and an inclination to the galactic plane of @xmath194 .
arrows show the relative direction of the stream s retrograde motion compared to the milky way .
the final columns of table 1 show the predicted proper motions ( @xmath97 ) for stars in the stream at each region 1 - 7 based on the distances in table 1 .
these predictions may be compared with actual observed proper motions for stream star candidates in table 2 at each region . in general
the agreement is quite good for regions 2 - 6 , given proper motion errors of @xmath195 in each coordinate .
spectral candidates more than @xmath196 away were excluded from figure 10 , dropping about 20% of the candidates , leaving a generally good fit to a shifted m92 fiducial sequence for these regions .
region 7 had fewer good proper motion matches , and it is possible that we are not seeing gd-1 stream candidates here .
figure 14 shows the photometrically chosen stars with proper motions available near regions 1,4,5 and 6 , along with an equivalent set of field stars ( chosen 5 degrees away ) for comparison .
there s a clear excess of ` on - stream ' stars in the lower right quadrant of each region on - stream these are likely stream members .
the estimated tangential velocities ( relative to the sun ) are given .
to search for a possible progenitor , we selected all milky way globular clusters from @xcite that had metallicities in the range @xmath197[fe / h]@xmath198
. only seven of these globular clusters ( terzan 8 , arp2 , ngc 6809 , ngc 6749 , ngc 6341 , ngc 6681 , and ngc 6752 ) are within @xmath76 of the gd-1 orbit .
additionally , we considered ngc 2298 , which is @xmath199 from the orbit , and has a metallicity of @xmath30 = -1.85 $ ] .
we compared the positions and velocities of these globular clusters with an orbital path that extends all the way around the milky way . to create a stream of
this length would require a globular cluster to orbit the milky way for on the order of gigayears , with the length depending on the concentration of the progenitor , as well as the shape and location of the progenitor s orbit .
of the eight globular clusters , ngc 6809 , ngc 6749 , and ngc 6752 are ruled out because their distances are more than a factor of two different from the distance to the orbit .
the remaining five clusters had radial velocities that are inconsistent with the predicted orbit by more than 50 km / s . we therefore conclude that the milky way globular cluster catalog published by @xcite does not contain the progenitor of this stream .
we use spectroscopic kinematic and abundance information to isolate stars in the gd-1 stream , and use the positions and velocities of those stars to derive orbital parameters for its orbit .
the gd-1 stream is moving very rapidly on a retrograde orbit around the milky way . in the region of the orbit which is detected
, it has a distance of about 7 - 11 kpc from the sun .
the stream s orbit takes it to apogalactic distances of @xmath200 kpc , and it has a perigalacticon of @xmath201 kpc , implying an eccentricity of @xmath202 .
the inclination to the galactic plane is about @xmath203 .
the metallicity of the stream is [ fe / h]@xmath204 plus systematic errors of a few tenths dex .
none of the known globular clusters in the milky way have positions , radial velocities , and metallicities that are consistent with being the progenitor of the gd-1 stream .
the consistency between the proper motions of these stream candidates and our best fit model gives us further confidence that we have identified stream members and that our model accurately represents the path on the sky of the stream stars .
while we claim only consistency here between the proper motion data and our model , we note that more detailed fits to the proper motion ( in addition to the radial velocities ) for such nearby streams can be a crucial tool in constraining the halo potential shape and other parameters .
we acknowledge a careful reading and several important suggestions from the anonymous referee which significantly improved the observational analysis section of this paper . b.a.w and h.j.n .
acknowledge support from the national science foundation , grant ast 06 - 06618 .
we gratefully acknowledge peter teuben for many useful nemo discussions and lee newberg for assisting us with calculating the measurement errors .
we also acknowledge useful discussion with linda sparke regarding streams in potentials . z.h.t .
acknowledges support from the national natural science foundation of china under grant no .
has received partial support for this work from grants phy 02 - 16783 and phy 08 - 22648 : physics frontiers center / joint institute for nuclear astrophysics ( jina ) , awarded by the u.s . national science foundation .
funding for the sdss and sdss - ii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the u.s .
department of energy , the national aeronautics and space administration , the japanese monbukagakusho , the max planck society , and the higher education funding council for england .
the sdss web site is http://www.sdss.org/. the sdss is managed by the astrophysical research consortium for the participating institutions .
the participating institutions are the american museum of natural history , astrophysical institute potsdam , university of basel , cambridge university , case western reserve university , university of chicago , drexel university , fermilab , the institute for advanced study , the japan participation group , johns hopkins university , the joint institute for nuclear astrophysics , the kavli institute for particle astrophysics and cosmology , the korean scientist group , the chinese academy of sciences ( lamost ) , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , ohio state university , university of pittsburgh , university of portsmouth , princeton university , the united states naval observatory , and the university of washington .
abazajian , k. et al . , 2009 ,
apjs , submitted , arxiv:0812.0649 adelman - mccarthy , j.k . , ageros , m.a . ,
allam , s.s .
et al . , 2008 , apjs , 175 , 297 allende prieto , c. et al .
2008 aj , 136 , 2070 . an , d. , johnson , j.a . ,
clem , j. l. , et al .
, 2008 , apjs , in press , arxiv:0808.0001 bellazzini , m. , ferraro , f. , and ibata , r. 2002 , aj , 125 , 188b belokurov , v. , et al .
2007 , , 658 , 337 bullock , j. s. , & johnston , k. v. 2005 apj 635 , 931 chapman , s. c. et al .
2008 , mnras , submitted , arxiv:0808.0755 clem , j. l. , vanden berg , d. a. , & stetson , p. b. 2008 aj 135 , 682 cole , nathan , et al .
2008 , , 683 , 750 duffau , s. , zinn , r. , vivas , a. k. , carraro , g. , mendez , r. a. , winnick , r. , & gallart , c. 2006 , , 636 , l97 fellhauer , m. , belokurov , v. , evans , w. et al .
2006 , , 651 , 167 ferguson , a. m. n. , johnson , r. a. , faria , d. c. , irwin , m. j. , ibata , r. a. , johnston , k. v. , lewis , g. f. , & tanvir , n. r. 2005 , , 622 , l109 fukugita , m. , ichikawa , t. , gunn , j.e . , doi , m. , shimasaku , k. , and schneider , d.p .
1996 , aj , 111 , 1748 grillmair , c. j. & dionatos , o. 2006 , , 643 , l17 ( gd ) grillmair , c. j. 2006 , , 645 , l37 grillmair , c. j. , & johnson , r. 2006 , , 639 , l17 grillmair , c. j. 2006 , , 651 , l29 grillmair , c. j. , carlin , j. l. & majewski , s. r. 2008 , , 689 , l117 grillmair , c. j. arxiv:0811.3965 , apj , in press .
gunn , j.e .
, et al . 1998 , aj , 116 , 3040 gunn , j.e .
2006 , aj , 131 , 2332 harris , w. e. 1996 , , 112 , 1487 helmi , a. , 2004 , , 610 , l97 hogg , d. w. , finkbeiner , d.p . ,
schlegel , d.j . , and gunn , j.e .
2001 , aj , 122 , 2129 ibata , r. a. , gilmore , g. , and irwin , m.j .
1995 , mnras , 277 , 781 ibata , r. , irwin , m. , ferguson , a. m. n. , lewis , g. f. , tanvir , n. 2001 nature 412 , 49 ibata , r. , lewis , g. f. , irwin , m. , totten , e. , and quinn , t. 2001 , , 551 , 294 ivezic , z. , et al . 2004 , an , 325 , 583 law , david r. , johnston , kathryn v. , and majewkski , steven r. 2005 , , 619 , 807 lee , y. s. , et al . 2008a , , 136 , 2022 lee , y. s. , et al . 2008b , , 136 , 2050 leon , s. , meylan , g. , and combes , f. 2000 , aap , 359 , 907 majewski , steven r. , skrutskie , m. f. , weinberg , martin d. , ostheimer , james c. 2003 , , 549 , l199 martinez - delgado et al .
2001 , , 549 , l199 martinez - delgado et al .
2008 , , submitted , arxiv:0801.4657 martinez - delgado , d. , gabany , r. j. , pennarrubia , j. , rix , h .- w . , majewski , s. r. , trujillo , i. , & pohlen , m. , in `` highlights of spanish astrophysics v '' , proceedings of the viii scientific meeting of the spanish astronomical society ( sea ) , springer , arxiv:0812.3219 munn , j. a. et al .
2004 aj 127 , 3034 newberg , h.j . , yanny , b. , et al .
2002 , apj , 569 , 245 newberg , h.j . ,
yanny , b. , cole , n. , et al . , 2007,apj , 668 , 221 odenkirchen , m. , et al .
2001 , , 548 , l165 odenkirchen , m. , et al .
2003 , aj , 126 , 2385 pier , j.r . ,
munn , j.a .
, hindsley , r.b . ,
hennessy , g.s .
, kent , s.m . ,
lupton , r.h . , and ivezic , z. 2003 , aj , 125 , 1559 rockosi , c.m . , odenkirchen , m . ,
grebel , e.k .
et.al.,2002,aj 124 , 349 schlegel , d.j . , finkbeiner , d.p .
, & davis , m. 1998 , apj , 500 , 525 sirko , e. et al . 2004 aj 127 , 899 smith , j.a . , et al .
2002 , aj , 123 , 2121 stoughton , c. , et al .
2002 , aj , 123 , 485 teuben , p.j .
the stellar dynamics toolbox nemo , in : astronomical data analysis software and systems iv , ed .
r. shaw , h.e .
payne and j.j.e .
( 1995 ) , pasp conf series 77 , p398 .
tucker , d. , et al .
2006 , an , 327 , 821 vivas , a. k. et al .
2001 , , 554 , l33 yanny , b. , newberg , h. j. , et al .
2000 , , 540 , 825 yanny , b. , rockosi , c. , newberg , h. , knapp , g. et al .
2009 aj , submitted .
york , d.g .
2000 , , 120 , 1579 rrrrrrrrrrrrrrrrrr 1&126.58&-0.22&224.47&0.5&20.88&0.5&108&259&5&11&-14.86&-6.72&3.66&10.4&1.2&7.0&-6.4 + 2&131.92&11.17&215.93&0.2&30.83&0.2&69&188 & & & -12.51&-3.27&3.32 & 6.5&0.6&8.6&-7.4 + 3&138.65&22.29&206.03&0.2&40.89&0.2 & & & & & -12.75&-2.31&4.57&7.0&0.4&10.0&-7.3 + 4&144.25&30.09&197.00&0.2&47.54&0.2&-7&39&1&3.9&-12.83&-1.47&5.52&7.5&0.3&10.9&-6.5 + 5&157.92&44.19&172.30&0.2&57.24&0.2&-71&-88&2&5.3&-12.30&0.58&6.74&8.0&0.5&11.8&-2.4 + 6&163.69&48.32&161.95&0.2&59.02&0.2&-87&-124&2&9.2&-12.30&1.41&7.54&8.8&0.8&11.5&-0.6 + 7&217.50&57.51&99.95&1.0&55.00&1.0 & & & & & -8.98&5.6&8.08&9.9&1.2&4.1&5.2 + rrrrrrrrrrrrrrrr 1154 - 53083 - 266&1&125.683891&-0.439561&113.4&7.4&264.7&19.107&0.909&0.275&-1.89&0.08&3.46&0.25&3.5&-4.3 + 1154 - 53083 - 145&1&126.577150&-0.439386&97.8&7.8&249.4&19.102&0.904&0.180&-2.14&0.07&3.62&0.23&12.9&-10.5 + 1154 - 53083 - 155&1&126.645407&-0.340477&109.0&3.6&260.3&18.117&0.935&0.275&-2.06&0.04&3.30&0.22&11.6&-3.9 + 1760 - 53086 - 339&2&131.406984&9.770413&65.8&14.5&186.9&18.389&0.909&0.227&-2.18&0.11&4.05&0.29&9.4&-8.4 + 2671 - 54141 - 432&2&132.157853&11.135851&77.3&6.0&193.9&18.780&0.841&0.406&-2.24&0.04&3.54&0.22&10.0&-5.2 + 2667 - 54142 - 427&2&132.367656&11.325495&70.5&4.0&186.5&17.433&0.987&0.420&-2.17&0.02&2.77&0.24&6.7&-7.4 + 2667 - 54142 - 604&2&133.073392&12.266884&62.3&2.5&175.1&15.859&1.159&0.486&-2.08&0.03&2.22&0.15&8.4&-6.1 + 2319 - 53763 - 347&3&138.399875&22.413022&38.1&11.6&114.5&19.451&0.843&0.243&-2.24&0.22&3.80&0.09&13.2&-7.7 + 2319 - 53763 - 358&3&138.450751&22.434572&40.4&4.7&116.7&17.996&0.865&0.259&-1.93&0.02&3.90&0.10&12.5&-9.7 + 2319 - 53763 - 387&3&138.669378&22.559736&48.9&11.8&124.7&19.603&0.874&0.385&-2.07&0.07&3.68&0.30&12.1&-7.9 + 2889 - 54530 - 311&4&142.787815&29.461085&-0.9&4.2&47.8&17.454&0.972&0.417&-1.98&0.04&3.23&0.28&10.7&-9.4 + 2914 - 54533 - 297&4&142.978804&29.136307&-3.0&3.9&46.9&18.139&0.925&0.189&-2.09&0.04&3.94&0.16&8.8&-9.3 + 2889 - 54530 - 240&4&143.339524&29.620398&-6.2&4.7&41.7&17.933&0.883&0.256&-2.14&0.03&3.63&0.17&9.6&-6.7 + 2889 - 54530 - 204&4&143.424098&29.036453&-3.6&5.1&46.5&17.822&0.872&0.271&-2.21&0.05&3.28&0.20&6.7&-4.5 + 2889 - 54530 - 215&4&143.453188&29.120632&-3.4&1.7&46.4&14.981&1.194&-0.237&-2.09&0.11&3.20&0.17&9.5&-10.9 + 2889 - 54530 - 238&4&143.552948&29.477508&-4.3&4.6&44.1&17.843&0.888&0.293&-2.21&0.01&3.60&0.18&8.4&-9.1 + 2889 - 54530 - 225&4&143.597836&29.802697&-6.1&1.7&41.0&14.845&1.196&-0.102&-1.98&0.06&3.34&0.09&13.7&-11.1 + 2914 - 54533 - 171&4&143.718962&29.874100&-3.6&8.0&43.2&19.432&0.747&0.285&-2.26&0.08&3.33&0.56&15.4&-8.7 + 2914 - 54533 - 458&4&144.107345&30.737924&-9.8&4.8&33.6&18.716&0.779&0.216&-2.26&0.05&4.41&0.10&13.1&-6.5 + 2914 - 54533 - 453&4&144.120941&30.947724&-1.4&4.4&41.2&17.955&0.827&0.220&-2.10&0.02&3.79&0.28&14.3&-6.3 + 2914 - 54533 - 509&4&144.275720&30.280468&-4.4&3.9&40.6&18.292&0.909&0.187&-2.25&0.01&3.46&0.31&9.5&-12.2 + 2914 - 54533 - 515&4&144.487234&30.410401&-4.7&3.6&39.8&18.247&0.855&0.232&-1.82&0.04&3.72&0.15&10.5&-1.6 + 2914 - 54533 - 540&4&144.561728&30.968385&-1.9&11.0&40.4&20.190&0.854&0.344&-1.95&0.10&3.46&0.21&12.6&-3.8 + 2567 - 54179 - 396&5&157.181542&44.514349&-67.9&6.8&-85.0&18.745&0.841&0.254&-2.08&0.04&3.78&0.19&13.9&1.8 + 2567 - 54179 - 246&5&157.565788&43.643316&-73.0&6.0&-87.1&18.630&0.934&0.167&-2.23&0.01&3.71&0.18&12.9&-0.9 + 2567 - 54179 - 238&5&157.691771&43.797748&-71.9&12.7&-86.7&19.546&0.908&0.206&-2.21&0.02&3.27&0.16&10.1&-5.6 + 2567 - 54179 - 189&5&157.817344&43.923322&-70.1&11.9&-85.5&19.761&0.772&0.342&-1.88&0.07&4.23&0.36&9.0&3.5 + 2567 - 54179 - 211&5&157.889540&43.500286&-65.2&10.0&-79.0&19.309&0.901&0.205&-1.89&0.15&3.74&0.07&9.2&-0.3 + 2567 - 54179 - 212&5&158.051293&43.644022&-67.6&5.2&-82.1&18.349&0.839&0.208&-1.90&0.07&3.67&0.26&9.7&-0.2 + 2567 - 54179 - 170&5&158.137638&44.086320&-58.2&9.8&-74.4&19.112&0.863&0.201&-2.04&0.06&3.64&0.17&14.4&-1.7 + 2567 - 54179 - 491&5&158.437074&44.466389&-69.2&9.1&-87.1&18.242&0.880&0.209&-2.02&0.09&3.95&0.16&8.9&-1.9 + 2567 - 54179 - 511&5&158.700621&44.530392&-68.9&4.4&-87.2&18.083&0.954&0.237&-2.22&0.01&3.18&0.25&12.1&1.3 + 2557 - 54178 - 498&5&158.747249&44.686776&-74.2&3.9&-93.2&16.794&1.048&0.476&-2.12&0.02&2.60&0.27&13.5&2.7 + 2410 - 54087 - 297&6&162.371533&47.207633&-87.0&6.9&-118.3&18.893&0.885&0.255&-2.26&0.02&3.46&0.20&15.4&-1.3 + 2410 - 54087 - 236&6&162.451620&48.002695&-85.2&4.2&-119.5&18.305&0.896&0.250&-1.71&0.04&3.51&0.21&8.6&4.8 + 2410 - 54087 - 288&6&162.547222&46.990942&-88.0&12.2&-118.7&19.344&0.893&0.164&-2.08&0.08&3.78&0.14&9.6&-3.8 + 2390 - 54094 - 256&6&162.557763&47.251097&-95.0&2.6&-126.6&16.945&1.048&0.036&-1.83&0.00&3.89&0.19&16.1&-1.5 + 2410 - 54087 - 380&6&162.743760&48.903984&-85.3&5.1&-123.1&18.632&0.840&0.269&-1.80&0.07&3.81&0.05&12.7&-1.6 + 2410 - 54087 - 173&6&163.799206&47.813588&-88.8&7.1&-123.6&18.828&0.845&0.200&-2.06&0.07&3.86&0.30&8.9&-1.8 + 2410 - 54087 - 539&6&164.064054&49.034036&-92.2&8.2&-131.6&19.293&0.925&0.169&-1.68&0.11&3.10&0.30&8.1&-1.7 + 2410 - 54087 - 501&6&164.334475&48.472850&-81.1&7.8&-118.8&19.401&0.925&0.202&-1.76&0.08&3.80&0.28&7.3&0.0 + 2410 - 54087 - 504&6&164.371325&48.224481&-90.6&8.8&-127.4&19.409&0.841&0.208&-1.75&0.07&4.30&0.14&11.3&-0.6 + 2390 - 54094 - 565&6&164.616981&49.202693&-81.1&2.0&-121.6&16.919&1.105&0.487&-1.86&0.01&2.93&0.19&6.9&3.7 + 2390 - 54094 - 615&6&165.199453&48.669262&-92.4&3.9&-131.5&17.895&0.896&0.385&-2.24&0.02&3.81&0.16&11.8&-1.7 + 2410 - 54087 - 637&6&165.381730&48.671045&-87.1&6.2&-126.4&18.928&0.887&0.206&-2.29&0.05&3.84&0.12&7.8&-0.2 + 2539 - 53918 - 406&7&217.430060&59.067783&-163.8&3.0&-297.5&15.805&1.226&-0.133&-2.11&0.03&3.29&0.12&5.9&3.7 + 2539 - 53918 - 070&7&219.681591&57.897316&-148.9&10.7&-283.7&17.970&0.862&0.199&-2.15&0.02&3.73&0.04&2.5&10.8 + 2539 - 53918 - 596&7&219.898491&58.258368&-157.6&5.5&-293.3&17.035&1.197&-0.167&-2.10&0.26&2.92&0.43&1.0&2.9 + cc @xmath20 & @xmath205 + @xmath145 & @xmath206 + @xmath146 & @xmath207 + @xmath147 & @xmath208
kpc + @xmath148 & @xmath209 kpc + @xmath149 & @xmath210 kpc + @xmath151 & @xmath211 km / s + @xmath152 & 1.0 + @xmath150 & 12 kpc + [ parameters ] cccc @xmath212 & sun - centered distance of region 5 & 0.1 kpc & 5 : 13 kpc + @xmath213 & x velocity of region 5 & 1 km / s & -130 : -60 km / s + @xmath214 & y velocity of region 5 & 1 km / s & -320 : -230 km / s + @xmath215 & z velocity of region 5 & 1 km / s & -80 : -120 km / s + [ fitpars ]
|
we use velocity and metallicity information from sdss and segue stellar spectroscopy to fit an orbit to the narrow @xmath0 stellar stream of grillmair and dionatos .
the stars in the stream have a retrograde orbit with eccentricity @xmath1 ( perigalacticon of 14.4 kpc and apogalacticon of 28.7 kpc ) and inclination approximately @xmath2 . in the region of the orbit which is detected
, it has a distance of about 7 to 11 kpc from the sun . assuming a standard disk plus bulge and logarithmic halo potential for the milky way stars plus dark matter , the stream stars are moving with a large space velocity of approximately @xmath3 at perigalacticon . using this stream alone , we are unable to determine if the dark matter halo is oblate or prolate .
the metallicity of the stream is [ fe / h ] @xmath4
. observed proper motions for individual stream members above the main sequence turnoff are consistent with the derived orbit .
none of the known globular clusters in the milky way have positions , radial velocities , and metallicities that are consistent with being the progenitor of the gd-1 stream .
|
the much anticipated experiments at the large hadron collider ( lhc ) are expected to refine our understandings of the standard model ( sm ) further and also shed some light on physics beyond the sm . but most importantly
, the lhc is envisioned as the machine to complete the picture of the sm by discovering the higgs boson , or instead , give a hint into the mechanism responsible for electroweak symmetry breaking .
being a subject of so much speculation and study in particular , one would always want to find out if the study on the higgs sector itself may reflect information on any kind of new physics beyond the sm . in this work
, we explore this above possibility by studying the effects of new physics on the signal of higgs boson in the light of experiments at the lhc . within some models of extradimensions motivated from the framework of string theory @xcite ,
non - gravitational fields are also free to propagate in the bulk provided they do not disturb the experimental constraints .
one such scenario , referred to as the universal extra dimension ( ued ) model @xcite , allows all the sm fields to propagate in the extra dimension . in the effective four dimensional space - time
the effects of the extra dimension is felt through the kaluza - klein ( kk ) excitations of these bulk fields which interact with the sm particles ( identified as the zero modes of the excitations ) . at tree level
, the momentum along the extra dimensions is conserved , which requires pair production of these kaluza - klein ( kk ) modes at colliders and preventing tree level mixing effects from altering precision electroweak measurements .
values of the compactification scale are constrained , and it has a lower bound of about 300 gev @xcite .
the phenomenological implications of ued have been extensively studied in the literature @xcite .
direct detection of ued kk states at future colliders requires them to be pair produced due to the kk number conservation and hence already puts a limit on the minimum energy at which the collider should run to produce these particles .
the lhc will invariably be able to probe physics at the energy regime unconstrained by precision measurements , where such particle resonances are expected to occur .
the main experimental signal for the production and decay of kk excitations at hadron colliders will be the observation of events with multiple leptons and jets of moderately high energies in association with large missing energy @xcite .
this draws a lot of parallels with supersymmetric searches at the hadron colliders and it will prove to be a strong challenge to distinguish the signatures .
we refer the readers to @xcite where one gets a nice review of different interesting signatures at colliders
. however , it would be worthwhile to look for its effects in higgs boson studies whose signals would be extensively studied .
there would be possible modifications in the signal , through the modification of higgs decay properties due to the kk states contributing in the loop mediated decay modes of the higgs boson .
the partial decay widths for @xmath1 , @xmath2 and @xmath3 decay modes which are driven by loops can be substantially modified due to kk excited modes of sm particles running in the loops .
there is in fact remarkably significant enhancement in the partial decay width of the higgs in @xmath1 due to the excited top quark loops @xcite .
this can greatly enhance the higgs production at the lhc viz .
the @xmath4 mode of production .
the production mode @xmath5 is relevant for the higgs boson lying in the mass range of @xmath6 gev . due to the limitations in the resolution of the calorimeter
the measurement of the decay width of the higgs boson in this mass range will be impossible
. thus it would be impossible to study the partial widths @xmath7 and @xmath8 and look for any kind of ued effects .
it would require study of event rates for its production and try to extract the contributions of new physics through the analysis of the rates for the above mentioned process .
a study considering rates to identify ued effects in higgs signals at a linear @xmath9 collider has been looked into , in ref @xcite .
however , the effects would be masked by the different uncertainties which will affect measurements at the lhc and thus make it difficult to differentiate the contributions coming from ued . in this work
we look at the dominant mode of higgs production through the @xmath10-fusion for a higgs in the mass range of @xmath6 gev and its subsequent decay into two photons and try to identify the contribution coming from ued and the extent to which these can be identified over the uncertainties that would affect measurements at the lhc .
a very similar analysis has been recently carried out , in context of split supersymmetry , in identifying additional contributions to the higgs rate @xcite . in section
2 we give a very brief overview about the model in consideration . in section 3
we discuss the process under consideration and how the signals for the diphoton final states get modified due to ued contributions . in section 4
we discuss the different uncertainties that would affect the signals . in section 5
we present our numerical results and finally we summarise and conclude in section 6 .
the ued model , in its simplest form @xcite , has all the sm particles propagating in a single extra dimension , which is compactified on an @xmath11 orbifold with @xmath12 as the radius of compactification .
conservation of kk number which is a consequence of momentum conservation along the extra dimension forces the kk particles to be pair produced .
consequently , ued predicts a stable lightest kaluza - klein particle ( lkp ) which would be much like the lightest supersymmetric particle ( lsp ) and a prospective candidate for dark matter @xcite .
bulk and brane radiative effects @xcite however break kk number down to a discrete conserved quantity , the so called kk parity , @xmath13 , where @xmath14 is the kk level .
kk parity conservation in turn , implies that the contributions to various precisely measured low - energy observables only arise at loop level and are small @xcite .
the kk tower resulting on the four dimensional space - time has a tree level mass given by m_n^2 = m^2 + where @xmath14 denotes the @xmath15-level of the kk tower and @xmath16 corresponds to the mass of the sm particle in question .
this implies a mass degeneracy in the @xmath15-level of the spectrum at least for the leptons and lighter quarks .
this degeneracy is however removed due to radiative corrections to the masses @xcite .
if the higgs exists in the mass range of @xmath6 gev , then we should be able to see it during an early phase of the lhc .
if that is possible , then it will be interesting to see if there are any indications of new physics in higgs signal itself , even if the detection of any new particle beyond the sm might not be possible due to their high mass .
the most suggestive channel in this context , for a higgs boson in the mass range @xmath6 gev , is the production of the higgs through the @xmath10-fusion channel followed by its decay into the diphotons . in this mode , the ( partial ) decay width @xmath17 ,
gets additional contributions from the kk excitation of the top quark , while the ( partial ) decay width @xmath18 gets additional contributions from the kk excitation of both the top quark and the @xmath19boson alongwith its associated goldstone modes , ghost kk states , and in addition , also due to the charged higgs tower .
it has been shown in quite detail @xcite that these loop contributions alter the higgs decay widths , thus making it distinguishable from the sm higgs boson . in this work ,
we are mainly interested in the modification of these partial decay width of the higgs .
since the kk number is not violated at any of the vertices inside a loop , the contributions come from all the kk - excitations , with a decoupling nature for the higher modes .
the combined expressions for the partial decay width for @xmath20 and @xmath21 for both ued and sm contributions can be written down as , ( hgg ) = ( ) ^2 |i_q+_n i_q^(n)|^2 + ( h ) = |i_q + i_w + _ n i_q^(n ) + _ n i_w^(n)|^2 where @xmath22 is the fermi constant , @xmath23 is the running qcd coupling evaluated at @xmath24 , @xmath25 is the electromagnetic coupling and @xmath26 are the contributions of the loop integrals for the sm and ued case respectively .
we consider the contributions from the kk excitation of the top quark as well as the bottom quark as we wish to make precise estimates comparable to uncertainties .
we have to include the kk excitations of the @xmath19boson and its associated goldstone modes , ghost kk states and the charged higgs tower for the diphoton decay channel .
the ued contributions include the sum over the kk towers of the respective particle .
as the more massive modes in the loop will hardly make significant contributions , we ensure that the sum is terminated as the higher modes decouple .
we include all the decay modes affected by ued contributions in the decay package hdecay @xcite to evaluate the relative sensitivities to the branching ratios to the different decay channels of the higgs boson .
it has to be remembered , however , that the above decay width will not be a directly measurable quantity at the lhc .
this is because the width is of the order of kev in the relevant higgs mass range , which is smaller than the resolution of the electromagnetic calorimeters to be used @xcite .
here we try to estimate how the ued contributions may be extracted in this channel , given the rather sizable theoretical as well as experimental uncertainties in the various relevant parameters . we , therefore , have chosen to do a calculation involving the full process @xmath27 , that is to say , the production of the higgs followed by its decay into the diphoton final state . taking all uncertainties into account ,
we have tried to find the significance level at which the additional contributions can be differentiated in different regions of the parameter space which in this scenario is the compactification radius @xmath12 .
we have confined ourselves to the production of higgs via gluon fusion .
the other important channel , namely gauge boson fusion , has been left out of this study , partly because it is plagued with uncertainties arising , for example , from diffractive production , which may be too large for the small effects under consideration here . in the sm ,
the loop - induced decay widths of the higgs boson , including qcd as well as further electroweak corrections , are well - documented in the literature @xcite .
the rate for the inclusive process @xmath28 ( where higgs production takes place via gluon fusion ) can be expressed in the leading order as @xmath29 where @xmath30 and @xmath31 is the gluon distribution function evaluated at @xmath32 and parton momentum fraction @xmath33 .
@xmath34 , @xmath35 and @xmath36 stand respectively for the diphoton , two - gluon and total decay widths of the higgs .
the lowest order estimate given above is further multiplied by the appropriate k - factors to obtain the next - to - next leading order ( nnlo ) predictions in qcd . while the computation of the rate is straightforward , we realise that the various quantities used are beset with theoretical as well as experimental uncertainties @xcite .
we undertake an analysis of these uncertainties in the next section .
as has already been stated in the previous section , the rate for diphoton production through real higgs at lhc is given by n = ( p p h ) b = ( p p h ) we have performed a parton - level monte carlo calculation for the production cross - section , using the mrs @xcite parton distribution functions and multiplied the results with the corresponding nnlo k - factors @xcite . it may be noted that nnlo k - factors are not yet available for most other parameterizations . in estimating the statistical uncertainties in the experimental value @xcite , mrs ( at leading order ) distributions
have been used by the cms group while atlas uses cteq distributions .
we have obtained the aforesaid uncertainty by taking the estimate based on mrs and multiplying the corresponding event rate by the nnlo k - factor for mrs .
it may also be mentioned that the difference between the nlo estimates of higgs production using the mrs and cteq parameterizations is rather small ( @xmath37 ) , according to recent studies @xcite .
therefore , it is expected that the nnlo estimate of uncertainties ( where there is scope of further evolution in any case ) used by us will ultimately converge to even better agreement with other parameterizations and will not introduce any serious inaccuracy in our conclusions .
the programme hdecay @xcite , including @xmath38 contributions , has been used for higgs decay computations . [
cols="^,^,^,^",options="header " , ] the number of two - photon events seen is given by @xmath39@xmath40 where @xmath39 is the integrated luminosity .
@xmath39 is expected to be known at the lhc to within 2 % .
we include this uncertainty in our calculation , although it has a rather small effect on our conclusions .
the possible sources of theoretical uncertainties can be divided into two general classes : parametric uncertainties and intrinsic uncertainties .
the former are related to the fact that , within the sm , each quantity of interest is a function of a set of input parameters , which are known with a finite experimental precision .
any variation of the input parameters within the experimentally allowed range gives rise to an uncertainty on the observable considered .
on the other hand , the intrinsic uncertainties have to do with the perturbative treatment of the quantum corrections : scheme dependence , ignorance of higher orders in the perturbative expansion and so on .
we have included the nnlo k - factors for the production cross - section @xmath41 , available in the literature and assume that our ignorance of more higher order contributions will not introduce a very significant uncertainty . in order to estimate the total uncertainty in @xmath40 , one has to first obtain the spread in theoretically predicted value in the sm due to the uncertainty in the various parameters used .
in addition , however , there is an uncertainty in the experimental values , although the actual level of this will be known only after the lhc run begins , the anticipated statistical spread in the measured value can be estimated through simulations .
these two uncertainties , combined in quadrature , are indicative of the difference with central value of the sm prediction which is required to establish any non - standard effect at any given confidence level .
we have performed such an exercise , taking the standard model calculation and that with sm + ued contributions . & & & + * higgs mass ( gev)*&@xmath42&@xmath43 & @xmath44&@xmath45&@xmath46 + @xmath47&8.9&8.1&@xmath48&@xmath49&@xmath50 + @xmath51&8.1&6.9&@xmath52&@xmath53&@xmath54 + @xmath55&8.6&5.6&@xmath56&@xmath57&@xmath58 + @xmath59&11.3&4.6&@xmath60&@xmath61&@xmath62 + thus the total uncertainty in @xmath40 can be expressed as ^2 = ( ) _ _ th_^2 + ( ) _ _ exp_^2 where the theoretical component can be further broken up as _ _
th_^2 = _ i^2_n_i where @xmath63 stands for the spread in the prediction of n due to uncertainty in the @xmath64 parameter relevant for the calculation . the sum runs over @xmath24 , @xmath65 , @xmath66 , @xmath67 , @xmath68 and @xmath69 , in addition to the uncertainty in the strong coupling @xmath70 .
the spread in the predicted value is predicted in each case by random generation of values for each parameter ( taken to vary one at a time ) within the allowed range .
thus we obtain @xmath71 corresponding to each parameter .
this has been listed in table [ param.tbl ] for different choices of @xmath24 .
one has to further include qcd uncertainties arising via parameterization dependence of the parton distribution functions ( pdf ) @xcite and the renormalisation scale .
although nnlo calculation reduced such uncertainties , the net spread in the prediction due to them could be as large as @xmath72 percent @xcite in the higgs mass range @xmath73 gev .
the levels of uncertainties in the various parameters , are presented in table [ param.tbl ] . in that table
we have given the uncertainties , wherever they are available , from recent and current experiments like the lep and the tevatron .
in addition , whatever improved measurement , leading to smaller errors ( in , say , @xmath66 or @xmath65 ) are expected after the initial run of the lhc are also separately incorporated in the table .
we have used the estimates corresponding to lhc wherever they are available . in our calculation
, we have used three values of the combined uncertainty from pdf and scale - dependence , namely , 15% , 10% and 5% , the latter two with an optimistic view to likely improvement using data at the lhc .
table 2 contains the finally predicted values of @xmath74 , for the different values of the higgs boson mass . and
( b ) @xmath20.__,title="fig:",width=292,height=283 ] and ( b ) @xmath20.__,title="fig:",width=288,height=288 ] @xmath75 includes statistical uncertainties , as estimated in detector simulations with a luminosity of 100 @xmath76@xcite .
as has been already mentioned , we have obtained benchmark values of this quantity using the results for cms presented in ref @xcite for mrs distributions at the lowest order , and appropriately improving them with the nnlo k - factors available in the literature .
the resulting predictions are listed as @xmath77 in table [ total.tbl ] for different values of @xmath24 .
thus one is able to obtain the net ( @xmath78 level ) uncertainties in the standard model as shown in the last three columns of table [ total.tbl ] . .
the horizontal lines in each figure corresponds to the confidence levels as labeled .
the graphs are shown for different choices of the pdf + scale uncertainty , viz .
( a ) 5% , ( b ) 10% and ( c ) 15%.__,title="fig:",width=268,height=268 ] .
the horizontal lines in each figure corresponds to the confidence levels as labeled .
the graphs are shown for different choices of the pdf + scale uncertainty , viz .
( a ) 5% , ( b ) 10% and ( c ) 15%.__,title="fig:",width=268,height=268 ] .
the horizontal lines in each figure corresponds to the confidence levels as labeled .
the graphs are shown for different choices of the pdf + scale uncertainty , viz .
( a ) 5% , ( b ) 10% and ( c ) 15%.__,title="fig:",width=268,height=268 ] next , the ued contributions via kk excitations - induced diagrams are calculated and added to the standard model amplitude .
the observable decay rate obtained therefrom is compared with that predicted in the standard model taking the uncertainty into account at various confidence levels .
thus one is able to decide whether the ued contributions to the diphoton rate are discernible from the standard model contributions at a given confidence level for a particular @xmath12 .
the realistic estimate requires subjecting the predictions to some experimental cuts aimed at maximizing the signal - to - background ratio as well as focusing on kinematic regions of optimal observability .
we incorporate the effects of such cuts with the help of an efficiency factor which , on explicit calculation in representative cases , turns out to be approximately 47 - 53% .
the only assumptions required are that the percentage error due to various parameters are the same for uncut rates as those calculated with cuts , and that the standard model and ued contribution suffer the same reduction due to cuts .
we have checked that this holds true so long as the kinematic region is not drastically curtailed by the cuts .
our purpose is to see at what confidence levels one can distinguish the ued effects on @xmath79 . with this in view , we estimate the excess in the rates due to the ued contributions and calculate the fractional difference with that predicted for standard model .
it is worth noting that for the mass range of the higgs boson that we consider , the partial decay width of the @xmath2 mode falls below the sm value while that of the @xmath1 mode is greater than that of the sm contribution . to highlight the dependence , we plot the branching ratios for these two modes in figure [ bratio ] .
this suggests that the ued contribution will have a slight suppression as the rate for the process in consideration is proportional to the product of the above branching ratios . in figure [ diff ]
we show the contour plots for the three choices of pdf + scale uncertainty .
we can see from figure [ diff](a ) , which has the most optimistic choice of 5% for the pdf + scale uncertainty , that at the 2@xmath80 confidence level , one can see excess over the sm rate for values of compactification scale as large as @xmath81 gev for higgs mass @xmath82 gev respectively , while at 1@xmath80 confidence level , these go up to @xmath83 gev for higgs mass @xmath84 gev respectively . for the more conservative choices of pdf + scale uncertainty , these numbers for @xmath85 will go down and as shown in figure [ diff](c ) , where the choice for pdf + scale uncertainty is taken as 15% , the 2@xmath80 ( 1@xmath80 ) confidence level limits @xmath86 gev for higgs mass @xmath87 gev respectively . with improvements in the measurement resolutions and lower uncertainties ,
this reach can be improved further .
we must point out that although the better convergence of the nnlo result over the nlo calculations do suggest a better understanding of the theoretical result , our ignorance of corrections beyond nnlo limits our complete knowledge of the intrinsic theoretical error .
the updated lower bounds on the compactification scale @xcite ( @xmath88 gev for @xmath89 gev and top quark mass of 173 gev at 90% c.l . ) , which depends on the higgs mass and the top quark mass , however suggest that the visible effects at the lhc would be marginal and one really needs a better hold on the different uncertainties to highlight the large deviations that are expected in the ued predictions .
we have explored the signals for the intermediate mass range of the higgs boson production through the @xmath4 channel and its subsequent decay to two photons .
both the production and decay channel get contributions through loops due to the absence of tree level couplings . the excited kk modes of the sm particles would give additional contribution to these decay modes , thus altering the decay rates .
we have performed a detailed analysis of the rates for the inclusive process @xmath90 incorporating the different uncertainties that would affect measurements at lhc .
we find that one can have observable enhancements in the allowed range of the parameter space of ued to distinguish effects of ued contributions in the higgs signals by studying the rates for the inclusive process considered in this work .
we show that although the rates ( after including ued contributions ) differ from the standard model prediction substantially , large uncertainties , both theoretical ( we have neglected the intrinsic theoretical error in the higgs production cross section from corrections beyond nnlo assuming it to be small ) as well as experimental , have a large role to play .
these uncertainties combine to dilute the substantially significant deviations from sm coming from new physics contributions and are seen to be marginal @xcite .
however , with better luminosity and improvements in the theoretical calculations and experimental measurements , this channel can provide for new physics effects in the higgs signal itself .
i. antoniadis , phys .
b * 246 * , 377 ( 1990 ) .
t. appelquist , h. c. cheng and b. a. dobrescu , phys . rev .
d * 64 * , 035002 ( 2001 ) [ arxiv : hep - ph/0012100 ] . t. appelquist and h. u. yee , phys .
d * 67 * , 055002 ( 2003 ) .
k. agashe , n. g. deshpande and g. h. wu , phys .
b * 514 * , 309 ( 2001 ) [ arxiv : hep - ph/0105084 ] .
d. chakraverty , k. huitu and a. kundu , phys .
b * 558 * , 173 ( 2003 ) [ arxiv : hep - ph/0212047 ] .
a. j. buras , m. spranger and a. weiler , nucl .
b * 660 * , 225 ( 2003 ) [ arxiv : hep - ph/0212143 ] .
a. j. buras , a. poschenrieder , m. spranger and a. weiler , nucl .
b * 678 * , 455 ( 2004 ) [ arxiv : hep - ph/0306158 ] . j. f. oliver , j. papavassiliou and a. santamaria , phys .
d * 67 * , 056002 ( 2003 ) [ arxiv : hep - ph/0212391 ] .
t. appelquist and b. a. dobrescu , phys .
b * 516 * , 85 ( 2001 ) [ arxiv : hep - ph/0106140 ] .
k. agashe , n. g. deshpande and g. h. wu , phys .
b * 511 * , 85 ( 2001 ) [ arxiv : hep - ph/0103235 ] .
t. g. rizzo and j. d. wells , phys .
d * 61 * , 016007 ( 2000 ) [ arxiv : hep - ph/9906234 ] .
a. strumia , phys .
b * 466 * , 107 ( 1999 ) [ arxiv : hep - ph/9906266 ] . c. d. carone , phys .
d * 61 * , 015008 ( 2000 ) [ arxiv : hep - ph/9907362 ] . c. macesanu , c. d. mcmullen and s. nandi , phys .
d * 66 * , 015009 ( 2002 ) [ arxiv : hep - ph/0201300 ] .
c. macesanu , c. d. mcmullen and s. nandi , phys .
b * 546 * , 253 ( 2002 ) [ arxiv : hep - ph/0207269 ] .
h. c. cheng , int .
j. mod .
a * 18 * , 2779 ( 2003 ) [ arxiv : hep - ph/0206035 ] . c. d. carone , j. m. conroy , m. sher and i. turan , phys .
d * 69 * , 074018 ( 2004 ) [ arxiv : hep - ph/0312055 ] .
a. muck , a. pilaftsis and r. ruckl , nucl .
b * 687 * , 55 ( 2004 ) [ arxiv : hep - ph/0312186 ] .
g. bhattacharyya , p. dey , a. kundu and a. raychaudhuri , phys .
b * 628 * , 141 ( 2005 ) [ arxiv : hep - ph/0502031 ] .
m. battaglia , a. datta , a. de roeck , k. kong and k. t. matchev , jhep * 0507 * , 033 ( 2005 ) [ arxiv : hep - ph/0502041 ] . h. c. cheng , k. t. matchev and m. schmaltz , phys .
d * 66 * , 056006 ( 2002 ) [ arxiv : hep - ph/0205314 ] .
b. bhattacherjee and a. kundu , phys .
b * 627 * , 137 ( 2005 ) [ arxiv : hep - ph/0508170 ] .
j. m. smillie and b. r. webber , jhep * 0510 * , 069 ( 2005 ) [ arxiv : hep - ph/0507170 ] .
m. kakizaki , s. matsumoto , y. sato and m. senami , phys .
d * 71 * , 123522 ( 2005 ) [ arxiv : hep - ph/0502059 ] ; c. macesanu , int .
j. mod .
a * 21 * , 2259 ( 2006 ) [ arxiv : hep - ph/0510418 ] .
f. j. petriello , jhep * 0205 * , 003 ( 2002 ) [ arxiv : hep - ph/0204067 ] .
a. datta and s. k. rai , arxiv : hep - ph/0509277 .
s. k. gupta , b. mukhopadhyaya and s. k. rai , arxiv : hep - ph/0510306 .
g. servant and t. m. p. tait , nucl .
b * 650 * , 391 ( 2003 ) [ arxiv : hep - ph/0206071 ] .
d. majumdar , phys .
d * 67 * , 095010 ( 2003 ) [ arxiv : hep - ph/0209277 ] .
h. georgi , a. k. grant and g. hailu , phys .
b * 506 * , 207 ( 2001 ) [ arxiv : hep - ph/0012379 ] .
g. von gersdorff , n. irges and m. quiros , nucl .
b * 635 * , 127 ( 2002 ) [ arxiv : hep - th/0204223 ] . h. c. cheng , k. t. matchev and m. schmaltz , phys .
d * 66 * , 036005 ( 2002 ) [ arxiv : hep - ph/0204342 ] .
m. puchwein and z. kunszt , annals phys .
* 311 * , 288 ( 2004 ) [ arxiv : hep - th/0309069 ] .
v. barger and r.j.n .
phillips , collider physics , addison - wesley publishing company ( 1987 ) .
a. djouadi , j. kalinowski and m. spira , comput .
commun .
* 108 * , 56 ( 1998 ) [ arxiv : hep - ph/9704448 ] .
atlas collaboration , atlas detector and physics performance technical design report , report cern - lhcc-99 - 15 ( 1999 ) .
h. zheng and d. wu , phys .
d * 42 * , 3760 ( 1990 ) .
a. djouadi , m. spira , j. j. van der bij and p. m. zerwas , phys .
b * 257 * , 187 ( 1991 ) .
s. dawson and r. p. kauffman , phys .
d * 47 * , 1264 ( 1993 ) .
k. melnikov and o. i. yakovlev , phys .
b * 312 * , 179 ( 1993 ) [ arxiv : hep - ph/9302281 ] .
a. djouadi , m. spira and p. m. zerwas , phys .
b * 311 * , 255 ( 1993 ) [ arxiv : hep - ph/9305335 ] .
m. inoue , r. najima , t. oka and j. saito , mod .
lett . a * 9 * , 1189 ( 1994 ) .
y. liao and x. y. li , phys .
b * 396 * , 225 ( 1997 ) [ arxiv : hep - ph/9605310 ] .
m. steinhauser , arxiv : hep - ph/9612395 .
a. djouadi , p. gambino and b. a. kniehl , nucl .
b * 523 * , 17 ( 1998 ) [ arxiv : hep - ph/9712330 ] .
j. fleischer , o. v. tarasov and v. o. tarasov , phys .
b * 584 * , 294 ( 2004 ) [ arxiv : hep - ph/0401090 ] .
u. aglietti , r. bonciani , g. degrassi and a. vicini , phys .
b * 595 * , 432 ( 2004 ) [ arxiv : hep - ph/0404071 ] .
g. degrassi and f. maltoni , arxiv : hep - ph/0504137 .
m. duhrssen , s. heinemeyer , h. logan , d. rainwater , g. weiglein and d. zeppenfeld , phys .
d * 70 * , 113009 ( 2004 ) arxiv : hep - ph/0406323 .
a. djouadi and s. ferrag , phys .
b * 586 * , 345 ( 2004 ) [ arxiv : hep - ph/0310209 ] .
d. zeppenfeld , r. kinnunen , a. nikitenko and e. richter - was , phys .
d * 62 * , 013009 ( 2000 ) [ arxiv : hep - ph/0002036 ] .
a. d. martin , r. g. roberts , w. j. stirling and r. s. thorne , phys .
b * 531 * , 216 ( 2002 ) [ arxiv : hep - ph/0201127 ] .
v. ravindran , j. smith and w. l. van neerven , nucl .
b * 665 * , 325 ( 2003 ) [ arxiv : hep - ph/0302135 ] . c. anastasiou and k. melnikov , nucl .
b * 646 * , 220 ( 2002 ) [ arxiv : hep - ph/0207004 ] .
r. v. harlander and w. b. kilgore , phys .
lett . * 88 * , 201801 ( 2002 ) [ arxiv : hep - ph/0201206 ] .
a. belyaev , j. pumplin , w. k. tung and c. p. yuan , jhep * 0601 * , 069 ( 2006 ) [ arxiv : hep - ph/0508222 ] .
a. cafarella , c. coriano , m. guzzi and j. smith , eur .
j. c * 47 * , 703 ( 2006 ) [ arxiv : hep - ph/0510179 ] .
s. eidelman _ et al .
_ [ particle data group ] , phys .
b * 592 * , 1 ( 2004 ) .
[ cdf collaboration ] , arxiv : hep - ex/0507091 .
s. heinemeyer , w. hollik and g. weiglein , arxiv : hep - ph/0412214 .
t. flacke , d. hooper and j. march - russell , phys .
d * 73 * ( 2006 ) 095002 [ erratum - ibid .
d * 74 * ( 2006 ) 019902 ] [ arxiv : hep - ph/0509352 ] .
i. gogoladze and c. macesanu , phys .
d * 74 * ( 2006 ) 093012 [ arxiv : hep - ph/0605207 ] .
|
a major focus at the large hadron collider ( lhc ) will be on higgs boson studies and it would be an interesting prospect to simultaneously probe for physics beyond the standard model ( sm ) in the higgs signals . in this work
we show as to what extent , the effects of universal extra dimension ( ued ) can be isolated at the lhc through the higgs signals . by doing a detailed study of the different uncertainties involved in the measurement of the rates for the process @xmath0 we estimate the extent to which these uncertainties can mask the effects of the contributions coming from ued .
effects of universal extra dimensions on higgs signals at lhc + santosh kumar rai + _ high energy physics division , department of physical sciences , university of helsinki , + and helsinki institute of physics , p.o . box 64 , fin-00014 university of helsinki , finland _ 10 mm 0.2 true cm * keywords * higgs , universal extra dimension , kaluza - klein .
|
the existence of the heavy @xmath0 gauge boson is predicted by a number of grand unified theories ( gut s ) and superstring theories @xcite .
the mass of this particle is expected to be of order @xmath1 gev , and therefore it can not be produced at present day accelerators .
various strategies of searching for signals of @xmath0 as a virtual heavy state were developed and different observables convenient for its experimental detection have been introduced ( see the survey @xcite and references therein ) .
the model - dependent and model - independent @xmath0 searches at @xmath2 colliders are discussed ( see , for instance , the report @xcite ) .
a popular model assumes that at low energies the @xmath0 interactions with ordinary particles of the standard model ( sm ) can be described by the effective gauge group @xmath3 .
an alternative choice is the gauge group @xmath4 @xcite .
these models are considered as the remnants of underlying theories which are not specified .
the low - energy effective lagrangians ( el ) take into consideration the most general property of renormalizable theories , ensured by the decoupling theorem @xcite the dominance of renormalizable interactions at low energies .
the interactions of non - renormalizable types , being generated at high energies due to radiation corrections , are suppressed by the inverse heavy mass @xmath5 .
therefore , it is possible not to consider them in leading order at lower energies .
another popular description is the introduction of the el , considered as the sum of all effective operators with dimensions @xmath6 , constructed from the fields of light particles .
the coefficients at these operators are treated as independent unknown numbers to be determined in experiments . for more details
see ref . @xcite . in general
, the number of possible @xmath0 couplings is large .
so , it is difficult to introduce observables allowing a unique detection of @xmath0 signals . in this regard
, it is desirable either to decrease the number of the independent @xmath0 parameters on some reasonable grounds and to introduce observables most sensitive to the @xmath0 virtual states . in any case
, the main idea is to find correlations between the @xmath0 couplings at low energies .
a straightforward way to find the correlations is to specify the underlying theory describing interactions at energies @xmath7 and to consider running of the couplings from high to low energies @xmath8 by using the renormalization group ( rg ) equations . in this approach
, each underlying theory leads to the unique values of the parameters and , hence , the corresponding correlations are model dependent ones . another way is to specify a basis low - energy theory ( for instance , the sm can be chosen ) and to determine the relations between the @xmath0 parameters , following from some model independent arguments .
these correlations are to be model independent .
naturally , they remain dependent on the chosen basis low - energy theory . in refs .
@xcite the method for derivation model independent correlations between the parameters of physics beyond the sm has been developed , and new observables convenient in searching for the @xmath0 boson in four - fermion processes were introduced .
this approach is based on principles of the rg and the decoupling theorem @xcite .
as it was argued , any virtual heavy particle can be treated as an `` external field '' scattering the sm particles .
the vertex describing interaction with the field contains a numeric factor , which is considered as an arbitrary parameter .
actually , it is generated by the decoupling and therefore depends on the underlying model . due to renormalizability ,
the scattering amplitude in the `` external field '' satisfies some simple relation ( named rg relation ) , which includes the @xmath9 and @xmath10 functions entering the rg equation .
these functions have to be calculated with the light particles only , and the vertex factor .
hence , relations between different vertex factors follow .
then , they can be implemented in a number of model independent observables corresponding to the specific heavy virtual state , in particular , to the @xmath0 gauge boson @xcite . in ref .
@xcite as the low - energy basis model the minimal sm ( with one scalar doublet ) has been chosen .
however , at present there is a few information about the scalar fields . in this regard , the theory with two scalar doublets is intensively studied @xcite . the two - higgs - doublet model ( thdm )
is also known as the low - energy limit of some @xmath11 based gut s , which predict the @xmath0 gauge boson . in the present paper , the results of ref .
@xcite are generalized to the thdm case .
we analyse in detail both the abelian and the so called `` chiral '' types of the @xmath0 couplings to light particles .
as the latter type is concerned , it was derived as follows .
we first assumed the most general parametrization of @xmath0 interactions with the sm fields and then derived the generator structures , compatible with the renormalizability .
as it will be shown in what follows , there is an important difference between these two types of interactions .
thus , in order to derive the model independent constraints we choose the thdm as the low - energy basis theory ( notice , the minimal sm is a particular case of the thdm ) .
then , we introduce a general parametrization of linear in @xmath0 couplings , which is independent of the specific underlying theory . as a result , the derived rg correlations are model independent ones .
they hold for the thdm as well as for the minimal sm . moreover
, the existence of other heavy particles with masses @xmath12 does not affect these correlations .
as it will be shown , there are two completely different sets of the @xmath0 couplings to the sm fields compatible with renormalizability .
the first one describes the abelian @xmath0 , which respects the additional @xmath13 symmetry of the low energy el . in this case
the @xmath0 couplings to the axial - vector fermion currents have a universal absolute value .
the second set corresponds to the chiral @xmath0 , which interacts with the sm doublets , only .
one has to distinguish these neutral @xmath0 gauge bosons because they are described by different operators .
the content is as follows . in sec .
[ sec : model ] the general parametrization of interactions involving the @xmath0 and the sm fields is introduced .
the rg correlations between the @xmath0 couplings are derived in sec .
[ sec : rgrelations ] . in sec .
[ sec : e6 ] they are compared with the specific values of the @xmath0 couplings in the gut s based on the @xmath11 group . in sec .
[ sec : observ ] the observables convenient in detection of the @xmath0 signals are proposed .
the results of our investigation are discussed in sec .
[ sec : discussion ] .
in the present paper we analyze the four - fermion scattering amplitudes of order @xmath14 generated by the virtual @xmath0 states .
vertices of interactions with more than one @xmath0 field contribute to the amplitudes involving several virtual @xmath0 states .
the latter processes have order @xmath15 and higher .
therefore , in what follows we consider the linear in @xmath0 vertices , only . to introduce a general parametrization of the vertices involving the sm fields and being linear in the @xmath0 field , let us impose a number of natural conditions .
first of all , the renormalizable type interactions are dominant at low energies @xmath8 .
the non - renormalizable interactions generated at high energies due to radiation corrections are suppressed by the inverse heavy mass @xmath5 ( or by other heavier scales @xmath16 ) and therefore at low energies can be neglected in leading order .
we assume that the @xmath0 is the only neutral vector boson with the mass @xmath17 , and the @xmath0 gauge field enters the theory through covariant derivatives with a corresponding charge .
we also assume that the @xmath18 gauge group of the sm is a subgroup of the gut group . in this case , a product of generators associated with the sm subgroup is a linear combination of these generators . as a consequence , the all structure constants connecting two sm gauge bosons with @xmath0 have to be zero .
hence , the interactions of gauge fields of the types @xmath19 , @xmath20 , and other are absent at tree level .
let @xmath21 ( @xmath22 ) be two complex scalar doublets : @xmath23 where @xmath24 marks corresponding vacuum expectation values , @xmath25 are complex fields , and @xmath26 , @xmath27 are real fields . by diagonalizing the quadratic terms of the scalar potential @xmath28 one obtains the mass eigenstates : two neutral @xmath29-even scalar particles , @xmath30 and @xmath31 , the neutral @xmath29-odd scalar particle , @xmath32 , the goldstone boson partner of the @xmath33 boson , @xmath34 , the charged higgs field , @xmath35 , and the goldstone field associated with the @xmath36 boson , @xmath37 : @xmath38 where @xmath39 and the angle @xmath40 is determined by the explicit form of the potential @xmath28 . for instance
, the @xmath29-conserving potential , which has only @xmath29-invariant minima , can be used @xcite : @xmath41 + \lambda_3 ( \mbox{re}[\phi^\dagger_1 \phi_2])^2 \nonumber\\ & & + \lambda_4 ( \mbox{im}[\phi^\dagger_1 \phi_2])^2 + \lambda_5 ( \phi^\dagger_1 \phi_1 ) ( \phi^\dagger_2 \phi_2).\end{aligned}\ ] ] it is consistent with the absence of the tree - level flavor - changing neutral currents ( fcnc s ) in the fermion sector .
the corresponding value of @xmath40 is @xcite @xmath42 at low energies , when all heavy states are decoupled , the @xmath0 interactions with the scalar doublets can be parametrized in a model independent way as follows @xcite : @xmath43 where @xmath44 , @xmath45 , @xmath46 are the charges associated with the @xmath47 , @xmath48 , and the @xmath0 gauge groups , respectively , @xmath49 are the pauli matrices , @xmath50 is the generator corresponding to the gauge group of the @xmath0 boson , and @xmath51 is the @xmath48 hypercharge .
the condition @xmath52 guarantees that the vacuum is invariant with respect to the gauge group of photon . the vector bosons , @xmath53 , @xmath33 , and @xmath0 , are related with the symmetry eigenstates as follows : @xmath54 where @xmath55 is the adopted in the sm value of the weinberg angle , and @xmath56 as is seen , the mixing angle @xmath57 is of order @xmath58 .
that results in the corrections of order @xmath59 to the interactions between the sm particles . to avoid the tree - level mixing of the @xmath33 boson and the physical scalar field @xmath32 one has to impose the condition @xmath60
now , let us parametrize the fermion - vector interactions introducing the effective low - energy lagrangian @xcite : @xmath61 where the renormalizable type interactions are admitted and the summation over the all sm left - handed fermion doublets , @xmath62 , and the right - handed singlets , @xmath63 , is understood .
@xmath64 denotes the charge of @xmath65 in the positron charge units , @xmath66 and @xmath67 equals to @xmath68 for leptons and @xmath69 for quarks .
renormalizable interactions of fermions and scalars are described by the yukawa lagrangian . to avoid the existence of the tree - level fcnc s one has to ensure that at the diagonalization of the fermion mass matrix the diagonalization of the scalar - fermion couplings is automatically fulfilled . in this case
the yukawa lagrangian , which respects the @xmath18 gauge group , can be written in the form : @xmath70 \right.\nonumber\\&&\left .
+ g_{f_u , i}\left [ \bar{f}_l\phi^c_i(f_u)_r + ( \bar{f}_u)_r \phi^{c\dagger}_i f_l \right ] \right\},\end{aligned}\ ] ] where @xmath71 is the charge conjugated scalar doublet , and the cabibbo - kobayashi - maskawa mixing is neglected .
then , the fermion masses are @xmath72 as was shown by glashow and weinberg @xcite , the tree - level fcnc s mediated by higgs bosons are absent in case when all fermions of a given electric charge couple to no more than one higgs doublet .
this restriction leads to four different models , as discussed in ref .
@xcite . in
what follows , we will use the most general parametrization ( [ l : yuk ] ) including the models mentioned as well as other possible variations of the yukawa sector without the tree - level fcnc s . by using eqs .
( [ l : scalar ] ) , ( [ l : fermion ] ) , and ( [ l : yuk ] ) it is easy to derive the feynman rules which are collected in appendix [ sec : vertices ] .
in this section we consider the correlations between the parameters @xmath73 , @xmath74 , @xmath75 , and @xmath76 appearing due to the renormalizability of an underlying theory .
as is known , @xmath77-matrix elements are to be invariant with respect to the rg transformations , which express the independence of the location of a normalization point @xmath78 in the momentum space . in a theory with
different mass scales the decoupling of heavy loop contributions at the thresholds of heavy masses , @xmath79 , results in the important property of low energy amplitudes : the running of all functions is regulated by the loops of light particles . therefore , the @xmath9 and @xmath10 functions at low energies are determined by the sm particles , only .
this fact is the consequence of the decoupling theorem @xcite .
actually , the decoupling results in the redefinition of the parameters of the theory at the scale @xmath79 and removing the all heavy particle loop contributions proportional to @xmath80 from the rg equation @xcite : @xmath81 where we use the notation @xmath82 to refer to the charges , and @xmath83 represents the all fields and masses . hats over quantities mark the parameters of the underlying theory .
they include the loops of both the sm and the heavy particles , whereas the quantities without hats are calculated assuming that no heavy particles are excited inside loops .
the matching between the both sets of parameters ( @xmath82 , @xmath83 and @xmath84 , @xmath85 ) is chosen to be done at the normalization point @xmath86 , @xmath87 since the sets of parameters @xmath82 , @xmath83 and @xmath84 , @xmath85 differ at one - loop level , it is possible to substitute one set by another . as is shown in ref .
@xcite , the redefinition of fields and charges ( [ decoupling ] ) allows one to eliminate the one - loop mixing between heavy and light virtual states .
therefore , virtual states of heavy particles can be treated as the `` external fields '' scattering sm particles .
the renormalizability of the underlying theory leads to some relations for vertices describing this scattering , called the rg relations .
let us consider the four - fermion amplitudes caused by the @xmath0 boson exchange .
in the lower order in ratio @xmath88 the process @xmath89 can be presented as scattering of the initial , @xmath90 , and the final , @xmath91 , fermions in the `` external field '' @xmath5 with the corresponding vertex factors @xmath92 , @xmath93 .
the quantity @xmath94 contains no contributions of heavy particle loops .
thus , it can be computed as a linear combination of the parameters @xmath73 , @xmath74 , @xmath75 , and @xmath76 .
the rg invariance of the vertex leads to equation @xmath95 where the effective low - energy rg operator @xcite is defined as follows : @xmath96 and the coefficient functions @xmath97 and @xmath98 are computed taking into account the loops of light particles .
relation ( [ rge ] ) ensures that , as a consequence of renormalizability , the mathematical structure of the leading logarithmic terms of the vertices , calculated in one- and higher - loop approximations , coincides with that of the tree - level structures .
the standard usage of eq .
( [ rge ] ) is to improve scattering amplitudes calculated in a fixed order of perturbation theory .
in contrast , in what follows we will apply eq .
( [ rge ] ) to obtain an algebraic relation between the parameters @xmath73 , @xmath74 , @xmath75 , @xmath76 , which are to be considered as arbitrary numbers , since the underlying theory is not specified .
let us explain the idea in more detail . in case
when the underlying theory is specified ( @xmath73 , @xmath74 , @xmath75 , @xmath76 have to be computed as discussed before ) , and the @xmath9 and @xmath10 functions as well as the @xmath77-matrix elements are calculated in a fixed order of perturbation theory , eq .
( [ rge ] ) is just the identity .
if the underlying theory is not specified , whereas the @xmath9 , @xmath10 functions and @xmath77-matrix elements are computed in a fixed order of perturbation theory , equality ( [ rge ] ) may serve to correlate the unknown parameters @xmath99 . in case of the four - fermion processes mediated by the gauge @xmath0 boson ,
the number of independent @xmath9 functions is less than the number of rg equations .
therefore , the non - trivial system of equations correlating the originally independent parameters occurs .
the one - loop rg relation for the fermion-@xmath0 vertex is @xcite @xmath100 where @xmath101 and @xmath102 denote the tree - level and the one - loop level contributions to the fermion-@xmath0 vertex , and @xmath103 is the one - loop level part of the rg operator , @xmath104 as it follows from eq .
( [ rgrelation ] ) , only the divergent parts of the one - loop vertices @xmath102 are to be calculated .
the corresponding diagrams are shown in fig .
[ fig:1 ] .
the following expressions for the right - handed and the left - handed fermions , respectively , have been obtained , @xmath105 \nonumber\\ & & + g^2_{f,2}\left [ 2t^3_f\left(\tilde{y}_{\phi,2 } + \tilde{y}_{\phi_2,1}\right ) + \tilde{y}_{l , f } + \tilde{y}_{l , f^\star } \right ] \nonumber\\ & & \left .
+ o\left(\frac{m^2_w}{m^2_{z^\prime}}\right ) \right\ } , \nonumber\\ \frac{\partial\gamma^\mu_{f_l z^\prime}}{\partial\ln\kappa } & = & \frac{\gamma^\mu}{8 \pi^2 } \left\{\frac{g^2}{2}\tilde{y}_{l , f^\star } + \frac{4}{3}g^2_{s , f}\tilde{y}_{l , f}\right .
\nonumber\\ & & + g^2\tilde{y}_{l , f}\left [ \frac{1}{4\cos^2{\theta_w } } + \left(q^2_f -\left|q_f\right|\right)\tan^2{\theta_w } \right ] \nonumber\\ & & + \left(g^2_{f,1}+g^2_{f,2}\right ) \left(\tilde{y}_{r , f } -2t^3_f\tilde{y}_{\phi,2}\right ) \nonumber\\ & & + g^2_{f^\star,1}\left(2t^3_f\tilde{y}_{\phi_1,1 } + \tilde{y}_{r , f^\star}\right ) \nonumber\\ & & + g^2_{f^\star,2}\left(2t^3_f\tilde{y}_{\phi_2,1 } + \tilde{y}_{r , f^\star}\right ) \nonumber\\ & & \left .
+ o\left(\frac{m^2_w}{m^2_{z^\prime}}\right)\right\},\end{aligned}\ ] ] where @xmath65 and @xmath106 are the partners of a @xmath47 fermion doublet ( namely , @xmath107 , @xmath108 , @xmath109 , and @xmath110 ) , @xmath111 is the third component of the weak isospin , and @xmath112 is the qcd charge for quarks , and zero for leptons .
= 0.4 the fermion anomalous dimensions can be calculated by using the diagrams of fig .
[ fig:2 ] : @xmath113 , \nonumber\\ \gamma_{f_l}&=&\frac{1}{16 \pi^2}\left [ g^2\left(q^2_f -\left|q_f\right|\right)\tan^2{\theta_w } + \frac{4}{3}g^2_{s , f } + \frac{g^2}{2 } \right.\nonumber\\ & & + \frac{g^2}{4\cos^2{\theta_w } } + g^2_{f,1 } + g^2_{f,2 } + g^2_{f^\star,1 } + g^2_{f^\star,2 } \nonumber\\&&\left .
+ o\left(\frac{m^2_w}{m^2_{z^\prime}}\right ) \right].\end{aligned}\ ] ] = 0.4 rg relations ( [ rgrelation ] ) considered in a lower order in @xmath88 lead to the equations for the parameters @xmath73 , @xmath74 , @xmath75 , and @xmath76 : @xmath114 \nonumber\\&&\quad -g^2_{f,2}\left [ 2t^3_f\left(\tilde{y}_{\phi,2 } + \tilde{y}_{\phi_2,1}\right ) + \tilde{y}_{l , f } + \tilde{y}_{l , f^\star } -2\tilde{y}_{r , f } \right ] , \nonumber \\ & & 4\pi^2\tilde{y}_{l , f } \left(\frac{\beta^{(1)}_{\tilde{g}}}{\tilde{g}^2 } + \gamma^{(1)}_{m^2_{z^\prime}}\right)= \frac{g^2}{2 } \left(\tilde{y}_{l , f } -\tilde{y}_{l,{f^\star } } \right ) \nonumber\\&&\quad + \left(g^2_{f,1 } + g^2_{f,2}\right)\left ( 2t^3_f\tilde{y}_{\phi,2 } + \tilde{y}_{l , f } -\tilde{y}_{r , f } \right ) \nonumber\\&&\quad -g^2_{f^\star,1}\left(2t^3_f\tilde{y}_{\phi_1,1 } -\tilde{y}_{l , f } + \tilde{y}_{r , f^\star}\right ) \nonumber\\&&\quad -g^2_{f^\star,2}\left(2t^3_f\tilde{y}_{\phi_2,1 } -\tilde{y}_{l , f } + \tilde{y}_{r , f^\star}\right).\end{aligned}\ ] ] one has to derive two sets of relations , which ensure the compatibility of eqs .
( [ eq ] ) .
the first one is @xmath115 it describes the @xmath0 boson analogous to the third component of the @xmath47 gauge field .
the characteristic features of these interactions are the zero traces of generators and the absence of couplings to the right - handed singlets . in what follows
, we shall call this type of interaction the `` chiral '' @xmath0 . the second set
, @xmath116 corresponds to the abelian @xmath0 boson . in this case
the sm lagrangian appears to be invariant with respect to the @xmath117 group associated with the @xmath0 .
the first and the second relations in eqs .
( [ abelian ] ) mean that appropriate generators are proportional to the unit matrix , whereas the third relation ensures the yukawa terms to be invariant with respect to the @xmath13 transformations .
introducing the @xmath0 couplings to the vector and the axial - vector fermion currents , @xmath118 , @xmath119 , one can rewrite the second and the third of eqs .
( [ abelian ] ) in the following form : @xmath120 as is seen , the couplings of the abelian @xmath0 to the axial - vector fermion currents have a universal absolute value proportional to the @xmath0 coupling to the scalar doublets .
the solutions derived are the same as in case of the minimal sm considered in ref .
notice that both of correlations ( [ non - abelian ] ) and ( [ abelian ] ) lead to the same @xmath0 couplings to each of the scalar doublets .
notice , in case of the abelian @xmath0 boson the correlations ( [ abelian]),([abelian1 ] ) can be derived on related but formally different grounds .
the point is that the renormalizability and gauge invariance of interactions are closely connected .
therefore , the requirement of renormalizability can be substituted by the requirement of gauge invariance of the effective low - energy lagrangian . in general , the el respects by construction various [ and , in particular , @xmath121 symmetries .
but if non - renormalizable interactions are admitted , no relations between the arbitrary parameters can be found .
if only the renormalizable interactions are taken into account , as in eq .
( [ l : fermion ] ) , some correlations appear .
in fact , to obtain formulae ( [ abelian]),([abelian1 ] ) it is sufficient to require the @xmath122 gauge invariance of the yukawa terms .
note also that the correlations in eq .
( [ abelian1 ] ) are the same as in the sm for the specific values of the hypercharges @xmath123 and @xmath124 corresponding to the @xmath125 gauge transformations of fermion and scalar fields . on the other hand , we did not find any symmetry requirement describing the all possible relations following from eq .
( [ non - abelian ] ) .
therefore , the renormalizability requirement looks as more general one .
over last decades the gut s based on the @xmath11 gauge group @xcite are intensively studied .
they predict the abelian @xmath0 boson with the mass @xmath126 . since the low - energy limit of the @xmath11 gut s is the thdm considered , it is of interest to check whether relations ( [ abelian1 ] ) hold for the specific values of the @xmath0 couplings in these models .
there are different schemes of the @xmath11-symmetry breaking .
one of them is @xmath127 this leads to the so called left - right ( lr ) model .
another scheme , @xmath128 predicts the abelian @xmath0 , which is a linear combination of the neutral vector bosons @xmath129 and @xmath130 , @xmath131 where @xmath132 is the mixing angle . in table
i ( see ref .
@xcite ) we show the @xmath0 couplings to the sm fermions in models mentioned ( notice , the sign of axial - vector couplings in ref .
@xcite is opposite to the sign of @xmath133 ) . at first glance , some of the couplings in table
i are inconsistent with relations ( [ abelian1 ] ) .
this requires to be discussed in more detail .
first of all , let us consider the @xmath0 couplings to neutrinos .
it is usually supposed in theories based on the @xmath134 group that the yukawa terms responsible for generation of the dirac masses of neutrinos must be set to zero @xcite
. therefore , there are no rg relations for the @xmath0 interactions with the neutrino axial - vector currents , because the terms proportional to @xmath135 vanish in eq .
( [ eq ] ) . in this case
the couplings @xmath136 given in table i are not restricted by relations ( [ abelian1 ] ) .
now , let us discuss the @xmath0 couplings to charged leptons and quarks .
the values of the couplings satisfy relations ( [ abelian1 ] ) in case of the lr model . as for models described by the @xmath11 breaking scheme ( [ e6 ] ) , two possibilities of choosing @xmath132 are of interest .
first is if the @xmath129 boson is much heavier than the @xmath130 field . in general , this is a natural condition , since the fields @xmath129 and @xmath130 arise at different energy scales . as a consequence , the field @xmath129 is decoupled , and the mixing angle @xmath132 is small ( @xmath137 ) . in this case
rg relations ( [ abelian1 ] ) hold in lower order in @xmath132 for the @xmath0 couplings to quarks and charged leptons .
the second possibility is if the masses of @xmath130 and @xmath129 are of the same order .
this means the tuning of the vacuum expectation values generating the vector boson masses .
this case can not be treated straightforwardly on the basis of relations ( [ abelian1 ] ) since the mixed states of the @xmath0 bosons have to be included into consideration explicitly .
although our approach is applicable in this case , it requires additional investigation .
moreover , the @xmath0 mixed states cause some different exchange amplitudes , which have to be incorporated into low - energy observables . in
what follows , we will not discuss the case of two @xmath0 bosons having masses of the same order .
now , let us introduce the observables convenient for detection of the @xmath0 in electron - positron annihilation into fermion pairs @xmath138 ( @xmath139 ) . the center - of - mass energy
is taken in the range @xmath140 gev .
consider the case of non - polarized initial and final fermions .
since the @xmath141 quark is not considered , other fermions at these energies can be treated as massless particles , @xmath142 . in this approximation
the left - handed and the right - handed fermions can be substituted by the helicity states , which will be marked as @xmath143 and @xmath144 for the incoming electron and the outgoing fermion , respectively ( @xmath145 ) .
let @xmath146 be the born amplitude of the process @xmath147
( @xmath139 ) with the virtual @xmath148-boson state in the @xmath149 channel ( @xmath150 ) .
the @xmath0 boson existence leads to the deviation of order @xmath14 of the cross section from its sm value . in general
, the tree - level deviations originate from two types of contributions . the first is caused by the @xmath33-@xmath0 mixing . using the results of sec .
[ sec : rgrelations ] the mixing angle @xmath57 [ see eq .
( [ zzpmixing ] ) ] can be calculated as follows , @xmath151 because of the mixing there are corrections of order @xmath152 to the vertex describing interaction of @xmath33 boson and fermions .
hence , the amplitude @xmath153 deviates from its sm value @xmath154 .
the second type describes the interference between the sm amplitude , @xmath155 , and the @xmath0 exchange amplitude , @xmath156 .
thus , for the process @xmath157 the deviation of the cross section is @xmath158 } { 32\pi s } + o\left(\frac{s^2}{m^4_{z^\prime}}\right),\ ] ] where @xmath159 the quantity @xmath160 can be calculated in the form @xmath161 { \left(z + p_\lambda p_\xi\right)}^2,\ ] ] where @xmath162 , @xmath163 , @xmath164 ( @xmath165 is the angle between the incoming electron and the outgoing fermion ) , @xmath166 denotes the @xmath33-@xmath0 interference term , and @xmath167 accounts of the contributions from the @xmath33-@xmath0 mixing : @xmath168 , \nonumber\\ { \cal m}^{ef}_{\lambda\xi}&= & \frac{\alpha_{\rm em}g\tilde{g } t^3_f n_f \theta_0 } { 4\pi\cos{\theta_w}(s -m^2_z ) } \left[\tilde{y}_{\xi , f } \left(\delta_{\lambda , l}-2\sin^2{\theta_w}\right ) \right.\nonumber\\&&+\left .
2t^3_f\tilde{y}_{\lambda , e } \left(2|q_f|{\sin}^2{\theta_w}-\delta_{\xi , l}\right ) \right ] \left[|q_f| \right.\nonumber\\ & & + \left.\chi(s)\left(p_\lambda -\varepsilon\right ) \left(p_\xi -1 + |q_f| -|q_f|\varepsilon\right ) \right],\end{aligned}\ ] ] where @xmath169 is the fine structure constant , @xmath170 for quarks and @xmath171 for leptons , @xmath172 , @xmath173 , and @xmath174 is the kronecker delta .
the leading contribution comes from the @xmath33-@xmath0 interference term @xmath166 , whereas the @xmath33-@xmath0 mixing terms are suppressed by the additional factor @xmath175 . at energies
@xmath140 gev @xmath176 . to take into consideration the correlations ( [ non - abelian ] ) or ( [ abelian ] )
let us introduce the function @xmath177 defined as the difference of cross sections integrated in a suitable range of @xmath178 @xcite : @xmath179 the conventionally used observables the total cross section @xmath180 and the forward - backward asymmetry @xmath181 can be obtained by a special choice of @xmath182 [ @xmath183 , @xmath184 .
one can express @xmath177 in terms of @xmath180 and @xmath181 : @xmath185.\ ] ] then , the deviation @xmath186 can be written in the form : @xmath187 \nonumber\\&&\times \left(p_\lambda p_\xi -z -z^2
p_\lambda p_\xi -\frac{z^3}{3}\right).\end{aligned}\ ] ] let us compare the observable @xmath188 with the differential cross section ( [ deviation ] ) .
as is seen , the polynomial in the polar angle @xmath182 in eq .
( [ deviation ] ) is replaced by the function of the boundary angle @xmath182 in eq .
( [ observable ] ) .
the overall factor @xmath189 appears due to the angular integration . in
what follows , we consider the observable ( [ observable ] ) taking into account correlations ( [ non - abelian ] ) and ( [ abelian ] ) .
the case of the chiral @xmath0 corresponds to correlations ( [ non - abelian ] ) . because of absence of the @xmath0 couplings to right - handed fermions
the leading contribution to @xmath188 is proportional to the same polynomial in @xmath182 for any outgoing fermion @xmath65 : @xmath190 + o\left(\varepsilon\right)\right\}.\end{aligned}\ ] ] therefore , the differential cross section is completely determined by the total one : @xmath191.\end{aligned}\ ] ] comparing the observables for fermions of the same @xmath47 isodoublet , @xmath192 , it is possible to derive the correlation : @xmath193.\end{aligned}\ ] ] hence , the ratio @xmath194 is independent of @xmath182 .
it equals to 5/4 for quarks and 1/2 for leptons in lower order in @xmath195 , @xmath196 .
so , the values of the observables in the @xmath197
@xmath198 plane are at the same curve ( straight line in the approximation used ) for any @xmath182 specified .
it also follows from eq .
( [ nonabeliansigma ] ) that there is a value @xmath199 when @xmath200 .
as one can check , @xmath201 .
notice , the observable @xmath202 is just the variable @xmath203 proposed in ref .
this quantity is completely insensitive to the chiral @xmath0 signals . on the other hand , the deviation of the total cross section , @xmath204 , is more sensitive to signals of the chiral @xmath0 , since the maximum of the polynomial @xmath205 is at @xmath206 .
the abelian @xmath0 beyond the minimal sm was considered recently in ref .
@xcite , where sign - definite observables convenient for detection of the abelian @xmath0 have been introduced .
rg correlations ( [ abelian ] ) in sec .
[ sec : rgrelations ] coincide with that of ref .
therefore , the observables for abelian @xmath0 beyond the thdm are to be the same as in case of the minimal sm . in case of the chiral @xmath0 the rg correlations ( [ non - abelian ] )
suppress amplitudes corresponding to the processes with right - handed fermions .
this is not the case for the abelian @xmath0 .
however , one can switch off some contributions to observable ( [ observable ] ) by specifying the kinematic parameter @xmath182 . in
what follows , it will be convenient to use the @xmath0 couplings to vector and axial - vector fermion currents [ @xmath118 , @xmath119 ] . because of correlations ( [ abelian1 ] ) the absolute value of the axial - vector couplings is universal for the all types of sm fermions , @xmath208 .
so , the observable @xmath188 has the form @xmath209.\end{aligned}\ ] ] as it was argued in ref .
@xcite , one is able to choose the value of @xmath210 , which switches off the leading contributions to the leptonic factors @xmath211 , @xmath212 , and the factor @xmath213 . the appropriate value of @xmath214 can be found from the equation @xmath215&= & 0.\end{aligned}\ ] ] the solution @xmath216 is shown in fig .
[ fig:3 ] .
this switches off the factor at @xmath217 .
as is seen , @xmath214 decreases from 0.317 at @xmath218 gev to 0.313 at @xmath219 gev . in
what follows the value of @xmath220 is taken to be 500 gev , because @xmath214 and @xmath221 depend on the center - of - mass energy through the small quantity @xmath175 ( such contributions are of order 3% ) .
= 0.35 with the above discussed choice of @xmath214 made , one can introduce the sign definite observable @xmath222 : @xmath223<0.\end{aligned}\ ] ] notice , the value of @xmath222 is universal for the all types of sm charged leptons .
there are also sign definite observables for the quarks of the same generation : @xmath224 hence one can conclude that the values of @xmath225 and @xmath226 in the @xmath225
@xmath226 plane have to be at the line crossing the axes at the points @xmath227 and @xmath228 , respectively .
signals of the abelian and the chiral @xmath0 are compared in figs .
[ fig:4]-[fig:5 ] .
suppose for a moment that experiments give the non - zero values of leptonic observables @xmath222 ( @xmath229 ) .
if they correspond to the abelian @xmath0 , either of the observables has to be the same negative number .
let one also know the values of the neutrino observables @xmath230 ( @xmath231 ) . in case of the chiral @xmath0 the corresponding point in fig .
[ fig:4 ] has to be at the straight line shown ( with the accuracy of the approximation ) . for the abelian @xmath0 the shaded region as a whole is available .
now , let us consider observables for the quarks of the same generation ( see fig .
[ fig:5 ] ) . if the value of the leptonic observable @xmath222 is measured , one has to expect that the experimental points will be located at one of two possible curves corresponding either to the chiral or to the abelian @xmath0 .
the shaded range represents signals of the abelian @xmath0 for the all possible values of the leptonic observable .
so , by measuring the observables @xmath232 for fermions of the same @xmath47 isodoublet , one is able to distinguish the abelian and the chiral @xmath0 couplings .
= 0.35 = 0.35
in the present paper the method of rg relations @xcite , developed originally for the minimal sm , is extended to searching for signals of the heavy @xmath0 gauge boson beyond the thdm .
general conditions when our consideration is applicable are the following .
1 ) the mechanism generating the heavy particle masses is not specified , and the @xmath0 mass is treated as an arbitrary parameter .
2 ) the light particle masses are generated in a standard way via the non - zero vacuum values of the scalar fields of the low - energy basis theory .
interactions of light particles with heavy scalar fields , which are responsible for @xmath233 , are excluded at tree level .
the radiation corrections to the masses due to heavy particle loops are suppressed by factors @xmath234 , and therefore not taken into account .
this kind of the mass hierarchy corresponds to the case when the basis theory is a subgroup of the underlying high energy model remaining unknown .
as our consideration shown , only two types of the @xmath0 couplings to light particles are consistent with the renormalizability .
the first type corresponds to the abelian couplings respecting the @xmath13 symmetry of the effective lagrangian ( [ l : fermion ] ) . in this case , the rg correlations fix the gauge symmetry of the yukawa terms , which relates the fermion and the scalar hypercharges . as a consequence ,
the @xmath0 couplings to the axial - vector fermion currents are completely determined by the scalar field hypercharge and the fermion isospin .
the second set of solutions
chiral @xmath0 describes interactions with the sm particles similar to the third component of the @xmath47 gauge field .
the characteristic feature of the latter couplings is the zero traces of generators associated with the @xmath0 .
notice that the @xmath0 interactions of the chiral type result in the effective four - fermion couplings @xmath235 described by the operators @xmath236 , @xmath237 , and @xmath238 according to the classification in refs . @xcite .
since each type of the @xmath0 interactions corresponds to one of mentioned operators , there is a possibility to select interactions by constructing the proper observables . as was shown , the observables proposed in ref .
@xcite can be chosen in searching for the abelian @xmath0 boson .
thus , the bounds on the @xmath0 couplings calculated therein are also applicable in case of the thdm .
the above note is important for the model independent search for @xmath0 virtual states at lep2 and future colliders lhc and nlc . in the analysis of experimental data
no discriminations between these two cases have been discussed in literature ( see , for instance , recent survey @xcite or report @xcite ) .
this difference should be important for the model - dependent @xmath0 search when different scenarios of symmetry breaking are discussed .
we believe that the derived rg relations to be useful in improving of experimental bounds on either the parameters of the @xmath0 interaction with fermions and on the relations between the cross sections of various four - fermion scattering processes .
the authors thank s. v. peletminski and n. f. shulga for discussions .
in what follows we use the notation @xmath239 , and all the momenta in the vertices are understood to be incoming . the feynman rules for vertices of figs .
[ fig:1 ] , [ fig:2 ] are listed below : 1 .
fermion - vector vertices @xmath240 2 .
fermion - scalar vertices @xmath241 ; \nonumber\\ \bar{f}_u f_d h^+ : & \quad & \sqrt{2}\left [ \omega_r\left(g_{f_d,1}\sin\beta -g_{f_d,2}\cos\beta\right ) \right.\nonumber\\&&\left .
+ \omega_l\left(-g_{f_u,1}\sin\beta + g_{f_u,2}\cos\beta\right ) \right ] ; \nonumber\\ \bar{f}_d f_u \chi^- : & \quad & \sqrt{2}\left [ -\omega_l\left(g_{f_d,1}\cos\beta + g_{f_d,2}\sin\beta\right ) \right.\nonumber\\&&\left .
+ \omega_r\left(g_{f_u,1}\cos\beta + g_{f_u,2}\sin\beta\right ) \right ] ; \nonumber\\ \bar{f}_u
f_d \chi^+ : & \quad & \sqrt{2}\left [ -\omega_r\left(g_{f_d,1}\cos\beta + g_{f_d,2}\sin\beta\right ) \right.\nonumber\\&&\left .
+ \omega_l\left(g_{f_u,1}\cos\beta + g_{f_u,2}\sin\beta\right ) \right ] ; \nonumber\end{aligned}\ ] ] 3 .
@xmath0 scalar vertices @xmath242 99 a. leike , phys . rep . *
317 * , 143 ( 1999 ) .
m. cveti and b. w. lynn , phys .
d * 35 * , 51 ( 1987 ) .
s. riemann , in _ beyond the standard model v , balholm , norway , april - may 1997 _ ( aip conference proceedings 415 ) , p. 387 . g. degrassi and a. sirlin , phys .
d * 40 * , 3066 ( 1989 ) .
t. appelquist and j. carazzone , phys .
d * 11 * , 2856 ( 1975 ) ; + j. c. collins , f. wilczek , and a. zee , _ ibid .
_ * 18 * , 242 ( 1978 ) .
m. bando , t. kugo , n. maekawa , and h. nakano , progress of theor . phys . * 90 * , 405 ( 1993 ) ; phys .
b * 301 * , 83 ( 1993 ) .
j. wudka , int .
j. of modern phys .
a * 9 * , 2301 ( 1994 ) .
a. v. gulov and v. v. skalozub , hep - ph/9812485 .
a. v. gulov and v. v. skalozub , phys .
d * 61 * , 055007 ( 2000 ) .
j. gunion , h. haber , g. kane , and s. dawson , _ the higgs hunter s guide _ ( addison - wesley , reading , ma , 1990 ) r. santos and a. barroso , phys .
d * 56 * , 5366 ( 1997 ) . c. caso
_ et al . _ ,
j. c * 3 * , 1 ( 1998 ) .
s. glashow and s. weinberg , phys .
d * 15 * , 1958 ( 1977 ) .
a. v. gulov and v. v. skalozub , yad
. fiz . * 63 * , no.1 ( 2000 ) .
j. hewett and t. rizzo , phys . rep .
* 183 * ( 1989 ) , 193 .
p. osland and a. pankov , phys .
b * 406 * , 328 ( 1997 ) ; a. pankov and n. paver , _ ibid .
_ * 432 * , 159 ( 1998 ) ; a. babich , a. pankov , and n. paver , _ ibid . _
* 426 * , 375 ( 1998 ) .
a. v. gulov and v. v. skalozub , yad
. fiz . * 62 * , 341 ( 1999 ) [ phys . at
. nucl . * 62 * , 306 ( 1999 ) ] .
w. buchmller and d. wyler , nucl .
b * 268 * , 621 ( 1986 ) ; c. arzt , m. einhorn , and j. wudka , _ ibid . _
* 433 * , 41 ( 1995 ) .
|
model independent search for signals of heavy @xmath0 gauge bosons in low - energy four - fermion processes is analyzed .
it is shown that the renormalizability of the underlying theory containing @xmath0 , formulated as a scattering in the field of heavy virtual states , can be implemented in specific relations between different processes .
considering the two - higgs - doublet model as the low - energy basis theory , the two types of @xmath0 interactions with light particles are found to be compatible with the renormalizability .
they are called the abelian and the `` chiral '' couplings .
observables giving possibility to uniquely detect @xmath0 in both cases are introduced .
|
observational astronomy is undergoing a paradigm shift .
this revolutionary change is driven by the enormous technological advances in telescopes and detectors ( e.g. , large digital arrays ) , the exponential increase in computing capabilities , and the fundamental changes in the observing strategies used to gather the data . in the past
, the usual mode of observational astronomy was that of a single astronomer or small group performing observations of a small number of objects ( from single objects and up to some hundreds of objects ) .
this is now changing : large digital sky surveys over a range of wavelengths , from radio to x - rays , from space and ground are becoming the dominant source of observational data .
data - mining of the resulting digital sky archives is becoming a major venue of the observational astronomy .
the optimal use of the large ground - based telescopes and space observatories is now as a follow - up of sources selected from large sky surveys .
this trend is bound to continue , as the data volumes and data complexity increase .
the very nature of the observational astronomy is thus changing rapidly .
see , e.g. , szalay & gray ( 2001 ) for a review .
the existing surveys already contain many terabytes of data , from which catalogs of many millions , or even billions of objects are extracted . for each object ,
some tens or even hundred parameters are measured , most ( but not all ) with quantifiable errors .
forthcoming projects and sky surveys are expected to deliver data volumes measured in petabytes .
for example , a major new area for exploration will be in the time domain , with a number of ongoing or forthcoming surveys aiming to map large portions of the sky in a repeated fashion , down to very faint flux levels .
these synoptic surveys will be generating petabytes of data , and they will open a whole new field of searches for variable astronomical objects . this richness of information is hard to translate into a derived knowledge and physical understanding .
questions abound : how do we explore datasets comprising hundreds of millions or billions of objects each with dozens of attributes ?
how do we objectively classify the detected sources to isolate subpopulations of astrophysical interest ? how do we identify correlations and anomalies within the data sets ?
how do we use the data to constrain astrophysical interpretation , which often involve highly non - linear parametric functions derived from fields such as physical cosmology , stellar structure , or atomic physics ? how do we match these complex data sets with equally complex numerical simulations , and how do we evaluate the performance of such models ?
the key task is now to enable an efficient and complete scientific exploitation of these enormous data sets .
the problems we face are inherently statistical in nature .
similar situations exist in many other fields of science and applied technology today .
this poses many technical and conceptual challenges , but it may lead to a whole new methodology of doing science in the information - rich era . in order to cope with this data flood ,
the astronomical community started a grassroots initiative , the national ( and ultimately global ) virtual observatory ( nvo ) .
the nvo would federate numerous large digital sky archives , provide the information infrastructure and standards for ingestion of new data and surveys , and develop the computational and analysis tools with which to explore these vast data volumes . recognising the urgent need , the national academy of science astronomy and astrophysics survey committee , in its new decadal survey _ astronomy and astrophysics in the new millennium _
( mckee , taylor , 2001 ) recommends , as a first priority , the establishment of a national virtual observatory ( nvo ) .
the nvo would provide new opportunities for scientific discovery that were unimaginable just a few years ago .
entirely new and unexpected scientific results of major significance will emerge from the combined use of the resulting datasets , science that would not be possible from such sets used singly . in the words of a `` white paper '' prepared by an interim steering group
the nvo will serve as _ an engine of discovery for astronomy .
_ implementation of the nvo involves significant technical challenges on many fronts , and in particular the _ data analysis_. whereas some of the nvo science would be done in the image ( pixels ) domain , and some in the interaction between the image and catalog domains , it is anticipated that much of the science ( at least initially ) will be done purely in the catalog domain of individual or federated sky surveys .
a typical data set may be a catalog of @xmath0 sources with @xmath1 measured attributes each , i.e. , a set of @xmath2 data vectors in a @xmath3-dimensional parameter space .
dealing with the analysis of such data sets is obviously an inherently multivariate statistical problem .
complications abound : parameter correlations will exist ; observational limits ( selection effects ) will generally have a complex geometry ; for some of the sources some of the measured parameters may be only upper or lower limits ; the measurement errors may vary widely ; some of the parameters will be continuous , and some discrete , or even without a well - defined metric ; etc .
in other words , analysis of the nvo data sets will present many challenging problems for multivariate statistics , and the resulting astronomical conclusions will be strongly affected by the correct application of statistical tools .
we review some important statistical challenges raised by the nvo .
these include the classification and extraction of desired subpopulations , understanding the relationships between observed properties within these subpopulations , and linking the astronomical data to astrophysical models .
this may require a generation of new methods in data mining , multivariate clustering and analysis , nonparametric and semiparametric estimation and model and hypothesis testing .
the exploration of observable parameter spaces , created by combining of large sky surveys over a range of wavelengths , will be one of the chief scientific purposes of a vo .
this includes an exciting possibility of discovering some previously unknown types of astronomical objects or phenomena ( see djorgovski 2001a , 2001b , 2001c for reviews ) .
a complete observable parameter space axes include quantities such as the object coordinates , velocities or redshifts , sometimes proper motions , fluxes at a range of wavelength ( i.e. , spectra ; imaging in a set of bandpasses can be considered a form of a very low resolution spectroscopy ) , surface brightness and image morphological parameters for resolved sources , variability ( or , more broadly , power spectra ) over a range of time scales , etc .
any given sky survey samples only a small portion of this grand observable parameter space , and is subject to its own selection and measurement limits , e.g. , limiting fluxes , surface brightness , angular resolution , spectroscopic resolution , sampling and baseline for variability if multiple epoch observations are obtained , etc . a major exploration technique envisioned for the nvo will be unsupervised clustering of data vectors in some parameter space of observed properties of detected sources .
aside from the computational challenges with large numbers of data vectors and a large dimensionality , this poses some highly non - trivial statistical problems .
the problems are driven not just by the @xmath4 of the data sets , but mainly ( in the statistical context ) by the _ heterogeneity and intrinsic complexity of the data_. a typical vo data set may consist of @xmath2 data vectors in @xmath1 dimensions .
these are measured source attributes , including positions , fluxes in different bandpasses , morphology quantified through different moments of light distribution and other suitably constructed parameters , etc .
some of the parameters would be primary measurements , and others may be derived attributes , such as the star - galaxy classification , some may be `` flags '' rather than numbers , some would have error - bars associated with them , and some would not , and the error - bars may be functions of some of the parameters , e.g. , fluxes
. some measurements would be present only as upper or lower limits .
some would be affected by `` glitches '' due to instrumental problems , and if a data set consists of a merger of two or more surveys , e.g. , cross - matched optical , infrared , and radio ( and this would be a common scenario within a vo ) , then some sources would be misidentified , and thus represent erroneous combinations of subsets of data dimensions .
surveys would be also affected by selection effects operating explicitly on some parameters ( e.g. , coordinate ranges , flux limits , etc . ) , but also mapping onto some other data dimensions through correlations of these properties ; some selection effects may be unknown .
physically , the data set may consist of a number of distinct classes of objects , such as stars ( including a range of spectral types ) , galaxies ( including a range of hubble types or morphologies ) , quasars , etc . within each object class or subclass , some of the physical properties may be correlated , and some of these correlations may be already known and some as yet unknown , and their discovery would be an important scientific result by itself
. some of the correlations may be spurious ( e.g. , driven by sample selection effects ) , or simply uninteresting ( e.g. , objects brighter in one optical bandpass will tend to be brighter in another optical bandpass ) .
correlations of independently measured physical parameters represent a reduction of the statistical dimensionality in a multidimensional data parameter space , and their discovery may be an integral part of the clustering analysis
. typical scientific questions posed may be : * how many statistically distinct classes of objects are in this data set , and which objects are to be assigned to which class , along with association probabilities ? * are there any previously unknown classes of objects , i.e. , statistically significant `` clouds '' in the parameter space distinct from the `` common '' types of objects ( e.g. , normal stars or galaxies ) ?
an application may be separating quasars from otherwise morphologically indistinguishable normal stars . *
are there rare outliers , i.e. , individual objects with a low probability of belonging to any one of the dominant classes ? examples may include known , bur relatively rare types of objects such as high - redshift quasars , brown dwarfs , etc . , but also previously unknown types of objects ; finding any such would be a significant discovery . * are there interesting ( in general , multivariate ) correlations among the properties of objects in any given class , and what are the optimal analytical expressions of such correlations ?
an example may be the `` fundamental plane '' of elliptical galaxies , a set of bivariate correlations obeyed by this hubble type , but no other types of galaxies ( see , e.g. , djorgovski 1992 , 1993 , and djorgovski 1995 , for reviews ) .
the complications include the following : 1 .
construction of these complex data sets , especially if multiple sky surveys , catalogs , or archives are being federated ( an essential vo activity ) will inevitably be imperfect , posing quality control problems which must be discovered and solved first , before the scientific exploration starts .
sources may be mismatched , there will be some gross errors or instrumental glitches within the data , subtle systematic calibration errors may affect pieces of the large data sets , etc .
the object classes form multivariate `` clouds '' in the parameter space , but these clouds in general need not be gaussian : some may have a power - law or exponential tails in some or all of the dimensions , and some may have sharp cutoffs , etc .
3 . the clouds may be well separated in some of the dimensions , but not in others .
how can we objectively decide which dimensions are irrelevant , and which ones are useful ? 4 .
the _ topology _ of clustering may not be simple : there may be clusters within clusters , holes in the data distribution ( negative clusters ? ) , multiply - connected clusters , etc .
all of this has to take into the account the heterogeneity of measurements , censored data , incompleteness , etc .
the majority of the technical and methodological challenges in this quest derive from the expected heterogeneity and intrinsic complexity of the data , including treatment of upper an lower limits , missing data , selection effects and data censoring , etc .
these issues affect the proper statistical description of the data , which then must be reflected in the clustering algorithms .
related to this are the problems arising from the data modeling .
the commonly used mixture - modeling assumption of clusters represented as multivariate gaussian clouds is rarely a good descriptor of the reality .
clusters may have non - gaussian shapes , e.g. , exponential or power - law tails , asymmetries , sharp cutoffs , etc .
this becomes a critical issue in evaluating the membership probabilities in partly overlapping clusters , or in a search for outliers ( anomalous events ) in the tails of the distributions .
in general , the proper functional forms for the modeling of clusters are not known _ a priori _ , and must be discovered from the data .
applications of non - parametric techniques may be essential here . a related , very interesting problem is posed by the _ topology _ of clustering , with a possibility of multiply - connected clusters or gaps in the data ( i.e. , negative clusters embedded within the positive ones ) , hierarchical or multi - scale clustering ( i.e. , clusters embedded within the clusters ) etc .
the clusters may be well separated in some of the dimensions , but not in others .
how can we objectively decide which dimensions are irrelevant , and which ones are useful ?
an automated and objective rejection of the `` useless '' dimensions , perhaps through some statistically defined entropy criterion , could greatly simplify and speed up the clustering analysis .
once the data are partitioned into distinct clusters , their analysis and interpretation starts .
one question is , are there interesting ( in general , multivariate ) correlations among the properties of objects in any given cluster
? such correlations may reflect interesting new astrophysics ( e.g. , , the stellar main sequence , the tully - fisher and fundamental plane correlations for galaxies , etc . ) , but at the same time complicate the statistical interpretation of the clustering .
they would be in general restricted to a subset of the dimensions , and not present in the others .
how do we identify all of the interesting correlations , and discriminate against the `` uninteresting '' observables ?
here we describe some of our experiments to date , and outline some possible avenues for future exploration .
separation of the data into different types of objects , be it known or unknown in nature , can be approached as a problem in automated classification or clustering analysis .
this is a part of a more general and rapidly growing field of data mining ( dm ) and knowledge discovery in databases ( kdd ) .
we see here great opportunities for collaborations between astronomers and computer scientists and statisticians . for an overview of some of the issues and methods , see , e.g. , fayyad ( 1996b ) .
if applied in the catalog domain , the data can be viewed as a set of @xmath5 points or vectors in an @xmath6-dimensional parameter space , where @xmath5 can be in the range of many millions or even billions , and @xmath6 in the range of a few tens to hundreds .
the data may be clustered in @xmath7 statistically distinct classes , which could be modeled , e.g. , as multivariate gaussian clouds , and which hopefully correspond to physically distinct classes of objects ( e.g. , stars , galaxies , quasars , etc . ) .
this is schematically illustrated in figure 1 .
, @xmath8 , and @xmath9 ( e.g. , some flux ratios or morphological parameters ) , and most of the data points belong to 3 major clusters , denoted @xmath10 , @xmath11 , and @xmath12 ( e.g. , stars , galaxies , and ordinary quasars ) .
one approach is to isolate these major classes of objects for some statistical studies , e.g. , stars as probes of the galactic structure , or galaxies as probes of the large scale structure of the universe , and filter out the `` anomalous '' objects .
a complementary view is to look for other , less populated , but statistically significant , distinct clusters of data points , or even individual outliers , as possible examples of rare or unknown types of objects .
another possibility is to look for holes ( negative clusters ) within the major clusters , as they may point to some interesting physical phenomenon or to a problem with the data . ] if the number of object classes @xmath7 is known ( or declared ) _ a priori _ , and training data set of representative objects is available , the problem reduces to supervised classification , where tools such as artificial neural nets or decision trees can be used .
this is now commonly done for star - galaxy separation in sky surveys ( e.g. , odewahn 1992 , or weir 1995 ) .
searches for known types of objects with predictable signatures in the parameter space ( e.g. , high-@xmath13 quasars ) can be also cast in this way .
however , a more interesting and less biased approach is where the number of classes @xmath7 is not known , and it has to be derived from the data themselves .
the problem of unsupervised classification is to determine this number in some objective and statistically sound manner , and then to associate class membership probabilities for all objects .
majority of objects may fall into a small number of classes , e.g. , normal stars or galaxies .
what is of special interest are objects which belong to much less populated clusters , or even individual outliers with low membership probabilities for any major class .
some initial experiments with unsupervised clustering algorithms in the astronomical context include , e.g. , goebel ( 1989 ) , weir ( 1995 ) , de carvalho ( 1995 ) , and yoo ( 1996 ) , but a full - scale application to major digital sky surveys yet remains to be done .
intriguing applications which addressed the issue of how many statistically distinct classes of grbs are there ( mukherjee 1998 , rogier 2000 , hakkila 2000 ) . in many situations ,
scientifically informed input is needed in designing the clustering experiments .
some observed parameters may have a highly significant , large dynamical range , dominate the sample variance , and naturally invite division into clusters along the corresponding parameter axes ; yet they may be completely irrelevant or uninteresting scientifically .
for example , if one wishes to classify sources of the basic of their broad - band spectral energy distributions ( or to search for objects with unusual spectra ) , the mean flux itself is not important , as it mainly reflects the distance ; coordinates on the sky may be unimportant ( unless one specifically looks for a spatial clustering ) ; etc .
thus , a clustering algorithm may divide the data set along one or more of such axes , and completely miss the really scientifically interesting partitions , e.g. , according to the colors of objects .
one method we have been experimenting with ( applied on the various data sets derived from dposs ) is the expectation maximisation ( em ) technique , with the monte carlo cross validation ( mccv ) as the way of determining the maximum likelihood number of the clusters .
this may be a computationally very expensive problem .
for the simple @xmath14-means algorithm , the computing cost scales as @xmath15 , where @xmath14 is the number of clusters chosen _ a priori _ , @xmath16 is the number of data vectors ( detected objects ) , @xmath17 is the number of iterations , and @xmath18 is the number of data dimensions ( measured parameters per object ) . for the more powerful expectation maximisation technique , the cost scales as @xmath19 , and again one must decide _ a priori _ on the value of @xmath14 . if this number has to be determined intrinsically from the data , e.g. , with the monte carlo cross validation method , the cost scales as @xmath20 where @xmath21 is the number of monte carlo trials / partitions , and @xmath22 is the maximum number of clusters tried . even with the typical numbers for the existing large digital sky surveys ( @xmath23 , @xmath24 ) this is already reaching in the realm of terascale computing , especially in the context of an interactive and iterative application of these analysis tools .
development of faster and smarter algorithms is clearly a priority .
one technique which can simplify the problem is the multi - resolution clustering . in this regime ,
expensive parameters to estimate , such as the number of classes and the initial broad clustering are quickly estimated using traditional techniques , and then one could proceed to refine the model locally and globally by iterating until some objective statistical ( e.g. , bayesian ) criterion is satisfied .
one can also use intelligent sampling methods where one forms `` prototypes''of the case vectors and thus reduces the number of cases to process .
prototypes can be determined from simple algorithms to get a rough estimate , and then refined using more sophisticated techniques .
a clustering algorithm can operate in prototype space .
the clusters found can later refined by locally replacing each prototype by its constituent population and reanalyzing the cluster .
techniques for dimensionality reduction , including principal component analysis and others can be used as preprocessing techniques to automatically derive the dimensions that contain most of the relevant information .
given this computational and statistical complexity , blind applications of the commonly used ( commercial or home - brewed ) clustering algorithms could produce some seriously misleading or simply wrong results .
the clustering methodology must be robust enough to cope with these problems , and the outcome of the analysis must have a solid statistical foundation
. in our experience , design and application of clustering algorithms must involve close , working collaboration between astronomers and computer scientists and statisticians .
there are too many unspoken assumptions , historical background knowledge specific to the given discipline , and opaque jargon ; constant communication and interchange of ideas are essential .
the entire issue of discovery and interpretation of multivariate correlations in these massive data sets has not really been addressed so far .
such correlations may contain essential clues about the physics and the origins of various types of astronomical objects .
effective and powerful data visualization , applied in the parameter space itself , is another essential part of the interactive clustering analysis .
good visualisation tools are also critical for the interpretation of results , especially in an iterative environment . while clustering algorithms can assist in the partitioning of the data space , and can draw the attention to anomalous objects , ultimately a scientist guides the experiment and draws the conclusions .
it is very hard for a human mind to really visualise clustering or correlations in more than a few dimensions , and yet both interesting clusters and multivariate correlations with statistical dimensionality @xmath25 or even higher are likely to exits , and possibly lead to some crucial new astrophysical insights .
perhaps the right approach would be to have a good visualisation embedded as a part of an interactive and iterative clustering analysis .
another key issue is interoperability and reusability of algorithms and models in a wide variety of problems posed by a rich data environment such as federated digital sky surveys in a vo .
implementation of clustering analysis algorithms must be done with this in mind . finally , scientific verification and evaluation , testing , and follow - up on any of the newly discovered classes of objects , physical clusters discovered by these methods , and other astrophysical analysis of the results is essential in order to demonstrate the actual usefulness of these techniques for a vo or other applications .
clustering analysis can be seen as a prelude to the more traditional type of astronomical studies , as a way of selecting of interesting objects of samples , and hopefully it can lead to advances in statistics and applied computer science as well .
we wish to thank numerous collaborators , including r. gal , s. odewahn , r. de carvalho , t. prince , j. jacob , d. curkendall , and many others .
this work was supported in part by the nasa grant nag5 - 9482 , and by private foundations .
finally , we thank the organizers for a pleasant and productive meeting .
boller , t. , meurs , e. , & adorf , h .-
1992 , a&a , 259 , 101 djorgovski , s.g . , mahabal , a. , brunner , r. , gal , r.r . , castro , s. , de carvalho , r.r .
, & odewahn , s.c .
2001a , in : _ virtual observatories of the future _ , eds .
r. brunner , s.g .
djorgovski & a. szalay , aspcs , 225 , 52 [ astro - ph/0012453 ] djorgovski , s.g . , brunner , r. , mahabal , a. , odewahn , s.c .
, de carvalho , r.r . , gal , r.r . ,
stolorz , p. , granat , r. , curkendall , d. , jacob , j. , & castro , s. 2001b , in : _ mining the sky _ ,
banday , eso astrophysics symposia , berlin : springer verlag , p. 305
[ astro - ph/0012489 ] djorgovski , s.g . , mahabal , a. , brunner , r. , williams , r. , granat , r. , curkendall , d. , jacob , j. , & stolorz , p. 2001c
, in : _ astronomical data analysis _ ,
starck & f. murtagh , _ proc .
spie _ * 4477 * , in press [ astro - ph/0108346 ] _ note added in the preprint version of the paper : _
interested reader may find a lot of information about the vo concept , and some useful links at the nvo science definition team website , http://www.nvosdt.org
|
here has been an unprecedented and continuing growth in the volume , quality , and complexity of astronomical data sets over the past few years , mainly through large digital sky surveys .
virtual observatory ( vo ) concept represents a scientific and technological framework needed to cope with this data flood .
we review some of the applied statistics and computing challenges posed by the analysis of large and complex data sets expected in the vo - based research .
the challenges are driven both by the size and the complexity of the data sets ( billions of data vectors in parameter spaces of tens or hundreds of dimensions ) , by the heterogeneity of the data and measurement errors , the selection effects and censored data , and by the intrinsic clustering properties ( functional form , topology ) of the data distribution in the parameter space of observed attributes .
examples of scientific questions one may wish to address include : objective determination of the numbers of object classes present in the data , and the membership probabilities for each source ; searches for unusual , rare , or even new types of objects and phenomena ; discovery of physically interesting multivariate correlations which may be present in some of the clusters ; etc .
|
in principle , a significant role of mergers implied by the hierarchical paradigm for the galaxy evolution must result in frequent visible misalignments of rotation momentum between various stellar and gaseous galactic subsystems . especially it must be true for non - cluster lenticular galaxies whose origin should be probably due to minor merger events .
however , findings of extended counterrotating gaseous disks are still rare . in the sa - galaxy ngc
3626 @xcite , in the s0 ngc 4546 @xcite , and in the sb - galaxy ngc 7742 @xcite all the gas counterrotates the stars . in the sa - galaxies
ngc 3593 @xcite , ngc 7217 @xcite , ngc 5719 @xcite , and ngc 4138 @xcite the counterrotating gas is already partly processed into stars , so these galaxies have two stellar counterrotating disks one of which corotates the gas . in ngc 4550
one can see already the full - size counterrotating stellar disk whereas the counterrotating gas is mainly exhausted @xcite ; the similar situation may be suspected in ngc 7331 @xcite . in the sab - galaxy ngc 4826 @xcite and in the s0 ngc 1596
@xcite the outer gas counterrotates the inner parts of the galaxies , certainly being accreted quite recently .
these few examples include almost all known extended counterrotating subsystems .
statistical estimates by @xcite and @xcite put an upper limit of 8% 12% of all spiral , s0/a - scd , galaxies to possess such structures . for s0 galaxies the appearance of counterrotating gas may be more frequent , @xcite gives the estimate of @xmath0% ; it can be consistent with the idea of s0 galaxy ( trans-)formation from a spiral by minor merger : in such event some external gas with decoupled momentum must be accreted . but in s0 galaxies the counterrotating gas is observed to be mostly confined to the very central part of the galaxies as it can be seen in the sample by @xcite ; the extended counterrotating gaseous disks are rare in s0s as extended gaseous disks in general .
another related phenomenon is inner gaseous polar disks in disk galaxies @xcite .
we found them as well in s0 galaxies with generally small amount of gas @xcite as in spiral galaxies with normal extended hi disks
ngc 2841 @xcite , ngc 7217 @xcite , ngc 7468 @xcite . for the inner polar disk origin
the most popular hypothesis is also external gas accretion ; however in some cases dynamical simulations predict strongly inclined circumnuclear gaseous disks produced by secular evolution processes .
the simulations by @xcite of the isolated stellar - gaseous disk evolution gave such ` polar disks ' as a result of gas redistribution in the global disk of a galaxy , if initially all the gas in the disk counterrotated the stars .
interestingly , the old question , what is the primary , a hen or an egg , is still actual concerning this problem . @xcite obtained the similar configuration , the inner polar disk plus the outer counterrotating gas , starting from a single inner polar disk : in a tumbling triaxial potential the outer parts of the gaseous polar disk warped in their model so that the outer gas counterrotated the stars almost in the main symmetry plane . by paying attention to the outer extension of the inner gaseous polar disks found by us ,
we have revealed indeed some cases of the configuration required , the inner polar disk plus the more outer counterrotation : these are the lenticular galaxies ngc 7280 and ngc 7332 @xcite , ic 1548 @xcite , and the late - type spiral galaxy ngc 7625 @xcite .
an exceptionality of the extended counterrotating disks means that there must exist some additional conditions for a disk galaxy to retain large masses of accreted counterrotating gas ; perhaps , it may be a very low rate of the acquisition process @xcite , or an absence of large amount of initial , ` own ' galactic gas in the recipient galaxy @xcite . every new disk galaxy with a globally counterrotating gas component may in principle help to determine these conditions .
in this paper we report a discovery of two more extended counterrotating gaseous disks in early - type disk galaxies .
we present the results of the kinematical study of the nearby s0 galaxies ngc 2551 and ngc 5631 .
ngc 2551 and ngc 5631 considered in this paper are early - type disk galaxies of intermediate luminosity .
both belong to spiral - dominated groups @xcite , and both have a substantial amount of rotating neutral hydrogen with unknown sense of rotation , according to single - dish radioobservations at 21 cm @xcite . their main parameters retrieved in databases and from literature are presented in the table [ tab1 ] .
the layout of the paper is the following . in section 2
we describe our observations , our data reduction and some additional information . in section 3
we present the counterrotating gaseous disk in ngc 2551 , and in section 4 the complex kinematics including inclined counterrotating stellar - gaseous disk in ngc 5631 .
section 5 contains some discussion of the origin and possible future fate of the counterrotating gas in these two galaxies .
lcc ngc & 2551 & 5631 + type ( ned@xmath1 ) & sa(s)0/a & sa(s)@xmath2 + @xmath3 , kpc ( leda@xmath4 ) & 8.5 & 9.3 + @xmath5 ( rc3@xmath6 ) & 12.78 & 12.35 + @xmath7 ( leda ) & 20.0 & 20.2 + @xmath8 ( rc3 ) & 0.92 & 0.90 + @xmath9 ( ned ) , @xmath10 & 2344 & 1979 + distance , mpc ( leda ) & 37.1 & 32.1 + inclination ( leda ) & @xmath11 & @xmath12 + _ pa_@xmath13 ( leda ) & @xmath14 &
+ @xmath15 , @xmath10 ( leda , hi ) & @xmath16 & @xmath17 + @xmath18 , @xmath19 & 1.4 & 1.4 + + + +
to study the rotation of stars and ionized gas , we use spectral data obtained for ngc 2551 and ngc 5631 with three different spectrograph .
the multi - pupil fiber spectrograph ( mpfs ) of the russian 6 m telescope @xcite is an integral - field unit constructed following the fiber - lens principle ; due to this feature it allows to obtain panoramic spectral data over a wide spectral range ( in our case , over 1500 with the spectral resolution of 3 ) .
the field of view is @xmath20 , with the sampling of @xmath21 per microlens .
we observed ngc 2551 and ngc 5631 in the green spectral range containing a lot of absorption lines and calculated the line - of - sight ( los ) stellar velocities by cross - correlating continuum - substracted and logarithmically - binned galactic spectra with the similarly prepaired spectra of the twilight ( the sun spectrum , of g2 spectral type ) and of g- and k - giant stars observed the same nights as the galaxies hd 19476 ( k0iii ) for ngc 2551 and hd 135722 ( g8iii ) and hd 167042 ( k1iii ) for ngc 5631 .
for ngc 2551 , also the red spectral range has been exposed to make gauss - fitting of the [ nii]@xmath226583 emission line and to estimate the los velocities of the ionized gas .
the statistical accuracy of one - element los velocity and velocity dispersion estimates with the mpfs data is about 10 km / s . in 2007
the galaxy ngc 5631 was also observed with another integral - field spectrograph , the sauron of the william herschel telescope at la palma @xcite .
we have retrieved these data from the open ing archive of the cambridge astronomical data center and have reduced them in our manner calculating the stellar los velocities by cross - correlation with the spectrum of a star observed the same night ( hd 72184 , k2 iii , this time ) and by calculating the gas los velocities by measuring the baricenter positions of the [ oiii]@xmath225007 emission line in the continuum - subtracted spectra . the field of view of the sauron is @xmath23 , with the sampling of @xmath24 ; the spectral resolution is 4 , and the spectral range is narrow , 48005350 , because it is a tiger - mode integral - field spectrograph . after discovering central gas counterrotation in both galaxies with the integral - field spectroscopic data , we have wanted to know a full extension of the counterrotating gas systems . to check it ,
we have observed the galaxies with the spectrograph scorpio of the 6 m telescope in the long - slit mode @xcite .
the slit , which length is about 6 arcmin , has been aligned with the kinematical major axis of the central los velocity fields . the red spectral range , 61007100 , with a @xmath25 spectral resolution , has been exposed to measure first of all the los velocities of the strongest emission line in the optical spectral range , [ nii]@xmath226583 , which is free of the underlying absorption contamination unlike the h@xmath26 .
however , the los velocity and stellar velocity dispersion profiles for the stellar components have been also estimated by cross - correlating galactic spectra binned along the slit with a template star spectrum from the library miles @xcite . through the library we chose spectra of the stars
hd 48433 ( k1iii ) and hd 10380 ( k3iii ) which provided the largest amplitudes of the cross - correlation function with the spectra of ngc 2551 and ngc 5631 respectively .
we measured the errors of the stellar velocity and velocity dispersion by using the formulae from the classical paper by @xcite .
the uncertainties of the ionized - gas kinematical parameters were estimated by monte carlo simulations of artificial spectra with the noise distribution similar to that of the original data .
the log of observations is given in table [ tab2 ] . [ cols="<,<,<,<,<,>,^,^",options="header " , ] + + + to analyze the morphological structure of the galaxies , we have also used surface photometric data obtained by reducing the digital images from the hst archive ( ngc 2551 , acs / f625w ) and from the sdss / dr6 @xcite data collection ( ngc 5631 , @xmath27-filters ) .
figure 1 presents the los velocity fields for the stellar and gaseous components in the center of ngc 2551 which we have obtained with the mpfs .
the seeing quality has been much worse during the red - range mpfs exposure , so the visible gas rotation seems to be slower than that of the stars ; it is an effect of smearing the steep velocity gradient by poor spatial resolution . in general , fig .
1 is intended only to demonstrate gas counterrotation with respect to the stars in the center of the galaxy .
the directions of the kinematical major axis which is defined as a direction of the maximum los velocity gradient are @xmath28 for the stellar component and @xmath29 for the ionized gas .
the former value coincides with the photometrical major axis direction , @xmath30 at @xmath31 according to the hst data , implying an axisymmetric character of the galaxy kinematics and structure .
indeed , ngc 2551 is known to be unbarred , and its photometric major axis direction , @xmath32 , is constant along the full radial extension @xcite .
the fact that the isophote ellipticity reaches its maximum , @xmath33 , already at @xmath34 ( @xmath35 ) implies that the galaxy is disk - dominated .
this conclusion is confirmed by the major - axis surface brightness profile decomposition by @xcite : according to their model , the regular exponential disk dominates in the brightness profile of ngc 2551 starting from @xmath36 .
the next question which arose after the gas counterrotation was found with the mpfs , was if we deal with the central decoupled gas subsystem , or the counterrotating gas is extended over the whole galaxy .
the scorpio gas and stellar los velocity profiles obtained at @xmath37 ( fig . 2 ) demonstrate persistence of the gas - star counterrotation up to @xmath38 at least .
the gas rotation curve is rather flat and extended to @xmath39 .
the projected gas rotation velocity , 150 km / s , exceeds even the aperture hi value from @xcite ( table 1 ) so @xmath37 may be well the global disk line - of - nodes direction . in general , the data favor coplanar stellar and gaseous disks in ngc 2551 though rotating in opposite senses .
figure 3 presents the stellar and gas los velocity fields for the center of ngc 5631 constructing by using the data of the mpfs and the sauron ; the los velocities of the ionized gas have been calculated by measuring the baricenter positions of the emission line [ oiii]@xmath225007 . the stars and the ionized gas counterrotate over the whole field of view of the mpfs , and there is a hint of the rotation reverse for the stars at the edges of the sauron field of view .
the orientations of the kinematical major axes within @xmath40 from the center , @xmath41 for the stars and @xmath42 for the ionized gas are consistent with each other and with the photometric major axis orientation in the central part of the galaxy , @xmath43 , implying coplanar axisymmetric rotation .
+ the long - slit cross - sections made with the scorpio have shown that the gaseous disk which rotation we observe in the center of ngc 5631 is rather extended : we see the measurable emission lines up to @xmath44 ( @xmath45 ) ( fig .
the gas excitation is shock - like over the full extension of the visible emission : the [ nii]@xmath226583 emission line is everywhere stronger than the h@xmath26 one .
the maximum projected rotation velocity reaches about 170 km / s being again consistent with the integrated hi data @xcite .
the stellar component counterrotates the ionized gas up to @xmath46 . at @xmath47
the projected rotation velocity of the stars falls to zero .
interestingly , the same character of the stellar los velocity profile was found by @xcite who obtained a long - slit cross - section at @xmath48 : the maximum velocity , @xmath49 km / s , was reached at @xmath50 , then the velocity curve falled , passed through zero at @xmath51 , and reversed its sense at @xmath52 .
the stellar velocity dispersion profile at @xmath53 demonstrates some rise at @xmath54 .
this feature becomes understandable if we look directly at the losvd shape ( fig .
5 ) . at @xmath55 and at @xmath56
the losvd becomes asymmetric with the hint on two peaks and remains two - peaked up to the limit of our measurements , @xmath57 .
this fact results in the visible increase of the stellar velocity dispersion and in the visible fall of the projected rotation velocity to zero : two counterrotating stellar components compensate each other being approximated by a single gaussian losvd .
the isophote behavior is not so simple in ngc 5631 as in ngc 2551 : though the galaxy is unbarred , the isophotes of the central part , within @xmath58 from the center , are more elliptical than those of the outer disk , and their major axis is turning perhaps between @xmath59 and @xmath60 from @xmath61 to @xmath62 ( fig .
the shape of the brightness profiles which we have derived from the sdss data by averaging the counts per arcsec over the ellipses with the parameters found by the isophote analysis allows to divide the whole galaxy into three main components ( fig .
the outer exponential stellar disk dominating at @xmath63 is seen face - on : the ellipticity of the isophotes is less than 0.05 , and the orientation of the major axis can not be determined . the inner exponential stellar component ( inner disk ? ) is clearly seen in the radius range of @xmath64 ; the orientation of the isophote major axis is @xmath65 , and the isophote ellipticity after subtracting the outer disk stays at 0.170.18 implying an inclination of @xmath66 under the assumption of a rather thin disk .
the surface brightness profile of the bulge , @xmath67 , can be approximated by a sersic law with @xmath68 with a high accuracy ; however , the isophote ellipticity and major - axis orientation within this central zone do not stay constant along the radius .
the recent work by @xcite suggests the brightness profile decomposition of ngc 5631 into only two components , the sersic bulge and the single exponential disk ; but as one can see in fig . 7
, their use of the shallow 2mass photometry does not allow them to measure the outer stellar disk in ngc 5631 .
the change of the stellar rotation direction takes place within the inner disk ; the isophote ellipticity does not fall to zero at @xmath69 , on the contrary , it stays constant at 0.170.18 .
so we conclude that the observed stellar los velocity behavior can not be due to a rotation plane warp in the nearly face - on galaxy , but is indeed a manifestation of the switch of the mean stellar rotation direction inside the zone of the photometric dominance of the inner stellar disk .
fortunately , we can say something about the gas plane orientation too . figure 8
presents a color map of ngc 5631 that we have constructed by using the sdss data ; for the sdss survey description
see @xcite .
at the radius of @xmath70 one can see a broad red ( dust ) ring .
more exactly we see a half of the dust ring , the other half being hidden behind the bulge . since the gas is thought to be coupled with the dust , we can estimate an inclination and line - of - nodes orientation of the gas plane at @xmath70 under the assumption of its circular shape .
it appears to be @xmath71 ( just as the inner stellar isophotes and the kinematical major axes are directed ! ) and @xmath72 just as the stellar isophotes within the inner disk .
then the deprojected gas rotation velocity under the assumption of @xmath73 is around 360390 km / s within the model of circular rotation that is high but not exceptional .
some asymmetry of the stellar los velocity profiles in fig . 4 could be then explained if we assume that the inclined gaseous disk with the orientation parameters deduced above contains also some stellar component , and both are coupled with the inner stellar disk derived from the surface brightness profile decomposition .
then the dusty stellar disk dominating photometrically in the radius range of @xmath74 and inclined with respect to the face - on main stellar disk would completely hide the main stellar component to the north of the center ; just this picture is observed in the cross - section at @xmath75 .
the slit at @xmath53 projected to the west below the line of nodes of this dusty disk , catches the rotation of the main stellar component ( a long western receding velocity branch , fig .
4 _ a _ ) . and
both cross - sections in their eastern parts demonstrate the zero mean los velocities and the visible stellar velocity dispersion raising up to 180 km / s at @xmath76 which can be treated as a superposition of two comparable stellar components counterrotating each other .
both ngc 2551 and ngc 5631 possess the extended ( up to 0.71.0@xmath3 , or up to 57 kpc from the center ) counterrotating gaseous disks .
both galaxies belong to loose groups dominating by spiral galaxies @xcite .
however either ngc 2551 nor ngc 5631 have a close neighbor within the circle of 100 kpc radius , to provide interaction and smooth gas accretion that is suggested by @xcite to be the most probable mechanism of massive counterrotating disk formation .
the only alternative which is available for ngc 2551 and ngc 5631 is a minor merger with a gas - rich satellite .
we do not know what morphological type the galaxies ngc 2551 and ngc 5631 had before their merging , and if they have had their own ( corotating ) gas .
but in any case the accreted gas had to suffer instaneous star formation triggered by shock compression during the merging .
perhaps , in ngc 2551 and ngc 5631 we see different stages of the same process .
ultraviolet imaging with the uit and later on with the galex has revealed an extended , up to @xmath77 , star - forming disk in ngc 2551 @xcite .
an intensity ratio h@xmath26/[nii ] observed by us with the scorpio implies an excitation by young stars up to @xmath78 ; however between @xmath77 and @xmath79 we see only one emission line , [ nii]@xmath226583 , so in this ring the gas excitation may be of shock origin . in ngc 5631
there is no current star formation in the counterrotating gaseous disk , and the gas is excited by shock over the full extension of the gaseous disk .
perhaps , it is the preceding evolutionary stage with respect to ngc 2551 , the accreted gas disk has not yet settled into the symmetry plane of the main galaxy , and star formation is only going to start in the compressed inner dusty ring at @xmath80 . and
what may be the subsequent stage ?
perhaps , it is ngc 4138 where a counterrotating extended gas is supplemented by the substantial counterrotating young stellar component @xcite the evident consequence of the star formation in the counterrotating gaseous disk . to give a conclusion
, we summarize that by applying complex spectral methods including integral - field spectroscopy to the central parts of the galaxies and long - slit deep spectroscopy to probe the external parts , we have found two more global gas counterrotating systems in non - interacting early - type disk galaxies . in ngc 2551 two counterrotating disks , gaseous and stellar ones ,
may be coplanar : the orientation parameters of the optical - band image and of the uv - band , star - formation related image are similar . in ngc 5631 the gaseous disk
is inclined by some @xmath81 to the main stellar disk ; it may contain also significant coupled stellar component .
the totality of the spectral and photometric data give evidence for the minor merging as the most probable origin of the counterrotating gas in these galaxies .
perhaps , we observe two different stages of the process of lenticular galaxy formation in rather sparse group environments .
the 6 m telescope is operated under the financial support of the ministry of science and education ( registration number 01 - 43 ) .
our study of the galaxies with the multi - tiers disks , such as ngc 5631 , is supported by the grant of the russian foundation for basic researches no .
07 - 02 - 00229 .
a. v. m. acknowledges a grant from the president of the russian federation ( mk1310.2007.2 ) . during the data analysis we have used the lyon - meudon extragalactic database ( hyperleda ) supplied by the leda team at the cral - observatoire de lyon ( france ) and the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration
this research is partly based on data obtained from the isaak newton group archive which is maintained as part of the casu astronomical data centre at the institute of astronomy , cambridge , on observations made with the nasa / esa hubble space telescope , obtained from the data archive at the space telescope science institute , which is operated by the association of universities for research in astronomy , inc .
, under nasa contract nas 5 - 26555 , and on sdss data .
funding for the sloan digital sky survey ( sdss ) and sdss - ii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the u.s .
department of energy , the national aeronautics and space administration , the japanese monbukagakusho , and the max planck society , and the higher education funding council for england .
the sdss web site is http://www.sdss.org/.
|
we report a discovery of extended counterrotating gaseous disks in early - type disk galaxies ngc 2551 and ngc 5631 . to find them
, we have undertaken complex spectral observations including integral - field spectroscopy for the central parts of the galaxies and long - slit deep spectroscopy to probe the external parts .
the line - of - sight velocity fields have been constructed and compared to the photometric structure of the galaxies . as a result , we have revealed full - size counterrotating gaseous disks , the one coplanar to the stellar disk in ngc 2551 and the other inclined to the main stellar disk in ngc 5631 .
we suggest that we observe the early stages of minor - merger events which may be two different stages of the process of lenticular galaxy formation in rather sparse environments .
|
a boolean function with @xmath0-arguments , @xmath1 , is a map @xmath2 where @xmath3 is the field with two elements .
the @xmath4-algebra @xmath5 of boolean functions on @xmath0-arguments , with pointwise sum and multiplication , is isomorphic to the boolean algebra @xmath6 $ ] of sets of subsets of @xmath7=\{1, ...
indeed , we identify a vector in @xmath8 with an element of @xmath9 $ ] via the characteristic function , and we identify a map @xmath9 \longrightarrow \mathbb{z}_2 $ ] with a subset of @xmath9 $ ] again with the help of characteristic functions .
the sum and product of boolean functions correspond to the symmetric difference and the intersection of subsets of @xmath9,$ ] respectively .
the canonical isomorphism @xmath10 $ ] just described establishes the link between classical propositional logic and set theory @xcite .
+ the partial derivative @xmath11 , for @xmath12 $ ] , of a boolean function @xmath13 , see @xcite , is given by @xmath14 where @xmath15 and @xmath16 is the vector with @xmath17 at the @xmath18-th position and @xmath19 at the other positions .
+ we define the @xmath4-algebra @xmath20 of boolean differential operators on @xmath8 in analogy with the definition of differential operators on the affine space @xmath21 , for a field @xmath22 of characteristic zero , i.e. @xmath20 is the subalgebra of @xmath23 generated by the operators of multiplication by boolean functions , and the partial derivative operators @xmath24 defined in ( [ e1 ] ) .
+ it turns out that @xmath25 , see @xcite , i.e. any @xmath4-linear operator from @xmath5 to itself is actually given by a boolean differential operator .
therefore a boolean differential operator @xmath26 is just a map @xmath27 we are interested in finding a suitable set theoretical interpretation for the algebras @xmath20 that extends the above mentioned interpretation of @xmath5 as the boolean algebra @xmath6 $ ] , and may shed a light towards a logical understanding of the @xmath4-algebras @xmath20 .
indeed , we believe that the @xmath4-algebras @xmath20 may play a semantic role , analogous to that played by truth functions in classical logic , within the context of a `` quantum like '' operational logic yet to be fully understood .
a few steps in that direction are taken in @xcite .
+ our main goal in this work is to find suitable matrix representations for the @xmath4-algebras @xmath20 . by dimension counting
@xmath25 is non - canonically isomorphic to the @xmath4-algebra @xmath28 of square matrices of size @xmath29 with @xmath30-@xmath17 entries .
note that @xmath28 may be identified , via characteristic functions , with @xmath31\times \mathrm{p}[n])$ ] the set of subsets of @xmath9\times \mathrm{p}[n]$ ] , or equivalently , with the set @xmath32}$ ] of simple directed graphs ( possibly with loops ) with vertex set @xmath9.$ ] a matrix @xmath33\times \mathrm{p}[n])$ ] is regarded as a directed graph with vertex set @xmath9 $ ] by drawing an edge from @xmath34 $ ] to @xmath35 $ ] if and only if @xmath36 . the sum and product of matrices in @xmath28 induce operations of sum and product of digraphs in @xmath32}$ ] .
the sum on @xmath32}$ ] is the symmetric difference .
the product @xmath37 of digraphs @xmath38}$ ] is such that the pair @xmath39 if and only if there is and odd number of sets @xmath40 $ ] such that @xmath41 and @xmath42 + to define an explicit isomorphism @xmath43 a choice of basis for @xmath5 must be made . in this work
we only consider the basis @xmath44\ \}$ ] for @xmath5 , where the boolean function @xmath45 is given on @xmath34 $ ] by : @xmath46 we let @xmath47 \in \mathrm{m}_{2^n\times 2^n}(\mathbb{z}_2)$ ] be the matrix of the boolean differential operator @xmath26 in the basis @xmath44\ \}$ ] .
+ we are going to use the following simple algebraic construction .
let @xmath48 be a @xmath4-algebra , @xmath49 a @xmath4-vector space , and @xmath50 be a @xmath4-linear bijective map .
we use @xmath51 to pullback the product on @xmath48 to a product on @xmath49 given for @xmath52 by @xmath53 with this product on @xmath49 the map @xmath51 becomes an algebra isomorphism .
+ as we shall see each of our choices of bases for @xmath20 induces a @xmath4-linear bijective map from @xmath32}$ ] to @xmath20 . in @xcite
we use four such bijections to pullback the composition product on @xmath20 to @xmath32}$ ] , thus we obtain four products on @xmath32}$ ] denoted , respectively , by @xmath54 having various presentations for the product on @xmath55 is desirable , just as it is useful to generate truth functions by several types of logical connectives . +
our main goal in this work is to explicitly describe matrix representations for the products @xmath56 on @xmath32}$ ] .
it turns out that the product @xmath57 is the easiest to handle , in section 2 we discuss some of its basic properties and describe an explicit isomorphism with @xmath28 . in the remaining sections , we present explicit isomorphisms between the products @xmath58 and the product on @xmath28 , the algebra of square matrices of size @xmath29 with entries in @xmath4 .
+ let us comment on some conventions assumed in this work . in the figures we draw a subset of @xmath9\times
\mathrm{p}[n]$ ] as a subset of the real plane , using the bijective correspondence between @xmath9=\mathbb{z}_2^n$ ] and the natural numbers in the interval @xmath59 $ ] resulting of ordering @xmath9 $ ] by cardinality and lexicographic order within a given cardinality . for example @xmath60
$ ] and @xmath61 $ ] are in correspondence as follows @xmath62 when drawing a product , the elements of the first factor are drawn as triangles ; the elements of the second factor are drawn as circles ; and the elements in the product are drawn as stars .
we identify matrices in @xmath28 with maps @xmath9 \times \mathrm{p}[n ] \longrightarrow \mathbb{z}_2 $ ] using again the cardinality - lexicographic order on @xmath63.$ ] we use juxtaposition for the product of matrices , and @xmath64 for the rank of matrix @xmath48 .
as mentioned in the introduction we are going to consider four different bases for the @xmath4-algebra @xmath20 of boolean differential operators on @xmath8 . in this section we consider the @xmath65-basis @xmath66\ \},\ ] ] where the boolean functions @xmath67 were described in the introduction , and the shift operators @xmath68 are given by @xmath69 , \ f \in \mathrm{bf}_n .\ ] ] note that @xmath70 and @xmath71 , where @xmath17 stands for the identity operator ; thus one can move back and forward from the shift operators @xmath72 to the partial derivatives operators @xmath73 indeed , it is easy to check that @xmath74 consider the identifications @xmath75 } \ \ \simeq \ \
\mathrm{map}(\mathrm{p}[n]\times \mathrm{p}[n ] , \mathbb{z}_2 ) \
\ \simeq \ \ \mathrm{bdo}_n,\ ] ] where the identification on the left is given by characteristic functions and we use it freely without change of notation ; the non - canonical identification on the right is obtained via the bijective @xmath4-linear map @xmath76 } \rightarrow \mathrm{bdo}_n$ ] sending a directed graph @xmath77}$ ] to the boolean differential operator given by @xmath78}a(c , d)m^cs^d.\ ] ] the @xmath57-product on @xmath32}$ ] is the pullback via the map @xmath79 of the composition product on @xmath20 .
the @xmath57-product , see @xcite , is given for @xmath80}$ ] by the equivalent identities : @xmath81 @xmath82}a(c , e)b(c+e , d+e ) ; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ ] ] @xmath83\times \mathrm{p}[n]\ \ | \ \ o\{e\in \mathrm{p}[n ] \ | \
( c , e ) \in a , \ \
( c+e , d + e ) \in b \ } \ \ } , \ \ \ \ \\ ] ] where the notation @xmath84 means that the finite set @xmath85 has odd cardinality .
+ we proceed to introduce a matrix representation for the algebra @xmath86 } , \star)$ ] .
consider the map @xmath87 } \longrightarrow \mathrm{m}_{2^n\times 2^n}(\mathbb{z}_2 ) $ ] sending a directed graph @xmath77 } $ ] to the matrix of the operator @xmath88}a(c , d)m^cs^d\ ] ] in the basis @xmath89\},$ ] i.e. we have that @xmath90.\ ] ] note that @xmath91 is a @xmath4-linear map since it is the composition of two @xmath4-linear maps . _
the map @xmath92 } , \star ) \longrightarrow ( \mathrm{m}_{2^n\times 2^n}(\mathbb{z}_2 ) , \ . )
$ ] is an algebra isomorphism given for @xmath77}$ ] by @xmath93 the inverse map @xmath94 } , \star ) $ ] sends a matrix @xmath95 to the directed graph @xmath96 with characteristic function given by @xmath97 _ first note that @xmath98 where @xmath99 is the kronecker delta function .
therefore the matrix @xmath100 \in \mathrm{m}_{2^n\times 2^n}(\mathbb{z}_2)$ ] is such that : @xmath101_{a , b}\ = \ \delta(a , c)\delta(b , c+d).\ ] ] thus for @xmath77}$ ] we have that : @xmath102}a(c , d)m^cs^d \ \right]_{a , b } \ = \ \sum_{c , d \in \mathrm{p}[n]}a(c , d)[m^cs^d]_{a , b}\ = \ ] ] @xmath103}a(c , d)\delta(a , c)\delta(b , c+d)\ = \ a(a , a+b).\ ] ] the maps @xmath91 and @xmath104 are inverse of each other , indeed we have that @xmath105 @xmath106 for @xmath77}$ ] and @xmath95 . + next we show that @xmath91 is an algebra morphism .
thus we have that : @xmath107 \ = \ [ l_1(a)l_1(b ) ) ] \
= \ [ l_1(a)][l_1(b ) ) ] \ = \
m_1(a ) m_1(b).\ ] ] explicitly , for @xmath38 } $ ] , we have using ( [ e5 ] ) and ( [ e3 ] ) that : @xmath108 @xmath109}a(a_1,b)b(a_1+b , a_1 + a_2+b)\ = \ \sum_{b\in \mathrm{p}[n]}m_1(a)(a_1,a_1 + b)m_1(b)(a_1+b , a_2)=\ ] ] @xmath109}m_1(a)(a_1,b)m_1(b)(b , a_2)\ = \ \left(m_1(a)m_1(b)\right)_{a_1 , a_2 } .\ ] ] _ @xmath110 and @xmath111 , where @xmath112 .
_ _ consider the jordan - like matrices @xmath28 having ones on the principal diagonal and on the diagonal directly above the principal . the associated boolean differential operators in the @xmath65-basis , for @xmath113 $ ] , are given in table [ jordanbdo ] .
_ _ the multiplication table of @xmath114 } , \star)$ ] is given in table [ table_product ] .
_ it would be nice to have an intuitive understanding of the @xmath57-product , say in the spirit of venn diagrams . in order to gain a better understanding of the meaning of the @xmath57-product we consider several examples .
the first example is the simple case of the product of graphs with a unique edge .
_ let @xmath115 $ ] , then we have that : @xmath116 _ let @xmath117 , then there is an odd number of sets @xmath118 $ ] such that @xmath119 thus @xmath120 so , there is no such @xmath121 if @xmath122 . in the case
@xmath123 , we have that @xmath124 in the next examples we consider the @xmath57-product on graphs using suitable decompositions of the graphs .
_ let @xmath125 } \ $ ] be given by @xmath126}a_b\times \{b\ } \ \ \ \ \
\mbox{and } \ \ \ \ \ b = \sum_{c\in \mathrm{p}[n]}b_c\times \{c\},\ ] ] where @xmath127 is the set endpoints of edges in @xmath48 starting at @xmath128 , and @xmath129 is similarly defined .
then we have that : @xmath130}\left(a_b \cap ( b_c + b)\right)\times \{b+c \ } .\ ] ] _ using the distributive property , recall that @xmath131 } , \star ) \ \simeq \ ( \mathrm{m}_{2^n\times 2^n}(\mathbb{z}_2 ) , \ . ) , $ ] we get @xmath130}\left(a_b\times \{b\}\right ) \star \left(b_c\times \{c\}\right).\ ] ] suppose @xmath132 , then there is an odd number of sets @xmath118 $ ] such that @xmath133 and @xmath134 .
clearly , the only possible set @xmath121 with those properties is @xmath135 , and furthermore @xmath136 , @xmath137 , and @xmath138 therefore , we have that : @xmath139 yielding the desired result . _ to contrast the classical intersection @xmath140 with the @xmath57-product note that the intersection of graphs @xmath80}$ ] can be written as @xmath141}\left(a_b \cap b_b \right)\times \{b\ } \ = \ \sum_{b , c \in \mathrm{p}[n]}\delta(b , c)\left(a_b
\cap b_b\right)\times \{b\}.\ ] ] _ [ e4 ] _ let @xmath142 $ ] , then @xmath143 since @xmath144 @xmath145 is equal to @xmath146 @xmath147 figure [ fig : star_uno ] shows a graphical representation of the product @xmath148 . _ , example [ e4 ] .
] _ let @xmath38}$ ] be written as @xmath126}a_b\times \{b\ } \ \ \ \ \ \ \mbox{and } \ \ \ \ \ \ b = \sum_{c\in \mathrm{p}[n]}\{c\ } \times b_c.\ ] ] then we have that : @xmath149 _ using the distributive property for the @xmath57-product we get : @xmath150}\left(a_b\times \{b\}\right ) \star \left(\{c\ } \times b_c\right ) \ = \
\sum_{b+c \in a_b}\{b+c \ } \times ( b_c + b).\ ] ] we show the second identity .
suppose @xmath151 then there is a odd number of sets @xmath152 $ ] such that @xmath133 and @xmath153 . clearly , the only possibility is @xmath135 , and furthermore @xmath154 , @xmath155 [ hoy ] _ let @xmath156 and @xmath157 in @xmath158 $ ] be given , respectively , by @xmath159 and @xmath160 then we have , see figure [ fig : star_dos ] , that : @xmath161 _ from example [ hoy ] . ]
_ let @xmath38}$ ] be written as @xmath126}\{b\}\times a_b \ \ \ \ \
\mbox{and } \ \ \ \ \ b = \sum_{c\in \mathrm{p}[n]}\{c\}\times b_c.\ ] ] then we have that : @xmath162 _ using the distributive property for the @xmath57-product we have that : @xmath163}(\{b\}\times a_b ) \star ( \{c\}\times b_c ) .\ ] ] a pair @xmath164 \times \mathrm{p}[n]$ ] belongs to @xmath165 if there is an odd number of sets @xmath118 $ ] such that @xmath166 and @xmath167 .
thus we have that @xmath168 the desired result follows .
[ mana ] _ let @xmath169 and @xmath170 in @xmath158 $ ] be given by @xmath171 and @xmath172 then , see figure [ fig : star_tres ] , we have that : @xmath173 _ , example [ mana ] . ] [ e2 ] _ let @xmath174}$ ] be given by @xmath175 @xmath176 @xmath177 @xmath178 the product @xmath179 , see figure [ fig : star_cuatro ] , is given by @xmath180 _ . ]
in this section we consider the @xmath183 basis for @xmath20 , thus we regard a directed graph @xmath77}$ ] as a boolean differential operator via the map @xmath184 } \longrightarrow \mathrm{bdo}_n$ ] given by @xmath185}a(c , d)m^c\partial^d,\ ] ] where for @xmath186 $ ] we set @xmath187 + the @xmath182-product on @xmath32}$ ] is the pullback via the map @xmath188 of the composition product on @xmath20 , i.e. the @xmath182-product is given for @xmath80}$ ] by @xmath189 .
explicitly , see @xcite , we have that @xmath190 equivalently , a pair @xmath191\times \mathrm{p}[n]$ ] belongs to @xmath192}$ ] if and only if there is an odd number of sets @xmath193\ $ ] and @xmath194 such that @xmath195 note that we can go back and forward from the @xmath65-basis to the @xmath196-basis for boolean differential operators as follows : @xmath197}a(c , d)m^c\partial^d\ = \ \sum_{c , d \in \mathrm{p}[n]}\widehat{a}(c , d)m^c s^d,\ ] ] indeed with the help of the identities ( [ e6 ] ) we get that @xmath198 consider the map @xmath199 } \longrightarrow \mathrm{m}_{2^n\times 2^n}(\mathbb{z}_2 ) $ ] sending a directed graph @xmath77 } $ ] to the matrix of the operator @xmath200}a(c , d)m^c\partial^d \ = \ \sum_{c , e \subseteq d } a(c , d)m^cs^e\ ] ] in the basis @xmath89\},$ ] i.e. we have that @xmath201 $ ] . _
the map @xmath202 } , \circ ) \longrightarrow ( \mathrm{m}_{2^n\times 2^n}(\mathbb{z}_2 ) , \ .)$ ] is an algebra isomorphism given for @xmath77}$ ] by @xmath203 the inverse map @xmath204 } , \circ ) $ ] sends @xmath95 to the directed graph @xmath205 with characteristic function given by @xmath206 _ for @xmath77}$ ] , identifying @xmath48 with @xmath207 and using ( @xmath208 ) , we get that @xmath209}a(c , d)m^c\partial^d\right]_{a , b } \ = \ \left [ \sum_{c , e \subseteq d } a(c , d)m^cs^e \right]_{a , b}\ ] ] @xmath210_{a , b}\ = \ \sum_{c , e \subseteq d } a(c , d)\delta(a , c)\delta(b , c+e)=\sum_{a+b \subseteq d } a(a , d),\ ] ] since @xmath211 , and @xmath212 implies that @xmath213 + we show that @xmath214 is indeed the inverse of @xmath215 .
we have for @xmath77}$ ] that : @xmath216 @xmath217 where the last identity is shown as follows : @xmath218 @xmath215 is an algebra morphism since @xmath219 $ ] , and thus we have that : @xmath220 \ = \ [ l_2(a)l_2(b ) ) ] \
= \ [ l_2(a ) ] [ l_2(b ) ) ] \ = \
m_2(a ) m_2(b).\ ] ] _ @xmath221 and @xmath222 , where @xmath223 .
for @xmath35 $ ] consider the boolean function @xmath225 given on @xmath34 $ ] by : @xmath226 one can go back and forward from the @xmath227 basis to the @xmath228 basis as follows : @xmath229 indeed , the identity on the right follows directly from definitions ( [ e20 ] ) and ( [ e16 ] ) ; the identity on the left follows from the m@xmath230bius inversion formula for modules over a @xmath4-ring , see @xcite , which states that for arbitrary maps @xmath231 \longrightarrow m,\ ] ] with @xmath232 a module over a @xmath4-ring , the identities @xmath233 note that @xmath234 for all @xmath235 .
the m@xmath236bius inversion formula follows from the identities ( valid for @xmath40 $ ] fixed ) : @xmath237 in this section we regard elements of @xmath32}$ ] as boolean differential operators via the map @xmath238 } \rightarrow \mathrm{bdo}_n$ ] given by @xmath239}a(c , d)x^cs^d.\ ] ] the @xmath224-product on @xmath32}$ ] is the pullback via the map @xmath240 of the composition product on @xmath20 , i.e. for @xmath80}$ ] we have that @xmath241 . explicitly , see @xcite , we have that latexmath:[\[a \ast b ( c , d ) \ = \
\sum_{e \subseteq c , g , h } equivalently , a pair @xmath191\times \mathrm{p}[n]$ ] belongs to @xmath243}$ ] if and only if there is an odd number of sets @xmath244 $ ] such that @xmath245 note that we can go back and forward from the @xmath65-basis to the @xmath246-basis for differential operators as follows : @xmath197}a(c , d)x^cs^d\ = \ \sum_{c , d \in \mathrm{p}[n]}\widehat{a}(c , d)m^cs^d,\ ] ] indeed using ( [ e10 ] ) we have that @xmath247 consider the map @xmath248 } \longrightarrow \mathrm{m}_{2^n\times 2^n}(\mathbb{z}_2 ) $ ] sending a directed graph @xmath77 } $ ] to the matrix of the operator @xmath249}a(c , d)x^cs^d \ = \
\sum_{c \subseteq e , d } a(c , d)m^es^d\ ] ] in the basis @xmath89\},$ ] thus we have that @xmath250 _ the map @xmath251 } , \ast ) \longrightarrow ( \mathrm{m}_{2^n\times 2^n}(\mathbb{z}_2 ) , \ .)$ ] is an algebra isomorphism given for @xmath77}$ ] by @xmath252 the inverse map @xmath253 } , \ast ) $ ] sends @xmath95 to the directed graph @xmath254 with characteristic function given by @xmath255 _
we have that @xmath256}a(c , d)x^cs^d \right]_{a , b}\ = \ \left[\sum_{c \subseteq e , d } a(c , d)m^es^d\right]_{a , b}\ ] ] @xmath257_{a , b}\ = \
\sum_{c \subseteq e , d } a(c , d)\delta(e , a)\delta(e , b+d)\ = \
\sum_{c\subseteq a } a(c , a+b).\ ] ] the map @xmath258 is inverse to @xmath259 since for @xmath77}$ ] we have that : @xmath260 @xmath261 we show that @xmath259 is an algebra morphism . by definition @xmath262,$ ] thus : @xmath263 \ = \ [ l_3(a)l_3(b ) ] \ = \ [ l_3(a)[l_3(b ) ] \ = \
m_3(a ) m_3(b).\ ] ] _ @xmath264 and @xmath265 , where @xmath266 .
in this section we regard elements of @xmath32}$ ] as boolean differential operators via the bijective map @xmath268 } \longrightarrow \mathrm{bdo}_n$ ] given by @xmath269}a(c , d)x^c\partial^d.\ ] ] the @xmath267-product on @xmath32}$ ] is the pullback via @xmath270 of the composition product on @xmath20 , thus for @xmath80}$ ] we have that @xmath271 .
explicitly @xcite we have that : @xmath272 equivalently , a pair @xmath191\times \mathrm{p}[n]$ ] belongs to @xmath273}$ ] if and only if there is an odd number of sets @xmath274 $ ] such that @xmath275 note that we can go back and forward from the @xmath65-basis to the @xmath276-basis for boolean differential operators , using equations ( [ e6 ] ) and ( [ e10 ] ) , as follows : @xmath197}a(c , d)x^c\partial^d\ = \ \sum_{c , d \in \mathrm{p}[n]}\widehat{a}(c , d)m^c s^d,\ ] ] where @xmath277 consider the map @xmath278 } \longrightarrow \mathrm{m}_{2^n\times 2^n}(\mathbb{z}_2 ) $ ] sending a directed graph @xmath77 } $ ] to the matrix of the operator @xmath279}a(c , d)x^c\partial^d \ = \
\sum_{c \subseteq e , \ f
\subseteq d } a(c , d)m^es^f\ ] ] in the basis @xmath89\}.$ ] _ the map @xmath278 } \longrightarrow \mathrm{m}_{2^n\times 2^n}(\mathbb{z}_2)$ ] defines an algebra isomorphism between @xmath86 } , \bullet)$ ] and @xmath28 .
explicitly , for @xmath77}$ ] we have that : @xmath280 the inverse map @xmath281}$ ] sends @xmath95 to the graph @xmath282}$ ] with characteristic function given by : @xmath283 _ since @xmath284 and @xmath285
we have that @xmath286_{a , b } \ = \
\left[\sum_{c \subseteq e , \ f \subseteq d } a(c , d)m^es^f\right]_{a , b } \ = \ ] ] @xmath287_{a , b}\ = \
\sum_{c \subseteq e , \ f \subseteq d } a(c , d)\delta(e , a)\delta(e , f+b)=\sum_{c \subseteq a , \
\subseteq d}a(c , d),\ ] ] as @xmath288 , and @xmath289 implies that @xmath290 + the map @xmath291 is inverse of @xmath292 since applying @xmath293 we have that : @xmath294 @xmath295 the map @xmath296 $ ] is an algebra morphism since : @xmath297 \ = \ [ l_4(a ) l_4(b ) ) ] \
= \ [ l_4(a ) ] [ l_4(b ) ) ] \ = \
m_4(a ) m_4(b).\ ] ] _ @xmath298 and @xmath299 where @xmath300 . _
in this work we considered four combinatorial interpretations using directed graphs for the composition ( together with the symmetric difference ) of boolean differential operators , and provided a matrix representation for each of these interpretations .
therefore our work provides set theoretical interpretations for the algebra of boolean differential operators .
it would be nice to find logical interpretations as well , i.e. some sort of non - commutative logic where boolean differential operators play the role played by boolean functions in classical propositional logic .
partial results along this line are developed in @xcite , where a couple of explicit presentations by generators and relations of the algebra of boolean differential operators are provided .
+ we thank an anonymous referee for many valuable suggestions .
[email protected] + departamento de matemticas + universidad nacional de colombia , manizales , colombia + [email protected] + instituto de matemticas y sus aplicaciones + universidad sergio arboleda , bogot , colombia +
|
we consider four combinatorial interpretations for the algebra of boolean differential operators and construct , for each interpretation , a matrix representation for the algebra of boolean differential operators .
+ keywords : boolean algebras , differential operators , directed graphs .
+ msc : 05e15 , 05c76 , 03g05 .
|
the hubbard hamiltonian @xcite is certainly the most important model in the field of strongly correlated electrons . the spectral function for the addition or removal of a single electron near half filling serves as a paradigm for the excitation spectrum of highly correlated electrons in the vicinity of a mott transition .
it was a great success of the dynamical mean field theory @xcite ( dmft ) to connect the high- and low - energy parts of the spectral function in a non - perturbative solution for arbitrary interaction strength .
especially , it was confirmed that the coherent low energy excitations in the metallic phase follow the same dynamics as the kondo resonance in the anderson impurity - model , the other generic hamiltonian for correlated electrons that is much better understood .
the analog to the kondo resonance in the impurity model is a quasiparticle ( qp ) band in the lattice model .
both straddle the chemical potential @xmath0 , i.e. the lowest excitations are gapless . in the strong coupling limit , the spectral weight @xmath1 of the qp band is small , relative to two sidebands , further removed from @xmath0 .
these sidebands are called the hubbard bands because they are roughly reminiscent of the hubbard - i solution .
@xcite in the doping controlled regime , @xcite one sideband always overlaps with the qps , the other represents true high energy excitations across the correlation gap .
hubbard - i is an approximation close to the atomic limit , but nevertheless taking exact spectral moments of the itinerant propagator up to the second order into account .
it can be considered the ancestor of the projection technique @xcite which systematically incorporates spectral moments of higher order .
these approximations have severe deficiencies in the low energy sector , unless the moment series can be effectively summed up .
in particular , generating a third pole in the spectral function from the high energy side alone leads to uncontrolled results . when the density of states ( dos ) , as obtained within dmft , is resolved with respect to the wavenumber @xmath2 , more details about the coexistence of this kondo resonance with atomic like features in a lattice system are revealed .
the lowest excitations are true fermi liquid ( fl ) qp s : ( i ) the finite dos at @xmath3 corresponds to longlived excitations .
( ii ) these are located in @xmath2-space on a fermi surface ( fs ) that satisfies luttinger s theorem .
@xcite ( iii ) as function of the distance @xmath4 from the fs , the excitation energy has a linear , strongly reduced dispersion .
( iv ) the damping is quadratic in @xmath4 but strongly enhanced , meaning that the linear and quadratic term are of the same order at a very small energy scale , the coherence energy @xmath5 .
the two atomic like excitations turn out to be strongly damped , even when @xmath2 is on the fs .
their peaks disperse with @xmath2 but spectral tails spread over the entire bandwidth .
the dmft thus unites atomic and itinerant features in a non perturbative approximation .
it is exact only in dimension @xmath6 . as a generic scenario ,
it is expected to hold down to @xmath7 , albeit with the caveat that the dmft suppresses additional structure due to bosonic couplings .
earlier approximations at finite @xmath8 already yielded qps@xcite and established a connection to the kondo effect .
@xcite the high prestige of the dmft is due to its ability to produce a selfconsistent , numerically manageable approximation to the spectral function for all energies , in particular to the parameter @xmath1 that governs the low energy sector .
this has opened a path to realistic modeling of correlated materials beyond the hubbard model in such methods as lda+dmft .
@xcite it is nevertheless desirable for several reasons to pursue alternative methods in parallel .
firstly , a @xmath2-independent selfenergy , which is the proper result at @xmath9 , does not allow to explain phenomena that depend on the different symmetry directions in the brillouin zone , especially in high temperature superconductors and other low dimensional systems .
cluster extensions of the dmft go in the direction of lifting this restriction , @xcite but these generalizations are numerically even more demanding than the dmft itself .
the precise solution of a manybody kondo problem is required at each iteration step towards selfconsistency . in practice
, when designing the `` impurity solver '' , a trade - off exists between improving the low energy , low temperature solution and exactly satisfying global sumrules .
such numerical problems are presently a bottleneck for extensions of the dmft to larger clusters or to lda+dmft with charge transfer into ligand bands .
a variational aspect was recently found , which may allow to circumvent some of the numerical problems .
@xcite in this paper , we present a continued fraction method ( cfm ) and implement it for the doping controlled metallic regime near the mott transition .
similar to other recent attempts , @xcite we start from the projection technique , applied to the @xmath2-resolved single particle green function ( gf ) .
the notations are introduced in section [ model ] . in section [ highe ] ,
the connections between the moment- and continued fraction- ( cf ) expansion as well as the pad approximant ( pa ) are established . in the pa , qualitatively important features of the macroscopic system , such as damping , are missing .
they can only be captured by resummation of the cf to infinite order .
the concept of a terminator function ( tf ) , by which an approximate resummation is achieved , is common to many methods based on the cf . as a general scheme ,
we define our cfm by allowing only such tf s that preserve the structure of a truncated cf , however with complex coefficients .
useful recursion relations , that are properties of pas , can thus be carried over and the solution for the gf can be constrained by high as well as low energy sumrules .
previous solutions obtained with this ansatz @xcite were partly phenomenological , because the strong coupling renormalization @xmath1 needed to be inferred from a separate gutzwiller approximation , or else was left open for fitting to experiments .
@xcite a closed solution is now achieved by minimizing the total energy in the presence of sumrules , for which the necessary selfconsistency loops are introduced .
the selfconsistent @xmath1 falls below the gutzwiller value and has a doping dependence close to that for the exact kondo scale .
@xcite this is now a true microscopic approximation , depending only on the parameters in the hamiltonian . in sections
[ lowe ] , [ numer ] and [ impgap ] we have investigated the tf s that correspond to adding one or two stages with complex coefficients to the cf . we show how the fs singularity , the enclosed luttinger volume in @xmath2-space and the fl damping can be modeled rigorously .
we assess the quality of our approximations by comparing the dos with the dmft result for two variants of the impurity solver , namely the numerical renormalization group ( nrg ) @xcite and the non - crossing approximation ( nca ) .
@xcite the success of the cfm with respect to the hubbard model allows to draw some optimistic conclusions about possible generalizations towards more realistic models , describing correlation effects in a multiband electronic environment .
this will be outlined as part of the conclusions .
the hubbard model for a grand canonical ensemble of electrons on a lattice of @xmath10 sites ( @xmath11 ) is written in the usual notations @xmath12 the kinetic energy consists of itinerant bloch states with energies @xmath13 and wavenumber @xmath2 , running through one brillouin zone .
the bandwidth is @xmath14 and , when not specified otherwise , @xmath15 is used as unit of both energy and frequency @xmath16 .
we formulate the method for an arbitrary density of bloch states .
numerical examples later on will be calculated for a semi - elliptic density .
the chemical potential @xmath0 is selfconsistently determined to satisfy the condition @xmath17 where the filling factor @xmath18 @xmath19 is part of the input .
the chemical potential for the @xmath20 limit is designated as @xmath21 . for @xmath22 ,
the right hand side is calculated with our method .
the overline and the bracket signify brillouin zone average and ensemble average , respectively .
the filling factor per spin direction in the spin degenerate phase is @xmath23 .
we approximate the advanced single particle gf @xmath24\rangle , \label{gfeq}\ ] ] from which the momentum distribution @xmath25 and other observables are calculated .
the spin index is dropped in the unpolarized phase .
the time dependent fermionic destruction operator @xmath26 is in the heisenberg representation with @xmath27 and the square bracket is the anticommutator .
the complex frequency @xmath28 has @xmath0 as origin .
asymptotically , for large @xmath29 , we have @xmath30 .
the coefficient 1 reflects the moment @xmath31 or spectral norm , as required by the pauli principle . for this relation between the leading coefficient and the norm to remain valid in an approximation ,
it is necessary and sufficient to conserve the herglotz property . in the case of the advanced gf , it means that the relation @xmath32 must be obeyed throughout the entire halfplane @xmath33 .
the physical meaning of the herglotz property is causality and it automatically entails the existence of kramers - kronig relations between the real and imaginary parts . a great advantage of our method is the possibility to make straightforward evaluations along the real axis . since this limit has to be approached from within the domain of analyticity , the notation @xmath34 with real energy @xmath35 is introduced .
the @xmath2-resolved spectral function is @xmath36 at @xmath20 it has a single sharp peak at the excitation energy @xmath37 which also serves to measure distance in @xmath2-space , at least in the vivinity of the fs .
the cf expansion , on which our method is based in a crucial way , has already a long tradition in solid state physics , in the one electron problem with disorder @xcite as well as in the many electron problem .
@xcite the cf is generated by various procedures like tridiagonalization , recursion- or lanczos - methods .
the hubbard - i gf is the simplest example of a cf that has been truncated at low order .
the exact gf for the hubbard model on finite clusters is a cf which naturally ends at very high order .
the cf for the infinite system does not end .
properties of the thermodynamic limit , such as damping due to electron - electron scattering , emerge only after resummation of the cf .
approximate resummation is achieved by the tf , an analytic function which also has the herglotz property . a well chosen tf is thus expected to bring two improvements to the approximation for the gf in dimensions @xmath38 : ( i ) from a set of discrete , more or less intense and more or less densely spaced dirac peaks emerges the final shape of the continuous spectral density ( see ref . for tight - binding like models and ref . for strongly correlated electrons ) .
( ii ) a fermi surface ( fs ) discontinuity emerges in the momentum distribution @xmath39 at temperatures below the strong coupling energy scale @xmath5 .
the fl discontinuity and the correct fs volume will be incorporated in our ansatz .
this means , we take the luttinger theorem for granted and use it as a principle , even for strong coupling where there is no rigorous proof .
the energy @xmath5 then comes out as part of the selfconsistent solution .
the first moment or center of gravity of @xmath40 is @xmath41 it disperses like the unrenormalized bloch energy @xmath42 . in models with a more general interaction ,
a @xmath43dependent hartree - fock shift is also present which , for onsite repulsion , reduces to a constant hartree shift @xmath44 .
the selfconsistent @xmath0 is the only unknown .
the high energy expansion about the center of gravity is @xmath45 its coefficients @xmath46 are called the central moments ( @xmath47 , by definition ) .
they can be related to correlation functions which occur in the short time , or liouville expansion of the operator @xmath26 and are evaluated in the limit @xmath48 .
it is remarkable that the variance @xmath49 of @xmath50 , defined by the second central moment @xmath51 is @xmath43independent in any dimension @xmath8 , not only @xmath9 .
all the terms in the high energy expansion are sensitive to the low energy sector , be it only via the selfconsistent @xmath0 .
we now turn to the cf expansion which is closely related to the moment expansion .
formally , it is initiated by using @xmath52 and @xmath49 to write the gf as @xmath53 in this identity , @xmath54 is again a herglotz function with asymptotics @xmath55 .
iterations , pushing the cf further down step by step , require knowledge of the center of gravity @xmath56 and the variance @xmath57 of @xmath58 , to write @xmath59 the two new expansion coefficients depend only on the central moments @xmath60 up to the order @xmath61 and @xmath62 of their respective index . by truncating the cf , i.e. by setting @xmath63 , an approximation to the gf
is obtained that has @xmath64 zeros and @xmath65 poles on the real axis .
this is defined @xcite as the pa @xmath66 .
it represents the optimal use one can make of a set of known spectral moments up to @xmath67 .
the present task , constructing the gf , is rendered essentially more difficult , because the moments themselves are not yet known . a solution based on a pa
can be made selfconsistent but the moments turn out to be numerically quite inexact .
this fact is often ignored when it is claimed that a certain high energy approximation obeys a set of `` exact '' sumrules .
we now discuss some well known results concerning approximations at the second stage of the cf . as a still exact representation of the gf we have @xmath68 the relations between the first few terms are : @xmath69 besides the variance , quantities used to further characterize the internal shape of a spectrum are the skewness @xmath70 and the kurtosis @xmath71 . in terms of these , we have @xmath72 and @xmath73 . from the third moment one
finds the coefficient @xmath74 this cf coefficient is the first quantity in the expansion with a non trivial @xmath2-dependence .
the full correlation function appearing in the third moment was first derived in ref . and determined selfconsistently for a short linear chain in ref .. the shift in the spectral skewness , caused by @xmath75 , regulates the dynamical weight transfer between the hubbard peaks at finite @xmath76 .
@xcite one can decompose @xmath77 in such a way that the term @xmath78 vanishes in high dimensions . for making contact with the dmft
we will presently neglect it and adopt the expression @xcite @xmath79 by which it is linked selfconsistently to the expectation value of the kinetic energy @xmath80 .
concerning the behavior of the fourth moment , not even the correlation functions involved in its selfconsistency loop have as yet been evaluated .
again , the actual numerical value of @xmath81 is also expected to be sensitive to the low energy sector and , in low dimensional systems , @xmath2-dependent .
given this situation , approximations on the level of eq.([eq3a ] ) are at present inevitable .
straightforward truncation , @xmath82 , leads to the pa @xmath83 .
this solution with two dirac peaks goes beyond hubbard - i , because the dynamical weight transfer is taken into account .
the first example of an approximate resummation of the cf is the alloy analogy , developed in the paper called hubbard - iii .
@xcite following hubbard s notation , we approximate @xmath84 by a @xmath2-independent tf @xmath85 , which has to be a herglotz function .
the alloy analogy satisfies at least the task ( i ) of a tf , namely to generate finite damping . far away from @xmath0 , where the excitations are incoherent
, it actually represents a physically correct picture .
we therefore keep the result @xmath86 for large @xmath29 from hubbard - iii .
the physical reason , why the damping is of the order of the bare bandwidth @xmath15 is that the mean free path is as short as one lattice constant . in practice
, we incorporate the high energy damping in an effective @xmath87 @xmath88 and henceforth deal with a terminator that decays as @xmath89 .
this way , we conserve the sumrules , encapsuled in the central moments @xmath90 to @xmath91 .
since hubbard - iii is unrealistic at low energies , we do not pursue it any further .
nevertheless , it should be noted that hubbard - iii generates a branchcut in @xmath85 , causing the imaginary part to drop back to zero and a correlation gap with sharp edges to appear , at least in the zero temperature limit .
this property of hubbard - iii is also not expected to survive in improved approximations for the metallic phase .
we will address the consequences that the absence of a branchcut has for the shape of the dos , both in the cfm and in the dmft . to sum up
, our approximation to the gf is formally similar to hubbard - iii , @xmath92 but with a tf , @xmath93 , that retains the strong damping of the alloy analogy only at high energy .
two successive implementations of the tf with appropriate fl properties at low energy , are the subject of the following sections .
a fs discontinuity is strictly realized only in the zero temperature limit and in a system with no residual disorder .
since @xmath94 solutions are hardest to obtain with dmft and , on the contrary , easily implemented with our method , we concentrate in the following on this limit . we write the standard microscopic definition of a selfenergy as a complex correction to the bare excitation @xmath95 : @xmath96 and compare with the inverse of eq .
( [ eq3b ] ) .
the high energy limit @xmath97 is the difference between two dispersive quantities . in the present case , eqs .
( [ eta ] ) and ( [ eq1c ] ) have identical dispersion and @xmath98 is , in fact , constant . within the other approximations , discussed in the preceding section , we then obtain the @xmath2-independent selfenergy @xmath99 in this case , as in the dmft , the fs has the exact shape of the uncorrelated system .
it is given by all @xmath43points where @xmath100 in eq .
( [ eta ] ) .
the qp peak of weight @xmath1 at the fermi level and the step of amplitude @xmath1 in the momentum distribution are fixed by the conditions @xmath101 and @xmath102 at finite @xmath103 or in the presence of a residual diffusive mean free path , @xmath104 remains finite .
guided by the insight that the strong coupling peak is distinct from the hubbard peaks , we can formulate a minimal ansatz for the tf @xcite as @xmath105 adding a new stage to the cf is the proper way to `` add '' a pole to the gf .
when this tf is inserted in eq .
( [ eq3b ] ) , it generates a gf with three complex zeros in the denominator and two zeros in the nominator , i.e. the same structure as the pa @xmath106 .
the connection of the parameters @xmath107 and @xmath108 to central moments @xmath109 and @xmath110 is lost .
in fact , the very existence of moments beyond @xmath91 has been sacrificed by admitting @xmath111 , @xmath112 , and @xmath113 as complex quantities .
they now have to be determined from conditions ( [ flcond1 ] ) and ( [ flcond2 ] ) . for the herglotz property one
finds @xmath114 as a necessary and sufficient condition .
this causes all three poles to lie in the upper half - plane .
further , it guarantees a normalized , positive semidefinite @xmath50 , which also implies quite intricate relations between the complex residues .
now , the important point is the following : this simple ansatz is so heavily constrained by sumrules that it offers a selfconsistent solution of the problem , without any free parameters .
it remains to substantiate this claim and then to discuss the quality of the solution . after inserting eq .
( [ tf1 ] ) in eq .
( [ eqse ] ) , the conditions ( [ flcond1 ] ) and ( [ flcond2 ] ) can be brought into a system of two linear equations for the unknowns @xmath107 and @xmath108 .
the determinant of this system is @xmath115 and the herglotz property requires @xmath116 .
this is a constraint on the qp weight @xmath1 : in stead of @xmath117 ( pauli principle ) we have @xmath118 .
closer inspection reveals that it means the qp can not take more spectral weight than the peak in the pa @xmath83 that is nearest to @xmath0 . since around
half filling this weight stays above @xmath119 , it is indeed only a weak constraint .
the solution @xmath120 and @xmath121 is expressed in terms of the complex quantity @xmath122 it fulfills the herglotz condition ( [ herglotz1 ] ) with the equality sign .
this is a consequence of our strong @xmath94 constraint @xmath123 , concerning both the real and imaginary part .
the selfenergy is now parametrized , up to @xmath1 , which remains free within a restrained interval and will be determined by minimizing the total energy .
we note , before closing this section , that ref .
allows to define one - pole tf s for the more general case of a truncated gf that is expressed as a higher order pa .
the general algorithm is given , by which the eqs .
( [ flcond1 ] ) and ( [ flcond2 ] ) can be fulfilled .
the uncorrelated chemical potential as function of the filling , @xmath124 , depends only on the kinetic energy part and is determined once for all .
the dos per lattice site in the @xmath20 limit @xmath125 is obtained from the onsite gf @xmath126 a factor two comes from summing over spin directions .
the dos of the correlated system @xmath127 is obtained from @xmath128 , the on - site gf in real space , which is independent of the site index . for a @xmath2-independent selfenergy such as eq .
( [ eqse ] ) , the on - site gf s @xmath129 and @xmath130 are related to each other by @xmath131 the @xmath2-summations can then be carried out by using the analytic function that represents the solution for @xmath130 in the limit @xmath11 .
we now turn to the discussion of the selfconsistency loops .
the condition for @xmath0 is implemented at @xmath94 by the integral @xmath132 according to eq .
( [ eqb3 ] ) , the term @xmath133 from the third moment requires the selfconsistent determination of the kinetic energy , @xmath134 .
one finds @xmath135 finally , the total energy is @xmath136 the integrals in ( [ sceq1 ] ) - ( [ eqetot ] ) are carried out numerically .
for the calculations in this paper we took the on - site gf @xmath137 in the context of @xmath6 , it is the gf for a bethe lattice .
a halfwidth @xmath138 is now used as energy unit . for the herglotz property
, it is important to choose the square root with a positive real part .
the model dos belonging to this gf , @xmath139 is the semi - elliptic function which was also used by hubbard .
while searching for the selfconsistent @xmath0 and @xmath140 at a given input @xmath18 and @xmath76 , the renormalization @xmath1 is still kept as a parameter , only limited by the condition @xmath141 . with these constrained solutions for the gf , we calculate the total energy @xmath142 . as shown in the example of fig .
[ fig1 ] , @xmath142 has a well defined minimum as a function of @xmath1 . taking the value which minimizes @xmath142 fixes the last parameter @xmath1 and defines our solution for the gf . the dos obtained for the same input as in fig .
[ fig1 ] is shown in fig .
[ fig2 ] , together with the @xmath20 limit .
the qp band has the same intensity at the fermi level as the uncorrelated band , @xmath143 .
this invariance signals the unitary limit for the kondo resonance in the limit @xmath94 .
thus , the reduction of the qp weight does not show up in @xmath144 but in the bandwidth , which is scaled down by @xmath1 . in a lattice system , this one - to - one relationship between qp weight and bandwidth only holds when the selfenergy is local ( @xmath2-independent ) . on the @xmath2-resolved level , near the fs
, the qp - pole in the complex plane has a parabolic trajectory parametrized by @xmath95 , eq.([eta ] ) : @xmath145 with a scattering rate @xmath146 the halfwidth for coherent states within the qp band is @xmath147 this formula is well behaved also in the weak coupling limit , in fact for all possible metallic , unpolarized regimes of the hubbard model . in the strongly correlated regime , the energy scale @xmath5 is smaller than @xmath148 , so that the excitations in the wings of the qp band cease to be coherent .
since we have modeled the ballistic limit ( residual diffusive scattering rate @xmath149 ) the qp resonance in @xmath50 is a dirac peak for @xmath150 and , for @xmath151 , it has the so called breit - wigner lineshape ( see hedin and lundquist @xcite for a generic plot ) .
this shape is due to an interference between the qp residue and the other residues .
the lineshape becomes approximately lorentzian whenever a @xmath152 is present .
returning to the dos , we note that the global shape of the valence spectrum for a hole doped mott insulator , i.e. qp band and lower hubbard band , is well rendered by our present approximation .
the sumrules up to @xmath91 are exactly satisfied and their interplay regulates the overall skewness and the relative weight of all three features .
the one - pole tf has the drawback of being unable to reproduce a sharp gap formation .
the high level of intensity between the qp band and the upper hubbard band shows that the dynamical spectral transfer @xcite is not realized completely , at least for @xmath153 .
the intensity at the minimum decays like @xmath154 , so that this spurious effect disappears for larger @xmath76 .
we shall discuss the presence of residual intensity in the gap region in more detail when we compare with our second ansatz and with the dmft .
two remarks to conclude this section : ( i ) the limit @xmath155 describes spin- and charge - excitations in the subspace of singly occupied sites .
it is equivalent to the @xmath156-model with @xmath157 .
the one - pole tf thus allows to project out a quantitatively valid gf for the valence sector near this limit , up to terms of order @xmath154 .
( ii ) ratios @xmath158 are relevant for the doping controlled mott transition in real materials close to criticality . our main motivation to pursue the cfm was to investigate whether by simply adding a second complex stage to the tf we could handle this regime in a semiquantitative way .
the derivation of the two - pole tf and its application to @xmath159 are presented in the next section .
to generalize our ansatz , we introduce algebraic expressions for @xmath160 , such that eq .
( [ eq3b ] ) can be cast into the form of a truncated cf with complex coefficients .
this defines the cfm , provided the herglotz condition is satisfied .
the @xmath2-resolved gf has then the structure of a generalized higher order pa . by terminating the pa @xmath83
, we still retain the important sumrules that govern the dynamical weight transfer .
spectral moments beyond @xmath91 can not be recovered , but this may not be a great sacrifice , given the difficulties known from the projection method to obtain correct values for higher moments . what can be gained by using complex coefficients is the possibility to model constructive and destructive interference phenomena in the gf at intermediate energies .
a single feature in the spectral function can be built up by the contributions of several poles , resulting in uncommon lineshapes .
an ansatz frequently employed in the phenomenological interpretation of spectra is the superposition of complex poles with real residues ( superposition of lorentzians in the spectrum ) .
although this allows several peaks to coalesce , it still eliminates interference .
one striking example of a fano like interference within the coherence range of halfwidth @xmath5 is the breit - wigner lineshape of the qp . @xcite as discussed in the preceding section , the complex residues of the two valence poles in the gf ( arising from the one - pole tf in the large @xmath76 limit ) are enough to obtain this lineshape .
likewise , the dynamical weight transfer and the formation of the correlation gap can be interpreted as a destructive interference in the intermediate energy range between the hubbard bands .
when the interference is complete the function @xmath161 in eq.([eq3a ] ) should acquire a branchcut and a gap interval with zero dos and sharp edges should result .
this may be possible only on the insulating side of the mott transition and strictly at @xmath94 .
when the system is metallic and the chemical potential falls in a region of high dos , it is satisfactory to model the correlation gap by a deep minimum .
we demonstrate here that this situation is captured by a two - pole tf of the form @xmath162 the new degrees of freedom are given by @xmath163 and @xmath108 .
these will be found due to some qualitative arguments , restricting the ansatz from the start .
then , @xmath164 and @xmath165 can again be eliminated by the fl conditions of eqs .
( [ flcond1 ] ) and ( [ flcond2 ] ) , using the next iteration of the algorithm in ref . .
the gf now has four poles and the herglotz condition becomes a crucially important issue . to formulate it , for arbitrary complex values of @xmath166 to @xmath165 ,
seems at first sight rather difficult .
the gf on the fs ( eq . ( [ eq9 ] ) with @xmath100 ) has additive coherent and incoherent contributions , @xmath167 the decomposition is possible , because one pole lies exactly on the real axis .
this will enable us to manage the herglotz condition for @xmath168 more easily : from eq .
( [ eqtf2 ] ) we obtain a background function @xmath169 with three poles that can be written @xmath170 the new coefficients are designated by capital greek letters .
systematically , they depend on @xmath1 and , at order @xmath171 , on all coefficients in eqs .
( [ eq3b ] ) and ( [ eqtf2 ] ) with index @xmath172 .
explicitly , the first three are @xmath173 in terms of the previously defined quantities , eqs.([variance ] ) , ( [ eqp1 ] ) , ( [ eqdet2 ] ) , and ( [ eq17 ] ) .
the high energy damping in the background function is @xmath174 between the coherence energy @xmath5 in eq .
( [ gamma ] ) and the background function at the fermi edge there is the relation @xmath175 for @xmath176 , we recover the one - pole tf and eq .
( [ eq * ] ) for @xmath5 .
since @xmath177 and @xmath178 are real , there are now only three complex quantities and the herglotz condition can be specified exhaustively , analogous to eq .
( [ herglotz1 ] ) : @xmath179 the foregoing analysis suggests that @xmath180 and @xmath181 are more useful than @xmath112 and @xmath113 as control parameters . to obtain the selfenergy , one
can then use eqs .
( [ eqgb1 ] ) and ( [ eqgb2 ] ) . for further discussion
we parametrize @xmath182 a minimal requirement for properly defining the dynamical weight transfer is vanishing @xmath183 in one other point @xmath184 on the real axis , apart from @xmath185 .
it happens if ( and only if ) the equality sign applies in ( [ herglotz2 ] ) .
this leads to the condition @xmath186 and to @xmath187 for the position .
the herglotz property guarantees that it is in fact a minimum .
the influence of this interference on the shape of the valence spectrum is weakest for @xmath188 . for simplicity
, we also need to set @xmath189 , where @xmath190 is already defined in eq .
( [ eqgb4 ] ) .
the last parameter @xmath191 is then fixed by the point with lowest intensity inside the correlation gap . before continuing with this ansatz ,
it is important to realize that it can not apply exactly at half filling .
there , the metallic phase is obtained by driving @xmath192 below the critical ratio ( so called bandwidth controlled transition ) .
@xcite the particle - hole symmetric dos has a quite different morphology than what is shown in fig .
[ fig2 ] : the qp s are in the center and the correlation gap is split in two symmetric gaps of order @xmath193 .
@xcite in our approach , it can be envisaged to use one additional complex stage to model two symmetric destructive interferences . in the doping controlled regime , there is only one large correlation gap and we have a good qualitative argument for @xmath194 : the strong skewness ( large @xmath195 ) causes the qp band and the minimum position @xmath194 to always be on opposite sides of the center of gravity .
this is well satisfied by setting @xmath196 the remaining free parameter @xmath1 is determined again by minimizing the total energy .
the numerical procedure is as described before . in fig .
[ fig3 ] , results with the one- and two - pole tf s ( cfm1 and cfm2 ) are compared to the gutzwiller approximation ( ga ) at constant @xmath76 , as function of the filling .
the upper curve is the well known lower bound for the ga , @xmath197 , obtained by excluding double occupancy . by projecting out the background ,
the ga is known to systematically overestimate the coherent weight .
the behavior that results from the cfm , i.e. : lowering of @xmath1 and upward curvature at the approach of zero doping ( @xmath198 ) , is close to that of the exact kondo scale in the bethe ansatz solution for the anderson impurity
with the two - pole tf , realistic results for the dos in the doping controlled regime can be obtained , even close to the critical @xmath76 . to illustrate this
, we compare our cfm with the dmft for two different impurity solvers .
the impurity solvers perform the crucial step in mapping the hubbard lattice model onto an anderson impurity model . the effective medium surrounding a given site is determined self - consistently , still a formidable manybody problem .
the nrg , @xcite used to solve it at the lowest temperatures and energies , requires a heavy amount of computer time .
the nca @xcite is an alternative , more analytic method , less reliable for @xmath199 , but obeying high energy sumrules well .
therefore , nrg and nca are expected to be complementary . a comparison for the same parameters as before , i.e. @xmath200 and @xmath201 , is shown in fig . [ fig4 ] .
the nrg data are taken from ref .
, nca is our own unpublished calculation , cfm1 is again the dos from fig .
[ fig2 ] and cfm2 the result with eq .
( [ eqtf2 ] ) .
all four solutions obey the @xmath143 condition .
this confirms that temperatures in the dmft solutions are sufficiently low to warrant a comparison with our @xmath94 results . as manifest in the width of the qp band ,
the selfconsistent @xmath1 obtained for cfm2 coincides with both versions of dmft . since nrg is expected to determine essentially the exact low energy scale , this is a good point for both the nca and the cfm2 results .
[ ht ] the solutions start to differ somewhat in the gap region .
neither dmft version shows a gap with sharp edges that would correspond to a branchcut in the selfenergy .
a real benchmark for low @xmath103 impurity solvers in the doping controlled regime does not yet exist . from the nca
, we can confirm that some very low residual density inside the gap seems to be the generic situation . in the ansatz for cfm2 , the existence of a point with zero dos is postulated .
determining its position according to eq .
( [ eqzero ] ) involves the selfconsistency conditions for @xmath0 and @xmath75 .
the quantitative agreement with the dmft in the qp band and good overall agreement in the entire valence sector is due to this built in interference . in comparing cfm1 and cfm2 ,
one notices a feedback of the improved gap region on the qp band : the sumrules up to @xmath91 are satisfied for both approximations , but the dynamical weight transfer is more complete within cfm2 . removing the spurious intensity inside the gap slightly raises the qp weight ( compare fig .
[ fig3 ] ) , bringing it in agreement with the nrg .
the rather large variation among the different solutions in the region of the upper hubbard band is remarkable and still deserves more detailed investigations . at higher temperatures , @xmath202 , where quantum monte carlo is available as benchmark ,
the nca was found to be satisfactory .
@xcite in the present comparison , the nca comes closer to obeying the sumrules than the nrg .
as far as numerical effort is concerned , the nrg is the most demanding , followed by the nca .
the cfm2 stands up quite honorably in this comparison , especially when considering that the sumrules are rigorously incorporated , no `` technical '' broadening needs to be introduced and the required computer time to achieve selfconsistency is in fact negligible .
we present the cfm as a new method to calculate the selfenergy , as well as various @xmath43resolved and ( partially or fully ) @xmath43integrated spectral functions of the hubbard model in the correlated metal phase .
we expand the single particle green function as a continued fraction , as far as moment sumrules are exploitable , and then use a properly chosen terminator function . in this paper , moment sumrules up to @xmath91 are implemented and the `` terminator '' is a @xmath43independent complex function with one or two poles that obeys the correct fermi liquid properties at low energies . in this local approximation to the selfenergy , we compare our results for the density of states with the dmft .
our method has a precision comparable to state - of - the - art impurity solvers nrg and nca .
it covers all energy scales reliably , whereas the low t impurity solvers each have their strengths and weaknesses . with the second stage in the terminating function
we are able to improve the dynamical weight transfer between the upper and lower hubbard peak and thereby obtain very good agreement with dmft for the qp weight @xmath1 or low energy scale .
this is significative , because nrg - dmft yields the exact result for this quantity .
unlike the time consuming dmft calculations , the cfm uses simple , algebraic functions , for which selfconsistency conditions are rapidly found .
the cfm is generalizable in many directions .
however , the possibility to circumvent heavy manybody calculations by such a simplified ansatz seems too attractive to be true .
thus , before advocating possible extensions , we need to analyze the reason for the quantitative success of the cfm in the strong coupling limit .
the hubbard model with a local selfenergy is , admittedly , only a toy model but nevertheless an obligatory testing ground for this important issue .
the algebraic terminator functions were already introduced earlier .
their fermi liquid properties , essential for circumventing the explicit manybody calculations , are determined by using the luttinger sumrule as an input .
their phenomenological possibilities could be demonstrated by leaving @xmath1 as a free parameter.@xcite the cornerstone of the cfm as a microscopic method is now the variation of the total energy to obtain @xmath1 .
given @xmath40 , we calculate the total energy from the exact manybody expression , actually another sumrule first found by galitski .
however , without an explicit wavefunction , we have no rigorous variational principle . in making the gutzwiller approximation , beyond the gutzwiller ansatz for the wavefunction ,
one is also abandoning the rigorous variational principle but one keeps @xmath1 as variational parameter . the answer to the question ,
why our method is variational , is probably that we are using a gf , fully constrained by sumrules , that leaves no other free parameter but @xmath1 . to obtain the kondo effect , we need degeneracy .
our model has spin degeneracy , @xmath203 , for its flavors .
a clue , why including the incoherent background spectrum , instead of projecting it out , improves the outcome for @xmath1 comes from the limit of large @xmath204 .
@xcite the low energy scale ( kondo temperature ) in the anderson impurity model can be obtained exactly , as function of @xmath18 and @xmath205 .
also , coherent spectral weight is of order zero in @xmath206 , the leading background contribution starts at first order . neglecting background , as for instance in the slave boson method at mean - field level , yields @xmath197 , as plotted for @xmath203 in fig.[fig3 ] .
compared with the bethe ansatz , this renormalization is not enough .
now , the influence of the background is strikingly illustrated by solving for the kondo temperature only to the first order in @xmath206.@xcite this causes indeed a substantial decrease , bringing the result close to the exact value .
the correct doping dependence displays the upward curvature , as also seen in our approximations cfm1 and cfm2 .
finally , the improvement from cfm1 to cfm2 shows a delicate interplay between the dynamical weight transfer , related to the double occupancy , and the low energy scale .
if determining @xmath1 by varying the total energy is indeed a valid variational principle , it makes the cfm independent of the limit @xmath6 , thus giving it high flexibility and a large field of applications .
it is straightforward and , for low dimensional systems , potentially very important to incorporate the @xmath43dependence in the moment @xmath91 .
the term @xmath207 in eq .
( [ eq6a ] ) was already identified in the exact diagonalization of a short linear chain , @xcite as causing a coupling of the qp to antiferromagnetic fluctuations .
this can be generalized to fluctuations above other possible groundstates and the selfconsistent determination of @xmath207 thus offers a path to describing the feedback of bosonic fluctuations on the low energy sector .
up to now , the treatment of low energy effects within the projection method was based more on physical intuition , or guesswork for the more critical observer , than on an objective procedure .
the extension of the cfm to higher moments becomes possible due to its close relationship with the numerical lanczos procedure for finite lattices .
all what is missing is a proper termination of the continued fraction with a tf representing the low energy sector and the dissipation .
the general algorithm for calculating the coefficients in the tf is given in ref . .
as an outlook , we enumerate other possibilities that are inherent in the cfm , beyond the results of this paper .
they are listed roughly according to increasing effort that will be required to implement them .
( i ) a more detailed exploitation of spectral functions on the @xmath2-resolved level : e.g. the interpretation of raman , arpes , or tunneling data requires the partial summation of @xmath50 over selected spots in the brillouin zone , weighted by matrix elements .
( ii ) hubbard lattice models with a more realistic kinetic energy part , including van hove singularities .
( iii ) the generalized periodic anderson model ( pam ) : lattice models with hubbard repulsion among transition orbitals but , in addition , hybridization with ligand orbitals .
( iv ) implementation of lda+cfm .
the algebraic simplicity of the cfm allows to calculate the charge transfer effects , present in model hamiltonians of the pam type , on an `` ab - initio '' level .
these effects , important for many real materials , could not yet be handled successfully by lda+dmft .
( v ) not difficult to implement , but leaving the strict framework of the cfm as an algebraic method , is the inclusion of non fermi liquid effects on a phenomenological level .
@xcite in conclusion , we have attempted to demonstrate by means of the hubbard model that the cfm is a powerful method .
numerically simple , due to its algebraic structure , it is still sufficiently rigorous to deal with strongly correlated electrons in mesoscopic and macroscopic samples of condensed matter .
90 j. hubbard , proc .
a * 276 * , 238 ( 1963 ) .
a. georges , g. kotliar , w. krauth , and m.j .
rozenberg , rev .
mod.phys .
* 68 * , 13 ( 1996 ) .
hewson , _ the kondo problem to heavy fermions _ ( cambridge university press , cambridge , 1997 ) .
m. imada , a. fujimori , and y. tokura , rev .
* 70 * , 1039 ( 1998 ) .
h. mori , progr .
phys . * 33 * , 423 ( 1965 ) ; see also : p. fulde , _ electron correlations in molecules and solids _
( springer - verlag , 1995 ) .
luttinger , phys .
rev . * 119 * , 1153 ( 1960 ) .
d. m. edwards and j. a. herz , physica b * 163 * , 527 ( 1990 ) .
gutzwiller , phys .
137 , a1726 ( 1965 ) ; for a review , cf .
d. vollhardt , rev .
phys . * 56 * , 99 ( 1984 ) .
k. held et al .
, psi - k newsl . *
56 * , 65 ( 2003 ) + ( psi-k.dl.ac.uk/newsletters/news_56/highlight_56.pdf ) th .
maier , m. jarrell , th .
pruschke , and m. hettler , rev .
, 1027 ( 2005 ) .
m. potthoff , m. aichhorn , and c. dahnken , phys .
lett . * 91 * , 206402 ( 2003 ) .
m. potthoff , t. herrmann , and w. nolting , eur .
j. b * 4 * , 485 ( 1998 ) .
k. matho , j. electron .
spectr . & rel . phenom .
* 117 - 118 * , 13 ( 2001 ) .
d. villani , e. lange , a. avella , g. kotliar , phys .
lett . , 804 ( 2000 ) .
a. avella , f. mancini , and r. hayn , eur .
j. b * 37 * , 465 ( 2004 ) .
y. kakehashi and p. fulde , phys .
, 156401 ( 2005 ) .
k. matho , molec .
reports * 17 * , 141 ( 1997 ) .
k. byczuk , r. bulla , r. claessen , and d. vollhardt , int .
journ . of mod .
b * 16 * , 3759 ( 2002 ) .
j. w. rasul and a. c. hewson , j. phys .
c * 17 * , 3337 ( 1984 ) .
r. bulla , phys .
* 83 * , 136 ( 1999 ) .
pruschke , d.l .
cox , and m. jarrell , phys .
b * 47 * , 3553 ( 1993 ) .
pruschke and n. grewe , z. phys .
b * 74 * , 439 ( 1989 ) .
r. haydock , v.heine , m.j .
kelly , j. phys .
c : solid state phys .
* 5 * , 2845 ( 1972 ) .
e. dagotto , rev .
phys . * 66 * , 763 ( 1994 ) .
p. turchi , f. ducastelle , and g. treglia , j. phys .
c : solid state phys .
* 15 * , 2891 ( 1982 ) .
kuzian , r. hayn , and j. richter , eur .
j. b * 35 * , 21 ( 2003 ) .
baker jr . , _ essentials of pad approximants _ , academic press , new york ( 1975 ) .
w. nolting and w. borgiel , phys .
b * 39 * , 6962 ( 1989 ) .
b. mehlig , h. eskes , r. hayn , and m.b.j .
meinders , phys .
b * 52 * , 2463 ( 1995 ) .
m. b. j. meinders , h. eskes , and g. a. sawatzky , phys .
b * 48 * , 3916 ( 1993 ) .
j. hubbard , proc .
a * 277 * , 237 ( 1963 ) .
l. hedin and s. lundquist , _ solid state physics _ ,
f. seitz , d. turnbull , and h. ehrenreich ( eds ) , vol .
* 23 * , 1 ( 1969 ) .
|
we present the continued fraction method ( cfm ) as a new microscopic approximation to the spectral density of the hubbard model in the correlated metal phase away from half filling . the quantity expanded as
a continued fraction is the single particle green function .
leading spectral moments are taken into account through a set of real expansion coefficients , as known from the projection technique .
the new aspect is to add further stages to the continued fraction , with complex coefficients , thus defining a terminator function .
this enables us to treat the entire spectral range of the green function on equal footing and determine the energy scale of the fermi liquid quasiparticles by minimizing the total energy .
the solution is free of phenomenological parameters and remains well defined in the strong coupling limit , near the doping controlled metal - insulator transition .
our results for the density of states agree reasonably with several variants of the dynamical mean field theory .
the cfm requires minimal numerical effort and can be generalized in several ways that are interesting for applications to real materials .
|
one of the purposes of this paper is to bring together fuzzy logic and quantum mechanics . here
we extend our prior analysis @xcite of a fuzzy logic interpretation of quantum mechanics by demonstrating that the schroedinger equation can be deduced from the assumptions of the fuzziness underlying not only quantum but also classical mechanics .
a pedestrian way of defining fuzziness was given by kosko @xcite who wrote that the fuzzy principle states that everything is a matter of degree .
more rigorously , fuzziness can be defined as multivalence .
+ interestingly enough , even separation between classical and quantum domains is somewhat fuzzy since there is no crisp boundary separating them ( see , for example , @xcite ) .
moreover , we can even claim that the difference between these domains is only in a degree of fuzziness .
in fact , both classical and quantum mechanics make predictions based on repetitive measurements which imply a certain spread of results
. + the crisp character of the formal apparatus of classical mechanics hides this important fact by a seemingly absolute character of a single measurement . from this point of view
the ultimate statements of classical mechanics are nothing but the results a certain averaging ( or defuzzification , meaning the elimination of the spread ) with some weight which we call the `` fuzziness density '' .
the latter can be represented by some function .
the fuzziness density then varies from `` sharp ( in classical mechanics ) to ' ' diffuse `` ( in quantum mechanics ) .
+ if we consider a concept of a ' ' thing in itself `` and assume(quite plausibly ) that it has a fuzzy and deterministic character , then in a series of experiments designed to elicit its properties to the outside observers it appears as a random set thus disguising its deterministic nature .
as we have already indicated , we consider classical and quantum mechanics as having common fuzzy roots and no sharp dividing boundary .
they can be viewed as different realizations of a fuzzy ' ' thing in itself `` .
this can explain why some phenomena in a strongly fuzzy domain of quantum mechanics can not be realized in a weakly fuzzy ( more precisely , zero fuzzy ) domain of classical mechanics .
+ thus if we accept quantum mechanics as a more general theory than classical mechanics , then it seems reasonable to expect that the former could be constructed independently from the latter .
however the basic postulates of quantum mechanics can not be formulated , even in principle , without invoking some concepts of classical mechanics .
both theories share some basic common feature , namely that they are rooted in the fuzzy reality .
this somehow justifies a paradoxical statement by goldstein ( as quoted in @xcite ) that quantum mechanics is a repetition of classical mechanics suitably understood .
+ our basic assumption is that reality is fuzzy and nonlocal not only in space but also in time . in this sense
idealized pointlike particles of classical mechanics corresponding to the ultimate ' ' sharpness `` of the fuzziness density emerge in a process of interaction between different parts of fuzzy wholeness .
this process is viewed as a continuous process of defuzzification .
it transforms a fuzzy reality into a crisp one .
it is clear that the emerging crisp reality ( understood as a final step of measurements which we call detection ) carries less information that the underlying fuzzy reality .
this means that there is an irreversible loss loss of information usually called a collapse of the wave function within a context of quantum mechanics . from our point of view
it is not so much a ' ' collapse as a realization of one of many possibilities existing within a fuzzy reality .
any measurements ( viewed as a process ) rearranges the fuzzy reality leading to different detection outcomes according to the changed fuzziness .
+ therefore it seems quite reasonable to expect that classical theory bears some traces of quantum theory underlying ( and connected to)it . in view of this
we would like to recall the words by bridgeman who remarked that the seeds and the sources of the ineptness of our thinking in the microscopic range are already contained in our present thinking applied to a large - scale regions .
one should have been capable of discovery of the former by a sufficiently critical analysis of our ordinary common sense thinking .
as we have already indicated , both classical and quantum mechanics can be viewed as statistical theories ( cf . @xcite)with
respect to an ensemble of repetitive measurements where each measurement must be carried out under the identical conditions .
the latter is a very restrictive requirement dictated by a crisp - logical world view and therefore not attainable even in a more general setting of fuzzy reality . on the other hand , if we assume a fuzzy nature of `` things '' then the apparent statistical character of physical phenomena would follow not from their intrinsic randomness but from their fuzzy - deterministic nature .
outwardly the latter expresses itself as randomness .
clearly , this definition of the apparent statistical nature of classical and quantum mechanics is applicable even to one measurement .
let us elaborate on this.conventionally , statistical theories are tied to randomness .
however recent results in the theory of fuzzy logic provided a deterministic definition of the relative frequency count of identical outcomes by expressing it as a measure of a subsethood @xmath0 , that is a degree to which a set @xmath1 is a subset of a set @xmath2 @xcite . to make it more concrete suppose that set @xmath2 contains @xmath3 trials and set @xmath1 contains @xmath4 @xmath5 trials .
then @xmath6 .
+ we would like to extend this concept to experimental outcomes of measurements performed on a classical particle .
this would be possible if we were to to consider the classical particle to be located simultaneously on all possible paths connecting two spatial points . in a sense
it is not so far fetched since it is analogous to the idea used by the least action principle .
+ to adapt the concept of fuzziness to a spatial localization of a particle we introduce the notion of the particle s membership in a spatial interval ( one- , two , or three - dimensional ) .
this membership , generally speaking , is going to vary from one interval to another .
we define the membership as follows .
let us say that we perform @xmath3 measurements aimed at detecting the particle in a certain spatial interval .
it turns out that the particle is found in this interval @xmath4 times .
the membership of the particle in the interval is then defined as @xmath7 and can be formally described with the help of zadeh @xcite sigma - function . + as a next step , this approach allows us to formally introduce the _ membership density _ defined as the derivative of the membership function .
if we denote the membership density by @xmath8 , then a degree of membership of the particle say in an elemental volume @xmath9 is @xmath10 . according to this definition
the particle has a @xmath11 membership in a spatial interval of measure @xmath12 , that is at a point .
such an apparently paradoxical result indicates that in general we should base our estimation of fuzziness on the relative degree of membership instead of the absolute one . + in other words , given a degree of membership @xmath13 of a particle in a volume @xmath14 containing the point @xmath15 and a degree of membership @xmath16 of the same particle in a volume @xmath14 containing the point @xmath17 , we find the @xmath18 of membership of the particle in both volumes : @xmath19 .
the same expression represents also the relative degree of membership of the particle in two points @xmath15 and @xmath17 despite the fact that the absolute degree of membership of the particle in either point is @xmath12 .
+ an importance of the relative degree of membership is due to the fact that experimentally a location of the particle is evaluated on the basis of its detection at a certain location in @xmath20 experimental trials out of @xmath3 trials .
as was shown by kosko @xcite , the ratio @xmath21 then measures the degree to which a sample of all elementary outcomes of the experiments is a subset of a space of the successful outcomes .
in other words , this ratio represents a degree of membership of the sample space in the space of the successful outcomes . in our case the relative degree of membership @xmath19 of a particle in two points can be identified as the relative count of the successful outcomes ( in a series of measurements ) of finding the particle at points @xmath15 and @xmath17 . + in view of these definitions the classical mechanical sigma - curve of particle s membership in a spatial interval is nothing but a step function .
this simply means that up to a certain spatial point @xmath22 the degree of particle s membership in an interval ( @xmath23 $ ] is @xmath12 , and for any value @xmath24 the degree of particle s membership in the interval ( @xmath25 $ ] is @xmath26 .
the corresponding membership density is the delta function .
thus the idealized picture of classical - mechanical phenomena with particles occupying intervals of measure zero corresponds to the statement that these particles are @xmath27 non - fuzzy , their behavior is governed by a crisp bivalent logic , and the respective membership density is the delta - function . + in reality , any physical `` particle '' occupies a small but nonzero spatial interval .
this means that the membership density is a sharp ( but not delta - like ) function corresponding to a minimum fuzziness . at the other end of the spectrum , in the microworld ,
the fuzziness is maximal .
in fact , if we accept the idea that a quantum mechanical `` particle '' ( a microobject ) `` resides '' in different elemental volumes @xmath14 of a three - dimensional space with the varying degrees of residence ( membership ) , then we can apply to such a microobject our concept of the membership density .
in general , this density can not be made arbitrarily narrow as is the case for a classical particle .
the latter can be considered as the limiting case of the former when the membership density becomes delta - function - like . moreover ,
the fuzziness in the microworld is even more subtle since mathematically it is described with the help of complex - valued functions .
+ the latter results in the emergence of the interference phenomenon for microobjects , which in the classical domain is an exclusive property of waves . therefore , mutually exclusive concepts of particles and waves in classical mechanics become inapplicable in the realm of fuzzy reality where `` particles '' and `` waves '' are not mutually exclusive concepts , but rather different expressions of fuzziness . for example , the double - slit experiment can be interpreted now as a microobject s `` interference with itself '' because it has a simultaneous membership in all parts of space including elemental volumes containing both slits .
since the total membership of a microobject in a given finite volume is fixed , any change of its membership in one of the slits affects the membership everywhere leading to the interference effects .
+ in the following we `` recover '' the fuzziness of the quantum world by deriving the schroedinger equation from the hamilton - jacobi equation , where the latter can be viewed as the result of the fuzziness reduction ( destruction ) of the quantum world .
first , we show how the hamilton - jacobi equation for a classical particle in a conservative field can be derived from newton s second law , thus connecting it to the destruction of fuzziness . in principle , a particle s motion between two fixed points , a and b , can occur along any conceivable path ( a `` fuzzy '' ensemble in a sense that a particle has membership in each of the paths ) connecting these two points . in the observable reality these paths `` collapse '' onto one observable path .
mathematically , this reduction is achieved by imposing a certain restriction on a certain global quantity ( the action @xmath28 ) , defined on the above family of paths .
+ let us consider newton s second law and assume that trajectories connecting points @xmath1 and @xmath2 comprise a continuous set .
this means in particular that the classical velocity is now a function of both the time and space coordinates , @xmath29,@xmath30 .
now we fix time @xmath31 . since on the above set for the fixed @xmath32 the correspondence @xmath33 to @xmath34 is many to one ,
@xmath33 is not fixed ( as was the case for a single trajectory ) , and therefore the velocity would vary with @xmath33 . physically this is equivalent to considering points on different trajectories at the same time .
our assumption means that now we must use the total time derivative : @xmath35 by applying the curl operation to newton s second law for a single particle and performing elementary vector operations we obtain @xmath36 where @xmath37 is the particle s momentum .
if we view ( 2 ) as the equation with respect to @xmath38 , then one of its solutions is @xmath39 where @xmath40 is some scalar function to be found . since @xmath28 is defined on the continuum of paths it can serve as a function related to the notion of fuzziness ( here a continuum of possible paths ) .
+ note that the spatial and time variables enter into @xmath28 on equal footing .
upon substitution of ( 2 ) back in newton s second law , @xmath41 , where @xmath42 is understood in the sense of ( 1 ) , we obtain @xmath43 = 0\ ] ] . integrating this equation and incorporating the constant of integration ( which , generally speaking , is some function of time ) into the function @xmath28
, we arrive at the determining equation for the function @xmath28 which is the familiar hamilton - jacobi equation for a classical particle in a potential field @xmath44 : @xmath45 by using eqs .
( 1 ) and ( 3 ) we can represent @xmath28 as a functional defined on the continuum of paths connecting two given points , say @xmath12 and @xmath26 , corresponding to the moments of time @xmath32 and @xmath46 . to this end
we rewrite ( 4 ) : @xmath47 integrating ( 5 ) we obtain the explicit expression of @xmath28 in the form of the following functional : @xmath48 which is the well - known definition of the action for a particle moving in the potential field @xmath44 .
thus we have connected the concept of fuzziness in classical mechanics with the action @xmath28 .
if we consider @xmath28 as a measure of fuzziness in accordance with our previous discussion , then by minimizing this functional ( i.e. , by postulating the principle of least action ) we `` eliminate '' ( or rather minimize ) fuzziness by generating the unique trajectory of a classical particle . in a certain sense the principle of least action serves as a defuzzification procedure .
+ now we proceed with the derivation of the schroedinger equation .
there are two basic experimental facts that make microobjects so different from classical particles .
first , all the microscale phenomena are linear .
second ( which is a corollary of the first ) , these phenomena obey the superposition principle . here
it would be useful to recall that even at the initial stages of development of quantum mechanics dirac formulated its fuzzy character , albeit without using the modern - day terminology .
he wrote : `` ... whenever the system is definitely in one state we can consider it as being partly in each of two or more other states''@xcite .
this is as close as one can come to the concepts of fuzzy sets and subsethood @xcite without directly formulating them . in view of this
it does not seem strange that a microobject sometimes can exhibit wave properties . on the contrary ,
they arise quite naturally as soon as we accept the fuzzy basis ( meaning `` being partly in ... other states '' ) of microscale phenomena which implies , among other things , the above - mentioned `` self - interference . ''
+ how can we derive the equation that would incorporate these essential features of microscale phenomena and , under certain conditions , would yield the hamilton - jacobi equation of classical mechanics ?
we start with the hamilton - jacobi equation ( not newton s second law ) because of its connection to the hidden fuzziness in classical mechanics .
we consider the simplest classical object that would allow us to get the desired results that will account for the two experimental facts mentioned earlier . + we choose a free particle by setting @xmath49 in eq .
our problem is somewhat simplified now .
we are looking for a linear equation whose wave - like solution is simultaneously a solution of the hamilton - jacobi equation .
since the mechanical phenomena behave differently at microscales and macroscales , the linear equation should contain a scale factor ( that is to be scale - dependent ) , such that in the limiting case corresponding to the macroscopic value of this factor we get the nonlinear hamilton - jacobi equation for a free particle .
+ a nonlinear equation admits a wave - like solution ( for a complex wave ) if this equation is homogeneous of order @xmath50 . since eq .
( 4 ) does not satisfy this criterion , we can not expect to find a wave solution for the function @xmath28 . however , this turns out to be a blessing in disguise , because by employing a new variable in place of the action @xmath28 , we can both convert this equation into a homogeneous ( of order 2 ) equation ( thus allowing for a wave - like solution ) and simultaneously introduce the scaling factor .
it is easy to show that there is one and only one transformation of variables that would satisfy both conditions : @xmath51 where the scaling factor @xmath52 is to be found later .
+ upon substitution of ( 7 ) in ( 4 ) , with @xmath53 , we obtain the following homogeneous equation of the second order with respect to the new function @xmath54 : @xmath55 equation ( 8) is easily solved by the separation of variables , yielding @xmath56\ ] ] where the vector @xmath57 of length @xmath58 is another constant of integration . since solution ( 9 ) must be a complex - valued wave , the argument of @xmath54 must satisfy two conditions : i)it must be imaginary , and ii)the factors at the variables @xmath34 and @xmath59 must be the frequency @xmath60 and the wave vector @xmath61 , respectively .
this results in the following : @xmath62 and @xmath63 where @xmath2 is a real - valued constant .
now the solution ( 9 ) is @xmath64\ ] ] since both functions @xmath28 and @xmath54 are related by eq .
( 7 ) , we can easily establish the connection between the kinematics parameters of the particle and the respective parameters @xmath65 and @xmath61 , which determine the wave - like solution of the hamilton - jacobi equation for the new variable @xmath54 . according to classical mechanics
, @xmath66 is the particle energy @xmath67 , and @xmath68 is the particle momentum @xmath69 . on the other hand , these quantities can be expressed in terms of the new variable @xmath54 with the help of eqs .
( 7 ) and ( 12 ) , yielding @xmath70 .
+ from these relations we see that for a free particle its energy ( momentum ) is proportional to the frequency @xmath65 ( wave vector * k * ) of the wave solution to the `` scale - sensitive '' modification of the hamilton - jacobi equation .
the constant @xmath2 is found by invoking the experimental fact that @xmath71 ( where @xmath72 is planck s constant ) .
this implies @xmath73 or @xmath74 , and as a byproduct , the de broglie equation @xmath75 . inserting solution ( 12 ) into the original nonlinear equation ( 8) , we arrive at the dispersion relation @xmath76
now we can find the linear wave equation whose solution and the resulting dispersion relation are given by eqs.(12 ) and ( 13 ) respectively .
using an elementary vector identity , we rewrite eq .
( 8) : @xmath77\psi - \frac{i \hbar}{2 m \psi}[div(\psi \nabla \psi)-2 \psi { \nabla}^2\psi]=0\ ] ] + equation ( 14 ) is the sum of the two parts , one linear and the other nonlinear in @xmath54 .
the solution ( 12 ) makes the nonlinear part identically zero , and this solution , together with the dispersion relation ( 13 ) , must also satisfy the linear part of eq .
therefore we have proven the following : if the wave - like solution ( 12 ) satisfies eq .
( 8) , then it is necessary and sufficient that it must be a solution of the following linear partial differential equation , the schroedinger equation : @xmath78\psi=0\ ] ] + now we return to the variable @xmath28 according to @xmath79 and introduce the following dimensionless quantities : time @xmath80 , spatial coordinates @xmath81 , the parameter @xmath82 ( which we call the schroedinger number ) , and the dimensionless action @xmath83 . here ,
@xmath84 , @xmath85 is the characteristic length , and @xmath32 is the characteristic time . as a result , we transform ( 15 ) into the following dimensionless equation : @xmath86 + this equation is reduced to the classical hamilton - jacobi equation ( or , equivalently , the equation corresponding to the minimum fuzziness ) if its right - hand side goes to 0 .
this is possible only when the schroedinger number @xmath87 goes to 0 .
therefore , at least for a free particle , this number serves as a measure of fuzziness of a microobject . since @xmath72 is a fixed number , the limit @xmath88 is possible only if @xmath89 , thus confirming our earlier assumption that action @xmath28 represents a measure of fuzziness of a microobject . for a free particle
this means that with the decrease of @xmath90 the fuzziness of the particle increases .
+ interestingly enough , the question of fuzziness ( although not in these terms ) was addressed in one of the first six papers on quantum mechanics written by schroedinger @xcite .
he wrote , `` ... the true laws of quantum mechanics do not consist of definite rules for the single path , but in these laws the elements of the whole manifold of paths of a system are bound together by equations , so that apparently a certain reciprocal action exists between the different paths . ''
+ it turns out that by using the same reasoning as for a free particle we can easily derive the schroedinger equation from the hamilton - jacobi equation for a piece - wise constant potential .
if we replace in the resulting schroedinger equation the function @xmath54 by @xmath28 according to ( 7 ) , and introduce the dimensionless variables used for a free particle we obtain @xmath91 where @xmath92 is the dimensionless potential .
once again , the schroedinger number serves as the indicator of the respective fuzziness , yielding the classical motion ( a zero fuzziness ) for @xmath93 + a more general case of a variable potential @xmath94 can not be derived from the hamilton - jacobi equation with the help of the technique used so far , since there are no monochromatic complex wave solutions common to the nonlinear hamilton - jacobi equation and the linear schroedinger equation .
therefore we postulate that the schroedinger equation describing a case of an arbitrary potential @xmath94 should have the same form as for a piece - wise constant potential .
this postulate is justified by the fact that , apart from the experimental confirmations , in the limiting case of a very small schroedinger number , @xmath95 ( minimum fuzziness ) , we recover the appropriate classical hamilton - jacobi equation . in
what follows we will describe this process of recovering classical mechanics from quantum mechanics ( which we dubbed `` defuzzification '' ) in a different fashion that will require a study of a physical meaning of the function @xmath54 .
earlier , by considering the schroedinger number @xmath87 , we saw that the action @xmath28 represents some measure of fuzziness .
therefore , it is reasonable to expect that the function @xmath96 is also related to the measure of fuzziness .
since the fuzziness is measured by real - valued quantities ( degree of membership , membership density ) , a possible candidate for such a measure would be some function of various combinations of @xmath54 and @xmath97 there is an infinite number of such combinations .
however , it is easy to demonstrate @xcite that the schroedinger equation is equivalent to the two nonlinear coupled equations with respect to the two real - valued functions constructed out of @xmath98 and @xmath99 therefore , our choice of all possible real - valued combinations is reduced to only two functions .
however , in the limiting transition to the classical case , @xmath100 is related to the classical velocity .
therefore we are left with only one choice : @xmath98 .
+ the easiest way to find a physical meaning of @xmath98 is to consider some simple specific example that can be reduced to a respective classical picture . to this end
we consider a solution of the schroedinger equation for a free particle passing through a gaussian slit @xcite : @xmath101\end{aligned}\ ] ] where @xmath102 is the initial moment of time , @xmath34 is any subsequent moment of time , @xmath103 is the half - width of the slit , @xmath104 , and @xmath105 is the coordinate of the center of the slit .
+ using ( 18 ) we immediately find that @xmath106 is @xmath107\ ] ] where now @xmath108 executing the transition to the case of a classical particle passing through an infinitesimally narrow slit , we set both @xmath109 and @xmath110 . as a result , ( 19 ) will become the delta function .
recalling that we define a classical mechanical particle as a fuzzy entity with a delta - like membership density , we arrive at the conclusion that the real - valued quantity @xmath111 can be identified as the membership density for a microobject . +
this allows one to ascribe to @xmath112 the physical meaning of the degree of membership of a microobject in an infinitesimal volume @xmath14 ( cf . the analogous statement postulated in ref .
@xcite ) . this in turn implies a nice geometrical interpretation with the help of a generalization of kosko s multi - dimensional cube . any fuzzy set a ( in our case a fuzzy state )
is represented ( see fig . 1 for a two - dimensional cube ) by point a inside this cube .
following kosko , we use the sum of the projections of vector a onto the sides of the cube as the cardinality measure .
let us consider the following integral : @xmath113 if this integral is bounded , then we can normalize it . as a result , we can treat the right - hand side of ( 20 ) as the sum of the projections of the `` vector '' @xmath114 onto the sides @xmath115 of the infinitely dimensional hypercube .
this allows us to represent the integral as the vertex a along the major diagonal of this hypercube .
+ according to the subsethood theorem @xcite each side of the hypercube represents the degree of membership of the microobject ( viewed as a deterministic fuzzy entity ) in any given elemental volume @xmath116 built around a given spatial point @xmath15 .
respectively the relative membership of the microobject in two different spatial points @xmath15 and @xmath17 , that is , @xmath117 is equal to the ratio of the respective numbers of the successful outcomes in a series of experiments aimed at locating the microobject ( or rather its part ) at the respective elemental volumes .
hence we can conclude that the membership density at a certain point is proportional to the number of successful outcomes in repeated experiments aimed at locating the fuzzy microobject at the respective elemental volumes .
+ if the integral on the right - hand side of ( 20 ) is divergent , this does not change our arguments , since @xmath118 is a measure of the successful outcomes in a series of experiments that do not depend on the convergence of the integral .
thus we see that the fuzziness , via its membership density , dictates the number of successful outcomes in experiments aimed at locating the fuzzy microobject .
continuing this line of thought we see that any physical quantity associated with the fuzzy microobject is not tied to a specific spatial point .
this indicates a need to introduce a process of defuzzification with the help of the membership density which would serve as the `` weight '' in this process .
such defuzzification is different from what is usually understood by this term , that is , a process of `` driving '' a fuzzy point to a nearest vertex of a hypercube .
instead , we take the degree of and @xmath17.,width=226,height=226 ] membership @xmath119 at each vertex of the infinite - dimensional hypercube and multiply it by the value of the physical quantity at the respective point @xmath15 . summing over all these products results in the averaged ( defuzzified ) value of the quantity .
thus , instead of averaging over the distribution of random quantities , we introduce the defuzzification of deterministic quantities . mathematically both processes are identical , but physically they are absolutely different .
we do not need any more the probabilistic interpretation of the wave function @xmath54 , which implies that there is another , more detailed level of description that would allow us to get rid of uncertainties introduced by randomness .
now it is clear that , within the framework of the fuzzy interpretation , we can not get rid of the uncertainties intrinsic to fuzziness ( and not connected to randomness ) . from this point of view
quantum mechanics does not need any hidden variable to improve its predictions .
they are precise within the framework of the fuzzy theory . + moreover ,
since quantum mechanics is a linear theory , one can speculate that according to the fuzzy approximation theorem @xcite the linearity and fuzziness of quantum mechanics are the best tools to approximate ( with any degree of accuracy ) any macrosystem ( linear or nonlinear ) .
the linearity of quantum mechanics is responsible for the uncertainty relations which are present in any linear system . therefore ( as was demonstrated long time ago @xcite ) , these relations enter quantum mechanics even before any concept of measurement . + let us consider the membership density of a free microobject ( a progenitor of a classical free particle ) .
it is obvious that @xmath120 .
this means that the relative degree of membership for any two points in space is 1 . in other words ,
the free microobject is `` everywhere , '' the same property that is characteristic for a three - dimensional standing wave . in particular
, this example shows that the wave - particle duality is not necessarily a duality but rather an expression of the fuzzy nature of things quantum .
+ in fact , we can even go that far as to claim that the complementarity principle is a product of a compromise between the requirements of the bivalent logic and the results of quantum experiments . within the framework of the fuzzy approach
there is no need to require complementarity , since the logic of a fuzzy microobject transcends the description of its properties in terms of either / or and , as a result , is much more complete , probably the most complete description under the given experimental results . + it turns out that the membership density has something more to offer than simply a degree to which a fuzzy microobject has a membership in a certain elemental volume dv .
in fact , using the expansion of the wave amplitude ( we could call it `` fuzziness amplitude '' ) in its orthonormal eigenfunctions @xmath54 and assuming that the integral in ( 20 ) is bounded , we write the well - known expression @xmath121 equation ( 21 ) allows a very simple geometric interpretation with the help of a @xmath122-dimensional simplex .
a fuzzy state @xmath54 is represented as a point @xmath1 at the boundary of this simplex .
( figure 2 shows this for a one - dimensional simplex , @xmath123 ) its projections onto the respective axes correspond to the values @xmath124 .
+ now applying the subsethood theorem @xcite , we interpret the values of @xmath124 as the degree to which the state a is contained in a particular eigenstate @xmath125 .
using fig .
2 we can clearly see that @xmath126 .
moreover , the same figure shows that the lengths of projections of a onto the respective axes ( namely , oa and oc ) are nothing but the cardinality sizes @xmath127 and @xmath128 . on the other hand , the cardinality size of a is @xmath129
. therefore , the respective subsethood measures are @xmath130 and @xmath131 at the same time , both of these measures provide a number of detections ( successful outcomes ) of the respective states @xmath132 or @xmath133 in the repeated experiments . + our discussions is applicable to a particular case of a state a described by a wave ( fuzziness ) amplitude @xmath54 corresponding to a pure state .
however it is general enough to describe a mixed state characterized by the density matrix @xmath134 .
the integral of @xmath134 over all @xmath135 yields the sum @xmath136 which is the generalization of a measure of containment of the fuzzy state a in the discrete states k. + as a point in a one - dimensional fuzzy simplex , width=226,height=226 ] by preparing a certain state , which is now understood to be a fuzzy entity , we fix the frequencies of the experimental realizations of this fuzzy state in its substates @xmath125 . if the fuzzy state a undergoes a continuous change , which corresponds in fig . 2 to motion of point a along the hypotenuse
, then its subsethood in any state @xmath125 changes .
this implies the following : if the eigenfunctions of a fuzzy set stay the same , the degree to which the respective eigenstates represent the fuzzy state varies .
the variation can occur continuously despite the fact that the eigenstates are discrete .
+ this indicates an interesting possibility that quantum mechanics is not necessarily tied to the hilbert space .
such a possibility was mentioned long ago by von neumann @xcite and recently was addressed by wulfman @xcite .
one of the hypothetical applications of this idea is to use quantum systems as an infinite continuum state machine in a fashion that is typical for a fuzzy system : small continuous changes in the input from some `` ugly '' nonlinear system will result in small changes at the output of the quantum system which in turn can be correlated with the input to produce the desired result .
+ concluding our introduction to a connection between fuzziness and quantum mechanics , we prove a statement that can be viewed as a generalized ehrenfest theorem .
we will demonstrate that defuzzification of the schroedinger equation ( with the help of the membership density @xmath98 ) yields the hamilton - jacobi equation .
this will provide @xmath137 derivation of the schroedinger equation for an arbitrary potential @xmath138 .
we assume that the fuzzy amplitude @xmath139 as @xmath140 and rewrite the schroedinger equation as follows : @xmath141 integrating ( 22 ) with the weight @xmath98 ( i.e. , `` defuzzifying '' it ) , we obtain @xmath142d^3x = 0\ ] ] integrating the second term by parts and taking into account that the resulting surface integral vanishes because @xmath143 at infinity , we obtain the following equation : @xmath144 where @xmath145 denote defuzzification with the weight @xmath98 , and @xmath146 .
this equation is analogous to the classical hamilton - jacobi equation ( 4 ) .
+ the generalized ehrenfest theorem shows that the classical description is true only on a coarse scale generated by the process of `` defuzzification , '' or measurement .
the `` classical measurement '' corresponds to the introduction of a non - quantum concept of the potential @xmath138 serving as a shorthand for the description of a process of interaction of a microobject ( truly quantum object ) with a multitude of other microobjects .
this process destroys a pure fuzzy state ( a constant fuzziness density ) of a free quantum `` particle . ''
paraphrasing peres , @xcite we can say that a classical description is the result of our `` sloppiness , '' which destroys the fuzzy character of the underlying quantum mechanical phenomena .
this means that , in contradistinction to peres , we consider these phenomena `` fuzzy '' in a sense that the respective membership distribution in quantum mechanics does not have a very sharp peak , characteristic of a classical mechanical phenomena . note that we exclude from our consideration the problem of the classical chaos , assuming that our repeated experiments are carried out under the absolutely identical conditions .
this work represents a continuation of our previous effort @xcite to understand quantum mechanics in terms of the fuzzy logic paradigm .
we regard reality as intrinsically fuzzy . in spatial terms
this is often called nonlocality .
reality is nonlocal temporarily as well , which means that any microobject has membership ( albeit to a different degree ) in both the future and the past . in this sense
one might define the present as the time average over the membership density .
a measurement is defined as a continuous process of defuzzification whose final stage , detection , is inevitably accompanied by a dramatic loss of information through the emergence of locality , or crispness , in fuzzy logic terms .
+ we have attempted to provide a description of quantum mechanics in terms of a deterministic fuzziness .
it is understood that this attempt is inevitably incomplete and has many features that can be improved , extended , or corrected .
however , we hope that this work will inspire others to start looking at the quantum phenomena through `` fuzzy '' eyes , and perhaps something practical ( apart from removing wave - particle duality and complementarity mysteries ) will come out of this . + * acknowledgment * one of the authors ( ag ) wishes to thank v. panico for very long and very illuminating discussions , which helped to shape this work , and for reading the manuscript .
hjc s work was supported by the air force office of scientific research 99 h. caulfield and a. granik , spec .
18 , 61 ( 1995 ) .
b. kosko , fuzzy thinking ( hyperion , ny , 1993 ) , p. 18 .
i. percival , in quantum chaos - quantum measurement , edited by p. cvitanovic , i. percival , and a. wirzbe ( kluwer academic , dordrecht , 1992 ) , p. 199 .
m. krieger , doing physics ( indiana university press , 1992 ) , p. xx .
l. mandelshtam , lectures on optics , relativity , and quantum mechanics ( nauka , moscow , 1972 ) , p. 332 .
b. kosko , neural networks and fuzzy systems ( prentice - hall , englewood cliffs , nj , 1992 ) , chap . 1
. l. zadeh , informal .
control 8 , 338 ( 1965 ) .
p. dirac , the principles of quantum mechanics ( oxford , 1957 ) , p. 10 .
b. kosko , int .
17 , 211 ( 1990 ) .
e. schroedinger , collected papers on quantum mechanics ( chelsea , ny , 1978 ) , p. 26 .
d. bohm , in problems of causality in quantum mechanics ( nauka , moscow , 1955 ) , p. 34 .
r. feynman and a. hibbs , quantum mechanics and path integrals ( mcgraw - hill , ny , 1965 ) , chap .
b. kosko , in proceedings of the first i.e.e.e .
conference of fuzzy systems ( ieee proceedings , san diego , 1992 ) , p. 1153 .
g. birkgoff and j. von neumann , ann .
37 , 823 ( 1936 ) . c. wulfman , int .
j. quantum chem .
49 , 185 ( 1994 ) .
a. peres , in ref . 4 , p. 249 .
|
it is shown that quantum mechanics can be regarded as what one might call a `` fuzzy '' mechanics whose underlying logic is the fuzzy one , in contradistinction to the classical `` crisp '' logic .
therefore classical mechanics can be viewed as a crisp limit of a `` fuzzy '' quantum mechanics . based on these considerations
it is possible to arrive at the schroedinger equation directly from the hamilton - jacobi equation .
the link between these equations is based on the fact that a unique ( `` crisp '' ) trajectory of a classical particle emerges out of a continuum of possible paths collapsing to a single trajectory according to the principle of least action .
this can be interpreted as a consequence of an assumption that a quantum `` particle '' `` resides '' in every path of the continuum of paths which collapse to a single ( unique ) trajectory of an observed classical motion .
a wave function then is treated as a function describing a deterministic entity having a fuzzy character . as a consequence of such an interpretation
, the complimentarity principle and wave - particle duality can be abandoned in favor of a fuzzy deterministic microoobject .
|
observations of starburst galaxies have found that relatively bright diffuse soft x - ray emission is always present ( e.g. * ? ? ? * and references therein ) .
these regions of hot ( @xmath3 ) gas have been observed to extend far beyond the starburst itself . in starbursts located in edge - on disk galaxies , the emission
can be traced many kpc out into the galactic halo .
this diffuse x- ray emission is thought to be a prime manifestation of the effects of `` feedback '' from star formation
. short lived massive stars in the starburst inject kinetic energy and highly metal - enriched gas into their surroundings through stellar winds and supernovae .
the collective effect is to drive a galaxy - scale outflow known as a `` superwind '' @xcite out of the starburst and into the galaxy halo and perhaps beyond .
these outflows could play a major role in the evolution of galaxies and the intergalactic medium ( igm ) .
for example , by propelling metals out of star - forming galaxies , they could explain both the mass - metallicity relation of galaxies ( e.g. * ? ? ? * ) and the presence of substantial amounts of metals in the intergalactic medium ( e.g. * ? ? ?
* ; * ? ? ?
* ) and in the intracluster medium ( e.g. * ? ? ?
spectroscopy in the rest - frame ultraviolet has shown that starburst - driven outflows are a characteristic feature of starburst galaxies at high redshift @xcite . however , the best laboratories for studying the astrophysics of these winds are in local galaxies in which their complex multi - phase nature can be investigated in detail in the x - ray , ultraviolet , optical , and radio regimes ( e.g. * ? ? ?
x - ray observations with high spatial resolution are particularly crucial , as they most directly trace the hot gas that powers the wind . in recognition of this ,
the _ chandra _ x - ray observatory has devoted significant amounts of time to the observation of starburst galaxies and their winds .
the galaxies studied span a broad range from dwarf starbursts ( e.g. * ? ? ?
* ; * ? ? ?
* ) , to edge - on disk galaxies ( e.g. * ? ? ?
* b ) , to interacting and merging systems @xcite , to the ultra luminous infrared galaxies ( ulirgs , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
halos of diffuse x - ray emission are seen in all these types of starburst , and in many cases the properties of the hot gas have been shown to be consistent with the superwind model .
however , investigations to date have mostly been of individual starburst systems , or ( in few cases ) of small samples of similar objects .
the time is now ripe for an investigation that exploits the _ chandra _ archive to uniformly analyze the properties of the diffuse x - ray emission in starburst galaxies spanning the widest possible range in the properties of both the starburst itself and of its `` host '' galaxy . in the present paper ,
we use _ chandra _ acis - s data to study the impact of starbursts in three distinct samples of starbursts : seven dwarf starbursts , six edge - on starburst galaxies of intermediate luminosity , and nine ulirgs . by combining the data from the three galaxy types we cover a range of roughly @xmath4 in x - ray luminosity , and several thousand in star formation rate and k - band luminosity ( a proxy for stellar mass ) .
note that the edge - on starburst sample has been previously examined extensively in two papers by ( * ? ? ?
the properties of the nuclear x - ray emission in the ulirgs have been discussed in @xcite .
each of the three samples was selected to be complete and unbiased .
the ulirg sample consists of the eight nearest such objects ( @xmath5 15,000 km / sec ) .
following @xcite we have additionally included ngc 6240 , whose far - ir luminosity is just below the ulirg range .
the members of the edge - on sample were selected to be the nearest ( closer than 20 mpc ) starbursts of intermediate far - ir luminosity ( @xmath6 few @xmath7 ) in moderately massive disk galaxies ( @xmath8 to @xmath9 ) viewed nearly edge - on ( inclinations greater than 60 degrees ) .
this range in luminosity and mass is representative of typical infrared selected starbursts in the present universe .
finally , our sample of dwarf starbursts includes the five brightest ( sum of ultraviolet plus far infrared flux ) starbursts in low mass galaxies ( @xmath10 below @xmath11 ) .
we have added two similar galaxies he 2 - 10 and ngc 3077 .
the samples are listed along with their most salient quantities in table [ lumin ] .
the x - ray data for our sample is based on _ chandra _ acis - s observations of 22 galaxies . in order to obtain a consistent data sample we have reanalyzed these observations for all of the ulirgs and all but one of the dwarf starbursts . for the edge - on starbursts we have used the luminosities , temperatures , and images derived in @xcite .
these were created in manner similar to that described below for our analysis .
the data and analysis for the dwarf starburst ngc 4214 was provided by @xcite . from their background subtracted image and exposure maps
we have calculated the appropriate surface brightness and radial values .
we have also used their derived flux and mekal temperature .
exposure times , pointing information , and observation dates for all of the galaxies are summarized in table [ obsdata ] . for our analysis ,
data reprocessing and flare removal was done as described in the ciao threads from the cxc .
ciao 3.01 with caldb 2.23 was used throughout the processing and analysis .
the two observations of iras 05189 - 2524 were combined using the ciao task reproject_events .
as the two observations are only separated by 3 months we have ignored differences in time dependent calibrations such as the acis optical blocking filter contamination .
only data from the s3 chip has been included so we have not attempted any charge transfer inefficiency corrections . to isolate the central sources in the ulirgs we have split each observation into an inner ( nuclear ) and outer ( halo ) region . in @xcite 2-d elliptical gaussians
were fit to the inner source of each ulirg .
we have doubled the minor and major axis of the elliptical gaussian fits and used that to define our inner region ( table [ innerregion ] ) .
the outer regions were manually chosen to include the rest of the galaxies extent ( from broad band smoothed images ) and exclude the central region .
extraneous point sources detected by wavdetect are excluded from the regions during spectral extraction . for the dwarf starbursts
we have defined a single region that includes all the x - ray emission from the galaxy and excludes all point sources .
accurate background subtraction is important when studying diffuse x - ray emission .
spatial , spectral , and time variations in the x - ray background complicate background analysis .
therefore a region enclosing the entire s3 chip but excluding extended and point sources was defined for each observation .
the count rate in these regions was then calculated over several energy bands and for the appropriate cxc - provided blank field datasets @xcite . if the count rates and spectral shapes were similar ( @xmath12 change )
the blank field background was renormalized by the ratio of the background rates in the 4 - 7 kev range .
the renormalized blank sky background was then used for both the spectral and image analysis . the background rates for iras 05189 - 2524 , iras 17208 - 0014 , iras 23128 - 5919 , mkn231 , ngc 1705 , and ngc 5253 were all above normal quiescent levels even after solar flare removal .
several of these observations were affected by extreme solar activity which multiply impacted chandra observations during october 2001 . due to the background spectral changes during periods of high solar activity it is inappropriate to rescale the quiescent blank field background datasets .
we have therefore used local backgrounds for these observations .
the local backgrounds are centered on the galaxies but exclude sources and the diffuse emission around the galaxy .
this local background was then used for spectral background subtraction .
the image background however was created by rescaling the blank field background images as described above for the galaxies with quiescent background levels . using the renormalized quiescent background
, background subtracted images were produced for every object in the sample .
the images were created in a variety of energy bands and then adaptively smoothed using the ciao task csmooth .
adaptively smoothed images for all of the ulirgs and dwarf starbursts in the 0.3 - 8 kev energy band can be found in figures [ smoothulg ] and [ smoothdwarf ] . in order to create representative color images for every galaxy we first divided the 0.3 - 1.0 kev , 1.0 - 2.0 kev , and 2.0 - 8.0 kev images by the monoenergetic exposure map for each observation .
the images were adaptively smoothed and combined into a single representative color image . to derive the radial extent of the galaxies we focused on the 0.3 - 1.0
kev background subtracted image .
this energy range was chosen as it is dominated by the diffuse thermal emission .
circular annuli starting at the galaxy center and radiating outward to the end of the diffuse x - ray emission were created for each observation . excluding all point sources except the central source , the number of counts in each annulus was extracted .
the 50% , 75% , 90% , and 95% counts enclosed radii were then calculated ( table [ obsradii ] ) .
the same annular regions were also used on an exposure map corrected image of the galaxy to calculate the surface brightness at the various radii .
an isometric radius was defined as the radius that the surface brightness was @xmath13 . the agn ulirg iras
05189 - 2524 is essentially pointlike in the chandra observations .
therefore we are unable to determine iras 05189 - 2524 s spatial extent and have not used it in our analyses of galaxy size and surface brightness . in the previous analysis of edge - on starburst galaxies @xcite , distances and surface brightness computations
are computed separately for both directions along the major and minor axis .
as the other galaxies in our sample represent a variety of orientations we have reanalyzed the edge - on starburst data using the same simple radial geometry we used in analyzing the ulirgs and dwarf starbursts above .
spectra were extracted for all of our defined regions .
after background subtraction the spectra were rebinned to a minimum of 20 counts per bin to allow use of the @xmath14 statistic . for every fit we have included galactic absorption and an acis contamination model @xcite .
the galactic absorber column was fixed to the value obtained from colden using the nrao h i dataset and found in table [ obsdata ] @xcite . allowing the absorption column to vary did not significantly improve fits and generally was consistent within one sigma of the colden value .
the extracted spectra are plotted in figures [ agnulgspectra ] , [ ulgspectra ] , and [ dwarfspectra ] .
a thermal component is clearly present in all but the lowest quality data .
as our objective is to compare the global properties of our galaxies in a consistent manner , we have used a single thermal model in all of our fits ( vmekal in xspec ) .
although there is strong evidence for spatial variations in temperature for several of our galaxies we have found generally , that a single thermal model adequately represents each galaxy s flux averaged properties . while other spectral models were examined , including two temperature models , intrinsically absorbed temperature models , and simpler thermal models
, these models generally resulted in higher values of the reduced @xmath14 .
as any thermal model would be a simplification of the true multi - temperature non - ionizational equilibrium physical conditions , we have used the simplest acceptable spectral fits throughout our analysis .
this allows us to analyze all of the galaxies in as uniform manner as possible .
this consistency is important when we compare the global properties of the x - ray gas throughout our sample .
the vmekal model allows for variations in elemental abundances .
however low signal to noise data and weak or unresolved emission lines make it impossible to constrain many of the elemental abundances .
following @xcite we have therefore tied many of the weakly constrained abundances to other similarly evolving elements .
mg , si , ne , s , and ar are then tied to o to form our @xmath2-element abundance .
we also tie the fe abundance to ni , ca , al , and na .
ca has been included with the fe elements as it is similarly depleted onto dust grains and has a negligible contribution to the x - ray emission at these temperatures .
although tied to other elements in the vmekal model , the fe and @xmath2 abundances are dominated by the contributions from fe and o respectively .
the absolute @xmath2 and fe abundances are degenerate and relatively unconstrained for many of the galaxies .
as we are primarily interested in @xmath2-element enrichment relative to fe we have only listed the @xmath15 ratio for which we can derive limits .
the spectra from the outer regions of the ulirgs and most of the the dwarf starbursts have significantly less high energy ( @xmath16 ) emission than the inner regions of the ulirgs . for several
there is no high energy emission detected . for this sample
we have restricted our fits to only include the 0.3 - 3.0 kev energy band .
a single thermal plasma ( vmekal ) model , without an additional powerlaw component , provides an excellent fit for the 8 ulirgs with detectable outer region emission and 5 of the dwarf starbursts .
fitting models to the outer regions of the ulirgs is complicated by low number counts in several of the ulirgs . in the case of iras
05189 - 2524 there are too few counts ( if any ) to attempt spectral fitting .
the other eight ulirgs do have detectable diffuse emission ranging from 76 counts above background in ugc 05101 to over 4100 counts for ngc 6240 .
an absorbed powerlaw was required to fit the inner regions of the ulirgs and the dwarf galaxies he 2 - 10 and ngc 4449 .
the redshift of the absorption column was set to that of the host ulirgs and the column density was allowed to vary freely . as the flux above 3.0 kev is dominated by the powerlaw emission we have fit these spectra in the 0.35 - 8.0 kev energy region .
this allows a much more accurate determination of the powerlaw parameters . in @xcite @xmath17 emission
was strongly detected in ngc 6240 and mkn 273 .
for the inner region in ngc 6240 we have added a gaussian component centered at 6.4 kev .
this provides an excellent fit for the spectrum of ngc 6240 .
the fit to the inner region of mkn 273 however is more complicated . to fit the @xmath17 line we have also added a gaussian .
however we have found that an additional unabsorbed powerlaw is required .
we have set the unabsorbed powerlaw to have the same slope as the absorbed powerlaw .
the additional powerlaw is motivated by examining representative color images of mkn 273 . the central source is strongly absorbed along some lines of sight but not others .
the final fit to mkn 273 gives us an acceptable reduced @xmath14@xmath18 .
the inner spectra of iras 20551 - 4250 suggests the need for a multiple temperature model .
although a two temperature model improves the reduced @xmath14 from 1.6 to 1.4 the derived parameters are not significantly affected .
the derived flux weighted temperature from the two temperature model is consistent with the single temperature model .
the powerlaw , redshifted hydrogen absorption column , @xmath15 , and flux are also found to be similar for both fits . for consistently with the other spectral fits we have used the results from the one temperature model for iras 20551 - 4250 .
the set of spectral fitting parameters for both the inner and outer regions for all the ulirgs and for the dwarf starbursts can be found in table [ spectra ] . to characterize the basic properties of the starbursts and their host galaxies , we have assembled a combination of near - infrared ( k - band ) , far - infrared ( iras ) , and vacuum ultraviolet data from the literature ( see table [ lumin ] ) .
we will use the near - ir luminosity as a rough tracer of the stellar mass of the galaxy , since the k - band mass - to - light ratio is only a weak function of the age of a stellar population ( e.g. * ? ? ?
* ; * ? ? ?
we note however that this approximation may not be valid for all the ulirgs , in which the extremely high star formation rate and possible presence of an agn mean that the k - band light will not necessarily be dominated by the older stars that dominate the galaxy mass ( e.g. * ? ? ?
in fact , the agn is known to dominate the k - band luminosity of mkn 231 , so we have dropped mkn 231 from our results involving its k - band luminosity .
the k - band cousins - glass - johnson luminosities were derived for every galaxy in our sample using the 2mass large galaxy atlas @xcite .
the k and j band fluxes given in the catalog were converted to k - band cgj using the transformations from @xcite .
the bulk of the bolometric luminosity of the ulirgs and edge - on starbursts emerges in the far - infrared .
@xcite argue that the hard x - ray properties of the ulirgs ( with the exception of the near - ulirg ngc 6240 ) do not support the idea that an agn makes a dominant contribution to the far - infrared luminosity .
we will therefore use the far - infrared luminosity as a proxy for the star formation rate in the ulirgs and edge - on starbursts .
the dwarf starbursts are substantially less dusty than the other two classes , and a significant amount of the starburst luminosity escapes in the uv . following @xcite
, we will use the sum of the uv and far - infrared luminosities as a proxy for the star formation rate in the dwarf starburst .
the _ iras _ fluxes @xcite and transformations from @xcite were used to calculate fir luminosities .
note that no iras color correction term has been applied to our fir luminosities .
the uv flux measurements we used have mean wavelengths in the range @xmath19 to @xmath20 .
uv fluxes are defined as @xmath21 .
the uv flux measurement for ngc 1569 comes from @xcite , for ngc 1705 from @xcite , for ngc 3077 from @xcite , for ngc 4214 and ngc 5253 from c. hoopes ( private communication , based on uit observations ) , and for ngc 4449 and he 2 - 10 from ned ( based on oao and iue observations ) .
the uv fluxes were corrected for foreground galactic extinction at the effective wavelength of the appropriate filter using the extinction law of @xcite and the values of the galactic optical extinction from @xcite .
luminosities are listed in table [ lumin ] .
in this section we will compare the principal properties of the diffuse x - ray emission in our sample of dwarf starburst , edge - on starbursts , and ulirgs .
we begin by comparing the structure and morphology of the diffuse x - ray emission and its morphological connection to the warm optically - emitting plasma .
we will then examine how the size and luminosity of the source of diffuse emission scales with the luminosity ( the star formation rate ) of the starburst .
finally , we will use the spectra to examine the thermal and chemical properties of the hot gas . a simple visual comparison of the morphology suggests a similar physical origin for the diffuse x - ray emission in dwarf starbursts , starbursts , and ulirgs ( figure [ 3types ] ) .
these three representative color images are very similar to one another , not only in morphology , but in `` color '' ( the broad - band x - ray spectral shape ) .
although the physical scales differ by almost a factor of 50 from dwarf to ulirgs , it is difficult to separate these galaxies based on the morphology of their diffuse x - ray emission .
the biggest visible difference is in the edge - on starburst galaxy ngc 3628 , but this is due to photoelectric absorption from the edge - on large scale ism of the galaxy ( i.e. it is simply an orientation effect ) .
we can examine the structure of the diffuse x - ray emission by using the images to define its characteristic x - ray surface brightness .
specifically , we have compared the mean surface brightness of the soft ( 0.3 - 1 kev ) x - ray emission interior to the radius enclosing 90% of the total soft x - ray flux .
figure [ s90vlfir ] shows that this surface brightness is relatively constant over the almost four orders of magnitude of sfr spanned by our samples .
a comparison of the diffuse x - ray and @xmath22 images also shows morphological relations between these two gas phases in all three images .
this can be seen in in figure [ ha ] .
although the x - ray images are adaptively smoothed and have a lower spatial resolution than the @xmath22 images , there is a relationship between the morphologies of the @xmath22 and diffuse x - ray emission in all three classes of galaxies . for every galaxy , including the ulirgs , in our sample where extended @xmath22 emission has been detected there is a qualitative physical correspondence in size and shape between the extended soft x - ray and @xmath22 emission .
similarly , regions lacking in extended x - ray emission lack extended @xmath22 emission . because of the large distances of the ulirgs even x - ray observations with _
chandra _ lack the spatial resolution to determine whether the soft x - ray and @xmath22 emission are correlated ( as seen in the edge - on spirals ngc 253 and ngc 3079 , @xcite ) or anti - correlated ( as seen in the dwarf starburst ngc 3077 , @xcite ) .
our point is merely that the hot and warm ionized gaseous phases around these galaxies are physically related . in the superwind model ,
the strong morphological connection between the x - ray and @xmath22 emission is most likely a consequence of the hydrodynamical interaction between the hot outflowing wind and ambient gas in the disk and halo of the galaxy ( see * ? ? ?
* ; * ? ? ?
we now examine how the basic physical properties of the diffuse x - ray emission ( luminosity and size ) scale with the fundamental properties of the starbursts and their host galaxies ( luminosity , mass ) . to do so , we have compiled information about the x - ray emission for different spatial regions in each galaxy type . for reference , for the ulirgs we have previously defined inner and outer regions . for the edge - on starbursts we have three regions , nucleus , disk , and halo @xcite . the dwarf starbursts however have only a single spatial region which includes the entire galaxy .
this variety of spatial regions complicates comparison between the galaxy types .
an additional issue is that some regions also include powerlaw components in the x - ray spectral fits . for simplicity
, we have just added the total luminosities of the thermal component in each region together to get the total luminosity of the hot gas for each galaxy . to obtain an average gas temperature
we have luminosity - averaged the temperatures of the separate spatial regions . in figure [ lxrayvslfir ]
we have plotted the thermal x - ray luminosity versus far infrared luminosity ( including the uv luminosity for the dwarfs ) . as the fir luminosity is proportional to the star formation rate
, we see a correlation between the thermal x - ray emission and the star formation rate . while the good correlation between star formation rate and x - ray luminosity has been widely noted ( e.g. * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) , our result differs from these in that we have isolated the x - ray emission due to hot diffuse gas ( excluding the contribution to the total x - ray luminosity from x - ray binaries and -in some cases- agn ) . on the right side of figure [ lxrayvslfir ]
we also show the ratio of the thermal x - ray luminosity to the far infrared luminosity , so that we could examine the relation more closely . over nearly four orders - of - magnitudes of far infrared luminosity ,
this ratio is roughly constant ( @xmath23 ) .
quantitatively , we find a scatter of 0.38 dex for the x - ray luminosity / fir luminosity ratio .
the rate at which a starburst generates mechanical energy is about 1% of its bolometric luminosity ( e.g. * ? ? ?
figure [ lxrayvslfir ] thus implies that typically only about 1% of the mechanical energy is lost in the form of x - ray emission from hot gas .
this fraction evidently does not depend on the star formation rate itself .
note that in figure [ lxrayvslfir ] we have separately color encoded those ulirgs having agn ( based on their optical emission line properties ) .
see @xcite for details .
apart from the `` nearly - a - ulirg '' ngc6240 ( in which an agn likely contributes to the soft x - ray emission ) , the agn ulirgs have ratios of thermal x - ray to fir luminosities that are similar to the other ulirgs , edge - on starbursts , and dwarf starbursts .
the correlation between fir and thermal x - ray luminosities is not simply an artifact of plotting luminosity vs. luminosity ( e.g. bigger galaxies have more of everything ) . in figure [ div_l_k ]
we have divided both the thermal x - ray and fir luminosities by the k - band luminosity . by renormalizing this way
, we see a clear correlation between the sfr per unit galaxy stellar mass and the thermal x - ray luminosity per unit mass .
the size of the x - ray emitting region is also closely correlated with both the fir and k - band luminosities .
we have plotted the 90% flux enclosed radii vs luminosities in figure [ r90 ] .
a correlation analysis using kendall s @xmath24 rank order correlation coefficient finds the probability of a spurious correlation of @xmath25 for the fir to 90% flux enclosed x - ray radii and @xmath26 for the k - band . therefore both the k - band and fir luminosity
are strongly correlated to the size of the x - ray emitting region .
the k - band relationship is marginally stronger than that in the fir .
this relation was seen in the starburst sample by @xcite .
we conclude that although the host galaxy affects the spatial extent of the x - ray emission , the power of the x - ray emission is determined primarily by the star formation rate .
the spectral degeneracy of the @xmath2 and fe abundances in the vmekal models makes it difficult to determine their absolute abundances .
however , we are able to determine @xmath15 ratios for the majority of the ulirgs and dwarf starbursts ( table [ spectra ] )
. taken as a whole we find clear evidence of enhanced @xmath15 ratios relative to solar abundances . excluding a few measurements with extremely large errors we find a fairly constant ratio of about @xmath27 across the sample .
we find this enrichment even in the outer halos of the ulirgs .
similar results have been previously noticed in other starbursting galaxies ( see * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* and references therein ) .
the enhanced @xmath15 ratios in the diffuse x - ray emission would be expected if the x - ray emitting gas has been significantly enriched by the metals returned by the starburst s supernovae .
it is also possible that the subsolar ratio is caused by significant depletion of fe onto dust grains in the hot gas .
we have previously concluded that star formation rate is correlated with the x - ray luminosity .
this suggests that the gas mass transported by the outflow is determined by the mechanical energy supplied by the supernovae and stellar winds .
however the gas temperature might be affected by other factors . to test this ,
we have plotted the @xmath28 relation for our sample in figure [ kt ] .
the luminosity - weighted mean temperatures are @xmath6 0.3 to 0.7 kev , with no strong correlation between x - ray luminosity and gas temperature .
it does appear that the ulirgs as a class tend to preferentially occupy the high end of this temperature range .
there are reasons to suggest this is not caused by contamination from the nuclear sources in the ulirgs .
first , the halos of the ulirgs also have higher gas temperatures even at large physical distances from the nuclear sources . and second
, the ulirgs with the lowest gas temperature is iras 05189 - 2524 .
as it is the most agn - dominated of the ulirgs , it provides a counterexample to attributing the higher gas temperatures to agn contamination .
also , as the outer halos are well fit by just a thermal model it is also probably not caused by a biased fit due to unresolved point sources in the outer regions of the ulirgs .
one physical difference between the ulirgs and the other starbursts ( apart from simply their larger star formation rates ) , is their larger star formation rates per unit area inside the starburst .
the star formation rate per unit area in starbursts is known to be a key parameter in determining the observed properties of their superwinds @xcite . as shown by @xcite the luminosity - weighted dust temperature ( as probed by the ratio of iras 60 and 100 @xmath29 m fluxes ) correlates strongly with the star formation rate per unit area in starbursts .
a comparison of the dust temperature and the luminosity averaged temperatures of the x - ray emitting gas appears in figure [ f60o100vkt ] .
kendall s tau statistic shows a rough correlation does exist which a chance of being spurious of less than @xmath30 .
we begin by summarizing our primary conclusions , and then briefly discuss their implications .
our general conclusion is that the properties of the diffuse thermal x - ray emission in starbursts are remarkably homogeneous and follow simple scaling relations over ranges of nearly four orders - of - magnitude in x - ray luminosity and over three orders - of - magnitude in sfr and galaxy mass . more specifically : * the soft x - ray morphology is independent of the star formation rate , and the same morphological relationship between the hot x - ray emitting gas and warm optical line emission is seen in the dwarf starbursts , the edge - on starbursts , and the ulirgs . *
the x - ray luminosity is linearly proportional to the star formation rate ( estimated from the sum of the far - ir and uv luminosity ) .
this simple scaling holds between the star formation rate and x - ray luminosity when both are normalized by the galaxy mass , so it is not simply a matter of big galaxies having more of everything . *
the characteristic surface brightness of the diffuse thermal x - ray emission ( defined as the mean surface brightness interior to the radius enclosing 90% of the flux ) occupies a relatively narrow range and is independent of the star formation rate . *
the characteristic size of the diffuse x - ray emission ( the radius enclosing 90% of the flux ) increases systematically with increasing star formation rate . these radii range from @xmath60.5 to 2 kpc in the dwarf starbursts , to 3 to 10 kpc in the edge - on starbursts , to 5 to 30 kpc in the ulirgs .
however , the correlation of this size is slightly stronger with k - band luminosity ( a proxy for galaxy mass ) than with the star formation rate . *
the emission - weighted mean temperature of the diffuse x - ray is @xmath1 0.25 to 0.75 kev .
there is a tendency for the ulirgs to occupy the high end of this range ( above @xmath60.6 kev ) . *
the ratio of the abundances of the @xmath2-elements to fe in the diffuse gas is several times the solar value , with no dependence on star formation rate .
these results strongly support the idea that the diffuse thermal x - ray emission in starburst galaxies has a common physical origin . in particular , we can briefly discuss the above results in the context of the model of a galactic superwind driven by the collective energy input from supernovae and stellar winds in the starburst .
the dynamical evolution of a starburst - driven outflow has been extensively discussed ( e.g. * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
briefly , the deposition of mechanical energy by supernovae and stellar winds results in an over - pressured cavity of hot gas inside the starburst .
this hot gas will expand , sweep up ambient material and thus develop a bubble - like structure .
if the ambient medium is stratified ( like a disk ) , the superbubble will expand most rapidly in the direction of the vertical pressure gradient .
after the superbubble size reaches several disk vertical scale heights , the expansion will accelerate , and it is believed that raleigh - taylor instabilities will then lead to the fragmentation of the bubble s outer wall ( e.g. * ? ? ?
this allows the hot gas to `` blow out '' of the disk and into the galactic halo in the form of a weakly collimated bipolar outflow .
the strong morphological relationship between the x - ray and optical emission in galactic winds has led to a picture in which both trace the collision of the wind with denser ambient material in the disk and halo of the starburst galaxy .
the simplest picture is one in which the optical line emission is produced by a shock driven into the ambient gas by the wind , and the x - rays arise in the `` reverse shock '' in the wind where its kinetic energy is transformed to thermal energy and its density increases ( e.g. * ? ? ?
* ; * ? ? ?
this picture is consistent with the results summarized above .
the proportionality of the x - ray luminosity and star formation rate simply means that a roughly constant fraction ( @xmath61% ) of the kinetic energy supplied by the starburst is converted into soft x - ray emission in the wind / cloud shocks .
this is consistent with the simplest relevant theoretical model of a galactic wind , namely a spherically symmetric wind - blown bubble in which the ratio of the x - ray to mechanical energy input is given by @xmath31 @xcite . here
@xmath32 is the rate of mechanical energy injection in units of @xmath33 erg / s ( characteristic of edge - on starbursts ) , @xmath34 is the ambient particle density ( @xmath35 ) , and @xmath36 is the age of the bubble in units of @xmath37 years .
the lack of a strong dependence of temperature on star formation rate implies that the wind s dynamics are relatively insensitive to the star formation rate .
note that @xmath38 0.25 to 0.75 kev corresponds to shock speeds of @xmath6 450 to 800 km / s .
if these speeds are identified with that of the wind fluid , then that implies that the wind is strongly `` mass - loaded '' . this is consistent with the modest enhancements seen in the abundance ratio of the @xmath2-elements relative to iron , since this ratio would be boosted by the contribution from the core - collapse supernovae that drive the wind ( e.g. *
* ) . finally , since the x - ray emission traces the interaction between the wind and the halo / ism , its size should be related both the size / mass of the galaxy and to the star formation rate ( a higher star formation rate leads to a wind with more kinetic energy which can plough its way further into the gaseous halo of the starburst galaxy ) . in summary , we believe that not only are our the results consistent with the expectations for galactic superwinds , they also provide an important benchmark against which increasingly detailed and more physically realistic hydrodynamically simulations of galactic superwinds can be judged . lccccccccc + & & inner & outer + arp 220 & & @xmath39 & @xmath40 & & & @xmath41 & 0.92 & @xmath42 + iras 05189 - 2524 & & @xmath43 & & & & @xmath44 & 0.82 & @xmath45 + iras 17208 - 0014 & & @xmath46 & @xmath47 & & & @xmath48 & 0.89 & @xmath49 + iras 20551 - 4250 & & @xmath50 & @xmath51 & & & @xmath52 & 1.28 & @xmath53 + iras 23128 - 5919 & & @xmath50 & @xmath50 & & & @xmath54 & 0.98 & @xmath55 + mkn 231 & & @xmath56 & @xmath57 & & & @xmath58 & 1.06 & @xmath59 + mkn 273 & & @xmath60 & @xmath61 & & & @xmath62 & 1.02 & @xmath63 + ngc 6240 & & @xmath61 & @xmath64 & & & @xmath65 & 0.57 & @xmath66 + ugc05101 & & @xmath67 & @xmath68 & & & @xmath69 & 1.11 & @xmath66 + + & & nucleus & disk & halo + ngc 253 & & @xmath70 & @xmath71 & @xmath72 & & @xmath73 & 0.5 & @xmath74 + ngc 1482 & & @xmath75 & @xmath76 & @xmath75 & & @xmath77 & 0.7 & @xmath78 + m 82 & & @xmath76 & @xmath76 & @xmath79 & & @xmath80 & 0.97 & @xmath81 + ngc 3079 & & @xmath82 & @xmath83 & @xmath84 & & @xmath77 & 0.49 & @xmath85 + ngc 3628 & & @xmath86 & @xmath87 & @xmath88 & & @xmath89 & 0.48 & @xmath90 + ngc 4631 & & @xmath91 & @xmath92 & @xmath93 & & @xmath94 & 0.4 & @xmath95 + + & & all + he 2 - 10 & & @xmath96 & & & & @xmath97 & 0.91 & @xmath98 + ngc 1705 & & @xmath99 & & & & @xmath100 & 0.38 & @xmath101 + ngc 1569 & & @xmath102 & & & & @xmath103 & 0.96 & @xmath104 + ngc 3077 & & @xmath99 & & & & @xmath105 & 0.54 & @xmath100 + ngc 4214 & & @xmath106 & & & & @xmath107 & 0.62 & @xmath108 + ngc 4449 & & @xmath109 & & & & @xmath110 & 0.49 & @xmath111 + ngc 5253 & & @xmath112 & & & & @xmath113 & 0.97 & @xmath114 lrllllll + + iras 05189 - 2524 & 05 21 01.5 & -25 21 45 & 200 & 0.82 & 10/30/2001 & 5.3 & 1.93 + & & & & & 01/30/2002 & 13.1 + ugc 05101 & 09 35 51.6 & + 61 21 11 & 185 & 0.76 & 05/28/2001 & 46.4 & 2.68 + mkn 231 & 12 56 14.2 & + 56 52 25 & 200 & 0.82 & 10/19/2000 & 25.3 & 1.25 + mkn 273 & 13 44 42.1 & + 55 53 13 & 176 & 0.74 & 04/19/2000 & 38.8 & 1.09
+ arp 220 & 15 34 57.1 & + 23 30 11 & 84 & 0.36 & 06/24/2000 & 52.8 & 4.29 + ngc 6240 & 16 52 58.9 & + 02 24 03 & 113 & 0.49 & 07/29/2001 & 35.6 & 5.69 + iras 17208 - 0014 & 17 23 21.9 & -00 17 00 & 200 & 0.83 & 10/24/2001 & 43.8 & 9.96 + iras 20551 - 4250 & 20 58 26.9 & -42 39 00 & 200 & 0.83 & 10/31/2001 & 32.2 & 3.82 + iras 23128 - 5919 & 23 15 47.0 & -59 03 17 & 210 & 0.86 & 09/30/2001 & 26.8 & 2.74 + + + m 82 & 09 55 55.1 & + 69 40 47 & 3.6 & 0.018 & 6/18/2002 & 18.0 & 4.0 + ngc 1482 & 03 54 39.9 & -20 30 42 & 22 & 0.107 & 2/5/2002 & 23.5 & 3.7 + ngc 253 & 00 47 21.3 & -25 13 00 & 2.6 & 0.013 & 12/27/1999 & 38.3 & 1.4 + ngc 3628 & 11 20 18.4 & + 13 35 54 & 10 & 0.049 & 12/12/2000 & 54.7 & 2.2 + ngc 3079 & 10 01 53.5 & + 55 40 42 & 17 & 0.083 & 3/7/2001 & 26.5 & 0.8 + ngc 4631 & 12 41 56.9 & + 32 37 34 & 7.5 & 0.036 & 4/16/2000 & 55.7 & 1.3 + + + he 2 - 10 & 08 36 15.2 & -26 24 34 & 9 & 0.044 & 3/23/2001 & 17.6 & 9.7 + ngc 1569 & 04 30 49.0 & + 64 50 54 & 2.2 & 0.010 & 4/11/2000 & 78.7 & 22.6 + ngc 1705 & 04 54 13.7 & -53 21 41 & 5.1 & 0.025 & 9/12/2003 & 32.1 & 3.85 + ngc 3077 & 10 03 21.0 & + 68 44 02 & 3.6 & 0.017 & 3/08/2001 & 45.5 & 3.9 + ngc 4214 & 12 15 38.7 & + 36 19 42 & 2.9 & 0.014 & 10/16/2001 & 26.4 & 1.5 + ngc 4449 & 12 28 12.0 & + 44 05 41 & 2.9 & 0.014 & 2/05/2001 & 21.2 & 1.5 + ngc 5253 & 13 39 56.0 & -31 38 24 & 3.2 & 0.015 & 1/14/2001 & 39.9 & 3.9 lrrccc arp 220 & + 15:34:57.254 & + 23:30:11.56 & 7.72 & 5.23 & 9.95 + iras 05189 - 2524 & + 05:21:01.393 & -25:21:45.37 & 3.52 & 3.52 & 0 + iras 17208 - 0014 & + 17:23:21.984 & -00:17:00.38 & 8.46 & 4.55 & 118.28 + iras
20551 - 4250 & + 20:58:26.778 & -42:39:00.19 & 5.16 & 5.16 & 0 + iras 23128 - 5919 & + 23:15:46.725 & -59:03:15.50 & 3.42 & 3.42 & 0 + mkn 231 & + 12:56:14.206 & + 56:52:25.33 & 3.34 & 3.34 & 0
+ mkn 273 & + 13:44:42.054 & + 55:53:12.73 & 4.75 & 3.74 & 319.58 + ngc 6240 & + 16:52:58.897 & + 02:24:03.62 & 10.8 & 7.67 & 103.14 + ugc 05101 & + 09:35:51.602 & + 61:21:11.98 & 4.28 & 4.28 & 0 lccccccccccc arp 220 & 0.39 & 3.90 & 8.32 & 11.9 & 13.6 & 2.56e-08 & 1.12e-08 & 7.17e-09 & 5.88e-09 & @xmath115 & 1.99 + iras 05189 - 2524 & 0.89 & 0.17 & 1.13 & 1.71 & 3.48 & 8.07e-07 & 8.07e-07 & 8.07e-07 & 5.19e-07 & @xmath116 & 672 + iras 17208 - 0014 & 0.89 & 2.84 & 5.40 & 8.09 & 9.10 & 4.08e-08 & 2.39e-08 & 1.40e-08 & 1.17e-08 & @xmath117 & 12.0 + iras 20551 - 4250 & 0.89 & 1.32 & 4.35 & 7.72 & 10.8 & 1.38e-07 & 9.64e-08 & 4.52e-08 & 2.59e-08 & @xmath118 & 15.2 + iras 23128 - 5919 & 0.93 & 2.11 & 3.72 & 6.72 & 8.26 & 1.49e-07 & 1.03e-07 & 5.12e-08 & 2.72e-08 & @xmath119 & 24.1 + mkn 231 & 0.89 & 3.63 & 16.0 & 28.4 & 35.9 & 8.97e-08 & 2.07e-08 & 8.87e-09 & 6.11e-09 & @xmath120 & 1.97 + mkn 273 & 0.79 & 3.12 & 19.8 & 31.3 & 39.9 & 1.31e-07 & 1.79e-08 & 9.44e-09 & 6.36e-09 & @xmath121 & 1.59 + ngc 6240 & 0.52 & 4.71 & 14.3 & 25.1 & 32.2 & 2.72e-07 & 6.07e-08 & 2.60e-08 & 1.71e-08 & @xmath122 & 6.56 + ugc05101 & 0.83 & 0.93 & 3.24 & 6.20 & 8.75 & 5.95e-08 & 4.81e-08 & 1.97e-08 & 1.34e-08 & @xmath117 & 8.28 + + m 82 acis - s & .0175 & 0.989 & 1.78 & 2.78 & 3.55 & 9.82e-08 & 6.00e-08 & 3.53e-08 & 2.53e-08 & @xmath123 & 24.6 + ngc 1482 & .1071 & 0.927 & 1.74 & 3.02 & 3.69 & 1.45e-07 & 7.96e-08 & 3.56e-08 & 2.60e-08 & @xmath124 & 12.8 + ngc 253 acis - s & .0126 & 0.822 & 1.88 & 2.99 & 3.70 & 3.53e-08 & 1.50e-08 & 8.82e-09 & 7.13e-09 & @xmath125 & 1.94 + ngc 3079 & .0829 & 2.71 & 6.05 & 8.50 & 10.0 & 2.20e-08 & 8.66e-09 & 5.68e-09 & 4.50e-09 & @xmath126 & 1.42 + ngc 3628 & .0485 & 3.38 & 4.60 & 5.47 & 5.90 & 2.81e-09 & 2.39e-09 & 2.10e-09 & 1.92e-09 & @xmath127 & 0.57 + ngc 4631 & .0364 & 3.55 & 5.64 & 6.96 & 7.56 & 4.40e-09 & 2.92e-09 & 2.39e-09 & 2.16e-09 & @xmath128 & 0.51 + + he 2 - 10 & 0.0435 & 0.182 & 0.490 & 0.759 & 0.906 & 4.02e-07 & 1.13e-07 & 6.00e-08 & 4.52e-08 & @xmath129 & 16.3 + ngc 1569 & 0.0107 & 0.297 & 0.509 & 0.680 & 0.739 & 1.08e-08 & 5.84e-09 & 4.03e-09 & 3.62e-09 & @xmath130 & 1.30 + ngc 1705 & 0.0247 & 0.538 & 0.870 & 1.01 & 1.06 & 1.16e-09 & 7.47e-10 & 6.90e-10 & 6.71e-10 & @xmath131 & 0.36 + ngc 3077 & 0.0174 & 0.231 & 0.384 & 0.595 & 0.701 & 1.37e-08 & 8.26e-09 & 4.42e-09 & 3.43e-09 & @xmath132 & 1.10 + ngc 4214 & 0.0142 & 0.271 & 0.549 & 0.735 & 0.841 & 1.43e-08 & 6.57e-09 & 4.74e-09 & 4.08e-09 & @xmath133 & 1.78 + ngc 4449 & 0.0142 & 0.711 & 1.10 & 1.55 & 1.77 & 1.42e-08 & 9.50e-09 & 5.88e-09 & 4.80e-09 & @xmath134 & 1.43 + ngc 5253 & 0.0152 & 0.151 & 0.356 & 0.571 & 0.676 & 4.14e-08 & 1.30e-08 & 6.41e-09 & 4.94e-09 & @xmath135 & 1.46 + lcccrrrrr arp 220 & inner & @xmath136 & @xmath137 & @xmath138 & @xmath139 & @xmath140 & @xmath141 & 18.4/23 + & outer & @xmath142 & @xmath143 & @xmath144 & & & & 39.3/37 + iras
05189 - 2524 & inner & @xmath145 & @xmath146 & & @xmath147 & @xmath148 & @xmath149 & 128/100 + & outer & & & + iras
17208 - 0014 & inner & @xmath150 & @xmath151 & & @xmath152 & @xmath153 & @xmath154 & 5.6/9 + & outer & @xmath155 & @xmath156 & & & & & 1.1/0 + iras 20551 - 4250 & inner & @xmath157 & @xmath158 & @xmath159 & @xmath160 & @xmath161 & @xmath162 & 17.1/12 + & outer & @xmath163 & @xmath164 & @xmath165 & & & & 6.0/4 + iras
23128 - 5919 & inner & @xmath166 & @xmath167 & @xmath168 & @xmath169 & @xmath170 & @xmath171 & 12.5/11 + & outer & @xmath172 & @xmath173 & @xmath174 & & & & 3.3/4 + mkn 231 & inner & @xmath175 & @xmath176 & @xmath177 & @xmath178 & @xmath179 & @xmath180 & 30.1/40 + & outer & @xmath181 & @xmath182 & @xmath183 & & & & 16.4/18 + mkn 273 & inner & @xmath184 & @xmath185 & @xmath186 & @xmath187 & @xmath188 & @xmath189 & 61.3/53 + & outer & @xmath190 & @xmath191 & @xmath192 & & & & 39.5/36 + ngc 6240 & inner & @xmath193 & @xmath194 & @xmath195 & @xmath196 & @xmath197 & @xmath198 & 272/177 + & outer & @xmath199 & @xmath200 & @xmath201 & & & & 99.4/84 + ugc 05101 & inner & @xmath202 & @xmath203 & @xmath204 & @xmath205 & @xmath206 & @xmath207 & 10.2/9 + & outer & @xmath208 & @xmath209 & & & & & 0.3/0 + + + he 2 - 10 & all & @xmath210 & @xmath211 & @xmath212 & @xmath213 & @xmath214 & @xmath215 & 40/41 + ngc 1569 & all & @xmath216 & @xmath217 & @xmath218 & & & & 111/83 + ngc 1705 & all & @xmath219 & @xmath220 & & & & & 9.2/5 + ngc 3077 & all & @xmath221 & @xmath222 & @xmath223 & & & & 31/20 + ngc 4214 & all & @xmath224 & & & & & & + ngc 4449 & all & @xmath225 & @xmath226 & @xmath227 & @xmath228 & @xmath229 & @xmath230 & 182/133 + ngc 5253 & all & @xmath231 & @xmath232 & @xmath233 & & & & 56/39 ) , the edge - on starburst ngc 3628 ( @xmath234 ) , and an agn ulirg mkn 231 ( @xmath235 ) .
the dimensions in the parenthesis are the physical size of the viewable area for each galaxy .
also , for comparison between the panels , in the middle plot and right panels we have placed a box of size @xmath236 and in the panel on the right we have additionally placed a box of size @xmath234 .
there are strong morphological similarities between the galaxy types even though they span almost a factor of 50 in physical scales and nearly four orders of magnitude in star formation rate .
[ 3types],width=576 ] and adaptively smoothed representative color x - ray images ( 0.3 - 1 kev red , 1 - 2 kev green , 2 - 8 kev blue ) ; these images show a morphological relationship between the hot gas probed by the diffuse x - ray emission and the warm gas probed by the @xmath22 emission .
[ ha],width=576 ] hoopes , c. g. , heckman , t. m. , strickland , d. k. , seibert , m. , madore , b. f. , rich , r. m. , bianchi , l. , gil de paz , a. , burgarella , d. , thilker , d. a. , friedman , p. g. , barlow , t. a. , byun , y. , donas , j. , forster , k. , jelinsky , p. .a . ,
lee , y. , malina , r. f. , martin , c. , milliard , b. , morrissey , p. f. , neff , s. g. , schiminovich , d. , siegmund , o. h. w. , small , t. , szalay , a. , welsh , b. y. , wyder , t. k. 2004 , , in press
|
we have analyzed observations with the chandra x - ray observatory of the diffuse emission by hot gas in 7 dwarf starburst galaxies , 6 edge - on starburst galaxies , and 9 ultra luminous infrared galaxies .
these systems cover ranges of @xmath0 in x - ray luminosity and several thousand in star formation rate and k - band luminosity ( a proxy for stellar mass ) . despite this range in fundamental parameters , we find that the properties of the diffuse x - ray emission are very similar in all three classes of starburst galaxies .
the spectrum of the diffuse emission is well fit by thermal emission from gas with @xmath1 0.25 to 0.8 kev and with several - times - solar abundance ratios of @xmath2 elements to fe .
the ratio of the thermal x - ray to far - infrared luminosity is roughly constant , as is the characteristic surface brightness of the diffuse x - ray emission .
the size of the diffuse x - ray source increases systematically with both far - infrared and k - band luminosity .
all three classes show strong morphological relationships between the regions of hot gas probed by the diffuse x - ray emission and the warm gas probed by optical line emission .
these findings suggest that the same physical mechanism is producing the diffuse x - ray emission in the three types of starbursts , and are consistent with that mechanism being shocks driven by a galactic `` superwind '' powered by the kinetic energy collectively supplied by stellar winds and supernovae in the starburst .
|
magnetars , comprising soft gamma repeaters ( sgrs ) and anomalous x - ray pulsars ( axps ) , have been brought to great attention because they are likely to have super strong surface magnetic fields reaching @xmath8 g @xcite .
this exceeds the critical field strength @xmath9 g , where @xmath10 , @xmath11 , @xmath12 , and @xmath13 are the electron mass , the light velocity , the electron charge and the reduced planck constant , respectively . to understand radiation processes in such an environment , it is necessary to fully take into account non - perturbative effects in quantum electrodynamics .
outstanding properties of magnetars include burst activity , observed from all sgrs and some axps .
a typical `` short burst '' has a duration of @xmath1100 ms , and a 2100 kev energy release by 10@xmath14 erg ( e.g. , @xcite ) . among a variety of burst activities ,
the most energetic ones are the giant flares , which were so far detected from sgr0526@xmath1566 ( e.g. , @xcite ) , sgr1900@xmath014 ( e.g. , @xcite ) , and sgr1806@xmath1520 ( e.g. , @xcite ) .
x - ray spectra of short bursts provide useful diagnostics of their emission mechanism .
wide - band spectra of short bursts from sgr1806@xmath1520 and sgr1900@xmath014 , detected by high energy transient explorer 2 ( hete-2 ; @xcite ) , are generally described by a photoelectrically absorbed two - blackbody ( 2bb ) model @xcite , even though this could be a phenomenological description .
spectra of bursts @xcite and intermidiate flares @xcite from sgr 1900@xmath014 also favor the 2bb modeling .
in addition , bursts from the new magnetar sgr0501@xmath04516 , detected by suzaku @xcite and swift @xcite , also exhibited 2bb - type spectra ( @xcite , hereafter paperi ; @xcite ) . in terms of this modeling
, these short bursts all exhibit an interesting scaling as @xmath16 @xcite , where @xmath17 and @xmath18 are the higher and lower temperatures of the 2bb model , respectively .
it has long been known that sgrs and axps show not only burst activities but also persistent emission in energies below @xmath110 kev , of which the spectra are generally reproduced by two phenomenological models ; 2bb ( e.g. , @xcite ) or a photoelectrically absorbed blackbody plus power law model ( bb@xmath0pl ; e.g. , @xcite ) .
moreover , recent studies of sgr0501@xmath04516 and other objects propose a `` blackbody plus comptonized blackbody '' model ( @xcite , hereafter paperii ) and a resonant cyclotron scattering model @xcite as alternative possibilities .
although the spectral modeling is thus ambiguous , the persistent x - ray emission interestingly shows the same @xmath16 relation as those of the bursts @xcite if we employ the 2bb representation .
this suggests a common radiation mechanism between the bursts and persistent emission , further leading to a possibility that the persistent x - ray emission may consist of numerous micro bursts @xcite .
recent studies using integral ( e.g. , @xcite ) and suzaku ( @xcite ; paperii ; @xcite ) revealed an extremely hard x - ray component above @xmath110 kev in persistent emission spectra of a significant fraction of sgrs ( including sgr0501@xmath04516 : paperii ) and axps . the hard x - ray component , which is thought to be distinct from the blackbody - like soft component , can be reproduced by a power law ( pl ) model with an extremely hard photon index of @xmath19 .
as reported in paperii for sgr0501@xmath04516 and in @xcite for some other sources , the 2100 kev luminosity of the hard x - ray component is often comparable to that of the soft blackbody component . considering these properties , as well as a clear dependence of the hardness ratio between the hard and soft luminosities on the characteristic age
as revealed with suzaku @xcite , the hard x - ray component is expected to provide an important clue to the nature of magnetars . theoretically , the hard x - ray production mechanism is extensively discussed @xcite , but it is not yet conclusive . if there is a common radiation mechanism between the bursts and persistent emissions , the hard x - ray component may also be seen in burst spectra . however , short bursts of magnetars so far studied , with fluence @xmath20 erg @xmath3 , generally have @xmath21715 kev ( e.g. , @xcite ) . as a result
, their 2bb spectra , extending well up to @xmath22 kev , would mask any hard x - ray component .
this raises a possibility that bursts with considerably lower fluence , which have remained not much studied , may have lower values of @xmath17 , e.g. , close to those found in the persistent emission ( e.g. , @xmath210.43.9 kev ; @xcite ; paperii ) , and would allow more sensitive searches for the hard - tail component . considering this
, we focus on wide - band spectroscopy of low - fluence bursts .
observations with suzaku are suitable for this purpose , because of its high sensitive over a broad energy band , realized by the x - ray imaging spectrometer ( xis ; 0.212 kev ; @xcite ) and the hard x - ray detector ( hxd ; 10600 kev ; @xcite ) .
we have hence revisited the suzaku data of sgr0501@xmath04516 , acquired during its 2008 august activity . as a third publication ( after paperi and paperii ) from this observation , the present paper reports on our successful detection of a hard component , in an hxd spectrum which sums over 31 short bursts from this new magnetar .
the new soft gamma repeater sgr0501@xmath04516 was discovered on 2008 august 22 by the burst alert telescope on - board swift , when it displayed sgr - like burst activity @xcite .
soon after the discovery , a spin period of @xmath23 was reported based on an observation by the rossi x - ray timing explorer @xcite .
as described in paperi and paperii , we triggered a suzaku target - of - opportunity ( too ) observation , which started at 00:05 on 2008 august 26 and ended at 08:25 on 2008 august 27 ( ut ) .
the xis was operated with a 1/4 window option which yields a 2 s time resolution , while the hxd was operated in the standard mode .
the acquired data were already utilized in paperi and paperii ; the former described a strong short burst and persistent soft x - ray emission , while the latter focused on the detection of a hard component in the persistent emission .
the present paper , utilizing the same too data , deals with broad - band spectra of 31 smaller short bursts .
the distance to sgr0501@xmath04516 , though estimated to be 1.5 kpc based on its directional proximity to the young supernova remnant hb9 @xcite , is actually very uncertain . in this paper ,
the distance is hence assumed to be 4 kpc , which is similar to the value of @xmath15 kpc employed by @xcite .
the reduction of the xis and hxd event data ( v2.2 ) were made using heasoft6.6.1 software .
the latest calibration database ( caldb:20090402 ) was applied to unfiltered xis event data using _ xispi _ ( v2008 - 04 - 10 ) .
then , using _ xselect _ ( v2.4a ) , we extracted a new set of filtered xis events with the standard criteria and a grade selection `` grade = ( 0,2 - 4,6 ) '' .
after that , hot and flickering pixels were removed using _ cleansis _ ( v1.7 ) .
telemetry - saturated time intervals , estimated by _ xisgtigen _ ( v2007 - 05 - 14 ) , were removed from the xis data using _
xselect_. we created light curves and spectra from the cleaned xis event data using _
xselect_. response matrix files were generated by _
( v2007 - 05 - 14 ) , and ancillary response function files by _ xissimarfgen _ ( v2008 - 04 - 05 ) .
the obtained net exposure is @xmath160 ks . using _ hxdpi _ and _ hxdgrade _ ( v2008 - 03 - 03 )
, we applied the latest calibration database ( caldb:20090902 ) to the unfiltered hxd event data . cleaned pin and gso events
were extracted from these newly calibrated data with the standard criteria using _
xselect_. again , we created light curves and spectra using _
xselect_. dead time corrections were applied to the spectra using _
hxddtcor_. response matrix files of version 2008 - 01 - 29 were used .
this yielded a net exposure of @xmath24 ks for the hxd data .
as shown in figure1 of paperi , a number of visually obvious bursts are found in a 0.212 kev xis light curve with 2-s resolution , obtained by summing the data from the two fi sensors ( xis0 and xis3 ) and the one bi sensor ( xis1 ) .
at least three of them , including the strongest one analyzed in paperi , were also noticed in the 1020 kev hxd - pin light curve with a 500 ms time resolution .
following our preliminary attempt in paperi , we conducted a quantitative burst search using the 0.212 kev light curve of the xis .
after visually eliminating 8 obvious bursts which have @xmath25 cts(2s)@xmath26 , the light curve was converted to a count - rate ( per 2s ) histogram as shown in figure [ fig : xis_hist ] ; this includes the background , the persistent signal emission , and short bursts .
the histogram has an average of @xmath27 and a standard deviation of @xmath28 , both in units of cts(2s)@xmath26 , where the quoted errors refer to 90% confidence levels , and can be approximated by a poissonian distribution .
we searched the xis count - rate histogram for those 2-s bins where the count rate exceeds @xmath29 cts(2s)@xmath26 .
this selection has yielded 35 time bins with significant excess counts . regarding a set of consecutive such bins as representing a single burst
, we thus detected 32 short bursts altogether . among them ,
the strongest one was already analyzed in paperi .
below , we therefore analyze the remaining 31 bursts , which are summarized in table [ tab : burst_summary ] .
they are hereafter identified sequentially as # 01 , # 02 , @xmath30 , and # 31 .
these 31 short bursts are considered to be free from event pile - up effects in the xis , because their count rates were less than 107 cts(2s)@xmath26xis@xmath26 above which the effect becomes significant .
light curves of typical short bursts ( # 03 , # 13 , # 14 , # 22 and # 23 ) are presented in figure [ fig : burst_lc ] . among them , two ( # 03 and # 14 ) are accompanied by significant emissions in the hxd - pin and/or hxd - gso energy bands .
since the present paper puts its focus on burst spectra , we must subtract the persistent emission of sgr0501@xmath04516 , as well as the non x - ray background and the cosmic x - ray background .
for each burst , we therefore accumulate the xis and hxd data over a time region ( see below ) containing the burst , and subtract the corresponding background spectra which are acquired before and after the burst period .
the on - burst and background data of the xis were both extracted from box regions with sizes of ( detx , dety ) = ( @xmath31 , @xmath32 ) for xis1 , and ( @xmath32 , @xmath31 ) for xis0 and xis3 , where detx and dety are detector coordinates .
each on - burst spectrum was made using a 2-s or 6-s time interval , depending on the burst duration in the xis .
the corresponding background spectra were extracted from two 10-s time intervals , one before and the other after the burst , both separated by 2 s from the on - burst time region . the result does not change if we instead employ 15 s for the background time intervals .
therefore , the pulsed persistent emission with a period of @xmath33 s ( paperi ) does not affect the background spectrum .
if the background time intervals contained other bursts , they were eliminated from the background spectra .
figure [ fig : lc ] shows 6-band synthetic light curves summed over the 31 short bursts , obtained by stacking their individual light curves in reference to the 0.212 kev xis data .
thus , the burst emission is clearly seen in the hxd - pin data up to 40 kev , and possibly in the 50100 kev hxd - gso band .
average on - burst and background count - rates of the xis are @xmath153 cts(2s)@xmath26 and @xmath114 cts(2s)@xmath26 , respectively .
since the xis have a time resolution of 2 s , the burst profiles in the hxd energy bands in figure [ fig : lc ] must be considerably smeared out .
figure [ fig : fgbg_comp ] shows on - burst ( _ green _ ) , background ( _ red _ ) , and background subtracted ( _ black _ ) spectra of the xis and the hxd , summed over the 31 bursts . in agreement with figure [ fig : lc ] ,
the burst emission is significantly detected with hxd - pin up to @xmath140 kev .
also , the burst signal may be detected marginally in the 50100 kev gso data . before quantifying the burst spectrum ,
let us compare it with that of the persistent emission of sgr0501@xmath04516 . to do this in a model - independent manner
, we directly divided the summed burst spectrum to the background - subtracted persistent - component spectrum derived in paperii .
the ratio in the xis range was estimated using the two fi sensors .
the results , presented in figure [ fig : spc_ratio ] , indicate that the burst spectrum is clearly harder than the persistent emission spectrum .
in addition , the ratio in the hxd range is approximately flat , implying that in this energy range the burst and persistent emission have approximately the same spectral shape . using xspec 12.5.0 @xcite , we fitted the summed burst spectrum with a photoelectrically absorbed 2bb model , which has been most successful on the short bursts from sgr1806@xmath1520 and sgr1900@xmath014 ( section [ intro ] ) .
the photoelectric absorption was fixed to @xmath34 @xmath3 , as estimated from the persistent x - ray emission observed by the xis ( paperi ) . according to a cross - calibration between the xis and the hxd described in @xcite , the hxd normalization is typically 13% higher than that of the xis for crab data acquired at the xis nominal position .
therefore the relative normalization of the hxd above the xis was fixed to 1.13 .
as shown in figure [ fig : spc ] ( b ) , this 2bb fits leaves significant positive residuals in energies above @xmath120 kev , which makes the fit unacceptable with @xmath35 .
although the 2bb fit could be improved to @xmath36 by allowing to vary the hxd vs. xis relative normalizaiton , the obtained normalization ratio , @xmath37 , is far outside the value of @xmath38 obtained from crab observations @xcite , making the fit unrealistic .
conversely , the 2bb fit did not improve significantly if the hxd normalization is kept within this uncertainty range .
given figure [ fig : spc_ratio ] , as well as the failure of the 2bb model , we fitted the burst spectrum with a 2bb plus power law model ( 2bb@xmath0pl ) .
the fit was then improved to @xmath39 .
the pl component is considered to be significant , because an f - test indicates a probability of @xmath10.1% for the fit improvement ( by adding pl ) to arise by chance .
the best - fit spectral parameters are summarized in table [ tab : spc_summary ] , and the @xmath40 form of the 2bb@xmath0pl fit is given in figure [ fig : spc ] ( d ) .
as already expected from figure [ fig : spc_ratio ] , the power - law component indeed exhibits a photon index of @xmath19 , which is comparable to that of the hard x - ray component of the persistent emission ( e.g. , paperii ) .
consequently , we conclude that the summed short burst spectrum has a hard - tail component , which has never been detected in the burst spectra of any other magnetar . using the best - fit 2bb@xmath0pl spectral parameters , a bolometric fluence of the 2bb component and a 240 kev fluence of the pl component
are calculated as shown in table [ tab : spc_summary ] .
those fluences refer to average values of the 31 short bursts , and are lower by two orders of magnitude than a typical 2100 kev fluence of @xmath41 erg@xmath3 for short bursts from sgr1806@xmath1520 and sgr1900@xmath014 ( e.g. , @xcite ) studied so far .
thus , the high sensitivity of suzaku allowed us to study , for the first time , the wide - band properties of these low - fluence bursts . assuming the effective duration of the 31 short bursts from sgr0501@xmath04516 to be 0.1 s , which is a typical value for this type of events ( e.g. , @xcite ) , we calculated the flux ( luminosity ) and the blackbody radii , all averaged over the 31 short bursts , and show the results in table [ tab : spc_summary ] .
the effective emission radii of the two bb components , @xmath114 km and @xmath11.9 km ( assuming a distance of 4 kpc ) , are comparable to typical values found in short bursts from sgr1806@xmath1520 and sgr1900@xmath014 @xcite , and from sgr0501@xmath04516 @xcite , although the distance uncertainty remains .
using the suzaku too observation of sgr0501@xmath04516 conducted in 2008 august , we studied relatively dim 31 short bursts from this new magnetar .
their average fluence , @xmath2 erg@xmath3 in 240 kev , is 12 orders of magnitude lower than those of typical short burst studied so far ( e.g. , @xcite ) . following the detection of a hard component from the persistent emission of sgr0501@xmath04516 ( paperii ; @xcite ) , the data have allowed a clear detection of a similar hard - tail component in the spectrum summed over the 31 short bursts .
these results for the first time reveal spectral properties of such dim bursts , and provide a new clue to the formation mechanisms of persistent and burst emissions from magnetars .
as already reported in paperi , the spectrum of the strongest burst ( actually its precursor ) from sgr0501@xmath04516 was well reproduced by a 2bb model , without indication of an additional hard x - ray component . however
, this could be due to the effect mentioned in section [ intro ] , namely , obscuration by the high 2bb temperatures ( @xmath42 kev ) ; the data of this strong precursor are worth searching for a hard tail component .
therefore , we re - analyzed the same pile - up and dead - time corrected spectrum of the precursor as studied in paperi , using the 2bb@xmath0pl model .
the photon index was fixed to @xmath43 to emulate the results obtained in subsection [ sec : spc_ana ] , and the photoelectric absorption was again fixed to @xmath34 @xmath3 after paperi .
the fit resulted in @xmath4/d.o.f .
= 40.1/37 = 1.09 , which is no better than the value of 41.2/38 = 1.08 using the 2bb model .
therefore , the data do not require any excess hard - tail component with @xmath44 .
the best - fit spectral parameters are summarized in table [ tab : spc_summary ] , which are consistent with the results in paperi after renormalizing to the distance of 4 kpc and the duration of 0.2 s. there , the 2100 kev flux of the pl component is given as a 90% upper limit .
given the gross spectral similarity between the short bursts and the persistent emission of sgr0501@xmath04516 ( subsection [ sec : spc_ana ] ) , let us perform more quantitative comparison among their soft components , referring to figure [ fig : comp_kt_r ] which summarizes three sets of 2bb parameters of sgr0501@xmath04516 ; the persistent emission , the 31 short bursts , and the precursor of the strongest burst .
there , we find three properties that are common to all the three spectra .
one is that the cooler and hotter blackbodies have comparable luminosities , and another is that @xmath17 is @xmath45 times higher than @xmath18 .
these 2bb properties are considered rather intrinsic to magnetars , because they also apply to more energetic ( typically by an order of magnitude in bolometric fluence ) bursts from sgr1806@xmath1520 and sgr1900@xmath014 observed with hete-2 ( figure 5 of @xcite ) , sgr1900@xmath014 observed with swift @xcite , and sgr0501@xmath04516 observed with swift @xcite , as well as to persistent emission from some other magnetars ( @xcite and references therein ) . the remaining property found in figure [ fig : comp_kt_r ]
is that the two temperatures increase with the luminosity .
in fact , the temperature of the 31 short bursts are by a factor of 49 lower than those of the typical bursts ( e.g. , @xcite ) .
this justifies a posteriori our conjecture made in section [ intro ] , i.e. , a positive temperature - luminosity correlation of the 2bb component .
incidentally , the ratio increase in figure [ fig : spc_ratio ] , from a few kev to @xmath46 kev , is at least partially due to the higher 2bb temperatures of the dim bursts than those of the persistent emission .
in contrast to the present results , some published results ( e.g. , @xcite ) suggest a weak negative correlation between the 2bb temperatures and the burst fluence . however , these results are usually limited to rather strong bursts with fluence @xmath20 erg @xmath3 .
then , the temperature vs. fluence might change at about this fluence .
alternatively , weaker bursts in these studies may have actually contained hard - tail components , and hence their spectra appeared rather hard .
in addition to these similarities in the soft component , the presence of a distinct pl - shaped hard component , found in the present work , provides a novel resemblance between the 31 short bursts and the persistent emission . moreover ,
the photon index of the former , @xmath47 , is consistent with the latter , @xmath48 ( paperii ) .
however , as already visualized by figure [ fig : spc_ratio ] , the two phenomena can differ in their ratios between the 240 kev hard - component luminosity @xmath49 and the bolometric soft - component luminosity @xmath50 , even excluding the effect caused by different bb temperatures .
we observed @xmath51 from the 31 short bursts , which is possibly higher than that of @xmath52 for the persistent emission at 2.6@xmath7 level . here ,
quoted errors are 68% confidence levels for the ratios , and 90% confidence levels for the photon indices . in order to visualize the wide - band spectral hardness , we compare in figure [ fig : lbb_lpl ] the relations between @xmath53 and @xmath54 .
there , @xmath49 was calculated again in the 240 kev range . even though the @xmath55 ratio could vary to some extent
, a fact of basic importance is that @xmath49 and @xmath50 increases , by about 2 orders of magnitude in an approximate proportion , from the persistent emission to the dim short bursts .
this , together with the spectral similarities discussed above , suggests that common emission mechanisms operate between these short bursts and the persistent emission .
this in turn gives a support to our idea that persistent x - ray emission of magnetars may in general consist of numerous micro - bursts @xcite .
in contrast , the upper limit on the @xmath55 ratio obtained for the strongest burst is considerably lower than is extrapolated from the other two less luminous data points , suggesting some changes in the emission mechanism from weaker to stronger short bursts .
arnaud , k. a. 1996 , in astronomical data analysis software and systems v , eds .
g. jacoby and j. barnes , asp conference series , 101 , 17 baring , m. g. , & harding , a. k. 2007 , , 308 , 109 barthelmy , s. d. 2008 , grb coord .
, 8113 duncan , r. , & thompson , c. 1992 , , 392 , l9 enoto , t. 2010 , ph.d .
thesis , university of tokyo enoto , t. 2009 , , 693 , l122 enoto , t. 2010a , , 715 , 665 enoto , t. 2010b , , 62 , 475 enoto , t. 2010c , , 722 , l162 esposito , p. 2007
, , 476 , 321 feroci , m. , calliandro , g. a. , massaro , e. , mereghetti , s. , & woods , p. m. 2004 , , 612 , 408 gaensler , b. m. , 2008 , grb coord .
, 8149 g , e. , woods , p. , & kouveliotou , c. 2008 , grb coord
8118 gehrels , n. , 2004 , , 611 , 1005 heyl , j. s. , & hernquist , l. 2005 , , 362 , 777 hurley , k. , 1999 , , 397 , 41 israel , g. l. , 2008 , , 685 , 1114 kokubun , m. , 2007 , , 59 , 53 koyama , k. , 2007 , , 59 , s23 kuiper , l. , hermsen , w. , & mendez , m. 2004 , , 613 , 1173 kumar , h. s. , ibrahim , a. i. , & safi - harb , s. 2010 , , 716 , 97 leathy , d. a. , & aschenbach , b. 1995 , , 293 , 853 marsden , d. , & white , n. e. 2001 , , 551 , l155 mazets , e. p. , golenetskii , s. v. , & guryan , yu . a. 1979 , soviet astron .
lett . , 5 , 343 mereghetti , s. , esposito , p. , & tiengo , a. 2007 , , 308 , 13 mitsuda , k. , 2007 , , 59 , s1 molkov , s. , hurley , k. , sunyaev , r. , shtykovsky , p. , revnivtsev , m. , & kouveliotou , c. 2005 , , 433 , l13 nakagawa , y. e. , 2007 , , 59 , 653 nakagawa , y. e. , 2008 , grb coord .
circ . , 8265 nakagawa , y. e. , 2009 , , 61 , s387 olive , j. -f .
2004 , , 616 , 1148 paczyski , b. 1992 , acta astron . , 42 , 145 palmer , d. m. , 2005 , , 434 , 1107 rea , n. , zane , s. , turolla , r. , lyutikov , m. , gtz , d. 2008 , , 686 , 1245 rea , n. , 2009 , , 396 , 2419 ricker , g. , 2003 , in gamma - ray bursts and afterglow astronomy , ed .
g. r. ricker & r. vanderspek ( melville : aip ) , 662 , 3 takahashi , t. , 2007 , , 59 , s35 tanaka , y. t. , terasawa , t. , yoshida , m. , horie , t. , hayakawa , m. 2008 , , 113 , a07307 thompson , c. , & beloborodov , a. m. 2005 , , 634 , 565 thompson , c. , & duncan , r. 1995 , , 275 , 255 thompson , c. , & duncan , r. 1996 , , 473 , 322 tiengo , a. , esposito , p. , & mereghetti , s. 2008 , , 680 , 133 llr burst & date ( ut ) & xis counts + # 01 & 2008 - 08 - 26 01:05:44 & 35 + # 02 & 2008 - 08 - 26 01:07:02 & 30 + # 03 & 2008 - 08 - 26 01:23:20 & 309 + # 04 & 2008 - 08 - 26 01:24:08 & 42 + # 05 & 2008 - 08 - 26
01:30:22 & 38 + # 06 & 2008 - 08 - 26 02:43:44 & 34 + # 07 & 2008 - 08 - 26 02:46:02 & 61 + # 08 & 2008 - 08 - 26 02:46:48 & 30 + # 09 & 2008 - 08 - 26 02:50:24 & 32 + # 10 & 2008 - 08 - 26 02:50:38 & 36 + # 11 & 2008 - 08 - 26 03:16:32 & 33 + # 12 & 2008 - 08 - 26 03:16:54 & 37 + # 13 & 2008 - 08 - 26 03:21:00 & 240 + # 14 & 2008 - 08 - 26 03:21:08 & 99 + # 15 & 2008 - 08 - 26 03:21:26 & 30 + # 16 & 2008 - 08 - 26 04:38:10 & 35 + # 17 & 2008 - 08 - 26 08:00:58 & 30 + # 18 & 2008 - 08 - 26 09:25:06 & 31 + # 19 & 2008 - 08 - 26 10:47:32 & 42 + # 20 & 2008 - 08 - 26 12:45:34 & 66 + # 21 & 2008 - 08 - 26 12:45:42 & 50 + # 22 & 2008 - 08 - 26 14:22:50 & 66 + # 23 & 2008 - 08 - 26 14:22:58 & 33 + # 24 & 2008 - 08 - 26 15:27:48 & 30 + # 25 & 2008 - 08 - 26 15:27:56 & 33 + # 26 & 2008 - 08 - 26 15:44:14 & 32 + # 27 & 2008 - 08 - 26 22:26:44 & 30 + # 28 & 2008 - 08 - 27 06:07:02 & 33 + # 29 & 2008 - 08 - 27 07:55:40 & 38 + # 30 & 2008 - 08 - 27 07:55:48 & 31 + # 31 & 2008 - 08 - 27 07:55:56 & 34 + lllllllllll model & @xmath18 & @xmath56 & @xmath17 & @xmath57 & @xmath58 & @xmath59 & @xmath60 & @xmath61 & @xmath62 & @xmath4/d.o.f .
+ & ( kev ) & ( km ) & ( kev ) & ( km ) & & & & & & + + 2bb & @xmath63 & ( @xmath64)@xmath65 & @xmath66 & ( @xmath67)@xmath65 & @xmath30 & @xmath68 & @xmath30 & ( @xmath68)@xmath69 & @xmath30 & 74/50 + 2bb@xmath0pl & @xmath70 & ( @xmath71)@xmath65 & @xmath72 & ( @xmath73)@xmath65 & @xmath74 & @xmath75 & @xmath76 & ( @xmath75)@xmath69 & ( @xmath76)@xmath69 & 56/48 + + 2bb@xmath0pl & @xmath77 & @xmath78 & @xmath79 & @xmath80 & 1.0 ( fixed ) & @xmath81 & @xmath82 & @xmath83 & @xmath84 & 40/36 +
|
using data with the suzaku xis and hxd , spectral studies of short bursts from the soft gamma repeater sgr0501@xmath04516 were performed . in total ,
32 bursts were detected during the @xmath160 ks of observation conducted in the 2008 august activity . excluding the strongest one , the remaining 31 bursts showed an average 240 kev fluence of @xmath2 erg@xmath3 .
a 140 kev spectrum summed over them leaves significant positive residuals in the hxd - pin band with @xmath4/d.o.f .
= 74/50 , when fitted with a two - blackbody function . by adding a power law model ,
the fit became acceptable with @xmath4/d.o.f .
= 56/48 , yielding a photon index of @xmath5 .
this photon index is comparable to @xmath6 @xcite for the persistent emission of the same object obtained with suzaku .
the two - blackbody components showed very similar ratios , both in the temperature and the emission radii , to those comprising the persistent emission .
however , the power - law to two - blackbody flux ratio was possibly higher than that of the persistent emission at 2.6@xmath7 level . based on these measurements , average wide - band properties of these relatively weak bursts are compared with those of the persistent emission .
|
4c+74.26 ( @xmath9 ; @xcite ) is a luminous broad - line radio galaxy ( blrg ) most notable for its large radio lobes , extending 10 arcminutes ( tip - to - tip ) on the sky @xcite .
a one - sided jet has been observed with the vla @xcite and on pc scales with vlbi @xcite .
the flux limit on a counterjet gives a limit to the inclination angle of @xmath10 degrees @xcite , resulting in a physical size for the radio source of @xmath11 mpc ( using a @xmath12 cosmology : @xmath13 km s@xmath14 mpc@xmath14 , @xmath15 , @xmath16 ; @xcite ) , well within the range to be classified as a giant radio galaxy @xcite .
the observed morphology of the radio jets clearly places it as a frii source , although the radio luminosity of 4c+74.26is on the border between fri and frii @xcite .
the bolometric luminosity has been estimated to be @xmath17 ergs s@xmath14 @xcite , indicating that 4c+74.26 is also close to the seyfert - quasar border .
the host galaxy has a size and luminosity typical of other giant ellipticals associated with radio sources @xcite .
optical spectra reveal very broad permitted lines with measurements of the h@xmath18 fwhm ranging from @xmath19 km s@xmath14 @xcite to 11,000 km s@xmath14 @xcite . using this last value , @xcite
employ the broad - line region radius - luminosity relation @xcite to obtain a black hole mass of @xmath20 m@xmath21 .
x - ray observations of 4c+74.26 began with a 23 ks _ asca _ observation in 1996 .
@xcite presented the first analysis of these data , as well as _
the data from the _ rosat _ all - sky survey yielded a very hard photon - index for the 0.32 kev band , @xmath22 , with a cold absorption column slightly above the galactic value of @xmath23 @xmath1 @xcite .
_ rosat _ pspc data of 4c+74.26 was available from a 20 ks observation of the cataclysmic variable vw cep during which the blrg was in the field - of - view .
an absorbed power - law fit to this dataset by @xcite also revealed a hard power - law ( @xmath24 ) with higher than galactic absorption .
a later analysis of this pspc data by @xcite found that a dusty warm absorber model provided a good fit and increased the photon - index to values closer to those found in the _ asca _ data ( @xmath25 ) and in _ rosat _ samples of radio - loud quasars ( e.g. * ? ? ? * ) .
the _ asca _ data has subsequently been re - analyzed by @xcite and @xcite .
all groups found a best fit model that included a power - law with a photon index @xmath25 and cold absorption in excess of the galactic value .
however , @xcite preferred a solution with a warm absorber , while the best fit of @xcite did not require one ( @xmath26 ) .
similarly , @xcite and @xcite find a gaussian fe k@xmath7 line at the 97 per cent confidence level , but @xcite find one about twice as strong and at @xmath27 per cent confidence .
the _ asca _ data also show a hardening at high energies , which @xcite and @xcite model with a reflection continuum , but find very high reflection fractions ( @xmath28 and @xmath29 , respectively ) .
@xcite argue that such a large reflection fraction should yield a much stronger fe k@xmath7 line , and propose a model where the hardening is due to a second , very hard power - law with @xmath30 , possibly arising from the radio jet .
this confusing situation was improved by a 100 ks _ bepposax _ observation presented by @xcite .
no significant variability was detected from 4c+74.26 even in the high - energy pds band .
this fact provides strong evidence that jet emission is not significantly contributing to the hard x - rays .
indeed , @xcite find that compton reflection provides the best fit to their broadband data with @xmath31 .
these authors also find cold absorption in excess of the galactic value , but a warm absorber is not required by the data .
a significant , but unresolved , fe k@xmath7 line was also detected in the _ bepposax _ spectrum .
more recently , very preliminary results from a 70 ks _ chandra _ gratings observation of 4c+74.26 have been presented by @xcite . a highly ionized and weak warm absorber
was detected by _
, including emission and absorption lines from h - like and he - like mg , al and si . moreover , the low number of counts at energies less than 0.8 kev could indicate o vii and o viii edges ( kaspi , private communication ) . here ,
we present the results of a 35 ks _ xmm - newton _ observation of 4c+74.26which will for the first time properly characterize the x - ray spectrum of this source .
the detection of a relativistically broadened fe k@xmath7line has already been reported elsewhere ( @xcite ; hereafter paper i ) .
this paper therefore concentrates on the timing analysis ( sect .
[ sect : timing ] ) and fitting the broadband spectrum ( sect . [ sect : spectral ] ) .
the paper concludes by discussing the results in sect .
[ sect : discuss ] .
we begin in the next section by describing the details of the observation and data reduction .
4c+74.26 was observed by _ xmm - newton _
@xcite for 35 ks during revolution 762 starting at 2004 february 6 13:57:42 .
data were collected using the single pn @xcite and two mos @xcite detectors in the european photon imaging camera ( epic ) system , both reflection grating spectrometers ( rgs ; @xcite ) and the optical monitor ( om ; @xcite ) .
the epic instruments were operated in large - window mode with the medium optical filter in place .
the rgs was operated in standard spectroscopy mode .
data reduction was performed using the _ xmm - newton _ science analysis system ( sas ) v.6.1 .
the analysis chains epchain and emchain were run on the observation data files to produce calibrated event lists for the mos and pn detectors by removing bad pixels and applying both gain and charge transfer inefficiency ( cti ) corrections to the data .
source spectra were extracted using circles of radius 115 arcseconds ( for the pn ) , 119 arcseconds ( mos1 ) and 122 arcseconds ( mos2 ) .
background spectra were extracted from source free areas on the same ccd using circles with radii of 60 arcseconds ( pn and mos1 ) and 50 arcseconds ( mos2 ) .
the extracted pn spectrum included both single and double events , while the mos spectra were comprised of events with all patterns .
the background was negligible during the observation except for a minor enhancement at @xmath32 ks into the integration .
a good - time interval file was constructed as described in 4.4.3 of the _ xmm - newton _ sas user s
guide to remove any potential contamination by background events .
the sas task epatplot was used to check the pattern distributions in the epic data , and both mos spectra were found to suffer from a non - negligible amount of pileup @xcite . to correct this ,
the mos spectra were re - extracted using only single ( i.e. , pattern 0 ) events .
following background subtraction , the final pn spectrum has 28.8 ks of good exposure time and consists of over @xmath33 counts , giving a mean count rate of 8.6 s@xmath14 .
the final background subtracted mos-1 and mos-2 spectra each had mean count rates of 2.2 s@xmath14 , and contained @xmath34 and @xmath35 counts , respectively .
prior to spectral analysis , all data were grouped using grppha to have a minimum of 20 counts per bin .
finally , the sas tasks rmfgen and arfgen were utilized to produce the response matrix and ancillary response files .
we note that the background extraction regions used above are smaller than the source extraction regions .
this was done in order that the background was taken from either the same ccd or same window as the majority of the source counts . to check the spectral analysis described in the following section ,
larger background regions from a different ccd with radii equal to or larger than the source regions were also extracted .
the derived spectral parameters were not significantly changed by using the larger background regions , therefore we report the results from the original data extraction . the rgs data were reduced using the rgsproc chain in the sas .
the observed count rate was 0.15 s@xmath14 for rgs1 and 0.19 s@xmath14 for rgs2 .
this yielded only @xmath36 and @xmath37 counts in the rgs1 and rgs2 spectra , respectively .
given the small number of counts it was decided not to pursue a detailed analysis of the rgs spectra , although the continua predicted by the broadband pn models ( including the warm absorption edges ) were checked against the observed rgs spectral shape .
figure [ fig : lightcurve ] shows the 0.210 kev pn lightcurve of 4c+74.26 in 100 s time bins .
the sas task lccorr was used to background subtract the data , as well as provide corrections for vignetting and deadtime . as with the earlier _ asca _ @xcite and _ bepposax _
@xcite observations , no significant variability was observed from 4c+74.26
. a constant fit to the pn lightcurve resulted in a @xmath38d.o.f.@xmath39 ( d.o.f.=degrees of freedom ) , with a best fit of 8.57 s@xmath14 .
in this section we present the results of fitting the _ xmm - newton _ spectrum of 4c+74.26 between 0.3 and 12 kev in the observed frame . since it has the largest number of counts and higher spectral resolution , the pn spectrum was initially analyzed on its own , but we include the mos data ( from 0.310 kev ) at the end of this section to check for any differences . as mentioned above , results for the fe k@xmath7 line region and reflection parameters are presented in paper i. here , we will concentrate on the absorption characteristics in the observed spectrum .
xspec v.11.3.1p @xcite was used for the spectral fitting , and a @xmath40 criterion was used to determine the 2@xmath41 errorbars on the best - fit parameters .
galactic absorption , modeled using the tbabs code in xspec @xcite , is included in all fits .
unless stated otherwise , the figures are plotted in the observed frame , while the fit parameters are quoted in the rest frame .
we begin the analysis with the best - fitting model found in fitting the 212 kev data .
this model , denoted iondisk*blr+iondisk+g in paper i , consists of a relativistically blurred ionized disk ( employing the models of @xcite ) , an unblurred neutral reflector ( also using the @xcite models ) , and a narrow gaussian emission line .
the data - to - model ratio when the 0.32 kev data are included is shown in figure [ fig : ratio ] .
the plot clearly shows that a significant amount of absorption is required at energies @xmath42 kev .
this opacity could be in the form of extra cold absorption ( as favored by the earlier analyses of @xcite and @xcite ) , warm absorption ( as indicated by @xcite ) , or a combination of the two . to test the different absorption models
, we first froze the parameters in the model related to the fe k@xmath7 lines , and then added additional intrinsic absorption via the ztbabs model in xspec . a good fit
was found ( @xmath38d.o.f.=1643/1643 ) to the 0.312 kevspectrum with an intrinsic column @xmath43 @xmath1 . when the cold absorption model was replaced with the warm absorber model absori @xcite ,
the ionization parameter of the absorber went to zero and basically the same fit was uncovered as before ( @xmath38d.o.f.=1638/1642 ) .
however , the _ chandra _ data of 4c+74.26 show evidence for a weak warm absorber with possible ionized oxygen edges . to check this , an o viii edge at 0.87 kev and an o vii edge at 0.739
kev were added to the cold absorption model above .
this new model resulted in a significant ( @xmath44 with only 2 additional degrees of freedom ) improvement over the purely cold absorber case , bolstering the case for a warm absorber toward 4c+74.26 .
since the previous model excluding the warm absorber resulted in a statistically acceptable fit , it is not possible to conclude that the data require warm absorption .
the results of the fit are shown in the first line of table [ table : fit1 ] . [
cols="^,^,^,^,^,^,^,^,^",options="header " , ] _ notes _ : @xmath45 parameter fixed at value there is strong evidence for an o vii edge in the data with a maximum optical depth of @xmath46@xmath47 , consistent with the limit from the _ asca _ data .
a further improvement to the fit was made when the fe k@xmath7 line parameters were allowed to vary ( second line in table [ table : fit1 ] ) . except for @xmath48 , the absorption parameters did not change significantly .
the residuals to this last fit ( our best - fit model ) are shown in figure [ fig : ratio2 ] , while the model is displayed in figure [ fig : model ] .
the residuals show no evidence for a soft excess .
the neutral reflector predicts soft x - ray emission lines from o vii and o viiithat do seem to be accounted for in the data .
it would be interesting to obtain a long gratings observation with either _ xmm - newton _ or _
chandra _ to confirm this possibility .
the above fits give strong indications that a warm absorber is present in 4c+74.26 , but unfortunately can not give much further information . in order to try to further characterize the warm absorber
, we performed a fit where we replaced the absorption edges with the absori model ( the extra cold absorption was still included in the model ) . to simplify the procedure ,
the fit was performed with the fe k@xmath7 line parameters frozen as in the first line of table [ table : fit1 ] .
the fit assumed solar abundances and an absorber temperature of @xmath49 k ( higher temperatures did not improve the results ) .
a good fit was obtained ( @xmath38d.o.f.=1616/1641 ) with a warm absorber column of @xmath50 @xmath1 and an ionization parameter of 56@xmath51 .
the intrinsic cold absorption column remained at @xmath52 @xmath1 .
while this parameterization is clearly inferior compared to a gratings analysis , it is consistent with kaspi s early inferences of the warm absorber in 4c+74.26 . using the best - fit model from table [
table : fit1 ] , we find a 0.52 kev flux of @xmath53 erg @xmath1 s@xmath14 and a 0.510 kev flux of @xmath54 erg @xmath1 s@xmath14 .
the total unabsorbed luminosity between 0.5 and 2 ( 10 ) kev is @xmath55 ( @xmath56 ) ergs s@xmath14 . employing the black hole mass estimate of @xmath20 m@xmath21 @xcite , and assuming that the x - ray luminosity is @xmath57 per cent of the bolometric luminosity , then 4c+74.26 has an eddington ratio of @xmath58 . as a final check on the results , mos data between 0.3 and 10
kevwere added and joint pn - mos spectral fits were performed .
the normalizations of the mos spectra were allowed to float relative to the pn spectrum to account for any slight calibration errors . also , a separate photon - index was fit to the mos data , as it is known that mos spectra are slightly harder than ones observed from the pn @xcite .
the joint pn - mos fits resulted in a slightly steeper @xmath59 than previously found , and consequently more absorption in the edges and intrinsic cold absorber was required .
the parameter values from our best fit model are shown in the last line in table [ table : fit1 ] .
one can see there is only minor changes in the fit parameters .
when the absori model was used , the warm absorber column reduced to @xmath60 @xmath1 , and the ionization parameter also fell to @xmath61 .
both these values are well within the errorbars of the previous ones found using only the pn data .
paper i presented evidence for a relativistically broadened fe k@xmath7 in the 212 kev spectrum of 4c+74.26 .
the broad line was the statistically preferred fit , but as seen in other agn @xcite the breadth of the fe k@xmath7 may also be explained by a complicated absorption model . in this section
we test if complex absorption models can provide a better fit to the broadband data than one involving a broad fe k@xmath7 line .
the pn data are employed between 0.312 kev to make use of its superior sensitivity at 6 kev .
in contrast to ngc 3783 @xcite and ngc 3516 @xcite , 4c+74.26 does not have a very significant warm absorber .
the highly ionized component detected by _
chandra _ was termed as `` weak '' ( kaspi , private communication ) .
furthermore , the column estimated from the absori model is one to two orders of magnitude lower than the columns found in ngc 3783 or ngc 3516 .
therefore , it seems unlikely that curvature caused by ionized absorption will significantly effect the spectrum at energies close to the iron line . as a result ,
we concentrate on partial covering models @xcite and use the zpcfabs model within xspec to simulate the effect of the primary x - ray continuum passing through a neutral absorber with hydrogen column density @xmath62 that covers a fraction @xmath63 of the x - ray source .
the attenuated continuum is assumed to consist of a reflection spectrum , modeled with the pexrav code of @xcite ( same parameter values as in paper i ) , and an intrinsically narrow ( @xmath64 kev ) neutral fe k@xmath7 line .
the warm absorber is modeled with the o vii and o viii edges , and galactic absorption was also included .
the subsequent fit was poor with @xmath38d.o.f.=1752/1640 , @xmath65 @xmath1 , and @xmath66 .
an improved fit was found by adding additional cold absorption with the ztbabs model . in this case ,
@xmath38d.o.f.=1674/1639 , @xmath67 @xmath1 and @xmath68 .
this fit also gave a photon - index of @xmath69 and a reflection fraction of 2.5 .
the column of extra cold absorption was @xmath70 @xmath1 .
a further improvement to @xmath38d.o.f.=1631/1640 was found by replacing the pexrav and gaussian models with a solar abundance @xcite reflector that had its ionization parameter frozen at its lowest value . in this case , @xmath71 @xmath1 and @xmath72 .
the photon - index was @xmath73 and a low reflection fraction of @xmath74 was also found . in summary ,
all the partial covering models considered here provide poorer fits to the 0.312 kev pn spectrum of 4c+74.26 than the ones that included a relativistically broadened fe k@xmath7 line .
the line should still be confirmed with a higher signal - to - noise observation , but these results help strengthen the conclusions presented in paper i. figures [ fig : ratio ] and [ fig : ratio2 ] both show a possible absorption line in the residuals at @xmath75 kev .
a narrow ( @xmath76 ) gaussian absorption line added to the best - fit model described above resulted in a @xmath77 with 2 extra degrees of freedom , significant at the @xmath78 per cent level according to the f - test . however , this line is not of an astrophysical origin .
it arises from the subtraction of a strong cu k@xmath7 emission line in the background spectrum produced by the circuit board supporting the pn detector in the spacecraft ( @xcite and references therein ) .
it is important to note that the strength of the this line is not constant over the instrument .
in fact , it is practically absent near the center where the image of the target source falls .
but , if the background accumulation region is closer to the edge of a ccd ( as it was in this case ) , it will include this line which will then produce a spurious absorption feature in the source data .
interestingly , if the line was interpreted as fe xxvi ly@xmath7 , whose rest energy is 6.97 kev , then one would conclude that it was blueshifted to a velocity of @xmath79 . thus , this feature could easily be misidentified as a highly ionized and rapidly outflowing absorption feature .
as the cu k@xmath7 line is very narrow , when our 4c+74.26 spectral analysis was repeated with a different background region that omitted the line , the results did not significantly change .
we therefore included the affected plots in this paper to illustrate the potential danger to future authors .
this paper presented the first detailed characterization of the broadband x - ray spectrum of 4c+74.26 . at energies less than 2
kev , the spectrum is dominated by cold absorption in excess of the galactic column in this direction . if this extra attenuation is at the redshift of 4c+74.26 , then the column required is @xmath0 @xmath1 .
this value is about two times smaller than the previous estimates based on either _ asca _ or _ bepposax _ data @xcite , although it is not too different from the @xcite value .
differing spectral models , analysis techniques and the superior quality of the _ xmm - newton _ data are most likely responsible for these disagreements .
an excess column of cold absorption appears to be common in radio - loud agn @xcite , and 4c+74.26 is no different despite exhibiting typical seyfert properties in almost every other regard . in this case ,
perhaps the larger inclination angle of @xmath80@xmath81 degrees ( as inferred from the fe k@xmath7 emission line ; paper i & table [ table : fit1 ] ) implies that the line - of - sight to the central engine has a longer path length within the host galaxy .
alternatively , the host galaxy of 4c+74.26 , like most radio - loud agn , is an elliptical ; therefore , unlike spiral galaxies , random lines - of - sights into the galaxy are unlikely to have a clear path to the center .
in addition to the excess cold absorption , 4c+74.26 also exhibits a weak warm absorber , as primarily evidenced by a significant o viiedge . while a proper parametrization of the warm absorber awaits a long gratings observation , we estimated the absorbing column and ionization parameter to be @xmath82 @xmath1 and @xmath83 , respectively .
the high - throughput capabilities of _ xmm - newton _ are allowing the discovery of warm absorbers in many more quasars than before @xcite , emphasizing that they are a common occurrence in all accreting supermassive black holes .
this analysis of the soft x - ray spectrum of 4c+74.26 does offer one surprise that will need to be followed up with a longer observation .
the best fit broadband spectral model shown in fig .
[ fig : model ] includes a very weakly ionized reflector which predicts a number of recombination lines a low energies ( e.g. * ? ? ?
* ) , in particular from o vii and o viii .
these lines do seem to fit the data , but whether they originate in the accretion disk or warm absorber ( or both ) is unknown . this may be elucidated with a long gratings observation of 4c+74.26 which would unravel the properties of the warm absorber .
based on observations obtained with _ xmm - newton _ , an esa science mission with instruments and contributions directly funded by esa member states and the usa ( nasa ) .
the author acknowledges financial support from the natural sciences and engineering research council of canada , and thanks j. golding for help with the data processing and analysis .
arnaud k.a . , 1996 , in jacoby g.h . , barnes j. , eds . , asp conf .
ser . 101 , astronomical data analysis software and systems v ( astron .
soc . pac .
: san francisco ) , 17 ballantyne d.r .
, fabian a.c . , 2005 , apj , 622 , l97 ( paper i ) ballet j. , 1999 , a&as , 135 , 371 brinkmann w. , yuan w. , siebert j. , 1997 , a&a , 319 , 413 brinkmann w. , otani c. , wagner s.j . , siebert , j. , 1998 , a&a , 330 , 67 corbin m.r . , 1997 , apjs , 113 , 245 den herder j.w .
et al . , 2001 , a&a , 365 , l7 dickey j.m . , lockman f.j . , 1990 ,
ara&a , 28 , 215 done c. , mulchaey j.s . , mushotzky r.f . , arnaud k.a . , 1992 , apj , 395 , 275 gallo l.c . , tanaka y. , boller th . , fabian a.c .
, vaughan s. , brandt w.n . , 2004 , mnras , 353 , 1064 hasenkopf c.a .
, sambruna r.m . , eracleous m. , 2002 , apj , 575 , 127 holt s.s . ,
mushotzky r.f . ,
boldt e.a .
, serlemitsos p.j . ,
becker r.h . ,
szymkowiak a.e .
, white n.e . , 1980 , apj , 241 , 13 jansen f. et al . , 2001 , a&a , 365 , l1 kaspi s. , 2004 , in bergmann th . s. , ho l.c .
& schmitt h.r . ,
, iau symp .
221 , the interplay among black holes , stars and ism in galactic nuclei ( cambridge univ .
press : cambridge ) , 41 kaspi s. , smith p.s .
, netzer h. , maoz d. , jannuzi b.t . , giveon u. , 2000 , apj , 533 , 631 katayama h. , takahashi i. , ikebe y. , matsushita k. , freyberg m.j . , 2004 , a&a , 414 , 767 komossa s. , meerschweinchen j. , 2000 , a&a , 354 , 411 lara l. , cotton w.d . , feretti l. , giovannini g. , marcaide j.m . , mrquez i. , venturi t. , 2001 , a&a , 370 , 409 lara l. , giovannini g. , cotton w.d . , feretti l. , marcaide j.m . , mrquez i. , venturi t. , 2004 , a&a , 421 , 899 magdziarz p. , zdziarski a.a .
, 1995 , mnras , 273 , 837 mason k.o .
et al . , 2001 , a&a , 365 , l36 pearson t.j . , blundell k.m .
, riley j.m . ,
warner p.j . , 1992 , mnras , 259 , 13p piconcelli e. , jinenez - bailn e. , guainazzi m. , schartel n. , rodrguez - pascual p.m. , santos - lle m. , 2005 , a&a , 432 , 15 reeves j.n . , turner m.j.l . , 2000 , mnras , 316 , 234 reeves j.n
, nandra k. , george i.m . ,
pounds k.a . ,
turner t.j . , yaqoob t. , 2004 , apj , 602 , 648 riley j.m . ,
warner p.j . , 1990 , mnras , 246 , 1p riley j.m . , warner p.j
. , rawlings s. , saunders r. , pooley g.g . , eales s.a . , 1988 , mnras , 236 , 13p rinn a.s . , sambruna r.m . , gliozzi m. , 2005 , apj , 621 , 167 robinson a. , corbett e.a .
, axon d.j . , young s. , 1999 , mnras , 305 , 97 ross r.r . , fabian a.c .
, young a.j .
, 1999 , mnras , 306 , 461 sambruna r.m . , eracleous m. , mushotzky , r.f . , 1999 ,
apj , 526 , 60 schurch n.j .
, warwick r.s . ,
griffiths r.e . , sembay s. , 2003 , mnras , 345 , 423 spergel d.n .
et al . , 2003 ,
apjs , 148 , 175 strder l. et al . , 2001 , a&a , 365 , l18 turner m.j.l .
et al . , 2001 , a&a , 365 , l27 turner t.j . , kraemer s.b .
, george i.m .
, reeves j.n .
, bottorff m.c . , 2005 , apj , 618 , 155 vaughan s. , fabian a.c . , 2004 ,
mnras , 348 , 1415 wilms j. , allen a. , mccray r. , 2000 , apj , 542 , 914 woo j .- h . , urry c.m
. , 2002 , apj , 579 , 530
|
this paper presents a timing study and broadband spectral analysis of the broad - line radio galaxy 4c+74.26 based on a 35 ks _ xmm - newton_observation .
as found in previous datasets , the source exhibits no evidence for rapid variability , and its 0.210 kev lightcurve is well fit by a constant .
an excellent fit to the pn 0.312 kevspectrum was found using a continuum that combines an ionized and a neutral reflector , augmented by both cold and warm absorption .
there is no evidence for a soft excess .
the column of cold absorption was greater than the galactic value with an intrinsic column of @xmath0 @xmath1 .
evidence for the warm absorber was found from o vii and o viii absorption edges with maximum optical depths of @xmath2 and @xmath3 , respectively .
a joint pn - mos fit increased the o viii optical depth to @xmath4 .
a simple , one - zone warm absorber model yielded a column of @xmath5 @xmath1 and an ionization parameter of @xmath6 .
partial covering models provide significantly worse fits than ones including a relativistically broadened fe k@xmath7 line , strengthening the case for the existence of such a line . on the whole , the x - ray spectrum of 4c+74.26exhibits many features typical of both a radio - loud quasar ( excess absorption ) and radio - quiet seyfert 1 galaxies ( fe k@xmath7 emission and warm absorption ) .
we also show that a spurious absorption line at @xmath8 kev can be created by the subtraction of an instrumental cu k@xmath7 emission line .
[ firstpage ] galaxies : active galaxies : individual : 4c+74.26 x - rays : galaxies
|
the clump of core - helium burning stars is a prominent feature in the colour - magnitude diagrams of open clusters .
cannon ( 1970 ) predicted that the red clump stars should also be abundant in the solar neighbourhood .
many photometric studies have tried with varying success to identify such stars in the galactic field ( see tautvaiien 1996 for a review ) , however we had to wait for the _ hipparcos _ mission .
the presence of red clump stars in the solar neighbourhood was clearly demonstrated in the hr diagrams by perryman et al .
the _ hipparcos _ catalogue ( perryman et al .
1997 ) contains about 600 clump stars with parallax error lower than 10% , and hence an error in absolute magnitude lower than 0.12 mag .
this accuracy limit corresponds to a distance of about 125 pc within which the sample of clump stars is complete .
now it is important to investigate their distributions of masses , ages , colours , magnitudes and metallicities , which may provide useful constraints to chemical evolution models of the local galactic disk .
moreover , clump stars may be useful indicators of ages and distances for stellar clusters and the local group galaxies ( cf . hatzidimitriou & hawkins 1989 , hatzidimitriou 1991 , udalski 1998 , girardi & salaris 2001 ) .
in this paper we report on the primary atmospheric parameters and the abundances of iron group elements in the 62 clump stars of the galactic field obtained from the high - resolution spectra .
the results are discussed in detail together with results of other studies of the clump stars .
preliminary results of this study were published by tautvaiien et al .
( 2005 ) and tautvaiien & puzeras ( 2008 ) . in fig . 1
, we show a hr diagram constructed for the _ hipparcos _ stars with @xmath5 and @xmath6 mag . on the giant branch
is a distinct red clump at @xmath7 mag .
the sample of 63 stars investigated in our study is indicated by open circles .
the red clump stars were selected from the _ hipparcos _ catalogue ( perryman et al .
spectra for partially overlapping star samples were observed on several telescopes , described below .
spectra for 17 stars were obtained at the nordic optical telescope ( not , la palma ) with the sofin chelle spectrograph ( tuominen et al .
1999 ) . the 2nd optical camera ( @xmath8 ) was used to observe simultaneously 13 spectral orders , each of @xmath9 in length , located from 5650 to 8130 .
reduction of the ccd images , obtained with sofin , was done using the _ 4a _ software package ( ilyin 2000 )
. procedures of bias subtraction , cosmic ray removal , flat field correction , scattered light subtraction , extraction of spectral orders were used for image processing .
a th - ar comparison spectrum was used for the wavelength calibration .
the continuum was defined from a number of narrow spectral regions , selected to be free of lines .
the spectra of 14 stars were observed with the hires spectrograph on the 10-m keck telescope . a 1.1 x 7 arcsec slit ( @xmath10 )
was used , and 19 spectral orders located from 5620 to 7860 were extracted .
the spectra were reduced using _ iraf _ and _ makee _ packages .
the spectra of 17 stars were observed with the long camera of the 1.22 m dominion astrophysical observatory telescope s coud spectrograph ( @xmath11 ) .
the interactive computer graphics program _ reduce _ by hill et al .
( 1982 ) was used to rectify them .
the scattered light was removed during the extraction procedure by the program _ ccdspec _ described in gullivier & hill ( 2002 ) .
the spectra for 18 stars in the spectral interval from 6220 to 6270 were obtained at the elginfield observatory ( canada ) with the 1.2 m telescope and the high - resolution coud spectrograph ( @xmath12 ) .
spectra were recorded using an 1872 diode reticon self - scanned array light detector , mounted in a schmidt camera , with focal length of 559 mm .
see brown et al .
( 2008 ) for further discussion concerning the equipment and operation .
this observational data was supplemented by spectroscopic observations ( @xmath13 ) of red clump stars obtained on the 2.16 m telescope of the beijing astronomical observatory ( china ) taken from the literature ( zhao et al .
2001 ) . in fig . 2
, we show examples of observed spectra for several common stars using different instruments
. a careful selection of spectral lines for the analysis has allowed us to avoid systematic differences in analysis results obtained from the different instruments .
e.g. , the star hd 216228 has been observed on four telescopes , a comparison of the measured equivalent widths ( ew ) of its fei lines is shown in fig .
the spectra were analysed using a differential model atmosphere technique .
the programme packages , developed at the uppsala astronomical observatory , were used to calculate the theoretical equivalent widths and the line profiles .
a set of plane parallel , line - blanketed , constant - flux lte model atmospheres was computed with an updated version of the _ marcs _ code ( gustafsson et al .
2003 ) . the vienna atomic line data base ( vald , piskunov et al .
1995 ) was extensively used in preparing the input data for the calculations .
atomic oscillator strengths for the spectral lines analysed this study were taken from an inverse solar spectrum analysis done in kiev ( gurtovenko & kostik 1989 ) .
because of the asymmetric nature of line measurement errors ( i.e. problems such as blending and telluric line superposition always increase measured line width ) , we used a `` quality over quantity '' approach when selecting lines for abundance calculations .
all lines used for calculations were carefully selected . inspection of the solar spectrum ( kurucz et al .
1984 ) and the solar line identifications of moore et al .
( 1966 ) were used to avoid blends and lines blended by telluric absorption lines .
all line profiles in all spectra were hand - checked requiring that the line profiles be sufficiently clean to provide reliable equivalent widths . only lines with equivalent widths between 20 m and 150 m were used for abundance determinations .
spectral lines systematically producing outlier abundances in a number of stars , indicating spectral ( observational ) defect , undetected blends or erroneous atomic data , were rejected as well .
the equivalent widths of the lines were measured by fitting of a gaussian profile using the _ 4a _ software package ( ilyin 2000 ) .
effective temperature , gravity and microturbulence were derived using traditional spectroscopic criteria .
the preliminary effective temperatures for the stars were determined using the @xmath14 and @xmath15 colour indices and the temperature calibrations by alonso et al .
for some stars the averaged temperatures also include the values obtained from the infrared flux method ( irfm ) .
all the effective temperatures were carefully checked and corrected if needed by forcing fe i lines to yield no dependency of iron abundance on excitation potential by changing the model effective temperature .
surface gravity was obtained by forcing fe i and fe ii lines to yield the same [ fe / h ] value by adjusting the model gravity .
microturbulence value corresponding to minimal line - to - line fe i abundance scattering was chosen as correct value .
depending upon the telescope , the number of fe i lines analysed was up to 65 and of fe ii up to 12 .
epsf = epsf = using the @xmath16 values and solar equivalent widths of analysed lines from gurtovenko & kostik ( 1989 ) we obtained the solar abundances , used later for the differential determination of abundances in the programme stars .
we used the solar model atmosphere from the set calculated in uppsala with a microturbulent velocity of 0.8 @xmath17 , as derived from fe i lines .
in addition to thermal and microturbulent doppler broadening of lines , atomic line broadening by radiation damping and van der waals damping were considered in the calculation of abundances .
radiation damping parameters of lines were taken from the vald database . in most cases
the hydrogen pressure damping of metal lines was treated using the modern quantum mechanical calculations by anstee & omara ( 1995 ) , barklem & omara ( 1997 ) and barklem et al .
( 1998 ) . when using the unsld ( 1955 ) approximation , correction factors to the
classical van der waals damping approximation by widths @xmath18 were taken from simmons & blackwell ( 1982 ) .
for all other species a correction factor of 2.5 was applied to the classical @xmath19 @xmath20 ) , following mckle et al .
( 1975 ) . for lines stronger than @xmath21 m
the correction factors were selected individually by inspection of the solar spectrum .
cobalt abundances were investigated with hyperfine structure ( hfs ) effects taken into account .
the hfs corrections for every line in every star we calculated with the lte spectral synthesis program _ moog _ ( sneden 1973 ) and the hfs input data adopted from prochaska et al . ( 2000 ) .
the sources of uncertainty can be divided into two distinct categories .
the first category includes the errors that affect a single line ( e.g. random errors in equivalent widths , oscillator strengths ) , i.e. uncertainties of the line parameters .
other sources of observational error , such as continuum placement or background subtraction problems also are partly included in the equivalent width uncertainties .
the second category includes the errors which affect all the lines together , i.e. mainly the model errors ( such as errors in the effective temperature , surface gravity , microturbulent velocity , etc . ) .
the scatter of the deduced line abundances @xmath22 , presented in table 2 and table 3 , gives an estimate of the uncertainty due to the random errors in the line parameters ( the mean value of @xmath22 is @xmath23 ) .
thus the uncertainties in the derived abundances that are the result of random errors amount to approximately this value .
typical internal error estimates for the atmospheric parameters are : @xmath24 k for @xmath25 , @xmath26 dex for log @xmath27 and @xmath28 for @xmath29 .
the sensitivity of the abundance estimates to changes in the atmospheric parameters by the assumed errors is illustrated for the star hd 218031 in table 1 .
it is seen that possible parameter errors do not affect the abundances seriously ; the element - to - iron ratios , for which we use the neutral species for both and use in our discussion , are even less sensitive .
.the sensitivity of stellar atmosphere abundances to changes in atmospheric parameters .
example for the star hd 218031 . [ cols=">,^,^,^,>,^,^,^ " , ]
there have been few attempts to derive typical metallicities for _ hipparcos _ clump stars .
the first was an indirect method by jimenez et al .
( 1998 ) , they obtained @xmath30 } < 0.0 $ ] . however , they modelled the clump with star formation rate ( sfr ) strongly decreasing with galactic age .
consequently , they were considering , essentially , the behaviour of the old clump stars , with masses of about 0.8 1.4 @xmath31 .
intermediate - mass clump stars with mass more than 1.7 @xmath31 were absent in their simulations ( c.f .
girardi & salaris 2001 ) . thus their description of the clump stars was incomplete .
epsf = epsf = girardi et al .
( 1998 ) considered the full mass range of clump stars , and models with constant sfr up to 10 gyr ago .
they demonstrated that the best fit is achieved with a galaxy model which in the mean has solar metallicity , with a very small metallicity dispersion of about 0.1 dex .
girardi & salaris ( 2001 ) collected the spectroscopic abundance determinations for the _ hipparcos _ clump stars in the catalogue by cayrel de strobel et al .
they found that the histogram of [ fe / h ] values is fairly well represented by a quite narrow gaussian curve of mean @xmath3}\rangle=-0.12 $ ] dex and standard deviation of 0.18 dex , as derived by means of a least - squares fit . in the same paper a theoretical simulation of the _ hipparcos _
clump was made .
girardi & salaris ( 2001 ) found that the total range of metallicities allowed by their model is quite large ( @xmath32 } \leq 0.3 $ ] ) , however the distribution for clump stars is very narrow : a gaussian fit to the [ fe / h ] distribution produces a mean @xmath3}\rangle= + 0.03 $ ] dex and dispersion @xmath4}=0.17 $ ] .
actually , the [ fe / h ] distribution presents an asymmetric tail at lower metallicities , which causes the straight mean of [ fe / h ] to be @xmath33 dex .
the near - solar metallicity and so small @xmath4}$ ] imply that nearby clump stars are ( in the mean ) relatively young objects , reflecting mainly the near - solar metallicities developed in the local disk during the last few gyrs of its history . from the same simulation they determined that the peak of age distribution in the _ hipparcos _ clump is at about 1 gyr . in our study ,
the results of [ fe / h ] determinations in clump stars of the galaxy confirm the theoretical model by girardi & salaris ( 2001 ) .
the metallicity range in our study is from @xmath34 to @xmath35 dex , however the majority of stars concentrate near the mean value @xmath3}\rangle = -0.04\pm0.15 $ ] .
a gaussian fit to the [ fe / h ] distribution produces the mean @xmath3}\rangle= -0.01 $ ] and very small dispersion @xmath4}=0.08 $ ] . in order to see what the metallicity distribution is in all the sample of galactic clump stars investigated to date using high resolution spectra
, we present in fig . 5 metallicity distributions for the samples of galactic clump stars investigated in this study ( 62 stars ) , by mishenina et al .
( 177 stars ) , liu et al .
( 63 stars ) and luck & heiter ( 138 stars ) ; and in fig .
6 , the metallicity distribution in the entire sample of 342 galactic clump stars is presented .
the metallicity values were averaged for the stars with multiple analyses . in the study of mishenina et al .
a special attempt was made to include the metal - deficient stars , so the distribution slightly reflects this selection effect .
the [ fe / h ] values of clump stars in fig .
6 range from + 0.4 to @xmath36 dex .
a gaussian fit to the [ fe / h ] distribution produces a mean @xmath3}\rangle= -0.02 $ ] and dispersion @xmath4}=0.13 $ ] dex , which is in agreement with the theoretical model by girardi & salaris ( 2001 ) .
epsf = fig .
7 presents the observed [ el / fe ] ratios for iron group elements in our sample of stars , always using the neutral species .
nickel abundances always closely follow solar nickel to iron ratios in the galactic disk . in our study @xmath37 } = 0.06\pm 0.07 $ ] , in mishenina et al .
@xmath37 } = 0.11\pm 0.03 $ ] , in liu et al .
( 2007 ) @xmath37 } = 0.02\pm 0.05 $ ] , and in luck & heiter ( 2007 ) @xmath37 }
= 0.01\pm 0.03 $ ] , so it can be said that the [ ni / fe ] ratios in clump stars are approximately solar . vanadium was investigated in our and two other studies .
we obtain the mean value of @xmath38}=0.11\pm 0.12 $ ] , liu et al . derived @xmath39 and luck & heiter found @xmath40 , which means that this element also has solar [ v / fe ] ratios .
chromium and cobalt were investigated in our work and by luck & heiter .
the mean [ cr / fe ] ratios are exactly solar in both studies .
[ co / fe ] is enhanced by about @xmath41 dex in the work by luck & heiter . in our study ,
cobalt abundances were investigated with hyperfine structure effects taken into account ; we find @xmath42}=0.02\pm 0.11 $ ] , which is close to solar .
the main atmospheric parameters @xmath25 , log @xmath27 , @xmath43 , [ fe / h ] and abundances of vanadium , chromium , cobalt and nickel were determined for 62 red clump stars revealed in the galactic field by the _ hipparcos _
orbiting observatory .
the stars form a homogeneous sample with the mean value of temperature @xmath0 k , of surface gravity log @xmath1 and the mean value of metallicity @xmath2}=-0.04\pm0.15 $ ] .
it is especially interesting to note that metallicities of stars in the galactic clump lie in quite a narrow interval .
a gaussian fit to the [ fe / h ] distribution produces the mean @xmath3}\rangle= -0.01 $ ] and dispersion @xmath4}=0.08 $ ] .
the near - solar metallicity and small dispersion of @xmath4}$ ] of clump stars of the galaxy obtained in this work confirm the theoretical model of the _ hipparcos _ clump by girardi & salaris ( 2001 ) which suggests that nearby clump stars are ( in the mean ) relatively young objects , reflecting mainly the near - solar metallicities developed in the local disk during the last few gyrs of its history .
the iron group element to iron abundance ratios in the investigated clump giants are close to solar .
this allows us to use the clump stars to study the chemical and dynamical evolution of the galaxy .
clump giants may provide a very useful information on mixing processes in evolved low mass stars .
we plan to address this question in our further study of the galactic clump .
this project has been supported by the european commission through the baltic grid project as well as through access to research infrastructures action " of the improving human potential programme " , awarded to the instituto de astrof ' isica de canarias to fund european astronomers access to the european nordern observatory , in the canary islands .
ep and gt acknowledge support from the lithuanian national science and studies foundation through gridtechno project .
jgc acknowledges the inspiration of the late bohdan paczynski for stimulating her interest in this area and is grateful to nsf grant ast-0507219 and ast-0908139 for partial support .
dfg is grateful for financial support from the natural sciences and engineering research council of canada .
sja thanks dr .
james e. hesser , director of the dominion astronomical observatory for the observing time used for this project and dr .
austin gulliver for help in reducing the dao spectrograms .
financial support was provided to sja by the citadel foundation .
alonso a. , arribas s. , mart ' inez - roger c. 1999 , a&as , 140 , 261 alonso a. , arribas s. , mart ' inez - roger c. 2001 , a&a , 376 , 1039 anstee s. d. , & omara b. j. 1995 , mnras , 276 , 859 barklem p. s. , & omara b. j. 1997 , mnras , 290 , 102 barklem p. s. , omara b. j. & ross j. e. 1998 , mnras , 296 , 1057 brown k. i. t. , gray d. f. , & baliunas s. l. 2008 , apj , 679 , 1531 cannon r. d. 1970 , mnras , 150 , 111 cayrel de strobel g. , soubiran c. , friel e. d. , ralite n. , francois p. 1997
, a&as , 124 , 299 girardi l. , groenewegen m. a. t. , weiss a. , salaris m. 1998 , mnras , 301 , 149 girardi l. , salaris m. 2001 , mnras , 323 , 109 gulliver a. f. , hill g. 2002 , in bohlender d. a. , durand d. & handley t. h. , eds .
, astronomical data analysis software and systems xi , aspc , 108 , 232 gurtovenko e. a. & kostik r. i. 1989 , fraunhofer s spectrum and a system of solar oscillator strengths , kiev , naukova dumka , p. 200
gustafsson b. , edvardsson b. , eriksson k. , jrgensen u. g. , mizuno - wiedner m. , plez b. , 2003 , iaus , 210 , a4 hatzidimitriou d. 1991 , mnras , 251 , 545 hatzidimitriou d. , hawkins m. r. s. 1989 , mnras , 241 , 667 hill g. , fisher w. a. , poeckert r. 1982 , pub .
dominion astrophys .
victoria , 16 , 27 ilyin i. v. 2000 , high resolution sofin ccd chelle spectroscopy , phd dissertation , univ .
oulu , finland jimenez r. , flynn c. , kotoneva e. , 1998 , mnras , 299 , 515 kurucz r. l. , furenlid i. , brault j. , & testerman l. 1984 , solar flux atlas from 296 to 1300 nm , national solar observatory , sunspot , new mexico liu y. j. , zhao g. , shi j. r. , pietrzynski g. , gieren w. 2007 , mnras , 382 , 553 luck r. e. , heiter u. 2007 , aj , 133 , 2464 mckle r. , holweger h. , griffin r. , griffin r. 1975 , a&a , 38 , 239 mcwilliam a. 1990 , apjs , 74 , 1075 mcwilliam a. , rich r. m. 1994 , apjs , 91 , 749 mishenina t. v. , bienaym o. , gorbaneva t. i. , charbonnel c. , soubiran c. , korotin s. a. , kovtyukh v. v. 2006 , a&a , 456 , 1109 moore c. e. , minnaert m. g. j. , & houtgast j. 1966 , the solar spectrum 2935 to 8770 , nbs monogr .
61 perryman m. a. c. 1997 , esasp 404 , 231 perryman m. a. c. , lindegren l. , kovalevsky j. 1995 , a&a , 304 , 69 piskunov n. e. , kupka f. , ryabchikova t. a. , weiss w. w. , & jeffery c. s. , 1995 , a&as 112 , 525 prochaska j. x. , naumov s. o. , carney b. w. , mcwilliam a. , wolfe a. m. , 2000 , aj , 120 , 2513 simmons g. j. , & blackwell d. e. 1982 , a&a 112 , 209 physics , 14 , 4015 sneden c. , 1973 , phd thesis , univ .
of texas tautvaiien g. 1996 , baltic astronomy , 5 , 503 tautvaiien g. , stasiukaitis e. , puzeras e. , gray d. f. , ilyin i. 2005 , esa sp-560 , eds .
f. favata , g. hussain , b. battrick , 989 tautvaiien g. , puzeras e. 2008 , in andersen j. , bland - hawthorn j. & nordstrm b. , eds . , the galaxy disk in cosmological context , iau symp .
254 tuominen i. , ilyin i. , petrov p. 1999
, in karttunen h. & piirola v. , eds . , astrophysics with the not , p. 47
udalski a. 1998 , acta astronomica , 48 , 383 unsld a. 1955 , physik der stern atmosphren ( zweite auflage ) .
springer - verlag , berlin zhao g. , qiu h. m. , mao s. 2001 , apj , 551 , l85
|
the main atmospheric parameters and abundances of the iron group elements ( vanadium , chromium , iron , cobalt and nickel ) are determined for 62 red giant clump " stars revealed in the galactic field by the _ hipparcos _
orbiting observatory .
the stars form a homogeneous sample with the mean value of temperature @xmath0 k , of surface gravity log @xmath1 and the mean value of metallicity @xmath2}=-0.04\pm0.15 $ ] dex .
a gaussian fit to the [ fe / h ] distribution produces the mean @xmath3}\rangle= -0.01 $ ] and dispersion @xmath4}=0.08 $ ] .
the near - solar metallicity and small dispersion of @xmath4}$ ] of clump stars of the galaxy obtained in this work confirm the theoretical model of the _ hipparcos _ clump by girardi & salaris ( 2001 ) .
this suggests that nearby clump stars are ( in the mean ) relatively young objects , reflecting mainly the near - solar metallicities developed in the local disk during the last few gyrs of its history .
we find iron group element to iron abundance ratios in clump giants to be close to solar .
[ firstpage ] stars : abundances stars : atmospheres stars : horizontal - branch .
|
quantum systems can exhibit non - classical correlation by a different way from quantum entanglement , which is well - known quantum discord ( qd ) .
qd characterizes the minimal perturbation induced by single - party von neumann measurement @xcite .
thus there exists non - entangled state with non - zero qd .
recently qd has been shown as a resource to speed up the quantum information processing .
for instance the determined quantum computation with one qubit @xcite and quantum metrology with noised states @xcite have been demonstrated the advantage over classical computation , in which the quantum entanglement is not involved .
in addition there exists extensive interest for qd in other diverse contexts @xcite .
however qd is difficult to determine analytically because of the optimization in the definition .
there exist very few exact results , even for the two - qubit cases @xcite .
( recently an exact evaluation of qd is proposed in ref @xcite .
however it is pointed out in @xcite that this approach is not completely correct . ) in addition the computation of quantum discord is shown to be np - complete @xcite ; the running time of the computation of qd is increased exponentially with the dimension of hilbert space .
thus one has to find an efficient way to calculate qd .
recently geometric discord ( gd ) is introduced by daki and the coauthors , which is defined as the square form of the shortest distance between the measured state @xmath0 and zero - discord state @xmath1 in hilbert space @xcite .
hence the optimization in the definition of geometric discord ( gd ) can be reduced greatly by the geometry of @xmath0 .
moreover a tight lower bound of gd can also been obtained for arbitrary states @xcite .
unfortunately the square form of gd is not monotonic under local operations ; the value of gd can increase by local operations @xcite .
this deficit raises the question whether gd or qd could unambiguously manifest the non - classical correlation in quantum systmes @xcite . in order to solve this problem ,
many generalizations of gd have been proposed .
for instance the rescaled gd is defined by rescaling the density operator with its norm @xcite .
furthermore the so - called schatten @xmath2-norm is also introduced to qualify the distance @xcite , instead of the @xmath3-norm form used in ref .
in addition the bures distance is also introduced @xcite .
however it is still difficult generally to find the analytical expression for gd since the comple evaluation or optimization .
recently the hellinger distance is introduced to measure qd @xcite .
this definition has a simple structure and can be evaluated readily .
moreover it is monotonically nonincreasing ( contractivity ) under local operations @xcite .
it is an interesting issue how to generalize qd into multipartite case .
a direct way is to introduce the three - tangle of qd in tripartite case @xcite and four - qubit case @xcite .
however it still is difficult to determine analytically and furthermore to generalize into the case with more parties .
another alternative way is to find the minimum qd between any single party and the others @xcite , named as the global qd .
however it does not include the other possible bipartite correlation and thus is not a comprehensive measurement of qd .
moreover the exact treatment of global qd is also difficult since one has to find optimal single - party von neumann measurements for every party . a geometric generalization of global qd
is introduced by finding the shortest distance from the zero global qd state @xcite .
however the author adopt the @xmath3-norm of distance , which suffers from the problem of non - contractivity under local operations @xcite .
with respect of these facts , we present an alternative approaching to qd in this article by a generalization of hellinger distance @xcite .
the main idea is to find the shortest distance from the completely classical state @xcite .
hence this approach is independent on the local measurements .
it should be pointed out that our way is a multipartite generalization of the method in ref .
@xcite , with consideration of the recently arising critiques about qd @xcite . by this way the exact evaluation can be reached for any bipartite pure state and also for some special mixed states . as for multipartite case the exact results can also be founded for symmetric states .
this article is divided into several sections .
the definition is presented in section ii , and a general expression is also presented .
then section iii presents the exact evaluation for bipartite pure states and for a special type of mixed states . in section
iv the situation for multipartite states is discussed and the exact result can be obtained for symmetric state .
in addition we also show its ability of marking the quantum phase transition in many - body systems in section v. in section vi a discussion of multilevel case is presented in comparison with the studies in ref.@xcite .
conclusion and discussion are presented in final section .
we first present the definition .
[ def ] given arbitrary state @xmath0 and completely classical state @xmath4 , we define geometric measure of qd as @xmath5 where @xmath6 is the hilbert - schmidt norm and the superscript means the hellinger distance .
@xmath4 can be written as the probabilistic mixture of local distinguishable states @xmath7 in which @xmath8 is a joint probability distribution and local states @xmath9 span an orthonormal basis @xcite .
the correlations in @xmath4 are identified as classical @xcite . for qubit case ,
a general orthonormal basis can be constructed by @xmath10 and @xmath11 with @xmath12 $ ] and @xmath13 . by eq . ,
@xmath14.\end{aligned}\ ] ] then the evaluation of @xmath15 is reduced to find the maximal overlap of @xmath16 and @xmath17 . for clarity @xmath4
is rewritten as @xmath18 in which @xmath19 denotes the local basis @xmath20 and the probability @xmath21 is to be determined . in the following we first present the general expressions for pure and mixed state cases respectively .
the detailed discussion is presented in the next section . for convenience
we set @xmath22 as the dimension of the local basis and @xmath23 with normalization .
-_pure state_- for a pure state @xmath24 , one gets @xmath25 with respect of @xmath21 , the extremal values of @xmath26 appear when @xmath27 it is not difficult to find @xmath28 . then the minimal extreme satisfies the relation @xmath29 consequently @xmath30 reduces to @xmath31 -_mixed state_- with the spectrum decomposition @xmath32 , eq . can be written as @xmath33 with respect of @xmath34 , the extremal points can be decided by @xmath35 directly @xmath36 . similarly one can obtain the relation @xmath37 then @xmath30 reduces to @xmath38 the determination of @xmath39 depends on @xmath0 .
it should be pointed out that the expressions above are general .
however for simplicity the following studies would focus mainly on qubit system since the simplicity and interest in quantum information processing .
the extension into multi - level state will be presented in sec .
_ -pure case- _ it is well known that any bipartite pure state @xmath40 can be written in a concise form @xmath41 by schmidt decomposition .
the schmidt bases @xmath42 and @xmath43 span a special space , of which the dimension is the minimal needed to expand @xmath40 by local orthonormal states . hence in order to find the maximal overlap of @xmath40 and @xmath4 , it is a natural conjecture that @xmath4 should belong to the special space too , i.e. , @xmath44 in which @xmath45 by eq
.. however we do not know how to prove this conjecture exactly .
an example is presented in order to display validity of the statement .
we try to find @xmath30 for @xmath46 of which schmidt decomposition is @xmath47 by eq .
, it is reduced to find the maximum of the overlap @xmath48 in which , @xmath49\right\}\nonumber\\ c_1&=&\tfrac{\sqrt{3}}{8}\sin\theta_1 \cos\phi_1\nonumber\\ c_2&=&\tfrac{\sqrt{3}}{8}\sin\theta_2 \cos\phi_2\end{aligned}\ ] ] by analysis @xmath50 have the maximal value of 1/16 when @xmath51 and @xmath52
. meanwhile @xmath53 and @xmath54 has the maximal values too .
it is obvious that the formula is invariant for @xmath55 .
consequently its extremal value happens when @xmath56 .
then @xmath57 which obviously is the sum of the fourth power of the schmidt coefficients .
so @xmath58 .
the `` nearest '' @xmath4 is @xmath59 in which @xmath60 and @xmath61 . by this example
we obtain the first conjecture [ c1 ] for arbitrary pure bipartite state , which has schmidt decomposition @xmath62 the `` nearest '' completely classical state @xmath4 can be written as @xmath63 in which @xmath64 . then @xmath65
_ -mixed case- _ as for spectrum decomposition @xmath32 , we can not find a general result for @xmath30 since the eigenstates of @xmath66 does not necessarily share the same schmidt bases .
however it is still possible for a exact treatment when @xmath0 shows `` x '' form @xmath67 in which the orthonormal basis is spanned by local states @xmath68 .
consequently for @xmath69 sub - matrix @xmath70 the eigenstates are schmidt decompositions in their own form .
more importantly since there is no overlap between different sub - matrices , then the `` nearest '' @xmath4 would be the probabilistic combination of the bases of all sub - matrices .
we work out two important examples to display the validity of statement _ example 1 . _
consider the werner state @xmath71 in which @xmath72 $ ] and @xmath73 .
it is obvious that @xmath74 has a `` x '' form on the basis @xmath75 @xmath76 which can be decomposed into two sub - matrices @xmath77 defined on the bases @xmath78 and @xmath79 respectively .
then there are four eigenstates @xmath80 of which the schmidt bases are @xmath81 and @xmath82 respectively . by eq .
, one obtains @xmath83=2\max\sqrt{c_0 ^ 2+c_1 ^ 2}\end{aligned}\ ] ] in which @xmath84 , @xmath85 and @xmath86 .
then one has @xmath87 when @xmath88 , which can occur , for example when @xmath89 and @xmath90 .
the corresponding `` nearest '' @xmath4 has the form @xmath91 which just is a mixed combination of the schmidt bases of @xmath92 and @xmath93 .
it should be pointed out the the choice of @xmath94 and @xmath95 is not unique .
_ example 2 .
_ consider the bell - diagonal state @xmath96 in which @xmath97 , @xmath98 . in matrix ,
@xmath99 on local basis @xmath75 . or equivalently @xmath100 on local basis @xmath101 .
then by eq .
@xmath102 = \tfrac{1}{4 } \max \sqrt{2(c_1 ^ 2+c_2 ^ 2)},\end{aligned}\ ] ] in which @xmath103 it is not difficult to find that the extremum happens when @xmath104 . then dependent on @xmath105 and @xmath106 , one
has @xmath102 = \tfrac{1}{2}\sqrt{h^2+\max\{d^2_1 , d^2_2 , d^2_3\}},\end{aligned}\ ] ] in which @xmath107 @xmath4 can be obtained by eq . ; when @xmath108 , @xmath109
. then @xmath110.\end{aligned}\ ] ] when @xmath111 , @xmath112
. then @xmath113.\end{aligned}\ ] ] as for @xmath114 , there are two cases ; when @xmath115 , @xmath116.\end{aligned}\ ] ] when @xmath117 or @xmath118 , @xmath119.\end{aligned}\ ] ] in both of cases , @xmath104 should be satisfied .
it is obvious that @xmath4 is detemined completely by the lcoal bases .
finally we should point out that our result is compatible with @xmath120 , defined by eq .
( 10 ) in ref .
@xcite . moreover @xmath121 is less than @xmath120 .
this compatibility display strongly the validity of our statement .
then we obtain the second conjecture [ c2 ] for @xmath122-type density operator defined by eq . , the `` nearest ''
completely classical state @xmath4 is necessarily the mixed combination of the bases @xmath68 with the joint probability distribution decided by eq ..
since the absence of schmidt decomposition in multipartite case , we focus on the states of symmetry in this section , which is defined as the invariance under the permutation or translation of single - party states . through several examples
we would demonstrate that the `` nearest '' @xmath4 necessarily displays the same invariance as that of the measured state . _
-ghz state- _
@xmath123 is obviously invariant under permutation .
then @xmath124 of which the maximal value is determined by relations @xmath125 the values for @xmath126 and @xmath127 are not unique ; for @xmath128 and @xmath129 , @xmath130 .
so @xmath131 , which obviously is invariant under permutation .
_ -@xmath132 state- _
@xmath133 is also invariant by permutation . by explicit calculation
, one finds @xmath134 in which @xmath135 it is not difficult to find that the extremum appears when @xmath136 . as for @xmath127 , one can find by thorough calculation that @xmath137 when @xmath138 .
thus @xmath139 and @xmath140 when , for instance , @xmath141 .
obviously @xmath4 is permutationally invariant too .
as for 4-qubit states , there exist a different invariance from the permutational , termed as translational invariance .
its meaning is similar to that in solid systems ; the difference is that it refer to single - party state in hilbert space in this place , instead of single particle in real lattice configuration @xcite .
we will display by two examples that the `` nearest '' @xmath4 is necessarily translationally invariant too . _
-@xmath142 state- _ , which is defined as @xmath143 .
it is obvious that the state is actually constructed by cyclic permutation of @xmath144 , which is named as _ cyclic unit_. it is not difficult to find @xmath145 in which @xmath146 which has maximal value @xmath147 and then @xmath148 when @xmath149 and @xmath150 .
consequently @xmath151 by setting @xmath152 , which is also translationally invariant .
_ -@xmath153- _
, which is defined as @xmath154 the state is actually constructed by cyclic unit @xmath155 .
moreover it is bi - seperable , @xmath156 .
thus the `` nearest '' @xmath4 can also be factorized into two parts , i.e. @xmath157 , in which @xmath158 and @xmath159 are the `` nearest '' completely classical states for @xmath160 and @xmath161 respectively . by conjecture [ c1 ]
, one can obtain @xmath162 and @xmath163 which is obviously translationally invariant .
we can obtain the third conjecture [ c3 ] for multipartite state with permutational or translational invariance , the `` nearest '' @xmath4 necessarily has the same invariance .
it should be pointed out that the form of @xmath4 can not be obtained directly from the measured state in general .
for example , we try to find the @xmath4 for dicke state @xmath164 . by explicit calculation , one
obtain @xmath165 which has maximal value when @xmath166 .
then @xmath167 and @xmath168 in which @xmath169 denotes the permutations of the density operators with two qubits being state @xmath170 and the other two being state @xmath171 .
in this section , we show that @xmath26 can also mark the quantum phase transition in many - body systems . for clarity , this discussion focuses on two popular models , lipkin - meshkov - glick ( lmg ) @xcite and dicke models @xcite , of which the ground states can be determined analytically .
the lmg model describes a set of spin - half particles coupled to all others with an interaction independent of the position and the nature of the elements .
the hamiltonian can be written as @xmath172 in which @xmath173 and the @xmath174 denotes the pauli operator , and @xmath175 is the total particle number in this system .
the prefactor @xmath176 is essential to ensure the convergence of the free energy per spin in the thermodynamic limit .
it is known that there is a second - order transition at @xmath177 for the ferromagnetic case ( @xmath178 ) and a first - order one at @xmath179 for the antiferromagnetic case ( @xmath180 ) @xcite .
the following discussion is divided into two parts by @xmath181 or not . _
-@xmath181- _ in this case the model can be solved exactly ; the eigenstate is @xmath182 , in which @xmath183 denote the quantum numbers of the total angular momentum @xmath184 and @xmath185 , and the corresponding eigenenergy is @xmath186 . for @xmath178 ,
the minimal value of @xmath187 appears when @xmath188 $ ] .
then the ground state is @xmath189 for @xmath190 .
the state is dicke state @xmath191 , for which obvioulsy @xmath192 . as for @xmath193 ,
the ground state is @xmath194\rangle}$ ] , which can be rewritten as dicke state@xmath195 in which @xmath196 $ ] .
it is obvious that the ground state is permutationally invariant .
thus with respect of dicke stat @xmath197 @xmath4 can be written directly as by conjecture [ c3 ] @xmath198 in which @xmath199 and @xmath200 with @xmath201 $ ] and @xmath202 .
@xmath203 can be determined by eq . .
it should be pointed out that because of permutational invariance the probability @xmath203 is same for the local states @xmath204 with the same @xmath205 . by numerical evaluation one has @xmath206 which is plotted for @xmath207 in fig.[fig : lmg1 ] . of the ground state for @xmath181 and @xmath178 in lmg model .
] as for @xmath180 , the minimal value of @xmath187 appears when @xmath208 $ ] .
then the ground state is @xmath189 for @xmath209 and @xmath210 for @xmath211 in angular moment picture , which are dicke states @xmath191 and @xmath212 respectively .
thus @xmath192 . _
-@xmath213- _ the ground state is @xcite @xmath214}(-1)^n\sqrt{\tfrac{(2n-1)!!}{2n!!}}\tanh^nx{|n , n-2n\rangle}_{\text{dicke}}\nonumber\\ c^2&=&\sum_{n=0}^{[n/2]}(-1)^n\tfrac{(2n-1)!!}{2n!!}\tanh^{2n}x,\end{aligned}\ ] ] in which @xmath215 for @xmath178 and @xmath216 for @xmath180 . in this case
@xmath4 has the same form as eq .. in fig.[fig : lmg23 ] , @xmath26 is plotted , in which the critical points can be identified , @xmath217 for @xmath178 and @xmath218 for @xmath180 . of the ground state for @xmath219 in lmg model.,title="fig : " ] of the ground state for @xmath219 in lmg model.,title="fig : " ] a generalization of lmg is the so - called uniaxial model , @xmath220 the ground state has the same form to eq . with @xmath221 ,
in which @xmath222 and @xmath223 is determined by the equation @xmath224 there are two critical points , @xmath225 for @xmath226 , which corresponds to a second order quantum phase transition and @xmath225 for @xmath227 , a first order one . as shown in fig .
[ fig : lmg4 ] , @xmath26 can unambiguously manifest the appearance of critical points . of the ground state for uniaxial model
when @xmath228.,title="fig : " ] of the ground state for uniaxial model when @xmath228.,title="fig : " ] of the ground state for uniaxial model when @xmath228.,title="fig : " ] dicke model @xciteis related to many fundamental issues in quantum optics , quantum mechanics and condensed matter physics , such as the coherent spontaneous radiation@xcite , the dissipation of quantum system @xcite , quantum chaos@xcite and atomic self - organization in a cavity@xcite .
the multipartite entanglement in dicke model has also been discussed @xcite .
the hamiltonian for single - model dicke model reads @xmath229 where @xmath230 and @xmath231 are the collective angular momentum operators .
there are two distinct phases for ground state , normal phase and superradiant phase , separated by critical point @xmath232 . of the ground state for dicke model
when @xmath228 , in which we has set @xmath233 and the critical point @xmath234 . ] by the method in ref .
@xcite , the reduce density operator of atom system for the ground state can be obtained analytically , which has the form @xmath235 in which @xmath236 .
it is obvious that @xmath0 is invariant under permutation .
then @xmath4 is eq . , and @xmath237 can be evaluated by numerical way , as shown in fig .
[ fig : dickemodel ] .
obviously it clearly marks the appearance of quantum phase transition .
the conjectures in this article can be generalized directly into multilevel case . in this section ,
we try to show the validity in comparison with two exactly solved examples in ref .
@xcite .
_ example 1 .
_ @xmath238-dimensional werner state @xmath239 in which @xmath240 $ ] and @xmath241 . in matrix , @xmath242 is the direct sum of the following sub - matrices @xmath243 in which @xmath244 thus @xmath242 is actually `` x '' form , defined in eq . .
it is easy to find @xmath245 in which @xmath246 and @xmath247 . by conjecture 2 ,
the nearest neighbor @xmath4 is @xmath248 in which @xmath249 and @xmath250 is determined by eq . .
thus by calculations , @xmath251\nonumber \\ & = & \left\ { m \tfrac{1+x}{m+1 } + \tfrac{m^2 - m}{4}\left [ \sqrt{\tfrac{1+x}{m^2 + m}}\ + \sqrt{\tfrac{1-x}{m^2 -m}}\ \right]^2 \right\}^{1/2}\end{aligned}\ ] ] thus one has @xmath252^{1/2}.\end{aligned}\ ] ] compared with the eq .
( 15 ) in ref .
@xcite @xmath253,\end{aligned}\ ] ] it is not difficult to find that the two results are compatible and @xmath254 . _ example 2 .
_ @xmath238-dimensional isotropic state @xmath255 in which @xmath256 . in matrix , @xmath257 is direct sum of two submatrices , spanned by the local orthonormal bases @xmath258 and @xmath259 respectively .
however @xmath257 does not show a `` x '' form so that an independent discussion is needed .
it should be pointed out that because of the isotropic feature , @xmath260 is already in the schmidt decomposition .
the other orthonormal states in the subspace spanned by @xmath258 is written as @xmath261 thus @xmath262 consequently the nearest neighbor @xmath4 should be the following form @xmath263 in which @xmath249 and @xmath250 can be decided by eq . . by calculation
one can obtain @xmath264\nonumber \\ & = & \sqrt{m \left [ \tfrac{\sqrt{x}}{m}+\tfrac{m-1}{m}\sqrt { \tfrac{1-x}{m^2 -1}}\right]^2+(m^2-m ) \tfrac{1-x}{m^2 -1 } } .\end{aligned}\ ] ] thus @xmath265 which is obviously compatible with eq .
( 17 ) in ref .
@xcite @xmath266 in this section we demonstrate the generality and popularity of our conjectures by two examples .
we also note that @xmath26 is always less than @xmath267 , defined by eq .
( 2 ) in ref .
in conclusion , a generalization of the geometric measure of quantum discord is introduced in this article .
our definition can be generalized readily into multipartite case .
moreover since the adopted hellinger distance and the uninvolved of local measurements , it does not suffered from the critiques raised recently in refs .
an important conclusin in this article is that in order to determine the optimal value of eq . , it is necessary to find the schmidt decomposition for the measured state .
then the optimal completely classical state @xmath4 is a joint distribution of the corresponding schmidt basis with the probability decided by eqs . and . in section
iii we display the validity of the result by exactly solving several examples .
then two conjectures are presented . up to our knowledge , it is the first general exact result for the geometric measure of quantum discord . for multipartite states , the geometric discord can also be evaluated exactly if the state possesses the invariance under permutation or translation , as shown in section iv .
furthermore it is pointed out in conjecture [ c3 ] that the optimal @xmath4 necessarily have the same invariance . in section
v we show by two models that our new definition can be used to mark the quantum phase transitions in many - body systems . a discussion of multilevel case is also presented in section vi .
two examples are worked out exactly , which also are studied in ref .
the fact that our results are compatible with that in ref .
@xcite unambiguously shows the validity and generality of our conjectures in this article .
finally we provide a further discussion on our conclusion .
as claimed in this article that the optimal @xmath4 is determined by the schmidt decomposition of bipartite state , it seems a natural hypothesis that one could found the generalized schmidt decomposition for multipartite state based on the `` nearest '' @xmath4 .
then a geometric understanding of schmidt decomposition can be constructed by this way , which is inevitably interesting , e.g. , in the measure of quantum correlation .
although this approaching is instructive , there are some problems to answer at first .
first as for pure bipartite state , the schmidt decomposition can be used to quantify the quantum entanglement in the state .
however it is unclear that this feature is preserved or not when generalized into multipartite .
second as for mixed state , what the meaning of schmidt decomposition is .
we do not know how to understand this point by now .
however it is still an interesting way to found the geometric understanding of quantum correlation .
this work is supported by nsf of china , grant no .
11005002 ( cui ) and 11475004 ( tian ) , new century excellent talent of m.o.e ( ncet-11 - 0937 ) , and sponsoring program of excellent younger teachers in universities in henan province of china ( 2010ggjs-181 ) .
k. modi , a. brodutch , h. cable , t. paterek , and v. vedral , rev .
phys . * 84 * , 1655 ( 2012 ) ; special issue `` classical vs quantum correlations in composite systems '' edited by l. amico , s. bose , v. korepin and v. vedral , int
b * 27 * , nos.1 and 3 , 1345011 - 1345055 ( 2013 ) .
paula , thiago r. de oliveira , and m.s .
sarandy , phys .
a * 87 * , 064101 ( 2013 ) ; s.j .
akhtarshenas , h. mohammadi , s. karimi , and z. azmi , arxiv : 1303.5570 ; f. ciccarello , t. tufarelli , and v. giovannetti , new j. phys * 16 * , 0103038 ( 2014 ) .
n. lamber , c. emary , and t. brandes , phys .
* 92 * , 073602(2004 ) ; phys .
rev . a * 71 * , 053804(2005 ) ; j. vidal , s. dusuel , t. barthel , j. stat .
p01015(2007 ) ; s. campbell , m. s. tame , and m. paternostro , new j. phys . * 11 * , 073039 ( 2009 ) ; h. t. cui , phys
a * 81 * , 042112 ( 2010 ) .
|
a generalization of the geometric measure of quantum discord is introduced in this article , based on hellinger distance .
our definition has virtues of computability and independence of local measurement .
in addition it also does not suffer from the recently raised critiques about quantum discord .
the exact result can be obtained for bipartite pure states with arbitrary levels , which is completely determined by the schmidt decomposition . for bipartite mixed states the exact result
can also be found for a special case .
furthermore the generalization into multipartite case is direct . it is shown that it can be evaluated exactly when the measured state is invariant under permutation or translation .
in addition the detection of quantum phase transition is also discussed for lipkin - meshkov - glick and dicke model .
|
the adiabatic theorem of quantum mechanics insures that an eigenstate of a system whose hamiltonian evolves sufficiently slowly in time ( as determined by criteria for the applicability of the theorem ) will remain in the same eigenstate , even though the eigenstate evolves in time @xcite .
hence , a slowly evolving system which is initially in its ground state will remain in the ground state throughout the course of its evolution .
the adiabatic theorem relies heavily on the superposition principle of quantum mechanics ( although in classical mechanics similar theorems are valid for nonlinear systems @xcite ) .
it is of interest to determine to what extent adiabaticity carries over to _ nonlinear _ quantum systems , such as bose - einstein condensates ( becs ) in the region where the mean - field description is appropriate .
well below the critical temperature , the mean - field description is based on the gross - pitaevskii equation ( gpe ) , @xmath0 \psi , \label{gp}\ ] ] and this approximation often yields excellent results for the system dynamics , even when the external potential @xmath1 varies with time . in eq .
( [ gp ] ) , @xmath2 is the atom - atom interaction strength that is proportional to the @xmath3-wave scattering length @xmath4 , and @xmath5 is the atomic mass .
the parameter @xmath6 in eq .
( [ gp ] ) is the total number of atoms , and the wave function @xmath7 is subject to the normalization @xmath8 ( the normalization integral is a dynamical invariant of gpe ) .
adiabatic considerations regarding the gpe dynamics have been applied to cold bosonic atoms trapped in optical lattices @xcite , and the formation of optical lattice gates for quantum computing from atomic becs @xcite . however , the applicability of the adiabaticity concept to becs does not follow from the above - mentioned adiabatic theorem of quantum mechanics , since the nonlinearity does not allow applicability of the superposition principle to the gpe .
on the other hand , adiabaticity of nonlinear wave equations , and in particular , of soliton solutions to such equations , have been extensively studied ( for a review , see ref .
nevertheless , bec dynamical problems based on the gpe have their own specific features , so that this case can be different from that studied in the framework of perturbed soliton solutions to the nonlinear schrdinger equation ( nlse ) in other contexts . here
we develop a physically relevant one - dimensional bec model which we study in detail by means of analytical and numerical methods to determine the nature of adiabaticity in nonlinear quantum systems within mean - field theory .
there are several regimes in which adiabaticity can be experimentally and theoretically probed for nonlinear systems .
the simplest regime is one for which the characteristic dynamical time scale ( i.e. , the time during which parameters of the hamiltonian undergo an essential change ) , @xmath9 , satisfies conditions @xmath10 here , @xmath11 is the quantum - mechanical linear adiabatic time scale determined in terms of the inverse of the difference of the energy eigenvalues at different values of time , @xmath12 \}$ ] , where the maximum is taken with respect to a given time interval @xcite , while the nonlinear time scale is @xmath13 , with @xmath14 being the instantaneous chemical potential @xcite . in this case
, the applicability of the adiabatic theorem of linear quantum mechanics @xcite is insured by the first inequality in ( [ applicability ] ) , and nonlinearity can not play a significant role in the dynamics due to the second inequality . therefore , the dynamics must be adiabatic in this case .
this regime applies to the nist optical - lattice experiments wherein microsecond duration light pulses are applied to a sodium bec @xcite .
a more intriguing and more problematic regime is when the dynamical time scale is _ large _ , i.e. , @xmath15 .
this case applies , e.g. , to the bec experiments reported in refs .
here , the nonlinearity plays an essential role in the dynamics , and a relevant question is whether the dynamics can be adiabatic .
generally , the answer is no , since further time scales may appear in the multidimensional gpe , viz .
, a diffraction time , @xmath16 , where @xmath17 is the length of the system ( see below ) , or a tunneling time scale , @xmath18 ( which itself may vary in the course of the system s evolution ) @xcite .
if @xmath9 is larger than _ all _ these time scales , clearly the dynamics will be adiabatic . in what follows ,
we show that the gpe does allow for adiabaticity when @xmath19 , and we give explicit criteria for the validity of adiabaticity in the gpe for the specific problem considered here .
we do so by means of an analytical estimate of corrections to ( deviations from ) the adiabatic approximation .
we also present numerical results for the dynamics of this system when this condition does not apply .
we consider a model based on the 1d gpe , in which the external potential @xmath20 is an infinitely deep well , @xmath21 the size of the well @xmath22 slowly varies with time , and we are interested in determining the behavior of the system in this case , to determine the applicability of adiabaticity .
the 1d gpe takes the form @xmath23 with the boundary conditions @xmath24 and normalization of the wave function , @xmath25 the nonlinearity parameter @xmath26 appearing in this 1d gpe is related to the nonlinearity parameter @xmath27 in its 3d counterpart , eq .
( [ gp ] ) , and is determined so that @xmath28 , where @xmath29 and @xmath30 are the maximum values of the 3d and 1d wave functions , respectively .
this condition insures that the time scales for the nonlinear interaction in the 3d and 1d cases are equal ( see below and ref .
@xcite ) .
this is the generalization of the particle in a box problem to the case where ( a ) the size of the box is varying with time , and ( b ) there are many bosonic particles in the box that are interacting via a mean field .
a typical situation in which the dynamics may be adiabatic is when the function @xmath22 takes on constant values as @xmath31 , and slowly varies in between on a long time scale @xmath9 .
we aim to find the final state @xmath32 into which an initial state @xmath33 will be transformed if @xmath9 is sufficiently large , and to check whether the wave function @xmath34 remains adiabatic during the course of the evolution , provided that the function @xmath22 varies slowly enough ( in practice , of course , the evolution time interval is large but finite ) . to determine what `` sufficiently slow '' means
, we define the nonlinear time scale obtained directly from the gpe as @xcite @xmath35 where @xmath36 and @xmath37 are the chemical potential and maximum of the wave function in the initial configuration .
the evolution is is slow as compared to nonlinear time scale if @xmath38 . in many bec systems ,
the nonlinear time scale is large compared with the diffraction time scale @xmath16 , also obtained directly from the gpe @xcite , so we should have @xmath39 .
it is convenient to transform the variables @xmath40 , @xmath41 and @xmath42 to new ( dimensionless ) variables @xmath43 , @xmath44 and @xmath45 : @xmath46 @xmath47 @xmath48 note that the problem is mapped onto a fixed spatial interval @xmath49 $ ] of the dimensionless spatial variable @xmath44 , and the boundary condition is therefore not time dependent when the problem is reformulated in terms of these variables .
the 1d gpe ( [ gpe ] ) takes the following form in terms of the new variables : @xmath50 where @xmath51 .
equation ( [ eq_u ] ) is supplemented by boundary conditions following from eq .
( [ bcl ] ) , @xmath52 the norm defined in terms of the transformed wave function @xmath45 , @xmath53\equiv \int_{0}^{1}\left| u(\xi , \tau ) \right|^{2}d\xi \ , , \label{normn}\ ] ] is _ not _ conserved in time , unlike the original norm , @xmath54 . indeed , as follows from the substitution of eq .
( [ u ] ) for @xmath45 into eq .
( [ normn ] ) , the @xmath45-norm is an explicit function of time : @xmath53=\frac{2gm}{\hbar^{2}}l(\tau ) \equiv n_{0}\frac{l(\tau ) } { l_{0}}\ , \label{nn}\ ] ] where @xmath55 ( or alternatively @xmath56 if the initial moment in time is @xmath57 ) .
the dimensionless nonlinear - strength parameter , @xmath58 introduced in eq .
( [ nn ] ) will play an important role below . when the system size @xmath59 is a slowly varying function of time , the right - hand side ( rhs ) of eq .
( [ eq_u ] ) is small , being proportional to the logarithmic derivative of the slowly varying function .
therefore eq .
( [ eq_u ] ) may be naturally considered as a perturbed self - defocusing nlse , and the adiabatic methods for nonlinear wave equations reviewed in ref .
@xcite might be applied . however , the perturbation term on the rhs of eq .
( [ eq_u ] ) need not allow straightforward application of the perturbation theory to the present problem since this term does _ not _ vanish at @xmath60 and @xmath61 when a general solution found in the zeroth - order approximation ( the expression ( [ zeroth ] ) below ) is inserted into it .
one can easily check that , as a consequence of this , a perturbative expansion generated by the term on rhs of eq .
( [ eq_u ] ) is incompatible with the boundary conditions ( [ bc1 ] ) . to resolve the problem ,
we transform the wave function once again , defining @xmath62 the transformation ( [ uv ] ) generates a more convenient form of the perturbed nls equation , @xmath63 which is subject to the same boundary conditions as in eq .
( [ bc1 ] ) , @xmath64 . an obvious advantage of having the perturbed nls equation in the form ( [ v ] ) is that now the perturbation vanishes at @xmath60 and @xmath61 , once a solution found in the zeroth - order approximation vanishes at these points .
note that the first term on the right hand side of eq .
( [ v ] ) is non - conservative .
accordingly , it is straightforward to see that this term leads to the exact relation ( [ nn ] ) for the norm evolution . another
important fact is that the second term on the rhs of eq .
( [ v ] ) , unlike the first term , is _ second - order small _ with regard to derivatives of the slowly varying functions . in the perturbation - theory section that follows below
, we will not consider effects produced by the second - order term , focusing solely on the most important first - order effects .
in zeroth - order approximation of the perturbation theory ( neglecting the rhs of eq .
( [ v ] ) ) , an exact stationary solution satisfying the zero boundary conditions at @xmath60 and @xmath61 is given by @xcite : @xmath65 here @xmath66 is the doubly periodic jacobi elliptic sine function , @xmath67 is the corresponding elliptic modulus , @xmath68 is the complete elliptic integral of the first kind , and the chemical potential @xmath36 is related to @xmath67 as follows : @xmath69 the modulus @xmath67 , which takes values @xmath70 , determines the strength of the nonlinearity : it is weak if @xmath71 , and strong if @xmath72 . in fact , @xmath67 is related directly to the dimensionless nonlinearity - strength parameter @xmath73 defined in eq .
( [ n0 ] ) as follows : @xmath74 = n_{0}\ , \label{k_n0_basic}\ ] ] where @xmath75 is the complete elliptic integral of the second kind .
thus , @xmath67 completely determines the normalization of the initial wave function , and visa versa . to illustrate the zeroth - order solution , plots of @xmath76 $ ] versus @xmath67 ( see eq .
( [ k_n0_basic ] ) ) , and @xmath77 versus @xmath44 for three different values of @xmath67 in the regime of strong nonlinearity , @xmath78 ( see eq .
( [ zeroth ] ) ) , are displayed in figs .
[ fig1](a ) and [ fig1](b ) .
we remark that an exact solution that can be expressed in terms of the jacobi elliptic functions is frequently called a _
cnoidal wave _ , which stems from the notation @xmath79 for the jacobi s elliptic cosine , related to the elliptic sine . the first - order perturbation term on the rhs of eq .
( [ v ] ) can be treated in terms of nonlinear adiabatic perturbation theory @xcite .
we stress that , unlike the perturbation term in the intermediate equation ( [ eq_u ] ) , which is `` abnormal '' in the sense that it is not compatible with the necessary boundary conditions , as it was explained above , the `` normal '' perturbation in eq .
( [ v ] ) satisfies the boundary conditions .
the applicability of simple perturbative techniques for this class of models can be proved using a rigorous expansion based on the inverse scattering transform for the unperturbed nls equation ( i.e. , one can prove that the `` simple techniques '' yield , in the lowest - order nontrivial approximation , exactly the same results as the rigorous methods , see ref .
@xcite and references therein ) .
the first standard step of the perturbative analysis is to apply the lowest - order adiabatic approximation .
this approximation takes the unperturbed solution ( [ zeroth ] ) , which contains the parameter @xmath67 , and makes it the first - order approximate solution to the perturbed equation , assuming that the modulus @xmath67 is slowly varying in time , rather than remaining constant . the slow dependence of the parameter(s )
is introduced so as to cancel the secular divergence(s ) in the perturbation theory .
an important case is when the unperturbed solution contains a single nontrivial parameter ( @xmath67 , in the present case ) , and the perturbed equation gives rise to an exact relation replacing a conservation law existing in the unperturbed version of the equation ( this exact relation is usually called a _ balance equation _ for the ( former ) conserved quantity ) .
this is the case in eq .
( [ nn ] ) . then , the time dependence of the parameter , i.e. , @xmath80 , can be found in a very simple way by substitution of the zeroth - order approximation for the solution into the balance equation @xcite . in the present case
, this condition amounts to evaluation of the actual value of the norm ( [ normn ] ) , inserting the solution ( [ zeroth ] ) into it , and then substituting the result into the exact relation ( [ nn ] ) .
the final result is @xmath74 = n_{0}\,l(\tau ) /l_{0}\ .
\label{basic}\ ] ] eq .
( [ basic ] ) is a transcendental equation to determine @xmath81 for a given function @xmath59 and @xmath73 ( recall that @xmath73 is a constant ) .
an essential ingredient of the adiabatic approximation is a consistent definition of the phase @xmath82 for the first - order solution with variable @xmath81 . indeed , substituting @xmath81 back into the general expressions ( [ zeroth ] ) and ( [ mu ] ) for the wave function , it is easy to see that the consistently defined phase is @xmath83 k^{2}\left ( k(\tau^{\prime } ) \right ) \,d\tau^{\prime } \ , \label{phi}\ ] ] @xmath84 being the initial time ( @xmath85 in the usual formulation of the adiabatic approximation ) .
thus , the full expression for the lowest - order perturbative solution obtained in the adiabatic approximation is @xmath86 where @xmath87 and @xmath88 are given by eqs .
( [ zeroth ] ) and ( [ phi ] ) , respectively .
note that expression ( [ first ] ) automatically satisfies the zero boundary conditions at the points @xmath60 and @xmath61 . knowing a particular form of the slow temporal dependence @xmath81 obtained from eq .
( [ basic ] ) , one can find the temporal dependence of the solution s amplitude , @xmath89 the temporal dependence of state s width ( which , for instance , can be defined as the full width at half - maximum of @xmath90 ) can similarly be obtained in the adiabatic approximation from the above expressions . using eqs .
( [ mu ] ) and ( [ basic ] ) , it is also possible to predict the evolution of the instantaneous value of the chemical potential @xmath91 .
once the slow time dependence of @xmath81 has been determined as described above , one can look for perturbation - induced corrections to the state s shape , which was not taken into account in the first - order adiabatic approximation . a solution to eq .
( [ v ] ) including the corrections can be sought in the form of an expansion compatible with the zero boundary conditions , namely , @xmath92
\exp \left ( i\phi ( \tau ) \right ) \ , \label{corrections}\ ] ] where the functions @xmath1 and @xmath93 are those which were obtained in the previous subsection .
the simplest way to derive evolution equations for the amplitudes @xmath94 is to directly substitute the expansion ( [ corrections ] ) into eq .
( [ v ] ) , multiply the resulting equation by @xmath95 , and integrate from @xmath60 to @xmath61 , carrying out this procedure for each integer @xmath5 .
the correction terms are neglected when substituting the expression ( [ corrections ] ) into the first perturbation term on the rhs of eq .
( [ v ] ) , as they would give rise to higher - order perturbations . implementing this procedure
, we use the classical fourier expansion for the function @xmath96 , @xmath97 \ , .
\label{q}\end{aligned}\ ] ] a complicated system of inhomogeneous linear evolution equations for @xmath98 ensues .
if @xmath5 is odd , i.e. , @xmath99 , we obtain @xmath100 and , if @xmath5 is even , i.e. , @xmath101 , @xmath102 here @xmath103 is the _ jacobi parameter _ defined in eq .
( [ q ] ) , and the coefficients appearing on the left - hand sides of eqs .
( [ odd ] ) and ( [ even ] ) are @xmath104 d\xi \ . \label{m}\end{aligned}\ ] ] in fact , all the coefficients @xmath105 with @xmath5 and @xmath106 having opposite parities are zero , hence we may set @xmath107 , and we are left with the system of equations ( [ odd ] ) . recall that one should substitute the time - dependent modulus @xmath80 as found from eqs .
( [ basic ] ) ) into the above expressions .
this cumbersome system can be simplified if @xmath80 does not take on values too close to unity .
then , @xmath96 remains close to the usual sine function ( for instance , at @xmath108 , the jacobi parameter , which determines the anharmonicity of the expansion ( [ jacobi ] ) , is @xmath109 , which may be regarded as a sufficiently small expansion factor ) .
thus , to obtain a simple approximation for the coefficients @xmath105 defined in eq .
( [ m ] ) , one may simply set @xmath110 . within this approximation the only nonzero components of in the matrix @xmath111 are @xmath112 furthermore , the rhs of eq .
( [ odd ] ) also greatly simplifies in the same approximation .
it is different from zero solely for @xmath113 , being equal to @xmath114 .
thus , the approximation which replaces the @xmath96 function by the usual sine leads to the following equations , instead of eqs .
( [ odd ] ) and ( [ even ] ) : @xmath115 b_{1}+3k^{2}k^{2}(k)b_{1}^{\ast } + k^{2}k^{2}\left ( 2b_{3}-b_{3}^{\ast } \right ) = \frac{il_{\tau } } { 2l}\ , , \label{p=1}\ ] ] @xmath116^{2}}{2}\right ] b_{2p-1}+2k^{2}k^{2}(k)b_{2p-1}^{\ast } \ ] ] @xmath117 where @xmath118 .
recall that all the amplitudes @xmath119 with even values of @xmath5 are zero .
despite the fact that the approximate system consisting of eqs .
( [ p=1 ] ) and ( [ 2p-1 ] ) is considerably simpler than the exact eqs .
( [ odd ] ) , it can only be solved numerically by truncating the system of the linear equations at some finite integer .
nevertheless , some qualitative generic features of the solution can be determined .
the general structure of the system is of the form @xmath120 where @xmath121 is a column vector of the variables @xmath122 , @xmath123 is a matrix of coefficients multiplying the variables @xmath124 , and @xmath125 is the vector column of free terms on the left - hand side , with the single nonzero entry @xmath126 . both @xmath125 and @xmath123 slowly depend upon time - the former directly , the latter via @xmath127 . solutions to the system ( [ general ] )
consist of terms of the type @xmath128 where @xmath129 is an eigenvalue of the matrix @xmath130 , and @xmath131 are slowly varying functions similar to the above - mentioned @xmath132 .
note that the time scales @xmath133 , determined by different eigenfrequencies @xmath134 , are , in fact , a mixture of the adiabatic and nonlinear time scales , @xmath11 and @xmath135 , defined in the introduction .
the following conclusion can be made concerning the size of the _ nonadiabatic _ effects ( shape corrections ) considered above . if the function @xmath136 slowly depends on @xmath43 with a characteristic time scale @xmath9 ( as defined in the introduction ) , and if a characteristic value of @xmath137 is @xmath138 ( within the limits of its slow evolution on the time scale @xmath139 ) , the criterion for the applicability of the adiabatic approximation is @xmath140 we stress that , as the characteristic times @xmath141 taken for the different eigenfrequencies constitute a set including the adiabatic and nonlinear time scales @xmath11 and @xmath135 ( see above ) , the inequality ( [ condition ] ) is exactly the condition for the applicability of the adiabatic approximation conjectured in the introduction .
the evolution equations ( [ general ] ) for the shape - correction amplitudes are to be solved for an initial state without shape corrections , i.e. , @xmath142 @xmath143 . if one takes the initial moment as @xmath144 ( as mentioned above , this is the standard assumption in the treatment of adiabatic processes @xcite ) , one can determine eventual values of the shape - correction amplitudes as @xmath98 at @xmath145 .
classical estimates for integrals involving products of rapidly and slowly varying functions @xcite show that the values of @xmath146 are _ exponentially small _ when condition ( [ condition ] ) is satisfied : @xmath147 a particular value of the constant in this expression depends on the choice of the unperturbed state and on the form of the function @xmath136 .
hence , in analogy with the well - known theorems estimating nonadiabatic corrections to the adiabatic approximation in ( nonlinear ) classical mechanics @xcite , the @xmath148 values are exponentially small .
we first present results for the amplitudes @xmath149 in eq .
( [ corrections ] ) , obtained by numerically solving eqs .
( [ p=1 ] ) and ( [ 2p-1 ] ) .
we take @xmath150 , @xmath151 , and @xmath152 \,,\ ] ] with @xmath153 .
figure [ fig2 ] shows the computed excited - state probability , @xmath154 versus time @xmath43 , with the number of modes kept in the truncated calculation being @xmath155 , and @xmath156 , for @xmath157 . except for @xmath158 ,
all the curves lie on top of each other , hence the results do converge very quickly as a function of the number of the modes .
we see from fig .
[ fig2 ] that the probability of finding excited states for all times is below @xmath159 , and for @xmath160 the probability is exceedingly small , i.e. , the process is almost completely adiabatic .
a minimum of @xmath161 is expected from the general form of the perturbation equations since the derivative of @xmath136 vanishes at @xmath162 . for @xmath163 ,
the excited - state probability ( [ excited ] ) begins to be large ( @xmath164 ) , and the adiabatic - theory results are no more reliable .
for example , fig .
[ fig3 ] shows the results for @xmath165 .
again the convergence as a function of the number of the modes is very fast , but the excited state probability is not small . for times
@xmath166 , @xmath167 oscillates with time . for stronger nonlinearity , @xmath168 ,
perturbative methods can not be used ( in particular , the approximation based on the replacement of the elliptic sine by the ordinary sine , as described in the above section , does not apply ) , so we directly solved eq .
( [ v ] ) using a split - step fast fourier transform method , in order to check if adiabaticity still takes place in this regime .
figure [ fig4 ] shows the results for the calculated wave function @xmath169 versus @xmath44 in the box at the completion of the dynamical process for small ( non - adiabatic ) time - scale , @xmath170 , and for @xmath171 .
also shown for comparison is the initial eigenstate magnitude @xmath172 .
the magnitude of the wave function in the final state is not too different from that in the initial eigenstate , and the spatial variation of the phase is fairly flat .
figure [ fig5 ] pertains to the same case , but with a larger time scale , @xmath173 .
now , the magnitudes of the wave function in the finite and initial state are barely distinguishable on the scale of the figure , and the spatial profile of the phase is almost flat .
thus , the process is largely adiabatic in the latter case .
we have presented a consistent derivation of the nonlinear dynamics for a simple model describing a bec confined in a box with a temporally - varying size @xmath136 .
the speed of the variation of @xmath136 determines whether the dynamics is adiabatic .
a `` trivial '' regime of adiabaticity is that for which @xmath174 ; in this work , we have shown that adiabaticity can also be maintained when @xmath19 , where the various time scales have been defined in the introduction .
if other time scales appear , the condition @xmath175 may not be sufficient to insure adiabaticity .
for example , if a barrier is present in the middle of the box - e.g. , a repulsive delta - function at @xmath176 - then another time scale , corresponding to the time of tunneling under the barrier , @xmath177 , is present in the problem .
if this time scale is long compared with @xmath135 , adiabaticity will not be maintained , and a non - vanishing spatially - varying phase will develop across the condensate wave function @xcite . hence , the issue of adiabaticity in nonlinear problems must be investigated carefully ; the perturbative techniques reviewed in ref .
@xcite may be applicable , but additional considerations may play a role . the particle in a box is a paradigm problem in one - body quantum mechanics .
we have extended it to the many - body regime at least within a mean - field approach , and studied the adiabaticity for such a system when the size of the box varies with time .
specifically , we formulated the criteria for the validity of the adiabatic approximation for a bec in a box whose size varies with time , developed the analytical and numerical tools for investigating adiabaticity in the dynamics within quantum mean - field theory , and presented results of calculations for this system .
this work was supported in part by grants from the us - israel binational science foundation , the israel science foundation , the james franck binational german - israeli program in laser - matter interaction , and the polish kbn 62/p03/2000/18 .
an alternative notation for the jacobi elliptic functions and the elliptic integrals is sometimes used .
see for example , _ handbook of mathematical functions _ , edited by m. abramowitz and i.a .
stegun ( national bureau of standards , washington , dc , 1964 ) , and ref . @xcite . in this alternative notation , @xmath178 , where @xmath179 and @xmath180 .
also , @xmath181 .
$ ] versus the elliptic modulus @xmath67 .
( b ) the zeroth - order analytic solution @xmath182 for three different values of @xmath67 .
the normalization of these soliton solutions can be read off the curve in ( a).,width=288 ]
|
a simple model of an atomic bose - einstein condensate in a box whose size varies with time is studied to determine the nature of adiabaticity in the nonlinear dynamics obtained within the gross - pitaevskii equation ( the nonlinear schrdinger equation ) .
analytical and numerical methods are used to determine the nature of adiabaticity in this nonlinear quantum system .
criteria for validity of an adiabatic approximation are formulated .
|
the electroweak form factors are sets of functions that are used to parameterize the structure of the nucleon as seen by the electromagnetic and the weak probes .
while a wealth of data and theoretical predictions exist for the electromagnetic form factors ( see , e.g. , @xcite and references therein ) , the nucleon form factors of the isovector axial - vector current , the axial form factor @xmath5 and , in particular , the induced pseudoscalar form factor @xmath6 , are not as well - known ( see , e.g. , @xcite for a review ) . however , there are ongoing efforts to increase our understanding of these form factors .
the value of the axial form factor at zero momentum transfer is defined as the axial - vector coupling constant @xmath7 and is quite precisely determined from neutron beta decay .
the @xmath8 dependence of the axial form factor can be obtained either through neutrino scattering or pion electroproduction ( see @xcite and references therein ) .
the second method makes use of the so - called adler - gilman relation @xcite which provides a chiral ward identity establishing a connection between charged pion electroproduction at threshold and the isovector axial - vector current evaluated between single - nucleon states ( see , e.g. , @xcite for more details ) .
the induced pseudoscalar form factor @xmath6 is even less known than @xmath5 .
it has been investigated in ordinary and radiative muon capture as well as pion electroproduction .
analogous to the axial - vector coupling constant @xmath7 , the induced pseudoscalar coupling constant is defined through @xmath9 , where @xmath10 corresponds to muon capture kinematics and the additional factor @xmath11 stems from a different convention used in muon capture . for a comprehensive review on the experimental and theoretical situation concerning @xmath6 see for example @xcite .
a discrepancy between the results in ordinary and radiative muon capture has recently been addressed in @xcite .
theoretical approaches to the axial and induced pseudoscalar form factors include the early current algebra and pcac calculations @xcite , various quark model ( see , e.g. , @xcite ) and lattice calculations @xcite .
for a recent discussion on extracting the axial form factor in the timelike region from @xmath12 ( @xmath13 or @xmath14 ) see @xcite .
chiral perturbation theory ( chpt ) @xcite is the low - energy effective theory of the standard model and as such allows model - independent calculations of nucleon properties ( see @xcite for an introduction ) .
the axial form factor has been addressed in the framework of heavy - baryon chpt @xcite . in principle , when considering a charged transition there is a third form factor , the induced pseudotensorial form factor @xmath15 . as will be explained below , this form factor vanishes when combining isospin symmetry and charge - conjugation invariance and therefore is not considered in this work @xcite .
experimentally the induced pseudotensorial form factor is found to be small @xcite .
finally , defining the pion - nucleon form factor in terms of the pseudoscalar quark density and using the partially conserved axial - vector current ( pcac ) relation allows one to determine the pion - nucleon form factor , once the axial and induced pseudoscalar form factors are known . in this paper
we calculate the axial , the induced pseudoscalar , and the pion nucleon form factors of the nucleon in manifestly lorentz - invariant chpt up to and including order @xmath3 .
the renormalization procedure is performed in the framework of the infrared renormalization of @xcite . in its reformulated version @xcite
, this renormalization scheme allows for the inclusion of further degrees of freedom . in the following
we will include the @xmath4 axial - vector meson as an explicit degree of freedom .
it needs to be pointed out that in a strict chiral expansion up to order @xmath3 the results will not differ from the ones obtained in the standard framework .
however , explicitly keeping all terms generated from the considered diagrams involving the axial - vector meson amounts to a resummation of higher - order contributions .
this phenomenological approach has shown an improved description of the electromagnetic form factors of the nucleon @xcite when the @xmath16 , @xmath17 , and @xmath18 mesons are included .
this paper is organized as follows : in sec .
[ sec : def ] the definitions and some important properties of the relevant form factors are given . section [ sec : lag ] contains the effective lagrangians used in the present calculation .
we present and discuss the results for the form factors with and without the inclusion of the axial - vector meson @xmath4 in sec .
[ sec : results ] .
section [ sec : sum ] contains a short summary .
in qcd , the three components of the isovector axial - vector current are defined as @xmath19 the operators @xmath20 satisfy the following properties relevant for the subsequent discussion : 1 .
hermiticity : @xmath21 2 .
equal - time commutation relations with the vector charges : @xmath22=i\epsilon^{abc}a^{\mu , c}(t,\vec x).\ ] ] 3 .
transformation behavior under parity : @xmath23 4 .
transformation behavior under charge conjugation : @xmath24 5 .
partially conserved axial - vector current ( pcac ) relation : @xmath25 where @xmath26 is the quark mass matrix . assuming isospin symmetry , @xmath27 , the most general parametrization of the isovector axial - vector current evaluated between one - nucleon states in terms of axial - vector covariants is given by @xmath28 \frac{\tau^a}{2}u(p),\ ] ] where @xmath29 and @xmath30 denotes the nucleon mass .
@xmath5 is called the axial form factor and @xmath6 is the induced pseudoscalar form factor . from the hermiticity of eq .
( [ aherm ] ) , we find that @xmath0 and @xmath1 are real for space - like momenta ( @xmath31 ) . in the case of perfect isospin symmetry
the strong interactions are invariant under @xmath32 conjugation , which is a combination of charge conjugation @xmath33 and a rotation by @xmath34 about the 2 axis in isospin space ( charge symmetry operation ) , @xmath35 the presence of a third so - called second - class structure @xcite of the type @xmath36 in the charged transition would indicate a violation of @xmath37 conjugation . as there seems to be no clear empirical evidence for such a contribution @xcite we will omit it henceforth .
similarly , the nucleon matrix element of the pseudoscalar density @xmath38 can be parameterized as @xmath39 where @xmath40 is the pion mass and @xmath41 the pion decay constant .
equation ( [ gpin ] ) _ defines _ the form factor @xmath42 in terms of the qcd operator @xmath43 .
the operator @xmath44 serves as an interpolating pion field and thus @xmath42 is also referred to as the pion - nucleon form factor for this specific choice of the interpolating pion field @xcite . the pion - nucleon coupling constant @xmath45 is defined through @xmath42 evaluated at @xmath46 . as a result of the pcac relation , eq .
( [ pcac ] ) , the three form factors @xmath0 , @xmath1 , and @xmath2 are related by @xmath47
the calculation of the isovector axial - vector current form factors of the nucleon requires both the purely mesonic as well as the one - nucleon part of the chiral effective lagrangian up to order @xmath3 , @xmath48 here , @xmath49 collectively stands for a `` small '' quantity such as the pion mass , a small external four - momentum of the pion or of an external source , and an external three - momentum of the nucleon .
the pion fields are contained in the @xmath50 matrix @xmath51 , @xmath52 and the purely mesonic lagrangian at order @xmath53 is given by @xcite @xmath54 + \frac{f^2}{4}\mbox{tr}\left[\chi u^\dagger + u \chi^\dagger\right].\ ] ] the covariant derivative @xmath55 with a coupling to an external axial - vector field @xmath56 only is given by @xmath57 while @xmath58 is defined as @xmath59 with @xmath60 and @xmath49 the scalar and pseudoscalar external sources , respectively .
@xmath61 denotes the pion decay constant in the chiral limit , @xmath62 = 92.42(26)$ ] mev @xcite .
we work in the isospin - symmetric limit @xmath63 , and the lowest - order expression for the squared pion mass is @xmath64 , where @xmath65 is related to the quark condensate @xmath66 in the chiral limit @xcite , @xmath67 . for the mesonic lagrangian at order @xmath3 we only list the term that contributes to our calculation , @xmath68\mbox{tr}\left[\chi u^\dagger + u \chi^\dagger\right ] + \cdots\,.\ ] ] the complete list for the @xmath69 case can be found in @xcite .
the lowest - order pion - nucleon lagrangian is given by @xcite @xmath70 with @xmath71 the nucleon mass and @xmath72 the axial - vector coupling constant both evaluated in the chiral limit .
for the nucleonic lagrangians of higher orders we only display those terms that contribute to our calculations .
a complete list of terms at orders @xmath53 and @xmath73 can be found in @xcite . at second order
the lagrangian reads @xmath74 + \frac{c_3}{2}\,\bar{\psi}\mbox{tr}\left(u_\mu u^\mu\right)\psi\nonumber\\ % & & + \bar{\psi}\left [ i\frac{c_4}{4 } \left[u_\mu , u_\nu \right ] \right ] & & -\frac{c_4}{4}\bar\psi\gamma^\mu\gamma^\nu [ u_\mu , u_\nu ] \psi + \cdots , \end{aligned}\ ] ] while at order @xmath73 we need @xmath75\psi + \cdots\,.\ ] ] there are no contributions from @xmath76 in our calculation .
the lagrangians contain the building blocks @xmath77,\\ u_\mu & = & i\left [ u^\dagger \partial_\mu u -u \partial_\mu u^\dagger - i ( u^\dagger a_\mu u + u a_\mu u^\dagger)\right],\\ \chi_+&=&u^\dagger\chi u^\dagger+u\chi^\dagger u,\\ f^-_{\mu\nu } & = & u^\dagger(\partial_\mu a_\nu-\partial_\nu a_\mu -i[a_\mu , a_\nu])u + u(\partial_\mu a_\nu-\partial_\nu a_\mu + i[a_\mu , a_\nu])u^\dagger,\end{aligned}\ ] ] where we only display the external axial - vector source @xmath78 . in order to include axial - vector mesons as explicit degrees of freedom we consider the vector - field formulation of @xcite in which the @xmath79 meson is represented by @xmath80 .
the advantage of this formulation is that the coupling of the axial - vector mesons to pions and external sources is at least of order @xmath73 .
a complete list of possible couplings at this order can be found in @xcite .
the calculation of the contributions to the isovector axial - vector form factors only requires the term @xmath81 where @xmath82 with @xmath83.\ ] ] the coupling of the axial - vector meson to the nucleon starts at order @xmath84 .
the corresponding lagrangian reads @xmath85 a calculation up to order @xmath3 would in principle also require the lagrangian of order @xmath86 . however , there is no term at this order that is allowed by the symmetries .
in addition to the usual counting rules for pions and nucleons ( see , e.g. , @xcite ) , we count the axial - vector meson propagator as order @xmath84 , vertices from @xmath87 as order @xmath73 and vertices from @xmath88 as order @xmath84 , respectively @xcite .
the axial form factor @xmath5 only receives contributions from the one - particle - irreducible diagrams of fig .
[ irreduc ] .
the unrenormalized result reads @xmath89 -c_3 i_{{\pi}n}^{(00)}(m_n^2 ) \right\}\nonumber \\ & & -\frac{\texttt{g}_a^3}{4f^2 } \left [ i_{\pi}-4m_n^2 i_{{\pi}n}^{(p)}(m_n^2 ) + 4m_n^2(n-2)i_{{\pi}nn}^{(00)}(q^2)\right.\nonumber\\ & & \left .
+ 16m_n^4 i_{{\pi}nn}^{(pp)}(q^2 ) + 4m_n^2 t i_{{\pi}nn}^{(qq)}(q^2 ) \right].\end{aligned}\ ] ] the definition of the integrals can be found in the appendix . to renormalize the expression for @xmath5
we multiply eq .
( [ ga ] ) by the nucleon wave function renormalization constant @xmath90 @xcite , @xmath91 + \frac{9 \texttt{g}_a^2 m^3}{64\pi f^2 m},\ ] ] and replace the integrals with their infrared singular parts . the axial - vector coupling constant @xmath7 is defined as @xmath92 @xcite and we obtain for its quark - mass expansion @xmath93 with @xmath94 where all coefficients are understood as ir renormalized parameters .
these results agree with the chiral coefficients obtained in hbchpt @xcite as well as the ir calculation of @xcite .
it is worth noting that an agreement for the analytic term @xmath95 can not be expected in general .
for example , when expressed in terms of the renormalized couplings of the extended on - mass - shell ( eoms ) renormalization scheme of @xcite , the @xmath95 coefficient is given by @xcite @xmath96 such a difference is not a surprise , because the use of different renormalization schemes is compensated by different values of the renormalized parameters . for a similar discussion regarding the chiral expansion of the nucleon mass , see @xcite .
the axial form factor can be written as @xmath97 where @xmath98 is the axial mean - square radius and @xmath99 contains loop contributions and satisfies @xmath100 .
the low - energy coupling constants ( lecs ) @xmath101 and @xmath102 are thus absorbed in the axial - vector coupling constant @xmath7 and the axial mean - square radius @xmath98 .
the numerical contribution of @xmath99 is negligible which can be understood by expanding @xmath103 in a taylor series in @xmath8 .
such an expansion generates powers of @xmath104 where the individual coefficients have a chiral expansion similar to eq .
( [ gaexpand ] ) . for the analysis of experimental data
, @xmath5 is conventionally parameterized using a dipole form as @xmath105 where the so - called axial mass @xmath106 is related to the axial root - mean - square radius by @xmath107 . the global average for the axial mass extracted from neutrino scattering experiments given in @xcite
is @xmath108 whereas a recent analysis @xcite taking account of updated expressions for the vector form factors finds a slightly smaller value @xmath109 on the other hand , smaller values of @xmath110 gev and @xmath111 gev have been obtained in @xcite as world averages from quasielastic scattering and @xmath112 gev from single pion neutrinoproduction .
finally , the most recent result extracted from quasielastic @xmath113 in oxygen nuclei reported by the k2k collaboration , @xmath114 gev , is considerably larger @xcite .
the extraction of the axial mean - square radius from charged pion electroproduction at threshold is motivated by the current algebra results and the pcac hypothesis . the most recent result for the reaction
@xmath115 has been obtained at mami at an invariant mass of @xmath116 mev ( corresponding to a pion center - of - mass momentum of @xmath117 mev ) and photon four - momentum transfers of @xmath118 , @xmath119 and 0.273 gev@xmath120 @xcite . using an effective - lagrangian model an axial mass of @xmath121 was extracted , where the bar is used to distinguish the result from the neutrino scattering value . in the meantime
, the experiment has been repeated including an additional value of @xmath122 gev@xmath120 @xcite and is currently being analyzed .
the global average from several pion electroproduction experiments is given by @xcite @xmath123 it can be seen that the values of eqs .
( [ mav1 ] ) and ( [ mav2 ] ) for the neutrino scattering experiments are smaller than that of eq .
( [ mapi ] ) for the pion electroproduction experiments .
the discrepancy was explained in heavy baryon chiral perturbation theory @xcite .
it was shown that at order @xmath73 pion loop contributions modify the @xmath124 dependence of the electric dipole amplitude from which @xmath125 is extracted .
these contributions result in a change of @xmath126 bringing the neutrino scattering and pion electroproduction results for the axial mass into agreement .
using the convention @xmath127 the result for the axial form factor @xmath5 in the momentum transfer region @xmath128 is shown in fig .
[ gawithout ] .
the parameters have been determined such as to reproduce the axial mean - square radius corresponding to the dipole parameterization with @xmath129 gev ( dashed line ) .
the dotted and dashed - dotted lines refer to dipole parameterizations with @xmath130 gev and @xmath131 gev , respectively . as anticipated , the loop contributions from @xmath99 are small and the result does not produce enough curvature to describe the data for momentum transfers @xmath132 .
the situation is reminiscent of the electromagnetic case @xcite where chpt at @xmath3 also fails to describe the form factors beyond @xmath133 .
the one - particle - irreducible diagrams of fig .
[ irreduc ] also contribute to the induced pseudoscalar form factor @xmath6 , @xmath134 furthermore , @xmath6 receives contributions from the pion pole graph of fig .
[ pipoledia ] .
it consists of three building blocks : the coupling of the external axial source to the pion , the pion propagator , and the @xmath135-vertex , respectively .
we consider each part separately . the renormalized coupling of the external axial source to a pion up to order @xmath3 is given by @xmath136 where the diagrams in fig .
[ apidia ] have been taken into account and the renormalized pion decay constant reads @xmath137.\ ] ] we have used the pion wave function renormalization constant @xmath138,\ ] ] with @xmath139 the renormalized coupling of eq .
( [ l4 ] ) and @xmath140 .
the renormalized pion propagator is obtained by simply replacing the lowest - order pion mass @xmath141 by the expression for the physical mass @xmath40 up to order @xmath3 , @xmath142.\ ] ] the @xmath135 vertex evaluated between on - mass - shell nucleon states up to order @xmath3 receives contributions from the diagrams in fig .
[ pindia ] and the unrenormalized result for a pion with isospin index @xmath143 is given by @xmath144 -c_3 i_{{\pi}n}^{(00)}(m_n^2 ) \right\ } \nonumber \\ & & \left .
+ \frac{\texttt{g}_a^3}{4f^3}m_n\left [ i_{\pi } + 4mn^2 i_{nn}(q^2 ) + 4m_n^2 m^2 i_{{\pi}nn}(q^2 ) \right]\right)\gamma_5 \tau_i\,.\end{aligned}\ ] ] to find the renormalized vertex one multiplies with @xmath145 and replaces the integrals with their infrared singular parts .
however , the renormalized result should not be confused with the pion - nucleon form factor @xmath42 of eq .
( [ gpin ] ) . in general
, the pion - nucleon vertex depends on the choice of the field variables in the ( effective ) lagrangian . in the present case ,
the pion - nucleon vertex is only an auxiliary quantity , whereas the `` fundamental '' quantity ( entering chiral ward identities ) is the matrix element of the pseudoscalar density . only at @xmath46
, we expect the same coupling strength , since both @xmath44 and the field @xmath146 of eq .
( [ u ] ) serve as interpolating pion fields .
after renormalization , we obtain for the pion - nucleon coupling constant the quark - mass expansion @xmath147 with @xmath148 where all coefficients are understood as ir renormalized parameters .
these results agree with the chiral coefficients obtained in @xcite . in the chiral limit ,
( [ gpinexpand ] ) satisfies the goldberger - treiman relation @xmath149 @xcite .
the numerical violation of the goldberger - treiman relation as expressed in the so - called goldberger - treiman discrepancy @xcite , @xmath150 is at the percent level , @xmath151 % for @xmath152 mev , @xmath153 , @xmath154 mev , and @xmath155 @xcite . using different values for the pion - nucleon coupling constant such as @xmath156 @xcite , @xmath157 @xcite , and @xmath158 @xcite results in the gt discrepancies @xmath159 % , @xmath160 % , and @xmath161 % , respectively . the chiral expansions of @xmath7 etc .
may be used to relate the parameter @xmath162 to @xmath163 @xcite , @xmath164 note that @xmath163 of eq .
( [ gtdiscrepancy ] ) and @xmath165 of @xcite are related by @xmath166 . in particular
, the leading order of the quark mass expansions of @xmath163 and @xmath165 is the same .
the induced pseudoscalar form factor @xmath6 is obtained by combining eqs .
( [ gpirr ] ) , ( [ frenorm ] ) , ( [ mphys ] ) and the renormalized expression for eq . ( [ pinvertex ] ) . with the help of eqs .
( [ gtdiscrepancy ] ) and ( [ delta_d_18 ] ) it can entirely be written in terms of known physical quantities as @xcite @xmath167 the @xmath168 behavior of @xmath1 is not in conflict with the book - keeping of a calculation at chiral order @xmath3 , because the external axial - vector field @xmath78 counts as @xmath86 , and the definition of the matrix element contains a momentum @xmath169 and the dirac matrix @xmath170 so that the combined order of all ingredients in the matrix element ranges from @xmath86 to @xmath3 .
the terms that have been neglected in the form factor @xmath1 are of order @xmath171 , @xmath104 and higher . using the above values for @xmath30 , @xmath7 , @xmath41 as well as @xmath172 , @xmath173 gev , @xmath174 mev and @xmath175 mev @xcite we obtain for the induced pseudoscalar coupling @xmath176 which is in agreement with the heavy - baryon results @xmath177 @xcite and @xmath178 @xcite , once the differences in the coupling constants used are taken in consideration . the first error given in eq .
( [ gp ] ) stems only from the empirical uncertainties in the quantities of eq .
( [ gpworesult ] ) . as an attempt to estimate the error originating in the truncation of the chiral expansion in the baryonic sector we assign a relative error of @xmath179 , where @xmath180 denotes the diffence between the order that has been neglected and the leading order at which a nonvanishing result appears .
such a ( conservative ) error is motivated by , e. g. , the analysis of the individual terms of eq .
( [ gaexpand ] ) as well as the determination of the lecs @xmath181 at @xmath53 and to one - loop accuracy @xmath73 in the heavy - baryon framework @xcite .
for @xmath182 we have thus added a truncation error of 0.52 .
figure [ gpwithout ] shows our result for @xmath6 in the momentum transfer region @xmath183 .
one can clearly see the dominant pion pole contribution at @xmath184 which is also supported by the experimental results of @xcite . using eq .
( [ ff_relation ] ) allows one to also determine the pion - nucleon form factor @xmath42 in terms of the results for @xmath5 and @xmath6 .
when expressed in terms of physical quantities , it has the particularly simple form @xmath185 we have explicitly verified that the results agree with a direct calculation of @xmath42 in terms of a coupling to an external pseudoscalar source .
observe that , with our definition in terms of qcd bilinears , the pion - nucleon form factor is , in general , _ not _ proportional to the axial form factor .
the relation @xmath186 which is sometimes used in pcac applications implies a pion - pole dominance for @xmath6 of the form @xmath187 .
however , as can be seen from eq.([gpinresult ] ) , there are deviations at @xmath53 from such a complete pion - pole dominance assumption .
the difference between @xmath188 and @xmath189 is entirely given in terms of the gt discrepancy @xcite @xmath190 parameterizing the form factor in terms of a monopole , @xmath191 eq . ( [ gpndiff ] ) translates into a mass parameter @xmath192 mev for @xmath193 % .
the contributions of the axial - vector meson to the form factors @xmath0 and @xmath1 at order @xmath3 stem from the diagram in fig .
[ avmdia ] .
we do not consider loop diagrams with internal axial - vector meson lines that do not contain internal pion lines , as these vanish in the infrared renormalization employed in this work . with the langrangians of eqs .
( [ lagavmmeson ] ) and ( [ lagavmnuc ] ) the axial form factor receives the contribution @xmath194 while the result for the induced pseudoscalar form factor reads @xmath195 the lagrangians for the axial - vector meson contain two new lecs , @xmath196 and @xmath197 , respectively .
however , we find that they only appear through the combination @xmath198 , effectively leaving only one unknown lec . performing a fit to the data of @xmath5 in the momentum region @xmath128 the product of the coupling constants is determined to be @xmath199 fig .
[ gawith ] shows our fitted result for the axial form factor @xmath5 at order @xmath3 in the momentum region @xmath128 with the @xmath4 meson included as an explicit degree of freedom .
as was expected from phenomenological considerations , the description of the data has improved for momentum transfers @xmath200 .
we would like to stress again that in a strict chiral expansion up to order @xmath3 the results with and without axial vector mesons do not differ from each other . the improved description of the data in the case with the explicit axial - vector meson is the result of a resummation of certain higher - order terms . while the choice of which additional degree of freedom to include compared to the standard calculation is completely phenomenological ,
once this choice has been made there exists a systematic framework in which to calculate the corresponding contributions as well as higher - order corrections .
it can be seen from eq .
( [ gaavm ] ) that in our formalism the axial - vector meson does not contribute to the axial - vector coupling constant @xmath7 .
the pion - nucleon vertex also remains unchanged at the given order , while the axial mean - square radius receives a contribution .
the values for the lecs @xmath101 and @xmath162 therefore do not change , while @xmath102 can be determined from the new expression for the axial radius using the value of eq .
( [ constantfitted ] ) for the combination of coupling constants . in fig .
[ gpwith ] we show the result for @xmath6 in the momentum transfer region @xmath183 . also shown for comparison is the result without the explicit axial - vector meson .
one sees that the contribution of the @xmath4 to @xmath6 for these momentum transfers is rather small and that @xmath6 is still dominated by the pion pole diagrams .
the form factors @xmath0 and @xmath1 are related to the pion - nucleon form factor via eq .
( [ ff_relation ] ) . for the contributions of the axial - vector meson
we find @xmath201 so that the pion - nucleon form factor is not modified by the inclusion of the @xmath4 meson .
we have discussed the nucleon form factors @xmath0 and @xmath1 of the isovector axial - vector current in manifestly lorentz - invariant baryon chiral perturbation theory up to and including order @xmath3 .
the main features of the results are similar to the case of the electromagnetic form factors at the one - loop level .
as far as the axial form factor is concerned , chpt can neither predict the axial - vector coupling constant @xmath7 nor the mean - square axial radius @xmath98 . instead , empirical information on these quantities is used to absorb the relevant lecs @xmath101 and @xmath102 in @xmath7 and @xmath98 .
moreover , the use of a manifestly lorentz - invariant framework does not lead to an improved description in comparison with the heavy - baryon framework , because the re - summed higher - order contributions are negligible .
the induced pseudoscalar form factor @xmath1 is completely fixed from @xmath202 up to and including @xmath86 , once the lec @xmath162 has been expressed in terms of the goldberger - treiman discrepancy . using @xmath203 for the pion - nucleon coupling constant ,
we obtain for the induced pseudoscalar coupling @xmath204 .
the first error is due to the error of the empirical quantities entering the expression for @xmath182 and the second error represents our estimate for the truncation in the chiral expansion .
defining the pion field in terms of the pcac relation allows one to introduce a pion - nucleon form factor which is entirely determined in terms of the axial and induced pseudoscalar form factors . assuming this pion - nucleon form factor to be proportional to the axial form factor leads to a restriction for @xmath1 which is not supported by the most general structure of chpt .
in addition to the standard treatment including the nucleon and pions , we have also considered the axial - vector meson @xmath4 as an explicit degree of freedom .
this was achieved by using the reformulated infrared renormalization scheme .
the inclusion of the axial - vector meson effectively results in one additional low - energy coupling constant which we have determined by a fit to the data for @xmath0 .
the inclusion of the axial - vector meson results in a considerably improved description of the experimental data for @xmath0 for values of @xmath205 up to about @xmath206 gev@xmath120 , while the contribution to @xmath1 is small .
m.r.s . and s.s .
would like to thank h.w .
fearing and j. gasser for useful discussions and the triumf theory group for their hospitality .
this work was made possible by the financial support from the deutsche forschungsgemeinschaft ( sfb 443 ) , the government of canada , and the eu integrated infrastructure initiative hadron physics project ( contract number rii3-ct-2004 - 506078 ) .
for the definition of the loop integrals in the expressions for the form factors we use the notation @xmath207 using dimensional regularization @xcite the loop integrals with one or two internal lines are defined as @xmath208[(k+q)^2-m^2+i\epsilon]},\\ i_{{\pi}n}(p^2)&= & i\int\frac{d^nk}{(2\pi)^n } \frac{1}{[k^2-m^2+i\epsilon][(k+p)^2-m^2+i\epsilon]},\\ p^{\mu}i_{{\pi}n}^{(p)}(p^2 ) & = & i\int\frac{d^nk}{(2\pi)^n } \frac{k^{\mu}}{[k^2-m^2+i\epsilon][(k+p)^2-m^2+i\epsilon]},\\ g^{\mu\nu}i_{{\pi}n}^{(00)}(p^2)+p^\mu p^\nu i_{{\pi}n}^{(pp)}(p^2 ) & = & i\int\frac{d^nk}{(2\pi)^n } \frac{k^{\mu}k^{\nu}}{[k^2-m^2+i\epsilon][(k+p)^2-m^2+i\epsilon]}.\end{aligned}\ ] ] for integrals with three internal lines we assume on - shell kinematics , @xmath209 , @xmath210[(k+p_i)^2-m^2+i\epsilon][(k+p_f)^2-m^2+i\epsilon ] } , \\
p^{\mu}i_{{\pi}nn}^{(p)}(q^2 ) & = & i\int\frac{d^nk}{(2\pi)^n}\frac{k^{\mu } } { [ k^2-m^2+i\epsilon][(k+p_i)^2-m^2+i\epsilon][(k+p_f)^2-m^2+i\epsilon ] } , \\
\lefteqn{g^{\mu\nu}i_{{\pi}nn}^{(00)}(q^2)+p^{\mu}p^{\nu}i_{{\pi}nn}^{(pp)}(q^2 ) + q^{\mu}q^{\nu}i_{{\pi}nn}^{(qq)}(q^2 ) } \\ & = & i\int\frac{d^nk}{(2\pi)^n}\frac{k^{\mu}k^{\nu } } { [ k^2-m^2+i\epsilon][(k+p_i)^2-m^2+i\epsilon][(k+p_f)^2-m^2+i\epsilon]}.\end{aligned}\ ] ] the tensorial loop integrals can be reduced to scalar ones @xcite and we obtain @xmath211 , \\
i_{{\pi}n}^{(00)}(p^2 ) & = & \frac{1}{2(n-1)}\left[i_n + 2 m^2
i_{{\pi}n}(p^2 ) + \frac{(p^2-m^2+m^2)}{p^2}i_{\pi n}^{(p)}(p^2)\right ] , \\
i_{{\pi}nn}^{(p)}(q^2 ) & = & \frac{1}{4m_n^2-q^2 } \left[i_{{\pi}n}(m_n^2)-i_{nn}(q^2)-m^2i_{{\pi}nn}(q^2)\right ] , \\
i_{{\pi}nn}^{(00)}(q^2)&=&\frac{1}{n-2}\left\{\left[i_{{\pi}nn}(q^2 ) + i_{{\pi}nn}^{(p)}(q^2)\right]m^2+\frac{1}{2}i_{nn}(q^2)\right\},\\ i_{{\pi}nn}^{(pp)}(q^2 ) & = & \frac{1}{(n-2)(4m_n^2-q^2)}\left\{\left[(n-1)i_{{\pi}nn}^{(p)}(q^2 ) + i_{{\pi}nn}(q^2)\right]m^2\right . \\ & & \left.-\frac{n-2}{2}i_{{\pi}n}^{(p)}(m_n^2 ) -\frac{n-3}{2}i_{nn}(q^2)\right\ } , \\
i_{{\pi}nn}^{(qq)}(q^2 ) & = & -\frac{1}{(n-2)q^2}\left\{\left[i_{{\pi}nn}^{(p)}(q^2)+i_{{\pi}nn}(q^2)\right]m^2 + \frac{n-2}{2}i_{{\pi}n}^{(p)}(m_n^2)+\frac{1}{2}i_{nn}(q^2)\right\}. \\\end{aligned}\ ] ] defining @xmath212\right\},\ ] ] and @xmath213 the scalar loop integrals are given by @xcite @xmath214 @xmath215 @xmath216,\ ] ] @xmath217\ ] ] @xmath218,\ ] ] where @xmath219\\ & = & \left \
{ \begin{array}{ll } -2-\sigma\ln\left(\frac{\sigma-1}{\sigma+1}\right),&x<0,\\ -2 + 2\sqrt{\frac{4}{x}-1}\,\mbox{arccot } \left(\sqrt{\frac{4}{x}-1}\right),&0\le x<4,\\ -2-\sigma\ln\left(\frac{1-\sigma}{1+\sigma}\right)-i\pi\sigma , & 4<x , \end{array } \right.\end{aligned}\ ] ] with @xmath220,\ ] ] and @xmath221 integrals with three propagators were analyzed numerically using a schwinger parametrization .
for purely mesonic integrals only the terms proportional to @xmath222 have to be subtracted . to determine the infrared regular parts @xmath223 of the scalar loop integrals , we use the method described in @xcite . on -
shell - kinematics are assumed for the subtraction terms .
note that we also list divergent terms , as they might give finite contributions in the expressions for tensor integrals .
@xmath224 -\frac{1}{16\pi^2}-\frac{m^2}{32\pi^2m^2}\left(3 + 8c_1m\right)\\ & & -\frac{3 g_a^2 m^3}{512 \pi^3 f^2 m}+{\cal o}(p^4 ) , \\
r_{\pi nn } & = & \frac{\bar\lambda}{m^2}\left[1 + \frac{q^2}{6m^2}+8c_1\frac{m^2}{m } + \frac{3 g_a^2 m^3}{16 \pi f^2 m}\right ] + \frac{1}{32\pi^2m^2}-\frac{m^2}{32\pi^2m^4}\left(1 - 16c_1m\right)\\ & & + \frac{3 g_a^2 m^3}{256 \pi^3 f^2 m^3}+{\cal o}(p^4).\end{aligned}\ ] ] h. y. gao , int .
j. mod .
e * 12 * ( 2003 ) 1 [ erratum - ibid .
e * 12 * ( 2003 ) 567 ] .
j. friedrich and th .
walcher , eur .
phys . j. a * 17 * ( 2003 ) 607 .
c. e. hyde - wright and k. de jager , ann .
nucl . part .
* 54 * ( 2004 ) 217 .
v. bernard , l. elouadrhiri and u .- g .
meiner , j. phys .
g * 28 * ( 2002 ) r1 .
t. gorringe and h. w. fearing , rev .
* 76 * ( 2004 ) 31
. s. scherer and j. h. koch , nucl .
a * 534 * ( 1991 ) 461 ; t. fuchs and s. scherer , phys .
c * 68 * ( 2003 ) 055501 .
j. h. d. clark _ et al .
_ , phys .
* 96 * ( 2006 ) 073401 .
y. nambu and e. shrauner , phys . rev . * 128 * ( 1962 ) 862 .
r. tegen and w. weise , z. phys .
a * 314 * ( 1983 ) 357 .
w. y. p. hwang and d. j. ernst , phys .
d * 31 * ( 1985 ) 2884 .
s. boffi , l. y. glozman , w. klink , w. plessas , m. radici and r. f. wagenbrunn , eur .
j. a * 14 * ( 2002 ) 17 .
d. merten , u. loring , k. kretzschmar , b. metsch and h. r. petry , eur .
j. a * 14 * ( 2002 ) 477 . b. q. ma , d. qing and i. schmidt , phys .
c * 66 * ( 2002 ) 048201 .
k. khosonthongkee , v. e. lyubovitskij , t. gutsche , a. faessler , k. pumsa - ard , s. cheedket and y. yan , j. phys .
g * 30 * ( 2004 ) 793 .
a. silva , h. c. kim , d. urbano and k. goeke , phys .
d * 72 * ( 2005 ) 094011 .
k. f. liu , s. j. dong , t. draper , j. m. wu and w. wilcox , phys .
d * 49 * ( 1994 ) 4755 ; k. f. liu , s. j. dong , t. draper and w. wilcox , phys .
* 74 * ( 1995 ) 2172 . c. adamuscin , e. a. kuraev , e. tomasi - gustafsson and f. e. maas , arxiv : hep - ph/0610429 .
s. weinberg , physica a * 96 * ( 1979 ) 327 .
j. gasser and h. leutwyler , annals phys .
* 158 * ( 1984 ) 142 .
j. gasser and h. leutwyler , nucl .
b * 250 * ( 1985 ) 465 .
j. gasser , m. e. sainio , and a. varc , nucl .
* b307 * ( 1988 ) 779 .
v. bernard , n. kaiser and u .- g .
meiner , int .
j. mod . phys .
e * 4 * ( 1995 ) 193 .
s. scherer , in _ advances in nuclear physics , vol .
27 _ , edited by j. w. negele and e. w. vogt ( kluwer academic / plenum publishers , new york , 2003 ) .
v. bernard , n. kaiser and u .- g .
meiner , phys . rev
. lett .
* 69 * ( 1992 ) 1877 .
v. bernard , n. kaiser , t. s. h. lee and u .- g .
meiner , phys .
* 246 * ( 1994 ) 315 .
h. w. fearing , r. lewis , n. mobed and s. scherer , phys .
d * 56 * ( 1997 ) 1783 .
v. bernard , h. w. fearing , t. r. hemmert , and u .- g .
meiner , nucl .
* a635 * ( 1998 ) 121 [ erratum - ibid .
* a642 * ( 1998 ) 563 ] .
g. colangelo , j. gasser and h. leutwyler , phys .
* 86 * ( 2001 ) 5008 .
n. fettes , u .- g .
meiner , m. mojzis and s. steininger , annals phys .
* 283 * ( 2000 ) 273 [ erratum - ibid .
* 288 * ( 2001 ) 249 ] .
g. ecker , j. gasser , h. leutwyler , a. pich and e. de rafael , phys .
b * 223 * ( 1989 ) 425 .
t. fuchs , m. r. schindler , j. gegelia and s. scherer , phys .
b * 575 * ( 2003 ) 11 .
j. kambor and m. moji , jhep * 9904 * ( 1999 ) 031 .
v. bernard and u. -g .
meiner , phys .
b * 639 * ( 2006 ) 278 .
s. i. ando and h. w. fearing , arxiv : hep - ph/0608195 .
t. fuchs , j. gegelia , g. japaridze and s. scherer , phys .
d * 68 * ( 2003 ) 056005 .
a. liesenfeld _ et al .
_ [ a1 collaboration ] , phys .
b * 468 * ( 1999 ) 20 .
h. budd , a. bodek and j. arrington , proceedings of the 2nd international workshop on neutrino - nucleus interactions in the few gev region ( nuint 02 ) , irvine , california , 12 - 15 dec 2002 , arxiv : hep - ex/0308005 .
k. s. kuzmin , v. v. lyubushkin and v. a. naumov , acta phys .
b * 37 * ( 2006 ) 2337 .
r. gran _
[ k2k collaboration ] , phys .
d * 74 * ( 2006 ) 052002 .
d. baumann , phd thesis , johannes gutenberg - universitt , mainz ( 2004 ) .
t. fuchs , j. gegelia and s. scherer , j. phys .
g * 30 * ( 2004 ) 1407 . t. becher and h. leutwyler , jhep * 0106 * ( 2001 ) 017 .
m. l. goldberger and s. b. treiman , phys .
* 110 * ( 1958 ) 1178 ; phys .
rev . * 111 * ( 1958 ) 354 ; y. nambu , phys .
lett . * 4 * ( 1960 ) 380 .
h. pagels , phys .
* 179 * ( 1969 ) 1337 .
h. c. schrder _ et al .
_ , eur . phys .
j. c * 21 * ( 2001 ) 473 .
v. g. j. stoks , r. timmermans and j. j. de swart , phys .
c * 47 * ( 1993 ) 512 .
t. e. o. ericson , b. loiseau and a. w. thomas , phys .
c * 66 * ( 2002 ) 014005 .
r. a. arndt , w. j. briscoe , i. i. strakovsky and r. l. workman , arxiv : nucl - th/0605082 .
v. bernard , n. kaiser and u .- g .
meiner , phys .
d * 50 * ( 1994 ) 6899 .
v. bernard , n. kaiser and u .- g .
meiner , nucl .
* a615 * ( 1997 ) 483 .
s. choi _ et al .
_ , phys .
* 71 * ( 1993 ) 3927 .
g. t hooft and m. j. g. veltman , nucl .
* b44 * ( 1972 ) 189 .
|
we calculate the nucleon form factors @xmath0 and @xmath1 of the isovector axial - vector current and the pion - nucleon form factor @xmath2 in manifestly lorentz - invariant baryon chiral perturbation theory up to and including order @xmath3 .
in addition to the standard treatment including the nucleon and pions , we also consider the axial - vector meson @xmath4 as an explicit degree of freedom .
this is achieved by using the reformulated infrared renormalization scheme .
we find that the inclusion of the axial - vector meson effectively results in one additional low - energy coupling constant that we determine by a fit to the data for @xmath0 .
the inclusion of the axial - vector meson results in an improved description of the experimental data for @xmath0 , while the contribution to @xmath1 is small .
|
first of all , let us recall the following classical result by bogomolov and gieseker : _ ( @xcite , @xcite ) _ let @xmath0 be a smooth projective complex variety and @xmath1 an ample divisor in @xmath0 . for any torsion free @xmath1-slope stable sheaf @xmath2 on @xmath0
, we have @xmath3 here @xmath4 is the discriminant @xmath5 it has been an interesting problem to improve the bg inequality for higher rank stable sheaves ( cf .
@xcite , @xcite ) .
so far such an improvement is only known for some particular surfaces , e.g. k3 surfaces or del pezzo surfaces , which easily follows from riemann - roch theorem and serre duality ( cf .
lemma [ strong : k3 ] , ( * ? ? ?
* appendix a ) ) . in the 3-fold case
, such an improvement is only known for rank two stable sheaves on @xmath6 by hartshorne @xcite . in a case of other 3-fold ,
even a conjectural improvement is not known .
the purpose of this note is to propose a conjectural improvement of bg inequality for stable sheaves on quintic 3-folds , motivated by an idea from mirror symmetry and matrix factorizations .
we first state the resulting conjecture : [ intro : strong ] let @xmath7 be a smooth quintic 3-fold and @xmath8 .
then for any torsion free @xmath1-slope stable sheaf on @xmath0 with @xmath9 , we have the following inequality : @xmath10 the rhs of ( [ intro : bound ] ) is a certain irrational real number contained in @xmath11 , and the detail will be discussed in conjecture [ strong ] .
our conjecture is derived from an attempt to construct a bridgeland stability condition on @xmath12 corresponding to the gepner point in the stringy khler moduli space of @xmath0 .
the rhs of ( [ intro : bound ] ) is related to the coefficient of the corresponding central charge .
it seems that conjecture [ intro : strong ] does not appear in literatures even in the rank two case , which we will give a proof in this note : conjecture [ intro : strong ] is true if @xmath13 . the above result will be proved in subsection [ rank : two ] .
based on a similar idea , we also propose a conjectural clifford type bound for stable coherent systems on quintic surfaces ( cf . section [ sec : cli ] ) .
below we discuss background of the derivation of the above conjecture .
the notion of stability conditions on triangulated categories introduced by bridgeland @xcite has turned out to be an important mathematical object to study .
however it has been a problem for more than ten years to construct bridgeland stability conditions on the derived categories of coherent sheaves on quintic 3-folds . from a picture of the mirror symmetry
, the space of stability conditions on a quintic 3-fold is expected to be related to its stringy khler moduli space , which is described in figure [ fig : one ] . in figure
[ fig : one ] , we see three special points , large volume limit , conifold point and gepner point .
a conjectural construction of a bridgeland stability condition near the large volume limit was proposed by bayer , macri and the author @xcite , and we reduced the problem to showing a bg type inequality evaluating @xmath14 for certain two term complexes . the main conjecture in @xcite is not yet proved except in the @xmath6 case @xcite , and we face our lack of knowledge on the set of chern characters of stable objects . in this note , we focus on the gepner point .
a corresponding stability condition is presumably constructed as a gepner type stability condition @xcite with respect to the pair @xmath15 where @xmath16 is the seidel - thomas twist @xcite associated to @xmath17 .
combined with orlov s result @xcite , as discussed in @xcite , such a stability condition is expected to give a natural stability condition on graded matrix factorizations of the defining polynomial of the quintic 3-fold .
one may expect that constructing a gepner point also requires such a conjectural inequality .
it seems worth formulating a conjectural bg type inequality which arises in an attempt to construct a gepner point , so that making it clear what we should know on chern characters of stable sheaves .
our conjecture [ intro : strong ] is the resulting output .
the inequality ( [ intro : bound ] ) itself is interesting since there have been several attempts to improve the classical bg inequality .
assuming conjecture [ intro : bound ] , we construct data which presumably give a bridgeland stability condition corresponding to the gepner point .
compared to the lower degree cases studied in @xcite , constructing gepner type stability conditions is much harder in quintic cases , and most of the attempts are still conjectural .
this is the reason we have separated the arguments for the quintic case from the previous paper @xcite .
we hope that the arguments in this note lead to future developments of the study of chern characters of stable objects on 3-folds .
the author would like to thank kentaro hori , kyoji saito and atsushi takahashi for valuable discussions .
the author also would like to thank johannes walcher for pointing out the reference @xcite .
this work is supported by world premier international research center initiative ( wpi initiative ) , mext , japan .
this work is also supported by grant - in aid for scientific research grant ( 22684002 ) from the ministry of education , culture , sports , science and technology , japan .
all the varieties or polynomials are defined over complex numbers . for a smooth projective variety @xmath0 of dimension @xmath18 and @xmath19
, we write its chern character as a vector @xmath20 for @xmath21 . for a triangulated category @xmath22 and a set of objects @xmath23 in @xmath22 , we denote by @xmath24 the smallest extension closed subcategory in @xmath22 which contains @xmath23 .
let @xmath22 be a triangulated category and @xmath25 its grothendieck group .
we first recall bridgeland s definition of stability conditions on it .
[ defi : stab ] _ ( @xcite ) _ a stability condition @xmath26 on @xmath22 consists of a pair @xmath27 @xmath28 where @xmath29 is a group homomorphism ( called central charge ) and @xmath30 is a full subcategory ( called @xmath26-semistable objects with phase @xmath31 ) satisfying the following conditions : * for @xmath32 , we have @xmath33 . * for all @xmath34 , we have @xmath35 $ ] . * for @xmath36 and @xmath37 , we have @xmath38 . * for each @xmath39 , there is a collection of distinguished triangles @xmath40 , \quad e_n = e , \ e_0=0\end{aligned}\ ] ] with @xmath41 and @xmath42 .
the full subcategory @xmath43 is shown to be an abelian category , and its simple objects are called @xmath26-stable . in @xcite , bridgeland shows that there is a natural topology on the set of ` good ' stability conditions @xmath44 , and its each connected component has a structure of a complex manifold .
let @xmath45 be the group of autoequivalences on @xmath22 .
there is a left @xmath45-action on the set of stability conditions on @xmath22 . for @xmath46
, it acts on the pair ( [ pair2 ] ) as follows : @xmath47 there is also a right @xmath48-action on the set of stability conditions on @xmath22 . for @xmath49
, it acts on the pair ( [ pair2 ] ) as follows : @xmath50 the notion of gepner type stability conditions is defined as follows : _ ( @xcite ) _ a stability condition @xmath26 on @xmath22 is called gepner type with respect to @xmath51 if the following condition holds : @xmath52 let @xmath53 be a homogeneous element @xmath54\end{aligned}\ ] ] of degree @xmath55 such that @xmath56 has an isolated singularity at the origin . for a graded @xmath57-module @xmath58 ,
we denote by @xmath59 its degree @xmath60-part , and @xmath61 the graded @xmath57-module whose grade is shifted by @xmath62 , i.e. @xmath63 . a graded matrix factorization of @xmath53 is data @xmath64 where @xmath65 are graded free @xmath57-modules of finite rank , @xmath66 are homomorphisms of graded @xmath57-modules , satisfying the following conditions : @xmath67 the category @xmath68 is defined to be the triangulated category whose objects consist of graded matrix factorizations of @xmath53 ( cf .
the grade shift functor @xmath69 induces the autoequivalence @xmath70 of @xmath68 , which satisfies the following identity : @xmath71 . \end{aligned}\ ] ] the following is the main conjecture in @xcite : [ conj : main ] there is a gepner type stability condition @xmath72 with respect to @xmath73 , whose central charge @xmath74 is given by @xmath75 the definition of the central charge @xmath74 first appeared in @xcite .
it is more precisely written as follows : since @xmath65 are free @xmath57-modules of finite rank , they are written as @xmath76 then ( [ z_g ] ) is written as @xmath77 so far conjecture [ conj : main ] is proved when @xmath78 @xcite , @xmath79 @xcite , and @xmath80 @xcite .
the most important unproven case is when @xmath81 , in which the variety @xmath0 is a quintic calabi - yau 3-fold .
we recall orlov s theorem @xcite relating the triangulated category @xmath68 with the derived category of coherent sheaves on the smooth projective variety @xmath82 we only use the results for @xmath83 case , i.e. @xmath0 is a calabi - yau manifold , and @xmath84 case , i.e. @xmath0 is general type . _
( ( * ? ? ?
* theorem 2.5 ) , ( * ? ? ?
* proposition 5.8))_[thm : orlov ] if @xmath83 , there is an equivalence of triangulated categories @xmath85 such that the following diagram commutes : @xmath86^{\psi } \ar[d]_{f } & { \mathrm{hmf}^{\rm{gr}}}(w ) \ar[d]^{\tau } \\
d^b { \mathop{\rm coh}\nolimits}(x ) \ar[r]^{\psi } & { \mathrm{hmf}^{\rm{gr}}}(w ) . }
\end{aligned}\ ] ] here @xmath87 is the autoequivalence given by @xmath88 .
recall that @xmath16 is the seidel - thomas twist @xcite , given by @xmath89 _ ( ( * ? ? ?
* theorem 2.5 ) , ( * ? ? ?
* proposition 3.22))_[thm : orlov2 ] if @xmath84 , then there is a fully faithful functor @xmath90 such that we have the semiorthogonal decomposition @xmath91 where @xmath92 is a certain exceptional object .
moreover the subcategory @xmath93 is the heart of a bounded t - structure on @xmath68 , and there is an equivalence of abelian categories @xmath94 here @xmath95 is the abelian category of coherent systems on @xmath0 .
recall that a coherent system on @xmath0 consists of data @xmath96 where @xmath97 is a finite dimensional @xmath48-vector space , @xmath98 and @xmath99 is a morphism in @xmath100 .
the set of morphisms in @xmath95 is given by the commutative diagrams in @xmath100 @xmath101^{s } \ar[d ] & f \ar[d ] \\
v'\otimes { \mathcal{o}}_x \ar[r]^{s ' } & f ' . }
\end{aligned}\ ] ] the equivalence @xmath102 sends @xmath103 to @xmath92 and @xmath104 for @xmath98 to @xmath105 .
in this section , we take @xmath53 to be a quintic homogeneous polynomial with five variables @xmath106 , \quad \deg(w)=5 . \end{aligned}\ ] ] the variety @xmath107 is a smooth quintic calabi - yau 3-fold .
this is the most interesting case in the study of conjecture [ conj : main ] .
we have an equivalence by theorem [ thm : orlov ] @xmath108 the goal of this section is to translate conjecture [ conj : main ] in terms of @xmath12 , and relate it to a stronger version of bg inequality for stable sheaves on @xmath0 .
let us first recall a mirror family of a quintic 3-fold @xmath0 and its stringy khler moduli space .
the mirror family of @xmath0 is a simultaneous crepant resolution @xmath109 of the following one parameter family of quotient varieties @xcite : @xmath110 here @xmath111 $ ] is the homogeneous coordinate of @xmath112 , and @xmath113 acts on @xmath112 by @xmath114 = [ \xi_1 y_1 \colon \xi_2 y_2 \colon \xi_3 y_3 \colon \xi_1^{-1 } \xi_2^{-1 } \xi_3^{-1 } y_4 \colon 1 ] \end{aligned}\ ] ] for @xmath115 .
let @xmath116 be the root of unity @xmath117 note that we have the isomorphism @xmath118 by @xmath119 for @xmath120 and @xmath121 .
also @xmath122 is a non - singular calabi - yau 3-fold if and only if @xmath123 .
hence the mirror family @xmath122 is parametrized by the following quotient stack ( see figure [ fig : one ] ) @xmath124\end{aligned}\ ] ] where the generator of @xmath125 acts on @xmath48 by the multiplication of @xmath116 .
the stack @xmath126 is called the stringy k@xmath127hler moduli space of @xmath0 .
we see that there are 3-special points in figure [ fig : one ] : * the point @xmath128 , called _
large volume limit_. * the point @xmath129 , called _
conifold point_. * the point @xmath130 , called _
gepner point_. the mirror variety @xmath122 is non - singular except at the first two special points .
it is also non - singular at the gepner point , but there admits a non - trivial @xmath131-action by the isomorphism ( [ isom : mirror ] ) .
we discuss a relationship between the space @xmath126 and the bridgeland s space @xmath132 based on the papers @xcite , @xcite .
let @xmath133 be the group of autoequivalences of @xmath12 .
it is expected that there is an embedding @xmath134 \end{aligned}\ ] ] such that , if we write @xmath135 then the central charge @xmath136 for @xmath137 is a solution of the picard - fuchs ( pf ) equation which the period integrals of the mirror family @xmath122 should satisfy . using the following notation @xmath138 the pf equation
is given by @xmath139 the solution space of the above pf equation is known to be four dimensional . in the @xmath140-variable
, the basis is given by ( cf .
@xcite ) @xmath141 for @xmath142 . for an object @xmath137 , the central charge
@xmath136 should satisfy the above pf equation , hence is written as @xmath143 where @xmath144 and @xmath145 is a linear combination of the basis @xmath146 which is independent of @xmath2 .
here we have identified @xmath147 with @xmath148 via the integration map .
on the other hand , around the large volume limit and the conifold point , the monodromy transformations induce linear isomorphisms @xmath149 , @xmath150 on the solution space of the pf equation ( [ pf ] ) . hence that monodromy transformations act on the central charge @xmath136 , which are expected to coincide with the actions of autoequivalences @xmath151 , @xmath16 respectively .
namely we should have the following identities : @xmath152 the coefficients of @xmath145 are uniquely determined by the above matching property of the monodromy transformations on both sides of ( [ i ] ) .
indeed , the above idea is used to give an embedding similar to ( [ i ] ) when @xmath0 is the local projective plane in @xcite . in the quintic 3-fold case , based on a similar idea as above ,
the central charges @xmath136 for line bundles @xmath153 are computed by aspinwall ( * ? ? ? * equation ( 217 ) ) : @xmath154 since @xmath155 for @xmath156 span @xmath157 , the above formula uniquely determines @xmath145 .
a direct computation shows that @xmath158 as a result , @xmath136 is written as @xmath159 let us consider a conjectural stability condition @xmath160 satisfying @xmath161= i(\psi^5=0 ) \in \left [ { \mathop{\rm auteq}\nolimits}(x ) \backslash { \mathop{\rm stab}\nolimits}(x)/ \mathbb{c } \right]\end{aligned}\ ] ] where @xmath162 is an expected embedding ( [ i ] ) . since the point @xmath163 ( gepner point ) in @xmath126 is an orbifold point with stabilizer group @xmath131 , the stability condition @xmath164 should also have the stabilizer group @xmath131 with respect to the @xmath165 action on @xmath166 . under a suitable choice of @xmath164 , the generator of the above stabilizer group
should be given by @xmath167 since the action of @xmath168 on @xmath157 corresponds to the composition of monodromy transformations at the large volume limit and the conifold point under the embedding ( [ i ] ) , and the five times composition of @xmath168 coincides with @xmath169 $ ] .
( this is a consequence of theorem [ thm : orlov ] and the identity ( [ taud ] ) . )
the property of @xmath164 fixed by ( [ st : stab ] ) is nothing but the gepner type property with respect to @xmath170 . by the above argument and theorem [ thm : orlov ]
, a stability condition corresponding to the gepner point gives a desired stability condition in conjecture [ conj : main ] via orlov equivalence ( [ or : quin ] ) .
as for the central charge at the gepner point , we consider the normalized central charge @xmath171 so that @xmath172 holds for any @xmath173 . under this normalization , @xmath171 is given by @xmath174 indeed , the coefficients @xmath175 of @xmath176 at @xmath177 are checked to form the unique solution of the linear equation @xmath178 where @xmath179 is given by the composition of matrices ( cf .
* subsection 4.1 ) ) @xmath180 here @xmath181 is the @xmath182-component of @xmath183 .
the above matrix @xmath179 induces the isomorphism on @xmath184 , which is identified with the action of @xmath168 on it . by (
* proposition 4.4 ) , the central charge @xmath171 is related to the central charge @xmath74 on @xmath68 given by ( [ z_g ] ) as @xmath185 for any @xmath137 .
here @xmath186 is the equivalence ( [ or : quin ] ) . by applying @xmath48-action on @xmath166 , conjecture [ conj : main ] for the polynomial ( [ poly : w ] ) leads to the following conjecture : [ conj : quintic ]
let @xmath187 be a smooth quintic 3-fold , @xmath8 and @xmath188 .
then there is a gepner type stability condition @xmath189 with respect to @xmath170 , whose central charge @xmath171 is given by @xmath190 let us try to construct a desired stability condition in conjecture [ conj : quintic ] . by (
* proposition 5.3 ) , giving data ( [ gep : quintic ] ) is equivalent to giving the heart of a bounded t - structure @xmath191 satisfying @xmath192 \}\end{aligned}\ ] ] and any object @xmath193 admits a harder - narasimhan filtration with respect to the @xmath171-stability .
we propose that a desired heart @xmath194 is constructed as a double tilting of @xmath100 , similar to the one in @xcite .
this is motivated by the following observations : firstly in @xcite , we constructed a gepner type stability condition for a quartic k3 surface @xmath195 via a tilting of @xmath196 .
the construction is similar to the one near the large volume limit in @xcite , @xcite .
a different point is that , although we only need a classical bg inequality to construct a stability condition near the large volume limit , a construction at the gepner point requires a stronger version of bg inequality given as follows : let @xmath195 be a k3 surface and @xmath2 a torsion free stable sheaf @xmath2 on @xmath195 with @xmath197 .
then we have the following inequality @xmath198 the above lemma is an easy consequence of the riemann - roch theorem and serre duality ( cf .
* corollary 2.5 ) ) and a similar improvement is not known for other surfaces except del pezzo surfaces . by the above observation , we expect that a desired gepner type stability condition on a quintic 3-fold is also constructed in a way similar to the one near the large volume limit , after an an improvement of bg inequality .
secondly we can rewrite the central charge @xmath176 in the following way : @xmath199 here @xmath200 and @xmath201 is the twisted chern character @xmath202 in ( [ rewrite ] ) , @xmath203 are some real numbers in @xmath204 , given by @xmath205 they are approximated by @xmath206 the expression ( [ rewrite ] ) is very similar to the central charge near the large volume limit , given by @xmath207 for @xmath208 .
the above integration is expanded as @xmath209 by comparing ( [ rewrite ] ) with ( [ expand ] ) , although they are in a similar form , we see that some signs of the coefficients are different . in @xcite
, we constructed a double tilting of @xmath100 which , together with the central charge ( [ expand ] ) , conjecturally gives a bridgeland stability condition near the large volume limit .
we propose to construct the heart @xmath194 via a double tilting of @xmath100 in a way similar to @xcite , by taking the difference of the signs of the coefficients into consideration .
we imitate the argument in @xcite to construct @xmath194 . in what follows
, we fix @xmath200 .
let @xmath210 be the twisted slope function on @xmath100 defined by @xmath211 here we set @xmath212 if @xmath2 is a torsion sheaf .
the above slope function defines the classical slope stability on @xmath100 .
we define the pair of full subcategories @xmath213 of @xmath100 to be @xmath214 the above subcategories form a torsion pair in @xmath100 .
the associated tilting @xmath215 is defined to be @xmath216 , { \mathcal{t}}_{b , h } \rangle_{\rm{ex}}. \end{aligned}\ ] ] the category @xmath215 is the heart of a bounded t - structure on @xmath12 . in ( * ? ? ?
* lemma 3.2.1 ) , it is observed that the central charge ( [ expand ] ) satisfies the following condition : an object @xmath217 with @xmath218 satisfies @xmath219 .
the classical bg inequality is used to show the above property .
we propose that a similar property also holds for the central charge @xmath171 , i.e. an object @xmath217 with @xmath218 satisfies @xmath220 .
note that such an object @xmath2 is contained in the category @xmath221 , { \mathop{\rm coh}\nolimits}_{\le 1}(x ) : f \mbox { is } \mu_{b , h } \mbox{-stable with } h^2{\mathop{\rm ch}\nolimits}_1^b(f)=0 \rangle_{\rm{ex}}\end{aligned}\ ] ] where @xmath222 is the category of coherent sheaves @xmath223 with @xmath224 . also noting the equality @xmath225 the above requirement leads to the following conjecture : [ strong ] let @xmath226 be a smooth quintic 3-fold and @xmath2 a torsion free slope stable sheaf on @xmath0 with @xmath9 .
then we have the following inequality : @xmath227 the rhs of ( [ bound ] ) is irrational , hence the equality is not achieved .
note that the rhs in ( [ bound ] ) is very close to the rhs in ( [ strong : k3 ] ) for the k3 surface case .
a stronger bg inequality similar to ( [ bound ] ) is predicted by @xcite without the condition @xmath228 .
the prediction in @xcite is shown to be false in @xcite , @xcite .
conjecture [ strong ] does not contradict to the results in @xcite , @xcite since we restrict to the sheaves with fixed slope @xmath228 .
there are few examples of stable sheaves on quintic 3-folds in literatures .
the following example is taken in @xcite : let @xmath2 be the kernel of the morphism @xmath229 given by the matrix @xmath230 here @xmath231 $ ] is the homogeneous coordinates in @xmath112 . by @xcite , @xmath2 is a stable vector bundle on @xmath0 with @xmath232 then we have @xmath233 the rank two case will be treated in subsection [ rank : two ] .
we now give a conjectural construction of a desired @xmath194 assuming conjecture [ strong ] .
similarly to ( * ? ? ?
* lemma 3.2.1 ) , we have the following lemma : [ lem : property ] suppose that conjecture [ strong ] is true .
then for any non - zero @xmath217 , we have the following : * we have @xmath234 . * if @xmath218 , then we have @xmath235 . *
if @xmath236 , then @xmath237 . the same argument of ( * ?
* lemma 3.2.1 ) is applied by using conjecture [ strong ] instead of the classical bg inequality .
the above lemma shows that the triple @xmath238 should behave like @xmath239 on coherent sheaves on algebraic surfaces . similarly to the slope function on coherent sheaves , we consider the following slope function on @xmath215 @xmath240 here we set @xmath241 if @xmath218 . if we assume conjecture [ strong ] , then lemma [ lem : property ] shows that the slope function @xmath242 satisfies the weak see - saw property .
an object @xmath217 is @xmath242-(semi)stable if , for any non - zero proper subobject @xmath243 in @xmath215 , we have the inequality @xmath244 we have the following lemma : suppose that conjecture [ strong ] is true .
then the @xmath242-stability on @xmath215 satisfies the harder - narasimhan property .
although the central charge @xmath245 is irrational , the values @xmath246 are contained in @xmath247 , hence they are discrete .
this is enough to apply the same argument of ( * ? ? ?
* lemma 3.2.4 ) , ( * ? ? ?
* proposition 7.1 ) to show the existence of harder - narasimhan filtrations with respect to @xmath242-stability .
assuming conjecture [ strong ] , we define the full subcategories in @xmath215 @xmath248 as before , the pair @xmath249 forms a torsion pair on @xmath215 . by taking the tilting , we obtain the heart of a bounded t - structure @xmath250 , { \mathcal{t}}_g \rangle_{\rm{ex}}. \end{aligned}\ ] ] we propose the following conjecture : [ conj : const ] let @xmath226 be a smooth quintic 3-fold and assume that conjecture [ strong ] is true . then the pair @xmath251 determines a gepner type stability condition on @xmath12 with respect to @xmath170 . by the construction and the irrationality of @xmath171 , the pair ( [ pair : gep ] ) satisfies the condition ( [ positivity ] ) . on the other hand ,
the irrationality of @xmath171 makes it hard to prove the harder - narasimhan property of the pair ( [ pair : gep ] ) .
we show that conjecture [ strong ] is true in the rank two case .
conjecture [ strong ] is true when @xmath13 .
since we have the inequality @xmath252 we may assume that @xmath2 is reflexive .
since @xmath13 , we have @xmath253 and @xmath254 the classical bg inequality implies that @xmath255 , i.e. @xmath256 . the conjectural inequality ( [ bound ] ) is equivalent to that @xmath257 .
it is enough to exclude the case @xmath258 , or equivalently @xmath259 .
suppose by contradiction that @xmath260 .
let us set @xmath261 , which is also a torsion free slope stable sheaf .
since @xmath87 is reflexive , we have @xmath262 and @xmath263 is a zero dimensional sheaf by ( * ? ? ?
* proposition 1.1.10 ) .
this implies that there is a distinguished triangle @xmath264\end{aligned}\ ] ] where @xmath265 is the derived dual @xmath266 .
therefore if we write @xmath267 then we have @xmath268 and @xmath269 here @xmath270 is the length of the zero dimensional sheaf @xmath271 . on the other hand ,
since @xmath87 is a rank two reflexive sheaf , we have the isomorphism ( cf .
* proposition 1.10 ) ) @xmath272 noting that @xmath273 , and ( [ chf ] ) , ( [ chf2 ] ) , we have @xmath274 the above equality and the assumption @xmath275 imply that @xmath276 noting that @xmath277 , the riemann - roch theorem and ( [ ch_3 ] ) imply that @xmath278 we divide into two cases : @xmath279 . by the serre duality and stability
, we have @xmath280 therefore , by the assumption @xmath279 and ( [ q ] ) , we have @xmath281 let us take the universal extension @xmath282 then by ( * ? ? ?
* lemma 2.1 ) , the sheaf @xmath283 is a torsion free slope stable sheaf . applying the bg inequality to @xmath283 , we obtain the inequality @xmath284 the above inequality implies that @xmath285 , which contradicts to ( [ h2(xf ) ] ) .
@xmath286 .
let us take a non - zero element @xmath287 , and an exact sequence @xmath288 by ( * ? ? ?
* lemma 2.2 ) , the sheaf @xmath179 is a torsion free slope stable sheaf .
therefore it is written as @xmath289 for some subscheme @xmath290 with @xmath291 .
we have the equalities of chern characters @xmath292 \\ { \mathop{\rm ch}\nolimits}_3(f ) & = \frac{1}{6}h^3 - h \cdot [ z ] -\chi({\mathcal{o}}_z ) .
\end{aligned}\ ] ] because @xmath293 , we have @xmath294=2 $ ] . hence we obtain @xmath295 on the other hand , ( [ ch_3 ] ) implies that @xmath296 , hence we have @xmath297 . by taking the generic projection of the one dimensional subscheme @xmath298 to @xmath6 , the castelnuovo inequality implies @xmath299 -1 ) ( h \cdot [ z]-2 ) .
\end{aligned}\ ] ] since @xmath294=2 $ ] , we have @xmath300 , which contradicts to @xmath297 .
in this section , we take @xmath301 to be a quintic homogeneous polynomial with four variables @xmath302 , \quad \deg(w')=5.\end{aligned}\ ] ] we consider conjecture [ conj : main ] in this case .
we relate it with some clifford type bound for stable coherent systems on the smooth quintic surface @xmath303 the surface @xmath195 is a hyperplane section @xmath304 of a quintic 3-fold @xmath305 , where @xmath53 is defined by @xmath306 .
\end{aligned}\ ] ] by theorem [ thm : orlov2 ] , there is the heart of a bounded t - structure @xmath307 , and an equivalence @xmath308 below we abbreviate @xmath102 and regard a coherent system @xmath309 as an object in @xmath310 .
there is a natural push - forward functor ( cf .
@xcite ) @xmath311 such that by ( * ? ? ?
* lemma 3.12 ) and ( * ? ? ?
* lemma 4.5 ) , we have @xmath312 here @xmath313 is the usual sheaf push - forward for the embedding @xmath314 , @xmath315 an equivalence in theorem [ thm : orlov ] and @xmath316 is an object in the derived category with @xmath313 located in degree zero .
let us consider the central charge @xmath317 on @xmath318 defined by @xmath319 where @xmath171 is the central charge ( [ rewrite ] ) on @xmath12 considered in the previous section . by the argument in (
* section 4 ) , the central charge @xmath317 on @xmath318 differs from ( [ z_g ] ) only up to a scalar multiplication . for @xmath320 ,
let us write @xmath321 with @xmath322 and @xmath323 . by setting @xmath324 and @xmath200
, we have @xmath325 applying the computation of @xmath171 in the previous section , we have @xmath326 here @xmath327 and @xmath203 are irrational numbers given in ( [ rewrite ] ) .
we expect that a desired gepner type stability condition in this case is constructed via double tilting of @xmath310 , similarly to the previous section .
let @xmath328 be the slope function on @xmath310 , given by ( using the notation in the previous subsection ) @xmath329 here we set @xmath330 if @xmath331 .
( also see ( * ? ? ? * subsection 5.4 ) . )
the above slope function defines the @xmath328-stability on @xmath310 , which satisfies the harder - narasimhan property ( cf .
* lemma 5.14 ) ) .
following the same argument in the previous section , we expect that any @xmath328-stable object @xmath332 with @xmath333 satisfies @xmath334 .
it leads to the following conjecture : if we assume the above conjecture , we are able to construct a double tilting @xmath339 of @xmath310 , such that the pair @xmath340 satisfies @xmath341 \}. \end{aligned}\ ] ] we conjecture that the pair @xmath340 gives a gepner type stability condition on @xmath318 with respect to @xmath342 . the construction of @xmath343 is similar to @xmath194 in the previous section , and we leave the readers to give its explicit construction .
we just check the easiest case of conjecture [ conj : clifford ] : let @xmath345 be a @xmath328-stable coherent system on @xmath195 with @xmath346 . the inequality in conjecture [ conj : clifford ]
is equivalent to @xmath347 .
it is enough to show that @xmath348 .
let @xmath349 be a torsion free quotient .
there is a surjection in @xmath310 @xmath350 whose kernel is of the form @xmath351 for a torsion sheaf @xmath352 on @xmath195 . obviously @xmath353 is also @xmath328-stable , and @xmath354 .
hence we may assume that @xmath87 is torsion free .
also note that @xmath355 , since otherwise there is an injection in @xmath310 @xmath356 satisfying @xmath357 which contradicts to the @xmath328-stability of @xmath358 .
let us set @xmath359 , and take a smooth member @xmath360 .
note that @xmath361 is a line bundle satisfying @xmath362 , and @xmath363 is a smooth quintic curve in @xmath364 .
suppose by contradiction that @xmath365 .
we have the exact sequence @xmath366 since @xmath367 by our assumption , we have @xmath368 and @xmath369 . on the other hand , clifford s theorem on @xmath363 yields ( cf .
* theorem 5.4 ) ) @xmath370 furthermore , the first inequality is strict since @xmath371 , and @xmath363 is not hyperelliptic .
therefore we obtain a contradiction .
|
we propose a conjectural stronger version of bogomolov - gieseker inequality for stable sheaves on quintic 3-folds .
our conjecture is derived from an attempt to construct a bridgeland stability condition on graded matrix factorizations , which should correspond to the gepner point via mirror symmetry and orlov equivalence .
we prove our conjecture in the rank two case .
|
this is a review talk to present the main ideas of some line of research in quantum gravity , namely the spin foam approach , that has been explored by a great number of physicists and mathematicians and has attracted much attention .
the three main lines of research in quantum gravity are denoted as `` canonical '' , `` covariant '' or `` sum over histories '' @xcite the canonical line of research is a theory in which the hilbert space carries a representation of the quantum operators corresponding to the full metric without background metric to be fixed .
it can be considered as a quantum field theory on a differentiable manifold .
the basis of the hilbert space are cilindrical functions defined on a graph ( wilson loops ) depending on ashtehar variables @xcite . a very important result of this approach was the discrete eigenvalues for the area and volumen operators .
the covariant line of research is the attempt to built the theory as a quantum field theory of the fluctuations of the metric over a flat minkowski space , or some other background metric space .
the theory has been proved to be renormalizable and finite order by order .
@xcite the sum over histories line of research uses the feymann path integral to quantize the einstein hilbert action .
there exist a duality between this model and group field theories .
the sum over spin foam can be generated as the feymann perturbative expansion of the group field theories .
each space - time appears as the feymann graph of the auxiliary groups field theory .
@xcite our presentation is going to be concentrated on this third line of research , namely , the spin network and the spin foam models , from an historical point of view .
the regge s paper @xcite was a pionnier work in the discretization of gr , that was motivated by the need to avoid coordinates , because the physical prediction of the theory was coordinates independent .
for that purpose he discretizes a continuous manifold by @xmath0-simplices , that are glued together by identification of their @xmath1-simplices .
the curvature lies on the @xmath2- dimensional subspaces , known as hinges or bones . from pedagogical reasons
we take a triangulation of a 2-dimensional surface .
when a collection of triangle meeting at a vertex is flattened there will be a gap or deficit angle @xmath3 , indicating the presence of curvature . using the gauss - bonet formula we can calculate the excess angle by @xmath4 , where @xmath5 is the curvature at that vertex and @xmath6 the area of the triangles around the vertex .
if the number of vertices increases we can take @xmath7 , where @xmath8 is the density of vertices in the triangulation (= number of vertices by unit area ) .
this method is easily enlarged to higher dimensions . in order to have connecton with gr we traslate into the triangulated surface ( the skeleton ) the hilbert - einstein action @xmath9 where @xmath5 is the scalar curvature in 4-dimensions .
the discrete version for a 4-dimensional skeleton is given in terms of the deficit angle in each bone where the curvature @xmath5 is calculated and some measure function @xmath10 is defined : @xmath11 here the summation extends to all the bones in the skeleton . in the continuous case the einstein s equations are derived from a stationary action , varying @xmath12 with respect to the metric . in the discrete version one derives the action with respect to the edge lengths , because in the simplicial decomposition all the properties can be derived from these edges . using schlaefli differential identity one finds @xmath13 which is the discrete version of einstein s equations @xcite
some years later ponzano and regge @xcite made use of @xmath14 symbols attached to the tetrahedra decomposition in order to calculate the state sum and were able to connect it to the feymann integral corresponding to the hilbert - einstein action .
the @xmath14 wigner simbols is a generalization of the clebsch - gordan coefficients that appear in the coupling of two angular momenta @xmath15 the new basis is given in term of the old basis : @xmath16 if we couple @xmath17 with a new angular momentum @xmath18 we have two possibilities @xmath19 in the first case the new basis is given ( in obvious notation ) @xmath20 in the second case the new basis is given by @xmath21 the transformating matrix between the two basis is given precisely by the @xmath14 symbols , namely , @xmath22 given a tetrahedra decomposition of a 3-dimensional surface we can attach a @xmath14 symbol to each tetrahedra , the edges of which have the length equal to the numerical values of the angular momenta appearing in the @xmath23-symbol .
this choice is consistent with the inequalities @xmath24 and the equalities @xmath25 the @xmath14 symbols are proportional to the racah polynomials @xcite @xmath26 from this equality and the assymptotic properties of racah polynomials one can derive very important limit @xmath27 where @xmath28 is the volume of the tetrahedra and @xmath29 the exterior dihedral angle adjoint to the edge @xmath30 in order to see the conection between @xmath14 symbols and quantum gravity we take a tetrahedra decomposition and take external edges @xmath31 of the bounding surface and internal edges @xmath32
. then ponzano and regge construct the state sum as follows @xmath33 where @xmath34 is a phase factor . substituting the @xmath14 symbols by their assymptotic values and the function cosine by the euler expression we arrive at @xmath35x_i } \right . } } \ ] ] we may replace the summation with an integral . then the most important contribution comes from the stationary phase that is to say when @xmath36 introducing this value in the state sum we obtain @xcite @xmath37 where @xmath38 the summation @xmath39 approaches the hilbert - einstein action that was given in the regge calculus , therefore in the limit the state sum strongly resembles a feymann summation over histories with density of lagrangian @xmath40 @xmath41namely @xmath42
penrose was interested in the interpretation of space - time @xcite by purely combinatorial properties of some elementary units that are connected among themselves by some interactions that follow the angular momentum quantum rules , and form some network of elementary units with assigned spins .
soon it was realized that the spin network was analog to simplicial quantum gravity , in particular the ponzano - regge model @xcite .
his networks had trivalent vertices and the edges were labeled with spin , satisfying the standard conditions at the vertices .
the model was generalized to any group different from the rotation group .
formaly a spin network ia a triple @xmath43 where 1 .
@xmath44 is a graph with a finite set of edges @xmath45 and a finite set of vertices , @xmath46 , 2 . to each edge
@xmath45 we attach an irrep of a group @xmath47 , @xmath48 3 . to each vertex
@xmath46 we attach an intertwiner .
when we take the dual of an spin network we obtain a triangulated figure , which , after embedding in a 3-dimensional manifold becomes the triangulation of regge calculus .
they defined a state sum for triangulated 3-manifold ( as in the ponzano - regge model ) that was independent os the triangulation and finite @xcite . for this purpose
they assign a value from the set @xmath49 , integer , to each edge of the triangulation , subject to the condition that the `` coloring '' of the three edges forming a triangle should satisfy the triangle inequalities and their sum should be and integer less than or equal to @xmath50 .
define the quantum object @xmath51 where @xmath52 is an admisible coloring of the edge @xmath53 , @xmath54^{1/2 } \quad , \quad \omega _ { } ^2 = \sum\limits_{j \in i_r } { \omega _ j^4 } \ ] ] and @xmath55 is the quantum @xmath23-simbol corresponding to the tetrahedron @xmath56 with coloring @xmath57 , such that @xmath58 } & { \left [ j \right ] } & { \left [ k \right ] } \cr { \left [ l \right ] } & { \left [ m \right ] } & { \left [ n \right ] } \cr \end{array } \end{matrix } \right\}\ ] ] where [ @xmath0 ] is the quantum number satisfying @xmath60 \to n$ ] .
summing @xmath61 over all admisible coloring we obtain an expresion in the limit @xmath62 or @xmath63 that becomes identical to the ponzano - regge state sum .
turaev and viro proved that their expression is manifold invariant ( or independent of triangulation ) under alexander moves , and also finite .
the ponzano - regge state sum and the turaev - viro model are defined over 3-dimensional manifold . to enlarge the model to four dimensions it vas necessary to increase the wigner symbols to @xmath64 .
the key to this approach was given by boulatov @xcite by the use of topological lattice gauge theories , taking group elements as variables ( matrix models ) .
the basic objects is a set of real functions of 3 variables @xmath65 ( where @xmath66 invariant under simultaneous right shift of all variables by @xmath67 and also by cyclic permutation of @xmath68 .
this function @xmath69 can be expanded , by peter - weyl theorem , in terms of representations of @xmath70 and 3j - symbols .
an action of interest can be constructed with those functions as follows @xmath71 if we attach the variable to the edges , the first term ( the kinematical term ) represent two glued triangle and the second one ( the interacting term ) four triangles forming a tetrahedron . substituting the fourier expansion of funcion @xmath72 , and integrating out group variables we obtain an action depending on the fourier coefficientes and 6j - simbols . from this result
we calculate the partition function as a feymann path integral with respect to the fourier coefficients @xmath73 where the products extend to all tetrahedra t , all edges @xmath74 , and the summation extend to all the representations @xmath75 , all the simplicial complexes @xmath76 and @xmath77 is the number of tetrahedra in complex @xmath78 .
this partition function is equivalent , up to renormalization , to ponzano - regge state sum applied to some triangulation of 3-dimensional manifold .
the underlying mathematical structure is a topological lattice gauge theory , it has the advantage that is topological invariant . in order to prove it ,
boulatov used the alexander moves , by which one complex , and the corresponding partition function is topological invariant
the 3-dimensional boulatov model paved the way to the ooguri s model in four dimensions .
@xcite let @xmath72 be a real valued function of four variables @xmath79 on @xmath80 a compact group . for simplicity
we take @xmath81 .
we require @xmath72 to be invariant under the right action of @xmath47 , and by cyclic permutation of these variables . the peter - weyl theorem , we can expand @xmath72 in terms of these representations and the 3j - symbols .
we define the action @xmath82 where the first term ( the kinematical term ) represents the coupling of a tetrahedrum with itself because each element @xmath83 is associated to each face of the tetrahedrum , and the second term ( the interacting term ) represents gluing faces of five tetrahedra to make a four - simplex . substituting the fourier expansion into the action we can integrate out the group variable , and then the action can be used to calculate a partition function as a feyman path integral with respect to this action : @xmath84 where the integral is defined in terms of the fourier coefficients @xmath85 , appearing in the action and in the measure , the first sum is over all complexes @xmath78 ( four - dimensional combinatorial manifolds ) , @xmath86 is the number of 4-simplices in @xmath78 , the second summation is over all irreducible representations os @xmath70 with angular momentum @xmath87 and @xmath88 are the triangles , tetrahedra and 4-simplexes respectively apearing in the complex .
ooguri also proved that the partition function is topological invariant under the alexander moves .
as in the boulatov model two complexes are combinatorially equivalent if and only if they are connected by a sequence of transformations called alexander moves .
a more abstract approach was taken by barrett and crane @xcite generalizing penrose s spin networks to four dimensions .
the novelty of this model consists in the association of representations of @xmath89 group to the faces of the tetrahedra , instead of the edges .
they decompose a triangulation of a 4-dimensional manifold into 4-simplices , the geometrical properties of which are characterized in terms of bivectors .
a geometric 4-simplex in euclidean space is given by the embedding of an ordered set of 5 points in @xmath90 which is required to be non - degenerate ( the points should not lie in any hyperplane ) .
each triangle in it determines a bivector constructed out of the vectors for the edges .
barrett and crane proved that classically , a geometric 4-simplex in euclidean space is completely characterized ( up to parallel translation an inversion through the origin ) by a set of 10 bivectors @xmath91 , each corresponding to a triangle in the 4-simplex and satisfying the following properties : 1 .
the bivector changes sign if the orientation of the triangle is changed ; 2 .
each bivector is simple , i.e. is given by the wedge product of two vectors for the edges ; 3 .
if two triangles share a common edge , the sum of the two bivector is simple ; 4 .
the sum ( considering orientation ) of the 4 bivectors corresponding to the faces of a tetrahedron is zero ; 5 . for six triangles sharing the same vertex ,
the six corresponding bivectors are linearly independent ; 6 .
the bivectors ( thought of as operators ) corresponding to triangles meeting at a vertex of a tetrahedron satisfy tr @xmath92>0 $ ] i.e. the tetrahedron has non - zero volume . then barrett and crane define the quantum 4-simplex with the help of bivectors thought as elements of the lie algebra @xmath89 , associating a representation to each triangle and a tensor to each tetrahedron .
the representations chosen should satisfy the following conditions corresponding to the geometrical ones : 1 .
different orientations of a triangle correspond to dual representations ; 2 .
the representations of the triangles are `` simple '' representations of @xmath89 , i.e. @xmath93 ; 3 . given two triangles , if we decompose the pair of representations into its clebsch - gordan series , the tensor for the tetrahedron is decomposed into summands which are non - zero only for simple representations ; 4 .
the tensor for the tetrahedron is invariant under @xmath89 .
+ now it is easy to construct an amplitude for the quantum 4-simplex .
the graph for a relativistic spin network is the 1-complex , dual to the boundary of the 4-simplex , having five 4-valent vertices ( corresponding to the five tetrahedra ) , with each of the ten edges connecting two different vertices ( corresponding to the ten triangles of the 4-simplex each shared by two tetrahedra ) .
now we associate to each triangle ( the dual of which is an edge ) a simple representation of the algebra @xmath89 and to each tetrahedra ( the dual of which is a vertex ) a intertwiner ; and to a 4-simplex the product of the five intertwiner with the indices suitable contracted , and the sum for all possible representations . the proposed state sum suitable for quantum gravity for a given triangulation ( decomposed into 4-simplices ) is @xmath94 where the sum extends to all possible values of the representations @xmath17 . in order to know the representation attached to each triangle of the tesselation , we take the unitary representation of so(4 ) in terms of euler angles . @xmath95 where @xmath96 is the rotation matrix in the @xmath97 plane , @xmath98 the rotation matrix in the @xmath99 plane and @xmath100 the rotation ( `` boost '' ) in the @xmath101 plane . in the angular momentum basis ,
the action of @xmath100 is a follows @xmath102 where @xmath103 is the biedenharn - dolginov function @xcite
in order to evaluate the state sum for a particular triangulation of the total @xmath104 space by 4-simplices , we assign an element @xmath105 to each tetrahedrum of the 4-simplex @xmath106 and a representation @xmath107 of @xmath89 to each triangle shared by two tetrahedra . from this triangulation
we obtain an 2-complex by the dual graph where one vertex corresponds to a tetrahedrum and an edge corresponds to triangle , with the ends of the edges identified with the vertices .
then we attach a representation of @xmath70 , @xmath108 and @xmath109 to the vertices @xmath56 and @xmath74 and contract both representation along the edges @xmath110 , giving @xmath111 where @xmath112 is the representation of @xmath89 corresponding to the product @xmath113 , the left and right components of the @xmath89 group .
the state sum for the 2-dimensional complex ( the feymann graph of the model ) is obtained by taking the product for all the edges of the graph and integrating for all the copies of @xmath70 @xmath114 due to the trace condition this expression is invariant under left and right multiplication of some elements of @xmath70 .
@xcite for the representation @xmath115 we choose the spherical function with respect to the identity representation . given a completely irreducible representation of the group @xmath116 on the space @xmath117
, we define the spherical function with respect to the finite irrepr . of the subgroup @xmath5 @xmath118 where @xmath119 is a projector of @xmath117 onto the space @xmath120 of @xmath5 we take for @xmath121 the simple representation @xmath122 and for the subgroup @xmath70 the identity representation @xmath123 . since @xmath124 is invariant under @xmath5 we can restrict the unitary representations to those of the boost @xmath125 . with the help of the biedenharn - dolginov function
it can be proved @xmath126 with this formula it is still possible to give a geometrical interpretation of the probability amplitude encompassed in the trace .
in fact the spin dependent factor appearing in the exponential of the spherical function @xmath127 corresponding to the two tetrahedra @xmath128 intersecting the triangle @xmath129 , can be interpreted as the product of the angle between the two vectors @xmath130 , perpendicular to the triangle , and the area @xmath131 of the intersecting triangle , @xmath132 being the spin corresponding to the representation @xmath115 associated to the triangle @xmath128 . substituting this value in the state sum , we obtain @xmath133 where the product extend to all tetrahedra with the vector @xmath134 perpendicular to the subspace where the tetrahedra is embedded , and summation is extended to all the triangle @xmath128 intersected by two tetrahedra @xmath56 and @xmath74 .
the exponential term correspond to the regge action , that in the assymptotic limit becomes the hilbert - einstein action @xcite because we are interested in the physical and mathematical properties of the barrett - crane model , we mention some recent work about this model combined with the matrix model approach of boulatov and ooguri.@xcite in this work the 2-dimensional quantum space - time emerges as a feymann graph , in the manner of the 4 dimensional matrix models . in this way a spin foam model is connected to the feyman diagram of quantum gravity .
now we apply the same technique to calculate the state sum invariant under the lorentz group that we have used in the case of the @xmath89 group for the barrett - crane model the unitary irreducible representation of the @xmath135 group for the principal series is given by the formula [ 7 ] @xmath136 } \psi } \right)\left ( z \right ) = \left ( { \beta z + \delta } \right)^{m + i\rho - 2 } \left ( { \beta z + \delta } \right)^ { - m } \psi \left ( { { { \alpha z + \gamma } \over { \beta z + \delta } } } \right)\ ] ] where @xmath137 , @xmath138 , integer , @xmath8 ral @xmath139 .
the numbers @xmath140 determine the eigenvalues of the representation @xmath141 in order to calculate the state sum we need the spherical functions of the irrep of @xmath135 .
these are given in terms of the biedenharn - dolginov function that correponds to the boost operator @xmath142 } \left ( \tau \right ) & = \int\limits _ { - \infty } ^\infty { d_{j - m}^ { - 1 } p_{j - m}^{\left ( { m - m , m + m ) } \right ) } } \left ( { \lambda , \rho } \right)e^ { - i\tau \lambda } \times \\ & \times\;d_{j ' - m}^ { - 1 } p_{j ' - m}^{\left ( { m - m , m + m ) } \right ) } \left ( { \lambda , \rho } \right)\omega \left ( \lambda \right)d\lambda \end{aligned}\ ] ] where @xmath143 are the angular momentum eigenvalues , @xmath144 is some normalization constant , and @xmath145 are the hahn polynomials of imaginary argument @xcite . given the unitary representation @xmath146 of the group @xmath135 and the identity representation of @xmath70 , the spherical functions is defined as in the case of @xmath89 @xmath147 where the last step has been calculated with the residue theorem .
as in the case of euclidean @xmath89 invariant model , we take a non degenerate finite triangulation of a 4-manifold .
we consider the 4-simplices in the homogeneous space @xmath148 @xmath149 , the hyperboloid @xmath150 and define the bivectors @xmath151 on each face of the 4-simplex , that can be space - like , null or timelike @xmath152.@xcite in order to quantize the bivectors , we take the isomorphism @xmath153 with @xmath154 a minkowski metric . the condition for @xmath151 to be a simple bivector @xmath155 , gives @xmath156 we have two cases : \1 ) @xmath157 ; @xmath10 , space - like , @xmath151 time - like , \2 ) @xmath158 ; @xmath10 , time like , @xmath151 space like ( remember , the hodge operator @xmath159 changes the signature ) in case 2 ) @xmath151 is space - like , @xmath160 . expanding this expression in terms of space like vectors , @xmath161 @xmath162 where @xmath163 is the lorentzian space - like angle between @xmath164 and @xmath165 ; this result gives a geometric interpretation between the parameter @xmath8 and the area expanded by the bivector @xmath166 , namely , @xmath167 .
( this result is equivalent to that obtained in in the euclidean case where the area of the triangle expanded by the bivector was proportional to the value @xmath168 , @xmath53 being the spin of the representation ) . in order to construct the lorentz invariant state sum we take a non - degenerate finite triangulation in a 4-dimensional simplices in such a way that all 3-dimensional and 2-dimensional subsimplices have space - like edge vectors which span space - like subspace .
we attach to each 2-dimensional face a simple irrep . of @xmath169 characterized by the parameter @xmath170 $ ] .
the sate sum is given by the expression @xcite @xmath171 where @xmath8 refers to all the faces in the triangulation , @xmath172 corresponds to the simple irrep attachec to 4 triangles in the tetrahedra and @xmath173 corresponds to the simple irrep attached to the 10 triangles in the 4-simplices .
the functions @xmath174 and @xmath175 are defined as traces of recombination diagrams for the simple representations .
the traces are explicitely given as multiple integrals on the upper sheet @xmath176 of the 2-sheeted hyperboloid in minkowski space .
for the integrand we take the spherical function @xmath177 where @xmath178 is the hyperbolic distance between @xmath164 and @xmath165 the trace of a recombination diagram is given by a multiple integral of products of spherical functions . for a tetrahedrum
we have @xmath179 where we have dropped one integral for the sake of normatlization without loosing lorentz symmetry . for a 4-simplex
we have @xmath180 the last 4 equation defines the state sum completely , that has been proved to be finite @xcite the assymptotic properties of the spherical functions are related to the einstein - hilbert action giving a connection of the model with the theory of general relativity .
the author wants to express is gratitude to the organizers of the workshop in particular , to professors odzijewicz and golinsk for the invitation to give this review talk .
this work was supported partially by m.e.c .
( spain ) through a grant bfm 2003 - 00313/fis .
barut a. and raczka r. , _ theory of group representations and applications _ , pwn polish scientific publishers , warsaw , 1977 .
naimark , _ linear representations of the lorentz group _ , pergamon press .
oxford 1964 . di pietri r. , reidel l. , k. krasnov k. , and rovelli c. , _ barrett - crane model from a boulatov - ooguri field theory over an homogeneous space _
b * 574 * ( 2000 ) 785806 .
m. reisemberger , c. rovelli .
, _ spin foams as a feymann diagram _ gr - qc/0002083 .
lorente m. and kramer p. , _ discrete quantum gravity :
i. spherical functions of the representations of so(4 ) group with respect to the su(2 ) subgroup and its applications to the euclidean invariant weight for the barrett - crane model_.
|
this is a review paper about one of the approaches to unify quantum mechanics and the theory of general relativity .
starting from the pioneer work of regge and penrose other scientists have constructed state sum models , as feymann path integrals , that are topological invariant on the triangulated riemannian surfaces , and that in the continuous limit become the hilbert - einstein action .
[ first ]
|
one of the interesting subjects for recent gravitational studies is the investigation of three dimensional black holes astorino1,astorino2,astorino3,astorino4,astorino5,astorino6,astorino7,astorino8,astorino9 .
considering three dimensional solutions helps us to find a profound insight in the black hole physics , quantum view of gravity and also its relations to string theory @xcite .
moreover , three dimensional spacetimes play an essential role to improve our understanding of gravitational interaction in low dimensional manifolds @xcite .
due to these facts , some of physicists have an interest in the @xmath0-dimensional manifolds and their attractive properties btz1,btz2,nojiri19981,nojiri19982,nojiri19983,nojiri19984,nojiri19985,nojiri19986,nojiri19987,nojiri19988,nojiri19989,nojiri199810 .
the maxwell theory is in agreement with experimental results , but it fails regarding some important issues such as self energy of point - like charges which motivates one to regard nonlinear electrodynamics ( ned ) .
there are some evidences that motivate one to consider ned theories : solving the problem of point - like charge self energy , understanding the nature of different complex systems , obtaining more information and insight regarding to quantum gravity , compatible with ads / cft correspondence and string theory frames , description of pair creation for hawking radiation and the behavior of the compact astrophysical objects such as neutron stars and pulsars chen , fukuma , aros . therefore , many authors investigated the black hole solutions with nonlinear sources nonsources1,nonsources2,nonsources3,nonsources4,nonsources5,nonsources6,nonsources7,nonsources8,nonsources9,nonsources10,nonsources11,nonsources13,nonsources14,nonsources15,nonsources16,nonsources17,nonsources18,nonsources19,nonsources20
. on the other hand , thermodynamical structure of the black holes , has been of great interest .
it is due to the fact that , according to ads / cft correspondence , black hole thermodynamics provides a machinery to map a solution in ads spacetime to a conformal field on the boundary of this spacetime @xcite .
also , it was recently pointed out that considering cosmological constant as a thermodynamical variable leads to the behavior similar to the van der waals liquid / gas system pvpaper1,pvpaper2,pvpaper3,pvpaper4,pvpaper5,pvpaper6,pvpaper7,pvpaper8 . in addition , phase transition of the black holes plays an important role in exploring the critical behavior of the system near critical points .
there are several approaches that one can employ to study the phase transition .
one of these approaches is studying the behavior of the heat capacity .
it is argued that roots and divergence points of the heat capacity are representing two types of phase transition @xcite . in addition , studying the heat capacity and its behavior , enable one to study the thermal stability of the black holes @xcite .
another approach for studying phase transition of black holes is through thermodynamical geometry .
the concept is to construct a spacetime by employing the thermodynamical properties of the system .
then , by studying the divergence points of thermodynamical ricci scalar of the metric , one can investigate phase transition points .
in other words , it is expected that divergencies of thermodynamical ricci scalar ( trs ) coincide with phase transition points of the black holes .
firstly , weinhold introduced differential geometric concepts into ordinary thermodynamics @xcite .
he considered a kind of metric defined as the second derivatives of internal energy with respect to entropy and other extensive quantities for a thermodynamical system .
later ruppeiner @xcite introduced another metric and defined the minus second derivatives of entropy with respect to the internal energy and other extensive quantities .
the ruppeiner metric is conformal to the weinhold metric with the inverse temperature as the conformal factor .
it is notable that , both metrics have been applied to study the thermodynamical geometry of ordinary systems janyszek1,janyszek2,janyszek3,janyszek4,janyszek5,janyszek6 . in particular
, it was found that the ruppeiner geometry carries information of phase structure of thermodynamical system .
for the systems with no statistical mechanical interactions ( for example , ideal gas ) , the scalar curvature is zero and the ruppeiner metric is flat . because of the success of their applications to ordinary thermodynamical systems , they have also been used to study black hole phase structures and lots of results have been obtained for different types of black holes ferrara1,ferrara2,ferrara3,ferrara4,ferrara5,ferrara6 .
it is notable that these two approaches fail in order to describe phase transition of several black holes @xcite . in order to overcome this problem ,
quevedo proposed new types of metrics for studying geometrical structure of the black hole thermodynamics @xcite .
this method was employed to study the geometrical structure of the phase transition of the black holes pt1,pt2,pt3,pt4,pt5,pt6,pt7,quevedop1,quevedop2,quevedop3,quevedop4,quevedop5 and proved to be a strong machinery for describing phase transition of the black holes . but this approach was not completely coincided with the results of classical thermodynamics arisen from the heat capacity @xcite . in ref .
@xcite , a new metric was proposed in which the denominator of its ricci scalar is only constructed of numerator and denominator of the heat capacity
. several phase transition of the black holes have been studied in context of the hpem ( hendi - panahiyan - eslam panah - momennia ) metric @xcite . in this paper
we study thermal stability and phase transition of the btz black holes in presence of several ned models in context of heat capacity .
then , we employ weinhold , ruppeiner and quevedo methods for studying geometrothermodynamics of these black holes .
we will see that weinhold and ruppeiner metrics fail to provide fruitful results and the consequences of the quevedo approach are not completely matched to the heat capacity results .
then , we employ the hpem metric and study the phase transition of these black holes in context of geometrothermodynamics .
we end the paper with some closing remarks .
the @xmath0-dimensional action of einstein gravity with ned field in the presence of cosmological constant is given by @xmath1 , \label{action}\]]where @xmath2 is the ricci scalar , the cosmological constant is @xmath3 in which @xmath4 is a scale factor . also , @xmath5 is the lagrangian of ned , in which we consider three models .
first model was proposed by hendi ( hned ) @xcite , second one is soleng theory ( sned ) @xcite and third one is correction form of ned ( cned ) @xcite @xmath6where @xmath7 and @xmath8 are called the nonlinearity parameters , the maxwell invariant @xmath9 in which @xmath10 is the electromagnetic field tensor and @xmath11 is the gauge potential .
we should note that for @xmath12 ( hned and sned branches ) and @xmath13 ( cned branch ) the maxwell lagrangian can be recovered .
the nonlinearly charged static black hole solutions can be introduced with the following line element @xmath14where the metric function @xmath15 was obtained in refs .
@xcite @xmath16 , & hned\vspace{0.3 cm } \\ 4\beta ^{2}r^{2}\left [ \ln \left ( \frac{\gamma + 1}{2}\right ) + 3\right ] -q^{2}% \left [ \ln \left ( \frac{\beta ^{2}r^{4}\left ( \gamma -1\right ) \left ( \gamma + 1\right ) ^{3}}{4q^{2}l^{2}}\right ) + \frac{6}{\gamma -1}-2\right ] , & sned% \vspace{0.3 cm } \\ -2q^{2}\ln \left ( \frac{r}{l}\right )
-\frac{2\alpha q^{4}}{r^{2}}+\mathcal{o}% ( \alpha ^{2 } ) , & cned% \end{array}% \right . , \label{metric3dim}\]]where @xmath17 , @xmath18 and @xmath19 are integrations constant which are related to mass parameter and the electric charge of the black hole , respectively . in addition , @xmath20 , @xmath21 and the special function @xmath22 .
the entropy and the electric charge of the obtained ned black hole solutions can be calculated with the following forms @xcite @xmath23where @xmath24 is the event horizon of black hole . on the other hand , the quasi - local mass , which is related to geometrical mass , can be obtained as @xcite @xmath25 regarding eqs .
( [ entropy ] ) and ( [ q ] ) with ( [ mass ] ) and obtaining @xmath18 by using @xmath26 , for these three cases of btz black holes we can write @xmath27 , & hned\vspace{0.4 cm } \\ \frac{4s^{2}}{l^{2}}+16\beta ^{2}s^{2}\left [ \ln \left ( \frac{\gamma ^{\prime } + 1}{2}\right ) + 3\right ] -4q^{2}\left [ \ln \left ( \frac{\beta ^{2}s^{4}\left ( \gamma ^{\prime } -1\right ) \left ( \gamma ^{\prime } + 1\right ) ^{3}}{q^{2}l^{2}\pi ^{2}}\right ) + \frac{6}{\gamma ^{\prime } -1}-2\right ] , & sned\vspace{0.4 cm } \\ \frac{4s^{2}}{l^{2}}-8q^{2}\ln \left ( \frac{2s}{l\pi } \right ) -\frac{8q^{4}}{% s^{2}}\alpha , & cned% \end{array}% \right . , \label{mass2}\ ] ] where @xmath28 and @xmath29 .
having conserved and thermodynamic quantities at hand , it was shown that the first law of thermodynamics may be satisfied @xcite .
the main goal of this paper is investigating thermal stability and phase transition for these black holes . in order to investigate thermal stability and phase transition
, one can usually adopt two different approaches to the matter at hand . in one method ,
the electric charge is considered as a fixed parameter and heat capacity of the black hole will be calculated .
the positivity of the heat capacity is sufficient to ensure the local thermal stability of the solutions and its divergencies are corresponding to the phase transition points .
this approach is known as canonical ensemble .
another approach for studying thermal stability of the black holes is grand canonical ensemble . in this approach
, thermal stability is investigated by calculating the determinant of hessian matrix of m(s , q ) with respect to its extensive variables .
the positivity of this determinant also represents the local stability of the solutions .
although these two approaches are different fundamentally , one expects that the results being the same for both ensembles ; i.e. ensemble independent .
here we use the first method for studying thermal stability . for this purpose ,
the system is considered to be in fixed charge and the heat capacity has the following form @xmath30 it is notable that , when we study heat capacity for investigating the phase transition , we encounter with two different phenomena . in one ,
the changes in the signature of the heat capacity is representing a phase transition of the system .
in other words , if the heat capacity is negative , then the system is in thermally unstable phase , whereas for the case of the positive @xmath31 , the system is thermally stable .
the roots of the heat capacity in this case are representing phase transition points which means one should solve the following equation @xmath32where hereafter we call this type of the phase transition as type one .
it is a matter of calculation to show that @xmath33 } { 2\pi sl^{2}\sqrt{l_{w}^{\prime } } \left ( 1+l_{w}^{\prime } \right ) } , & hned\vspace{0.4 cm } \\ \begin{array}{c } \frac{4\beta ^{2}l^{2}q^{2}\left ( q^{2}-s^{2}\beta ^{2}\right ) \ln \left ( \frac{1+\gamma ^{\prime } } { 2}\right ) + 8s^{2}q^{2}l^{2}\beta ^{4}+q^{4}\left ( 1+s^{2}+8\beta ^{2}l^{2}\right ) } { 2\pi s^{3}\beta ^{4}l^{2}\gamma ^{\prime } \left ( 1-\gamma ^{\prime } \right ) \left ( 1-\gamma ^{\prime 2}\right ) } + \vspace{0.25cm}\\ \frac{s^{4}\beta ^{2}\left ( 1-s^{2}\sqrt{\gamma ^{\prime } } \right ) -6q^{4}l^{2}}{2\pi s^{3}\beta ^{2}l^{2}\left ( 1-\gamma ^{\prime } \right ) \left ( 1-\gamma ^{\prime 2}\right ) } + \frac{2s^{2}\ln \left ( \frac{\gamma ^{\prime } + 1}{2}\right ) + 6\beta ^{2}+q^{2}}{\pi s\left ( 1-\gamma ^{\prime }
\right ) } , % \end{array } & sned\vspace{0.4 cm } \\ \frac{\left ( 2q^{4}\alpha -s^{2}q^{2}\right ) l^{2}+s^{4}}{\pi l^{2}s^{2 } } , & cned% \end{array}% \right . , \label{dmds}\ ] ] the other case of phase transition is the divergency of the heat capacity .
in other words , the singular points of the heat capacity are representing places in which system goes under phase transition .
this assumption leads to the fact that the roots of the denominator of the heat capacity are representing phase transitions .
therefore , we have the following relation for this type of phase transition @xmath34where we call this type of the phase transition as type two . due to economical reasons , we did not write the explicit relations of @xmath35 for different btz solutions . in order to have an effective geometrical approach for studying phase transition of a system
, one can build a suitable thermodynamical metric and investigate its ricci scalar .
thermodynamical metrics were introduced based on the hessian matrix of the mass ( internal energy ) with respect to the extensive variables .
therefore , although the electric charge is a fixed parameter for calculating the heat capacity in canonical ensemble , it may be an extensive variable for constructing thermodynamical metrics . in this method we expect that trs diverges in both types of the mentioned phase transition points .
in other words , the denominator of trs must be constructed in a way that contains roots of the denominator and numerator of the heat capacity . in what follows
, we will study the denominator of trs of the several geometrical approaches and follow the recently proposed thermodynamical metric which its denominator only contains numerator and denominator of the heat capacity , and therefore , divergencies of trs coincide with roots and divergences of the heat capacity . in order to find the roots and divergence points of heat capacity
, we should solve its numerator and denominator , separately . solving the mentioned equations with respect to entropy ,
leads to @xmath36and@xmath37 } } , & sned\vspace{% 0.3 cm } \\ \frac{q}{2}\sqrt{2l\left ( \sqrt{l^{2}+24\alpha } -l\right ) } , & cned% \end{array}% \right . \label{cqinf}\]]where@xmath38 , \]]@xmath39 .\ ] ]
it is evident from obtained equation for hned and sned that there is only one real positive entropy in which heat capacity vanishes .
interestingly , in case of cned theory we find two roots for heat capacity .
it is evident that the roots are increasing functions of the electric charge in these theories . as for nonlinearity parameter , in case of the hned and sned theories , the root is an increasing function of @xmath7 .
whereas for cned theory , the smaller root is an increasing function of @xmath40 whereas the larger root is a decreasing function of it .
now we are in position to study the existence of the type two phase transition point which is related to divergency of the heat capacity . considering hned branch of eq .
( [ cqinf ] ) , one finds @xmath41 is not real for all values of @xmath4 and @xmath7 .
therefore , there is no physical divergence point for the heat capacity of hned model .
next , for the case of sned model the same behavior is seen . in other words , since @xmath42 for @xmath43 , we can not obtain real @xmath41 for all values of @xmath4 and @xmath7 .
next , we should investigate cned model .
regarding eq .
( [ cqinf ] ) , we find that there is a divergence point for heat capacity ( we should note that in this paper we consider positive @xmath44 ) .
in other words , this theory of nonlinear electromagnetic field enjoys the phase transition of type two .
the divergence point is an increasing function of the nonlinearity and electric charge parameters .
it is worthwhile to mention that for the case of vanishing nonlinearity parameter in this theory , @xmath41 goes to zero . in other words
, there is no divergence point for heat capacity of maxwell theory .
it is also clear that in case of chargeless btz black holes , there is no root and divergence point for heat capacity . in other words , in case of chargeless btz black holes , there is not any kind of phase transition .
@xmath45 @xmath46 @xmath47 @xmath48 @xmath49 the weinhold metric was given in @xcite @xmath50 where @xmath51 and also @xmath52 , where @xmath53 denotes other extensive variables of the system . in case of weinhold approach , one is considering the mass of the system as potential , other parameters such as entropy and electric charge as extensive parameters and related quantities such as temperature and electric potential as intensive parameters .
the ruppeiner metric was defined as @xcite @xmath54 where @xmath55 and @xmath56 . in this case
the thermodynamical potential is entropy .
it is worthwhile to mention that according to the proposal of the quevedo , these two approaches are related to each other by a legendre transformation @xcite .
taking into account thermodynamical metrics of weinhold and ruppeiner , one can obtain their ricci scalars . since we would like to investigate divergence points of trs , @xmath57
, we focus on its denominator ( @xmath58 ) .
one finds @xmath59where @xmath60 and @xmath61 .
the quevedo metrics have two kinds with the following forms quevedo1,quevedo2 @xmath62where @xmath63 is@xmath64with@xmath65taking into account quevedo metrics , one can find that their related ( denomerator of ) ricci scalars can be written as @xmath66 in order to avoid any extra divergencies in trs which may not coincide with phase transitions of the type one and two , and also ensure that all the divergencies of the trs coincide with phase transition points of the both types , hpem metric was introduced @xcite @xmath67 in this case we have considered the total mass as thermodynamical potential , entropy and electric charge as extensive parameters .
calculations show that denominator of trs leads to @xmath68
here , we investigate phase transitions of black holes using geometrothermodynamics . for this purpose , we used thermodynamical metrics introduced in previous section for the black holes solutions obtained in the section [ sol ] . for weinhold metric , none of divergencies of the ricci scalar coincide with roots of the heat capacity in every theories of ned models that we have considered in this paper . on the other hand ,
one of the divergencies of trs and divergence point of the heat capacity in cned theory , coincide with each other .
it is notable that , in cases of the hned and sned theories , there is one divergence point for trs ( up panels of figs .
[ fig1 ] ) whereas for cned , there are two divergencies ( down panels of figs .
[ fig1 ] ) . in case of ruppeiner metric , for hned and sned models ,
there is a root for heat capacity in which ricci scalar of the ruppeiner metric has a divergency .
but there is also another divergence point for ricci scalar which does not coincide with any phase transition point ( up panels of figs .
[ fig2 ] ) . therefore , there is an extra divergence point . in case of the cned model , two roots and one divergence point
are observed for heat capacity in which ricci scalar has related divergencies .
in addition to these divergence points , one extra divergence point is also observed which is not related to any phase transition point ( down panels of figs .
[ fig2 ] ) . as for quevedo metrics , for case i , similar behavior as weinhold
is observed for all three theories of ned ( fig .
[ fig3 ] ) . on the other hand , in case of the other metric of quevedo , two divergence points for ricci scalar were observed for hned and sned theories .
one of these divergence points coincides with root of the heat capacity for these two nonlinear theories whereas the other one does not ( up panels of figs .
[ fig4 ] ) . in case of cned theory ,
all the divergence points of trs coincide with phase transition points except one ( down panels of figs .
[ fig4 ] ) . in other words ,
quevedo s metric predicts an extra divergence point , corresponding to the equation @xmath69 , which is not predicted in classical black hole thermodynamics .
it is evident that in case of hpem , all types of phase transition points of heat capacity coincide with divergencies of trs of hpem method ( figs .
[ fig5 ] ) . in other words ,
independent of the nonlinear theory under consideration , the hpem method provide a machinery in which no extra divergency for trs is observed and divergence points of trs and phase transition points coincide .
another interesting and important property of the hpem method is the behavior of trs near divergence point for different types of phase transition .
as one can see , the signature and behavior of trs near divergence point for phase transition type one and two are different .
therefore , independent of studying the heat capacity , one can distinguish these types of phase transition from one another only by studying the behavior of trs .
in this paper , we have considered btz black holes , in presence of three models of ned . we studied stability and phase transitions related to the heat capacity of the mentioned black holes .
next , we employed geometrical approach to study the thermodynamical behavior of the system .
in other words , we have studied phase transitions of the system through weinhold , ruppeiner and quevedo methods .
also , we used the recently proposed approach to study geometrical thermodynamics .
we found that the weinhold and ruppeiner metrics for studying these btz solutions fail to provide a suitable result .
in addition , the divergence points of the quevedo trs were not completely matched with the phase transition points of the heat capacity results .
in other words , in these approaches , the existence of extra divergencies were observed which were not related to any phase transition point in the classical thermodynamics . in some of these approaches
no divergency of trs coincided with phase transition points . in order to obtain a consistent results with the classical thermodynamic consequences ( the heat capacity ) , we employed a new thermodynamical metric . in this approach , all the divergencies of trs coincided with phase transition points . in other words , roots and divergence points of the heat capacity of the btz black holes in presence of each nonlinear models matched with divergencies of trs of this metric .
also , we found that in case of hned and sned theories , there is no divergency for heat capacity .
it means that , like maxwell theory , these two theories have no second type phase transition .
these two nonlinear theories of electrodynamics , preserved the characteristic behavior of the maxwell theory in case of heat capacity . on other hand , for cned model , two roots and one divergence point was found for the heat capacity . in other words , due to contribution of the nonlinear electromagnetic field , heat capacity enjoys the existence of one more phase transition point of type one and a phase transition of the type two . in essence
this theory is a generalization of the maxwell theory . but this generalization added another property to heat capacity that was not observed for the maxwell theory . finally it is worthwhile to mention a comment related to legendre invariancy .
it was shown that @xcite the legendre invariance alone is not sufficient to guarantee a unique description of thermodynamical metrics in terms of their curvatures .
in addition to legendre invariancy , one needs to demand curvature invariancy under a change of representation .
therefore , it will be worthwhile to investigate both legendre and curvature invariancies .
in addition , it will be interesting to think about the fundamental relation between the following two issues : ( i ) agreement of thermodynamical curvature results with usual thermodynamical approaches ( such as the heat capacity ) ; ( ii ) curvature invariancy in addition to the legendre invariancy .
it is also worthwhile to probe the fundamentality of cases ( i ) and ( ii ) to find that considering which one may leads to satisfy another one .
although first issue has been investigated for special cases @xcite , the second one has been remained unanalyzed yet .
we may address them in an independent work in the future .
we would like to thank the anonymous referee for valuable suggestions .
we also thank the shiraz university research council .
this work has been supported financially by the research institute for astronomy and astrophysics of maragha , iran .
|
in this paper we consider three dimensional btz black holes with three models of nonlinear electrodynamics as source . calculating heat capacity , we study the stability and phase transitions of these black holes .
we show that maxwell , logarithmic and exponential theories yield only type one phase transition which is related to the root(s ) of heat capacity . whereas for correction form of nonlinear electrodynamics , heat capacity contains two roots and one divergence point .
next , we use geometrical approach for studying classical thermodynamical behavior of the system .
we show that weinhold and ruppeiner metrics fail to provide fruitful results and the consequences of the quevedo approach are not completely matched to the heat capacity results .
then , we employ a new metric for solving this problem .
we show that this approach is successful and all divergencies of its ricci scalar and phase transition points coincide .
we also show that there is no phase transition for uncharged btz black holes .
|
although the standard model ( sm ) ckm picture of flavor and cp violation has been confirmed over the last years at the level of ( 10 - 20)% @xcite , there are in fact hints of discrepancies with respect to some sm expectations ( see e.g. @xcite ) : * the measured amount of cp violation in @xmath9 mixing ( @xmath10 ) seems insufficient to explain cp violation in @xmath11 mixing ( @xmath12 ) ; * the time - dependent cp asymmetries in the loop induced decays @xmath13 and @xmath14 ( @xmath15 and @xmath16 ) are measured to be considerably smaller than @xmath10 ; * recent analyses find a @xmath8 mixing phase much larger than the tiny sm prediction . taking these tensions seriously , a natural way to address them
is to go beyond the sm and to introduce new cp violating phases .
the mssm contains many free parameters that can provide additional sources of cp violation . once
( some of ) these parameters are assumed to be complex , in general several cp violating processes will receive np contributions simultaneously . in the following we consider observables that are sensitive to cp violation in * @xmath1 amplitudes , like the edms of the electron and neutron , @xmath17 and @xmath18 ; * @xmath2 amplitudes , like the direct cp asymmetry in the @xmath5 decay , @xmath19 or the cp asymmetries in the @xmath6 decay ; * @xmath7 amplitudes , like @xmath12 , that measures the amount of cp violation in @xmath11 mixing or the mixing induced cp asymmetries in @xmath20 and @xmath21 , @xmath10 and @xmath22 , that measure the @xmath9 and @xmath8 mixing phases ; * both @xmath2 and @xmath7 amplitudes , like the time dependent cp asymmetries in @xmath13 and @xmath14 , @xmath15 and @xmath16 . in the following we focus on the phenomenology of a so called flavor blind mssm , a rather restricted framework where the ckm matrix remains the only source of flavor violation , but additional cp violating phases are introduced in the soft sector . in particular , in @xcite we assumed flavor universal squark masses , flavor diagonal but hierarchical trilinear couplings , but allowed the trilinear couplings to be complex .
we find that in this framework non - standard effects in cp violating @xmath23 observables arise dominantly through the @xmath2 magnetic and chromomagnetic dipole operators .
the corresponding wilson coefficients get np contributions mainly through higgsino - stop loops that are @xmath24 enhanced and proportional to a single complex parameter combination @xmath25 . in fig .
[ fig : fbmssm ] we show three examples of the resulting highly correlated effects in low energy observables . with @xmath15 ( left ) , @xmath19 with @xmath15 ( middle ) and @xmath17 with @xmath15 ( right ) in the fbmssm .
the gray regions correspond to the experimental @xmath26 ranges for @xmath15 and @xmath16 .
, title="fig : " ] with @xmath15 ( left ) , @xmath19 with @xmath15 ( middle ) and @xmath17 with @xmath15 ( right ) in the fbmssm .
the gray regions correspond to the experimental @xmath26 ranges for @xmath15 and @xmath16 .
, title="fig : " ] with @xmath15 ( left ) , @xmath19 with @xmath15 ( middle ) and @xmath17 with @xmath15 ( right ) in the fbmssm .
the gray regions correspond to the experimental @xmath26 ranges for @xmath15 and @xmath16 .
, title="fig : " ] both @xmath15 and @xmath16 , that we evaluate following @xcite , can depart significantly from their sm expectations .
the effects in these cp asymmetries are strongly correlated and both observables can be brought simultaneously in agreement with the measurements . the direct cp asymmetry in @xmath5 @xcite is a very suitable observable to look for np effects @xcite , as it is predicted to be very small in the sm , @xmath27 @xcite . in the fbmssm
values up to @xmath28 can be reached .
furthermore , @xmath15 in agreement with the central experimental value unambiguously implies a positive value for @xmath19 .
the np effects in @xmath15 are also strongly correlated with the edms of the electron and neutron . in our framework , the dominant contributions to @xmath17 and @xmath18 arise from two loop barr - zee type diagrams @xcite that are also proportional to im@xmath29 .
the desire to explain the measured value of @xmath15 then implies lower bounds of @xmath17 and @xmath18 at the level of @xmath30 cm , only one order of magnitude below the current experimental constraints .
finally we also find large effects in several observables accessible in the @xmath6 decay @xcite .
in particular , as discussed in @xcite , the two t - odd cp asymmetries @xmath31 and @xmath32 are strongly enhanced with respect to their tiny sm predictions .
the effects in @xmath31 and @xmath32 are highly correlated among themselves and also with the other cp violating observables discussed so far .
concerning cp violation in @xmath7 on the other hand , the leading contributions to the mixing amplitudes for the @xmath11 , @xmath9 and @xmath8 systems are not sensitive to the new phases of the fbmssm .
in fact , only for an extremely light susy spectrum with higgsinos and stops lighter than 200 gev , @xmath12 can be modified by a positive np shift at the level of at most @xmath33 . also @xmath10 and @xmath34
remain essentially sm like , with @xmath35 . to summarize
, we stress that the combined study of the above considered observables and especially the characteristic pattern of correlations among them constitutes a very powerfull test of the fbmssm framework .
in particular , if a large @xmath8 mixing phase will be confirmed at lhcb , the fbmssm can eventually be ruled out . in order to generate sizeable effects in @xmath34 in the mssm
, one has to go beyond the minimal ansatz of the fbmssm and introduce not only additional sources of cp violation , but also of flavor violation .
the latter can be present both in the soft masses of the squarks and in the trilinear couplings and they are conveniently parameterized by so called mass insertions ( mis ) . in presence of complex mis , flavor and cp violating gluino - squark - quark interactions arise , that typically give the dominant contributions to fcncs . while left - right flipping mis are strongly constrained by the @xmath36 decay and
can hardly generate effects in @xmath8 mixing , @xmath34 can take values in the entire range @xmath37 if left - left and/or right - right mis are present .
in particular , if both left - left and right - right mis are present simultaneously , contributions to the @xmath8 mixing amplitude are generated that are strongly enhanced by renormalization group effects @xcite and a large loop function @xcite .
in such a situation , even for moderate , ckm like values of the mis @xmath38 , huge effects in @xmath34 can be achieved .
in a flavor blind mssm , sizeable non - standard effects in cp violating low energy observables are possible .
in particular cp violating @xmath1 and @xmath2 dipole amplitudes can receive large complex np contributions , leading to highly correlated modifications of the sm predictions of the edms , @xmath15 , @xmath16 , @xmath19 and cp asymmetries in @xmath6 .
cp violation in @xmath7 amplitudes however remains sm like , _
i.e. _ one gets only small effects in @xmath12 , @xmath10 and especially in @xmath34 . to generate large cp violating effects in @xmath7 amplitudes , additional flavor structures in the soft susy breaking
terms are required . as left - left mis
are always induced radiatively through renormalization group running , models predicting sizeable right - right mis are natural frameworks where a large @xmath8 mixing phase can occur . a detailed comparative study of the phenomenology of well motivated susy flavor models showing representative patterns of mass insertions can be found in @xcite .
i warmly thank the other authors of @xcite , @xcite and @xcite for the very pleasant collaborations .
this work has been supported by the german bundesministerium fr bildung und forschung under contract 05ht6woa and the graduiertenkolleg grk 1054 of dfg .
e. barberio _ et al .
_ [ heavy flavor averaging group ] , arxiv:0808.1297 [ hep - ex ] . e. lunghi and a. soni , phys
. lett .
b * 666 * ( 2008 ) 162 [ arxiv:0803.4340 [ hep - ph ] ] , jhep * 0908 * ( 2009 ) 051 [ arxiv:0903.5059 [ hep - ph ] ] , a. j. buras and d. guadagnoli , phys . rev .
d * 78 * ( 2008 ) 033005 [ arxiv:0805.3887 [ hep - ph ] ] , phys . rev .
d * 79 * ( 2009 ) 053010 [ arxiv:0901.2056 [ hep - ph ] ] , m. bona _ et al . _
[ utfit collaboration ] , arxiv:0803.0659 [ hep - ph ] , j. laiho , r. s. van de water and e. lunghi , arxiv:0910.2928 [ hep - ph ] .
w. altmannshofer , a. j. buras and p. paradisi , phys .
b * 669 * ( 2008 ) 239 [ arxiv:0808.0707 [ hep - ph ] ] .
s. baek and p. ko , phys .
rev . lett .
* 83 * ( 1999 ) 488 [ arxiv : hep - ph/9812229 ] , phys .
b * 462 * ( 1999 ) 95 [ arxiv : hep - ph/9904283 ] , a. bartl _ et al .
d * 64 * ( 2001 ) 076009 [ arxiv : hep - ph/0103324 ] , j. r. ellis , j. s. lee and a. pilaftsis , phys .
d * 76 * ( 2007 ) 115011 [ arxiv:0708.2079 [ hep - ph ] ] , l. mercolli and c. smith , nucl .
b * 817 * ( 2009 ) 1 [ arxiv:0902.1949 [ hep - ph ] ] , p. paradisi and d. straub , arxiv:0906.4551 [ hep - ph ] . g. buchalla , g. hiller , y. nir and g. raz , jhep * 0509 * ( 2005 ) 074 [ arxiv : hep - ph/0503151 ] .
l. hofer , u. nierste and d. scherer , arxiv:0907.5408 [ hep - ph ] .
j. m. soares , nucl .
b * 367 * ( 1991 ) 575 .
a. l. kagan and m. neubert , phys .
d * 58 * ( 1998 ) 094012 [ arxiv : hep - ph/9803368 ] .
t. hurth , e. lunghi and w. porod , nucl . phys .
b * 704 * ( 2005 ) 56 [ arxiv : hep - ph/0312260 ] .
d. chang , w. y. keung and a. pilaftsis , phys .
* 82 * ( 1999 ) 900 [ erratum - ibid .
* 83 * ( 1999 ) 3972 ] [ arxiv : hep - ph/9811202 ] .
c. bobeth , g. hiller and g. piranishvili , jhep * 0807 * ( 2008 ) 106 [ arxiv:0805.2525 [ hep - ph ] ] , u. egede , t. hurth , j. matias , m. ramon and w. reece , jhep * 0811 * ( 2008 ) 032 [ arxiv:0807.2589 [ hep - ph ] ] .
w. altmannshofer , p. ball , a. bharucha , a. j. buras , d. m. straub and m. wick , jhep * 0901 * ( 2009 ) 019 [ arxiv:0811.1214 [ hep - ph ] ] .
m. ciuchini , e. franco , v. lubicz , g. martinelli , i. scimemi and l. silvestrini , nucl .
b * 523 * ( 1998 ) 501 [ arxiv : hep - ph/9711402 ] , a. j. buras , m. misiak and j. urban , nucl .
b * 586 * ( 2000 ) 397 [ arxiv : hep - ph/0005183 ] .
f. gabbiani , e. gabrielli , a. masiero and l. silvestrini , nucl .
b * 477 * ( 1996 ) 321 [ arxiv : hep - ph/9604387 ] .
w. altmannshofer , a. j. buras , s. gori , p. paradisi and d. m. straub , arxiv:0909.1333 [ hep - ph ] .
|
we present an analysis of low energy cp violating observables in the minimal supersymmetric standard model ( mssm ) .
we focus on the predictions of cp violation in @xmath0 transitions in the framework of a flavor blind mssm , where the ckm matrix remains the only source of flavor violation , but additional cp violating phases are introduced in the soft susy breaking sector .
we find large and strongly correlated effects in @xmath1 observables like the electric dipole moments ( edms ) of the electron and the neutron , as well as in @xmath2 observables like the time dependent cp asymmetries in @xmath3 and @xmath4 , the direct cp asymmetry in @xmath5 and in several cp asymmetries in @xmath6 . on the other hand , observables that are only sensitive to cp violation in @xmath7 transitions , in particular the @xmath8 mixing phase , are found to be sm like in this framework .
we stress that only in presence of additional sources of _ flavor violation _ , sizeable new physics effects to _ cp violation _ in meson mixing can occur .
address = physik department , technische universitt mnchen , d-85748 garching , germany
|
turbulence , be it classical or quantum , remains as a fundamental problem in the dynamics of fluids .
its high complexity makes it a very difficult , yet a fascinating state of matter in which one of the challenges is to describe it in simple terms .
while there does not exist a clear definition or specification of what turbulence should be , there are several aspects or signatures that indicate whether or not one is facing a turbulent flow .
most notorious is the cascade of energy @xcite through vortices or eddies , in the classical case , or by a tangle of vortices @xcite in the quantum version . among the relevant questions one must deal with , specially in the quantum regime , is the initiation or generation of such a complex state@xcite and its characterization .
@xcite these issues motivate the study presented in this article .
we point out that there are already experimental realizations of quantum turbulence@xcite as well of studies of the route to turbulence.@xcite in this work , through the gross - pitaevskii ( gp ) model@xcite of an ultracold quantum gas , and assuming that arbitrary phase - imprinted states can be generated@xcite we present here a survey of macroscopic excitations that lead a bose - einstein condensate ( bec ) , confined in an external harmonic potential , to stationary agitated or chaotic states that , under appropriate conditions , may be considered to pass through turbulent transients or remain as such .
although the details of the model and of the different excitations that we explore are presented below , we want to advance here a general result that we find may be of relevance in a more complete study .
this is the fact that the gp model is capable of showing a clear transfer of energy , initiated with the large scale imprinted excitation by external means , and ending up in an effective stationary " state , sometimes with a stable array of vortices with large scale collective excitations@xcite , and other times without traces of the initial excitation , but with the presence of bogoliubov phonons.@xcite the former shows spectra close to kolmogorov law @xmath1 , while the latter a spectral law with a positive exponent .
we analyze the numerical solution of gp equation in terms of the spectra of the kinetic energy and in terms of power spectra of the time evolution of the steady state . while we do not claim that such an analysis permits discerning the presence of turbulence , we find that those spectra do serve as signatures of the reached stationary states as well of the evolution towards it .
the mentioned decay into a stationary or steady state is not a true irreversible@xcite behavior of gp but actually a complicated non - linear dephasing effect , since gp obeys time - reversal invariance .
the evolution appears as irreversible " but this is due to numerical rounding errors .
we present a detailed study of such an effect in order to understand these stationary states .
we believe it is relevant to take these aspects into account since dephasing contributions will compete with true irreversible development induced by actual dissipative effects - not considered in the present study . in this work ,
we limit ourselves to the numerical solutions of the time - reversal invariant gp equation , valid for temperatures as close as possible to absolute zero , but we recognize that in order to make a direct comparison with actual experiments , being these at finite temperature , one must take into account dissipative effects .
there are already efforts along these lines @xcite including the effects of the thermal cloud that surrounds a confined bec .
true stationary states , of course , are found in those cases
our model is summarized in the gross - pitaevskii equation@xcite for a three dimensional , one - component bec confined by an external potential , @xmath2 the mean - field interaction coupling term @xmath3 , where @xmath4 is the number of bosonic atoms of mass @xmath5 , and @xmath6 is the @xmath7-wave scattering length , assumed positive throughout .
the gp wave function @xmath8 is normalized to one .
although our numerical analysis will be limited to an isotropic harmonic potential @xmath9 , there are several comments and aspects that should be generic for any potential .
in particular , due to the problem at hand , we are interested in also posing relevant questions in terms of a hydrodynamic formulation@xcite .
this is done as follows .
using the transformation @xmath10 and the identification @xmath11 , where @xmath12 is the particle density and @xmath13 may be interpreted as the fluid velocity field , one finds the following hydrodynamic equations , @xmath14 and @xmath15 where the stress tensor is , @xmath16 and with the additional condition that the fluid is irrotational @xmath17 .
it is of interest to note that the stress tensor has a hydrostatic pressure @xmath18 , due to the interatomic interactions , and other volumetric and shear stresses of pure quantum origin , namely , all those proportional to @xmath19 .
the above hydrodynamic equations have been also previously derived by grant.@xcite as it is known , and we discuss and review it here , there is a ground stationary state , @xmath20 , with @xmath21 the chemical potential and @xmath22 the ground state macroscopic wave function , solution to , @xmath23 for @xmath24 , @xmath25 and @xmath5 given , the chemical potential @xmath21 and @xmath26 are uniquely given .
the wave function @xmath22 can be accurately found numerically.@xcite this is our starting point for all the forthcoming calculations . from the hydrodynamic point of view , this fluid has an inhomogenous density @xmath27 and zero velocity field everywhere , @xmath28 .
any macroscopic excitation on top of the state @xmath26 will necessarily evolve in time ( except perhaps a line vortex at an axis , but this is unstable@xcite ) . while there may be many more excitations , we consider for the moment three of them , vortices , collective modes , such as breathing or scissors modes,@xcite and sound waves , or bogoliubov phonons,@xcite and their interactions .
the line vortex , for @xmath29 , is the actual solution given by gross@xcite and pitaevskii@xcite , with zero density at the vortex and the velocity field yielding a non - zero circulation , @xmath30 with @xmath31 the topological charge . on the other extreme , we have that any small perturbation is composed of bogoliubov sound waves , as we now briefly review .
let us assume a stationary solution to the equations , which is a uniform one @xmath32 if @xmath29 , and the non - uniform solution @xmath27 for the harmonic trap , discussed above . in both cases , the velocity field is zero everywhere , @xmath28 .
let us now consider a perturbation in the free - field case , @xmath29 , @xmath33 then , a simple linearization of the hydrodynamic equations , eqs .
( [ nfield ] ) and ( [ vfield ] ) , yields , @xmath34 this is a fourth - order wave equation whose solution is @xmath35 , with @xmath36 an arbitrary complex amplitude , yielding , @xmath37 which , as expected , is bogoliubov dispersion relation for the elementary excitations in a weakly - interacting bose gas @xcite .
the velocity field is , in turn , found as @xmath38 , which shows that the waves ( phonons ) are longitudinal , also as expected .
the third important excitation are those that are referred as collective , namely , excitations that involve the whole cloud , and that occur from such a length scale down to perhaps the vortex size , such as breathing or scissors modes .
these can be studied in the thomas - fermi limit , namely , neglecting the kinetic energy in gp equation .
a clear exposition of these modes may be found in the works of stringari and coworkers.@xcite suffice to say here that these are not uncoupled to the other excitations , vortices and phonons , as we will exemplify below .
in this section we study the time evolution of several initial states , described in detail below . we analyze their decay to their respective
stationary " states as well as the time evolution of frequency and wavenumber spectra of the different energy components of the motion .
the different initial states are prepared , first , by allowing the system achieve its ground state @xmath22 ( for an isotropic harmonic potential of frequency @xmath39 ) and then by imprinting different types of vortex perturbations . in a typical case ,
the initial state is @xmath40 where the imposed phase functions @xmath41 are equivalent to the generation of a velocity field representing a given vortex excitation .
@xmath42 is the vortex topological charge .
we study the crossing of two orthogonal vortices ( 3.1 ) ; the collision of two anti - vortices ( 3.2 ) ; an off - center vortex ring ( 3.3 ) ; a vortex of charge @xmath0 ( 3.4 ) ; and a tangle of 4 vortices ( 3.5 ) . in the parenthesis we have indicated the section below in which we analyze them .
our calculations are performed using parallel computing with graphic processors units ( gpu ) , which allows us to develop large and fast calculations ; details of the numerical methods and programming will be reported elsewhere .
we use a spatial grid of size @xmath43 , in double - precision and with time steps of @xmath44 , in dimensionless units @xmath45 , and with a coupling @xmath46 .
this corresponds fairly well to a gas of @xmath47 atoms of @xmath48rb in an isotropic trap of frequency @xmath49 hz with a scattering length @xmath50 , and with a dimensionless equilibrium chemical potential of @xmath51 19.63 .
the unit of time corresponds to 0.00159 seconds .
while we monitor wave function normalization , energy and angular momentum conservation in order to partially ensure numerical convergence of our calculations , we have also used a further dynamical criterion since we are interested in time evolution behavior rather than in static properties .
this criterion arises from the observation that the prepared initial states all do seem to relax " to stationary " states .
this is an important aspect to analyze since the lack of dissipation in gp equation should prevent the system from truly relaxing to a thermal equilibrium state . in the present case ,
given that gp is time - reversal invariant , one faces a _ dephasing _ mechanism rather than a dissipative relaxing one .
this is the more important for non - linear equations solved by numerical means which , because of their concomitant round - off errors , very commonly give rise to a loss of the intrinsic reversibility of the dynamical equation , thus yielding an apparent irreversible behavior . to be more precise , gp as well as schrdinger equation , are time - reversal invariant under the transformation @xmath52 and @xmath53 .
hence , if at any time @xmath54 we change @xmath8 to @xmath55 , the system returns to its initial state at a time @xmath56 . on the one hand
, one expects that if time @xmath54 is very large , due to rounding errors , eventually the state will not return to its original one .
that is , for long times we can no longer assert that the observed evolution does correspond to the original initial state . on the other hand , however , due to the non - linearity of the equation , and aided by numerical errors , the state of the system can reach what one may call basins of attractions",@xcite such that , once the state enters one of those , it does not leave them anymore .
we call these states _ stationary_. we further find that , once the system reaches one of these states , time - reversal invariance is numerically restored .
this is just an indication that those states are very stable , even against round off errors .
this study is performed as follows .
let @xmath57 be any of the initial states at time @xmath58 , and let @xmath59 its gp solution at time @xmath54 . at time
@xmath60 we make the transformation @xmath61 , and evolve @xmath62 for another interval of time @xmath63 .
time reversal invariance demands that @xmath64 .
thus , for different times @xmath63 we calculate the so - called fidelity @xmath65 if there were no loss of time reversal invariance , @xmath66 for all times @xmath63 . in fig .
[ fidelity0all ] we show such a calculation for all the cases we study for up to time @xmath67 .
we see that case ( 3.1 ) is the most stable , followed by ( 3.3 ) , while ( 3.2 ) , ( 3.4 ) and ( 3.5 ) loose their time - reversal invariance for times longer than @xmath68 .
interestingly , all relax to a clear stable stationary state , as we now show .
however , while in the first two cases one can affirm that the stationary state corresponds to the initial state , for the latter three we can only say that the initial state is _ close _ to the basin of attraction of the reached stationary state . in order to verify the stability of the stationary state , we first let the system achieve it .
those states are typically reached by time @xmath69 .
therefore , we now calculate the fidelity at times @xmath70 with @xmath71 as the initial or reference state , that is , @xmath72 the result is shown in fig .
[ fidelity200all_v1 ] . with the exception of ( 3.4 ) that looses a bit of fidelity ,
the rest appear very stable . as part of the statistical analysis of both the time evolution and the stationary state properties , we present the spectra of the incompressible part of
the kinetic energy.@xcite the spectra is calculated as @xmath73 of particular interest is the search for situations whether the incompressible spectra shows a kolmogorov law @xmath1 or not .
in addition to the time evolution of the spectra , and in order to obtain a better understanding of the stationary state , we calculate the time evolution and the frequency spectrum of the modulus of the overlap between the state at the time @xmath71 and later times @xmath74 .
this is @xmath75 we denote its fourier transform as @xmath76 $ ] .
as we shall see , this function samples the contribution of different excitations in the stationary state .
this information can be correlated with the spectra @xmath77 to elucidate whether the excitations are vortices , collective modes and/or phonons . in this case , the initial state is given by two line vortices at orthogonal directions , see fig .
[ cruzados ] , one with @xmath78 parallel to the @xmath79-axis at @xmath80 and @xmath81 , the other with @xmath82 parallel to the @xmath83 axis at @xmath84 and @xmath85 , @xmath86 as one may expect , see the set of snapshots for the magnitude of the velocity field in fig .
[ cruzados ] , the vortices join and reconnect few times .
we have observed three of these reconnections within a time interval of 50 units of time .
then , the vortices stop crossing and become almost but not quite parallel and keep orbiting around each other , in a stationary state .
this late state shows , on top of the orbiting vortices , an agitated fluid that appears to be a superposition of collective modes at different lenght scales , with the presence of few phonons .
very interestingly , its incompressible energy espectra shows a scaling law quite close to kolmogorov law .
this is the most stable case since it reaches its stationary state preserving its time - reversal invariance , showing a stable fidelity times much longer than 200 units of time .
[ ene - cruzados ] shows the time evolution of the incompressible kinetic energy spectrum .
the compressible part is not shown being one order of magnitude smaller .
the spectrum does not show complicated features in its evolution , varying very little in time and showing that although the initial state is somewhat unstable , it does not differ much from the stationary one .
one sees that the spectra is quite parallel to the @xmath1 line in the segment of the spectrum between the thomas - fermi and the vortex size wavenumbers , @xmath87 and @xmath88 .
the average slope is @xmath89 , very close to @xmath90 .
as discussed further below , this appears as a signature of an algebraic cascade of energy between the vortices and collective excitations , with very little presence of phonons .
the above conclusion may be further understood from the overlap evolution @xmath91 and its fourier transform , see eq .
( [ overtime ] ) , as shown in fig .
[ overlapcross ] .
the largest peak corresponds to rotation frequency of the two vortices ( @xmath92 ) , while the other frequencies are collective modes .
the peaks in @xmath93 correspond also to motion of the whole cloud in the harmonic potential ; we have verified this by studying the center of mass motion of the cloud ( not shown here ) . as we shall verify below
, there seems not be an important contribution of phonons in this case .
[ collision ] shows the evolution of the collision of two parallel vortices , along the @xmath94-axis , one with topological charge @xmath95 , at @xmath85 , and the other with @xmath82 at @xmath80 , namely , @xmath96 the behavior of this case is quite the opposite of the two orthogonal vortices .
referring to fig .
[ collision ] , first , the vortices approach each other and reconnect ( @xmath97 ) forming two vortices at directions orthogonal to the original ones ( @xmath98 ) , but because they have opposite circulations , eventually join at their extrema forming a twisted vortex ring ( @xmath99 ) that further folds into itself .
then , it appears to break into smaller rings that eventually become what appears to be a very agitated state with excitations and phonons at all length scales and in all directions .
while this case seems similar to the crow instability in uniform systems,@xcite the presence of the trap may affect its evolution .
this case has the shortest dephasing time , see fig .
[ fidelity0all ] , yet , once it enters the stationary state remains there , see fig .
[ fidelity200all_v1 ] .
[ ene - collision ] shows the time evolution of the energy spectrum of the collision of two anti - vortices .
the figure shows that the system quickly evolves towards a stationary state , which for late times @xmath100 , is composed by a collective oscillation of the cloud , with the first peak about @xmath87 followed by phonons excitations .
the latter can be verified by the line indicated by @xmath101 which is the dispersion relation of bogolubov phonons , see eq .
( [ bogo ] ) ; that is , the kinetic energy spectra is essentially composed of phonons in this case .
compressible _ contribution of the kinetic energy ( not shown here ) is quite similar to the incompressible one of fig .
[ ene - collision ] , corroborating that the excitations are compressible , namely , acoustic excitations .
the overlap function @xmath102 in the stationary state , fig .
[ overlapcoll ] , shows no presence of collective excitations , except for two peaks at @xmath93 and @xmath103 , which are motions of the center of mass , as directly verified from the numerical solution .
this case corresponds to the evolution of a single vortex ring , shown in fig .
[ ring ] , initially placed at @xmath104 and @xmath105 , with @xmath106 we find that , due to its circulation , the ring starts to move to one of the poles of the condensate , see fig .
[ ring ] , expanding at the surface of the cloud ( @xmath107 ) , returning to the inside but then breaking into internal excitations and what appears to be a vortex at the edge of the condensate ( @xmath108 and @xmath109 ) . referring to fig .
[ fidelity0all ] , we find that this excitation does not loose completely its time - reversal invariance , but again , once it enters the stationary state remains there , see fig .
[ fidelity200all_v1 ] .
as we see below , the stationary state is composed of oscillations of the whole cloud , some collective excitations and certainly with the presence of phonons .
this can be concluded from the incompressible kinetic energy spectra and the overlap function , eq .
( [ overtime ] ) , see figs .
[ ene - ring ] and [ overlapringnc ] .
this stationary state appears to be close to the previous one of two colliding anti vortices , with a strong presence of phonons , yet with collective modes playing a role .
we now analyze the case in which the initial excitation is one that corresponds a single line vortex with topological charge @xmath0 , parallel to @xmath94 , at @xmath110 and @xmath111 , see fig .
[ q2 ] . as it is known,@xcite vortices of charge @xmath112 are unstable and decay into two vortices of the same charge @xmath95 .
this situation is exemplified in the snapshots of times @xmath113 , 25.0 and 40.0 of fig .
it is very interesting to see that right after its creation , the vortex separates into two braid - entangled vortices , then these unwind until two stable parallel vortices are formed .
the evolution dephases after 50 units of time , see fig .
[ fidelity0all ] , yet , as with the other cases , it enters a stationary state .
this stationary state , however , is the least stable , as can be seen from fig .
[ fidelity200all_v1 ] , where the fidelity appears to be lossing about 5 % of its value , and thus , for longer times the system may migrate to a different stationary state .
this remains to be verified . in any case , judging from figs .
[ q2 ] and [ overlapq2 ] , the stationary state shows mainly vortex excitations with collective excitations of the cloud , similar to the case of crossed vortices .
the large peak in the overlap spectrum in fig .
[ overlapq2 ] corresponds the fast orbiting of the two vortices around each other .
here we study a tangle " of 4 vortices , two anti - vortices parallel to the @xmath94-axis at random positions @xmath114 and @xmath115 , with @xmath116 and @xmath117 ; and two cross vortices , one parallel to the @xmath79-axis at @xmath118 with @xmath119 , and the other parallel to the @xmath120-axis at @xmath121 with @xmath122 , see fig .
this would be the simplest tangle of vortices that may lead to turbulence , as originally suggested by tsubota et al.@xcite and also already treated by white et al.@xcite the evolution of this case is very harsh , similarly to the case of the colliding anti - vortices ; here we see that by @xmath123 there have been several reconnections and there remain two vortices only .
then the system enters a very agitated flow , perhaps turbulent for times @xmath124 to @xmath125 .
unfortunately , the dephasing time is about 50 units of time , and therefore the true evolution is lost for longer times . nevertheless , after this agitated state , one of the vortices disappears , and the system enters a stable stationary state .
these stages can also be observed in the evolution of the incompressible kinetic energy spectrum , fig .
[ ene - tangle ] : the slope changes from nearly @xmath90 to @xmath126 to settle about @xmath127 .
this last stage corresponds to the stationary state which , from observing the overlap @xmath102 and its fourier transform in fig .
[ overlaprnd4 ] , appears to be composed of a fast vortex , the largest peak , with collective excitations and motion of the whole cloud , but with very few phonons .
in this article we have analyzed the evolution of several representative vortex states in an otherwise equilibrium bose - einstein condensate , using gross - pitaevskii equation as a model for a gas at very low temperatures .
our goal is to add to the understanding of evolution of complicated states in the search for different way to approach a quantum turbulent state . since the dynamics observed show different evolutions in wide time scales , and these types of studies can only be realized numerically
, one must elucidate the role of rounding errors in determining the fate of the evolution .
we have found that , depending on the initial state , the evolved state indeed looses its time - reversal invariance but it does so at different late times .
moreover , the system reaches a stationary state in which time - reversal invariance is robust . referring to the particular initial states considered in the present article
, we can say that they are widely divided in two types : ( i ) those that at the stationary state have one vortex or an array of vortices with collective excitations of different sizes , and ( ii ) one with an agitated state that appears dominated by bogoliubov phonons . in case ( i ) one can include the initial states with two crossed vortices ( 3.1 ) , the vortex of charge two ( 3.4 ) and the tangle of 4 vortices ( 3.5 ) ; in case ( ii ) , we find the two vortex - antivortex system ( 3.2 ) and the off - center ring vortex ( 3.3 ) .
this wide classification is based both on the spectra of the incompressible kinetic energy and in the overlap function in the stationary state , eq .
( [ overtime ] ) . in the relevant wavenumber range , from @xmath87 to @xmath88 ,
one corresponding to the size of the cloud , the other to the vortices cores , the kinetic energy spectra in case ( i ) appears to be close to the kolmogorov law @xmath1 , while in ( ii ) the spectra is very different even with positive power law , but that matches very well the bogoliubov phonon dispersion relation . while finding a similarity with kolmogorov law @xmath1
does not necessarily implies the presence of turbulence , it nevertheless suggests that there is a transfer or interchange of energy between excitations of different sizes , without a characteristic length scale , namely , with a scaling algebraic law .
we recall that kolmogorov derivation of his law@xcite involves two parts , one in which a scaling law is assumed for the velocity flow , and a second one , the elucidation of the energy transfer rate between excitations of different sizes .
the exponent @xmath90 is a consequence of the scaling , independent of the energy transfer mechanism .
thus , we hypothesize that the observed similarity with kolmogorov law in some cases , essentially the same for the two crossed vortices , indicates the presence of energy transfer among excitations at different length scales : typically a single vortex , that while rotates and orbits around the cloud , interacts with collective cloud excitations .
this gives rise clearly to a complicated , agitated flow . whether this may be considered turbulence " remains to be fully elucidated .
it is , therefore , the more important to better understand why colliding antivortices and ring vortices appear to be destroyed , with the accompanying creation of acoustic phonons , yielding also a chaotic state , but with a completely spectral law .
j. a. seman , e. a. l. henn , r. f. shiozaki , g. roati , f. j. poveda - cuevas , k. m. f. magalhaes , v. i. yukalov , m. tsubota , m. kobayashi , k. kasamatsu and v. s. bagnato , _ laser phys .
* 8 * , 691 ( 2011 ) .
|
we present a survey of macroscopic excitations of harmonically confined bose - einstein condensates ( bec ) , described by gross - pitaevskii ( gp ) equation , in search of routes to develop quantum turbulence .
these excitations can all be created by phase imprinting techniques on an otherwise equilibrium bose - einstein condensate .
we analyze two crossed vortices , two parallel anti - vortices , a vortex ring , a vortex with topological charge @xmath0 , and a tangle of 4 vortices . since gp equation is time - reversal invariant , we are careful to distinguish time intervals in which this symmetry is preserved and those in which rounding errors play a role .
we find that the system tends to reach stationary states that may be widely classified as having either an array of vortices with collective excitations at different length scales or an agitated state composed mainly of bogoliubov phonons .
pacs numbers : 67.85.fg , 03.75.lm,05.30.jp
|
the different response of a molecule to right- and left- circularly polarized light can be measured experimentally and can be explained with a semiclassical theory of multipole oscillators induced by the electric and magnetic fields of the optical wave @xcite .
thanks to advances in computational chemistry calculations of the optical activity of oriented molecules are today relatively easy to do using modern software packages that do quantum - mechanical calculations of molecular property tensors @xcite
. measurements of optical activity started more than two centuries ago and modern chiroptical spectroscopy techniques that exploit the intrinsic chirality of circular polarized light are widely available in chemistry laboratories .
however , the experimental study of the optical activity of oriented systems is still challenging , and active research is being carried to find reliable methods to measure the typically small chiroptical contributions embedded in the large optical anisotropy of oriented system .
in last few years mueller matrix spectroscopy has emerged as a promising and powerful technique for these type of measurements @xcite .
therefore , it seems that we have reached a point where the experimental and theoretical approaches to study the optical activity of oriented molecules are mature enough to be compared .
one additional factor that complicates the reconciliation between these two approaches is that theoretical calculations of optical activity are typically done assuming single molecules while , in most cases , measurements are performed in macroscopic media .
traditionally , measurements of ensembles of molecules were always done in solution , in which small molecules tend to adopt random orientations .
therefore calculations were always made under the assumption of a large collections of molecules randomly oriented .
however , in recent years there is a growing interest to study the richer anisotropic spectroscopic information provided by oriented molecules .
there are many strategies to orient a molecule , specially if the molecules are large .
the most evident strategy consist of crystallizing them and form a molecular crystal but , in some occasions , it is also possible to incorporate the molecules during the growth of a different crystal that acts as a host @xcite . in these cases the optical properties not only
depend on the individual molecules but also in the type of crystal lattice .
elongated molecular aggregates are typically orientable by flows @xcite and some molecules can also be oriented by simple mechanical actions such as rubbing @xcite .
other methods to orient molecules use lasers pulses or other forms of electric and magnetic fields @xcite .
however , in this work , we will consider that there are no additional external fields affecting the molecules during the optical activity measurements . in this paper
we discuss the theoretical basis that permits the correlation between computations of the averaged molecular polarizability tensors and measurements .
we show that the mathematical tools used to describe measurements and calculations naturally converge if a bianisotropic formulation of the material constitutive equations is considered .
the key element of the analysis is to refer any measured or calculated optical activity to elements of the magnetoelectric tensor of gyration that is included in the bianisotropic constitutive equations .
the results we find simplify the comparison between calculations and experiments .
the following form of bianisotropic constitutive equations was first given by tellegen @xcite : [ uno ] @xmath0 @xmath1 where frequency - domain fields are considered and the choice of the time dependence is given by @xmath2 .
@xmath3 is the permittivity dyadic , @xmath4 the permeability dyadic and @xmath5 and @xmath6 are the two magnetoelectric dyadics @xcite and transmit the relation between the electric and magnetic field quantities @xmath7 , @xmath8 and the flux quantities @xmath9 , @xmath10 . these four constitutive dyadics contain full information of the electromagnetic response of a bianisotropic medium . lately , this form about the constitutive equations has attracted a lot of attention for the optical characterization of metamaterials @xcite , because these materials typically have large magnetoelectric tensors .
one common simplification for eqs .
is that the specific medium should be lorentz - reciprocal @xcite .
this implies : @xmath11 where the superscript @xmath12 indicates transposition . in this case
the magnetoelectric dyadics , that in general are written as @xmath13 and @xmath14 , must satisfy @xmath15 and @xmath16 .
therefore , the constitutive equations for a bianisotropic reciprocal medium can be rewritten as : [ constitutivesimplied2 ] @xmath17 @xmath18 it should be indicated that , in the presence of absorption , all the tensors in this constitutive equations , i.e. @xmath3 , @xmath19 and @xmath4 , become complex . for systems
uni- and biaxial orthorhombic crystallographic symmetries the real and imaginary parts of the tensor have the same system of principal axes . however , the situation is more complicated if the tensors have no such system , as can happen for media with monoclinic and triclinic symmetries , as they can lead to apparent non - reciprocal optical response , despite being reciprocal media .
multipole theory can be used to calculate the reciprocal magnetolectric coupling ( now given by @xmath19 ) between the electric and magnetic fields @xcite .
note that eqs .
do not necessarily constraint @xmath19 to be a symmetric tensor . however , natural optical activity is a lorentz reciprocal effect and only it can be contributed by the symmetric part of the magnetoelectric tensor . in this work
we only consider this part of the tensor as we are interested in natural optical activity and non - magnetic media . for a complete description of natural optical activity
it has been shown that , within a semiclassical theory , in addition to the mean electric dipole induced by the electric field of light it is necessary to include : the electric dipole contribution by the time - derivative of the magnetic field of the light wave , the associated mean magnetic dipole induced by the time - derivative of the electric field , the electric dipole contribution induced by the electric field gradient of the electromagnetic wave and the electric quadrupole contribution induced by the electric field @xcite . when all these contributions are considered the oscillating induced moments ( the electric dipole @xmath20 , the magnetic dipole @xmath21 and the electric quadrupole @xmath22 ) are the real parts of the following complex expressions @xcite : [ moments ] @xmath23 @xmath24 @xmath25 where @xmath26 and @xmath27 the actual electric and magnetic fields of the optical wave and @xmath28 is the electric field gradient .
@xmath29 is the electric dipole - electric dipole polarizability tensor , @xmath30 is the electric dipole - magnetic dipole polarizability tensor and @xmath31 is the electric dipole - electric quadrupole polarizability tensor . higher order polarizabilities ( e. g. electric octopole , magnetic quadrupole , ... ) have not been specified because they do not contribute to optical activity @xcite .
the hats @xmath32 stress that these are complex quantities .
computational software has been developed to compute all these tensors quantum mechanically by applying time - dependent perturbation theory to molecular orbitals @xcite . in these calculations
the electric and magnetic dipole moments and the electric quadrupole moment are usually treated as microscopic quantities that apply to single molecules . however , in this paper we discuss the optical activity of a macroscopic medium and , therefore , all the moments included in eqs .
should be regarded as macroscopic moment densities , i. e. their statistical average multiplied by the number density .
we are also assuming that local field effects are small enough to be neglected . from this point , we refer always to moment densities instead of molecular moments and to macroscopic fields instead of local fields . in a nonmagnetic medium the molecular property tensors @xmath29 , @xmath30 and @xmath31 must define the constitutive dyadics of the bianisotropic eqs . so that the optical response of an oriented molecule or crystal is calculated from them . following the work of graham and raab @xcite we find the following relations : [ unodos ] @xmath33 @xmath34 @xmath35,\ ] ] where @xmath36 is the levi - cevita operator and @xmath37 is the kronecker delta .
we are using the sum over indices einstein convention and the notation is the same as in refs .
. [ factor ] can be found , apart from a numerical factor , in ref .
@xcite , but it is different from its equivalent appearing in ref .
@xcite because here we have disregarded nonreciprocal factors . as natural optical activity is a reciprocal optical phenomena only reciprocal contributions ( time - even ) need to be considered and the nonreciprocal factors ( time - odd ) can be safely neglected .
the condition @xmath38 is automatically satisfied because @xmath29 as well as the sums ( @xmath39 ) and ( @xmath40 ) are all symmetric contributions .
an additional difference compared to ref .
@xcite is that we get a 1/3 factor in front of the @xmath41 terms , as opposed to the 1/2 factor obtained by graham and raab .
this change arises because they used a so - called primitive definition of the electric quadrupole moment compared to the more extended traceless definition used for example by buckingham and dunn @xcite .
both , the traceless and primitive definitions , are applicable because it has been shown that they allow for an origin independent description of the theoretical optical activity @xcite , i.e. its final expression is not dependent on the arbitrary choice of coordinate origin .
eq . is specially important because provides the connection between the molecular polarizability tensors and the magnetoelectric tensor .
this equation appears in the seminal publication of buckingham - dunn ( eq . [ 19 ] in ref .
@xcite ) but , as far as we know , this work does not offer a clear interpretation about the meaning of @xmath42 .
apparently , it was introduced as an accessory equation for their calculation of optical activity but it was not highlighted as a main result of this publication .
the elements of the magnetoelectric tensor of gyrotropy @xmath42 ( from this point we omit the dyadic notation ) are calculated expanding eq .
: [ kap ] @xmath43 @xmath44 @xmath45
@xmath46 @xmath47 @xmath48 where @xmath31 holds the symmetry property @xmath49 @xcite . to relate this result with an experiment it is still necessary to investigate the relation between the optical activity measured in a given molecular direction and the components of @xmath42
. optical activity is usually expressed in terms of circular birefringence cb ( twice the optical rotation ) and circular dichroism ( cd ) .
both quantities can be measured independently and they can be combined into a complex expression given by @xmath50 where @xmath51 and @xmath52 are , respectively , the complex refractive indices for left an right circularly polarized light .
@xmath53 is the pathlength of the medium
. a straight calculation of the values of @xmath51 and @xmath52 from the tensors @xmath54 , @xmath55 and @xmath42 is in general a complicate task .
the problem of light propagation ( assuming planewaves ) through a reciprocal bianosotropic medium is usually treated by eigenanalysis of a wave equation , which provides the refractive indices of the eigenpolarizations that propagate through the medium . for chiral
isotropic medium or along the optic axis of a chiral uniaxial medium in which the eigenmodes are circularly polarized waves it can be easily shown that c depends only on the magnetoelectric parameter or tensor @xcite . however , the two eigenmodes , with complex refractive indices @xmath56 and @xmath57 , in general do not correspond with circularly polarized waves and , therefore , the values of @xmath51 and @xmath52 are not directly obtained . with the help of the jones formalism for polarization optics , we showed in ref .
@xcite that optical activity ( @xmath58 ) could be calculated from the complex retardation between the eigenmodes @xmath59 with the following equation : @xmath60 in which @xmath61 and @xmath62 are given by the polarization of each eigenmode , namely @xmath63 and @xmath64 , where @xmath65 are electric field amplitudes .
note that in simple case of a chiral isotropic medium the eigenmodes are circularly polarized waves @xmath66 and @xmath67 and , obviously , @xmath68 .
it is possible to calculate the dependence of optical activity with the material s constitutive tensors of eq . for any direction of light propagation . in appendix
a we show the results of this calculation for an uniaxial medium with crystallographic point group of symmetry 32 , 422 or 622 and in which the optic axis is perpendicular to the direction of propagation of light . in practice , for media with lower symmetry
it is very difficult to keep the eigenanalysis at an analytical level and it is more practical to do the calculation numerically . for constitutive tensors belonging to any crystal symmetry it is found , either analytically ( as in appendix a ) or numerically , that c depends _ only _ on the components of the tensor magnetoelectric tensor @xmath42 , so that this tensor fully describes the optical activity of the sample and that @xmath54 is completely irrelevant to optical activity . for any crystal class , the dependence of c with @xmath42 follows a general rule : c can be calculated from @xmath42 by adding the two components of the tensor that are perpendicular to the direction of propagation of the beam .
this simple result was already suggested for some particular cases in @xcite , but now we can generalized it with the following equations that apply to ensembles of molecules : [ orto ] @xmath69 @xmath70 @xmath71 @xmath72 @xmath73 @xmath74 where the angular brackets @xmath75 denote a statistical average
. @xmath76 is the number of molecules per unit volume and @xmath53 is the pathlength .
the subscript in @xmath77 indicates the direction of measurement of optical activity , e.g. @xmath78 corresponds to optical activity in the direction that bisects the @xmath79 and @xmath80 axes . as @xmath81
, it immediately follows that @xmath82 because natural optical activity is a reciprocal phenomenon .
a qualitative understanding of the dependence of c with @xmath42 is gained by realizing that a circularly polarized wave can be decomposed into the sum of two orthogonal linearly polarized components that are in the plane of polarization and have a phase difference of @xmath83 .
the sign of this phase difference determines the handedness of the wave .
only the tensor components that correspond to two these two orthogonal directions are able to contribute in a different way to left- and right- circularly polarized waves . the other non - vanishing tensor elements ( either in the magnetoelectric or dielectric tensors )
affect the absorption and refraction of light but they have exactly the same effect for left- and right- circularly polarized wave and , therefore , they do not contribute to c. for example , to calculate the optical activity corresponding to light propagating in the z or -z direction of an oriented medium eq .
is used in combination with eqs . and : @xmath84 where it has been used that @xmath85 . according to ref .
@xcite at transparent frequencies @xmath86 and @xmath87 , where @xmath88 and @xmath89 are real tensors , so it is possible to rewrite : @xmath90 which is the same result found in @xcite . as a practical demonstration of the implications of eqs .
, we have plotted in fig .
[ figtensor ] a visual representation of the value of cb as a function of the orientation of a molecule . in all cases
we have considered an incoming linearly polarized light beam propagating along the z axis . panels
a , b and c correspond to a molecule with crystallographic point group of symmetry @xmath91 or mm2 at three different orientations .
the water ( h@xmath92o ) molecule could be a good an example of this symmetry , which is characterized by a magnetoelectric tensor having only one independent component .
molecules belonging to this group do not have optical activity in solution because when all directions of the space are considered the magnetoelectric tensor averages to zero ( which happens because the trace of the tensor matrix is always zero ) @xcite .
however , for the orientations shown in b and c there exits cb and it , respectively , takes opposite signs .
panels d , e and f show a molecule with crystallographic point group of symmetry 3 , 32 or 622 that has a magnetoelectric tensor with two different tensor components . in this example
we have considered that these two components keep a relation -2:1 .
note that the trace of the magnetoelectric tensor for molecules holding this ratio would be zero and , consequently , they would neither be optically active in solution .
for each case the cb along the z axis is given by the real part of the sum of the tensor elements @xmath93 , which is theoretically calculated using eq . .
the correspondence between calculated of molecular tensor and optical activity values is treated in other works considering another tensor , usually called gyration tensor or optical activity tensor , and typically represented by @xmath94 @xcite .
this tensor is based in the so - called equation of the normals : @xmath95 where @xmath96 gives the possible values of the refractive of the refractive index , for a given direction of the wave normal , @xmath97 and @xmath98 are the refractive indices of the eigenwaves propagating in the crystal in the absence of optical activity , and @xmath99 is the scalar gyration parameter @xmath100 @xmath101 and @xmath102 are direction cosines of the wave normal .
is typically approximated to @xmath103 by assuming that the birefringence of the system is not too large ( @xmath104 ) .
then if @xmath99 is assumed to be very small , the two solutions of this equation can be written as @xmath105 that correspond to the refractive indices for left and right circularly polarized waves .
then the optical activity [ eq . ] at a given direction can be written as @xmath106 this equation is similar to eqs . , but here the magnetoelectric tensor does not appear at all and and it has been replaced by a different tensor . the description of optical activity based on the tensor @xmath94 is problematic not only because it works uniquely within the approximations that we have detailed above , but also because it arises from an incorrect formulation of the constitutive equations for optical activity . to the best of our knowledge eq .
was first proposed by szivessy in 1928 @xcite and it requires constitutive equations in which the permittivity tensor is perturbed by the gyration tensor .
this formulation corresponds to the classic treatment of optical activity by m. born @xcite , which neglects any magnetoelectric tensor and optical activity is then justified by modification of the permittivity tensor : @xmath107 where @xmath108 is the levi - civita symbol , @xmath109 are the components of the optical activity tensor and @xmath110 are the direction cosines of the wave normal .
this form of the constitutive equations is similar to the one used for describing magneto - optical phenomena ( faraday effect and magneto - optic kerr effect ) but , despite being quite widespread in classical crystal optics texts @xcite , it is not suitable for natural optical activity as it violates fundamental principles that natural optical activity should preserve @xcite .
the magnetoelectric tensor @xmath42 corresponding every crystal class takes the same forms of symmetry as the gyration tensor @xmath94 ( see for example the tables in refs .
@xcite ) because the same constraints based on neumann s principle are applicable .
however , the values of the non - vanishing elements of @xmath42 bear no relation with those of @xmath94 . in general , we strongly recommend to use @xmath42 instead of @xmath94 to express the experimental or calculated optical activity of oriented systems .
the optical activity of anisotropic crystals or oriented molecules is adequately described by a reciprocal bianisotropic formulation of the constitutive equations .
cb and cd for any molecular or crystallographic direction are defined by the symmetric magnetoelectric tensor of chirality @xmath42 which , in turn , is expressed in terms of two molecular property tensors .
these contain the effect of the mean electric and magnetic dipoles induced , respectively , by the time - derivative of the magnetic and electric fields of the light wave , as well as the electric dipole induced by the electric field gradient of the wave and the electric quadrupole induced by the electric field .
we have shown that the optical activity found by a light plane wave propagating along a certain direction of a molecule or a crystal is given by the sum of two components of @xmath42 that are perpendicular to this direction of propagation .
these two tensor components describe the change of polarization of light that is due to the special interaction between the electric and magnetic fields in systems with optical activity .
with bianisotropic constitutive equations this change in polarization is only described by the magnetoelectric tensor and it is separable from the change in polarization due to the linear birefringence or linear dichroism typical of oriented systems .
we expect that the use of the bianisotropic formalism will simplify the comparison between experiments and calculations and it will bring the field of optical activity in molecules to a closer connection with that of artificial metamaterials .
the author acknowledges financial support from a marie curie iif fellowship ( piif - ga-2012 - 330513 nanochirality ) .
he is also grateful to v. murphy and b. kahr for helpful discussions .
this appendix includes an analytical calculation of the dependence of optical activity with the magnetoelectric tensor for a medium with crystallographic symmetry 32 , 422 or 622 . for this example
we assume that light propagates in a direction perpendicular to the optic axis . for a medium with this symmetry ,
the dielectric tensor is @xmath111 and the magnetoelectric tensor is @xmath112 . after performing the eigenalysis of the wave equations using the produce described in ref.@xcite we find that the refractive indices of the eigenmodes are given by .
@xmath113^{1/2}/2,\ ] ] in which @xmath114 . when all these equations are combined according to eq .
there are many simplifications and the final result is extremely simple .
the optical activity for this direction of light propagation is given by : @xmath117 case e of fig .
1 shows a pictorial representation of this particular relationship between optical activity and the magnetoelectric tensor . or mm2 .
panels d , e and f correspond to a trigonal molecule with point group of symmetry 3 , 32 or 622 in which the two independent components of its magnetoelectric tensor have a relation -2:1 . in all cases the resulting cb
is given by the real part of the sum of the two components of the magnetoeletric tensor that are perpendicular to the direction of propagation : @xmath93.,width=642 ] m. j. frisch , g. w. trucks , h. b. schlegel , g. e. scuseria , m. a. robb , j. r. cheeseman , g. scalmani , v. barone , b. mennucci , g. a. petersson , h. nakatsuji , m. caricato , x. li , h. p. hratchian , a. f. izmaylov , j. bloino , g. zheng , j. l. sonnenberg , m. hada , m. ehara , k. toyota , r. fukuda , j. hasegawa , m. ishida , t. nakajima , y. honda , o. kitao , h. nakai , t. vreven , j. a. montgomery , jr .
, j. e. peralta , f. ogliaro , m. bearpark , j. j. heyd , e. brothers , k. n. kudin , v. n. staroverov , r. kobayashi , j. normand , k. raghavachari , a. rendell , j. c. burant , s. s. iyengar , j. tomasi , m. cossi , n. rega , j. m. millam , m. klene , j. e. knox , j. b. cross , v. bakken , c. adamo , j. jaramillo , r. gomperts , r. e. stratmann , o. yazyev , a. j. austin , r. cammi , c. pomelli , j. w. ochterski , r. l. martin , k. morokuma , v. g. zakrzewski , g. a. voth , p. salvador , j. j. dannenberg , s. dapprich , a. d. daniels , .
farkas , j. b. foresman , j. v. ortiz , j. cioslowski , and d. j. fox , `` gaussian 09 revision a.1 , '' gaussian inc .
wallingford ct 2009 .
l. wong , c. hu , r. paradise , z. zhu , a. shtukenberg , and b. kahr , `` relationship between tribology and optics in thin films of mechanically oriented nanocrystals , '' j. am .
* 134 * , 1224512251 ( 2012 ) .
r. e. raab and o. l. de lange , _ multipole theory in electromagnetism : classical , quantum , and symmetry aspects , with applications ( international series of monographs on physics ) _ ( oxford university press , usa , 2005 ) .
e. b. graham and r. e. raab , `` light propagation in cubic and other anisotropic crystals , '' proceedings of the royal society of london . series a : mathematical and physical sciences * 430 * , 593614 ( 1990 ) .
m. p. silverman , `` reflection and refraction at the surface of a chiral medium : comparison of gyrotropic constitutive relations invariant or noninvariant under a duality transformation , '' j. opt .
am . a * 3 * , 830837 ( 1986 ) .
|
the optical activity of oriented molecular systems is investigated using bianisotropic material constitutives for maxwell s equations .
it is shown that the circular birefringence and circular dichroism for an oriented system can be conveniently expressed in terms of the two components of the symmetric magnetoelectric tensor of gyrotropy that are perpendicular to this direction of light propagation .
this description establishes a direct link between measurable anisotropic optical activity and the tensors that describe the oscillating electric and magnetic dipole and electric quadrupole moments induced by the optical wave .
|
the numerical renormalization group method,@xcite originally developed by wilson@xcite for the kondo model , is by now an established technique for the solution of general quantum impurity problems .
it has been applied , e.g. , to magnetic atoms in metals , to quantum dots and magnetic molecules , and as an impurity solver within dynamical mean - field theory .
its generalization@xcite to bosonic baths has enabled the treatment of dissipative impurity models and those with both bosonic and fermionic baths.@xcite quite often , impurity quantum phase transitions@xcite are in the focus of interest .
the strengths of nrg in treating such critical phenomena lie in its ability to treat arbitrarily small energy scales and in its renormalization - group character which allows e.g. for the analysis of flow diagrams .
recently , conflicting results have been reported about the critical behavior of certain impurity models with a bosonic bath , in particular the spin - boson and the ising - symmetric bose - fermi kondo model.@xcite for a bosonic bath with power - law spectral density @xmath0 , these models display a quantum phase transition for @xmath2 .
statistical - mechanics arguments suggest that this transition is in the same universality class as the thermal phase transition of the one - dimensional ( 1d ) ising model with @xmath3 long - range interactions . at issue
is the validity of this quantum - to - classical correspondence for @xmath1 where the ising model is above its upper - critical dimension and displays mean - field behavior.@xcite initially , two of us claimed non - classical behavior with hyperscaling in the spin - boson model for @xmath1 , based primarily on nrg results.@xcite these results have been verified by others,@xcite and extended to the ising - symmetric bose - fermi kondo model.@xcite in contrast , subsequent quantum monte carlo ( qmc)@xcite and exact - diagonalization@xcite studies concluded that the critical behavior of the spin - boson model for @xmath1 is classical and of mean - field type .
we have recently retracted the claim@xcite of non - classical behavior , because we have realized two different sources of error of the nrg which spoil the determination of critical exponents.@xcite however , other authors continue to rely on nrg results in this context.@xcite , as function of temperature @xmath4 and tunneling strength @xmath5 ( keeping the dissipation strength @xmath6 fixed ) , showing a path with @xmath7 ( thick solid ) along which quantum - critical observables should be measured . due to the mass - flow effect ,
a temperature - dependent deviation of the order - parameter mass is induced , such that a system located at @xmath8 at @xmath9 follows a path with finite mass , @xmath10 , at any @xmath11 ( thick dashed ) .
the thin lines represent trajectories with ( dashed ) and without ( solid ) mass - flow effect for @xmath12 .
, width=240 ] in this paper , we investigate one of the error sources of the nrg in more detail , which we have dubbed the mass - flow effect .
it arises from the nrg algorithm which iteratively integrates out the impurity s bath . for a particle - hole asymmetric bath ,
the real part of the bath propagator generates a physical shift of impurity parameters ; for models with single - particle tunneling between impurity and bath it is simply the energy of the impurity level which is shifted due to the real part of the hybridization function . as a nrg calculation ignores the part of the bath spectrum below the current nrg scale ,
there is , at any nrg step , a _ missing _
parameter shift which is is set by the current nrg scale . near a quantum phase transition
, this implies an artificial scale - dependent shift of the order - parameter mass . for a nrg calculation with model parameter values corresponding to the critical point , the system is therefore _ not _ located at the critical coupling for any finite @xmath4 , but effectively follows a trajectory in the phase diagram as sketched in fig .
[ fig : pdflow ] .
this spoils the measurement of critical properties extracted in a nrg run as function of @xmath4 .
other sources of error within the nrg method are the discretization of the continuous bath density of states , the truncation of the eigenvalue spectrum in each nrg step to the lowest @xmath13 states , and the truncation of the bath hilbert space in the case of a bosonic bath , where only @xmath14 states are taken into account .
while the effects of discretization and spectrum truncation are well studied and understood within the fermionic nrg,@xcite hilbert - space truncation is more serious .
ref . pointed out that it precludes a correct representation of the ordered phase of the spin - boson model for @xmath15 at low energies or temperatures .
later , it was realized@xcite that it also leads to incorrect results for the order - parameter exponents @xmath16 and @xmath17 of the phase transition above the upper - critical dimension . in this paper , our focus will be on the mass - flow effect ; the other errors will be discussed when appropriate .
first , we shall demonstrate the mass - flow effect within a model of non - interacting bosons , namely the dissipative harmonic oscillator , for which all statements can be made exact . in this model
, the critical point translates into the instability point where the renormalized impurity energy is zero .
the mass flow will be shown to lead to qualitatively incorrect results ; this problem carries over to interacting models , like the anharmonic oscillator or the spin - boson model , if the critical point is gaussian ( i.e. above its upper - critical dimension ) .
second , we propose an extension of the iterative diagonalization scheme to cure the mass - flow error .
this extension solves the problem for the full parameter range of the non - interacting harmonic oscillator , while working asymptotically for models of interacting bosons .
third , we apply the extended nrg algorithm to the spin - boson model . for @xmath1 , we find results qualitatively different from those@xcite of the standard nrg implementation : our new results signify a flow towards a gaussian critical fixed point .
while the truncation of the bosonic hilbert space precludes calculations very close to this gaussian fixed point , we can identify a mean - field power law in the impurity susceptibility .
taken together , this shows that as other methods also the nrg predicts that the spin - boson model exhibits mean - field behavior for @xmath1 .
it is worth noting that an observation reminiscent of the mass - flow effect has been made in ref . :
the critical behavior of the classical long - range ising model with @xmath1 was found to change from mean - field - like to hyperscaling - like upon artificially truncating the `` winding '' of the long - range interaction ( i.e. upon violating the periodic boundary conditions in imaginary time ) .
this finding underscores that mean - field critical behavior in the long - range models under consideration can be easily spoiled by algorithmic errors .
note , however , that we disagree with the interpretation regarding the quantum - to - classical correspondence given in ref . , see below .
the remainder of the paper is organized as follows : in sec .
[ sec : models ] the model hamiltonians are introduced .
[ sec : chain ] explains how the mass - flow effect arises from the iterative diagonalization of the wilson chain .
the dissipative harmonic oscillator is subject of sec .
[ sec : dho ] , where the mass - flow error in the susceptibility is demonstrated analytically .
this knowledge is used in sec .
[ sec : cure ] to propose a modification of the iterative - diagonalization scheme , designed to cure the mass - flow error .
finally , the modified nrg algorithm is applied to the spin - boson model in sec .
[ sec : sb ] .
the nrg flow is discussed separately for @xmath18 and @xmath1 and compared to the results from standard nrg .
the results are interpreted in terms of a gaussian critical fixed point for @xmath1 .
conclusions close the paper .
various details , including a discussion of the mass - flow effect in fermionic impurity models with particle hole asymmetry , are relegated to the appendices .
the mass - flow effect can be most easily demonstrated using impurity models of non - interacting particles .
we shall consider the dissipative harmonic oscillator , with the hamiltonian @xmath19 where @xmath20 is the bare `` impurity '' oscillator frequency , @xmath21 is a field conjugate to the oscillator position , and the @xmath22 are the frequencies of the bath oscillators .
the bath is completely specified by its propagator at the `` impurity '' location @xmath23 with the spectral density @xmath24 universal properties of impurity phase transitions are determined by the behavior of the low - energy part of the bath spectrum @xmath25 . discarding high - energy details
, the common parametrization is @xmath26 where the dimensionless parameter @xmath6 characterizes the dissipation strength , and @xmath27 is a cutoff energy .
the value @xmath28 represents the case of ohmic dissipation .
the dissipative oscillator with a power - law bath spectrum is known to become unstable at large dissipation:@xcite the coupling to the bath renormalizes the oscillator frequency @xmath29 downwards , which becomes zero at some @xmath30 .
hence , the behavior of the model is not well - defined for @xmath31 .
the system at large dissipation may be stabilized by adding a local repulsive interaction to @xmath32 .
a symmetry - broken phase can emerge , with `` condensation '' of the @xmath33 bosons .
two possible routes are @xmath34 and @xmath35 the latter , @xmath36 , can be understood as a local @xmath37 impurity . on the other hand , @xmath38 in the limit
@xmath39 becomes equivalent to the standard spin - boson model @xmath40 where @xmath41 are the local impurity states , and @xmath29 is the tunneling rate .
the equivalence is seen by identifying the remaining oscillator states @xmath42 and @xmath43 in @xmath38 with the states @xmath44 . in all three models ( [ dao],[dao2],[sbm ] ) ,
the ordered phase at large dissipation breaks an ising symmetry , @xmath45 ( or @xmath46 ) , @xmath47 , and is associated with a non - zero expectation value @xmath48 ( or @xmath49 ) .
universality arguments suggest that the critical properties of the phase transitions are identical in the three models and coincide with those of a classical ising chain with @xmath3 interactions .
this quantum - to - classical correspondence trivially holds for @xmath36 in eq .
, as its imaginary - time path integral representation at @xmath9 is identical to the continuum limit of the one - dimensional ising model ( i.e. a scalar @xmath37 theory).@xcite for @xmath1 , the critical behavior is gaussian and mean - field like , with the quartic interaction being dangerously irrelevant at criticality . the quantum phase transition in the spin - boson model has been extensively studied : while the ohmic case , @xmath28 , has long been known to display a kosterlitz - thouless transition,@xcite the sub - ohmic case has only been investigated more recently.@xcite for @xmath50 , a continuous quantum phase transition emerges , with critical exponents depending on @xmath51 . while there is consensus that , for @xmath52 , those exponents are identical to the ones of the corresponding 1d ising model , a debate is centered around the issue of whether or not this continues to hold for @xmath53 where the ising model displays mean - field behavior .
alternatively , non - mean - field exponents obeying hyperscaling have been proposed on the basis of nrg calculations@xcite , and also carried over to the ising - symmetric bose - fermi kondo model.@xcite in particular , nrg has been used to calculate the local susceptibility @xmath54 at the critical coupling as function of temperature , which was found to follow a power law @xmath55 with @xmath56 .
in contrast , mean - field behavior implies@xcite @xmath57 , which has indeed been found e.g. using qmc simulations .
@xcite we shall argue here , expanding on our previous note,@xcite that the proposals of non - classical behavior are erroneous for the spin - boson model and questionable for the ising - symmetric bose - fermi kondo model . for the former , we show that the critical behavior instead is of mean - field type , consistent with numerical studies using qmc and exact - diagonalization methods.@xcite
within the nrg algorithm , the bath is represented by a semi - infinite ( `` wilson '' ) chain , fig . [
fig : itdiag ] , such that the local density of states at the first site of this chain is a discrete approximation to the bath density of states.@xcite due to the logarithmic discretization , the site energies @xmath58 and hopping matrix elements @xmath59 decay exponentially along the chain according to @xmath60 , where @xmath61 is the discretization parameter . .
the boxes indicate the iterative diagonalization scheme .
, width=278 ] let us denote by @xmath62 the hamiltonian of impurity plus @xmath63 sites of the wilson chain , and by @xmath64 the propagator at the impurity site of this @xmath63-site bath .
then , @xmath65 is the discretized version of the original problem . during the nrg run
, @xmath65 is diagonalized iteratively : first , @xmath66 is diagonalized and the lowest @xmath13 eigenstates are kept .
then , the next bath site is added to form @xmath67 , the new system is diagonalized , and again the lowest @xmath13 eigenstates are kept ( which are approximations to the lowest states of @xmath67 ) .
as the characteristic energy scale of the low - lying part of the eigenvalue spectrum decreases by a factor of @xmath61 in each step , this process is repeated until the desired lowest energy is reached .
temperature - dependent thermodynamic observables at a temperature @xmath68 are typically calculated via a thermal average taken from the eigenstates at nrg step @xmath63 . here
, @xmath69 is a parameter of order unity which is often chosen as @xmath70 .
the iterative diagonalization procedure implies that , at nrg step @xmath63 , the chain sites @xmath71 , @xmath72 , have not yet been taken into account , i.e. , the effect of those sites does not enter thermodynamic observables at temperature @xmath73 .
typically , this is a reasonable approximation , as the spectral density of the missing part of the chain , @xmath74 , has contributions at energies below @xmath75 only .
however , the missing chain also implies a missing contribution to the real part of the bath propagator .
this can be easily estimated : for a power - law bath spectrum , eq
. [ power ] , the zero - frequency real part @xmath76 is generated by frequencies @xmath77 and scales as @xmath78 , i.e. , up to numerical factors it scales as the nrg energy scale @xmath73 to the power @xmath51 . as we will show below , this missing real part implies a flow of the order - parameter mass and can spoil the analysis of critical phenomena . to support the above estimate
, we calculate the local green s function @xmath79 at the initial site , @xmath80 , of the wilson chain ( which is proportional to @xmath64 ) for different chain lengths @xmath63 . to this end
, we numerically diagonalize the single - particle problem corresponding to a wilson chain with parameters @xmath58 and @xmath59 chosen to represent a power - law bath spectrum , eq . , as in the nrg.@xcite explicit results for @xmath81
are shown in fig .
[ fig : reg]a . as expected
, @xmath82 approaches a finite ( negative ) value as @xmath83 , which depends on both @xmath51 and @xmath61 .
the missing real part @xmath84 is shown in panel b and scales as @xmath85 , with a prefactor which depends on @xmath51 , but only weakly on @xmath61 .
we shall discuss how the mass - flow effect influences observables for the simplest model , the dissipative harmonic oscillator .
it is important to distinguish the various methods to calculate observables in this non - interacting model : ( i ) for a continuous power - law spectrum , a number of quantities can be calculated analytically .
( ii ) for a discretized bath , represented by a semi - infinite wilson chain , the single - particle problem can be solved by exact diagonalization for long chains .
( iii ) as in nrg , one may use a truncated wilson chain with temperature - dependent length , and again diagonalize the single - particle problem .
( iv ) a true nrg calculation can be performed , which treats the full many - body problem . here
, we shall mainly be interested in comparing the results of ( ii ) and ( iii ) , which allows to assess the mass - flow error .
in contrast , the difference between ( i ) and ( ii ) can be used to quantify the discretization error , while the difference between ( iii ) and ( iv ) is due to spectrum and hilbert - space truncation of nrg .
the most interesting observable is the susceptibility associated with the oscillator position , defined according to @xmath86 which is the analogue of @xmath87 in the spin - boson model .
importantly , @xmath54 is given by a single - particle propagator , @xmath88 , with @xmath89 / 2 } , \nonumber\end{aligned}\ ] ] note the factors of @xmath90 in eq . .
this equation shows that the dissipative oscillator is unstable at and beyond the `` resonance '' which occurs at some dissipation strength @xmath30 , defined by @xmath91 . for @xmath92 ,
all eigenenergies of the system are positive , whereas the lowest one turns negative for @xmath31 .
thus , @xmath30 corresponds to a singularity of the dissipative harmonic oscillator , separating the stable from the unstable regime .
= 3.1 in returning to the susceptibility @xmath54 , its static limit evaluates to @xmath93 which is seen to be temperature - independent and only determined by @xmath29 and the real part of the bath propagator .
consequently , there is a strong mass - flow effect , as the renormalized oscillator frequency in the denominator of @xmath54 reads @xmath94 for a @xmath63-site chain .
this is illustrated in fig .
[ fig : chi_ho ] where we show the susceptibility as function of temperature , calculated using either a long wilson chain for all @xmath4 or an @xmath63-site wilson chain at temperature @xmath73 , i.e. , using methods ( ii ) and ( iii ) described above .
most importantly , @xmath54 calculated from the truncated wilson chain is temperature - dependent , in contrast to the exact result . for @xmath92 ,
the exact result is approached at low @xmath4 .
the error is most drastic at resonance , @xmath95 .
there , @xmath96 ( i.e. the system is unstable ) , whereas the calculation using a truncated wilson chain gives @xmath97 .
physicswise , the mass - flow effect introduces a finite and temperature - dependent oscillator frequency @xmath98 , thus artificially stabilizing the system at @xmath30 .
naturally , the same result is found using a full nrg calculation .
as the dissipative harmonic oscillator represents the fixed - point hamiltonian of the gaussian critical point of , e.g. , the anharmonic oscillator @xmath36 in eq . , it is straightforward to discuss the mass - flow effect there .
the renormalized @xmath99 can be identified with the order - parameter mass , and @xmath100 . along the flow towards the gaussian fixed point ,
the irrelevant interaction @xmath101 leads to an order - parameter mass @xmath102 for @xmath1 , and the physical susceptibility follows @xmath103 ( ref . ) .
however , the artificial mass @xmath104 caused by the mass - flow effect dominates the physical mass at low @xmath4 , leading again to the unphysical result @xmath97 .
as this coincides with the physical result for an interacting critical fixed point with hyperscaling , the unphysical result from the mass - flow effect could be mistaken as a signature of interacting quantum criticality .
we should emphasize that a renormalization - group scheme which successively integrates out the impurity s bath is perfectly valid .
however , it requires that the calculation of observables at some scale @xmath4 accounts for the remaining part of the bath .
the latter is not the case in the iterative diagonalization scheme of standard nrg .
the mass - flow error arises from the missing real part of the bath propagator , @xmath105 , which , for every step of the iterative diagonalization , is simply a number .
ideally , a general algorithmic solution of the mass - flow problem would directly correct @xmath106 .
however , this is limited by kramer - kronig relations , and we have not found a manageable implementation of this idea . in the following ,
we shall instead make use of physics arguments in order to ( approximately ) correct the mass - flow error .
for the harmonic oscillator , @xmath106 directly renormalizes the oscillator s energy , while things are conceptually more complicated for interacting models ( like the spin - boson or bose - fermi kondo models ) .
therefore , we shall separately discuss the non - interacting and interacting cases in the following .
a simple recipe can be used to correct the mass - flow error when diagonalizing a finite - length chain corresponding to the harmonic - oscillator @xmath62 .
we define a hamiltonian piece @xmath107 by @xmath108 with @xmath109 . as a result
, @xmath110 has the correct mass term , i.e. , the correct renormalized oscillator frequency , for any @xmath63 , and diagonalizing @xmath110 instead of @xmath62 in step @xmath63 removes the mass - flow problem .
one obtains the correct result for @xmath54 : thanks to @xmath107 , the denominator of @xmath54 in eq .
is replaced by @xmath111 which is the exact result for the semi - infinite wilson chain . a mass - flow correction via @xmath107
can be straightforwardly implemented into the iterative diagonalization scheme of the nrg method
. the modified nrg algorithm ( dubbed nrg@xmath112 in the following ) works as follows : ( i ) initially , one diagonalizes @xmath113 .
in addition to the usual observables , the matrix elements of the operator @xmath114 are stored as well .
then , the following steps are repeated : ( ii ) from the lowest @xmath13 states of the solution of nrg step @xmath63 and the states of the impurity site @xmath71 , one constructs @xmath115 .
in contrast to @xmath116 , the operator @xmath115 contains a mass - flow correction from the previous steps .
( iii ) using the matrix elements of @xmath117 , one constructs @xmath118 .
( iv ) one diagonalizes @xmath119 and re - calculates the matrix elements of the desired observables and of @xmath114 .
the correction of the mass - flow error , contained in steps ( i ) and ( iii ) which differ from the usual nrg algorithm , is implemented such that the frequency shifts cancel in the limit @xmath83 .
hence , runs of nrg and nrg@xmath112 with the same model parameters should target the same point in the phase diagram as @xmath120 ( although their finite - temperature trajectories are different , fig . [ fig : pdflow ] ) .
however , this is only true in the absence of spectrum truncation . for finite @xmath13 ,
the cancellation is only approximate , i.e. , there will be a small ( but unimportant ) parameter shift due to the mass - flow correction .
being interested in extracting critical properties , we identify the mass - flow effect as a scale - dependent shift of the order - parameter mass .
this suggests that the mass flow can be corrected by an appropriate shift in the phase transition s control parameter this is simply a generalization of eq . where @xmath107 shifts the oscillator frequency .
we thus propose to employ a correction of the form @xmath121 where @xmath117 is now a ( local ) operator which can be used to tune the phase transition , e.g. , the tunneling term @xmath122 in the spin - boson model or the kondo coupling term in a bose - fermi kondo model .
importantly , the required shift will no longer be identical to @xmath123 .
this is already clear for the dissipative anharmonic oscillators , eqs . and , where the quartic interaction will renormalize both the oscillator frequency and also its shift due to the bath , but in a different fashion .
hence , we have introduced the non - universal prefactor @xmath124 which we intend to determine by physical criteria .
two issues require special consideration : ( a ) is the linear relation between the required shift in the control parameter and the missing real part of @xmath125 , which is implied by eq .
, justified ? ( b ) how can one determine the prefactor @xmath124 ? the simplest argument for ( a ) is as follows : the phase transition s control parameter ( equivalently , the distance to criticality or the bare order - parameter mass ) depends on both the prefactor of @xmath117 and the real part of @xmath126 .
both dependencies have a regular taylor expansion at a given point in parameter space , hence , the leading terms are linear . as @xmath127 changes by a known amount in every step of the iterative diagonalization due to the mass - flow effect , this can be compensated by a change in the prefactor of @xmath117 which proportional to this amount , i.e. , a change of the form @xmath128 with some fixed @xmath124 .
this argument only relies on the taylor expansion and is thus asymptotically correct for small changes in @xmath127 , i.e. , for @xmath129 .
( for a given model , like the anharmonic oscillator , one can check the linear behavior by an explicit perturbative calculation . )
physically , it is clear that the linear term of the expansion will capture the correct behavior in the vicinity of a given renormalization - group fixed point , i.e. , the required @xmath124 depends on the fixed point of interest ( and on non - universal high - energy details ) .
note , however , that the procedure is more general than these considerations suggest : as both the gaussian critical fixed point and the delocalized fixed point are asymptotically non - interacting , a fixed @xmath124 can be used to capture the entire crossover from the quantum critical to the delocalized regime in this case .
question ( b ) will be discussed for different types of critical fixed points in turn .
we shall use the language of the dissipative anharmonic oscillator , where the critical theory is known.@xcite a gaussian critical fixed point , realized for @xmath1 , provides a simple criterion to find the correct value @xmath130 of the correction parameter @xmath124 , namely the temperature dependence of the order parameter mass . as emphasized in sec .
[ sec : dho ] , the artificial mass generated by the mass - flow effect follows @xmath131 , while the physical mass scales as @xmath132 .
thus , in general the mass at the critical coupling will be given by @xmath133 where @xmath134 are prefactors .
for @xmath135 ( undercompensation ) , the positive @xmath131 term will always dominate at low @xmath4 and mimic hyperscaling properties . for @xmath136 ( overcompensation ) , the mass will become negative at low @xmath4 , i.e. , the flow will be towards the localized phase .
an intermediate flow inside the localized phase will even occur if couplings are chosen to be slightly in the delocalized phase : for @xmath137 or @xmath138 the system flows from critical to localized and then back to delocalized upon lowering @xmath4 , accompanied by a non - monotonic behavior of @xmath54 .
this suggests the following simple recipe to determine @xmath130 : start with large @xmath124 such that non - monotonic flows are seen near the critical coupling .
decrease @xmath124 until those disappear and the susceptibility follows a power law different from hyperscaling at the critical coupling down to the lowest accessible temperatures . if @xmath124 is decreased too far , then @xmath139 is recovered . hence
, a clear signature of gaussian criticality is a qualitatively different behavior in @xmath54 for small and large @xmath124 . in sec .
[ sec : sb ] and app .
[ app : kappa ] , we shall demonstrate this for the spin - boson model at @xmath1
. in the case of an interacting critical fixed point , realized for @xmath18 , hyperscaling is fulfilled on physical grounds .
hence , the mass will invariably scale as @xmath131 at the critical coupling , both for @xmath135 and @xmath136 .
this simply reflects the fact that the mass - flow effect does not introduce qualitative ( but only quantitative ) errors here , in contrast to the case of gaussian criticality .
hence , the behavior in the quantum - critical regime does not provide a sharp physical criterion to determine @xmath130 .
we conclude that a clear signature of true interacting criticality is an insensitivity to the value of @xmath124 of the qualitative critical behavior . for the spin - boson model at @xmath28 , a comparison of observables to those from other solutions like bethe ansatz or bosonization
could be used to determine @xmath130 ( for either the localized or the delocalized phase ) . as @xmath28 plays the role of a lower - critical dimension , we have not followed this route further
we now apply the modified nrg@xmath112 algorithm , which includes the mass - flow correction , to the spin - boson model .
note that we will make no a - priori assumptions on the nature of the critical fixed points , but instead apply the strategies outlined in sec .
[ sec : cureint ] to determine the optimal @xmath130 within the nrg@xmath112 algorithm .
we have studied the flow diagrams for various values of the bath exponent @xmath50 and the mass - flow correction parameter @xmath124 . while a detailed set of data is displayed in app .
[ app : kappa ] , the main conclusion is that for @xmath1 the flow changes qualitatively as @xmath124 is varied , while this is not the case for @xmath18 .
the former fact can be used to determine @xmath130 for @xmath1 , while a rough estimate of @xmath130 for @xmath140 may be obtained from an extrapolation of @xmath141 .
doing so , we obtain the flow diagrams from the mass - flow corrected nrg@xmath112 algorithm , which represent a central result of this paper . those are shown in figs .
[ fig : flows04 ] and [ fig : flows06 ] for @xmath142 and 0.6 , respectively , together with the flow diagrams from standard nrg .
the latter are similar to the ones shown in earlier papers.@xcite = 3.6 in = 3.6 in let us start the discussion with fig .
[ fig : flows04]a , displaying the standard nrg flow for @xmath142 near the critical coupling strength . for @xmath138 ( left )
the flow reaches the delocalized fixed point , whereas it is directed towards the localized fixed point for @xmath143 ( right ) ; note that the latter is not correctly described due to hilbert - space truncation.@xcite the flow at @xmath30 ( dashed ) shows a different nrg fixed point , which has been identified with the critical fixed point . for both @xmath138 and @xmath143
this level structure is visible at intermediate stages of the flow , before the system departs towards one of the stable fixed points this crossover is usually identified with the quantum critical crossover scale @xmath144 above which the system is critical .
now consider the flow of nrg@xmath112 , fig .
[ fig : flows04]b , which includes the mass - flow correction of sec .
[ sec : cure ] .
while the asymptotic fixed points for both @xmath92 and @xmath31 are identical , the flow near criticality is strikingly different . in particular , no stable level pattern emerges , possibly corresponding to a critical nrg fixed point .
instead , all levels appear to converge toward zero energy before the critical regime is left .
note that the critical flow can not be followed to large @xmath63 ( the system is always localized or delocalized for @xmath145 ) . in fig .
[ fig : flows06 ] , the same comparison of flow diagrams is given for @xmath146 . here , no qualitative difference between the flows without and with mass - flow correction is seen .
a stable level pattern is visible near criticality in both cases , but the level energies differ slightly in fig .
[ fig : flows06]a and b. we found this behavior to be generic for @xmath52 , while the absence of a critical nrg fixed point as in fig .
[ fig : flows04]b is characteristic for all @xmath53 , if @xmath124 is chosen according to the criteria in sec .
[ sec : cureint ] .
it is straightforward to discuss what would be expected for a quantum phase transition above its upper - critical dimension .
the gaussian fixed point features free massless bosons , and interactions are required to stabilize the system at @xmath11 .
those are dangerously irrelevant and flow to zero in the critical regime , with a scaling dimension which is small near the upper - critical dimension . translated into a many - body spectrum , this implies that _ at _ the gaussian critical fixed point the spectrum consists of an infinite number of degenerate levels at zero energy , while the flow towards the critical fixed point is characterized by the level spacing flowing to zero as @xmath83 .
the latter is precisely what is seen in fig .
[ fig : flows04]b .
it is also clear that within nrg@xmath112 the fixed point itself can never be reached , because with decreasing interactions ( i.e. decreasing level spacing ) the error introduced by the hilbert - space truncation becomes more and more serious ( i.e. bosonic occupation numbers become large ) .
this implies that small values of @xmath144 can not be reached ( as the system always flows to either the localized or delocalized phase below some @xmath147 ) which also limits the precision with which we can determine @xmath30 .
a few remarks are in order : ( i ) during the flow towards the gaussian fixed point , fig .
[ fig : flows04]b , the rate of decrease in level spacing as function of @xmath63 depends strongly on @xmath51 , i.e. , the level spacing decays faster with smaller @xmath51 , qualitatively consistent with the scaling dimension of the interaction @xmath101 being@xcite @xmath148 .
correspondingly , the critical flows breaks down earlier for smaller @xmath51 .
( ii ) the value of the critical coupling @xmath30 differs between nrg and nrg@xmath112 . as discussed above
, this is a result of spectrum truncation within nrg@xmath112 .
we have checked that the difference decreases with increasing @xmath13 .
further the difference is larger for smaller @xmath51 , which follows from the mass - flow error itself being larger for smaller @xmath51 , see fig .
[ fig : reg]b .
we conclude that the critical behavior of the spin - boson model for @xmath1 is gaussian .
the stable critical fixed point in fig .
[ fig : flows04]a is then an artifact of the mass - flow error , where the system follows the thick solid trajectory in fig .
[ fig : pdflow ] .
in contrast , for @xmath18 the critical theory of the spin - boson model is interacting .
these conclusions are supported by the analysis of @xmath149 , see next subsection .
we continue with nrg results for the order - parameter susceptibility @xmath150 of the spin - boson model .
we will focus on the power - law behavior @xmath151 in the quantum - critical regime .
= 3.4 in for both @xmath142 and @xmath152 , data from both standard nrg and nrg@xmath112 are shown in figs .
[ fig : chi04 ] and [ fig : chi03 ] , respectively .
as reported before , @xmath56 is obtained from nrg , fig .
[ fig : chi04]a and fig .
[ fig : chi03]a , while the correct result near a gaussian fixed point is @xmath57 .
it should be noted that this @xmath153 power law requires the renormalized quartic interaction to be small .
however , once the effective interaction becomes small in the numerics , the nrg@xmath112 algorithm breaks down due to hilbert - space truncation .
thus , the weakly interacting gaussian critical regime can not be reached , and we can not expect to see an asymptotic @xmath153 susceptibility power law . for our parameter values , the truncation - induced lower cutoff scale , @xmath147 , for the critical regime is @xmath154 for @xmath142 and @xmath155 for @xmath152 .
notably , the nrg@xmath112 results in figs . [ fig : chi04]b and fig . [ fig : chi03]b _ do _ follow @xmath153 over two to three decades in temperature above @xmath147 , while @xmath156 is never seen . we are again forced to conclude that the @xmath156 behavior in standard nrg , figs . [ fig : chi04]a and [ fig : chi03]a , is an artifact of the mass - flow error . to support this
, we also show the susceptibility of the dissipative harmonic oscillator model , eq . ,
calculated using a truncated wilson chain with the _ same _ chain parameters as in the nrg run for the spin - boson model and @xmath29 tuned to resonance .
as explained in sec .
[ sec : dho ] , this model has @xmath157 , but a finite @xmath54 results exclusively from the mass - flow error . remarkably , this @xmath54 matches the @xmath54 from nrg for the spin - boson model at low temperatures to an accuracy of better than 15% this is consistent with the assertion that the latter reflects the physics of a gaussian fixed point artificially stabilized by the mass - flow effect .
= 3.4 in = 3.4 in susceptibility data for @xmath146 are shown in fig .
[ fig : chi06 ] . both nrg and nrg@xmath112 yield a power law with @xmath56 , albeit with prefactors differing by 10% . here ,
the harmonic - oscillator @xmath54 ( with mass flow ) and the nrg @xmath54 do _ not _ match , but instead differ by roughly a factor of 1.9 .
all this is consistent with a true interacting fixed point . from the discussion
, it is obvious that other observables _ at _ criticality will suffer the mass - flow error similar to @xmath149 .
this applies to thermodynamic quantities including entropy and specific heat , but also to zero - temperature dynamic quantities , like the susceptibility @xmath158 .
while the latter is defined from the ground state , the corresponding nrg evaluation is in fact done during the flow.@xcite hence , @xmath158 for @xmath1 is potentially incorrect as well .
however , @xmath158 can be proven to follow @xmath159 for all @xmath51 , irrespective of whether the fixed point is gaussian or interacting,@xcite such that the mass - flow effect only introduces quantitative deviations .
off - critical properties are to leading order _ not _ affected by the mass - flow error , because the artificial mass vanishes as @xmath129 while the physical mass remains finite . however ,
subleading corrections are subject to the mass - flow error .
the nrg calculations of ref .
did not only find the critical exponent @xmath160 to deviate from its mean - field value for @xmath1 , but also the order - parameter exponents @xmath16 and @xmath17 .
as discussed in detail in ref .
, this incorrect result is due to a _
different _ failure of the bosonic nrg , namely the fact that the hilbert - space truncation prevents an asymptotically correct representation of the localized fixed point for @xmath15 . for mean - field criticality ,
@xmath16 and @xmath17 are not properties of the critical fixed point , but instead of the flow towards the localized fixed point . as the latter suffers from the hilbert - space truncation , @xmath16 and
@xmath17 are unreliable .
however , large values of @xmath14 can be used to uncover the physical power laws at intermediate scales ( which are of mean - field type for @xmath1 ) , before truncation effects set in.@xcite
for both versions of the anharmonic oscillator , eqs . and , we have obtained results which are qualitatively similar to those for the spin - boson model . in particular , the standard nrg exhibits signatures of an interacting critical fixed point for all @xmath51 , as in figs . [
fig : flows04]a and [ fig : flows06]a . for @xmath161 ,
this result is obviously incorrect for @xmath1 , due to its equivalence to a local @xmath37 theory .
accordingly , the mass - flow corrected nrg@xmath112 algorithm yields gaussian behavior in both models for @xmath1 . hence , all models ( [ dao],[dao2],[sbm ] ) belong to the same universality class and follow the quantum - to - classical correspondence .
we have also investigated the mass - flow effect for particle - hole asymmetric fermionic impurity models . while generically present , its effects on observables turn out to be tiny , for details see app .
[ app : rlm ] .
finally , a remark on symmetries and the quantum - to - classical correspondence is in order : while all cases discussed so far feature ising - symmetric critical degrees of freedom , impurity spin models with higher symmetry [ e.g. su(2 ) ] have been discussed extensively in the literature as well . here ,
a direct quantum - to - classical mapping ( via a re - interpretation of the trotter - discretized action of the quantum model after integrating out the bath ) is usually not possible , due to the impurity spin s berry phase .
indeed , the so - called bose - kondo model with su(2 ) symmetry exhibits a stable intermediate - coupling fixed point@xcite ( unlike any classical 1d spin model ) , and the su(@xmath162)-symmetric bose - fermi kondo model has been shown to display a quantum critical point with hyperscaling for all @xmath51.@xcite
in this paper , we have investigated a source of error in wilson s nrg method which had received little attention before .
this mass - flow error is inherent to the iterative diagonalization scheme of nrg which neglects the low - energy part of the bath when calculating observables .
we have traced the mass - flow effect in the dissipative harmonic oscillator model , where results for the finite - temperature susceptibility turn out to be qualitatively incorrect in general . applied to quantum phase transitions in bosonic impurity models ,
we have argued that the mass - flow effect introduces qualitative errors in the critical regime of mean - field quantum phase transitions , while it only leads to quantitative errors for interacting quantum criticality .
a simple extension of the nrg algorithm allows to cure the mass - flow error asymptotically near the fixed points of interest .
we have applied this modified algorithm to the sub - ohmic spin - boson model and found unambiguous signatures of mean - field behavior for @xmath1 , including a flow towards a gaussian critical fixed point , fig . [
fig : flows04]b , and a susceptibility power law with mean - field exponent , figs .
[ fig : chi04]b and [ fig : chi03]b .
we have thus resolved the discrepancy between results from nrg and those from other numerical methods.@xcite as the conventional nrg is not capable of describing mean - field critical points , claims of non - mean - field behavior in related@xcite ising - symmetric impurity models with sub - ohmic bosonic bath@xcite need to be re - visited .
we thank s. florens , e. grtner , k. ingersent , s. kirchner , r. narayanan , h. rieger , q. si , and t. vojta for discussions .
this research was supported by the dfg through sfb 608 ( mv , rb ) , fg 538 ( mv ) , and fg 960 ( mv , rb ) .
mv also acknowledges financial support by the heinrich - hertz - stiftung nrw and the hospitality of the centro atomico bariloche where part of this work was performed .
= 6.7 in as discussed in sec .
[ sec : cureint ] , a general algorithmic solution to the mass - flow problem for interacting bosonic impurity models is not available .
instead , we have argued that an empirical correction via eq . within the nrg@xmath112 algorithm is appropriate , with a prefactor @xmath124 which depends on the fixed point of interest .
here we show the influence of @xmath124 on the nrg@xmath112 results for the sub - ohmic spin - boson model near criticality .
[ fig : flows04kappa ] displays nrg flows for @xmath138 ( top ) and local susceptibility data for various @xmath163 ( bottom ) for different values of @xmath124 for @xmath142 .
the central observation is that the behavior _ qualitatively _ changes when @xmath124 is varied from 0.3 to 1.0 .
in a ) both the critical and delocalized nrg fixed points are clearly visible in the flow , and the critical @xmath164 . in b ) , the flow in the critical regime displays a decreasing level spacing with increasing @xmath63 , and no asymptotic @xmath156 power law is observed .
panels c ) and d ) show clear signs of overcompensation as discussed in sec .
[ sec : cureint ] , i.e. , a non - monotonic flow ( critical localized delocalized ) and a corresponding non - monotonic @xmath149 near @xmath30 . here , a precise determination of @xmath30 is impossible , and no critical power law in @xmath149 emerges .
a detailed analysis of case a ) shows that the behavior is qualitatively similar to that of the standard nrg . for a gaussian fixed point
, this would imply undercompensation . together with the discussion in sec .
[ sec : cureint ] , these observations strongly suggest that the critical fixed point of the spin - boson for @xmath142 is gaussian , with @xmath165 , see also the data in fig . [
fig : flows04 ] .
indeed , the critical @xmath149 in fig .
[ fig : flows04kappa]b does not follow @xmath55 with @xmath166 at the lowest @xmath4 shown , but instead crosses over to larger @xmath160 . a similar procedure for other @xmath1 yields @xmath167 and @xmath168 .
= 3.6 in in contrast , data for @xmath146 , fig .
[ fig : flows06kappa ] , do _ not _ display a qualitative change when @xmath124 is varied from 0.5 to 2.0 .
( @xmath169 data from the standard nrg are in fig .
[ fig : flows06 ] . ) instead , for all @xmath124 a stable critical fixed - point spectrum emerges , and the critical susceptibility follows @xmath139 .
this implies an interacting critical fixed point in the spin - boson model for @xmath146 .
an extrapolation of @xmath141 suggests @xmath170 ( which , however , is not very accurate , see sec .
[ sec : cureint ] ) .
we note that figs .
[ fig : flows06kappa]a and b differ quantitatively : the level structure at the critical fixed point is somewhat shifted , and the prefactor of the critical power law of @xmath149 is 40% larger in b ) .
this trend simply reflects that larger @xmath124 reduces the order - parameter mass along the flow trajectory .
the mass - flow effect is in principle also present in fermionic impurity models if the bath is particle - hole asymmetric in the low - energy limit . consider the spinless resonant - level model @xmath171 with the bare level energy @xmath172 .
as in sec . [
sec : models ] , one can define a bath spectral density @xmath25 , which , however , now generically has contributions at both positive and negative frequencies .
the solution for the @xmath173 ( impurity ) green s function is @xmath174 the impurity properties of this model are non - singular except at resonance , @xmath95 , where @xmath175 , i.e. , where the renormalized @xmath173 level coincides with the fermi level . the properties near resonance have been studied extensively in refs . for the particle - hole symmetric case .
a mass - flow error arises only for bath spectra @xmath25 which are particle - hole asymmetric at low energies .
the low - energy asymmetry may be quantified by looking at @xmath176 a finite @xmath177 implies particle - hole asymmetry in leading order .
otherwise the mass - flow error vanishes in the low - energy limit , this applies e.g. to a metallic fermionic bath spectrum with different positive and negative band cutoff energies .
we have studied the mass - flow error for the resonant - level model with a maximally particle - hole asymmetric power - law bath , i.e. , @xmath178 and @xmath179 . in analogy with sec . [
sec : dho ] , we then expect a large mass - flow error at resonance .
indeed , the resonance position is shifted in a similar fashion as for the harmonic oscillator in sec .
[ sec : dho ] .
however , when comparing observables , like the @xmath173 level occupancy or its susceptibility , calculated at @xmath73 for the semi - infinite and the truncated chains with fixed @xmath172 , we find that the differences are tiny ( less than @xmath180 ) , in stark contrast to the bosonic case .
the reason for the small mass - flow error is rooted in both the character of the observables and the statistics of the particles .
first , in the fermionic case all observables are related to two - particle propagators , in contrast to the bosonic @xmath54 of eq . .
hence , the real part of @xmath181 never shows up as directly as in @xmath54 , due to a convolution integral .
second , the response of fermions at resonance is less singular than that of bosons . therefore , deviations from the exact resonance condition have less consequences as compared to the bosonic case . the bose - fermi kondo model with a metallic bath density of states of fermions and an ising - symmetric bosonic bath is believed to have the same critical properties as the spin - boson model , because the fermionic bath can be bosonized by standard techniques , leading to a dissipative ohmic bath in addition to the ising - symmetric bath .
if the latter has a sub - ohmic density of states , it dominates the critical properties , which then are those of the sub - ohmic spin - boson model .
results from nrg calculations@xcite are consistent with this expectation .
|
we discuss a particular source of error in the numerical renormalization group ( nrg ) method for quantum impurity problems , which is related to a renormalization of impurity parameters due to the bath propagator . at any step of the nrg calculation ,
this renormalization is only partially taken into account , leading to systematic variation of the impurity parameters along the flow .
this effect can cause qualitatively incorrect results when studying quantum critical phenomena , as it leads to an implicit variation of the phase transition s control parameter as function of the temperature and thus to an unphysical temperature dependence of the order - parameter mass .
we demonstrate the mass - flow effect for bosonic impurity models with a power law bath spectrum , @xmath0 , namely the dissipative harmonic oscillator and the spin - boson model .
we propose an extension of the nrg to correct the mass - flow error .
using this , we find unambiguous signatures of a gaussian critical fixed point in the spin - boson model for @xmath1 , consistent with mean - field behavior as expected from quantum - to - classical mapping .
|
the large volume detector ( lvd ) , in the infn gran sasso national laboratory ( italy ) , below an average 3600 m water equivalent rock overburden , is a 1 kt liquid scintillator detector whose major purpose is monitoring the galaxy to study neutrino bursts from gravitational stellar collapses @xcite .
lvd consists of an array of 840 scintillator counters , 1.5 m@xmath0 each , divided in three , identical and independent _
towers_. each counter is viewed from the top by three 15 cm photomultiplier tubes ( pmts ) @xcite .
+ the lvd liquid scintillator is a mixture of aliphatic and aromatic hydrocarbons ( c@xmath1h@xmath2 with @xmath3 = 9.6 ) also known as _
white spirit_. for historical reasons 73% of the lvd scintillator was produced in russia and has an aromatic hydrocarbons content of @xmath416@xmath5 , while the remaining amount has been produced in italy and contains @xmath48@xmath5 of aromatics .
the scintillator with the higher percentage of aromatics presents a light yield of about 20@xmath5 higher .
we do not expect this slight difference on the scintillator composition to have an impact on the quenching factors therefore we decided to measure the proton light output response in the liquid scintillator produced in russia that is present in the majority of counters ( lvd - scint ) .
+ in addition , since in 2005 one of the lvd counters ( with aromatic hydrocarbons content of @xmath48@xmath5 ) was doped with gadolinium @xcite to improve the signal to noise ratio in neutron detection , we decided to repeat the measurement also on this particular scintillator ( lvd - gd ) .
+ in organic scintillators the light output response , the energy emitted as fluorescence , is non - linear with respect to the energy deposited by ionizing particles ( this is particularly true for highly ionizing particles ) .
this behavior is attributed to quenching of the primary excitation by the high ionization density along the particle track which leads to a decrease of the scintillation efficiency . in the absence of quenching
the light yield is proportional to the energy released , which may be written in differential form as : @xmath6 where @xmath7 is the fluorescent energy emitted per unit path length , @xmath8 is the scintillation efficiency and @xmath9 is the specific stopping power . to describe the quenching effect , birks
@xcite proposed a semi - empirical relation in which the differential light output is reduced for high energy loss : @xmath10 where the product @xmath11 is usually treated as a single parameter and it is known as birks factor . + the quenching factor for ions is defined as the ratio of light yield of ions to that of electrons of the same energy . integrating eq .
[ eqbirks ] it can be written as : @xmath12
two scintillator detectors of the same dimensions of the counters of the lvd experiment , have been used to perform the measurements .
each detector consists of a 1 @xmath13 1 @xmath13 1.5 m@xmath0 stainless steel tank filled with 1.2 ton of liquid scintillator and viewed from the top by three pmts photonis xp3550 of 5 diameter .
the signals of each pmt are recorded by a caen v1724 waveform digitizer , with 100 mhz of sampling frequency and 137 @xmath14 of resolution per bit .
the board hosts a field programmable gate array ( fpga ) for each input channel .
the data stream is continuously written in a circular memory buffer , which is frozen by the fpga when the trigger occurs .
the board implement also the zero length encoding ( zle ) algorithm for data reduction .
this algorithm allows to set a threshold , which could be different from the trigger threshold , and discard the data under threshold saving time in both data transfer and processing .
the acquisition window length , the number of samples pre- and post - trigger , the threshold and the level of coincidence among different channels are programmable features .
+ in the center of the detector we have inserted an @xmath15am@xmath16be source encapsulated in a waterproof housing and held in place by a stainless steel rod .
+ the radionuclide @xmath15am decays emitting an alpha particle of 5.6 mev . under bombardment by alphas
, @xmath16be undergoes a nuclear reaction forming the excited compound nucleus @xmath17c@xmath18 , which then decays mostly producing either @xmath19c at ground state or at the first excited level @xcite : @xmath20 in the latter case the neutron is accompanied by a de - excitation gamma - ray of energy 4.44 mev .
the source used in these measurements emits about 10 neutrons s@xmath21 .
+ the emitted neutrons are thermalized by elastic scattering on the hydrogen atoms of the organic scintillator . during the slowing down process
, a fraction of the neutron kinetic energy is converted to photons which are detected by the pmts giving the prompt signal . after being thermalized ,
neutrons are eventually captured by @xmath22h , @xmath23gd ( when present ) and with lower probability by @xmath19c and @xmath24fe nuclei producing the delayed signal .
the mean neutron capture time depends on the target composition ( n - capture cross section ) and on the detector geometry because neutrons which migrate farther from the source can escape from the detector leading to a shorter mean capture time .
in the present work we will refer to the light output in terms of the scintillator response to electrons , as this can be taken to be linear ( within @xmath41@xmath5 for electrons of energy e @xmath25 2 mev ) .
the measured signals will then be expressed in mev electron recoils equivalent ( mev@xmath26 ) .
+ assuming that our scintillator has a response function to nuclear recoils consistent with that of similar organic scintillators , we should expect signals up to @xmath42 mev@xmath26 for neutrons of 6.5 mev which elastically scatter protons .
neutrons of lower energy are accompanied with a 4.44 mev gamma emission , according to equations [ eqn1 ] and [ eqn2 ] .
+ neutron captures on h are followed by the emission of the 2.22 mev @xmath27 quantum from the deuterium de - excitation ; while captures on gd originate @xmath27 cascades of total energy @xmath48 mev .
as we want to measure the prompt signals with a threshold as low as possible , we decided to trigger on the delayed pulse and then look backward in time to find the prompt one . the trigger condition to enable samples storage
was chosen to be a 3-fold coincidence of the pmts signal exceeding a threshold of 1.8 mev@xmath26 for the counter with lvd - scint and 3.2 mev@xmath26 for the counter with lvd - gd .
the background rate at these thresholds , for a counter placed underground , is about 50@xmath5 and 2@xmath5 of the neutron rate from the source .
+ neutron events are selected requiring that the delayed pulse must be accompanied , in a time window of 320 @xmath28s preceding the trigger , by a second signal ( prompt ) detected with a zle threshold of 0.8 mev@xmath26 for both the detectors .
this additional condition helps to get rid of the background enanching the signal to noise ratio of about a factor ten .
+ the highest - energy prompt signals are large enough to trigger by themselves , in this case the delayed pulse will be left out of the acquisition window .
to avoid this situation the acquisition window has been increased to 640 @xmath28s moving the trigger in the middle of it . in this way we can consider the trigger pulse as due to nuclear recoil or nuclear capture , depending on the presence of a pulse in the following or in the preceding 320 @xmath28s respectively . in turn
the pulse in the pre - trigger region can be regarded as a recoil signal and used to extend the recoil energy spectrum down to 0.8 mev@xmath26 .
[ dt ] we show the distribution of the absolute value of the difference between the trigger pulse and the pre - trigger or post - trigger pulse if present in the same event , for both the scintillators measured . fitting the distribution with an exponential function we found , for lvd - scint , a mean neutron capture time of 210 @xmath28s , well in agreement with previous measurements taking into account that the neutron source was placed in the center of the detector .
distribution of the absolute value of the time difference between the trigger and pre- or post - trigger in the same event ( blu points ) compared with the background level ( in gray ) .
shapes are , as expected , flat for the background ( fit function : y@xmath29 ) and exponential for the data with the neutron source ( fit function : y@xmath30 ) .
top and bottom panel are relative to lvd - scint and lvd - gd scintillator respectively.,title="fig : " ] distribution of the absolute value of the time difference between the trigger and pre- or post - trigger in the same event ( blu points ) compared with the background level ( in gray ) .
shapes are , as expected , flat for the background ( fit function : y@xmath29 ) and exponential for the data with the neutron source ( fit function : y@xmath30 ) .
top and bottom panel are relative to lvd - scint and lvd - gd scintillator respectively.,title="fig : " ] concerning the counter filled up with lvd - gd , doped at a gd concentration of 0.93
g / l , we found a mean neutron capture time of 27@xmath28s .
also in this case the measured value is in agreement with previous measurements and simulations @xcite . in the same figure ,
gray points represent the background level , obtained applying the same analysis to a dataset collected after removing the am - be source .
+ we developed a monte carlo simulation based on the geant4 toolkit @xcite implementing the detector description and the am - be source energy spectrum taken from @xcite .
+ geant4 has high precision transport models ( g4neutronhpelastic and g4neutronhpinelastic ) available for neutrons of energies up to 20 mev and for all materials .
the high precision models are based on the evaluated nuclear data files ( endf / b - vi ) @xcite , which contains tabulated cross - sections for elastic reactions and the different inelastic final states ( e.g. : n@xmath27 , np , nd , etc . ) .
the high - quality endf / b - vi data generally allows for accurate simulations of these low - energy neutrons @xcite .
+ the outcome of the simulation is shown in fig .
[ cap ] where the energy spectra of the de - excitation @xmath27-ray following neutron capture are compared with experimental ones .
energy spectrum of the neutron capture signals ( background subtracted ) ( in black ) , compared with simulation ( in red ) .
top and bottom panel are relative to lvd - scint and lvd - gd scintillator respectively.,title="fig : " ] energy spectrum of the neutron capture signals ( background subtracted ) ( in black ) , compared with simulation ( in red ) .
top and bottom panel are relative to lvd - scint and lvd - gd scintillator respectively.,title="fig : " ] the 2.22 mev gamma emitted after neutron capture on @xmath22h , clearly visible for both the detectors , has been used to determine the energy scale with an uncertainty of @xmath311.5@xmath5 .
+ the spectrum relative to the unloaded scintillator exhibits the peaks due to neutron captures on @xmath19c ( 4.95 mev ) , and @xmath24fe ( 7.65 mev ) , which are overwhelmed in the other detector by the presence of gadolinium .
+ the light output produced by the nuclear recoils was evaluated from the eq .
[ eqq ] , taking into account the contribution of both the ions @xmath22h and @xmath19c , even if the latter was rather negligible when compared to the former .
+ the stopping power @xmath32 in liquid scintillator , from which the quenching factor depends , was taken from the estar database @xcite for electrons , whereas for protons and heavier ions it was calculated with the srim code @xcite . + for a specific material , the eq .
[ eqq ] has only one free parameter , @xmath11 , the so - called birks factor , which has been determined by fitting , with the least - squares method , the simulated energy spectrum of nuclear recoils to the experimental one . in the fitting procedure
, the comparison between experimental data and simulation was performed by re - weighting monte carlo events for different choice of the @xmath11 parameter .
energy spectrum of the prompt signals after background subtraction ( in black ) , compared with the simulation ( in red ) .
the detection threshold is at about 0.8 mev@xmath26 .
the value of the birks factor which minimize the @xmath33 is @xmath34 for lvd - scint ( top panel ) and @xmath35 for lvd - gd ( bottom panel).,title="fig : " ] energy spectrum of the prompt signals after background subtraction ( in black ) , compared with the simulation ( in red ) .
the detection threshold is at about 0.8 mev@xmath26 .
the value of the birks factor which minimize the @xmath33 is @xmath34 for lvd - scint ( top panel ) and @xmath35 for lvd - gd ( bottom panel).,title="fig : " ] the parameter estimation and the goodness of fit are based on the @xmath36 test , modified for comparing weighted and unweighted events @xcite . + the result of the fitting procedure is shown in fig .
[ rec ] , where experimental and simulated date are represented with black and red line respectively .
the two spectra diverge at [email protected] mev@xmath26 because of the detection threshold .
the first peak at [email protected] mev@xmath26 is due to neutron elastic scattering , whereas the second peak is due to the 4.44 mev gamma from @xmath19c@xmath39 in addition to elastic scattering of lower energy neutrons ( e@xmath40 @xmath416.5 mev ) .
+ the probability to get by chance a @xmath36 higher than the observed is 34% for the russian scintillator and 17% for the italian one .
+ the minimum @xmath36 corresponds to a birks parameter value @xmath34 and @xmath35 for lvd - scint and lvd - gd respectively .
quenching factors for protons , alpha particles and carbon ions obtained from the eq .
[ eqq ] using the best fit parameter @xmath34 for lvd - scint ( solid lines ) and @xmath35 for lvd - gd ( dashed lines ) .
] by letting @xmath36 vary by one unit we found the standard error interval , which is @xmath42 in both cases .
in addition to the statistical error we estimated a systematic contribution of @xmath43 coming mostly from the uncertainty in the energy calibration . + according to the author of the work @xcite , the birks factor is independent of the particle type and
can then be used afterwords to calculate quenching factors for other particles and in other energy regions .
[ curve ] we show the resulting quenching curve for protons , alpha particles , and carbon ions .
two different detectors of the lvd experiment have been exposed to an am - be ( @xmath44,n ) source , with the purpose of measuring the light output response to nuclear recoils .
+ the delayed signal due to de - excitation @xmath27-ray following neutron capture has been used to tag the events and effectively reject the gamma background .
+ the analysis is based on comparison of the experimental nuclear recoil spectra with monte carlo simulation .
we found that the light output response is well described by the birks model and a birks factor kb = ( 0.0125 @xmath31 0.0004 ) @xmath45 and kb = ( 0.0140 @xmath31 0.0007 ) @xmath45 was obtained from the two scintillators lvd - scint and lvd - gd respectively .
+ quenching factors for ions in scintillator have then been calculated up to 100 mev .
+ this result can be of interest for fast and high - energy neutron spectroscopy performed with the lvd detector .
it can be directly applied to 73% of the whole lvd array ( russian scintillator ,
i.e. lvd - scint ) concerning the remaining amount of counters , filled up with italian scintillator , we have not yet made a direct measurement of the quenching factors .
nevertheless , we could probably consider the results coming from the italian gd - doped scintillator as a lower bound for the undoped one .
chadwick et al . , _
endf / b - vii.1 nuclear data for science and technology : cross sections , covariances , fission product yields and decay data _ , http://dx.doi.org/10.1016/j.nds.2011.11.002[_nuclear data sheets _ * 112 * , 12 , 2887 - 2996 ( 2011 ) ] z. kohley et al . , _ modeling interactions of intermediate - energy neutrons in a plastic scintillator array with geant4 _ , http://dx.doi.org/10.1016/j.nima.2012.04.060[_nucl .
_ * a682 * 59 - 65 ( 2012 ) ] j.w .
marsh et al . , _ high resolution measurements of neutron energy spectra from am - be and am - b neutron sources _ , http://dx.doi.org/10.1016/0168-9002(95)00613-3[_nucl .
. meth . _ * a366 * 340 - 348 ( 1995 ) ] g. bohm , g.zech , _ comparison of experimental data to monte carlo simulation
parameter estimation and goodness - of - fit testing with weighted events .
_ , http://dx.doi.org/10.1016/j.nima.2012.06.021[_nucl .
intrum . meth .
_ * a691 * 171 - 177 ( 2012 ) ]
|
the organic liquid scintillator used in the lvd experiment ( infn gran sasso national laboratory ) has been exposed to an am - be neutron source to measure the light response function for neutron energies in the region from about 4 to 11 mev . + a full monte carlo simulation , incorporating the detector response , is used to generate neutron scattering spectra which are matched to the observed ones to determine the quenching factors .
+ the obtained light output response is well described by the semi - empirical birks model .
+ the results , consistent with those obtained by other authors using similar hydrocarbonate scintillators , can be of interest for fast and high - energy neutron spectroscopy that could be performed with this detector .
|
the first discovered gravitational lens system , qso 0957 + 561 ( walsh , carswell , & weymann 1979 ) , has been the subject of a continuous and exhaustive monitoring in several bands since 1979 .
the special characteristics of this system made it very attractive for time delay determinations , and different values for this quantity were presented during the 1980s : @xmath8 days ( florentin - nielsen 1984 ) ; @xmath9 days ( schild & cholfin 1986 ) ; @xmath10 days ( gondhalekar et al . 1986 ) ; @xmath11 days ( lehr , hewitt , & roberts 1989 ) .
as can be seen , there was wide dispersion in the results obtained by different groups .
however , the monitoring campaigns carried out during the early 1990s led to quite an odd situation , as all the results concentrate around two different values for the time delay : @xmath12 days and @xmath13 days .
calculations leading to the first value were presented by vanderriest et al .
( 1989 , @xmath14 days ) ; schild ( 1990 , @xmath15 days ) ; and pelt et al .
( 1994 , @xmath16 days in the @xmath17 band and @xmath18 days in radio ) . on the other hand , a value close to 510 days was obtained by beskin & oknyanskij ( 1992 , @xmath19 days in the @xmath17 band and @xmath20 days in the @xmath4 band )
; roberts et al .
( 1991 , @xmath21 days ) ; and press , rybicki , & hewitt ( 1992a , @xmath22 days in the @xmath17 band ; 1992b , @xmath23 days in @xmath17+radio ) .
this situation abruptly changed when kundi et al .
( 1995 ) presented their observations in the @xmath3 and @xmath4 bands .
a sharp drop appeared in 1994 december and could be used to discern between the long " ( 510 days ) and the short " ( 420 days ) time delay , provided continuous monitoring of qso 0957 + 561 was carried out in the first half of 1996 .
this monitoring was performed , and the long time delay was rejected ( oscoz et al .
1996 ; kundi et al .
the controversy regarding the time delay seemed to be finally solved .
however , the results appearing in the literature since 1995 concentrate again around two values , 417 and 424 days .
these results are summarized as follows : @xmath24 days ( pelt et al . 1996 ) ; @xmath25 days ( kundi et al . 1997 ) ; @xmath26 days ( oscoz et al .
1997 ) ; @xmath27 ( pijpers 1997 ) ; @xmath28 days ( pelt et al . 1998 ) ; @xmath29 days ( serra - ricart et al . 1999 ) ; @xmath30 days ( colley & schild 2000 ) .
this difference is irrelevant in the hubble constant calculations , as the uncertainty introduced by the time delay is much lower than the uncertainty given by other factors ( e.g. , the main lens galaxy s velocity dispersion or the lens modeling ) .
however , the most accurate time delay should be used in the search for possible very rapid microlensing events in qso 0957 + 561 , and a week s difference in @xmath31 could lead to the detection of false events or failure to detect real ones . in this paper
we have compiled photometric data of qso 0957 + 561 from three different observing groups covering the period 198499 to obtain an estimate of the time delay by means of several statistical methods .
this includes new unpublished data corresponding to the last observational campaign ( 19989 ) at the iac80 telescope .
the data sets are presented in 2 , while the methods for obtaining the time delay appear in 3 .
a first check of the goodness of these methods applied to simulated data is calculated in 4 , and the best techniques are applied in 5 to real data .
finally , a discussion of our results appears in section 6 .
several monitoring campaigns of qso 0957 + 561 in different bands have been performed since 1979 . however , to obtain the time delay , only the observations obtained by groups from three different institutions will be considered here : princeton university ( hereafter pu ) , harvard - smithsonian center for astrophysics ( hereafter cfa ) , and the instituto de astrofsica de canarias ( hereafter iac ) .
images were obtained at the apache point observatory 3.5 m telescope in the @xmath3 and @xmath4 bands . qso 0957
+ 561 was monitored during several observational campaigns , although only data corresponding to the first two seasons have been published ( kundi et al .
1995 , 1997 ) : ( i ) from 1994 december to 1995 may ; and ( ii ) from late 1995 to mid 1996 .
the resulting data comprise 51 + 46 @xmath3-band points and 54 + 46 @xmath4-band points .
the light curves were calculated via aperture photometry , and have neither large error bars nor significant gaps .
their main characteristic is the presence of a sharp drop of about 0.1 mag in the @xmath32-component in late 1994 december , very useful for time - delay calculations .
this data set is the largest ever obtained for a gravitational lens system .
it consists of 1069 brightness measurements in the @xmath1 band , from late 1979 to mid 1996 .
the observations corresponding to the period 197989 were obtained at the whipple observatory 0.61 m telescope , while the remaining data were obtained with a 1.2 m telescope ( schild & thomson 1995 , and references therein ) .
the reduction procedure followed a basic aperture photometry scheme ( although a new automated photometry reduction code is now being applied by the authors , the results do not substantially differ from the old " photometry ) .
the error bars are not large , with the exception of the first five years .
the main problem with this data set is the scarcity of observations during the first 1800 days ( 81 brightness measurements ) .
moreover , those points coincide with the largest error bars .
so , we will consider the data from mid 1984 for time delay calculations .
lens monitoring was performed in four consecutive seasons ( 1996 february to june , 1996 october to 1997 july , 1997 october to 1998 july , and 1998 october to 1999 july ) using the ccd camera of the 82 cm iac-80 telescope ( iac80 hereafter ) , sited at the iac s teide observatory ( tenerife , canary islands , spain ) .
a thomson 1024@xmath331024 chip was used , offering a field of nearly 7@xmath345 .
standard @xmath35 broad - band filters were used for the observations , corresponding fairly closely to the landolt system ( landolt 1992 ) .
the final data set comprises 172 point in the @xmath0 band , 301 points in the @xmath1 band , and 112 points in the @xmath2 band .
accurate photometry was obtained by simultaneously fitting a stellar two - dimensional profile on each component by means of daophot software ( see details in serra - ricart et al .
a new , completely automatic iraf task has been developed demonstrating , using a sample of simulated data , that the proposed method can achieve high - precision photometry .
however , the errors bars obtained for iac80 data are slightly larger than those of pu and cfa data .
this could be explained by a decrease in chip sensibility due to the age of the ccd . in order to assess the reliability of our method using real data , simultaneous observations of qso 0957
+ 561 were undertaken on 1999 february 19 by using the iac80 and the 2.5 m nordic optical telescope ( not hereafter ) sited at the iac s roque de los muchachos observatory ( la palma , canary islands , spain ) .
the final reduced results are presented in figure 1 and table 1 ( photometric errors for comparison star h and d are also included ) .
the not light - curve errors ( a few millimagnitudes ) are much lower than the iac80 ones , and this difference could be explained in terms of the following : i ) the not has a larger aperture than the iac80 , and ii ) the not also has a better ccd chip .
however , the good agreement between the two curves demonstrates that our reduction method works with high degree of accuracy .
the photometric data are available at url http://www.iac.es/lent .
the large amount of data described in 2 adds new biases ( different telescopes , filters , reduction processes , and behavior ) to the inherent difficulty to analyse discrete , unevenly sampled temporary series .
these facts leaded us to employ several statistical methods to calculate the time delay to increase the robustness of the results thus obtained . as a first step , several techniques will be checked by using simulated data : the discrete correlation function , dispersion spectra , @xmath36 , @xmath36 modified , linear interpolation , and the @xmath37-transformed discrete correlation function . the dcf ( edelson & krolik 1988 ) is a technique valid for any physical quantity that is observed to vary in time . for two discrete data trains , @xmath38 and @xmath39 , the formula representing their dcf is @xmath40 averaging over the @xmath41 pairs for which @xmath42 , @xmath43 , @xmath44 , and @xmath45 being the bin semi - size , the measurement error associated with the data set @xmath46 , and the standard deviation , respectively . the maximum of the dcf is identified with the time delay .
the data model ( pelt et al . 1996 ) consists of two time series , @xmath47 , @xmath48 , and @xmath49 , @xmath50 , where @xmath51 represents the intrinsic variability of the quasar , @xmath52 accounts for the difference in magnitudes plus additional variability in time due to microlensing , and @xmath53 and @xmath54 are observational errors .
this two series are combined into one , @xmath55 , for every fixed combination [ @xmath56 , @xmath52 ] by taking all values of @xmath32 as they are and correcting the @xmath17 data by @xmath52 and shifting them by @xmath56 .
the dispersion spectra that will be used here are represented by @xmath58 where the @xmath59 are statistical weights of the combined light curves , and @xmath60 when @xmath61 and @xmath62 come from different curves ( @xmath32 or @xmath17 ) and 0 otherwise . from eq .
( 2 ) , we can consider two different approximations depending on the definition of @xmath63 : ( i ) @xmath64 if @xmath65 and 0 otherwise ; and ( ii ) @xmath66 if @xmath67 and 0 otherwise , @xmath6 being the maximum distance between two observations which can be considered as nearby . the minimum value of eq .
( 2 ) is assumed as the time delay .
the @xmath36 method ( serra - ricart et al . 1999 ) makes use of the similarity between the discrete autocorrelation function ( dac ) of the light curve of one of the components and the @xmath32@xmath17 discrete correlation function ( dcf ) . from the dac and dcf functions ,
one can define a function @xmath68 ^ 2\ ] ] for every fixed value @xmath69 ( days ) , where @xmath70 when both the dcf and dac are defined at @xmath71 and @xmath72 , respectively , and 0 otherwise . the most probable value for the time delay should correspond to the minimum of this function .
this modification of the @xmath36 method was suggested by schild ( 1999 , private communication ) .
it consists in comparing the dac and dcf curves by taking their ratio instead of by calculating their difference .
so , the final equation to obtain the time delay is @xmath73 ^ 2 \ , .\ ] ] the linear method is similar to that suggested by kundi et al .
one of the two light curves ( hereafter light curve 1 ) is selected , and the linear interpolation of data and their errors is considered as reference . the other light curve ( hereafter light curve 2 ) is then shifted in magnitude by just the difference between the means of both light curves .
after that , light curve 2 is shifted in time and chi - square per degree of freedom ( @xmath74 ) for each time delay is calculated .
the number of degrees of freedom is equal to the number of points of the light curve 2 in the overlapping interval minus 2 ( because we are fitting shifts in magnitude and time ) .
the time that minimizes @xmath74 is taken as a provisional time delay .
this procedure is followed by using as reference both the @xmath32- and @xmath17-component light curves , selecting then the time delay closest to 421 days ( the intermediate point between 417 and 425 days ) .
the uneven sampling of the light curves usually leads to a better time delay taking as a reference one of the two light curves .
the zdcf ( alexander 1997 ) is a new method for estimating the cross - correlation function ( ccf ) of sparse , unevenly sampled light curves .
fisher s @xmath37-transform of the linear correlation coefficient , @xmath4 , is used to estimate the confidence level of a measured correlation .
this technique attempts to correct the biases that affect the original dcf by using equal - population binning .
the zdcf involves three steps : \(i ) all possible pairs of observations , @xmath75 , are sorted according to their time - lag , @xmath76 , and binned into equal population bins of at least 11 pairs .
multiple occurrences of the same point in a bin are discarded so that each point appears only once per bin .
\(ii ) each bin is assigned its mean time - lag and the intervals above and below the mean that contain 1@xmath77 of the points each .
\(iii ) the correlation coefficients of the bins are calculated and @xmath37-transformed .
the error is calculated in @xmath37-space and transformed back to @xmath4-space .
the time - lag corresponding to the maximum value of the zdf is assumed as the time delay between both components .
the application of the statistical methods described in 3 to simulated data sets can serve to check the validity of their results under different conditions , always with discrete and irregularly sampled data sets .
the six selected data sets are quite similar to those presented in serra - ricart et al .
( 1999 ) , where several sets of simulated photometric data with similar irregularity in the observations ( time distribution of the data ) , magnitudes , and error bars ( i.e. , as large as or even larger than those of pu , cfa , and iac ; the worst situation is selected ) to that of the iac observations were created . first , a set of dates , @xmath78 , between 1800 and 2000 ( tjd = jd2449000 ) approximately , was generated with a pseudo - random separation , taken from a uniform distribution between 0 and 5 days .
these data were alternatively separated in two time series , corresponding to @xmath32- and @xmath17-component light curves , this last curve being shifted by 420 days to simulate the existence of a time delay .
a first magnitude was calculated for each date with the relationship @xmath79 [ see below for the different shapes of @xmath80 .
the probability of measuring a value @xmath81 for each @xmath78 is proportional to @xmath82}$ ] , and hence characterized by @xmath83 , or , equivalently , the variable @xmath84 is distributed as @xmath85}$ ] .
a @xmath83 taking pseudo - random values between 0.01 and 0.03 was generated for each @xmath78 . from here
the quantities @xmath86 , pseudo - random numbers obtained from a normal gaussian distribution with zero mean and standard deviation , @xmath83 , were calculated , allowing them to adopt positive or negative values .
finally , the magnitude was generated from the equation @xmath87 , with an error bar of @xmath83 .
the @xmath32 component was made brighter by adding 0.1 to the magnitudes of the @xmath17 component .
the first selected function was : @xmath88 this function represents light curves in which a sharp event similar to that reported by kundi et al .
( 1995 ) appears .
the second function is @xmath89 in this case , the light curves present several maxima and minima , although none of them is clearly remarkable .
an additional function , consistent with the actual variability of q0957 + 561 , was created ( the iac observational data from 9798 seasons , were selected as reference ) .
the light curves were then fitted by the function @xmath90 @xmath91 being the mean of the tjd in the selected range and @xmath92 its standard deviation .
the resulting simulated data show a lower variability to that obtained from @xmath93 and @xmath94 . besides the comparison between the @xmath32- and @xmath17-component for the three data sets , an additional test was performed by removing some data of the @xmath32-component . by doing this ,
we want to simulate the data sets corresponding to certain periods in which no valid data could be obtained during several days ( bad weather , problems with the telescope , etc . ) .
the time delay corresponding to the different statistical techniques were first calculated by allowing their free parameters to vary .
the best results were obtained with a @xmath95 days for the dcf , @xmath96 days for @xmath97 and @xmath98 , and @xmath99 days for @xmath36 and @xmath100 . in the dcf , @xmath36 and @xmath100 cases ,
the results were quite similar for the parameters varying between 4 and 12 days .
however , slightly better values were obtained for @xmath95 days in the dcf case , and for @xmath101 days in the @xmath36 and @xmath100 cases , so these quantities were selected .
the objective in this paper is to analyze the different methods with several data sets by using the same conditions .
an analysis of the best method and/or parameters to be used with a particular data set will be presented in a future paper .
the uncertainties in the time - delay estimates were computed by generating 1000 bootstrap samples and applying the statistical methods in each case .
some interesting consequences can be derived from the results in table 2 . as expected ,
the best estimate of the time delay is always obtained for the function @xmath93 , i.e. , when a sharp drop similar to the one appearing in 2.1 is present in the light curves .
all the methods give a good value for the delay . on the other hand , in the @xmath94 and @xmath102 cases
the error bar are too large for @xmath100 , @xmath97 , and @xmath98 when compared with the results obtained with the other methods .
so only the dcf , @xmath36 , zdcf , and li techniques will be applied to real data calculations , similar in most of the cases to the functions @xmath94 and @xmath102 .
notice that the true time delay is always within the error bar , even when a large gap is present in the light curves .
prior to applying the different methods to calculate the time delay from real data , some considerations have to be taken into account .
first , data should be checked to eliminate inconsistent measurements .
this modification of the raw data , based on a suggestion by falco ( 1997 , private communication ) , takes account of the possible existence of strong and simultaneous ( not time - shifted ) variations of some data point in both components .
the inclusion of such strange " brightness records in the final data sets probably originated from failures in the ccd or from bad weather conditions , creates artificial peaks or valleys in the light curve of one of the components .
these maxima / minima have no importance when a sharp change in the behavior of the quasar is being analyzed , but can lead to a completely wrong time delay estimate when dealing with a smoother season . to avoid these abnormal observations
we have removed the points with a simultaneous difference in magnitude in both components as compared with the previous and following records larger than 2.5 times their error bar .
this was done by considering only those points with a difference in the observation dates of less than 10 days .
the resulting data sets will be named bad point free ( bpf ) .
the 301 points of the iac data set in the @xmath1 band are reduced to 289 with this restriction , while less than 60 of the 1069 cfa observations have to be removed .
finally , no brightness measurement of the pu data seems to be wrong .
however , the bpf restriction could be applied neither to the iac @xmath2- and @xmath0-band data nor to the cfa 8990 and 9192 @xmath1-band data due to their low number of points , with a minimum distance between neighboring points more than 10 days in most of the cases .
another drawback when dealing with real data is the impossibility of observing the system during certain months of the year and the consequent lack of suitable edges .
once it is stated , see 1 , that the rough value of the time delay is around 420 days , the comparison between the @xmath32 and @xmath17 components should be made by previously selecting a clean " data set ( cd , see serra - ricart et al .
1999 ) , i.e. , homogeneous monitoring of both images during two active and clear ( free from large gaps ) epochs separated by @xmath103 days . finally , both types of corrections , bpf and cd , were combined to obtain the definitive data sets used in this paper . to see the importance of applying the cd
bpf corrections , we mention two extreme examples : 1 ) @xmath104 days with the raw iac data corresponding to the 967 season , while @xmath105 days with the cd bpf approximation ; and ( 2 ) @xmath106 days with the original 934 cfa data and @xmath107 days with the cd bpf approximation . to summarize , monte carlo calculations were applied to the four techniques ( dcf , @xmath36 , zdcf , and li ) with the cd and the cd - bpf restrictions .
the pu , iac , and cfa data sets were divided into observational seasons , leading to the 23 different time - delay estimates per method appearing in tables 3 to 6 .
the results obtained from the pu data in both the @xmath3 and @xmath4 filter are presented in table 3 .
notice that two of the methods ( dcf and li ) employed here are also used by kundi et al .
our results are quite similar to those obtained by these authors , and , moreover , they are always into their error .
the small differences come from the selection of clean data sets .
tables 4 and 5 offer the time delay for the iac data in the @xmath1 , and in the @xmath2 and @xmath0 bands , respectively .
finally , table 6 was obtained from the cfa @xmath1 band data .
the analysis of these delays are done by considering the cd bpf quantities , except in those cases in which only the cd results could be obtained .
the quantities obtained for the cfa 923 season with the dcf and @xmath36 methods can be discarded , as they appear to be clearly inconsistent ( 394 @xmath108 1 and 403 @xmath108 1 days , respectively ) .
a first step in discussing the value of @xmath5 consists in computing , for each of the techniques ( dcf , @xmath36 , zdcf , and li ) , the number of occurrences of each value of the time delay .
these quantities , obtained from tables 3 to 6 , are depicted by the black lines in figure 2 : dcf ( top panel , left ) ; @xmath36 ( top panel , right ) ; zdcf ( bottom panel , left ) ; and li ( bottom panel , right ) .
( the different values of @xmath5 have been grouped into two - day bins ) .
as can be seen , two remarkable characteristics can be deduced from figure 2 : ( i ) there is a small dispersion in the delays , as most of them are in the interval 415428 days ; and ( ii ) the centroids of the histograms , given by the average of the time delays derived in tables 3 to 6 , are always in the interval 420424 days .
these last quantities are represented in figure 2 ( open circles ) together with their uncertainty ( r.m.s . , see table 7 and discussion below ) .
note that the largest peak of each histogram coincides with this average value , except for the zdcf technique . in the histogram corresponding to @xmath36
, the maximum corresponds to the values 421424 , while the average is 421.8 @xmath108 1.3 days .
the dcf panel does not lead to a clear time delay , with two maxima in the intervals 417420 and 423424 days .
the average here is given by 423.3 @xmath108 1.4 days .
in the case of li , the peak is placed at @xmath5 = 423424 days , and the average is 424.3 @xmath108 1.2 days . finally , the zdcf method has a maximum around 423424 days .
however , the average is 420.6 @xmath108 1.1 days , which is slightly different .
the red histograms in figure 2 represent the total number of occurrences obtained by adding the results from the four techniques .
its center is again over 420 days , giving a maximum of 423424 days . to complement these calculations , which have been done by fixing the method and computing the probability of appearance of each delay
, we can now represent the number of times each value appears for each data set , independently of the method employed .
four different data sets have been selected : ( i ) pu @xmath4 and @xmath3 filters , figure 3a ; ( ii ) iac @xmath1 data , figure 3b ; ( iii ) iac @xmath0 and @xmath2 records , figure 3c ; and ( iv ) cfa @xmath1 data , figure 3d . in this case , the centroid of the distributions , represented by open circles in figure 3 , is again over 420 days : 421.4 @xmath108 1.1 days , 423.7 @xmath108 1.3 days , 423.7 @xmath108 1.2 days , and 421.8 @xmath108 1.0 , for pu @xmath4 and @xmath3 , iac @xmath1 , iac @xmath2 and @xmath0 , and cfa @xmath1 , respectively .
a first positive consequence of the pu results is their extremely short dispersion indicating the goodness of the data and the presence of the sharp event .
the maximum here is placed between 419 and 424 days .
however , the clearest peak appears from the iac @xmath1 values around 423424 days .
the iac @xmath2 and @xmath0 panel shows more dispersion , probably due to the higher error bars of the light curves .
there is not a unique maximum here , as two peaks appear around 423424 and 427428 days .
the highest dispersion in the results can be seen in figure 3d , corresponding to the cfa data .
the maximum would in this case be in the interval 417422 days .
a remarkable result here is that the average coincides with the maxima in all the panels .
once again , the red histogram gives the total number of occurrences .
the combination of the results derived from figures 2 and 3 ( centroids and maxima of the distributions ) support a @xmath5 in the range of 420424 days , although a time delay of around 417 days can not be totally discarded .
it is important to remark that the results are the same independently of the technique employed or of the data set selected .
the time delay between a and b components of q0957 + 561 does not depend on the filter ( as it is an acromatic effect ) and/or the time ( different campaigns ) , so it should be possible to merge the different sample results .
a very important point is to assess the statistical reliability of the different delay calculation methods in order to estimate final delay errors .
several statistics were calculated , mean delay ( @xmath109 ) , mean error ( @xmath110 , with @xmath111 individual errors ) , and dispersion ( @xmath112^{1/2}$ ] ) .
if the error estimate is correct , then @xmath113 , and the final error for the time delay will be given by the r.m.s .
( @xmath114^{1/2}$ ] ) ( see eadie et al . 1971 ) .
tables 7 to 10 ( see below ) show the final results for the four methods . in all cases , within the statistical errors , good agreement is found between the mean error and the dispersion . when this procedure is applied to the different time delays given by each technique , one obtains the results in the first four rows of table 7 , where the mean values of the time delay and the uncertainties ( r.m.s . )
are shown . according to these calculations
, the definitive time delay would be in the interval 420(zdcf)424(li ) days , with an uncertainty below 1.4 days .
this interval coincides with that derived from the analysis of figures 2 and 3 . when this calculation is done with only the @xmath1 band results ( table 8) , the results are almost identical .
the different analyses performed until now have been done considering all the results obtained in tables 3 to 6 .
however , the error bars of some of these values exclude the interval 420424 days , where , as shown before , there is the highest probability of finding the right @xmath5 .
the mean delays and uncertainties obtained when these values are removed ( first four rows of table 9 ) are very similar to those of table 7 .
@xmath5 is now restricted to the interval 420.8423.2 days , with an uncertainty below 1.3 days .
once again , the results derived from the @xmath1-filter data are almost the same ( table 10 )
. the validity of these statistical results leaded us to repeat the calculations of tables 7 to 10 , but this time having into account all the delays , i.e. , without considering the method ( dcf , @xmath36 , zdcf , or li ) .
this allows us to have four different time delays per year in most of the occasions , and so , a larger amount of data for the statistical analysis .
the results appear in the last row of tables 7 to 10 , although only the values of tables 7 and 9 , @xmath115 days and @xmath116 days , respectively , will be considered , as they were obtained from a larger amount of data . adopting a conservative point of view
, we will select the quantity with the higher uncertainty , @xmath7 days , as the final time delay .
different treatments of the time delays obtained in 5 have been performed .
the analysis of these results always points to a time delay in the interval 420424 days .
none of these methods clearly favors the values 416418 days as the right time delay .
moreover , the averages of the quantities of tables 7 and 9 give values around 422 days , coinciding with the maxima obtained in figures 2 and 3 . assuming this value as the time delay between the @xmath32 and @xmath17 components of q0957 + 561 ,
let us check which of the results given in tables 3 to 6 include 422 days in their the error bars . figure 4 offers the number of times , written as a percentage of the total number of time delays obtained for each method , that each value of @xmath5 is included in these error bars .
this probability is the same , 86% , for dcf , @xmath36 and zdcf , and 74% for li . in this case , the dcf and @xmath36 methods show the highest probability for a @xmath5 of 421422 days , while the peak in the li curve is placed at 423 days .
a wider maximum is obtained in the case of zdcf , with the same probability between 418 and 421 days .
as can be seen , not all the seasons in the different data sets are fully appropriate for calculating the time delay . however , our intention was to analyze all the data in the three different data sets with four different methods , and finally to restrict the uncertainty in the calculation of the time delay .
the statistical treatment of all the results confirms a time delay of @xmath117 days .
we are especially grateful to e. e. falco for advising us on the possible presence of strange points in our data sets , to r. schild for helpful comments on the statistical methods , and to t. alexander for providing us with his programs to calculate the zdcf method and his help in understanding it .
this work was supported by the p688 project of the instituto de astrofsica de canarias ( iac ) , universidad de cantabria funds , and dgesic ( spain ) grant pb97 - 0220-c02 .
alexander , t. 1997 , astronomical time series , maoz , d. , sternberg , a. , and leibowitz , e. m. ( eds ) , dordrecht : kluwer , 163 beskin , g. m. , and oknyanskij , v. l. 1992 , lecture notes in physics 406 , gravitational lenses , kayzer , r. , schramm , t. , and refsdal s. ( eds ) , springer verlag : heidelberg , 67 colley , w. n. , and schild , r. e. 2000 , , 540 , 104 eadie , w. t. , drijard , d. , james , f. e. , and roos , m. 1971 , statistical methods in experimental physics ( amsterdam : north - holland ) edelson , r. a. , and krolik , j. h. 1988 , , 333 , 646 florentin - nielsen , r. 1984 , a&a , 138 , 119 gondhalekar , p. m. , wilson , r. , dupree , a. k. , and burke , b.f .
1986 , london conference proceedings , new insights in astrophysics : eight years of astronomy with iue , sp-263 , 715 kundi , t. , colley , w. n. , gott iii , j. r. , malhotra , s. , pen , u. , rhoads , j. e. , stanek , k. z. , turner , e. l. , and wambsganss , j. 1995 , , 455 , l5 kundi , t. , turner , e. l. , colley , w. n. , gott iii , j. r. , rhoads , j. e. , wang , y. , bergeron , l. e. , gloria , k. a. , long , d. c , malhotra , s. , and wambsganss , j. 1997 , , 482 , 75 landolt , a. u. 1992 , , 104 , 340 lehr , j. , hewitt , j. n. , and roberts , d. h. 1989 , gravitational lenses , moran , j. m. , hewitt , j. n. , and lo , k. y. ( eds ) , gravitational lenses , dordrecht : reidel , 84 oscoz , a. , serra - ricart , m. , goicoechea , l. j. , buitrago j. , and mediavilla e. 1996 , , 470 , l19 oscoz , a. , mediavilla , e. , goicoechea , l. j. , serra - ricart , m. , and buitrago , j. 1997 , , 479 , l89 pelt , j. , hoff , w. , kayser , r. , refsdal , s. , and schramm , t. 1994 , a&a , 256 , 775 pelt , j. , kayser , r. , refsdal , s. , and schramm , t. 1996 , a&a , 305 , 97 pelt , j. , schild , r. , refsdal , s. , and stabell , r. 1998 , a&a , 336 , 829 pijpers , f. p. 1997
, , 289 , 933 press , w. h. , rybicki , g. b. , and hewitt , j. n. 1992a , , 385 , 404 press , w. h. , rybicki , g. b. , and hewitt , j. n. 1992b , , 385 , 416 roberts , d. h. , lehr , j. , hewitt , j. n. , and burke , b. f. 1991 , nature , 352 , 43 schild , r. 1990 , , 100 , 1771 schild , r. e. , and cholfin , b. 1986 , , 300 , 209 schild , r. , and thomson , d. j. 1995 , , 109 , 1970 serra - ricart , m. , oscoz a. , sanchs , t. , mediavilla , e. , goicoechea , l. j. , licandro , j. , alcalde , d. , and gil - merino , r. 1999 , , 526 , 40 vanderriest , c. , schneider , j. , herpe , g. , chevreton , m. , moles , m. , and wlrick , f. 1989 , a&a , 215 , 1 walsh , d. , carswell , r. f. , and weymann , r. j. 1979 , nature , 279 , 381 lcccccc method&@xmath93&@xmath93 ( gap in @xmath32)&@xmath94&@xmath94 ( gap in @xmath32)&@xmath102&@xmath102 ( gap in @xmath32 ) + dcf ( @xmath118)&421@xmath1082&421@xmath1084&419@xmath1081&420@xmath1082&415@xmath1089&412@xmath1089 + @xmath36 ( @xmath119)&422@xmath1082&421@xmath1082&419@xmath1081&420@xmath1081&420@xmath1085&417@xmath1085 + zdcf & 420@xmath1083&420@xmath1083&421@xmath1081&420@xmath1087&418@xmath10814&416@xmath10813 + li&421@xmath1082&421@xmath1082&420@xmath1081&421@xmath1081&419@xmath10812&418@xmath10813 + @xmath100 ( @xmath99 ) & 422@xmath1089&422@xmath10813&425@xmath10830&416@xmath10834&403@xmath10834&413@xmath10829 + @xmath97 ( @xmath120)&421@xmath1083&421@xmath1084&421@xmath10812&416@xmath10820&422@xmath10827&414@xmath10825 + @xmath98 ( @xmath120)&421@xmath1082&421@xmath1083&423@xmath10815&415@xmath10826&420@xmath10826&412@xmath10821 + ccccc & & dcf&&422@xmath1081 + _ g _ & @xmath32 ( 945 ) ; @xmath17 ( 956 ) & @xmath36 & 42 ; 39 & 423@xmath1081 + & & zdcf&&416@xmath1082 + & & li&&419@xmath1085 + & & dcf&&426@xmath1085 + _ r _ & @xmath32 ( 945 ) ; @xmath17 ( 956 ) & @xmath36 & 41 ; 41 & 423@xmath1082 + & & zdcf&&420@xmath1088 + & & li&&422@xmath1083 + cccccc & dcf&&428@xmath10812&&425@xmath10811 + @xmath32 ( 968 ) ; @xmath17 ( 979)&@xmath36&182 ; 220&421@xmath1085&173 ; 212&421@xmath1084 + & zdcf&&418@xmath10813&&426@xmath10813 + & li&&421@xmath10816&&424@xmath10817 + & dcf&&432@xmath1088&&436@xmath1088 + @xmath32 ( 96 ) ; @xmath17 ( 967)&@xmath36&31 ; 36&425@xmath1088&28 ; 33&428@xmath1089 + & zdcf&&422@xmath10813&&424@xmath10814 + & li&&424@xmath1087&&424@xmath1087 + & dcf&&436@xmath10810&&424@xmath10813 + @xmath32 ( 967 ) ; @xmath17 ( 978)&@xmath36&46 ; 86&431@xmath1084&45 ; 84&425@xmath1084 + & zdcf&&429@xmath10813&&424@xmath10812 + & li&&415@xmath10816&&426@xmath10816 + & dcf&&414@xmath10812&&416@xmath10812 + @xmath32 ( 978 ) ; @xmath17 ( 989)&@xmath36&79 ; 62&414@xmath10810&78 ; 61&413@xmath1088 + & zdcf&&420@xmath10812&&420@xmath10813 + & li&&427@xmath10813&&423@xmath10813 + ccccc & & dcf&&415@xmath10813 + _ i_&@xmath32 ( 968 ) ; @xmath17 ( 979)&@xmath36&65 ; 87&423@xmath10812 + & & zdcf&&417@xmath10814 + & & li&&419@xmath1089 + & & dcf&&424@xmath10816 + _ i_&@xmath32 ( 96 ) ; @xmath17 ( 967)&@xmath36&19 ; 18&424@xmath10816 + & & zdcf & & + & & li&&430@xmath1083 + & & dcf&&419@xmath1089 + _ i_&@xmath32 ( 967 ) ; @xmath17 ( 978)&@xmath36&18 ; 31&422@xmath1087 + & & zdcf & & + & & li&&427@xmath1087 + & & dcf&&414@xmath10815 + _ i_&@xmath32 ( 978 ) ; @xmath17 ( 989)&@xmath36&28 ; 39&417@xmath10818 + & & zdcf&&421@xmath10814 + & & li&&427@xmath10811 + & & dcf&&432@xmath10811 + _ v_&@xmath32 ( 968 ) ; @xmath17 ( 979)&@xmath36&68 ; 149&429@xmath10811 + & & zdcf&&418@xmath1089 + & & li&&428@xmath10814 + & & dcf&&433@xmath1087 + _ v_&@xmath32 ( 978 ) ; @xmath17 ( 989)&@xmath36&29 ; 84&432@xmath1086 + & & zdcf&&428@xmath1089 + & & li&&423@xmath1083 + cccccc & dcf&&420@xmath1086&&419@xmath1084 + @xmath32 ( 845 ) ; @xmath17 ( 856)&@xmath36&40 ; 49&420@xmath1085 & 39 ; 47&421@xmath1086 + & zdcf&&429@xmath1087 & & 428@xmath1087 + & li&&432@xmath1085 & & 431@xmath1086 + & dcf&&419@xmath10810&&417@xmath10812 + @xmath32 ( 856 ) ; @xmath17 ( 867)&@xmath36&49 ; 74&410@xmath10813 & 46 ; 72&407@xmath10815 + & zdcf&&420@xmath1089 & & 417@xmath10812 + & li&&409@xmath10814 & & 431@xmath1088 + & dcf&&431@xmath10812&&428@xmath10813 + @xmath32 ( 867 ) ; @xmath17 ( 878)&@xmath36&53 ; 58&434@xmath10810 & 52 ; 58&418@xmath10817 + & zdcf&&429@xmath10815 & & 430@xmath10812 + & li&&424@xmath1089 & & 420@xmath1085 + & dcf&&425@xmath10812 & & 428@xmath10813 + @xmath32 ( 878 ) ; @xmath17 ( 889)&@xmath36&60 ; 53&423@xmath10817&58 ; 53&422@xmath10818 + & zdcf&&419@xmath10817 & & 411@xmath10813 + & li&&421@xmath10811 & & 422@xmath10812 + & dcf&&417@xmath10821 & & + @xmath32 ( 889 ) ; @xmath17 ( 8990)&@xmath36&23 ; 19&420@xmath10810 & & + & zdcf&&412@xmath1088 & & + & li&&421@xmath1082 & & + & dcf&&440@xmath1085 & & 424@xmath10819 + @xmath32 ( 8990 ) ; @xmath17 ( 901)&@xmath36&40 ; 38&407@xmath1087&40 ; 34&420@xmath1083 + & zdcf&&417@xmath1086 & & 420@xmath1087 + & li&&410@xmath1086 & & 411@xmath1083 + & dcf&&435@xmath1088 & & + @xmath32 ( 901 ) ; @xmath17 ( 912)&@xmath36&15 ; 29&433@xmath10818 & & + & zdcf&&423@xmath10819 & & + & li&&426@xmath1086 & & + & dcf & & 394@xmath1080&&394@xmath1081 + @xmath32 ( 912 ) ; @xmath17 ( 923)&@xmath36&14 ; 72&402@xmath1081&14 ; 67&403@xmath1081 + & zdcf&&423@xmath10812 & & 416@xmath10813 + & li&&436@xmath1089 & & 437@xmath1088 + & dcf&&411@xmath1085 & & 419@xmath1086 + @xmath32 ( 923 ) ; @xmath17 ( 934)&@xmath36&70 ; 98&423@xmath1082&63 ; 93&423@xmath1082 + & zdcf&&411@xmath10811 & & 424@xmath10815 + & li&&423@xmath10813 & & 433@xmath1083 + & dcf&&422@xmath1084 & & 422@xmath1084 + @xmath32 ( 934 ) ; @xmath17 ( 945)&@xmath36&83 ; 111&421@xmath1082&78 ; 95&421@xmath1082 + & zdcf&&422@xmath1086 & & 422@xmath1087 + & li&&421@xmath1084 & & 418@xmath1086 + & dcf&&415@xmath1086 & & 418@xmath1087 + @xmath32 ( 945 ) ; @xmath17 ( 956)&@xmath36&101 ; 66&411@xmath1082&87 ; 57&415@xmath1083 + & zdcf&&416@xmath1082 & & 416@xmath1085 + & li&&421@xmath10812 & & 418@xmath1085 +
|
photometric optical data of qso 0957 + 561 covering the period 198499 are analyzed to discern between the two values of the time delay ( 417 and 424 days ) mostly accepted in the recent literature .
the observations , performed by groups from three different institutions princeton university , harvard - smithsonian center for astrophysics , and instituto de astrofsica de canarias and including new unpublished 19989 data from the iac80 telescope , were obtained in five filters ( @xmath0 , @xmath1 , @xmath2 , @xmath3 , and @xmath4 ) .
the different light curves have been divided into observational seasons and two restriction have been applied to better calculate the time delay : ( i ) points with a strange photometric behavior have been removed ; and ( ii ) data sets without large gaps have been selected .
simulated data were generated to test several numerical methods intended to compute the time delay ( @xmath5 ) .
the methods giving the best results the discrete correlation function , @xmath6-square , z - transformed discrete correlation function , and linear interpolation were then applied to real data .
a first analysis of the 23 different time delays derived from each technique shows that @xmath5 must be into the interval 420424 days . from our statistical study ,
a most probable value of @xmath7 days is inferred .
|
cosmic acceleration can arise either from an unknown component of dark energy with negative pressure or a modification to gravity that only appears at cosmological scales and densities .
additional terms in the einstein - hilbert action that are non - linear functions @xmath0 of the ricci scalar @xmath1 have long been known to cause acceleration @xcite and have been the subject of much recent interest @xcite . the main challenge for @xmath0 models as a complete description of gravity lies with the extremely tight constraints on such modifications placed by solar system and local tests of general relativity .
chiba @xcite showed that the fundamental problem is that @xmath0 models generically introduce a light scalar degree of freedom with a long compton wavelength at cosmological densities @xcite .
this problem can be mitigated by the so - called chameleon mechanism @xcite , where the local compton wavelength can decrease in high density environments @xcite .
the cosmological compton wavelength can then be a scale of cosmological interest .
nevertheless , it must typically be less than a few tens of mpc if the galactic gravitational potential is not substantially deeper than implied by local rotation curve measurements @xcite . in this _
paper _ , we take the perspective that @xmath0 models are effective theories that are valid at cosmological densities and scales and are not necessarily predictive at the high densities and small scales of local tests .
hence we explore models with compton wavelengths out to the horizon size and seek to constrain them from cosmological observables alone . at the very least ,
this exploration yields a cosmological test of general relativity that is independent of local constraints .
although @xmath0 models can change the expansion history during the acceleration epoch , there in principle always exists a dark energy model that provides the same history .
the unique and strongest signatures of @xmath0 modifications are on cosmological structure formation @xcite and can impact observables down to compton wavelengths of a few mpc @xcite .
we consider constraints arising from the cmb angular power spectrum measured by wmap @xcite , the linear matter power spectrum inferred from the sloan digital sky survey ( sdss ) luminous red galaxy ( lrg ) sample @xcite , the distance measures by the supernovae legacy survey ( snls ) @xcite , and the cross - correlation between the cmb and large scale structure as measured by wmap and a range of galaxy and quasar surveys @xcite .
the outline of the paper is as follows . in [ sec : methodology ] , we review the @xmath0 model and discuss our calculation and analysis methods . in [ sec : constraints ] we present the results of joint cosmological constraints on these models .
we discuss these results in [ sec : discussion ] .
in @xmath0 models of gravity , the einstein - hilbert action is supplemented by a term that is non - linear in the ricci scalar @xmath1 s = d^4x , where @xmath2 is the matter lagrangian .
the modified einstein and friedmann equations result from varying the action with respect to the metric .
given the freedom to choose a functional form for @xmath0 , any desired expansion history can be replicated @xcite . in particular
, one can choose the @xmath3cdm expansion history which is known to satisfy distance constraints from high redshift supernovae , baryon acoustic oscillations and the cmb .
even given the degeneracy with dark energy in the expansion history , @xmath0 models have distinguishable effects on the formation of structure .
the promotion of the ricci scalar to a dynamical degree of freedom modifies the force law between particles .
this modified force is mediated by a new scalar degree of freedom @xmath4 , which has a squared compton wavelength proportional to @xmath5 . below the compton wavelength scale gravity
becomes a scalar - tensor theory , leading to an enhancement in the growth of cosmological density perturbations and a corresponding suppression in the decay of gravitational potentials . for cosmological tests ,
it is convenient to express the squared compton wavelength in the background in units of the hubble length squared @xcite @xmath6 where @xmath7 is the scale factor and the ricci scalar is evaluated at the background density .
we will specialize our consideration to @xmath0 models that exactly reproduce the @xmath3cdm expansion history to test whether the unique signatures of @xmath0 gravity are seen in current cosmological data sets . under this assumption
, the additional degree of freedom in @xmath0 gravity is parameterized by the value of @xmath8 today , @xmath9 .
stability requires the mass squared of the scalar to be positive ( _ i.e. _ a prior of @xmath10 ) @xcite .
note that in the limit that @xmath11 , the phenomenology of @xmath3cdm is recovered in structure formation tests as well as expansion history tests .
more generally , the control parameter is the average compton wavelength through the acceleration epoch when gravity is modified .
the fundamental observables of our @xmath0 model are the same as in @xmath3cdm .
these include the redshift - distance relation , the cmb angular power spectrum @xmath12 , galaxy power spectra @xmath13 , and the angular correlation between galaxies and the cmb @xmath14 .
cosmic shear , dark matter halo profiles and masses from weak lensing as well as the cluster abundance are also potential observables but their utilization requires cosmological simulations that are beyond the scope of this work .
we modified the boltzmann code camb @xcite to calculate these observables in @xmath0 gravity .
in the camb code , the density perturbations are computed in the synchronous gauge .
we evolve the usual boltzmann code up to @xmath15 when deviations introduced by @xmath0 are still small . at this epoch
, we transform the matter perturbations from synchronous gauge to gauge invariant variables , and feed them as initial conditions to the linear perturbation equations for the density fluctuation and cmb sources ( see @xcite eqs .
28 - 35 ) .
once the initial conditions are specified , the power spectrum observables are computed from a separate @xmath0 routine that bypasses the usual camb code without significantly increasing the computational time .
as we shall discuss , the main modification made by our @xmath0 models on the cmb is a change in the evolution of gravitational potentials during the acceleration epoch .
the cmb power spectrum is modified at low multipoles through the so called integrated sachs - wolfe " ( isw ) effect , and the temperature field is correlated with tracers of gravitational potentials such as galaxies .
this correlation is strong if the redshift of the galaxies is matched to the epoch at which the gravitational potentials evolve .
following @xcite , we model the angular correlation between a set of galaxy surveys indexed by @xmath16 and the cmb , assuming a broad band selection function for the galaxies , n_i(z)=e^-(z / z_i)^1.5 , where the @xmath17 are chosen to match the median redshift in the surveys . while crude , the uncertainties introduced by this choice of selection function are smaller than the difference in model predictions that will be considered . the galaxy bias is assumed to be constant in each redshift bin .
we calculate the cross power spectrum between galaxy number density and cmb temperature following @xcite ( eqs .
55 - 60 ) and transform it to the angular power correlation function @xmath18 .
we use a markov chain monte carlo ( mcmc ) technique @xcite to evaluate the likelihood function of model parameters .
the mcmc is used to simulate observations from the posterior distribution @xmath19 , of a set of parameters @xmath20 given event @xmath21 , obtained via bayes theorem , @xmath22 where @xmath23 is the likelihood of event @xmath21 given the model parameters @xmath20 and @xmath24 is the prior probability density .
the mcmc generates random draws ( i.e. simulations ) from the posterior distribution that are a `` fair '' sample of the likelihood surface . from this sample
, we can estimate all of the quantities of interest about the posterior distribution ( mean , variance , confidence levels ) .
a properly derived and implemented mcmc draws from the joint posterior density @xmath25 once it has converged to the stationary distribution .
we use 16 chains and a conservative gelman - rubin convergence criterion @xcite to determine when the chains have converged to the stationary distribution . for our application
, @xmath20 denotes a set of cosmological parameters .
we then use a modified version of the cosmomc code @xcite to determine constraints placed on this parameter space by wmap , sdss lrg galaxy power spectrum , and supernova distance measures . for the lrg data ,
we only use the first 14 @xmath26-bins in our fits reflecting a conservative cut for linearity @xmath27/mpc . for the supernovae constraint , we take the supernovae legacy survey data set and its analysis remains unaffected by our @xmath0 modification .
likewise , its impact is mainly to help determine expansion history parameters . in our analysis
, we take the parameter set @xmath28 , @xmath29 , @xmath30 , where @xmath31 is the angular size of the acoustic horizon , and @xmath32 is the power in the primordial curvature perturbation at @xmath33 .
the universe is assumed to be to be spatially flat .
recall also that we fix the expansion history to be the same as a @xmath3cdm model .
hence @xmath34 is also a proxy for the hubble constant @xmath35 km / s / mpc or @xmath36 .
the last two parameters in the set are the linear galaxy bias @xmath37 and non - linearity parameter @xmath38 required for the interpretation of the galaxy power spectrum @xmath39 of sdss lrgs @xcite ( see also [ sec : pk ] ) , defined as @xmath40 the linear bias factor @xmath37 is defined with respect to the linear matter power spectrum @xmath41 at @xmath42 , the effective redshift of the lrg galaxies . since @xmath0 models predict a scale dependent growth rate that changes the shape of the power spectrum as a function of redshift , we do not define the bias as relative to the power spectrum at @xmath43 ( _ cf . _
the second factor on the right hand side accounts for the non - linear evolution of the matter power spectrum shape and the scale - dependent bias of the galaxies relative to dark matter .
we assume that this _ ansatz _ continues to be a valid approximation for @xmath0 theories .
we do not marginalize analytically over the @xmath37 and @xmath38 parameters as is done normally in the cosmomc code ; instead the parameters are marginalized numerically as independent nuisance parameters in the mcmc analysis . for all of our parameters except for @xmath44 , we employ flat linear priors in the stated parameters . for @xmath44 , we supplement the flat prior with the stability condition that @xmath45 .
finally , these constraints are projected onto the space of the galaxy - isw angular correlation function for comparison with the data .
we do not attempt here to include the correlation function data in the likelihood .
that would require a joint re - analysis of the various data sets to capture the covariance between the measurements .
it would also require an assessment of systematic errors in the correlation due to uncertainties in the selection functions and other effects .
instead we compare predictions with the individual measurements and their errors as quoted in the literature .
in @xmath0 models that follow the @xmath3cdm expansion history considered here , none of the cmb phenomenology at recombination is affected by the modification to gravity .
the compton wavelength parameter @xmath8 is driven rapidly to zero at high density and curvature . hence all of the successes of @xmath3cdm in explaining the acoustic peaks carries over to these @xmath0 models .
the impact of @xmath0 gravity on the cmb comes exclusively through the so - called integrated sachs - wolfe ( isw ) effect at the lowest multipole moments .
the isw effect arises from an imbalance between the blueshift a cmb photon suffers while falling into a gravitational potential well and the redshift while climbing out if the gravitational potential evolves during transit . with a cosmological constant , gravitational potentials decay during the acceleration epoch . for @xmath0 models ,
the enhancement of the growth rate below the compton scale can change the decay into growth ( see fig .
[ fig : pot ] ) .
this reversal changes the sign of the isw effect .
cmb photons then become colder along directions associated with overdense regions .
= 3.3truein fig .
[ fig : cl ] illustrates the impact of this effect on the cmb power spectrum .
as the compton wavelength approaches the gpc ( @xmath46 mpc@xmath47 ) scales associated with the low multipoles of the isw effect , the reduction in the decay of the potential also suppresses the isw effect . near @xmath48
the isw effect is almost entirely absent at the quadrupole .
reduction in the amplitude of the quadrupole is in fact weakly favored by the data but the large cosmic variance of the low multipoles prevents this from being a significant improvement .
moreover , the elimination of the isw effect at higher multipoles where the observed power is higher counteracts this improvement . for @xmath49
the compton wavelength exceeds the scales of interest and potential decay turns to potential growth .
by @xmath50 the isw effect has an equal amplitude to that of @xmath3cdm . for @xmath51
, it is too large to accommodate the wmap data .
these features drive the overall joint constraint on @xmath52 shown in fig .
[ fig : like ] .
the wmap data also serve to fix the parameters that control the high redshift universe . in particular
, it constrains the initial amplitude of power on scales that are observed in galaxy surveys such as the sdss lrg survey .
= 3.3truein the time - dependent compton wavelength of @xmath0 models induces a more dramatic effect in the matter power spectrum during the acceleration epoch .
in fact , average compton wavelengths down to a few mpc are potentially observable in the linear power spectrum . under this scale ,
the enhanced growth rate leads to excess power relative to the same initial power spectrum determined by the wmap data .
unfortunately the overall change in power is degenerate with the unknown galaxy bias .
however if the compton wavelength appears between the few to 100 mpc scale the distortion of the shape of the power spectrum is potentially distinguishable in current surveys . in our parameterization
this occurs for @xmath53 .
= 3.3truein the sdss lrg data set in fact favor enhanced power over the _ linear _
@xmath3cdm power spectrum . in fig .
[ fig : pk ] , we compare a linear @xmath3cdm power spectrum @xmath54 and a linear @xmath0 power spectrum with the same parameters and @xmath55 .
the @xmath0 model is in fact a better fit to the shape of the power spectrum , with @xmath56 for this specific choice of bias .
however in the @xmath3cdm model , we expect lrg galaxy clustering to be nonlinear in exactly the region where these changes of shape occur .
in fact a non - linearity parameter of @xmath57 , which represents a reasonable amount of non - linearity , produces an excellent fit to the data ( see fig .
[ fig : pk ] ) .
hence , @xmath0 enhancements of small scale power are degenerate with non - linear effects in @xmath3cdm and open up a corresponding degeneracy between @xmath38 and @xmath52 ( see fig .
[ fig : qnlb0 ] ) .
the non - linear modification also introduces a small suppression of power at intermediate @xmath26 for any value of @xmath38 which also marginally improves the fit .
= 3.3truein = 3.3truein furthermore if @xmath58 the compton scale exceeds the largest scales in the survey at @xmath59mpc .
the enhancement of small scale power , while pronounced , becomes degenerate with the galaxy bias @xmath37 . with the primordial amplitude fixed by wmap
, the bias must decrease as @xmath52 increases , leading to an anti - correlation between the two parameters ( see fig .
[ fig : bb0 ] ) .
even order unity @xmath44 is allowed after marginalization .
the lrg data do weakly disfavor even larger @xmath44 but the overall constraint is dominated by the cmb data ( see fig .
[ fig : like ] and compare @xmath60 relative to @xmath61 , where the cmb spectra are nearly identical ) .
it is the marginalization over the non - linear parameter @xmath38 and bias which substantially degrades the ability of the lrg data to constrain @xmath0 models .
this is a theoretical and not an observational limitation .
as we have seen , even the small value of @xmath62 substantially impacts the current data . with cosmological simulations that address the non - linear evolution of the matter power spectrum in @xmath0 and the association of lrg s with dark matter haloes in the simulation
, this uncertainty can be lifted leading to substantially tighter bounds on the compton wavelength .
= 3.3truein = 3.3truein = 3.5truein the angular correlation between the cmb temperature field and galaxy number density field induced by the isw effect places an interesting constraint on @xmath0 models with averaged compton wavelengths in the @xmath63 mpc regime . as discussed in [ sec
: cmb ] , the isw effect reverses sign for wavelengths smaller than the average compton wavelength during acceleration .
this reversal of sign causes galaxies to be anti - correlated with the cmb @xcite . for sufficiently large @xmath64 ,
the compton wavelength subtends a scale larger than the @xmath65 over which the correlation has been measured . in this regime ,
@xmath0 models predict that galaxies are anti - correlated with the cmb , in conflict with the data measured by 2mass , apm , sdss , nvss and qso @xcite .
figure [ fig : wth ] shows the angular correlation data at @xmath66 from a compilation in @xcite ( updated by gaztanaga , private communication ) .
we compare these data with predictions from models in the chain spanning the 68% and 95% allowed regions given by the wmap cmb , sdss lrg power spectrum and supernovae data sets .
the lack of anti - correlation in any given data point places an upper bound of @xmath67 at the significance level of the detection .
this is approximately a factor of 4 improvement in @xmath44 over the other constraints or a factor of 2 in the compton wavelength .
these individual constraints can be improved somewhat by a full joint analysis of the correlation measurements but that lies beyond the scope of this work .
we have analyzed current cosmological constraints on @xmath0 acceleration models from the cmb , sdss galaxy power spectrum , galaxy - isw correlations and supernovae . by choosing @xmath0 models with a @xmath3cdm expansion history , we have explored whether the unique signatures of the @xmath0 modification to gravity are seen in current data . despite the relatively large impact of @xmath0 models on the matter power spectrum ,
the strongest current constraints involve the modification these models induce on the isw effect in the cmb .
the growth of density perturbations is enhanced under the compton wavelength of the field @xmath68 by scalar - tensor modifications to the gravitational force law .
this enhancement turns the decay of gravitational potentials during the acceleration epoch in @xmath3cdm into growth .
the joint constraint on the compton wavelength parameter @xmath69 ( 95% cl ) , from the wmap cmb power spectrum , sdss galaxy power spectrum and supernovae , is driven by the cmb .
this constraint allows compton wavelengths as large as the current horizon and is a weak bound on @xmath0 models .
a stronger bound of @xmath70 can be inferred from the positive correlation between the cmb and a range of galaxy surveys from @xmath71 . in @xmath3cdm
this positive correlation is induced by the isw effect from a decaying potential . in @xmath0 models
growing potentials convert the positive correlation into anti - correlation in violation of the observations .
the weak impact of the sdss lrg galaxy power spectrum on our joint constraints is due to a theoretical rather than observational limitation .
intriguingly the _ linear _ @xmath0 power spectrum is a marginally better fit to the data than a _ linear _ @xmath3cdm one .
however the nonlinear corrections expected in @xmath3cdm bring the model back in agreement with the data .
@xmath72-body simulations have yet to determine the non - linear matter power spectrum and dark matter halo ( and hence galaxy ) correlations in @xmath0 models .
the likely impact of non - linear evolution is degenerate with the enhancement of small scale linear power seen in @xmath0 models .
when non - linearity of the form expected for @xmath3cdm is marginalized for @xmath0 models , the impact of the galaxy power spectrum on joint constraints becomes very weak , and compton wavelengths out to the horizon length are allowed . in the future , the window between the horizon length and a few tens of mpc can be tested by larger photometric surveys that probe the shape of the power spectrum across the whole range of scales . in principle
, compton wavelengths down to a few mpc can be tested by galaxy power spectra and cosmic shear from comparison of data with simulations of @xmath0 models .
we thank ignacy sawicki for help during the initial phases of this work , enrique gaztanaga for the compilation of galaxy - isw correlation data , and marilena loverde for useful discussions .
ys and wh are supported by the u.s . dept .
of energy contract de - fg02 - 90er-40560 .
hvp is supported by nasa through hubble fellowship grant # hf-01177.01-a awarded by the space telescope science institute , which is operated by the association of universities for research in astronomy , inc .
, for nasa , under contract nas 5 - 26555 .
wh is additionally supported by the david and lucile packard foundation .
this work was carried out at the kicp under nsf phy-0114422 .
|
models which accelerate the expansion of the universe through the addition of a function of the ricci scalar @xmath0 leave a characteristic signature in the large - scale structure of the universe at the compton wavelength scale of the extra scalar degree of freedom .
we search for such a signature in current cosmological data sets : the wmap cosmic microwave background ( cmb ) power spectrum , snls supernovae distance measures , the sdss luminous red galaxy power spectrum , and galaxy - cmb angular correlations .
due to theoretical uncertainties in the nonlinear evolution of @xmath0 models , the galaxy power spectrum conservatively yields only weak constraints on the models despite the strong predicted signature in the linear matter power spectrum . currently the tightest constraints involve the modification to the integrated sachs - wolfe effect from growth of gravitational potentials during the acceleration epoch .
this effect is manifest for large compton wavelengths in enhanced low multipole power in the cmb and anti - correlation between the cmb and tracers of the potential .
they place a bound on the compton wavelength of the field be less than of order the hubble scale .
|
statistical and dynamical properties of polymers with nontrivial topology such as ring polymers have attracted much interest in several branches of physics , chemistry and biology .
for instance , some fundamental properties of ring polymers in solution were studied many years ago@xcite circular dna have been found in nature in the 1960s and knotted dna molecules are synthesized in experiments in the 1980s ; @xcite looped or knotted proteins have been found in nature during the 2000s.@xcite there are many theoretical studies on knotted ring polymers in solution .
@xcite due to novel developments in synthetic chemistry , polymers with different topological structures are synthesized in experiments during the last decade such as not only ring polymers but also tadpole ( or lasso ) polymers , double - ring ( or di - cyclic ) polymers and even complete bipartite graph polymers .
@xcite lasso polymers are studied also in the dynamics of protein folding .
@xcite we call polymers with nontrivial topology _ topological polymers_. @xcite in order to characterize topological polymers it is fundamental to study the statistical and dynamical properties such as the mean - square radius of gyration and the diffusion coefficient .
they are related to experimental results such as the size exclusion chromatography ( sec ) spectrum . for multiple - ring ( or multi - cyclic )
polymers hydrodynamic properties are studied theoretically by a perturbative method .
@xcite for double - ring polymers they are studied by the monte - carlo ( mc ) simulation of self - avoiding double - polygons.@xcite the purpose of the present research is to study fundamental aspects of the statistical and dynamical properties of various topological polymers in solution such as the mean - square radius of gyration and the diffusion coefficient systematically through simulation .
we numerically evaluate them for two different models of topological polymers : ideal topological polymers which have no excluded volume and real topological polymers which have excluded volume , and show how they depend on the topology of the polymers . here
, the topological structures of topological polymers are expressed in terms of spatial graphs .
furthermore , we show that the short - range intrachain correlation is much enhanced for real topological polymers with complex graphs .
we numerically investigate fundamental properties of topological polymers in the following procedures .
firstly , we construct a weighted ensemble of ideal topological polymers by an algorithm which is based on some properties of quaternions .
@xcite by the algorithm we generate @xmath0-step random walks connecting any given two points , @xcite where the computational time is linearly proportional to the step number @xmath0 in spite of the nontrivial constraints on the sub - chains of the graphs .
we evaluate the statistical average of some physical quantities over the weighted ensemble .
secondly , we construct an ensemble of conformations of real topological polymers by performing the molecular dynamics of the kremer - grest model and evaluate the statistical average of the quantities over the ensemble .
thirdly , by comparing the results of the real topological polymers with those of the ideal ones , we numerically show that for the mean - square radius of gyration @xmath1 and the hydrodynamic radius @xmath2 the set of ratios of the quantities among different topological polymers is given by almost the same for ideal and real topological polymers if the valency of each vertex is equal to or less than three in the graphs , as far as for the polymers we have investigated .
it agrees with tezuka s observation that the sec results are not affected by the excluded volume if each vertex of a topological polymer has less than or equal to three connecting bonds .
@xcite here we evaluate the hydrodynamic radius @xmath2 through kirkwood s approximation .
@xcite it thus follows that the quaternion method for generating ideal topological polymers is practically quite useful for evaluating physical quantities .
in fact , we can estimate at least approximately the values of @xmath1 or @xmath3 for real topological polymers by making use of the ratios among the corresponding ideal ones , if the valency of vertices is limited up to three .
moreover , we show that the ratio of the gyration radius to the hydrodynamic radius of a topological polymer , @xmath4 , is characteristic to the topology of the polymer . here , the gyration radius @xmath5 is given by the square root of the mean - square radius of gyration .
we remark that the quaternion method has been generalized to a fast algorithm for generating equilateral random polygons through symplectic geometry .
@xcite we analytically show that the ratio of the gyration radius to the hydrodynamic radius of a topological polymer , @xmath4 , is characterized by the variance of the probability distribution function of the distance between two segments of the polymer .
in particular , we argue that the ratio decreases if the variance becomes small . here
we call the function the distance distribution function , briefly .
it is expressed with the pair distribution function , or the radial distribution function of polymer segments . the mean - square radius of gyration @xmath1 and the hydrodynamic radius @xmath3 correspond to the second moment and the inverse moment of the distance distribution function , respectively .
we numerically evaluate the distance distribution function of a topological polymer both for the ideal and real chain models .
for the ideal chain model , the numerical plots of the distance distribution function are approximated well by fitted curves of a simple formula .
they lead to the numerical estimates of the mean - square radius of gyration and the hydrodynamic radius consistent with those evaluated directly from the chain conformations . in the case of ideal ring polymers
, we exactly derive an analytic expression of the pair distribution function , which is consistent with the fitting formula . for real topological polymers
we show that the short - distance intrachain correlation of a topological polymer is much enhanced , i.e. the correlation hole becomes large , if the graph is complex .
the exponent of the short - range power - law behavior in the distance distribution function is given by 0.7 for a linear polymer and a ring polymer , while it is given by larger values such as 0.9 and 1.15 for a @xmath6-shaped polymer and a complete bipartite graph @xmath7 polymer , respectively .
we suggest that the estimate to the exponent of the short - distance correlation such as 0.7 for linear and ring chains is consistent with the estimate of exponent @xmath8 of the short - range correlation in a self - avoiding walk ( saw ) derived by des cloizeaux with the renormalization group ( rg ) arguments , @xcite as will be shown shortly . here
we remark that the estimate of @xmath8 has been confirmed in the mc simulation of saw .
@xcite in order to describe the intrachain correlations of a saw , @xcite we denote by @xmath9 the probability distribution function of the end - to - end distance @xmath10 of an @xmath0-step saw .
it was shown that the large - distance asymptotic behavior of @xmath9 for @xmath11 is given by @xcite @xmath12 where @xmath13 and @xmath14 does not change exponentially fast for large @xmath15 , while the short - distance behavior is given by @xmath16 with the exponent given by @xmath17 through asymptotic analysis @xcite and @xmath18 through the rg arguments .
@xcite the asymptotic properties of @xmath9 are studied theoretically @xcite and numerically .
@xcite in order to study the short - distance intrachain correlation between two segments of a long polymer in a good solvent ,
let us denote by @xmath19 and @xmath20 the probability distribution function between a middle point and an end point of an @xmath0-step saw and that of two middle points , respectively .
then , we define critical exponents @xmath21 for @xmath22 as follows .
assuming @xmath23 we have @xmath24 the rg estimates of the exponents for @xmath25 are given by @xmath26 and @xmath27 , respectively .
@xcite they are close to the mc estimates such as @xmath28 and @xmath29 , respectively .
@xcite thus , these estimates of @xmath8 are in agreement with the value 0.7 of the exponent for the short - range power - law behavior of the distance distribution functions of real linear and real ring polymers within errors .
here we remark that most of the pairs of segments in a saw are given by those between middle points .
the contents of the paper consist of the following . in section 2
we introduce the notation of graphs expressing topological polymers .
we explain the quaternion algorithm for generating topological polymers with given graph , and then the kremer - grest model of molecular dynamics and give estimates of the relaxation time . in section 3 we present the data of the mean - square radius of gyration and the hydrodynamic radius both for ideal and real topological polymers .
we numerically show that the ratios of the mean - square radius of gyration among such topological polymers having up to trivalent vertices are given by the same values both for ideal and real topological polymers .
we also evaluate the ratio of the gyration radius to the hydrodynamic radius .
we argue that the ratio of the root mean - square radius of gyration to the hydrodynamic radius for a topological polymer decreases if the variance of the distance distribution function decreases . in section 4
we evaluate the distance distribution functions both for ideal and real topological polymers .
for ideal ones they are consistent with the exact result for ideal ring polymers .
for the real topological polymers we numerically show that short - range intrachain correlation is much enhanced , i.e. the exponent of the short - range behavior in the distance distribution function becomes large , for topological polymers with complex graphs .
finally , in section 5 we give concluding remarks .
let us call a polymer of complex topology expressed by a spatial graph @xmath30 a _ topological polymer of graph @xmath30 _ or a _ graph @xmath30 polymer_. for an illustration , four graphs of topological polymers are given in fig .
[ fig : graph ] .
they are tadpole , @xmath6-shaped , double - ring and complete bipartite @xmath7 graphs , respectively . the graph of a tadpole , which we also call a lasso , corresponds to a tadpole polymer .
it is given by a polymer of ` a ring with a branch ' architecture .
@xcite the graph of a ` theta ' in fig .
[ fig : graph ] denotes a @xmath6-shaped curve , which we also call a theta curve .
it corresponds to a @xmath6-shaped polymer , which is given by a singly - fused polymer .
@xcite the graph of a ` double ring ' corresponds to a double - ring polymer , which we also call a di - cyclic or an 8-shaped polymer .
@xcite here we remark that a complete bipartite graph @xmath7 gives one of the simplest nonplanar spatial graphs .
-shaped polymer , ` double - ring ' for a double - ring , bicyclic or 8-shaped polymer , and ` @xmath7 ' for a complete bipartite graph @xmath7 polymer , respectively .
, width=283 ] we shall explain the method of quaternions for generating a large number of conformations of a topological polymer expressed by a graph .
@xcite let us introduce the basis @xmath31 , @xmath32 and @xmath33 of quaternions .
we assume that the square of each basis @xmath34 and @xmath35 is given by -1 : @xmath36 and they satisfy the anti - commutation relations @xmath37 any quaternion @xmath38 is expressed in terms of the basis with real coefficients @xmath39 and @xmath40 as @xmath41 if the real part of a quaternion is given by zero , i.e. @xmath42 in ( [ eq : quat ] ) , we call it a pure quaternion
. we identify it with a position vector @xmath43 in three dimensions . under the complex conjugate operation
each of the basis @xmath44 and @xmath35 changes the sign : @xmath45 , @xmath46 and @xmath47 .
here we remark that the complex conjugate of the product of two quaternions is given by the product of the two complex conjugates in the reversed order : @xmath48 .
we can express any given quaternion @xmath38 in terms of two complex numbers @xmath49 and @xmath50 as @xmath51 we define the hopf map by @xmath52 it is instructive to show that the real part of the hopf map vanishes @xmath53 thus , the hopf map of a quaternion gives a pure quaternion . for given @xmath0-dimensional complex vectors @xmath54 and @xmath55 we denote by @xmath56 the @xmath0-dimensional vector of quaternions as follows .
@xmath57 we define the hopf map for the @xmath0-dimensional vectors of quaternions @xmath58 for a given pair of complex vectors @xmath59 and @xmath60 we define bond vecters @xmath61 for @xmath62 by @xmath63 the hopf map ( [ eq : vechopf ] ) corresponds to the sum of the bond vectors @xmath64 s .
suppose that a pair of gaussian @xmath0-dimensional complex vectors @xmath65 and @xmath66 are given , randomly .
we define @xmath0-dimensional complex vectors @xmath59 and @xmath60 through the gram - schmidt method by @xmath67 they have the same length @xmath0 and are orthogonal to each other with respect to the standard scalar product among complex vectors : @xmath68 . here
@xmath69 denotes the @xmath0-dimensional complex vector where each entry is given by the complex conjugate of the corresponding entry of the vector @xmath65 : @xmath70 , where @xmath71 are the @xmath72th component of the vector @xmath65 . for a pair of complex vectors
@xmath59 and @xmath60 constructed by ( [ eq : gram - schmidt ] ) a series of the three - dimensional vectors @xmath73 defined by ( [ eq : jumpvector ] ) for @xmath74 , gives a random polygon of @xmath72 segments .
@xcite in order to construct such a random walk that has the end - to - end distance @xmath75 we introduce another complex vector @xmath76 by @xmath77 where weights @xmath78 and @xmath79 are given by @xmath80 here we remark that the vector @xmath76 satisfies @xmath81 we define another quaternion vector @xmath82 by @xmath83 through the hopf map it gives an @xmath0-step random walk with end - to - end distance @xmath75 .
we have @xmath84 thus , a series of the three - dimensional vectors @xmath73 constructed by ( [ eq : jumpvector ] ) with @xmath60 replaced by @xmath76 for @xmath74 , gives a random walk of @xmath72 segments which connects the origin and the point of @xmath85 on the @xmath40 axis .
we construct weighted ensembles of random configurations for the topological polymer of a graph and numerically evaluate the expectation value of a physical quantity by taking the weighted sum for the quantity .
we first generate random configurations for the open chains and closed chains which are part of the given graph , and we attach appropriate weights to the parts of the graph .
we determine the weight of the configuration for the whole graph by the product of all the weights to the parts of the graph .
* theta curve graph ( @xmath6-shaped graph ) * we generate weighted random configurations of a theta - curve graph polymer of @xmath86 segments as follows .
@xcite we first construct random polygons of @xmath87 segments by the method of quaternions . secondly , we take two antipodal points a and b on the polygon such that each of the sub - chains connecting a and b has @xmath72 segments .
thirdly , we connect the points a and b by an @xmath72-step random walk with end - to - end distance equal to the distance between the points a and b by making use of the method of quaternions . to the whole configuration we assign the weight @xmath88 which is given by the @xmath0-step gaussian probability density of the end - to - end vector between the points a and b. the expression of the probability density will be given by eq .
( [ eq : ete - gauss ] ) in section 4.2 , where the length of bond vectors is given by @xmath89 . 12 pt * complete bipartite graph @xmath7 * we generate weighted random configurations of the topological polymer of a complete bipartite graph @xmath7 with @xmath90 segments as follows .
firstly , we generate random polygons of @xmath91 segments .
we take antipodal points a and b on the polygon such that each sub - chain between a and b has @xmath87 segments ( see fig .
[ fig : graph - complete ] ) .
we take a point c on one of the sub - chains between a and b such that both the sub - chain between a and c ( sub - chain ac ) and that of c and b ( sub - chain bc ) have @xmath72 segments , respectively .
similarly , we take a middle point d on the other sub - chain between a and b so that sub - chains ad and bd have @xmath72 segments , respectively .
secondly , we generate an @xmath87-step random walk such that it has the end - to - end distance equal to the distance between points a and b , and put it so that it connects a and b. we take the point e on the middle point of it so that sub - chains ae and be have @xmath72 segments , respectively
. thirdly , we generate a @xmath87-step random walk connecting c and e. we take the middle point @xmath30 on the walk so that sub - chains cf and ef have @xmath72 segments , respectively .
finally , we generate an @xmath72-step random walk df which connects points d and f. we associate the weight @xmath88 , @xmath92 and @xmath93 for sub - chains ab , ce and fd , respectively .
we define the weight for the whole configuration by multiplying them as @xmath94 . .
, width=283 ] a polymer chain in the kremer - grest model has both the repulsive lennard - jones ( lj ) potentials and the finitely extensible nonlinear elongation ( fene ) potentials to prevent the bonds from crossing each other .
the lj potential is given by @xmath95 where @xmath96 is the distance between the @xmath97th and @xmath98th atoms and we set the lj parameters @xmath99 and @xmath100 as @xmath101 and @xmath102 .
the term of @xmath103 corresponds to the short - range repulsion , and that of @xmath104 the long - range attractive interaction .
the minimum of the lj potential is given by @xmath105 at @xmath106 .
we introduce cutoff in order to produce a repulsive lennard - jones potential as follows .
@xmath107 where the constant term @xmath108 is added to eliminate the discontinuous jump at @xmath109 .
the fene potential between a pair of bonded atoms @xmath110\ ] ] is employed to provide a finitely extensible and nonlinear elastic potential , @xmath111 is the maximum extent of the bond . here
we choose @xmath112 and @xmath113 .
we generate an ensemble of conformations of a topological polymer of graph @xmath30 by lammps : the initial conformation is given by putting the atoms on the lattice points along the polygonal lines of the given graph @xmath30 in a cubic lattice .
then by lammps we integrate newton s equation of motion for the atoms under the repulsive lennard - jones and fene potentials .
the topological type of the conformation of a topological polymer does not change during time evolution .
the bonds can hardly cross each other , since the atoms are surrounded by strong barriers which increase with respect to the inverse of @xmath114 while they are connected with nonlinear elastic springs of finite length .
let us denote by @xmath115 the position vectors of the @xmath98th segments of a polymer for @xmath116 .
we define the correlation between the conformation at time @xmath117 and that of the initial time @xmath118 by @xmath119 where the center of mass of the polymer , @xmath120 , is given by @xmath121 we define the relaxation time @xmath122 of the conformational correlation by the number of steps at which the conformational correlation decreases to @xmath123 .
we take independent conformations at every @xmath124 steps . for example
, we have evaluated @xmath125 for a linear polymer of @xmath126 segments , @xmath127 for a ring polymer of @xmath126 segments , and @xmath128 for a @xmath7 graph polymer of @xmath129 segments .
let us denote the distance distribution function by @xmath130 in terms of distance @xmath10 .
here we recall that it is the probability distribution function of the distance between any given pair of segments of a polymer . the probability that the distance between a given pair of segments in the polymer is larger than @xmath10 and less than @xmath131 is given by @xmath132 .
suppose that there are @xmath0 segments in a region of volume @xmath133 .
we denote the average global density by @xmath134 . in terms of the pair distribution function @xmath135
, it is expressed as @xmath136 .
we numerically evaluate the distance distribution function @xmath130 as follows .
firstly , we generate an ensemble of random conformations of a polymer with graph @xmath30 . secondly ,
for each conformation we choose a pair of segments randomly , and calculate the distance between them .
we repeat the procedure several times .
thirdly , we make the histogram of the estimates with the distance between two segments for all conformations in the ensemble .
we define the mean - square radius of gyration for a topological polymer of graph @xmath30 consisting of @xmath0 segments by @xmath137 here the symbol @xmath138 denotes the ensemble average of @xmath139 over all possible configurations of the topological polymer with graph @xmath30 .
we denote by @xmath140 the square root of the mean - square radius of gyration @xmath141 : @xmath142 we also call it the gyration radius of the polymer . versus the number of segments @xmath0 for ideal topological polymers with graph @xmath30 evaluated by the quaternion method for six graphs @xmath30 such as linear , tadpole ( lasso ) , ring , double - ring , @xmath6-shaped , and complete bipartite graph @xmath7 polymers , depicted by filled circles , filled diamonds , filled upper triangles , filled lower triangles , filled stars , and filled squares , respectively .
each data point corresponds to the average over @xmath143 samples .
, width=302 ] in fig .
[ fig : qrg ] we plot against the number of segments @xmath0 the numerical estimates of the mean - square radius of gyration @xmath141 for ideal topological polymers of graph @xmath30 for six different graphs .
they are evaluated by the quaternion method , and are given in decreasing order for a given number of segments @xmath0 as follows : those of linear polymers , tadpole ( lasso ) polymers , ring polymers , double - ring polymers , @xmath6-shaped polymers , and polymers with a complete bipartite graph @xmath7 . here
we remark that the markers in figures correspond to the same graphs throughout the paper .
we observe in fig .
[ fig : qrg ] that the estimate of the gyration radius for a double - ring polymer of @xmath0 segments , depicted by filled lower triangles , is close to that of a @xmath6-shaped polymer of @xmath0 segments depicted with filled stars .
the former is only slightly larger than the latter .
for various other graphs the estimates of the mean - square radius of gyration are distinct among the different graphs with same given number of segments @xmath0 .
the estimates of the mean - square radius of gyration for the ideal topological polymers linearly depend on the number of segments @xmath0 , as shown in fig .
[ fig : qrg ] .
they are fitted by the formula @xmath144 the best estimates of parameters @xmath145 and @xmath146 in ( [ eq : fit - ideal ] ) are listed in table [ tab : ideal - rg ] together with @xmath147 values per degree of freedom ( df ) .
the @xmath147 values per df for the topological polymers with the different graphs are at most 1.1 and are small .
.best estimates of the parameters in eq .
( [ eq : fit - ideal ] ) fitted to the estimates of the mean - square radius of gyration @xmath141 for ideal topological polymers of graph @xmath30 evaluated by the quaternion method and the @xmath147 values per degree of freedom ( df ) . [ cols="^,^,^,^",options="header " , ]
let us formulate the double integral in ( [ eq : def - ring ] ) into a single integral ( [ eq : pair - polygon ] ) .
we divide the square region @xmath72 and @xmath148 for @xmath149 into the region of @xmath150 and that of @xmath151 . in the former region
we define @xmath152 by @xmath153 , and we have @xmath154 in the latter we define @xmath152 by @xmath155 , and we have @xmath156 since @xmath157 and @xmath158 after taking the sum of ( [ eq : gt ] ) and ( [ eq : lt ] ) we derive the single integral ( [ eq : pair - polygon ] ) .
making use of the probability density of gaussian chains we have @xmath159 due to the symmetry of a ring the integral of @xmath72 from 0 to @xmath160 and from @xmath160 to @xmath0 give the same result , and hence we have @xmath161 we now define variable @xmath40 by @xmath162 in terms of variable @xmath40 parameter @xmath72 is expressed as @xmath163 where the minus symbol is for @xmath164 and the plus symbol for @xmath165 .
we then introduce variable @xmath117 by @xmath166 .
we thus have @xmath167 b. h. zimm and w. h. stockmayer , _ j. chem .
* 17 * , 1301 ( 1949 ) .
e. f. casassa ,
_ j. polym .
, part a _ , * 3 * , 605 ( 1965 ) . , ed .
j. a. semlyen , ( elsevier applied science publishers , new york , 1986 ) ; 2nd ed .
( kluwee academic publ . ,
dordrecht , 2000 ) m.a .
krasnow , a. stasiak , s.j .
spengler , f. dean , t. koller and n.r .
cozzarelli , _ nature _ , * 304 * , 559 ( 1983 ) .
f.b . dean , a. stasiak , t. koller and n.r .
cozzarelli , _ j. biol
_ , * 260 * , 4975 ( 1985 ) .
bates and a. maxwell , _ dna topology _ ( oxford univ .
press , 2005 ) .
|
for various polymers with different topological structures we numerically evaluate the mean - square radius of gyration and the hydrodynamic radius systematically through simulation .
we call polymers with nontrivial topology _
topological polymers_. we evaluate the two quantities both for ideal and real chain models and show that the ratios of the quantities among different topological types do not depend on the existence of excluded volume if the topological polymers have only up to trivalent vertices , as far as the polymers investigated .
we also evaluate the ratio of the gyration radius to the hydrodynamic radius , which we expect to be universal from the viewpoint of renormalization group .
furthermore , we show that the short - distance intrachain correlation is much enhanced for topological polymers expressed with complex graphs .
department of physics , faculty of core research + ochanomizu university , + 2 - 1 - 1 ohtsuka , bunkyo - ku , tokyo 112 - 8610 , japan
|
in the children s game of rock - paper - scissors , paper wraps rock , rock smashes scissors , and scissors cut paper .
theoretical biologists have used this game as a metaphor in ecology and evolutionary biology to describe interactions among three competing species in which each species has an advantage over one of its opponents but not the other @xcite , a situation that can result in cyclic dominance @xcite .
interactions of this type occur between three competing strains of _ e. coli _
@xcite as well as in the mating strategies of side - blotched lizards @xcite .
the rock - paper - scissors game also has applications outside of biology .
for example , it has been used to analyze the dynamics of a sociological system with three strategies in a public goods game @xcite .
mathematically , the dynamics of the rock - paper - scissors game is often studied using the replicator equations @xcite , a system of coupled nonlinear differential equations that have been applied in diverse scientific settings .
for example , replicator equations have been used to model the evolution of language , fashion , autocatalytic chemical networks , behavioral dynamics , and multi - agent decision making in social networks @xcite . in the specific context of the rock - paper - scissors game ,
the replicator equations are most often studied in the absence of mutation @xcite .
the dynamics in that case tend to exhibit one of three types of long - term behavior , depending on a parameter @xmath0 that characterizes how far the game is from being a zero - sum game .
the three types of behavior are : ( i ) stable coexistence of all three species , ( ii ) neutrally stable cycles that fill the whole state space , and ( iii ) large - amplitude heteroclinic cycles in which each species in turn almost takes over
the whole population and then almost goes extinct . in 2010 , mobilia @xcite broke new ground by asking what would happen if the strategies are allowed to mutate into one another . for simplicity he assumed that every species could mutate into every other at a uniform rate @xmath1 .
for this highly symmetrical case ( which we call global mutation ) , mobilia @xcite found a new form of behavior , not present in the mutation - free case : the system could settle into a stable limit - cycle oscillation , born from a supercritical hopf bifurcation .
in this paper we investigate what happens in other mutational regimes .
for example , we consider the dynamics when precisely one species mutates into one other , or when two species mutate into two others , as well as more complicated patterns of mutation . in every case
we find that stable limit cycles can occur , and we calculate the regions in parameter space where such attracting periodic behavior occurs .
our work was motivated by a question that arose in one of our previous studies @xcite .
there we had explored the dynamics of the repeated prisoner s dilemma game for three strategies : always defect ( alld ) , always cooperate ( allc ) , and tit - for - tat ( tft ) .
we incorporated mutations at a rate @xmath1 and a complexity cost @xmath2 of playing tft into the replicator equations and analyzed the six possible single - mutation cases ( exactly one strategy mutates into one other with rate @xmath1 ) as well as the global mutation case ( each strategy mutates into the two others at the same rate @xmath1 ) .
our results showed that stable limit cycles of cooperation and defection were possible for _ every _ pattern of mutation we considered .
what was particularly striking was that stable cycles occurred for parameter values arbitrarily close to the structurally unstable limiting case of zero mutation rate and zero complexity cost @xmath3 .
we conjectured , but were unable to prove , that stable limit cycles would continue to exist near this point for _ any _ pattern of mutation , not just the single - mutation patterns we looked at in ref .
@xcite . in hopes of finding a more tractable system where the same phenomena might occur
, we turned to the rock - paper - scissors game .
the analogous question is , does this system always exhibit stable limit cycles for parameters arbitrarily close to the zero - sum , zero - mutation - rate limit , for any possible pattern of mutation ?
for single mutations , the answer is yes , as we will show below . for arbitrary patterns of mutation
, the answer again appears to be yes , but we have only managed to prove this under a further constraint , namely that the mutation pattern preserves the symmetrical coexistence state where all three strategies are equally populated .
the standard rock - paper - scissors game is a zero - sum game with payoff matrix given in table [ tab : payoff0sum ] .
the entries show the payoff received by the row player when playing with the column player .
lcdr & & & + rock & 0 & -1 & 1 + paper & 1 & 0 & -1 + scissors & -1 & 1 & 0 + according to the payoff matrix in table [ tab : payoff0sum ] , each species gets a payoff @xmath4 when playing against itself .
when playing against a different species , the winner gets a payoff @xmath5 while the loser gets @xmath6 . a more general payoff matrix , considered in ref .
@xcite , allows for non - zero - sum games .
now the winner gets a payoff @xmath5 and the loser gets @xmath7 , as shown in table [ tab : payoffeps1 ] .
this game is zero sum if and only if @xmath8 .
lcdr & & & + rock & 0 & - & 1 + paper & 1 & 0 & @xmath7 + scissors & @xmath7 & 1 & 0 + for convenience , we redefine the entries of the payoff matrix so that the zero - sum case corresponds to @xmath9 rather than @xmath8 .
the payoff matrix for the rest of this paper is shown in table [ tab : payoffeps0 ] .
lcdr & & & + rock & 0 & -(+1 ) & 1 + paper & 1 & 0 & @xmath10 + scissors & @xmath10 & 1 & 0 +
the most symmetrical rock - paper - scissors game with mutation is the one with global mutation , where each species mutates into the other two with a rate @xmath1 , as shown in fig .
[ fig : arrows ] .
suppose that global mutation occurs in a well - mixed population with @xmath11 individuals .
let @xmath12 and @xmath13 denote the relative frequencies of individuals playing rock , paper and scissors , respectively
. then @xmath14 or @xmath15 . by eliminating @xmath13 in this fashion
, one can capture the dynamics of the three strategies by studying @xmath16 and @xmath17 alone .
following ref .
@xcite , the replicator - mutator equations can be written as @xmath18 where @xmath19 is the fitness of strategy @xmath20 , defined as its expected payoff against the current mix of strategies , and @xmath21 is the average fitness in the whole population .
for the payoff matrix defined above in table [ tab : payoffeps0 ] , and using @xmath15 , we find @xmath22 and @xmath23 a conceptual disadvantage of eliminating @xmath13 in favor of @xmath16 and @xmath17 is that the system s cyclic symmetry becomes less apparent . to highlight it , one can plot the phase portraits of the system on the equilateral triangle defined by the face of the simplex @xmath14 , where @xmath24 .
this mapping of the phase portrait onto the equilateral triangle can be achieved by using the following transformation : @xmath25 in what follows , all phase portraits will be plotted in @xmath26 space , but we will still indicate the values of @xmath27 at the vertices of the simplex , since these variables have clearer interpretations . equation has four fixed points : @xmath28 , and @xmath29 .
the inner fixed point undergoes a hopf bifurcation when @xmath30 , as shown in @xcite .
moreover , the system undergoes three simultaneous saddle connections when @xmath31 , meaning that when @xmath31 , there is a heteroclinic cycle linking the three corners of the simplex .
figure [ fig : rpsglobal ] shows the stability diagram of eq . as well as phase portraits corresponding to the different regions of fig .
[ fig : rpsglobal](a ) .
( color online ) stability diagram for eq . and its associated phase portraits , in the case of global mutation .
( a ) stability diagram .
locus of supercritical hopf bifurcation , solid blue curve ; saddle connection , dotted red line .
( b ) phase portrait corresponding to region 1 of the stability diagram .
the system has a stable interior point , corresponding to coexistence of all three strategies .
( c ) phase portrait corresponding to region 2 .
the system has a stable limit cycle . ]
incidentally , the occurrence of the heteroclinic cycle is not an artifact of the global mutation structure assumed here . for any mutation pattern
, @xmath31 will always ( trivially ) yield a heteroclinic cycle linking the three corners of the simplex , since that cycle is known to be present in the absence of mutation @xcite .
thus , all the subsequent stability diagrams in this paper will show a saddle connection curve along the line @xmath31 .
because of the cyclic symmetry of the rock - paper - scissors game , it suffices to consider two of the six possible single - mutation pathways . so without loss of generality , we restrict attention to rock ( @xmath16 ) @xmath32 paper ( @xmath17 ) and paper @xmath33 @xmath32 rock @xmath34 . the two cases are qualitatively different . in the first case ,
the direction of mutation reinforces the system s inherent tendency to flow from rock to paper .
( recall that paper beats rock , so trajectories tend to flow from rock to paper , as the paper population grows at rock s expense . ) by contrast , in the second case , the mutation pathway runs counter to this natural flow .
when rock mutates into paper , the system becomes @xmath36 figure [ fig : rp ] plots the stability diagram and phase portraits of eq . .
( color online ) stability diagram and phase portraits for eq . .
in this single - mutation case , rock @xmath16 mutates into paper @xmath17 at a rate @xmath1 : @xmath37 .
( a ) stability diagram .
supercritical hopf bifurcation , solid blue curve ; saddle connection , dotted red line .
( b ) phase portrait corresponding to region 1 of the stability diagram .
the system has a stable interior point , corresponding to coexistence of all three strategies .
( c ) phase portrait corresponding to region 2 .
the system has a stable limit cycle . ] to derive the results shown in fig .
[ fig : rp ] , we observe first that eq .
has three fixed points : @xmath38 and an inner fixed point @xmath39 , with @xmath40 where the quantity @xmath41 is given by @xmath42 these fixed points display some notable differences from those found above , when mutation was either absent or global .
for example , the state in which only rock exists , corresponding to the corner @xmath43 of the simplex , is no longer a fixed point , since @xmath16 is now constantly mutating into @xmath17 .
so paper must exist whenever rock does .
the complicated interior fixed point @xmath39 can be regarded as a perturbation of @xmath44 , in the sense that @xmath45 where both @xmath46 and @xmath47 approach 0 as @xmath48 .
equation produces stable limit cycles when the interior fixed point undergoes a supercritical hopf bifurcation .
this transition can be shown to occur at @xmath49 the stable limit cycle created in the hopf bifurcation grows into a heteroclinic cycle as @xmath50 . in the case where paper @xmath17 mutates into rock @xmath16 at a rate @xmath1 ,
the system becomes @xmath52 figure [ fig : pr ] plots the stability diagram and phase portraits of eq . .
note the existence of a new region in parameter space , bounded below by a transcritical bifurcation curve .
this region did not exist when the mutation was in the direction of the system s inherent flow , depicted earlier in fig .
[ fig : rp ] .
( the existence of such a region also holds if there are multiple mutations , unless the inner fixed point is @xmath53 , as we will see below . )
( color online ) stability diagram and phase portraits for eq .
, corresponding to a single mutation pathway in which paper @xmath17 mutates into rock @xmath16 at a rate @xmath1 .
note that this direction of mutation goes `` against the flow '' in the following sense : in the absence of mutation , paper beats rock and thus the flow normally tends to convert @xmath16 into @xmath17 .
thus , the direction of mutation assumed here opposes that flow .
( a ) stability diagram .
supercritical hopf bifurcation , solid blue curve ; saddle connection , dotted red line ; transcritical bifurcation , dashed green curve .
in the coexistence region above the transcritical bifurcation curve , rock coexists with paper at the stable fixed point .
( b ) phase portrait corresponding to region 1 of the stability diagram .
the system has a stable interior point , corresponding to coexistence of all three strategies .
( c ) phase portrait corresponding to region 2 .
the system has a stable limit cycle .
( d ) phase portrait corresponding to region 3 , where rock and paper coexist and scissors has gone extinct .
the system has a stable point on the boundary line joining rock and paper . ]
the results shown in fig . [ fig : pr ] can be derived analytically , as follows . because the direction of mutation @xmath54 here opposes the system s inherent flow , the fixed points of eq .
now include a new fixed point @xmath55 on the boundary line given by @xmath56 and @xmath57 .
the fixed points are thus @xmath58 an interior fixed point @xmath39 similar to that found earlier , and a fourth fixed point on the boundary .
the coordinates of the nontrivial fixed points are given by @xmath59 where @xmath41 is given by the expression obtained earlier .
as shown in fig .
[ fig : pr](a ) , the fixed point @xmath55 exists only for parameter values above the transcritical bifurcation curve . by linearizing about the fixed point and seeking a zero - eigenvalue bifurcation
, we find that the transcritical bifurcation curve is given by @xmath60 interestingly , the hopf bifurcation curve for eq .
is identical to that for eq .
, even though the two systems have qualitatively different dynamics .
we have no explanation for this coincidence .
it does not follow from any obvious symmetry , as evidenced by the fact that the inner fixed points @xmath39 are different in the two cases .
if we allow mutations to occur along two pathways instead of one , and assume that they both occur at the same rate @xmath1 , then by the cyclic symmetry of the rock - paper - scissors game there are four qualitatively different cases to consider . the analysis becomes more complicated than with single mutations , so we omit the details and summarize the main results in the stability diagrams shown in figure [ fig : double ] .
the key point is that in every case , stable limit cycles exist arbitrarily close to the origin @xmath61 in parameter space , consistent with the conjecture discussed in the introduction .
figure [ fig : double](a ) shows the stability diagram for the first case , defined by having one of the two mutations go in the direction of the system s inherent flow and the other against it .
an example is @xmath62 and @xmath37 . as shown in fig .
[ fig : double](a ) , the resulting stability diagram has three regions and resembles what we saw earlier in fig .
[ fig : pr](a ) .
nothing qualitatively new happens if both pathways go against the flow so this case is omitted .
the second case occurs when both mutation pathways go in the same direction relative to the flow , as in fig .
[ fig : double](b ) .
then the stability diagram has only two regions , similar to fig .
[ fig : rp](a ) . the final case , shown in fig .
[ fig : double](c ) , occurs when the two mutation pathways go in opposite directions between the same two species , as in @xmath37 and @xmath54 .
again , the stability diagram shows only two regions .
the boundary between the regions turns out to be exactly straight in cases like this . specifically , we find that the hopf curve here is given by the line @xmath63 .
the key to the analysis is the observation that the interior fixed point reduces to @xmath64 for this case .
this convenient symmetry property eases the calculation of the hopf bifurcation curve .
indeed , the hopf curve continues to be straight , even for more complex patterns of mutation , as long as all three species are equally populated at the interior fixed point , as we will show next .
( color online ) stability diagram of the replicator equations for different patterns of two mutations , both occurring at a rate @xmath1 .
supercritical hopf bifurcation , solid blue curve ; transcritical bifurcation , dashed green curve ; saddle connection , dotted red line .
( a ) opposing mutations : @xmath62 and @xmath37 .
( b ) mutations in the same direction of circulation : @xmath65 and @xmath37 .
( c ) bidirectional mutation between the same two species : @xmath37 and @xmath54.,width=188,height=132 ] ( color online ) stability diagram of the replicator equations for different patterns of two mutations , both occurring at a rate @xmath1 .
supercritical hopf bifurcation , solid blue curve ; transcritical bifurcation , dashed green curve ; saddle connection , dotted red line .
( a ) opposing mutations : @xmath62 and @xmath37 .
( b ) mutations in the same direction of circulation : @xmath65 and @xmath37 .
( c ) bidirectional mutation between the same two species : @xmath37 and @xmath54.,width=188,height=132 ] ( color online ) stability diagram of the replicator equations for different patterns of two mutations , both occurring at a rate @xmath1 .
supercritical hopf bifurcation , solid blue curve ; transcritical bifurcation , dashed green curve ; saddle connection , dotted red line .
( a ) opposing mutations : @xmath62 and @xmath37 .
( b ) mutations in the same direction of circulation : @xmath65 and @xmath37 .
( c ) bidirectional mutation between the same two species : @xmath37 and @xmath54.,width=188,height=132 ] ccccc mutation patterns + ( rate @xmath1 ) & # of mutations & hopf curve + + & 2 & @xmath66 + + & 3 & @xmath67 + + & 4 & @xmath68 + + & 6 & @xmath69 +
the analytical treatment of the model becomes prohibitively messy as one adds more mutation pathways . to make further progress ,
let us restrict attention to mutation patterns that preserve @xmath70 as the inner fixed point for all values of @xmath0 and @xmath1 .
these mutation patterns are shown in table [ table : hopf ] .
we computed the hopf curve analytically for all of them and noticed something curious : the equation of the hopf curve is always @xmath71 as shown in table [ table : hopf ] . to derive this formula ,
we write the replicator equations for all the mutation patterns in table [ table : hopf ] as a single system : @xmath72+\beta_x y+\gamma_x z ) \nonumber\\ \dot{y } & = & y ( f_y - \phi ) + \mu(-[\beta_x y+\beta_z y]+\alpha_y x+\gamma_y z ) \nonumber\\ \dot{z } & = & z ( f_z - \phi ) + \mu(-[\gamma_x z+\gamma_y z]+\alpha_z x+\beta_z y ) \label{eqn : all}.\end{aligned}\ ] ] here the indicator coefficients given by the various @xmath73 and @xmath74 are set to 0 or 1 , depending on which mutation pathways are absent or present .
for example , we set @xmath75 if @xmath16 mutates into @xmath17 . otherwise , we set @xmath76 .
the same sort of reasoning applies to the various @xmath77 and @xmath78 . to ensure that @xmath79 is a fixed point of eq .
, the indicator coefficients must satisfy certain algebraic constraints .
these are given by @xmath80 using eqs . and
, one can then show that the fixed point @xmath79 undergoes a hopf bifurcation at @xmath81 which yields the results in table [ table : hopf ] .
the upshot is that a curve of supercritical hopf bifurcations emanates from the origin in parameter space .
hence , for complex mutation patterns that preserve the fixed point @xmath70 , we have confirmed the conjecture that stable limit cycles exist for parameters arbitrarily close to the zero - sum , zero - mutation - rate limit of the replicator equations for the rock - paper - scissors game .
our main result is that for a wide class of mutation patterns , the replicator - mutator equations for the rock - paper - scissors game have stable limit cycle solutions . for this class of mutation patterns ,
a tiny rate of mutation and a tiny departure from a zero - sum game is enough to destabilize the coexistence state of a rock - paper - scissors game and to set it into self - sustained oscillations .
another caveat is that our results have been obtained for one version of the replicator - mutator equations , namely , that in which the mutation terms are _ added _ to the replicator vector field . in making this choice we are following mobilia @xcite , who investigated the effect of additive global mutation on the replicator dynamics for the rock - paper - scissors game .
but there is another way to include the effect of mutation in the replicator equations : one can include it _ multiplicatively _ as in , e.g. , ref .
@xcite . in biological terms
, the multiplicative case makes sense if the mutations occur in the offspring , whereas in the additive case they occur in the adults .
we found the additive case easier to work with mathematically , but it would be interesting to see if our results would still hold ( or not ) in the multiplicative case .
22ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12
& 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop _ _ ( , ) @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) in @noop _ _ ( , ) pp .
@noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( )
|
we analyze the replicator - mutator equations for the rock - paper - scissors game .
various graph - theoretic patterns of mutation are considered , ranging from a single unidirectional mutation pathway between two of the species , to global bidirectional mutation among all the species .
our main result is that the coexistence state , in which all three species exist in equilibrium , can be destabilized by arbitrarily small mutation rates .
after it loses stability , the coexistence state gives birth to a stable limit cycle solution created in a supercritical hopf bifurcation .
this attracting periodic solution exists for all the mutation patterns considered , and persists arbitrarily close to the limit of zero mutation rate and a zero - sum game .
|
sharing of quantum correlations among many parties is known to play an important role in quantum phenomena , ranging from quantum communication protocols @xcite to cooperative events in quantum many - body systems @xcite .
it is therefore important to conceptualize and quantify quantum correlations , for which investigations are usually pursued in two directions , viz .
the entanglement - separability @xcite and the information - theoretic @xcite ones .
any such measure of quantum correlation is expected to satisfy a monotonicity ( precisely , non - increasing ) under an intuitively satisfactory set of local quantum operations . in case
the quantum state is shared between more than two parties , one also expects that all measures of quantum correlation would additionally follow a monogamy property @xcite , which restricts the sharability of quantum correlations among many parties . in the case of three parties ,
say , alice , bob and charu , monogamy of a measure says that the sum of quantum correlations of the two - party local states between the alice - bob and the alice - charu pairs , should not exceed the quantum correlation of alice with bob and charu taken together .
alice is therefore alloted a special status , and is called the `` nodal observer '' .
the concept has also been carried over to more than two extra - nodal observers .
classical correlations certainly do not satisfy such a monogamy constraint .
the monogamous nature of quantum correlations plays a key role in the security of quantum cryptography @xcite . surprisingly however
, there are important and useful entanglement measures that do not satisfy monogamy for certain multiparty quantum states , an example being the entanglement of formation @xcite , which quantifies the amount of entanglement required for preparation of a given bipartite quantum state .
nevertheless , it was found that for three - qubit systems , the concurrence squared @xcite , a monotonically increasing function of the entanglement of formation is monogamous @xcite .
recently , it was shown that the information - theoretic quantum correlation measure , quantum discord @xcite , can violate monogamy @xcite ( cf .
@xcite ) , and again a monotonically increasing function of the quantum discord satisfies monogamy for three - qubit pure states @xcite . in this paper , we show that if any bipartite quantum correlation measure , of an arbitrary number of parties in arbitrary dimensions , is non - increasing under loss of a part of a local subsystem , any multiparty quantum state is either already monogamous with respect to that measure or an increasing function of the bipartite measure can make it so .
note that the result holds for both pure and mixed states .
it is interesting to note that the increasing function also satisfies all the properties for being a measure of quantum correlation , which include monotonicity under local operations and vanishing for `` classically correlated '' states ( which is the set of separable states for measures of entanglement ) . to illustrate the result ,
we show that although the quantum work - deficit @xcite , an information - theoretic quantum correlation measure , violates monogamy even for three - qubit pure states , the states become monogamous when one considers integer powers of the measure . in stark contrast to what happens for concurrence and quantum discord ,
we show that for the three - qubit generalized w states @xcite , the fourth power of quantum work - deficit is required to obtain monogamy for these states . in case of arbitrary three - qubit w - class states
@xcite and the ghz - class states @xcite , to obtain monogamy of quantum work - deficit , one requires higher polynomials .
we also find that three - qubit pure states that are monogamous with respect to quantum discord are also so with respect to quantum work - deficit . in sect .
ii , we prove the result about the transformability of all non - monogamous multiparty states into monogamous ones .
we illustrate this result in the next section ( sect .
iii ) by using the concept of quantum work - deficit , where we prove certain general results about monogamy of quantum work - deficit for arbitrary three - party quantum states .
we present a conclusion in sect .
iv . a brief introduction to quantum work - deficit
is given in appendix a.
in proving the results , we will work with three - party quantum states .
however , they can easily be generalized to an arbitrary number of parties .
let @xmath0 be a quantum correlation measure that is defined for arbitrary bipartite states ( pure or mixed ) in arbitrary dimensions .
consider a three - party quantum state ( pure or mixed ) , @xmath1 , in arbitrary dimensions , shared between three observers , alice @xmath2 , bob @xmath3 , and charu @xmath4 .
let @xmath5 denote the quantum correlation @xmath6 for the two - party reduced state @xmath7 .
@xmath8 is similarly defined .
let @xmath9 denote the quantum correlation for the state @xmath1 in the @xmath10 partition .
the measure @xmath6 is said to satisfy monogamy for the state @xmath1 if @xmath11 .
the idea is that a measure will be called monogamous for a certain shared quantum state if the amount of quantum correlations that alice has with bob and charu separately would be smaller than what she has with her partners taken together .
the measure will be called strictly monogamous for @xmath1 if @xmath12 . on the other hand , @xmath13 ,
will imply that the measure is non - monogamous for the corresponding state .
the following theorem demonstrates that the non - monogamous nature of any measure for any state can be transformed to a monogamous one ( in fact , strictly so ) , by considering an increasing function of the measure .
let @xmath14 be the set of all real numbers .
+ * theorem 1 : * if @xmath0 violates monogamy for an arbitrary three - party quantum state @xmath1 in arbitrary dimensions , there always exists an increasing function @xmath15 such that @xmath16 provided that @xmath0 is monotonically decreasing under discarding systems and invariance under discarding systems occurs only for monogamy - satisfying states .
+ ` proof : ` let us first rename @xmath17 for notational simplicity . then the constraints in the premise of the theorem ( non - monogamy and monotonicity of @xmath0 ) can be rewritten as @xmath18 hence it follows that @xmath19 and @xmath20 + this implies that @xmath21 hence @xmath22 , there exists positive integers @xmath23 such that @xmath24 let us now choose @xmath25 .
therefore , @xmath26 and @xmath27 , @xmath28 positive integers @xmath29 , where @xmath30 adding the inequalities , we have @xmath31 , @xmath28 positive integers @xmath29 . hence the proof .
@xmath32 note here that invariance under discarding part of a subsystem implying monogamy , holds for many quantum correlation measures , including entanglement of formation and concurrence for three - qubit systems and quantum discord in arbitrary - dimensional three - party states .
note also that the power of a measure vanishes for the same class of states for which the original measure vanishes , so that the set of states that is indicated to be `` classical '' by the original measure , is invariant after the transformation of the original measure into the new one .
let us also mention here that if a measure is monotonically non - increasing for a certain class of local operations ( possibly assisted by classical communication between the parties ) , a positive integer power of the measure also has the same property .
note that while the cases of vanishing @xmath33 have been ignored in the proof , they can be handled easily .
we now show that the class of monogamous states is closed under the operation of taking positive integral powers of the corresponding measure . + * theorem 2 : * if a quantum correlation measure is monogamous for a three - party quantum state , any positive integer power of the measure is also monogamous for the same state .
+ ` proof : ` the premise implies that @xmath34
. then for any positive integer @xmath35 , we have @xmath36 which in turn is @xmath37 , as @xmath38 , @xmath39 are non - negative .
hence the proof .
we will now consider the monogamy properties of the information - theoretic quantum correlation measure , called quantum work - deficit ( wd ) @xcite , for arbitrary three - qubit pure states . in particular , this will help to illustrate that powers of a measure can lead to monogamous nature for a state , when the measure itself is not so .
we begin by relating the monogamy properties of quantum discord , quantum work deficit , and entanglement of formation .
consider an arbitrary pure three - party state @xmath40 .
let us denote the quantum discord for the state @xmath41 by @xmath42 , where the measurement is performed by the observer @xmath43 .
@xmath44 is similarly defined , with the measurement being performed by the observer @xmath45 .
the entanglements of formation of @xmath46 and @xmath47 are denoted by @xmath48 and @xmath49 respectively .
similar notations are used for the different varieties of the quantum work - deficits , @xmath50 , @xmath51 , and @xmath52 .
see appendix a for the definition of wd .
+ * proposition 1 : * for an arbitrary three - party pure state , @xmath53 , where @xmath54 is the entropy produced by the measurement in @xmath43 , and similarly for @xmath55 .
+ ` proof : ` it can be obtained from ref .
@xcite that for an arbitrary pure state @xmath40 , @xmath56 where @xmath57 forms the optimal measurement by the observer @xmath45 and @xmath58 are the corresponding probabilities . here
@xmath59 denotes the von neumann entropy of its argument .
therefore , @xmath60 , where @xmath61 denotes the shannon entropy of the probability distribution in its argument . here
we assume that projective measurements attain optimality , which is indeed the case for rank-2 states @xcite .
consequently , @xmath62 .
hence the result .
@xmath32 performing measurements on the first parties will lead to @xmath63 , where @xmath64 ( @xmath65 ) is the entropy produced in the measurement at @xmath66 on @xmath46 ( @xmath47 ) . + * theorem 3 : * for an arbitrary pure three - party quantum state @xmath40 , quantum discord is monogamous whenever the quantum work - deficit , @xmath51 , is so .
+ ` proof : ` from the definitions of quantum discord and wd , we obtain @xmath67 where @xmath68 is the von neumann entropy of @xmath69 . since @xmath70 , @xmath71 . for states for which wd is monogamous , we have @xmath72 here we assume that the minimum of work - deficit and quantum discord are attained by the same measurement .
it is easy to see that the theorem holds even if the first parties perform the measurements .
we now consider the monogamy properties of quantum work - deficit for an important class of three - qubit pure states , viz .
the generalized w states @xcite , given by @xmath73 where @xmath74 $ ] and @xmath75 $ ] .
we find that quantum work - deficit is non - monogamous for almost all members of this class ( see fig . 1 ( left ) ) .
in other words , setting @xmath76 for an arbitrary bipartite quantum correlation measure @xmath6 and an arbitrary three - party state @xmath1 , we find that @xmath77 for about @xmath78 of randomly chosen generalized w states .
note here that another information - theoretic quantum correlation measure , the quantum discord , can also be non - monogamous for these states @xcite .
however , recently it has been shown that the square of ( one variety of ) quantum discord is a monogamous quantity for all three - qubit pure states @xcite .
this however is no longer valid for wd . as stated in theorem 1 ,
suitably chosen integral powers of wd will be monogamous for any given state .
and we find that for wd , monogamy for almost all generalized w states is obtained for the fifth power ( see fig .
1 ( right ) ) , i.e. @xmath79 for about @xmath80 of the generalized w states .
we have also considered the monogamy properties of general three - qubit pure states with respect to quantum work - deficit , @xmath52 .
a histogram showing the relative frequencies of non - monogamous states among randomly chosen pure three - qubit states , for different powers of quantum work - deficit , is given in fig .
quantum correlation measures can be monogamous or non - monogamous for multisite quantum states .
this can happen for measures within the entanglement - separability paradigm , as well as those in the information - theoretic one .
we demonstrated that any quantum correlation measure that is non - monogamous for a multiparty quantum state can be made monogamous for the same by considering an increasing function of the measure .
the transformed measure retains the important properties , like monotonicity under local operations and vanishing for `` classical '' states , of the original measure .
we illustrate the results by using the concept of quantum work - deficit , an information - theoretic quantum correlation measure .
we show that while the generalized w states are non - monogamous with respect to quantum work - deficit , the fourth power of the measure makes the states monogamous .
we also discuss the monogamy properties of quantum work - deficit , and its powers , for arbitrary three - qubit pure states .
let us mention here that in the literature , monotonically increasing functions of a quantum correlation measure is regarded with the same level of importance as the original measure .
so , for example , the nearest - neighbor entanglement of spin-1/2 systems @xcite is usually investigated by employing the measure , concurrence , although a more physically meaningful measure is the entanglement of formation , with concurrence being an increasing function of the latter .
rp acknowledges an inspire - faculty position at the harish - chandra research institute ( hri ) from the department of science and technology , government of india , and sk thanks hri for hospitality and support .
in this appendix , we briefly introduce the information - theoretic measure of quantum correlation , known as quantum work - deficit @xcite for an arbitrary bipartite quantum state @xmath81 .
let us begin by considering the number , @xmath82 , of pure qubits that can be extracted from @xmath81 by `` closed global operations '' , with the latter consisting of any sequence of unitary operations and dephasing .
it can be shown that @xmath83 where @xmath84 is the @xmath85 of the dimension of the hilbert space @xmath86 on which @xmath81 is defined .
this thermodynamic `` work '' that can be extracted from the quantum state @xmath81 may require to employ global operations , which are not accessible to observers who are situated in separated laboratories . to obtain a quantification of the amount of work that can be extracted from @xmath81 by local actions , we restrict to `` closed local quantum operations and classical communication ( clocc ) '' , which consists of local unitaries , local dephasings , and sending dephased states from one party to another . under these local actions ,
the number of pure qubits that can be extracted is given by @xmath87,\ ] ] where @xmath88 and @xmath89 .
for an arbitrary bipartite state @xmath90 , the quantum work - deficit is then defined as @xmath91 and is interpreted as an information - theoretic quantum correlation measure of @xmath81 .
the quantity is not efficiently computable for arbitrary bipartite states .
general clocc actions are also difficult to implement in an experiment
. therefore we will also consider the quantity @xmath92 , in which we restrict our attention to clocc consisting of projection measurements at the single party ( @xmath66 ) only for extracting work with local actions .
if the measurement is performed by @xmath43 , we denote it as @xmath93 .
bennett , g. brassard , c. crpeau , r. josza , a. peres , and w.k .
wootters , phys .
lett . * 70 * , 1895 ( 1993 ) .
w . pan , z .- b .
chen , m. ukowski , h. weinfurter , and a. zeilinger , arxiv:0805.2853 [ quant - ph ] .
h. hffner , c.f .
roos , and r. blatt , phys . rep .
* 469 * , 155 ( 2008 ) ; l .- m . duan and c. monroe , rev .
phys . * 82 * , 1209 ( 2010 ) ; k. singer , u. poschinger , m. murphy , p. ivanov , f. ziesel , t. calarco , and f. schmidt - kaler , rev .
phys . * 82 * , 2609 ( 2010);d .
jaksch , contemp . phys .
* 45 * 367 ( 2004 ) ; d. jaksch and p. zoller , ann . phys . * 315 * 52 ( 2005 ) ; l.m.k .
vandersypen and i. l. chuang , rev .
* 76 * , 1037 ( 2005 ) ; y. makhlin , g. schn , and a. shnirman , rev .
phys . * 73 * , 357 ( 2001 ) ; j.m .
raimond , m. brune , and s. haroche , rev .
mod . phys . * 73 * , 565 ( 2001 ) .
see also t.j . osborne and f. verstraete , phys .
96 * , 220503 ( 2006 ) ; g. adesso , a. serafini , and f. illuminati , phys .
a * 73 * , 032345 ( 2006 ) ; t. hiroshima , g. adesso , and f. illuminati , phys . rev . lett . * 98 * , 050503 ( 2007 ) ; m. seevinck , phys . rev .
a * 76 * , 012106 ( 2007 ) ; s. lee and j. park , _ ibid . _ * 79 * , 054309 ( 2009 ) ; a. kay , d. kaszlikowski , and r. ramanathan , phys .
* 103 * , 050501 ( 2009 ) ; f.f .
fanchini , m.f .
cornelio , m.c .
de oliveira , and a.o .
caldeira , phys .
a * 84 * , 012313 ( 2011 ) ; m. hayashi and l. chen , _ ibid . _ * 84 * , 012325 ( 2011 ) , and references therein .
j. oppenheim , m. horodecki , p. horodecki , and r. horodecki , phys .
rev . lett . * 89 * , 180402 ( 2002 ) ; m. horodecki , k. horodecki , p. horodecki , r. horodecki , j. oppenheim , a. sen(de ) , and u. sen , _ ibid . _ * 90 * , 100402 ( 2003 ) ; m. horodecki , p. horodecki , r. horodecki , j. oppenheim , a. sen(de ) , u. sen , and b. synak - radtke , phys .
a * 71 * , 062307 ( 2005 ) .
|
we show that arbitrary multiparty quantum states can be made to satisfy monogamy by considering increasing functions of any bipartite quantum correlation that may itself lead to a non - monogamous feature .
this is true for states of an arbitrary number of parties in arbitrary dimensions , and irrespective of whether the state is pure or mixed .
the increasing function of the quantum correlation satisfies all the expected quantum correlation properties as the original one .
we illustrate this by considering a thermodynamic quantum correlation measure , known as quantum work - deficit .
we find that although quantum work - deficit is non - monogamous for certain three - qubit states , there exist polynomials of the measure that satisfy monogamy for those states .
|
the _ swift _ satellite @xcite certainly revolutionised the grb field , but it has also shaken some of our pre_swift _ strong beliefs .
one of those was the reassuring division between the prompt and the afterglow phases understood in terms of internal shocks between relativistic shells followed by the external shock of the fireball with the circumburst medium .
this scenario could well explain the erratic and violent prompt emission lasting up to a few tens of seconds , and most of the ( then existing ) observations of the afterglow , typically starting a few hours after the trigger .
the afterglow phase was thought to be somewhat better understood , since it was constructed upon the very solid basis of the conservation of energy and momentum , shock acceleration physics and the synchrotron process as the main radiation mechanism ( that was even confirmed by the detection of a small , but significant , linear polarisation ; see @xcite for a recent review ) . although some problem remained to be solved for the prompt phase ( as the inherently low efficiency of internal shocks and the hardness of the observed spectra if the emission process is synchrotron by cooling electrons , @xcite ) ,
the overall scenario was satisfying , especially after the discovery that long grbs were indeed associated with powerful supernovae , confirming that the central engine had to be an hyper accreting stellar size black hole .
the _ swift _ capability of a very fast slew allowed to explore the very early phases , soon discovering the unexpected steep flat steep " behaviour of the light
curve , especially in the x - rays @xcite .
although interpreted in several ways ( see e.g. @xcite for a recent review ) , none seems conclusive . the spectral slope does not change across the temporal break from the shallow to the normal decay phase , ruling out a changing spectral break as a viable explanation .
an hydrodynamical or geometrical nature of the break is instead preferred .
specific models often focused only on the x ray behaviour , but there is another crucial information that can not be overlooked : _ the x ray and optical light
curves often do not track one another _
( e.g. @xcite ) , in a way that can not be accommodated by the standard external shock / fireball scenario .
it is this remarkable characteristic that suggests that there must be two components contributing to the observed flux .
[ f1 ] shows two examples of optical and x ray afterglows , to illustrate the different behaviour of the optical and x ray light curves in grb 061126 , while they track one another in grb 060614 .
we ( @xcite , see also @xcite ) then used a simple phenomenological 2component model ( briefly described in the following section ) to fit _ both _ the x ray and optical light curves of _ swift _ grbs with redshift and a well sampled optical light
curve , allowing to estimate the optical extinction due to dust in the host galaxies . as of march 2008
, they amount to 33 grbs .
the logic we follow is to see if with a minimum number of parameters we can fit the observed light curves of these bursts .
if so , we can find some hints for a more physical interpretation .
we assume that at all times the flux is the sum of two components : a standard afterglow plus ( for the time being ) a completely phenomenological second component . * component 1 : * this is the synchrotron radiation produced by the standard forward shock caused by the fireball running into the circum burst material . for this component
we follow the prescription of @xcite , that requires 6 free parameters , plus the assumption of an homogeneous or a wind like profile of the circumburst density .
these are : i ) the isotropic equivalent kinetic energy of the fireball @xmath1 ; ii ) its initial bulk lorentz factor @xmath2 ; iii ) the value of the circum burst medium density @xmath3 ;
iv)v ) the `` equipartition '' parameters @xmath4 and @xmath5 ; vi ) the slope of the relativistic electron energy distribution @xmath6 . *
component 2 : * it is treated phenomenologically , since its form / origin is not currently known , though it can be possibly ascribed to the extension in time of the early prompt emission ( as discussed below ) .
we parametrise this component with the only criterion of minimising the number of free parameters . for simplicity
we assume that its spectral shape is constant in time and is described by a broken power
law , with spectral indices @xmath7 and @xmath8 below and above the break frequency @xmath9 .
the temporal parameters are described by the flat and steep decay indices , @xmath10 and @xmath11 respectively , and the time @xmath12 at which the two behaviours join .
the time @xmath12 is the time at which the shallow phase ( of the steep flat
steep behaviour ) ends .
then we have 3 spectral and 3 temporal parameters , and we must add one normalisation .
component 2 is thus determined by 7 free parameters , that can be rather well constrained by observations when this component dominates the emission . in this case @xmath10 , @xmath11 and @xmath12 can be directly determined , as well as one spectral index ( usually @xmath13 , since the late prompt emission is usually dominating in the x ray range ) .
, as a function of the isotropic energy of the prompt emission , @xmath14 .
the dashed line corresponds to the least square fit , @xmath15 \propto e_{\rm \gamma , iso}^{0.86}$ ] ( chance probability @xmath16 , excluding the outlier grb 070125 ) .
right panel : the kinetic energy @xmath1 after the prompt emission as a function of @xmath17 .
the dashed line is a least square fit , yielding @xmath18 ( chance probability @xmath19 , excluding grb 070125 ) .
adapted from @xcite .
, title="fig:",height=321 ] , as a function of the isotropic energy of the prompt emission , @xmath14 .
the dashed line corresponds to the least square fit , @xmath15 \propto e_{\rm \gamma , iso}^{0.86}$ ] ( chance probability @xmath16 , excluding the outlier grb 070125 ) .
right panel : the kinetic energy @xmath1 after the prompt emission as a function of @xmath17 .
the dashed line is a least square fit , yielding @xmath18 ( chance probability @xmath19 , excluding grb 070125 ) .
adapted from @xcite .
, title="fig:",height=321 ]
fig . [ f1 ] shows two examples of our modelling ( in @xcite one can find all the 33 grbs we have analysed ) . in these two illustrative examples both grbs
are dominated by component 2 at late times in both bands , while the standard afterglow emission ( i.e. the synchrotron emission from the external shock ) is important only at very early times .
the complex shape of the optical light curve of grb 061126 is due to the emerging of component 2 at @xmath201000 s. the break at @xmath21 s has nothing to do with a jet break , but is the end of the shallow component ( i.e. it is @xmath12 , the time break of component 2 , see @xcite for more discussion ) .
if this burst ( of known @xmath22 ) had to follow the ghirlanda " relation , then it should have a jet break a few days after trigger ( rest frame time ) as indicated by the vertical line .
but at this time the emission is dominated by component 2 , _ thus the jet break is hidden , and can not be observed_. see also @xcite for more examples and discussion of this point . for grb 060614 the entire observed light curves are dominated by component 2 : also in this burst the very prominent and _ achromatic _ break at @xmath2040,000 s ( rest frame ) has again nothing to do with a jet break , being associated to @xmath12 .
note that in our modelling we do not treat the steep phase immediately following ( in most , but not all grbs ) the prompt phase , nor the x ray or optical flares .
consider also that the standard afterglow code we use assume isotropic emission , and so it can not produce any jet break ( i.e. for simplicity we do not use the opening angle of the jet as a free parameter ) . and
@xmath23 , as labelled .
the x ray luminosity profile is flatter than @xmath23 and closer to a @xmath24 decay .
however , this behaviour is due to the contribution in some grbs of the afterglow emission at late times , flattening the overall light curve .
in fact the right panel shows the modelled x ray light curves . on the left
we show component 2 ( i.e. the late prompt " ) , in the middle the afterglow and on the right their sum .
the dashed lines correspond to @xmath25 or @xmath26 , as labelled . adapted from @xcite .
, title="fig:",height=321 ] and @xmath23 , as labelled .
the x ray luminosity profile is flatter than @xmath23 and closer to a @xmath24 decay .
however , this behaviour is due to the contribution in some grbs of the afterglow emission at late times , flattening the overall light curve .
in fact the right panel shows the modelled x ray light curves . on the left
we show component 2 ( i.e. the late prompt " ) , in the middle the afterglow and on the right their sum .
the dashed lines correspond to @xmath25 or @xmath26 , as labelled . adapted from @xcite .
, title="fig:",height=321 ] considering now the entire sample of 33 grbs , we can see if there is some relation between the ( kinetic ) energy of the fireball @xmath1 and the energy emitted during the prompt phase ( i.e. @xmath14 ) .
similarly , we can see how the energy emitted by component 2 , approximated by @xmath27 , relates with @xmath14 .
we are aware of the danger of finding signs of a correlation when considering two energies ( or luminosities ) , since both quantities are function of redshift , but we can compare the two plots . fig .
[ f2 ] shows that @xmath27 correlates with @xmath14 better than @xmath1 .
this suggests that the early prompt and component 2 are related .
the most important results of our analysis is the found distribution of @xmath11 , the decay index of component 2 after @xmath12 . fig .
[ f3 ] shows a clustering around @xmath28 .
this value is equal to the time profile of the accretion rate of fallback material in supernovae ( @xcite , @xcite , @xcite ) , and to the average decay of the x ray flare luminosity , as analysed by @xcite .
this is _ not _ the average observed decay slope : the x ray light
curves are flatter than @xmath29 ( see fig . [ f4 ] ) , but this is due to the contribution , especially at early and late times , of the emission of the standard afterglow . the light
curves of component 2 are indeed steeper , on average , than the total emission that reproduces the data .
the fact that @xmath30 strongly suggests that component 2 can be interpreted as due to the late time accretion of fallback mass , namely material that failed to reach the escape velocity from the exploding progenitor star , and falls back .
this can continue for weeks , enough to sustain late prompt emission even at very late times .
furthermore , the fact that also x ray flares follow a similar behaviour suggests that both x ray flares and component 2 have a common origin , related to the accretion of the fallback material .
the puzzling features of the early afterglow disclosed by _ swift _ have revealed an unforeseen complexity , difficult to explain in terms of the standard internal / external shock scenario .
many alternatives have been proposed ( see the introduction of @xcite for a brief list of models and @xcite for a review ) , but only a few are able to explain a different behaviour of the optical and x ray light curves .
uhm & beloborodov @xcite and genet , daigne & mochkovitch @xcite suggested that the x ray plateau emission is not due to the forward , but to the reverse shock running into ejecta of relatively small ( and decreasing ) lorentz factors .
the optical can instead be due to the standard emission of the forward shock .
this scenario requires an appropriate @xmath31distribution of the ejecta , and also the suppression of the x ray flux produced by the forward shock .
we ( @xcite ) instead suggested that the plateau phase of the x ray emission ( and sometimes even of the optical ) is due to a prolonged activity of the central engine ( see also @xcite ) , responsible for a `` late
prompt '' phase : after the early standard " prompt the central engine continues to produce for a long time ( i.e. days ) shells of progressively lower power and bulk lorentz factor .
the dissipation process during this and the early phases occur at similar radii ( namely close to the transparency radius ) .
the reason for the shallow decay phase , and for the break ending it , is that the @xmath31factors of the late shells are monotonically decreasing , allowing to see an increasing portion of the emitting surface , until all of it is visible .
then the break occurs when @xmath32 , at @xmath12 .
the shallow phase is then the result of a balance : the total emitted luminosity is decreasing ( with a decay slope @xmath11 ) , but for some time ( before @xmath12 ) the surface visible to us is increasing ( because of the decreasing @xmath31 ) .
the combination of these two effects flattens the observed flux decay , that appears to be characterised by @xmath10 .
after @xmath12 the entire emitting surface is visible , and we see the real " decay slope @xmath11 . with respect to other models ,
late prompt " scenario is very economic , since the total extra energy needed is a relatively small fraction of what used during the real prompt phase .
finally , we would like to collect the different pieces of evidence found in these studies , to try to construct an heuristic simple scenario .
let us assume that , following the death of a massive star , a rapidly spinning black hole is formed .
the fast rotation of the equatorial material prevented it to immediately fall into the black hole .
this material forms a very dense torus .
viscosity and angular momentum conservation makes this torus to spread , and accretion takes place .
this accretion phase corresponds to the real prompt emission , possibly mediated by the strong magnetic field formed in the vicinity of the black hole , making the blandford & znajek @xcite mechanism at work .
the energy stored in a maximally spinning black hole of 2 solar masses , and that can be extracted , amounts to @xmath33 erg .
we need only a few per cent of it .
when the bulk of the dense torus has been accreted , there must be a discontinuity in the accretion rate , to explain the steep behaviour of the light
curve following the real prompt emission .
this discontinuity could be associated to the transition from the end of the accretion of the material of the dense torus and the beginning of the accretion of the fallback material .
after this transition phase accretion proceeds at a reduced rate : this phase is associated to fallback , with its typical time profile .
a reduced accretion most likely corresponds to a reduced magnetic field in the vicinity of the black hole , and so to a reduced capacity to extract the spin energy of the black hole .
occasionally , however , fragmentation of the accreting material leads to temporary enhanced accretion , that we can associate to the x ray flares .
9 n. gerhels , g. chincarini , p. giommi , et al . , _ apj _ , * 611 * , 1005 ( 2004 ) s. covino , proceedings of the conference `` the coming of age of x - ray polarimetry '' , rome , italy , 27 - 30 april 2009 ( astro ph.0906.5440 ) g. ghisellini g. , a. celotti , d. lazzati d. , _ mnras _ , * 313 * , l1 ( 2000 ) g. tagliaferri , m. goad , g. chincarini , et al . , 2005 , _ nature _ , * 436 * , 985 j.a .
nousek , c. kouveliotou , d. grupe , et al . , 2005 , _
apj _ , * 642 * , 389 b. zhang , 2007 , _ advances in space research _ , * 40 * , issue 8 , p. 1186
( astro ph/0611774 ) a. panaitescu , p. meszaros , d. burrows , n. nousek et al .
2006 , _ mnras _ , * 369 * , 2059 a. panaitescu , _ il nuovo cimento _ , * 121 * , 1099 ( 2006 ) g. ghisellini , m. nardini , g. ghirlanda , a. celotti , _ mnras _ , * 393 * , 253 ( 2209 ) m. nardini , g. ghisellini , g. ghirlanda , a. celotti , this volume ( 2009 ) l. nava , g. ghisellini , g. ghirlanda , et al .
, _ mnras _ , * 377 * , 1464 ( 2007 ) a. panaitescu , p. kumar , _ apj _ , * 543 * , 66 ( 2000 ) g. ghirlanda , g. ghisellini & d , lazzati , 2004 , _ apj _ , * 616 * , 331 g. ghirlanda , l. nava , g. ghisellini , c. firmani , _ a&a _ ,
* 466 * , 127 ( 2007 ) d. lazzati , r. perna , m.c .
begelman , _ mnras _ , * 388 * , l15 ( 2008 ) r.a .
chevalier , _ apj _ , * 346 * , 847 ( 1989 ) a.i .
macfadyen , s.e .
woosley , a. heger , _ apj _ , * 550 * , 410 ( 2001 ) w. zhang , s.e .
woosley , a. heger , 2007 , _ apj _ , * 679 * , 639 ( 2007 ) l.z .
uhm & a.m. beloborodov , _ apj _ , * 665 * , l93 ( 2007 ) f. genet , f. daigne , & r. mochkovitch , _ mnras _ , * 381 * , 732 ( 2007 ) g. ghisellini , g. ghirlanda , l. nava & c. firmani , _ apj _ , * 658 * , l75 ( 2007 ) d. lazzati & r. perna , _ mnras _ , * 375 * , l46 ( 2007 ) r.d .
blandford , r.l .
znajek , _ mnras _ , * 179 * , 433 ( 1977 )
|
we call prompt " emission of gamma ray bursts ( grbs ) the erratic and violent phase of hard x - ray and soft @xmath0ray emission , usually lasting for tens of seconds in long grbs
. however , the central engine of grbs may live much longer .
evidence of it comes from the strange behaviour of the early afterglow " , seen especially in the x ray band , characterised by a
steep flat steep " light curve ,
very often not paralleled by a similar behaviour in the optical band .
this difference makes it hard to explain _ both _ the optical and the x ray emission with a unique component .
two different mechanisms seem to be required .
one can well be the standard emission from the forward shock of the fireball running through the interstellar medium .
the second one is more elusive , and to characterise its properties we have studied those grbs with well sampled data and known redshift , fitting _ at the same time _ the optical and the x ray light
curves by means of a composite model , i.e. the sum of the standard forward shock emission and a phenomenological additional component with a minimum of free parameters .
some interesting findings have emerged , pointing to a long activity of the central engine powered by fallback material .
address = inaf osservatorio astronomico di brera , via e. bianchi , 46 i23807 , merate , italy address = sissa / isas , via beirut 24 , i34151 , trieste , italy , address = inaf osservatorio astronomico di brera , via e. bianchi , 46 i23807 , merate , italy
|
determining the energy levels of a quantum system was a desperate enterprise before the advent of the bohr sommerfeld quantization conditions .
max planck had emphasized that the energy of the harmonic oscillator is quantized in units of a fundamental energy given by the product of what we now call dirac s constant and the frequency of the oscillator .
however , this rule failed miserably when used in other quantum systems .
paul ehrenfest s adiabatic principle applied to the pendulum @xcite whose length slowly changes as a function of time , made clear that it is not the energy , but the action that is quantized , but why ?
when we change the length of the pendulum adiabatically , the amplitude of oscillation does not stay constant , neither does the frequency nor the energy .
what stays constant is the action , that is , the area in phase space . for the founding fathers of quantum mechanics
it must have been a miracle , an amazing fact symbolizing in post
schrdinger language that the number of nodes in an energy wave function stays constant under adiabatic changes .
the classical dynamics of the pendulum serves as an excellent guide for many quantum phenomena . for example , the upside down pendulum yields insight into the energy wave function of a periodic potential @xcite .
here we do not focus on the familiar bloch states which appear when the potential enjoys a strict periodicity over the whole space .
our states occur when the modulation of the potential extends only over a finite domain of space . due to the shape of this potential
we refer to it as the _ accordion potential_. extensions of these one dimensional considerations to two and three dimensions lead us to the effect of the quantum anti centrifugal force @xcite . in the present paper
we take seriously the joke ` physics takes mathematics and makes it understandable ' , therefore we shall suppress all the mathematics and highlight the essential ideas .
this approach is justified by the fact that we are not inventing or applying new mathematics , but attempt to draw together phenomena of different fields exposing a common thread .
we shall focus on a general point of view but emphasize that the modern tools of cold atoms in a standing electromagnetic wave can demonstrate these unusual bound or localized states predicted in this paper .
we follow one ` leitmotif ' , that is , a theme common to all phenomena discussed in this article : the classical physics of the upside down pendulum .
moreover , we develop a little side theme summarized by the phrase `` wave mechanics in quantum potentials '' .
our paper is organized as follows : in sec.2 we briefly summarize the classical physics of the upside down pendulum and then turn in sec.3 to related phenomena , such as the paul trap , the helium atom and field induced multipoles .
we devote sec.4 to the transition to quantum mechanics and in particular , discuss the shape of the ground state wave function in the accordion potential . in sec.5 we show that in three dimensions a localized ground state can exist even at zero energy . the quantum anti
centrifugal potential emerges when in sec.6 we focus on the schrdinger equation in two dimensions .
we conclude in sec.7 with an outlook and present extensions of the concepts discussed in this paper .
an appendix summarizes the mathematical aspects of the upside down pendulum and the schrdinger equation of the particle in a periodic potential .
in the mid - forties it was recognized @xcite that the cyclotron would never be able to accelerate particles to energies above a critical value .
the problem lies in the fact that in free space static electric or magnetic or both fields together can not focus a beam of charged particles in all planes through the axis of the beam .
fortunately , this difficulty , which applies to linear accelerators , can be circumvented to some degree in circular systems .
however , the degree of focusing permitted by the effect of the centrifugal force is extremely weak .
these considerations suggested that energies higher than 10 mev are not attainable without unreasonable expenditures of money and materials .
however , in 1949 nick christofilos from athens , who at that time was an elevator operator , dreamt up the principle of strong focusing .
although he got a patent , his work was unpublished and the principle was re discovered @xcite in 1952 .
the central idea of the principle of strong focusing is to use a sequence of strong focusing and defocusing fields to achieve a net focusing effect .
this idea is best illustrated using the principle of the upside down pendulum @xcite .
consider the motion of a point mass on one end of a massless rod .
the opposite end of the rod is connected via a hinge to another rod that is fixed to a support .
this pendulum experiences a constant gravitational force pulling downwards .
when the point of suspension is such that the pendulum hangs down and we concentrate on the limit of small displacements from the stable hanging position we can approximate the motion of the mass by that of a harmonic oscillator . however , when the pendulum is upside down and stands straight against the gravitational force , the motion is unstable and the mass tends to fall down . in this case
we face an inverted harmonic oscillator of negative steepness . in order to stabilize it ,
pyotr l. kapitza @xcite in 1951 suggested a rapid vertical modulation of the foundation of the pendulum .
when the modulation frequency is above a critical value determined by the length of the pendulum , the gravitational acceleration and the modulation amplitude , the motion of the pendulum becomes stable .
why is this so ?
many mathematical arguments offer themselves .
they range from the method of averaging @xcite via the secular growth theory @xcite to the floquet theorem @xcite .
however , none of these mathematical techniques provide deeper insight into the physics of the stabilizing mechanism . for the sake of argument and to illustrate this lack of insight , we follow the floquet reasoning .
the equation of motion for the phase angle @xmath0 of the pendulum , measured from its upside down unstable position , is the mathieu equation @xcite , as discussed in the appendix .
the floquet theorem @xcite determines the domains of stability and instability of this equation .
hence , it is the stability chart @xcite of the mathieu equation that governs the parameter regime for which the upside down pendulum is stable . what is the mechanism of this stabilization ? when the foundation accelerates the pendulum upwards , the gravitational force which pulls the pendulum down and makes its motion unstable , is reduced .
a push up therefore corresponds to a stabilizing force . on the other hand
, a motion of the foundation downwards increases the effective gravitational force and thereby enhances the instability of the pendulum : a pull down corresponds to a de stabilizing force . on the first sight
one might think that during a complete cycle of modulation , the two effects average out , however , during a majority of time there is a pushing up effect . why is this ? in the stable mode
the pendulum performs two types of motion , the secular and the micro motion .
the secular motion corresponds to a slow oscillation of the mass between two extremes of phase angles .
in addition , the mass undergoes a rapid motion with a frequency identical to the modulation frequency .
it is this micro
motion , superimposed on the secular motion , which makes the pendulum stable and which breaks the symmetry between the upward and the downward push .
indeed , due to the geometry of the pendulum , the angular acceleration is not only determined by the acceleration due to the modulation , but by the product of this acceleration and the instantaneous phase angle .
the corresponding force is therefore a tidal force : no displacement of the oscillator no force ; large displacement large force .
the modulation of the foundation translates into a modulation of the phase angle . since the effective acceleration is the product of the acceleration due to the modulation and the phase angle we find the product of two oscillatory functions with the same phase .
the time average of such a quantity is a positive constant providing a harmonic oscillator potential of positive steepness for the slow motion .
many other applications of the upside down pendulum come to mind , for example the paul trap @xcite that allows us to store and manipulate single ions in a controlled way .
it is worth mentioning that paul traps play an important role in the recent proposals for a quantum computer . in the paul trap
the need for a time dependent force is dictated by the poisson equation of electrostatics making it impossible to create binding forces in all three dimensions of space .
indeed , the poisson equation enforces the feature that even when two spatial directions enjoy a binding potential the third one is anti binding . to overcome the instability of the trap in the third direction two possibilities offer themselves :
apply a time independent homogeneous magnetic field that confines the ion in the third direction or apply an alternating voltage to the trap .
the first approach corresponds to the penning trap , the second to the paul trap . in the most elementary case
the paul trap consists of a quadrupole field , giving rise to harmonic oscillator potentials .
the alternating voltage applied to the electrodes creates a dynamical binding in three dimensions . in each dimension
the physics of this binding phenomenon is identical to the upside down pendulum .
the helium atom represents another interesting application of this concept of dynamical binding . for more than one hundred years
physicists have tried to understand the motion of two electrons around the nucleus .
a one dimensional model in which the electrons move along a line with the nucleus at the origin brings out the essential physics . on first sight ,
a situation in which the electrons are on different sides of the nucleus seems to be preferable since in this way the electrons can avoid each other in a most effective way .
however , an extremely interesting situation @xcite occurs when both electrons are on the same side .
indeed , here one electron is close to the nucleus and oscillates rapidly between two equilibrium points .
the second electron is far away from the nucleus and moves slowly feeling the attractive force of the nucleus and the repulsion from the inner electron .
indeed , the fast motion of the inner electron creates a time averaged repulsive potential , which superposes with the attractive coulomb potential of the nucleus , giving rise to a local potential minimum for the outer electron . in this case , the electron electron repulsion , together with the rapid motion of the inner electron , forms a time dependent barrier to stop the unstable motion of the outer electron caused by the attraction towards the nucleus .
this phenomenon is the upside down pendulum in disguise @xcite .
our last illustration of effective potentials , arising from time averaging , is the effect of an induced dipole or multipole . it helps us to make the transition to unusual bound states in quantum mechanics .
two examples offer themselves , ( _ i _ ) the binding of an electron in a negative ion and ( _ ii _ ) the atom in a standing electromagnetic wave .
the field of an electron can polarize a neutral atom even when the electron is at a distance large compared with the atomic dimensions .
this interaction between the electron and the atom leads to a force of attraction .
this attraction is the reason why some atoms , such as hydrogen or halogen , are able to form negative ions by an attachment of an electron .
we now consider the motion of an atom in a standing electromagnetic wave . indeed , in quantum optics many such experiments have been and are being performed .
the light induces a dipole moment in the atom and this dipole again interacts with the light field . since the interaction energy is the product of the induced dipole and the field , it enters quadratically .
consequently , the position dependence of the interaction energy follows from the position dependence of the square of the field that is , from the square of the mode function thereby creating the effective potential for the motion of the atom .
all examples in the previous sections illustrate how dynamical binding can occur in classical physics .
we now turn our attention from newton s equation of motion @xmath1\varphi(t)=0\ ] ] for a harmonic oscillator with time dependent steepness @xmath2 to the time independent schrdinger equation @xmath3 \
u(x)= 0 \label{schrodinger}\ ] ] for a point particle of mass @xmath4 in a position dependent potential @xmath5 ; both equations are identical in form .
the role of the angle coordinate of the classical upside - down pendulum is now played by the energy wave function @xmath6 .
moreover , the time variable @xmath7 is replaced by the coordinate variable @xmath8 . indeed ,
newton s equation of motion is an equation of second order in time for the position and the schrdinger equation is an equation of second order in position for the wave function .
we emphasize that this analogy between the two equations only holds true for the classical harmonic oscillator .
whereas the schrdinger equation is always linear , a feature independent of the form of the potential , the form of newton s equation strongly depends on the shape of the potential and in general is nonlinear .
we focus on a periodic potential in space
. the particular shape of this potential is of no importance as long as it is averages out to zero over space ; it is as often positive as it is negative . for our analysis
it is crucial that the oscillations of the potential start at a given point in space and end thereby connecting a flat space through a domain of wiggles back to a flat space .
we measure energy relative to flat space .
fig.1 shows an example of such a potential .
it is not surprising that under these circumstances we find bound states of negative energy , however , under appropriate conditions , this potential can also display a localized state of zero energy .
thus , it is again the physics of the upside down pendulum that allows us to understand this phenomenon .
we first concentrate on the one dimensional case .
guided by the analogy between the newton and the schrdinger equation we expect the corresponding ground state energy wave function to exhibit a slowly varying envelope that is modulated with the period of the periodic potential ; but where does the binding come from ?
the origin of this bound state lies in the modulation of the envelope without the modulation there is no binding .
the modulation can be easily understood : the particle is more likely to be in a stable minimum of the potential rather than in an unstable maximum .
the curvature of the wave function at a given position , expressed by its second derivative , is determined , not only by the potential , but also by the wave function at that point .
indeed , according to the schrdinger equation ( [ schrodinger ] ) , it is the product of the potential and the wave function that governs the curvature .
this quantity contains , apart from other terms , the product of the oscillatory potential and the modulations of the wave function .
the simplest case of a cosine modulation gives rise to the square of the cosine , thus creating a constant potential and a part that varies in space with twice the modulation frequency .
we focus on the constant part and note that this constant is always negative . in order to explain this feature we recall that the modulation of the envelope of the wave function is such that at the minima of the potential we have local maxima of the wave function and vice versa .
hence , the modulation and the potential defer in their sign and their product creating the potential well is negative .
the fact that the potential only wiggles over a finite domain makes this constant negative potential into an attractive potential well whose width is determined by the range of the oscillations .
the depth of the well involves , apart from other parameters , the square of the height of the periodic potential . for more mathematical treatment and explicit expressions for
the well , we refer to the appendix .
there is a close analogy between the ground state in the periodic potential and the problem of the upside down pendulum . in both cases
the binding effect is a consequence of the average of the product of two oscillatory functions : for the pendulum it is the time dependent modulation of the foundation and the time dependent angle of oscillation , which creates a harmonic oscillator of positive steepness . when considering the particle in the periodic potential
, it is the product of the position dependent potential and the wave function modulated in space that gives rise to a well .
however , there is a dramatic difference : whereas the modulation of the pendulum is on for all times , the spatial modulation is confined to a certain domain of space .
it is this confinement in space , together with the creation of an effective potential , which defines the bound state .
the spatial domain , over which the wiggles are present , and the depth of the potential well , determine the ground state energy .
is it possible to choose these parameters so as to achieve a ground state of zero energy ?
the answer is : it depends ! in one dimension the answer is a flat no , but in three dimensions it is possible , as we shall now show .
we start by considering the situation in one dimension .
as we decrease the depth of the well and the energy of the ground state approaches zero , the tails of the wave function reach more and more into the forbidden region outside of the well .
this feature is a consequence of the fact that the decay is governed by the square root of the absolute value of the energy . in the limit of zero energy
the wave function is no longer localized ; no mechanism prevents the wave from flooding all space .
consequently , there is no ground state of zero energy in one dimension .
is it possible to construct such a zero - energy bound state in higher dimensions ? in order to answer this question , we first concentrate on the three - dimensional case and consider a potential @xmath9 that wiggles along the radial direction . beyond a final radius @xmath10 , the potential vanishes
. the time independent schrdinger equation @xmath11 \ u_{0}(r)=0 \label{radial}\ ] ] for the radial energy wave function @xmath12 corresponding to vanishing angular momentum is then identical to the one - dimensional case .
due to the product of the oscillatory potential and the wave function , and the finite domain of oscillations , we find an attractive potential well of radius @xmath10 .
there is however , a subtle difference to the one - dimensional case : it is the origin where the radial wave function has to vanish @xcite corresponding to an infinitely high potential wall at the origin .
this feature is a consequence of the spherical symmetry .
therefore , the infinite wall at the origin provides one side of the effective well creating even a node of the wave function .
the other wall of finite height is due to the effective potential together with the finite range of the oscillatory domain . for appropriate parameters such as the final radius and the depth of the modulation of the potential we can fit a ground state of negative energy into this well , as shown in fig.2 .
the corresponding wave function displays a node at the origin and an exponential decay in the classically forbidden domain beyond the critical radius , with a single maximum lying in between .
can we now take the limit of zero energy and thus realize a zero - energy ground state ?
the answer is yes !
indeed , we can arrange the depth of the well such that exactly one quarter of a period of a sine oscillation fits into the potential well .
why a quarter of a period ?
since the decay of the wave function outside of the potential is governed by the square root of the energy we find that for zero energy the wave function is constant . due to the condition of continuity on the wave function it must take on the value at the wall .
the radial wave function displays a node at the origin and increases like a sine function to a constant value .
however , the sine function must merge smoothly into the constant , which is only possible at a point where the sine function has an extremum . since we want to have the ground state without a node , except the one at the origin , the only possibility for such a merger appears at one quarter of a period @xcite of the sine function .
obviously , this zero energy ground state radial wave function does not display any localization .
the localization becomes apparent when we recall that we have to divide the radial wave function by the radial variable in order to find the total wave function , that is @xmath13 indeed , now the total wave function enjoys a maximum at the origin and decays with the inverse power of the radius .
we emphasize however , that the total wave function is not square integrable since the volume element @xmath14 brings in the square of the radius .
this contribution cancels exactly the factor creating the localization in the probability density , that is , in the absolute value squared of the wave function . for this reason we refer to this state as localized state rather than bound state .
we conclude this discussion by briefly addressing the case of positive energies . here
, the exponential decay into the classically forbidden region of a wave function corresponding to negative energy turns into right and left running plane waves of positive energy .
obviously , plane waves are not localized wave functions , nevertheless , the total wave function is concentrated at the origin due to the inverse power of the radius ; in this sense we have localized wave functions in the continuum .
the physics of a bound state in two dimensions is very different from the one in other dimensions ; and this for many reasons . in @xcite
we have studied this case in more detail . in the present paper
we only highlight and motivate the results of @xcite . for detailed derivations
we refer to that article . to bring out the peculiarities of the two dimensional case we first consider the case of no external potential . in two dimensions a vanishing angular momentum
does not imply @xcite a vanishing centrifugal potential as in three dimensions .
indeed here the radial schrdinger equation reads @xmath15 \ u_{0}(r)=0 . \label{polar}\ ] ] we can trace the origin of the new potential @xmath16 back to the non - vanishing commutation relation between the operators of momentum and the radial unit vector .
this potential is , therefore , a true quantum potential . since , in addition , it is attractive rather than repulsive we have named it the quantum anti - centrifugal potential @xcite .
moreover , the potential is attractive only in a two dimensional world . for one or three dimensions it vanishes , but for higher than
three dimensions it becomes repulsive .
the quantum anti - centrifugal potential has brought us into a rather unusual situation .
usually , we perform wave mechanics in a classical external potential . in the present context however , we have a quantum potential .
this feature stems from the reduction of space from three to two dimensions , reminiscent of the born - oppenheimer approximation in molecular physics .
a molecule is a quantum system consisting of several degrees of freedom . in the framework of
oppenheimer approximation we solve the schrdinger equation for the electrons for a fixed position of the nuclei and then use the resulting potential curves , that is , the electronic energies as a function of the separation of the nuclei to determine their relative motion . in this sense
we indeed perform wave mechanics on potentials that are not due to external forces but due to quantum mechanics . in the example of the nuclear motion in a molecule ,
the appearance of wave mechanics in a quantum potential is a consequence of the reduction of the degrees of freedom @xcite .
the attractive quantum anti - centrifugal potential manifests itself even in the energy wave function of a free particle , that is in the absence of any other external potential .
the ordinary bessel functions @xmath17 and @xmath18 are two independent solutions of the time independent schrdinger equation for positive energy .
both show a bunching of nodes towards the origin , as suggested by an attractive potential leading to an acceleration of the particle towards the origin and demonstrated for the example of @xmath17 in fig.3 .
we have to compare and contrast this situation with the case of a particle with one unit of angular momentum , where the centrifugal potential is indeed repulsive .
it is worth noting that the quantum anti - centrifugal potential reduces the strength of repulsion of the centrifugal potential and the effective potential is not as repulsive as in the classical case . in the case of one unit of angular momentum ,
the solutions of the time independent schrdinger equation are the ordinary bessel functions @xmath19 and @xmath20 .
as expected by the repulsive potential enforcing a deceleration of the particle as it approaches the origin , the nodes of the bessel functions show anti - bunching towards the origin .
one might wonder if the quantum anti - centrifugal potential is strong enough to support a bound state with negative energy . in the transition from positive to negative energies
real wave numbers transform into purely imaginary ones as a consequence of the quadratic dispersion relation between energy and wave number .
therefore , the ordinary bessel functions @xmath17 and @xmath18 turn into modified bessel functions @xmath21 and @xmath22 . since @xmath21 explodes for large arguments @xcite , that is , large radial distances , this solution does not satisfy the requirement of exponential decay enforced by the attractive quantum anti - centrifugal potential .
only the function @xmath22 achieves this goal , but it has another disease : it explodes at the origin like a logarithm .
this singularity at the origin indicates that the solution @xmath23 of the schrdinger equation , shown in fig.4 , does not describe a truly free particle but that there is a delta function potential present .
the strength of this external potential determines @xcite the energy eigenvalue of this bound state .
moreover , due to the quantum anti - centrifugal potential the probability of finding the particle within a given area in space is concentrated along a band around the location of the delta function ; a remarkable feature unique to two dimensions . in a similar arrangement in one or three dimensions
the maximum of the probability is at the location of the potential minimum .
this property might have interesting applications for guiding atoms along wires @xcite or electro - magnetic waves in wave - guides @xcite .
we now return briefly to the case of an energy eigenstate in a two dimensional oscillatory potential of finite range . as emphasized in the previous paragraph , the case of vanishing angular momentum
contains the most quantum effects . here
, the ground state wave function of negative energy feels the combination of the quantum anti - centrifugal potential and the potential well created by the product of the wave function and the potential .
the examples discussed in the preceding sections represent but only a small subsection of the class of unusual bound states . they have been selected because they follow or are motivated by the physics of the upside - down pendulum . however , there are many more intriguing bound states in atomic and molecular physics @xcite .
we conclude our paper by briefly summarizing three examples starting with one that is still closely related to our main theme of the upside - down pendulum and gradually moving away towards new frontiers .
consider a diatomic molecule with various electronic states . within
oppenheimer approximation the electronic states provide the potentials for the relative motion of the two nuclei .
a strong external time dependent laser field interacts with the electrons and couples the individual electronic potential surfaces .
the wave functions for the vibratory motion of the nuclei are therefore coupled through a time dependent periodic interaction hamiltonian .
the product of the interaction hamiltonian and the nuclear wave functions govern the time evolution of the wave functions ; a feature that is again reminiscent of the upside - down pendulum . in the latter the product
consists of the time dependent steepness of the oscillator and the phase angle .
the only difference lies in the fact that in the classical case the newton equation is of second order , whereas the schrdinger equation is a first order .
however , this distinction is superficial since we are dealing with the nuclear wave functions corresponding to different electronic states and hence , with a system of differential equations of first order . due to this product of two periodic functions with the same period ( interaction hamiltonian and wave function )
we find again a constant which lifts up the electronic potential by integer multiples of the energy of the absorbed photon .
in mathematical terms this energy elevation is a consequence of floquet states .
however , it can be viewed as just one more illustration of the upside - down pendulum with counter - intuitive consequences . indeed , when we consider a binding and a repulsive potential this constant shift in energy can result in a crossing of the potential curves . since in a diatomic molecule
there exists a non - crossing rule we obtain an avoided crossing forming new potentials ; one is binding and one is repulsive .
however , these potentials are rather peculiar because each of them consists of parts of the two original potentials .
the formation of new bound states @xcite in this light induced molecule has been observed experimentally @xcite .
trojan asteroids are celestial bodies that are held in their positions by the gravitational forces of the sun and jupiter
. a similar phenomenon can occur @xcite in an atom where the electron plays the role of a trojan asteroid , and the nucleus substitutes for one of the two planets .
a circularly polarized electromagnetic field simulates the effect of the second planet .
indeed , the electromagnetic interaction of the electron with the field and the coulomb attraction between the electron and the nucleus replace the gravitational attraction of the celestial bodies . in a frame rotating with the electromagnetic field ,
the motion of the electron is in a plane and governed by three potentials which depend on the two - dimensional radial direction : the coulomb attraction of the nucleus , the centrifugal potential of the circular motion and the linear potential arising from the interaction of the electron with the electromagnetic field . for short distances
the coulomb potential dominates , whereas for large distances the repulsive potential of the circular motion prevails .
consequently , for intermediate distances , an unstable potential maximum occurs .
this feature is in complete analogy with the upside down pendulum .
but where does the stabilizing drive come from ?
the answer to this question lies in the fact that the radial motion is coupled to the angular motion .
the latter is locked to the rotating electromagnetic field and the electron performs angular vibrations around this stable point of equilibrium .
these vibrations couple into the radial motion creating a dynamically binding potential .
this classical picture also holds true for the quantum case as verified by extensive studies of wave packets propagating in this minimum . in our last example of unusual bound states
it is the balance of static forces that is responsible for the formation of a bound state of a lone electron in a rydberg atom in crossed electric and magnetic fields @xcite .
the crossed - field situation is of particular appeal in the field of quantum chaology since in this system energy is the only conserved quantity .
the classical system displays chaos and is therefore of interest in the search for fingerprints of chaos in the corresponding quantum system .
apart from these questions of quantum chaos there is another quite interesting aspect of this system .
the electron in the atom experiences , not only the attractive coulomb potential of the nucleus , but also a linear potential due to a constant electric field and a binding harmonic oscillator potential due to the magnetic field .
the superposition of all three potentials creates a local potential minimum far away from the nucleus .
an electron bound in this minimum displays a large dipole moment that can be observed by sending the atom through an inhomogeneous electric field .
indeed , the experiments @xcite have confirmed the existence of this far outside lying minimum . we have come a long way on our journey into unusual bound states .
starting from the classical physics of the upside - down pendulum we have been led to the phenomenon of a zero - energy ground state in a periodic potential in three dimensions .
the two - dimensional world has even more surprises in store : the attractive quantum anti - centrifugal force that manifests itself in a free particle through the bunching of the nodes or even in a bound state . here
the probability of finding the particle is concentrated in the domain where no force is acting .
this feature resulting from the action of the quantum anti - centrifugal potential could be useful in guiding atoms along wires or electromagnetic waves along fibers .
experiments on trapping cold atoms by single photons @xcite or along a whispering gallery mode of a glass sphere resonator @xcite are yet more realizations of the physics of the upside - down pendulum and give us confidence that our predictions of unusual bound and localized states can be verified experimentally in the near future .
this work started when i ( wps ) had the great privilege to be a postdoc with john a. wheeler more than a decade ago .
since that time , john and i have frequently returned to the topics addressed in the present paper .
these discussions took place in trains , planes and automobiles and at various locations such as hightstown , high island , and ulm .
i thank john for this wonderful time , his outstanding hospitality at hightstown and high island and especially for the unique experience to work with him .
the present paper is partially based on notes and sketches of figures prepared jointly .
i am grateful for many fruitful discussions and john s deep insights and , in particular , for allowing us to use the material that had originally been obtained in close collaboration with him .
the proceedings of the lake garda conference are a most welcome opportunity to finally summarize this project started many years ago .
moreover , we are grateful to i. biaynicki - birula for a critical reading of the manuscript .
two of us ( mac and wps ) thank r. bonifacio for his hospitality and for organizing a most interesting conference in the splendid surroundings of lake garda .
the work of wps is partially supported by the deutsche forschungsgemeinschaft , moreover , he gratefully acknowledges a travel grant from the universitt ulm which made part of this research possible .
in contrast to the main body of the paper , we pursue in the present appendix a more mathematical approach .
in particular , we emphasize the similarities between the classical equation of motion for the upside down pendulum and the schrdinger equation for a nonrelativistic particle in a periodic potential . for this purpose
, we first cast the corresponding equations into a dimensionless form .
this approach allows us to simultaneously derive equations for the dynamics contained in the macro and micro motion of the pendulum or the envelope function and the modulation of the ground state wave function .
we start by summarizing the classical equation of motion for the upside down pendulum . for the sake of simplicity
we consider the limit of small angles @xmath24 measured relative to the vertical position .
a vertical acceleration @xmath25 of the foundation translates into a vertical acceleration of the mass @xmath4 of the pendulum and adds to the gravitational acceleration @xmath26 pointing downwards . here ,
the sign of @xmath25 is crucial : when @xmath25 is positive , that is , when the pendulum is accelerated upwards , the gravitational acceleration pointing downwards is reduced . likewise ,
when @xmath25 is negative , that is , when the pendulum is accelerated downwards , the gravitational acceleration is increased .
consequently , the total acceleration is @xmath27 .
hence for small angles @xmath24 the force tangentially to the rod of the pendulum of length @xmath28 reads it is reasonable to assume that the modulation @xmath32 is a periodic function of period @xmath33 .
this time scale allows us to introduce a dimensionless variable @xmath34 and the differential equation of the upside down pendulum takes the form here we have introduced the abbreviations @xmath36 and @xmath37 for the steepness of the inverted harmonic oscillator and the acceleration due to the modulation , respectively .
moreover , prime denotes differentiation with respect to @xmath38 .
the two dimensionless equations of the driven pendulum ( [ eight ] ) and the quantum particle in the periodic potential ( [ ten ] ) have identical structure .
in secs.2 and 4 we have presented qualitative arguments to explain the motion of the pendulum as a superposition of a slow motion ( macromotion ) and a rapid motion ( micromotion ) or , in the language of wave functions , to decompose the energy wave function into an envelope and a modulation .
we now take advantage of the analogy between the two systems to support these qualitative arguments by rigorous mathematics . for this purpose
we start from the equation again , we emphasize that @xmath6 can either be the phase angle of the pendulum or the wave function .
likewise , @xmath38 represents time or position and the drive @xmath47 is either the acceleration of the foundation or the potential .
@xmath50{\cal a } \nonumber \\ & & -\left [ { \cal a}''+\left ( \eta + \frac{1}{\epsilon}+ \frac{v''}{v}\right ) { \cal a}\right ] \epsilon v \nonumber \\ & & -2\epsilon { \cal a } ' v'=0 .
\label{star}\end{aligned}\ ] ] so far , our analysis is exact .
we now solve the equation ( [ star ] ) in an approximate way .
for this purpose , we first neglect the last contribution in ( [ star ] ) involving the product of the first derivatives of the envelope function and the potential .
this approximation is justified since @xmath51 is out of phase with @xmath52 and can not therefore lead to a significant contribution .
moreover , we choose @xmath53 such that we emphasize that this is not possible in a strict sense , since @xmath55 is not necessarily a constant .
however , in the oscillatory domain of the potential the most elementary model @xmath56 suggests the estimate @xmath57 and hence we find for the case of the upside down pendulum , the parameter @xmath61 is negative . in the absence of the drive
, the negative value of @xmath61 leads to an exponential growth of @xmath59 .
however , the square of the drive is always positive and can create an overall positive coefficient in front of @xmath59 , providing an oscillation rather that an exponential explosion . in this case
, the envelope @xmath59 is an energy wave function of this potential well and the parameter @xmath61 is the dimensionless energy eigenvalue .
hence , @xmath61 is not determined from the outside , but rather by the well ( [ well ] ) itself . if the well is deep enough , it can support , apart from the ground state , also excited states with negative energy .
p. ehrenfest , naturwiss .
* 11 * , 543 ( 1923 ) ; for an excellent introduction into and comprehensive summary of the early quantum mechanics we refer to d. ter haar , the old quantum theory , pergamon press , oxford , 1967 , p. 44 . a classic in this field
is m. born , atommechanik , springer , heidelberg , 1925 ; see also j. duck and e. c. g. sudarshan , 100 years of planck s quantum , world scientific , singapore , 2000 . for a summary of the problem of acceleration of charged particles to high energies
see the article by j. p. blewett , in e. u. condon and h. odishaw ( eds ) , handbook of physics , mcgraw - hill , new york , 1958 , chapter 9 , p. 153 .
j. a. wheeler in many discussions with one of us ( wps ) repeatedly credited h. bethe with the discovery of the problems leading to the principle of strong focusing .
unfortunately , we have not been able to locate the appropriate reference .
we have talked to prof .
bethe , but also he was not able to point out the article .
m. abramowitz and i. a. stegun , handbook of mathematical functions , dover publications , new york , 1965 ; for a particularly illuminating method to motivate the stability chart of the mathieu equation using wkb wave functions , see m. j. richardson , am . j. phys . * 39 * , 560 ( 1971 ) .
w. paul , rev .
phys . * 62 * , 531 ( 1990 ) .
a pedagogical introduction into the physics of paul traps emphasizing the concept of the effective potential is given by p. e. toschek , in g. grynberg and r. stora ( eds ) , new trends in atomic physics , north - holland , amsterdam , 1984 , p. 390 .
for a summary of current activities in paul traps see h. walther , in b. bederson and h. walther ( eds ) , advances in atomic , molecular and optical physics , academic press , boston , 1995 .
k. richter and d. wintgen , phys .
* 65 * , 1965 ( 1990 ) ; j. phys .
b * 24 * , l565 ( 1991 ) : for an overview of the full classical and quantum dynamics of two - electron systems see k. richter , g. tanner , and d. wintgen , phys .
a * 48 * , 4182 ( 1993 ) .
we can also interpret the system of the two electrons in the helium atom as one realization of the fermi accelerator . in the most elementary version of this device
an oscillating wall confines the unbounded motion of a particle in a linear potential . for a review of the fermi accelerator see f. saif , i. biaynicki - birula , m. fortunato and w. p. schleich , physics reports , to be published .
l. d. landau and e. m. lifshitz , quantum mechanics , non - relativistic theory , pergamon press , oxford , 1965 , p. 267
; there exists a large amount of literature on the physics of negative ions .
see for example the classic book h. massey , negative ions , cambridge university press , cambridge , 1976 .
for the most recent activities in this field see the springer series , production and neutralization of negative ions and beams .
a similar effect appears also for an electron in the field of a super - heavy nuclei , see for example f. g. werner and j. a. wheeler , phys .
* 109 * , 126 ( 1958 ) .
for summary see c. n. cohen - tannoudji , in j. dalibard et al .
( eds ) , fundamental systems in quantum optics , elsevier science , amsterdam , 1992 ; see also the nobel lectures by s. chu , c.n .
cohen - tannoudji , and w. d. phillips , rev .
phys . * 70 * , 685 ( 1998 ) . for a most striking and counter - intuitive bound state resulting from the peculiar behavior of the radial wave function at the origin of three - dimensional space
we refer to j. rauch and m. reed , comm .
* 29 * , 105 ( 1973 ) ; and m. reed , and b. simon , methods of modern mathematical analysis ii , academic press , new york , 1975 . in this example
the potential is a sequence of appropriately constructed steps that lead continuously downwards as the radial variable increases .
classically a particle of given energy has to fall down the steps .
however , the reflections of the quantum wave at the individual steps interfere in a way as to localize the particle .
likewise , the same authors discuss a potential consisting of an infinite sequence of potential spikes that classically would keep a particle trapped , however , due to the tunneling effect the quantum particle escapes .
a similar reasoning appears @xcite in general relativity , that is in geometrodynamics , when we determine the metric coefficients of a system .
they follow from brill s equation @xcite , which is similar to the time independent schrdinger equation for zero energy .
in contrast to quantum mechanics where the schrdinger equation determines the energy eigenvalues we now have to solve the equation under the constraints that the wave is not allowed to have nodes and corresponds to zero energy .
this observation is in complete accordance with a recent paper arguing that the time dependent schrdinger equation is an approximation of the time independent schrdinger equation resulting from the elimination of degrees of freedom .
see for example j. s. briggs and j. m. rost , epjd * 10 * , 311 ( 2000 ) .
two examples illustrate the unusual bound states that originate from the mutual interaction of the electrons in a heavy atom : the thomas - fermi potential together with the centrifugal potential can form a second potential minimum , which is very deep and located close to the nucleus .
this effect occurs provided the atomic number is larger than 57 and we are dealing with an energy eigenstate corresponding to the angular momentum quantum number @xmath68 , see for example m. goeppert - mayer , phys . rev .
* 60 * , 184 ( 1941 ) . since a. sommerfeld we associate the motion of an electron in an atom with an ellipse or a circle .
however , the electron at the top of the sea of filled atomic states moves in an effective screened potential giving rise to a necklace orbit as pointed out by j. a. wheeler , in e. h. lieb et al .
( eds ) , studies in mathematical physics , princeton university press , princeton , 1976 , p. 383
|
we summarize unusual bound or localized states in quantum mechanics .
our guide through these intriguing phenomena is the classical physics of the upside down pendulum , taking advantage of the analogy between the corresponding newton s equation of motion and the time independent schrdinger equation .
we discuss the zero energy ground state in a three dimensional , spatially oscillating , potential .
moreover , we focus on the effect of the attractive quantum anti
centrifugal potential that only occurs in a two dimensional situation .
quantum mechanics ; bound states ; parametric oscillator ; periodic potential .
|
in a recent letter , m. sonnleitner _ et al . _
@xcite studied counterintuitive forces which arise between a black body and neutral atoms or molecules - named black body force ( bbf ) - due to the energy associated with shifts in the absorption lines of their spectra .
these shifts are induced by the so - called dynamical stark effect , in which the incoherent electromagnetic waves generated from the heated black body surface disturbs the electronic transitions in the atomic spectrum . in the above mentioned article ,
the authors have found that attractive forces undoubtedly arise from shifts in the electronic transitions involving the @xmath0 level of the hydrogen atom , and such forces would eventually be stronger than the repulsive force caused by the radiation pressure and the gravitational force due to the black body mass .
however , the energy associated to the spectral shift induced in other atom species could be positive , leading to repulsive forces . in the microscopic level
, this effect can arise from other incoherent electromagnetic emissions , strongly dependent on the shape of the emitting surface .
it is also noticeable in @xcite the emphasis put on the connection between the geometry of the bodies and the bbf , but the authors did not investigate the possible influences of both the topology and spacetime geometry on that force . in other words , those authors limited themselves to flat geometry with trivial topology .
astrophysical and cosmological scenarios as well as some models in condensed matter physics offer possible investigation lines for the bbf .
however , corrections to that force are necessary because even in a weak - field regime , in standard or alternative theories of gravitation , there are non - negligible modifications in the black body radiation law caused by the gravity @xcite , which must be taken into account for a complete understanding of the phenomena .
this is another sense in which geometry is fundamental in understanding of the bbf .
it is worth noticing that the authors of ref.@xcite show that the bbf depends on both the temperature and the solid angle , which are modified by the spacetime geometry .
therefore this work focus on the effects due to both the spacetime geometry and topology on the bbf .
this is considered for static and spherical gravitational sources as well as for infinitely long cylinders , including those ones like cosmic strings .
we compute these force corrections and study their behavior comparing them with the flat case , obtained in @xcite . in the case of the spherical symmetry we will particularize for the schwarzschild solution , making equally manifest the role of the local spacetime geometry in the intensity of the bbf .
we will numerically calculate this corrected force regarding typical neutron stars and the sun . regarding the cylindrical geometry ,
whose bbf is initially calculated for the infinitely long newtonian cylinder , an angular ( deficit ) parameter @xmath1 is appropriately introduced in order to investigate effects of the spacetime topology on that force .
such a parameter multiplies the azimuthal angle entering in the calculation of the solid angle subtended by the atom , and it characterizes the global properties of the cosmic string spacetime , which presents a locally flat geometry , but with global curvature .
this feature generates interesting effects , as self - interaction on particles and dipoles @xcite and gravitational lens @xcite .
in addition , we also consider the bbf caused by a cosmic string rotating stationarily around its symmetry axis , where effects due to the spacetime rotation are expected .
the cosmic string spacetime has a non - trivial topology which is encoded in the aforementioned angular parameter given by @xmath2 @xcite , where @xmath3 is the newtonian constant of gravity and @xmath4 is the linear mass density of the string .
the more recent observations of the cosmic microwave background ( cmb ) provide an upper bound on dimensionless parameter @xmath5 of the order of @xmath6 @xcite .
it is worth emphasize that cosmic strings can have played an important role in the primitive universe , and it is known that in its very early ages the temperature was extremely hot @xcite ; thus it is natural thinking that those objects were in constant interaction with the thermal radiation that bathed the universe .
indeed , the effects of the thermal equilibrium of cosmic strings with radiation were analyzed in @xcite .
although cosmic strings are still subjects of theoretical speculation , a concrete realization of them occurs in some condensed matter systems .
the static cosmic strings are emulated as topological defects termed disclinations , which lie within crystalline structures , including liquid crystals in the nematic phase [ 8,9 ] , while the avatars of the stationary cosmic strings are vortices in a superfluid [ 10,11 ] .
the paper is organized as follows . in section 2
, we analyze the bbf on a neutral atom due to the radiation emitted by spherical sources , studying changes in this force due to the gravitational field . in section 3
, we consider cylindrical sources in the calculation of that force , revealing the role of both the topology and spacetime rotation in the production of it and , finally , in section 4 we discuss the results .
units in which @xmath7 is taken in all calculations .
in this section we examine the influence of spacetime geometry on the bbf . as shown in ref .
@xcite , for the non - relativistic case , the attractive potential associated to this force exerted on the neutral hydrogen atom in its fundamental state , placed at a distance @xmath8 from the center of a spherical black body of radius @xmath9 at temperature @xmath10 is , in natural units , @xmath11 where @xmath12 is the fine structure constant , @xmath13 is the mass of the electron , and @xmath14 is the solid angle subtended by the atom .
about the above expression some points are worth comment . as pointed in ref .
@xcite , for the hydrogen atom in an isotropic thermal bath , the dynamical stark effect will mostly cause a negative energy shift for the ground state .
if the temperature of the thermal bath is @xmath10 , this shift is given by @xmath15 since the atom is in an isotropic thermal bath , the black body field can not have any directional effect on the atom motion , inducing only friction and diffusion .
those authors then argue that if a finite source is considered , this produces a net force .
the point is that in this case , the atom does not see radiation coming from all directions , but only from a solid angle determined by the source .
therefore , only a fraction of the energy shift in the isotropic bath would cause the bbf .
this fraction is given by the solid angle compared to @xmath16 , giving us the final potential energy of eq .
( [ potential ] ) . for the non - relativistic spherically symmetric case
this solid angle is given by @xmath17 from the above expression we can see that the radial dependence of the potential comes only from the solid angle .
however , when we consider curved static spacetimes , the effective temperature will also depend on the coordinates via tolman temperature @xcite . therefore , in order to find corrections to the bbf two aspects must be considered : the first is the curvature correction to the temperature and the second is the modification in the solid angle determined by the spacetime curvature .
this is the general procedure that we will follow throughout this manuscript .
now we must consider the specific case with spherical symmetry .
the cases in which global or topological properties are relevant will be left to the next section , where cylindrical symmetries play an important role .
the static and spherically symmetric exterior solution of einstein s equation of general relativity is considered , which is given by the metric @xmath18 where @xmath19 and @xmath9 is the radius of the source as seem from infinity .
the interior solution is also important to determine the solid angle as seem by the atom . for this
we use the schwarzschild one which is given by @xmath20 as said above , the spacetime modification to the temperature is given by the tolman temperature . at the position of the atom
this is given by @xmath21 where @xmath10 is the temperature of the isotropic thermal bath , as measured by an observer at the infinity .
the fact that the temperature depends on the radial coordinate will contribute to modify the bbf .
now we must consider the modification in the solid angle due to the spacetime curvature which occurs in the system investigated here .
two corrections in expression ( [ solido ] ) must be considered , since the size of the source and the distance of the atom to its center are modified by gravity .
these two quantities are calculated via @xmath22 where @xmath23 is the effective distance of the atom to the source center and the first integral in the rhs is its effective radius @xmath24 .
this latter can be obtained by using eq .
( [ metric < ] ) and is given by @xmath25 and the second integral is @xmath26 thus we obtain that the modification in the solid angle will be given by @xmath27 notice that the above expression reproduces the euclidean solid angle for large values of @xmath8 and at the surface of the source is given by @xmath28 .
plugging eq .
( [ teff ] ) and eq .
( [ solidomodificado ] ) into eq .
( [ potential ] ) we get the final corrected expression for the potential @xmath29 this potential corresponds to a force that also drops with the cube - inverse at large distances and is infinity when @xmath30 , as in the flat spacetime case . ,
gravitationally corrected and flat , both as functions of the radial coordinate.,width=321 ] in fig . 1
, we depict the graph of the relativistically corrected potential @xmath31 over the temperature function @xmath32 , against the same potential energy in the flat case , both depending on the radius @xmath8 , for a compact sphere with @xmath33 .
the difference between both is sensible , and the magnitude of the corrected one is considerably greater than the observed in the other case . at @xmath34 ,
the corrected potential is approximately 14% lower than the flat case . at smaller distances ,
the difference increases even more .
the calculation of the bbf itself is made by performing @xmath35 @xcite .
notice that by considering the sun , this reduction in the temperature - independent part of the bbf potential is approximately 0.001% at @xmath34 . for a quantum gravity correction to this force ,
see @xcite .
in this section we will consider the bbf caused by systems with cylindrical symmetries .
there are three cases to be considered here .
one is the non - relativistic radiating infinity cylinder , which was not considered in the original ref .
the other two cases consider the influence of the spacetime generated by both static and stationary cosmic strings , with and without thickness . from the expression without gravity ( [ potential ] ) , the only change in the potential will be due to the solid angle . supposing that the particle is at a distance @xmath36 of the rectangle center , thus the solid angle of the whole cylinder subtended by the atom is given by @xcite @xmath37 where @xmath38 , @xmath39 ; @xmath40 and @xmath41 are the length and width of the rectangle , respectively .
taking an infinitely long cylinder , @xmath42 , and writing the width in terms of the cylinder radius , @xmath9 , and of the distance of the particle from the cylinder symmetry axis , @xmath8 , which coincides with the @xmath43 axis , we have @xmath44 with @xmath45 .
the argument @xmath46 is just the half - angle measured from the atom and which embraces the cylinder in the plane @xmath47 . in the fig .
1 we depict the neutral atom ( a ) close to the cylindrical black body . according to it ,
the geometrical elements present in our analysis are : @xmath48 , @xmath49 , @xmath50 , @xmath51 , and @xmath52 .
we find that the attractive potential on the hydrogen atom in the ground state due to the cylindrical black body at absolute temperature @xmath10 is given by @xcite @xmath53 where @xmath54 is the thermal shift in the absorption lines of the spectrum of the unexcited hydrogen atom .
thus the magnitude of the radial force exerted on this atom is given by @xmath55 where @xmath12 is now the fine - structure constant .
notice that the bbf is attractive , falling with a square - inverse law at large distances ( or @xmath56 ) . that is different from what happens in the spherical black - bodies where the force drops with the cubic - inverse law at those distances @xcite .
the fact that the cosmic strings spacetime have a locally flat geometry but a non - trivial topology raises interesting points .
first of all , due to the local flatness , the effective temperature is not changed and therefore the only possible correction comes from the solid angle like in the previous case .
the interesting point is that the solid angle now depends on the deficit angular parameter @xmath1 , incorporating then topological features in the bbf .
let us model a static cosmic string , or its realization as a disclination in a liquid crystal , by the cylinder above considered .
first , the conical geometry of the space around of the defect has to be incorporated .
such a geometry is described through the metric @xmath57 this expression describes the space around an ideal cosmic string of zero thickness , or the exterior region of that one endowed with an internal structure @xcite . for this defect , @xmath58 ( angular deficit parameter ) , and @xmath59 . for a disclination , @xmath1 can also be between zero and one , meaning that we have an angular excess parameter .
then all we have to do is rewrite the half - angle from which the atom sees the cylinder , @xmath60 , in terms of the correspondent half central ( azimuthal ) angle , @xmath61 , such that @xmath62 and to insert the factor @xmath63 which multiplies the azimuthal angles , according to the metric ( [ 05 ] ) .
thus the expression ( [ potentialnrc ] ) becomes @xmath64}}}},\ ] ] and calculating the force via @xmath65 , we get @xmath66 it is worth notice that , for the neutral hydrogen atom in the minkowsky space ( _ i.e. _ , when @xmath67 ) we have the expression ( [ 04 ] ) again , and when one takes the limit @xmath68 valid for an ideal cosmic string , the bbf vanishes .
then , an important conclusion is that ideal cosmic strings irradiating do not exert bbfs . for non - zero thickness and @xmath69
, we can approximate eq .
( [ fnu ] ) by @xmath70 the force is attractive , due to the factor @xmath71 to be negative for neutral hydrogen atoms , and it still falls off with the inverse square of the radial distance . such a law is the same one that manages the self - force on neutral ( or charged ) massive particles situated in the spacetime of a gott - hiscock cosmic string , when the particle is far away from the string @xcite .
let us now analyze the bbf on a neutral atom due to a spinning cosmic string , which is the stationary counterpart of the static cosmic string . as seen in the introduction ,
these objects appear in condensed matter physics as vortices in superfluids .
the correspondent spacetime metric is given by @xcite @xmath72 where @xmath73 is the angular momentum per unit length of the string , such that when it vanishes , we recover eq .
( [ 05 ] ) .
from the above expression we can see that the effective angular deficit parameter is given by @xmath74 notice that this provides a consistent definition since it recover the static case for large @xmath8 .
the bbf on the neutral atom , when one modifies @xmath75 in eq .
( [ vnu ] ) is given by @xmath76 where @xmath77 .
we call attention to the fact that the force now is non - zero when the string radius vanishes , yielding @xmath78 while the string is spinning . for a neutral hydrogen atom very distant from the string
, the attractive bbf now falls off with the inverse cube of its radial distance , @xmath79 , and diverges at @xmath80 , _
i.e. _ , at the neighborhood of the boundary that defines the region of the closed time - like curves ( ctc s ) @xcite . in the interior of that region , @xmath81 ,
the bbf is not defined .
strictly speaking , the solid angle given in eq .
( [ cylindersolidangle ] ) is valid only for flat spaces .
however , the modification that must occur in a curved space was effectively implemented in the above systems through the azimuthal angle , since this latter is the only geometric parameter that is changed in fact , as we can see in the metrics ( 23 ) and ( 28 ) , altering solely the global properties of these space - times , which remain locally flat .
we have studied the effects of the black body radiation produced by spherical and cylindrical sources on a nearby neutral hydrogen atom in the ground state , which suffers , via dynamical stark effect , a shift in its spectrum absorption lines producing a attractive force ( bbf ) .
this was made by analyzing geometrical and topological influences of their respective space - times on it .
initially , we have calculated the effects due to a static and spherically symmetric gravitational background , starting by taking the tolman temperature and substituting it in bbf general expression , yielding a first correction to it . in order to arrive at the full correction , we taken also into account modifications in the solid angle subtended by the atom due to the curvature of the spacetime . to do this , we considered the radial proper distance to the center of the source , which was computed from both the exterior and interior schwarzschild solutions . when the results found here are applied to a compact object with this parameter measuring ten times the correspondent schwarzschild radius ,
the bbf potential gravitationally corrected is _ circa _ 14% lower than the flat case at @xmath34 . for the sun
, we verified that the einstein s gravity reduces the bbf potential about 0.001 per cent in relation to this quantity when calculated in the flat spacetime at the same distance , which is therefore a very tiny difference when compared to the previous case .
the analysis continued by taking into account the radiation emitted from an infinitely long cylinder with a diameter equal to @xmath82 , and calculated the attractive potential on the hydrogen atom related to that effect , taking into account just the source s shape . in this case , the bbf is attractive , falling with a square - inverse law at large distances , differently of that happens with spherical black - bodies , in which the force obeys the inverse cube law at those distances as shown in @xcite .
we must point out , as awaited , that the proper body geometry plays a central role in the bbf .
next , we have appropriately introduced the angular parameter @xmath1 in the expression correspondent to the attractive potential due to the cylindric black body radiation .
this parameter multiplies the azimuthal ( or central ) angle which enters in the calculation of the solid angle subtended by the atom , characterizing the conical geometry of the cosmic string ( or of its realization as a disclination in a crystal ) spacetime , and , thus , its topological properties .
the bbf vanishes when we consider an ideal static cosmic string , in which @xmath83 , but it is non - zero when we taken into account an ideal spinning cosmic string ( or vortices in superfluids ) , now falling off at large distances according to an inverse cube law , as in the spherical case .
we can conclude that such a force is weaker than that one presented in the static case because the non - inertial effects caused by the spacetime rotation itself , or , in other words , by the frame dragging associated with stationary cosmic strings @xcite .
we have also shown that this force is not defined in the interior of the ctc s domain , diverging at its boundary .
the authors acknowledge the financial support provided by fundao cearense de apoio ao desenvolvimento cientfico e tecnolgico ( funcap ) , by conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) , and funcap / cnpq / pronex .
m. sonnleitner , m. ritsch - marte and h. ritsch , phys .
lett . * 111 * , 023601 ( 2013 ) . c. r. muniz , v. b. bezerra and m. s. cunha , phys .
d * 88 * , 104035 ( 2013 ) .
bezerra , h. f. mota and c. r. muniz , phys . rev . *
d 89 * , 044015 ( 2014 ) .
m. r. anderson , _ the mathematical theory of cosmic strings : cosmic strings in the wire approximation _ , iop publishing ltd . , ( 2003 ) . c. r. muniz and v. b. bezerra , ann .
phys . * 340 * , 87 ( 2014 ) .
a. vilenkin and e.p.s .
shellard , _ cosmic strings and other topological defects _ , cambrigde university press ( 1994 ) .
p. a. r. ade et al . , arxiv:1303.5085 .
p. j. e. peebles , _ principles of physical cosmology _ , princeton university press , princeton , ( 1993 ) .
p. c. w. davies and v. sahni , class .
quantum grav . * 5 * , 1 ( 1988 ) .
a. l. muoz , _ liquid crystal dynamics : defects , walls and gels _ , cuvillier verlag , goetingen ( 2005 ) . c. stiro and f. moraes , mod .
a * 20 * , 2561 ( 2005 )
. r.l .
davis and e.p.s shellard , phys .
* 63 * , 2021 ( 1989 ) .
g. e. volovik , jetp lett . ,
* 67 * , 881 ( 1998 ) .
a. khadjavi , j. opt .
. am . * a58 * , 1417 ( 1968 ) .
b. allen , b. s. kay and a. c. ottewill , phys.rev . * d53 * , 6829 ( 1996 ) .
n.r . khusnutdinov and v.b .
bezerra , phys.rev .
* d64 * 083506 ( 2001 ) .
v. a. de lorenci , r. d. m. de paola and n. f. svaiter , class .
quantum grav .
* 16 * , 3047 ( 1999 ) . c. r. muniz , v. b. bezerra and m. s. cunha , ann
* 350 * , 105 ( 2014 ) .
b. jensen and h. h. soleng , phys .
rev . * d45 * , 3528 ( 1992 ) .
r. p. feynman , _ statistical mechanics , a set of lectures _ , levant books , kolkata ( 2008 ) .
r.c.tolman , _ relativity , thermodynamics and cosmology _ , dover publications , inc .
, new york ( 1987 ) .
w.rindler , _ relativity : special , general , and cosmological _ , oxford university press , 2nd ed .
( 2006 ) g. alencar , v.b .
bezerra , m.s .
cunha , and c.r .
muniz phys.lett.*b747 * , 536 ( 2015 ) .
h. culetu , j. phys .
68 , 012036 ( 2007 ) .
|
in this paper we compute the corrections to the black body force due to spacetime geometry and topology .
this recently discovered attractive force on neutral atoms is caused by the thermal radiation emitted from black bodies and here we investigate it in systems with spherical and cylindrical symmetries .
for some astrophysical objects we find that the corrected force is greater than the flat case , showing that this kind of correction can be quite relevant when curved spaces are considered .
then we consider four cases : the schwarzschild spacetime , the non - relativistic infinity cylinder , and both the static and stationary cosmic strings .
for the spherically symmetric case we find that two corrections appear : one due to the gravitational modification of the temperature and the other due to the modification of the solid angle subtended by the atom .
we apply the found results to a typical neutron star and to the sun . for the cylindrical case , which is locally flat , no gravitational correction to the temperature exists . however , we find the curious fact that the black body force depends on the topology of the spacetime through the modification of the azimuthal angle and therefore of the solid angle . for the static cosmic string
we find that the force is null for zero thickness case .
for the stationary one , we find that the force is non - null even for the thin case .
|
realistic modeling of first - order fermi cosmic - ray acceleration at relativistic shock waves is a difficult task due to the strong dependence of the resulting particle spectra on the essentially unknown local conditions at such shocks .
therefore , progress in this field mostly arises from the increasingly more realistic mhd conditions that are assumed to apply near the shocks . for mildly relativistic shocks in the _ test particle _
approach the advances that have been made in the past twenty years involve a semianalytic solution of the particle pitch - angle diffusion equation at parallel shocks @xcite , the effects of different shock compressions in parallel shocks @xcite , the impact of an oblique mean magnetic field for subluminal configurations @xcite , and the effects of oblique field configurations at sub- and superluminal shocks considered with an increasing degree of complication , applying successively more realistic physical approximations @xcite . for ultrarelativistic shocks
the notion of a `` universal '' spectral index was first described by @xcite and later found in a variety of studies @xcite , but more recent studies @xcite indicate basic problems with first - order fermi acceleration , in particular in generating wide - energy power - law spectra .
the latter work also contains a more systematic discussion of the subject literature . on the other hand ,
there have been attempts to self - consistently derive the electromagnetic shock structure .
one very simplified approach applied modeling of nonlinear effects in the first - order fermi acceleration at the shock ( e.g. , ellison & double 2002 ) .
the limitations of this technique arise from the approximate treatment of particle scattering near the shock and its simple scaling with particle energy instead of a realistic accounting for the microphysics of the process .
a more realistic microscopic and self - consistent description of the collisionless shock transition may be afforded by particle - in - cell ( pic ) simulations ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) , which follow the formation of a relativistic collisionless shock starting with processes acting on the plasma scale . in these simulations
one typically observes that the generation of magnetic and electric fields is accompanied by an evolution of the particle distribution function from the background plasma conditions up to highly superthermal energy scales .
although providing a wealth of information on the collisionless shock structure , the pic simulations are still very much limited in dynamical range , in particular for cosmic - ray particles whose energies are many orders of magnitude larger than the plasma particle energies .
the present paper is a continuation of the studies of @xcite .
here we attempt to incorporate the results of the pic simulations concerning the generation of magnetic fields at relativistic shocks into the test particle monte carlo simulations of cosmic - ray particle acceleration . as in the previous papers , we study ultrarelativistic shock waves propagating in a medium with a magnetic field that is perturbed over a wide range of macroscopic scales
. however , we also consider an additional short - wave isotropic turbulent field component downstream of the shock , analogous to the shock generated turbulent fields revealed by the pic simulations .
we treat the amplitude of the short - wave component as a model parameter and study the effects of the small - scale perturbations on the particle spectra and angular distributions derived in the previous work . in what follows , @xmath1 is the speed of light .
the integration of particle trajectories is performed in the respective local plasma ( upstream or downstream ) rest frame , and we use index 1 " ( 2 " ) to label quantities provided in the upstream ( downstream ) frame .
we consider ultrarelativistic particles with @xmath2 .
we use dimensionless variables , so a particle of unit energy moving in an uniform mean upstream magnetic field @xmath3 has the unit maximum ( for @xmath4 ) gyroradius @xmath5 and the respective resonance wavevector is @xmath6 .
as in @xcite we assume a relativistic shock wave to propagate with velocity * _ u_*@xmath7 ( or the respective lorentz factor @xmath8 ) in the upstream medium with a uniform magnetic field * _ b_*@xmath9 * _ b_*@xmath10 , inclined at an angle @xmath11 to the shock normal ( along * _ u_*@xmath7 direction ) , superimposed on which are isotropic 3d magnetic field perturbations .
these perturbations are characterized with a power - law wave power spectrum @xmath12 that is defined in a wide wavevector range ( @xmath13 , @xmath14 ) , where @xmath15 and @xmath16 .
specifically we consider the wave spectral indices @xmath17 , describing the flat power spectrum , and @xmath18 for the kolmogorov distribution .
the integral power of these perturbations is given by the ( upstream ) amplitude @xmath19 and @xmath20 is one of our model parameters . as described in detail in @xcite , the upstream field perturbations
are modeled as the superposition of static sinusoidal waves of finite amplitude .
the downstream magnetic field structure , including the turbulent component , is derived as the shock - compressed upstream field , and hence the downstream turbulent magnetic field is naturally anisotropic .
our method of applying the hydrodynamical shock jump conditions ( for the electron - proton plasma , * ? ? ?
* ) preserves the continuity of the magnetic field lines across the shock . in what follows
, we refer to the field component described above as the _ large - scale _ ( or _ long - wave _ ) _ background field_. the other physical magnetic field component considered in the present study is the short - wave turbulence , assumed to be a result of kinetic magnetic field generation processes acting at the shock .
this component , taken for simplicity to be isotropic and static , is imposed upon the nonuniform background magnetic field downstream of the shock .
the short - wave nonlinear field perturbations are introduced with a flat spectral distribution in the wavevector range ( @xmath21 , @xmath22 ) , where the shortest downstream waves are the shock - compressed shortest upstream waves : @xmath23 here @xmath24 is the compression ratio in the shock rest frame , and @xmath8 ( @xmath25 ) is the upstream ( downstream ) shock lorentz factor .
the wavevector range of the small - scale turbulent component is chosen to be one decade in @xmath26 apart from @xmath27 to separate the influence of these perturbations on the low - energy particle motion from that exerted by the short - wave component of the large - scale background field .
this choice also facilitates the use of a hybrid method for the calculation of particle trajectories ( see 2.2 ) .
note also , that the definition of the short - wave component depends on the lorentz factor of the shock .
below , we refer to the short - wave magnetic field component also as the _ shock - generated turbulence _
( `` _ sh _ '' ) , to distinguish it from the large - scale background field , that exists in the upstream region and is only compressed upon passage through the shock . in our simulations
we exclusively study the first - order fermi acceleration process and neglect second - order processes .
therefore , the turbulent magnetic field components ( both short - wave and large - scale background perturbations ) can be considered to be static in the respective plasma rest frames , both upstream and downstream of the shock , and electric fields that may exist in the shock transition layer are neglected .
the implementation of short - wave turbulence into the monte carlo simulations forces us to dispense with the direct integration of the particle equations of motion in the analytically modeled magnetic field ( see * ? ? ?
* ) . instead
, we resume the derivation of particle trajectories with the hybrid approach proposed in @xcite . thus , particle trajectories are directly calculated from the equations of motion in the large - scale background magnetic field only , whereas the trajectory perturbations due to the shock - generated small - scale turbulence are accounted for through a small - amplitude pitch - angle scattering term .
auxiliary simulations have been performed to determine the scattering amplitude distributions for various particle energies . in the scattering procedure , after each time step @xmath28 the particle momentum direction is perturbed by a small angle @xmath29 .
the time step itself scales with particle energy ( or gyroradius ) , @xmath30 , and is chosen so that the condition @xmath31 is always fulfilled ( @xmath32 is the wavelength of shock - generated perturbations ) , which means that the auxiliary simulations follow the particle scattering on the shock - generated turbulence well into the diffusive regime .
therefore , the scattering amplitude distributions also scale linearly with the short - wave turbulence amplitude , and the mean scattering angle variations are related to the particle energy as @xmath33 . an individual simulation run is performed as follows .
we start with injecting monoenergetic particles ( with the initial energy @xmath34 ) at random positions along the shock front , with their momenta isotropically distributed within a cone around the shock normal pointing upstream of the shock . in an initial simulation cycle , the injection process is continued until the required number of particles , @xmath35 , has been selected those that after being injected upstream and then transmitted downstream of the shock , succeed in recrossing the shock front again .
then the calculation of individual particle trajectories proceeds through all subsequent upstream - downstream cycles . in each cycle
, a fraction of the particles escapes through a free - escape boundary introduced `` far downstream '' of the shock , i.e. at a location from which there is only a negligible chance that particles crossing the boundary would return back to the shock . to the particles remaining in the simulations the trajectory - splitting procedure is applied ( see * ? ? ?
* ) , so the number of particles remains constant in the acceleration process , but the statistical weights of the particles are appropriately reduced .
the final spectra and angular distributions of accelerated particles , derived in the shock normal rest frame for particles crossing the shock front , are averaged over many statistically different simulation runs .
because of the limited capabilities of present - day computers , the long - time nonlinear development of plasma instabilities leading to the generation of the short - wave turbulent downstream magnetic field component can not yet be fully investigated , in particular for electron - ion plasma collision fronts
. therefore , the ultimate structure of the small - scale field and the global effectiveness of the generation mechanism remain uncertain . to estimate the role of the shock - generated turbulence in cosmic - ray acceleration at ultrarelativistic shocks , we introduce the small - scale wave component as isotropic 3d magnetic field perturbations and treat the amplitude of these perturbations as a model parameter .
the spectra of accelerated particles for oblique superluminal shocks are presented in figures [ obl1]-[obl4 ] , and those for parallel shocks are displayed in figures [ par10 ] and [ par30 ] .
the amplitude of the short - wave component , @xmath36 , as provided in the figures , is measured in units of the average downstream perturbed magnetic field strength @xmath37*_b_*@xmath38*_b_*@xmath39 .
the spectra presented with solid lines has been derived in the turbulence model without the @xmath40 term ( most of them are presented in * ? ? ?
some of the energetic particle distributions follow a power - law in a certain energy range . in these cases
linear fits to the power - law portions of the spectra are presented , and values of the phase - space spectral indices , @xmath41 , are given ( the equivalent _ number _ spectral index @xmath42 ) .
the spectral indices may help to make a quantitative comparison of the spectra , but care must be exercised , because the spectral indices depend on the energy range chosen for the fit on account of the curvature in most of the spectra . in the case of superluminal shocks , we use the same low - energy limit for the energy range selected for the fits
. accelerated particle spectra for oblique superluminal shocks are presented in figures [ obl1]-[obl3 ] for @xmath43 ( @xmath44 ) , and in figure [ obl4 ] for @xmath45 ( perpendicular shock ) .
one can see that increasing the amplitude of the shock - generated turbulence leads to a more efficient acceleration with particle spectral tails extending to higher energies .
however , in all cases , in which @xmath46 , the energetic spectral tails are convex , so the spectral index increases with particle energy .
in addition , all the spectra have cutoffs at an energy for which the resonance condition for interactions with the long - wave turbulence is fulfilled .
these features result from the fact that the influence on particle trajectories of the shock - generated small - scale turbulence decreases with increasing particle energy . in our numerical approach
this corresponds to a reduction of the scattering amplitude @xmath47 ( see 2.2 ) .
most published treatments of first - order fermi acceleration at ultrarelativistic shocks , that apply the pitch - angle diffusion approximation @xcite , assume scattering conditions that do not change with particle energy . as discussed by @xcite , these authors show that if the small - angle scattering dominates over the possible influence of the oblique mean magnetic field component and/or the long - wave perturbations in shaping the particle trajectories , then power - law particle distributions with
universal " spectral index @xmath48 may be formed . with the more realistic magnetic field model considered in the present paper ,
it is not possible to reproduce these results .
however , particle spectra can approach a power - law form in a limited energy range near the particle injection energy , as demonstrated by the fits to the low - energy parts of the spectra in figures [ obl1]-[obl4 ] .
the energy ranges for the fits are arbitrary selected but have the same low - energy bound .
note that in the figures the spectral indices , though depending on the background conditions , are _ always _ larger than @xmath49 .
if one chose higher energies to derive a power - law fit to the spectra , one would obtain a higher spectral index on account of the convex shapes of the spectra .
convex spectra could have been expected based on the earlier studies of @xcite , who showed how an increased pitch - angle scattering angle in superluminal shocks leads to a flattening of the particle spectrum to the limiting
universal " power law ( see the discussion of this effect in * ? ? ?
* ) . to further illustrate this behavior ,
we have recalculated the spectrum for @xmath50 in figure [ obl1 ] ( _ filled dots _ ) for a fixed scattering amplitude @xmath51 at particle energies @xmath52 ( @xmath53 ) .
then the pitch - angle scattering term dominates downstream of the shock at high particle energies , which leads to a power - law spectrum formation over a wide energy range without a steepening or a cutoff , but with @xmath54 . , @xmath55 , and a flat wave - power spectrum of large - scale background magnetic field turbulence .
the amplitude of the upstream field perturbations is @xmath56 .
the amplitudes of the downstream shock - generated turbulence , @xmath36 , are given near the respective spectra .
linear fits to some spectra are also presented with dashed lines .
the energy ranges selected for the fits have the same low - energy bounds to facilitate comparison of the spectral indices @xmath41 , given in italic .
the spectrum plotted with a solid line applies to the case without short - wave shock - generated perturbations . the spectrum shown with filled dots
is calculated with a fixed scattering term for @xmath52 , and the linear fit to the part of this spectrum above @xmath52 is presented with dotted line .
some spectra are vertically shifted for clarity .
particles in the energy range ( @xmath57 , @xmath58 ) can satisfy the resonance condition @xmath59 for some of the waves in the background turbulence spectrum .
[ obl1 ] ] , @xmath55 , and a kolmogorov power spectrum of background long - wave turbulence with @xmath56 has been assumed .
the spectrum shown with the solid line is derived in the limit @xmath60 ( see fig . 2 in * ? ? ?
[ obl2 ] ] , @xmath55 ) shocks with @xmath61 . filled and open symbols refer to results for a flat and a kolmogorov wave - power spectrum of large - scale background field perturbations , respectively .
the linear fit to the spectrum for the _ flat _ wave power spectrum is derived in an energy range with the same low - energy bound as used for the fits shown in figs .
[ obl1 ] and [ obl2 ] .
[ obl3 ] ] , @xmath62 ) shocks for the same parameter combinations as in fig .
[ obl4 ] ] , @xmath43 , and a kolmogorov wave power spectrum of large - scale background magnetic field turbulence .
particles with @xmath63 are directed upstream of the shock .
the distributions are calculated by summing the quantity @xmath64 in the respective @xmath65-bins at every particle shock crossing , where @xmath66 is the angle between the particle momentum and the shock normal , @xmath67 is the normal component of the particle velocity , and @xmath68 is the statistical weight of the particle . only particles with @xmath69 are included , and low - energy particles with large weights contribute most to the distributions . the angular distribution presented with filled dots is derived without short - wave perturbations ( @xmath60 ) .
the distributions for @xmath70 , and for @xmath56 and @xmath71 are presented with solid and dashed lines , respectively .
the angular distributions for @xmath72 , and @xmath56 and @xmath71 are shown with dash - dotted and dotted lines , respectively .
[ angsup ] ] in the case of superluminal shocks , the particle spectra calculated for a high amplitude of shock - generated turbulence , @xmath70 in figures [ obl1]-[obl4 ] , depend only weakly on the background large - scale magnetic field structure , in contrast to the @xmath73 limit ( see also * ? ? ?
the acceleration process is only slightly more efficient in the case of a large amplitude of long - wave perturbations ( @xmath56 in figs .
[ obl1 ] and [ obl2 ] ) , in which the spectral tails extend to marginally higher energies and show softer cutoffs , when compared to the cases with @xmath61 ( figs . [ obl3 ] and [ obl4 ] ) .
there are also only minor differences between the spectra derived for a flat spectrum of background perturbations , for which the wave power is uniformly distributed per logarithmic wavevector range , and for the kolmogorov distribution , for which most power is carried by long waves . in the latter case ,
one finds power - law particle spectra in a slightly wider energy range than in the case of flat distributions ( compare figs .
[ obl1 ] and [ obl2 ] and spectra in fig .
[ obl4 ] ) , but the high - energy tails and the cutoff shape remain nearly the same for both types of the wave power spectra .
the structure of the background large - scale magnetic field can , however , influence particle acceleration for smaller amplitudes of the short - wave turbulence . as one can see in figure [ obl2 ] , the power - law part of the spectrum for @xmath72 and kolmogorov - type long - wave turbulence spectrum , although steeper , continues to higher energies than that for @xmath70 , which is caused by the large - scale component providing particle scattering at higher energies ( see , e.g. , the spectrum for @xmath73 in fig .
[ obl2 ] ) .
this is not the case for either a flat spectrum ( fig . [ obl1 ] ) or a small - amplitude ( fig .
[ obl3 ] ) of long - wave turbulence , since in superluminal shocks particle acceleration processes in the absence of large - amplitude long - wave magnetic field perturbations are inefficient @xcite .
figure [ angsup ] presents particle angular distributions for superluminal shocks ( @xmath43 ) with the kolmogorov power spectrum of large - scale magnetic field perturbations .
note that the angular distributions for a large amplitude of the small - scale turbulence , @xmath70 , are closer to that for @xmath60 ( _ filled dots _ ) than angular distributions for the smaller amplitude , @xmath72 .
as the particle spectra , the distributions for @xmath70 do not depend on the amplitude of the background field perturbations , nor on their wave - power spectrum ( distributions for the flat power spectrum are not shown for presentation clarity ) .
similarity in shape of these distributions to the angular distribution derived without the @xmath40 scattering term is caused by efficient particle scattering on the downstream short - wave perturbations at low particle energies , which enables a considerable fraction of them to return upstream of the shock .
the difference in the distributions for the smaller short - wave amplitude , @xmath72 , illustrates the effects the long - wave turbulence has on the particle acceleration process , as discussed above .
a caveat is in order concerning the low - energy part of the simulated spectra . in our simulations with large - amplitude short - wave turbulence , the scattering amplitudes for low particle energies , @xmath74 for the spectra with @xmath70 and @xmath75 for @xmath50 , violate the pitch - angle diffusion requirement @xmath76 .
this is because in our approach the time between subsequent scatterings is fixed for a given particle energy , and the scattering angle scales linearly with @xmath40 ( see 2.2 ) providing large scattering amplitudes for large values of @xmath77 .
therefore , the simulated spectra may suffer a weak systematic flattening at low energies . as discussed by @xcite , the particle spectra with
universal " slope have been derived in conditions equivalent to a parallel ( @xmath78 ) mean shock configuration . in our earlier work @xcite we have demonstrated that the spectra formed at parallel relativistic shocks depend substantially on background conditions , and features such as a relation between the spectral index and the turbulence amplitude or a cutoff formation in the case of ultrarelativistic shocks can be observed . in this section ,
we investigate the role of shock - generated downstream turbulence in particle acceleration at parallel shocks .
for that purpose , we have performed simulations for highly relativistic shock waves with lorentz factor @xmath79 and @xmath80 , as presented in figures [ par10 ] and [ par30 ] , respectively . for parallel shocks
the amplitude of the compressed background field downstream of the shock , @xmath77 , is on average much smaller than for the oblique shocks discussed in 3.1 .
therefore , the numerical constraints are largely relaxed , and we can use larger values of @xmath36 , up to @xmath81 or even @xmath82 .
however , we would still slightly violate the pitch - angle scattering approximation at @xmath74 for the spectra with @xmath50 in figs .
[ par10]_a _ and [ par10]_b _ , and @xmath83 in figs .
[ par10]_c _ and [ par10]_d_. figures [ par10]_a _ and [ par10]_b _ show the results for shocks with @xmath84 and large - amplitude background field perturbations ( @xmath85 ) .
the qualitative similarity of the spectral features observed in this case and for oblique superluminal shocks is apparent ( see also * ? ? ?
* ) the spectra are convex and display cutoffs occurring much below @xmath86 .
this behavior proves that the large - amplitude long - wave perturbations locally form superluminal conditions at the shock , thus leading to a spectral cutoff as the pitch - angle scattering term in the downstream shock - generated turbulence substantially decreases with growing particle energy .
note , that in the case of a lower amplitude of the short - wave component , @xmath72 , the large - scale background turbulence with kolmogorov distribution provides additional particle scattering , allowing for the formation of a quasi power - law spectrum in a slightly wider energy range than for the larger @xmath36 ( fig .
[ par10]_b _ ) , again in correspondence to the analogous result for the superluminal shocks ( fig . [ obl2 ] ) . for the smaller amplitude of the large - scale magnetic field perturbations , @xmath61 ,
the role of the long - wave perturbations is less significant and the scattering on the short - wave turbulence can dominate up to the energies higher than the upper resonance energy @xmath87 .
though being gradually weaker with growing particle energy , in this case , as shown in figure [ par10]_c _ , pitch - angle diffusion can dominate over scattering by the long - wave perturbations without a limit for particle energy below @xmath87 .
the scattering mean free paths along the mean background magnetic field ( shock normal ) and , correspondingly , the mean acceleration timescale also increase with particle energy , but the particle spectra retain a power - law form up to the highest energies @xmath88 studied in our simulations .
however , the spectra are steeper than the expected `` universal '' spectrum , @xmath54 , and only weakly dependent on @xmath40 . to understand this feature ,
we have performed additional test simulations with the short - wave component introduced both downstream and _ upstream _ of the shock ( with constant upstream scattering amplitude @xmath89 ) .
this case , equivalent to the pure pitch - angle diffusion model , should lead to the formation of the universal " spectrum and , indeed , such the spectrum is produced in our simulations ( described as _ scatt up _ " in fig .
[ par10]_c _ ) .
thus , the slightly steeper spectra obtained in the models with only a downstream short - wave component can be understood solely as the result of the upstream long - wave turbulence providing a different distribution of pitch - angle perturbations to the particle orbits than the purely diffusive process ( see fig .
[ angpar ] ) .
this allows for the generation of a power - law particle spectrum , but with somewhat steeper index than the universal " value . with the kolmogorov background wave spectrum ( fig .
[ par10]_d _ ) there is enough power in long waves to influence the spectrum at high energies , leading to the noticable steepening of the particle distributions below @xmath87 .
however , also in this case the model with the short - wave field perturbations imposed upstream of the shock yields the universal " power - law spectrum . in figure [ angpar ]
we compare the angular distributions of particles at the shock in the cases of only downstream small - scale turbulence and with pitch - angle diffusion also upstream of the shock .
the spectra for weakly perturbed background magnetic fields , @xmath90 , presented in figures [ par10]_e _ and [ par10]_f _ , show a new characteristic feature , namely a spectral bump besides the power - law spectrum at lower particle energies .
the bump shape closely resembles the distribution of particles accelerated without the shock - generated field component ( see the discussion in * ? ? ?
* ) , and its presence indicates that at these energies the short - wave perturbations become negligible compared to the background long - wave component .
note , that @xmath40 is scaled with respect to @xmath77 , which is relatively small in comparison to the cases with larger @xmath91 discussed above .
we have also performed a series of simulations for shocks with a larger lorentz factor of @xmath92 .
the particle spectra presented in figure [ par30 ] for @xmath93 show a similarity of spectral features to those observed for shocks with @xmath84 .
however , there is an approximate scaling between the two cases , with characteristic spectral features appearing at respectively lower amplitudes of the long - wave magnetic field perturbations @xmath91 for shocks with @xmath92 than in the spectra for @xmath84 .
such a scaling may result from the stronger compression of these perturbations , as measured between the two plasma rest frames , and also from the larger particle anisotropy involved at the higher-@xmath94 shock . , @xmath95 ) for @xmath93 and a weakly perturbed large - scale background magnetic field ( @xmath61 ) with either the flat ( solid and dashed line ) or the kolmogorov ( dash - dotted and dotted line ) wave power spectrum of magnetic field perturbations .
the distributions indicated as `` _ scatt up _ ''
are derived in a model with short - wave turbulence imposed both downstream and upstream of the shock .
[ angpar ] ]
the present paper concludes our studies @xcite of the first - order fermi acceleration mechanism acting at relativistic and ultrarelativistic shock waves . we have attempted to consider as realistic models as possible for the perturbed magnetic field structures at the shock , which allow us to study all the field characteristics important for particle acceleration . in particular ,
we have investigated the dependence of particle spectra on the mean magnetic field inclination with respect to the shock normal and on the power spectra of magnetic field perturbations , including the long - wave background turbulence and the short - wave turbulence generated at the shock . in the present paper
we append the model of @xcite with downstream large - amplitude small - scale mhd turbulence component , analogous to those seen in pic simulations of collisionless relativistic shocks .
the form of these additional perturbations is arbitrarily chosen to consist of a very short sinusoidal waves , that form ( static ) isotropic highly nonlinear turbulence .
we have considered a wide variety of ultrarelativistic shock configurations for which , as presented in 3 , a number of different spectral features can be observed .
we show that with growing particle energy the role of short - wave ( @xmath96 ) magnetic field perturbations decreases , and spectra at high energies are shaped only by the mean magnetic field and the long - wave perturbations .
thus , in oblique superluminal shocks concave particle spectra with cutoffs can be generated even for @xmath97 . extended power - law particle distributions can be formed in parallel shocks propagating in a medium with low - amplitude long - wave field perturbations , but the spectral indices obtained are larger than the universal " spectral index @xmath98 , that is widely considered in the literature . the only case in which we have been able to obtain spectra with @xmath99 involved the unphysical assumption , that large - amplitude short - wave field perturbations also exist upstream of the shock . in summary ,
the accelerated particle spectral distributions obtained in this work and in our previous studies @xcite generally differ from the spectra of relativistic electrons ( power - laws with the spectral indices close to @xmath49 ) inferred from modeling the electromagnetic emission spectra of astrophysical sources hosting relativistic shocks ( e.g. , hot spots in extragalactic radio sources , quasar jets , gamma - ray burst afterglows ) .
our results thus provide a strong argument against considering the first - order fermi shock acceleration as the main mechanism producing the observed radiating electrons . in our opinion ,
other processes must be invoked to explain the observed emission spectra , for example second - order fermi processes acting in the regions of relativistic mhd turbulence downstream of the shock , which have hardly ever been discussed for relativistic conditions till now @xcite , or the collisionless plasma processes that have been studied in numerous pic simulations ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) , or other non - standard acceleration processes like those discussed by @xcite for electrons or @xcite for both electrons and protons
. our simulations also show , that relativistic shocks , being essentially always superluminal , possibly generate accelerated particle distributions with cutoffs below either the maximum resonance energy enabled by the _ high - amplitude _ background turbulence ( @xmath100 ) , or approximately at the energy of the compressed background plasma ions @xmath101 ( where @xmath102 is a mass of the heaviest ions present in the background medium ) .
thus , in conclusion , we maintain our opinion from the previous publications ( see also * ? ? ?
* ) that relativistic shocks are not promising sites as possible sources of ultra high - energy nuclei registered by the air shower experiments .
achterberg , a. , gallant , y. a. , kirk , j. g. , & guthmann , a. w. 2001 , , 328 , 393 bednarz , j. , & ostrowski , m. 1996 , , 283 , 447 bednarz , j. , & ostrowski , m. 1998 , , 80 , 3911 begelman , m. c. , & kirk , j. g. 1990 , , 353 , 66 derishev , e. v. , aharonian , f. , kocharovsky , v.v . , kocharovsky , vl .
v. 2003 , , 68 , 043003 ellison , d. c. , & double , g. p. 2002 , astroparticle phys . , 18 , 213 ellison , d. c. , & double , g. p. 2004 , astroparticle phys . , 22 , 323 frederiksen , j. t. , hededal , c. , haugblle , t. , & nordlund , .
2004 , , 608 , l13 gallant , y. a. , & achterberg , a. 1999 , , 305 , l6 heavens , a. , & drury , l. oc .
1988 , , 235 , 997 hededal , c. , haugblle , t. , frederiksen , j. t. , & nordlund , .
2004 , , 617 , l107 hededal , c. , & nishikawa , k .- i .
2005 , , 623 , l89 hoshino , m. , arons , j. , gallant , y. a. , & langdon , a. b. 1992 , , 390 , 454 jaroschek , c. h. , lesch , h. , & treumann , r. a. 2005 , , 618 , 822 keshet , u. , & waxman , e. 2005 , , 94 , 111102 kirk , j. g. , guthmann , a. w. , gallant , y. a. , & achterberg , a. 2000 , , 542 , 235 kirk , j. g. , & heavens , a. 1989 , , 239 , 995 kirk , j. g. , & schneider , p. 1987
, , 315 , 425 lemoine , m. , & pelletier , g. 2003 , , 589 , l73 medvedev , m. v. , & loeb , a. 1999 , , 526 , 697 niemiec , j. , & ostrowski , m. 2004 , , 610 , 851 niemiec , j. , & ostrowski , m. 2006 , , in press nishikawa , k .- i . ,
hardee , p. , richardson , g. , preece , r. , sol , h. , & fishman , g. j. 2003 , , 595 , 555 nishikawa , k .- i . ,
hardee , p. , richardson , g. , preece , r. , sol , h. , & fishman , g. j. 2005 , , 622 , 927 nishikawa , k .-
i . , hardee , p. , hededal , c. , & fishman , g. j. 2005 , , accepted ostrowski , m. 1991 , , 249 , 551 ostrowski , m. 1993 , , 264 , 248 ostrowski , m. , & bednarz , j. 2002 , , 394 , 1141 silva , l. o. , fonseca , r. a. , tonge , j. w. , dawson , j. m. , mori , w. b. , & m. medvedev , m. v. 2003 , , 596 , l121 stern , b.e .
2003 , , 345 , 590 virtanen , j. , & vainio , r. 2005 , , 621 , 313
|
the present paper is the last of a series studying the first - order fermi acceleration processes at relativistic shock waves with the method of monte carlo simulations applied to shocks propagating in realistically modeled turbulent magnetic fields .
the model of the background magnetic field structure of @xcite has been augmented here by a large - amplitude short - wave downstream component , imitating that generated by plasma instabilities at the shock front .
following @xcite , we have considered ultrarelativistic shocks with the mean magnetic field oriented both oblique and parallel to the shock normal . for both cases simulations
have been performed for different choices of magnetic field perturbations , represented by various wave power spectra within a wide wavevector range .
the results show that the introduction of the short - wave component downstream of the shock is not sufficient to produce power - law particle spectra with the universal " spectral index @xmath0 . on the contrary , concave spectra with cutoffs
are preferentially formed , the curvature and cutoff energy being dependent on the properties of turbulence .
our results suggest that the electromagnetic emission observed from astrophysical sites with relativistic jets , e.g. active galactic nuclei and gamma - ray bursts , is likely generated by particles accelerated in processes other than the widely invoked first - order fermi mechanism .
|
electric currents induce spin - transfer torques in heterogeneous or textured magnetic systems.@xcite in this context , magnetic insulators such as yttrium iron garnet ( yig ) combined with normal metal contacts exhibiting spin - orbit interactions , such as pt , have recently attracted considerable interest , both experimentally kajiwara : nat10,sandweg : apl10,sandweg : prl11,vilela - leao : apl11,burrowes : apl12,rezende : apl12,nakayama : prl13 and theoretically.xiao:prb10,slonczewski:prb10,xingtao:epl11,xiao:prl12,kapelrud:prl13,yan-ting:prb13,xiao:xxx13 since the discovery of non - local exchange coupling and giant magnetoresistance in spin valves , i.e. , a normal metal sandwiched between two ferromagnetic metals , these systems have been known to display rich physics .
some of these effects , such as the dynamic exchange interaction,@xcite should also arise when the magnetic layers are insulators .
the spin hall magnetoresistance ( smr ) is predicted to be enhanced in such spin valves , @xcite although experimental realizations have not yet been reported .
here , we consider multilayer structures with ferromagnetic but electrically insulating ( fi ) layers and normal metal ( n ) spacers .
in - plane electric currents applied to n generate perpendicular spin currents via the spin hall effect ( she ) .
when these spin currents are absorbed at the n@xmath0fi interfaces , the ensuing spin - transfer torques can induce magnetization dynamics and switching .
we consider ground state configurations in which the magnetizations are parallel or antiparallel to each other . for thin magnetic layers , even small torques can effectively modify the ( gilbert ) damping , which can be observed as changes in the line width of the ferromagnetic resonance ( fmr ) spectra .
we employ the macrospin model for the magnetization vectors that is applicable for sufficiently strong and homogeneous magnetic fields , while extensions are possible.@xcite our results include the observation of effective ( anti)damping resulting from in - plane charge currents in fi@xmath0n@xmath0fi trilayers , magnetic stability analysis in the current - magnetic field parameter space and a brief analysis of the dynamics for currents above the critical value .
we also consider current - induced effects in superlattices .
our paper is organized as follows . in section [
s : model ] , we present our model for a fi@xmath0n@xmath0fi spin valve including the she spin current generation and spin pumping , modeled as additional torques in the landau - lifshitz - gilbert equation . we proceed to formulate the linearized magnetization dynamics and the spin accumulation in n in section [ s : torques ] . in section [ s : eigenmodes ] , we calculate the eigenmodes and the current - controlled effective gilbert damping and determine the critical currents at which the magnetic precession becomes unstable . we discuss the current - induced dynamics of @xmath1 superlattices in section [ s : superlattice ] .
finally , we summarize our conclusions and provide an outlook in section [ s : conclusions ] .
fi1@xmath0n@xmath0fi2 denotes the heterostructure composed of a normal metal ( n ) layer sandwiched between two layers of ferromagnetic insulators ( fis ) ( see fig . [
fig : parallel ] ) .
we denote the thicknesses of fi1 , n and fi2 by @xmath2 , @xmath3 and @xmath4 , respectively .
we adopt a macrospin model of spatially constant magnetization @xmath5 in each layer .
the magnetization dynamics of the two layers are described by the coupled landau - lifshitz - gilbert - slonczewski ( llgs ) equations : @xmath6where @xmath5 is the unit vector in the direction of the magnetization in the left / right layer with indices @xmath7 ; @xmath8 is the saturation magnetization ; @xmath9 is the gyromagnetic ratio ; @xmath10 is the gilbert damping constant ; @xmath11 is the interlayer dipolar and exchange energy areal density , with @xmath12 when @xmath13 ; and @xmath14 is an effective magnetic field : @xmath15consisting of the external magnetic field @xmath16 as well as the anisotropy fields @xmath17 for the left / right layer . we distinguish direct ( dsp ) and indirect spin pumping ( isp ) .
dsp generates the spin angular momentum current @xmath18 through the interfaces of fi1(2 ) .
a positive spin current corresponds to a spin flow toward the fi from which it originates .
the dsp spin current is expressed as @xmath19where @xmath20 is the real part of the spin - mixing conductance of the n@xmath0fi1(2 ) interface per unit area for @xmath21 , respectively , and @xmath22 is the electron charge .
this angular momentum loss causes a damping torque ( here and below in cgs units ) : @xmath23 in ballistic systems , the spin current emitted by the neighboring layer is directly absorbed and generates an indirect spin torque on the opposing layer:@xcite @xmath24 in the presence of an interface or bulk disorder , the transport is diffuse , and the isp is @xmath25where @xmath26 is the spin pumping contribution to the spin accumulation ( difference in chemical potentials ) at the interface in units of energy , with @xmath27 for @xmath7 .
@xmath28 is the solution of the spin diffusion equation in n as discussed below . due to
the she , an in - plane dc charge current produces a transverse spin current that interacts with the fi@xmath0n interfaces . focusing on the diffusive regime , the areal density of charge current @xmath29 as well as the spin @xmath30 current@xmath31 in the @xmath32-direction , where @xmath33 is the spin polarization unit vector , can be written in terms of a symmetric linear response matrix:@xcite @xmath34where @xmath35 is the spin hall angle , @xmath36 is the electrical conductivity and @xmath37 is the charge chemical potential .
@xmath38 is the spin accumulation induced by reflection of the spin currents at the interfaces .
the spin transfer torques @xmath39 at the fi interfaces ( @xmath7 ) are then expressed as @xmath40the polarization of @xmath41 and thereby @xmath39 can be controlled by the charge current direction . in the following sections , we assume that the shape anisotropy and exchange coupling favor parallel or antiparallel equilibrium orientations of @xmath42 and @xmath43 . for small current levels , the torques normal to the magnetization induce tilts from their equilibrium directions and , at sufficiently large currents , trigger complicated dynamics , while torques directed along the equilibrium magnetization modify the effective damping and induce magnetization reversal . here
, we focus on the latter configuration , in which the spin accumulation in n is collinear to the equilibrium magnetizations . in the following equations , we take the thickness , saturation magnetization , gilbert damping and spin - mixing conductance to be equal in the two layers fi1 and fi2 , with an out - of - plane hard axis and an in - plane internal field : @xmath44 with @xmath45 and @xmath46 . pure dipolar interlayer coupling with @xmath47 favors an antiparallel ground state configuration , while the exchange coupling oscillates as a function of @xmath48 .
and @xmath43 are collinear , i.e. , parallel or antiparallel .
a spin - hall - induced spin current flows in the @xmath49-direction and is polarized along @xmath50.[fig : parallel],width=302 ]
the spin - pumping and spin - transfer torques @xmath51 and @xmath52 ( eqs . and )
cause dynamic coupling between the two magnetizations . to leading order
, these torques can be treated separately .
we now derive expressions for disordered systems that support spin accumulations @xmath53 @xmath54 governed by the spin - diffusion equation : @xmath55 here , @xmath56 is the diffusion constant , and @xmath57 is the spin - flip relaxation time .
the diffuse spin current in the @xmath49-direction related to this spin accumulation follows eq .
( [ sheishe ] ) : @xmath58where @xmath36 is the conductivity of n. the total spin current into an fi is the sum of the spin - transfer and spin - pumping currents . disregarding interface spin - flip scattering , the boundary conditions for the left / right layer are @xmath59the -(+ ) sign on the right - hand side is due to the opposite flow direction of the spin currents at the left ( right ) interface .
we expand the magnetization direction around the equilibrium configuration as @xmath60 as long as @xmath61 or @xmath62 .
the parameter @xmath63 when the equilibrium configuration is parallel ; @xmath64 when it is antiparallel .
the fmr frequency is usually much smaller than the diffuse electron traversal rate @xmath65 and spin - flip relaxation @xmath66 rate ; thus , retardation of the spin flow may be disregarded . in the steady state , the left - hand side of eq .
we solve eq .
for the adiabatic magnetization dynamics with boundary conditions eq . to obtain the spin accumulation : @xmath67 , \end{aligned}\ ] ] where @xmath68 is the spin - diffusion length and @xmath69 the torques are @xmath70 because the spin accumulation is generated by the dynamics of both ferromagnets , we obtain spin - pumping - induced dynamic coupling that is quenched when @xmath71 . in the limit of vanishing spin - flip scattering ,
the spin accumulation is spatially constant and is expressed as @xmath72the corresponding diffusive torque is then a simple average of the contributions from the two spin - pumping currents , in contrast to the ballistic torque that depends only on the magnetization on the opposite side .
a charge current in the @xmath73-direction causes a spin hall current in the @xmath49-direction that is polarized along the @xmath50-direction ( see fig . [
fig : parallel ] ) . at the interfaces ,
the current induces a spin - accumulation @xmath41 that satisfies the diffusion eq . and drives a spin current ( dropping the index @xmath49 from now on ) : @xmath74where @xmath75angular momentum conservation at the left / right boundaries leads to @xmath76when @xmath77 , the spin hall current is completely reflected and the spin current at the interface vanishes , while the absorption and torque are maximal when @xmath78 .
spin currents and torques at the interface scale favor @xmath79 for small magnetization amplitudes .
let us define a time - independent @xmath80 for collinear magnetizations and spin current polarization . for small dynamic magnetizations ,
then @xmath81where @xmath82 .
we will show that the spin - hall induced spin accumulation leads to a ( anti)damping torque in the trilayer , while it gives a contribution to the real part of the frequency for superlattices ( see sec .
[ s : superlattice ] ) . solving the diffusion eq .
( [ eq : spin - pumping diffusion ] ) with boundary conditions , eq .
( [ current conservation ] ) yields @xmath83the dynamic correction @xmath84 \label{eq : spin accumulation}\end{aligned}\]]leads to she torques [ eq .
( [ eq : stt ] ) ] : @xmath85 .\ \label{sh coupling}\]]eq . then reduces to four coupled linear first - order partial differential equations for @xmath79
after linearizing eq . and fourier transforming to the frequency domain @xmath86 , eq
. becomes @xmath87where @xmath88 and @xmath89 is a @xmath90 frequency - dependent matrix that can be decomposed as @xmath91with [ eq : matrices ] @xmath92here , @xmath93 describes dissipationless precession in the effective magnetic fields , and @xmath94 arises from gilbert damping and the direct effect of spin pumping with a renormalized damping coefficient @xmath95 and @xmath96 @xmath97 represents interlayer exchange coupling , @xmath98 represents spin - pumping - induced spin transfer , and @xmath99 represents the spin transfer caused by the spin hall current .
the external and possible in - plane anisotropy fields are modified by the interlayer coupling , @xmath100 , where @xmath101 .
the matrix elements @xmath102 , @xmath103 , @xmath104 and @xmath105 are generalized susceptibilities extracted from eqs . and : @xmath106 the explicit expressions given in appendix [ app : matrix elements ] are simplified for very thick and thin n spacers . _
thin n layer : _ when @xmath107 , the interlayer coupling @xmath103 due to spin pumping approaches @xmath108 , the intralayer coupling : @xmath109 which implies that the incoming and outgoing spin currents are the same .
this outcome represents the limit of strong dynamic coupling in which the additional gilbert damping due to spin pumping vanishes when the magnetization motion is synchronized.@xcite in this regime , the she becomes ineffective because @xmath104 and @xmath105 scale as @xmath110 .
@xmath111 because @xmath104 contains a contribution from both the static as well as the dynamic spin accumulation .
_ thick n layer : _ in the thick film limit , @xmath112 , the interlayer coupling vanishes as @xmath113 and @xmath114 , while @xmath115 introducing the spin conductance @xmath116 @xmath117 and @xmath118 , the total resistance of the interface and the spin active region of n , @xmath119 , represents the backflow of pumped spins .
the same holds for the part of @xmath104 that originates from the dynamic part of @xmath120 , while the static part approaches a constant value when @xmath121 becomes large ( see appendix [ app : matrix elements ] ) . in this limit
, the system reduces to two decoupled fi@xmath0n bilayers .
the eigenmodes of the coupled system are the solutions of @xmath122 = 0 $ ] with complex eigenfrequencies @xmath123 .
the she spin current induces spin accumulations with opposite polarizations at the two interfaces . in the parallel case ,
the torques acting on the two fis are exerted in opposite directions .
the torques then stabilize one magnetization , but destabilize the other .
when the eigenfrequencies acquire a negative imaginary part , their amplitude grows exponentially in time .
we define the threshold current @xmath124 by the value at which @xmath125 = 0.$ ] because the total damping has to be overcome at the threshold , @xmath126 .
we treat the damping and exchange coupling perturbatively , thereby assuming @xmath127 and @xmath128 where @xmath129 is the fmr frequency .
the spin hall angle is usually much smaller than unity ; thus , @xmath130 is treated as a perturbation for currents up to the order of the threshold current , implying that @xmath131 \right\vert \ll \left\vert \text{re}[\omega _ { n}\left ( j_{0}^{\mathrm{sh}}\right ) ] \right\vert $ ] .
the exchange coupling @xmath132 for yig@xmath0pt@xmath0 yig should be weaker than that of the well - studied metallic magnetic monolayers , where it is known to become very small for @xmath133.@xcite in the following sections , we assume that @xmath134 may be treated as a perturbation . to treat the damping , spin pumping , spin - hall - induced torques and static exchange perturbatively , we introduce the smallness parameter @xmath135 and let @xmath136 , @xmath137 , @xmath138 , @xmath139 . in the following sections ,
a first - order perturbation is applied by linearizing in @xmath135 and subsequently setting @xmath140 .
we transform @xmath89 by the matrix @xmath141 that diagonalizes @xmath93 with eigenvalues @xmath142 .
we then extract the part corresponding to the real eigenfrequencies , which yields the following equation : @xmath143 where @xmath144 .
we thus reduce the fourth - order secular equation in @xmath145 to a second - order expression . to the first order
, we find for the parallel ( @xmath63 ) case , @xmath146where we introduced a current - controlled effective gilbert damping : @xmath147the imaginary part of the square root in eq
. causes a first - order real frequency shift that we may disregard , i.e. , @xmath148 \approx \tilde{\omega}_{0}\approx \omega _ { 0}$ ] .
we thus find two modes with nearly the same frequencies but different effective broadenings .
the critical current @xmath149 is now determined by requiring that @xmath150 vanish , leading to @xmath151while the critical charge current is @xmath152 .
spin pumping and spin flip dissipate energy , leading to a higher threshold current , which is reflected by @xmath153 .
the reactive part of the she - induced torque ( @xmath154 ) suppresses the effect of the applied current and thereby increases the critical current as well .
the static exchange couples @xmath42 and @xmath43 , hence increasing @xmath149 .
the critical spin current decreases monotonically with increasing @xmath155 , implying that the spin valve ( with parallel magnetization ) has a larger threshold current than the fi@xmath0n bilayer ( with thick @xmath121 ) .
analogous to the parallel case , we find two eigenmodes for the antiparallel case ( @xmath64 ) , with eigenfrequencies @xmath156and corresponding effective gilbert damping parameters @xmath157which depend on the magnetic configuration because the dynamic exchange coupling differs , while the resonance frequency is affected by the static coupling . in the ap configurations , the spin hall current
acts with the same sign on both layers due to the increase / decrease in damping on both sides depending on the applied current direction .
the corresponding threshold current is expressed as @xmath158with @xmath159 .
again , the threshold for current - induced excitation is increased by the spin pumping .
to zeroth order in the smallness parameter @xmath135 , we find that the eigenvectors for the parallel configuration take the form @xmath160 , where @xmath161 is the 2-component vector @xmath162the imbalance in the amplitudes of both layers is parameterized by @xmath163where @xmath164 corresponds to the @xmath165 in eq . .
for the symmetric case , the applied current favors out - of - phase oscillations .
it can be demonstrated that in the limit of large currents and low spin - memory loss , the corresponding amplitude difference is @xmath166 , with @xmath167 , and an interlayer coupling dominated by either dynamic or static exchange @xmath168 , which correspond to an optical and an acoustic mode , respectively .
we use the labels acoustic and opticeven though the phase difference is not precisely 0 or @xmath169 due to the static exchange interaction . note that @xmath170 is required by symmetry ;
inverting the current direction is equivalent to interchanging fi1 and fi2 . for @xmath171
, @xmath172 is a pole or node depending on the current direction for the acoustic mode in which the magnetization in one layer vanishes . above this
current , @xmath173 change signs , and both modes have a phase difference of @xmath169 .
the critical current lies above the current corresponding to the node at which the acoustic mode becomes unstable .
the ballistic model also supports acoustic and optical modes , tserkovnyak : revmod05 with the optical mode being more efficiently damped . in the antiparallel case ,
acoustic and optical modes can are characterized by amplitudes @xmath174where the optical ( acoustic ) mode corresponds to the + ( - ) sign in eq . .
the labels optical and acoustic are kept because of the difference in effective damping ; a 180@xmath175 rotation about the @xmath73 axis of fi2 map these modes to the corresponding modes for the parallel case .
.physical parameters used in the numerical calculations [ cols="<,^,^",options="header " , ] \a ) ref .
[ ] , b ) ref .
[ ] , c ) ref .
[ ] when the composition of the spin valve is slightly asymmetric , the dynamics of the two layers can still be synchronized by the static and dynamic coupling
. however , at some critical detuning @xmath176 , this technique no longer works , as illustrated by the eigenfrequencies for the asymmetric spin valve in fig .
fig : synchronization .
here , we employ yig@xmath0pt@xmath0yig parameters but tune the fmr frequency of the right yig layer . in practice , the tuning can be achieved by varying the direction of the applied magnetic field.heinrich:prl03 when the fmr frequencies of the two layers are sufficiently close , the precessional motions in the two layers lock to each other .
the asymmetry introduced by higher currents is observed to suppress the synchronization .
n@xmath0fi2 spin valve as a function of the detuning of the fmr frequencies of the individual layers and for different currents @xmath177 for the solid , black dashed and grey dashed lines , respectively . at zero applied current the two layers lock when detuning is small .
the current suppresses synchronization almost completely when reaching the threshold value.the inset shows the corresponding broadenings.,width=302 ] [ fig : omegaparallel ] .
the effective damping is rescaled by letting @xmath178 and @xmath179 .
the numerical calculation was carried out by a 4th order runge - kutta method with a step size @xmath180 .
, title="fig:",width=302 ] .
the effective damping is rescaled by letting @xmath178 and @xmath179 .
the numerical calculation was carried out by a 4th order runge - kutta method with a step size @xmath180 .
, title="fig:",width=297 ] the non - linear large - angle precession that occurs for currents above the threshold is not amenable to analytical treatments ; however , numerical calculations can provide some insights . because the dissipation of yig is very low the number of oscillations required to achieve a noticable change in the precession angle is very large . to speed up the calculations and make the results more readable we rescale both @xmath181 and @xmath182 by a factor @xmath183 , in this way
the effective damping is rescaled .
fig : omegaparallel shows the components of the magnetization in the two layers as a function of time when a large current is switched on for an initially parallel magnetization along @xmath50 with a slight canting of @xmath184 for @xmath7 .
we apply a current @xmath185 at @xmath186 . for @xmath187 ,
the precession is out of phase , and the amplitude gradually increases . at @xmath188 ,
the applied current is ramped up to @xmath189 . at @xmath190 ,
the precession angle is no longer small , and our previous perturbative treatment breaks down .
however , we can understand that the right layer precesses with a large angle , while the left layer stays close to the initial equilibrium from the opposite direction of the interface spin accumulations @xmath80 .
a periodic stack of fis coupled through ns supports spin wave excitations propagating in the perpendicular direction .
the coupling between layers is described by eq . ; however , each fi is coupled through the n layers to two neighboring layers .
the primitive unit cell of the superlattice with collinear magnetization is the fi@xmath0n bilayer for the parallel configuration ( two bilayers in the antiparallel configuration ) . for equivalent saturation magnetizations in all fi layers
, we can write for @xmath191 @xmath192where @xmath193 for the parallel and @xmath194 for the antiparallel ground state
. we can then linearize the expression with respect to the small parameters @xmath79 .
an in - plane charge current causes accumulations of opposite sign in each n layer .
the long - wavelength excitations of the superlattice magnetization can be treated in the continuum limit . denoting the total thickness of a unit cell @xmath195
, we find for the parallel case ( @xmath193 ) @xmath196 .\end{aligned}\]]for @xmath197 , the linearized dispersion relation is @xmath198the applied current thus adds a term that is linear in @xmath199 to the real part of the frequency . the direct effect of the she now vanishes because the torques on both sides of any fi cancel .
however , when @xmath200 , a net spin current flows normal to the stack , which affects the dispersion . in the ferromagnetic layers ,
this phenomenon is equivalent to a pure strain field on the magnetization and is therefore non - dissipative .
while generating @xmath130 causes ohmic losses , the magnetization dynamics in this limit do not add to the energy dissipation , explaining the contribution to @xmath201.$ ] in this regime , there are no external current - induced contributions or instabilities .
antiferromagnetic superlattices appear to be difficult to realize experimentally because a staggered external magnetic field would be required .
the unit cell is doubled as is the number of variables in the equation of motion .
determining the coupling coefficients from eq . is straightforward but cumbersome and is not presented here .
naively , one could expect that the she - induced torque would act very differently in the antiferromagnetic case .
the she acts in a symmetric manner on the fi(@xmath202)@xmath0n@xmath0fi(@xmath203 ) system , stabilizing or destabilizing both layers simultaneously . however , similarly to the ferromagnetic superlattice , the direct she vanishes also in the antiferromagnetic superlattice .
each fi is in contact with an n , with spin accumulations of opposite sign on the left and right side of the interfaces , which leads to the same cancellation of the direct she - induced torque presented for the ferromagnetic superlattice .
we can also envision a multilayer in which individual metallic layers can be contacted separately and independently .
n@xmath0fi@xmath0n structures have been predicted to display a magnon drag effect through the ferromagnetic film,@xcite i.e. a current in one layer induces an emf in the other one . a drag effect does also exists in our macrospin model : if we induce dynamics by a current in one layer by the spin hall effect , the spin pumping and inverse spin hall effect generates a current in the other layer , but only above a current threshhold . with separate contacts to the layers
one may drive opposite currents through neighboring films . in that case , the spin currents absorbed by a ferromagnetic layer is relatively twice as large as in the fi@xmath0n bilayer , thereby reducing the critical currents for the parallel configuration , but of opposite sign for neighboring magnetic layers . a staggered current distribution in the superlattice destabilizes the ferromagnetic configuration , but it can stabilize an antiferromagnetic one even in the absence of static exchange coupling .
this leads to intricate dynamics when competing with an applied magnetic field .
we study current - induced magnetization dynamics in spin valves and superlattices consisting of insulating magnets separated by metallic spacers with spin hall effect .
the current - induced torques experienced by the two magnetic layers in an fi(@xmath204)@xmath0n@xmath0fi(@xmath204 ) spin valve caused by the spin hall effect are opposite in sign .
a charge current in n normal to the magnetization this leads to a damping and an antidamping , stabilizing one and destabilizing the other magnetization .
we calculate the magnetization dynamics when the two layers are exchange coupled and in the presence of the dynamic exchange coupling induced by spin pumping . in an antiparallel configuration fi(@xmath204)@xmath0n@xmath0fi(@xmath205 ) the interlayer couplings play a minor role in the current - induced effects .
the threshold currents at which self - oscillation occur are higher for parallel than antiparallel spin valves .
we predict interesting current - induced effects for superlattices and multilayers in which the metallic spacer layers can be individually contacted .
and a. b. acknowledge support from the research council of norway , project number 216700 .
this work was supported by kakenhi ( grants - in - aid for scientific research ) nos .
25247056 and 25220910 , fom ( stichting voor fundamenteel onderzoek der materie ) , the icc - imr , the eu - rtn spinicur , eu - fet grant inspin 612759 and dfg priority program 1538 `` spin - caloric transport '' ( go 944/4 ) . 99 a. brataas , a. d. kent , and h. ohno , nature mat . ,
* 373 * ( 2012 ) .
h. nakayama , m. althammer , y .- t .
chen , k. uchida , y. kajiwara , d. kikuchi , t. ohtani , s. geprgs , m. opel , s.takahashi , r. gross , g.e .
w. bauer , s. t. b. goennenwein and e. saitoh , phys .
* 110 * 206601 ( 2013 ) .
y. kajiwara , k. harii , s. takahashi , j. ohe , k. uchida , m. mizuguchi , h. umezawa , h. kawai , k. ando , k. takanashi , s. maekawa , and e. saito , nature * 464 , * 7269 ( 2010 ) . c. w. sandweg , y. kajiwara , k. ando , e. saitoh , and b. hillebrands , appl .
. lett . * 97 * , 252504 ( 2010 ) . c. w. sandweg , y. kajiwara , a. v. chumak , a. a. serga , v. i. vasyuchka , m. b. jungfleisch , e. saitoh , and b. hillebrands , phys .
* 106 * , 216601 ( 2011 ) .
l. h. vilela - leao , c. salvador , a. azevedo , s. m. rezende , appl . phys
. lett . * 99 * , 102505 ( 2011 ) . c. burrowes , b. heinrich , b. kardasz , e. a. montoya , e. girt , yiyan sun , young - yeal song , and mingzhong wu , appl .
. lett . * 100 * , 092403 ( 2012 ) .
s. m. rezende , r. l. rodriguez - suarez , m. m. soares , l. h. vilela - leao , d. ley dominguez , and a. azevedo , appl .
. lett . * 102 * , 012402 ( 2012 ) .
j. xiao , g. e. w. bauer , k .- c .
uchida , e. saitoh , and s. maekawa , phys .
b * 81 * , 214418 ( 2010 ) .
chen , s. takahashi , h. nakayama , m. althammer , s. t. b. goennenwein , e. saitoh , and g. e. w. bauer , phys .
b * 87 * , 144411 ( 2013 ) .
j. c. slonczewski , phys .
b * 82 * , 054403 ( 2010 ) .
j. xingtao , l. kai , k. xia , and g. e. w. bauer , epl * 96 * , 17005 ( 2011 ) .
a. kapelrud and a. brataas , phys .
lett . * 111 * , 097602 ( 2013 ) .
y. zhou , h. j. jiao , y. t. chen , g. e. w. bauer , and j. xiao , phys .
b * 88 * , 184403 ( 2013 ) . j. xiao and g. e. w. bauer , phys
. rev . lett . * 108 * , 217204 ( 2012 ) .
b. heinrich , y. tserkovnyak , g. woltersdorf , a. brataas , r. urban and g. e. w. bauer , phys .
90 , 187601 ( 2003 ) . z.
q. qiu , j. pearson and s. d. bader , phys .
b * 46 * , 8659 ( 1992 ) .
beaujour , w. chen , a. d. kent and j. z. sun , j. appl . phys . * 99 * , 08n503 ( 2006 ) .
y. tserkovnyak , a. brataas , g. e. w. bauer and b. i. halperin , rev .
* 77 * 1375 ( 2005 ) .
m. b. jungfleisch , v. lauer , r. neb , a. v. chumak and b. hillebrands , appl .
. lett . * 103 * , 022411 ( 2013 ) .
d. giancoli , `` 25 .
electric currents and resistance '' . in jocelyn phillips .
physics for scientists and engineers with modern physics ( 4th ed . ) , ( 2009 ) [ 1984 ] a. a. serga , a. v. chumak and b. hillebrands , j. phys .
phys . * 43 * , 264002 ( 2010 ) steven s .-
zhang and s. zhang , phys .
* 109 * , 096603 ( 2012 ) .
here , we derive the response coefficients @xmath104 , @xmath154 , @xmath102 and @xmath103 that determine the torques , depending on the properties of the normal metal .
let us first discuss the coefficients related to the torques induced by the she .
the functions @xmath104 and @xmath154 are extracted from the derivatives of eq . with respect to the transverse components of the dynamic magnetizations
@xmath104 governs the she - induced torque in one layer due to displaced magnetization in the same layer and can be computed as @xmath206thus , @xmath207 .\end{aligned}\]]similarly , we can identify @xmath154 , which governs the cross - correlation of the she - induced torque in one layer arising from a displaced magnetization in the other layer from@xmath208thus@xmath209 .\end{aligned}\]]torques generated by spin pumping contain terms of the form @xmath210 and couple the @xmath73- and @xmath49-components of the magnetization dynamics .
we find @xmath211where @xmath212similarly , @xmath213where @xmath214we finally note that some of the coefficients are related:@xmath215
|
we study the dynamics of spin valves consisting of two layers of magnetic insulators separated by a normal metal in the macrospin model . a current through the spacer generates a spin hall current that can actuate the magnetization via the spin - transfer torque .
we derive expressions for the effective gilbert damping and the critical currents for the onset of magnetization dynamics including the effects of spin pumping that can be tested by ferromagnetic resonance experiments .
the current generates an amplitude asymmetry between the in - phase and out - of - phase modes .
we briefly discuss superlattices of metals and magnetic insulators .
|
the large area telescope ( lat ; atwood et al . 2009 ) aboard the fermi gamma ray space telescope ( fermi ) has recently detected nearly 20 grbs ( e.g. abdo et al .
2009a , b ; abdo et al . 2010
; ackermann et al . 2010 , see zhang et al .
2011 for a synthetic study ) . among them , several bright grbs ( e.g. grbs 080916c , 090510 , 090902b and 090926a )
have well sampled long - term lat - band lightcurves . in logarithmic space , these grbs have count rates that rise , peak and begin decaying before the mev prompt emission is over , i.e. peaking at a time smaller than @xmath0 defined in the gamma - ray burst monitor ( gbm ; meegan et al .
2009 ) detector energy band .
the post peak lightcurve typically has a decay slope steeper than @xmath1 ( e.g. ranging from @xmath2 to @xmath3 , ghisellini et al .
2009 ; zhang et al .
the simple temporal behavior ( a broken power law lightcurve ) of lat emission led to the suggestion that grb gev emission is of an external forward shock origin ( kumar & barniol duran 2009 , 2010 ; ghisellini et al .
2009 ) , possibly from a highly radiative blastwave .
a simple broken power law lightcurve is expected from the blastwave evolution of an instantaneously injected fireball with fixed explosion energy .
such an approximation is valid if the analyzed time scale is much longer than @xmath0 , the duration of the prompt gamma - ray emission itself .
however , for the early blastwave evolution , especially during the epoch when the central engine is still active ( as is the case for the lat grbs discussed in this paper ) , one would not expect a simple lightcurve evolution , since the energy output from the central engine is continuously injected into the blastwave .
the high quality spectral and temporal data of grbs co - detected by fermi lat and gbm allow us to track the energy output from the central engine as a function of time .
recently we have developed a shell code to model the internal and external shock development for arbitrary central engine activities ( maxham & zhang 2009 ) . by processing the spectral and temporal evolution data of fermi grbs using the method described in zhang et al .
( 2011 ) , we can model the early development of the external shock based on first hand data .
we study four bright lat grbs ( 080916c , 090510 , 090902b , and 090926a ) .
gbm and lat data reduction was carried out using the data analysis script introduced in zhang et al .
this code uses the public fermi data and extracts time - resolved spectral information derived from a joint gbm / lat fit . for the gbm data , the background spectrum is extracted using the cspec data , while the source spectrum is extracted using the event ( tte ) data .
the lat background is different since only a few photons are detected by lat for most grbs , so on - source region data long after the gbm trigger when the photon counts merge into a poisson noise are used to derive the lat lightcurve background .
the gbm and lat data are then used to make dynamically time - dependent spectral fits .
the code refines the number of time slices as necessary to preserve adequate statistics in each bin , and a spectral fit is chosen among a list of spectral models , such as a single power - law , a power - law with exponential cut - off , a band function , a black body or a combination of these .
chi square statistics are performed to determine which fits are the best , and ockham s razor chooses the simplest spectral model between two statistically reasonable fits ( zhang et al .
2011 ) . for the 4 bright grbs in our sample ,
we adopt the following models ( for details , see zhang et al .
2011 ) . for grb 080916c and 090926a we adopt the band function model throughout the burst , with the spectral parameters evolving with time .
grb 090902b shows a blackbody thermal component plus a non - thermal single power law component , and the short burst grb 090510 is best - fit with a cutoff power law plus power law component .
similar to ghisellini et al .
( 2009 ) , we found that the long - term lat light curves decay before the end of @xmath0 with a slope steeper than -1 .
we model a grb as an explosion of many matter shells with some mass and lorentz factor ( rees & mszros 1994 ) .
as the first matter shell moves outward into the ambient medium , it slows down when sweeping up this medium ( mszros & rees 1993 ) .
as time goes by , more and more trailing shells pile up onto the leading decelerating shell ( rees & mszros 1998 ) . for an instantaneous explosion with constant energy , the motion of this decelerating ejecta along with the medium collected along the way , known as the blastwave " ,
is governed by three differential equations ( chiang & dermer 1999 ; huang et al .
2000 ) : radius changing with time , @xmath4 , a statement of conservation of energy and momentum across the blastwave @xmath5 ( blandford & mckee 1976 ) , and the amount of medium swept up as a function of radius @xmath6 . here
@xmath7 is the time since explosion in the rest frame of the central engine , @xmath8 is the distance from the central engine , @xmath9 is the density of the ambient medium , @xmath10 is the lorentz factor of the shell , @xmath11 is the swept - up mass , and @xmath12 is the total effective mass including the internal energy of the blastwave , where @xmath13 is the initial mass of the ejecta . as a result ,
one has another differential equation , @xmath14 where the value of @xmath15 determines the efficiency of the radiation , with 0 representing the purely adiabatic case and 1 representing the fully radiative condition .
the above set of differential equations can be solved analytically .
the adiabatic solution was presented as eq.(14 ) in maxham & zhang ( 2009 ) .
since the lat lightcurves decay with a slope steeper than -1 ( typical value for an adiabatic blastwave ) , e.g. in the range of -1.3 and -2 ( zhang et al . 2011 ) , it may be reasonable to assume a completely radiative blastwave ( ghisellini et al .
2009 ) . by adopting a value of @xmath16
, one can get a purely radiative solution for the blastwave , which reads @xmath17 in the deceleration regime , one has @xmath18 , and @xmath19 for @xmath20 ( which is relevant for lat band ) , which is @xmath21 for @xmath22
( e.g. sari et al . 1998 ) .
this is consistent with the rapid decay observations . during the prompt emission phase ( i.e. @xmath23 )
, the central engine continuously injects energy into the blastwave .
so the solution should take into account the progressively increasing total energy in the blastwave .
we apply the shell code developed and laid forth in maxham & zhang ( 2009 ) to this problem .
the code , which originally generated randomized matter shells with different mass , lorentz factor and ejection time , is here modified to use input values for these parameters which are taken from the data as follows .
the most important parameter affecting blastwave evolution is the total injection energy . in principle
the injected energy during each episode is the kinetic energy of the ejecta after energy dissipation during the prompt emission phase . lacking a direct measure of this energy , we hereby assume that the emitted @xmath10-ray energy is a good proxy of the kinetic energy , so that @xmath24 . in other words ,
we assume a constant radiative efficiency throughout the burst .
we take @xmath25 as the nominal value ( i.e. 50% radiative efficiency , which may be achieved for efficient magnetic energy dissipation , zhang & yan 2011 ) . in order to fit the data
, we also allow @xmath26 for the grbs , which corresponds to a less efficient dissipation mechanism ( e.g. in internal shocks , panaitescu et al . 1999 ; kumar 1999 ; maxham & zhang 2009 ) . to evaluate @xmath10-ray energy @xmath27 as a function of time , we divide the lightcurve into multiple time bins for each burst . for each time bin
( with uneven duration denoted as @xmath28 for @xmath29-th bin ) , we record its average flux @xmath30 in the gbm band , along with other useful information such as spectral parameters and the maximum photon energy .
the total gamma - ray energy released in this time bin ( @xmath29-th ) is therefore @xmath31 where @xmath32 is the redshift ( see table 1 for values of each burst ) , @xmath33 is the luminosity distance of the source , and the concordance cosmology with @xmath34 and @xmath35 is adopted in the calculation . adopting @xmath36 , we then progressively increase the total energy in the blastwave @xmath37 by adding @xmath38 in each step . for each time step , we calculate the lightcurve giving the available @xmath39 .
this results in a series of lightcurve solutions .
the final lightcurve is then derived by jumping to progressively higher level solutions due to additional energy injections in each time step ( see also maxham & zhang 2009 ) .
this would result in a series of `` glitches '' in the lightcurves , each representing injection of energy from @xmath29-th shell into the blastwave .
besides the energy , we also derive the ( lower limit ) lorentz factor @xmath40 of each shell .
this parameter is important , especially for early shells , since it determines the deceleration time of a certain shell .
this is particularly relevant for the first shell .
the lorentz factors of later shells are also relevant fot two reasons .
first , they can be used to calculate the effective lorentz factor of a `` merged '' shell after adding energy to an existing shell .
this is needed to calculate the deceleration time of the blastwave solutions .
second , since the observed time for a late energy injection is defined by ( maxham & zhang 2009 ) @xmath41 where @xmath42 and @xmath43 are the times of ejection and collision measured in the rest frame of the central engine .
the effect of @xmath10 becomes progressively less important , since at large @xmath42 s , the second term in eq.([collisiontime ] ) becomes negligible so that the observed collision time is essentially defined by the ejection time . in any case , we derive the constraints on @xmath10 for each time bin using the pair opacity argument as described below . to derive a constraint on the lorentz factor ,
we have collected the spectral parameters and the observed maximum photon energy @xmath44 for each time bin .
one can then derive the maximum photon energy in the cosmological local frame , i.e. @xmath45 . requiring the pair production optical depth to be less than unity for @xmath46 , we can write a general constraint in the parameter space of @xmath8 and @xmath10 ( where @xmath8 is the distance of the emission region from the central engine , gupta & zhang 2008 ; zhang & peer 2009 ) , i.e. @xmath47 where @xmath48 is the thompson cross section , @xmath49 represents the slope of the power law component for grbs 090902b and 090510 and the band function high energy spectral parameter for grbs 080916c and 090926a , and @xmath50 ( in units of @xmath51 ) can be written as @xmath52^{\alpha - \beta } \rm{exp}(\beta - \alpha)(100 \quad \rm{kev})^{-\alpha}$ ] for the band function model , and @xmath53 for the simple power law model , where @xmath54 and @xmath55 are normalization factors ( both normalized to 100 kev ) .
the approximation @xmath56 ( svensson 1987 ) is adopted to perform the calculation . in order to further constrain @xmath10 ,
one needs to make an assumption about @xmath8 . without other independent constraints ,
we apply the conventional assumption of internal shocks , so that @xmath57 , where @xmath58 is the observed minimum variability time scale . combining eq.([solveme ] ) , the lower limit for @xmath10 is derived for each time bin of each burst ( see also lithwick & sari 2001 , abdo et al .
2009 ) . in our calculation
, we generally adopt @xmath40 as the derived lower limit .
this is because the derived lorentz factors of other grbs using the afterglow deceleration constraint ( liang et al .
2010 ) or photosphere constraint ( peer et al . 2011 ) are all below or consistent with these lower limits derived from the opacity constraints ( abdo et al .
2009a , b , 2010 ; ackermann et al . 2010 ) . feeding this data into our shell model code , letting
each shell be ejected with energy @xmath38 and lorentz factor @xmath40 at time equal to that of the beginning of the bin time , we can calculate the early blastwave evolution and lat band ( integrated over @xmath59 mev ) lightcurve for the four grbs . to match the observed steep decay ( with slope @xmath60 ) , we adopt a radiative fireball solution or an adiabatic fireball solution with steep electron energy index .
even though each solution ( for a fixed kinetic energy ) has a steep decay slope , the overall lightcurve shows a shallower decay due to piling up of successive shells ejected later , with glitches introduced by jumping among the solutions . as an example , the radiative model lightcurve of grb 080916c as compared with observation is presented in fig.1 .
the top panel shows the long term evolution , while the bottom panel is the zoomed - in early afterglow lightcurve .
the dotted lines denote the blastwave solutions with progressively increasing total energy .
the lowest one corresponds to the first time bin , the second lowest corresponds to adding the energy of the second time bin , etc .
since the lightcurve is chopped into discrete time bins , the blastwave energy is added in discrete steps .
this introduces some artificial glitches in the lightcurve .
such an approximation is more realistic for grbs with distinct emission episodes . for grb 080916c ,
the lightcurve is more appropriately approximated as a continuous wind with variable luminosity .
the artificial glitches should appear to be more smeared .
for this reason , we have smoothed the glitches to make more natural transitions between solutions .
the model afterglow parameters ( the fraction of electron energy @xmath61 , the fraction of magnetic energy @xmath62 , and the number density @xmath63 ) are presented in table 1 .
these are in general consistent with the parameter constraints derived by kumar & barniol duran ( 2009 , 2010 ) .
in general , the model lightcurve of grb 080916c can not fit the early lat data .
making the model suitable to fit the late - time steep decay , the early model lightcurve level is too low to account for the observed data .
alternatively , one can make the early model lightcurve match the observed flux level .
then inevitably the late time afterglow level exceeds the observed level significantly due to the continuous energy injection .
we believe that if the lat band emission after @xmath0 originates from the external shock , then the lat emission during the prompt emission phase _ can not _ be solely interpreted by the external shock model .
the external shock contribution is relatively small , especially during early epochs when energy in the blastwave is small . as a result ,
the gev emission during the prompt phase must be of an internal origin .
this is consistent with the fact that the entire gbm / lat emission during the prompt phase can be well fit by a single band - function spectral model in all the time bins ( abdo et al .
2009a ; zhang et al .
2011 ) .
we have also modeled grbs 090510 , 090902b and 090926a .
the model parameters ( for both radiative and adiabatic solutions ) are listed in table 1 , and the results for radiative solution are shown in fig.[fluxlightcurves ] . in all cases , the slope and flux level of the data are matched in the latter part of the curve only . during the prompt emission phase , the data points rise above the flux prediction of the external shock model , suggesting that gev emission is a superposition of external and internal components during the prompt emission phase ( @xmath64 ) .
this conclusion is valid for both the adiabatic and radiative solutions .
the difference between the two is that the adiabatic model invokes a shallower @xmath65 but a larger @xmath66 ( and hence a larger energy budget ) to fit the same data .
.parameters used for the four sample bursts[tbl ] [ cols="^,^,^",options="header " , ] mev lightcurve of grb 080916c for a radiative blastwave solution ( yellow line ) as compared with the data ( blue points ) .
successive lightcurves that correspond to different total blastwave kinetic energy are shown as dashed lines .
the top panel shows the global lightcurve , while the bottom panel shows a zoom view where the flux deficit at early times can be clearly seen.,title="fig : " ] mev lightcurve of grb 080916c for a radiative blastwave solution ( yellow line ) as compared with the data ( blue points ) .
successive lightcurves that correspond to different total blastwave kinetic energy are shown as dashed lines .
the top panel shows the global lightcurve , while the bottom panel shows a zoom view where the flux deficit at early times can be clearly seen.,title="fig : " ] mev lightcurve ( for a radiative blastwave solution ) vs. observed data for grbs 090510 , 090902b , and 090926a .
the conventions are similar to fig.1 , but without successive solutions specifically plotted.,title="fig : " ] mev lightcurve ( for a radiative blastwave solution ) vs. observed data for grbs 090510 , 090902b , and 090926a .
the conventions are similar to fig.1 , but without successive solutions specifically plotted.,title="fig : " ] mev lightcurve ( for a radiative blastwave solution ) vs. observed data for grbs 090510 , 090902b , and 090926a .
the conventions are similar to fig.1 , but without successive solutions specifically plotted.,title="fig : " ]
using the first - hand fermi data , we have tracked the energy output from the central engine and modeled the early blastwave evolution of four bright lat grbs .
the predicted @xmath59 mev lightcurve is found unable to account for the observed lat emission during the prompt emission phase .
the main reason is that during the phase when the central engine is still active , the forward shock is continuously refreshed by late energy injection , so that the afterglow decays much slower than the case predicted by an instantaneously ejected constant energy fireball .
this suggests that at least during the prompt emission phase , the lat band emission is not of external forward shock origin .
this is in contrast to the suggestion of ghisellini et al .
( 2009 ) , kumar & barniol duran ( 2009 ) and feng & dai ( 2010 ) , who did not consider the energy accumulation during the prompt emission phase and interpreted the entire gev emission as due to the external shock origin . our conclusion is based on the assumption that grb radiative efficiency is essentially a constant throughout the burst . in order to interpret the entire afterglow as due to the external forward shock origin
, one needs to artificially " assume that the grb efficiency increases with time , so that the late time central engine activity , even though producing bright @xmath10-ray emission , adds little kinetic energy into the blastwave .
we believe that such an assumption is contrived . our conclusion is consistent with some independent arguments . from data analysis , zhang et al .
( 2011 ) showed that during the prompt emission phase the gev emission and mev emission traces each other well .
for grb 080916c , the entire gbm / lat emission can be modeled by a single band function component in all the time bins ( see also abdo et al .
2009a ) . for grb 090902b ,
even though gev emission belongs to a distinct spectral component , its flux seems to track the flux of the mev component nicely , suggesting a connection in the physical origin ( see peer et al .
2011 for modeling ) .
a more definite argument in favor of an internal origin of gev emission in grb 080916c is that the gev lightcurve peak coincides the second peak in the gbm lightcurve , suggesting that gev emission is the spectral extension of mev emission to higher energies ( zhang et al .
. also individual case studies of grb 090902b ( peer et al .
2011 ; liu & wang 2011 ) and grb 090510 ( he et al .
2011 ) all suggest that the external shock model can not interpret the prompt gev data .
in general , our modeling suggests that it is possible to use the external shock model to interpret gev emission after the prompt emission phase , but not during the prompt emission phase ( see also kumar & barniol duran 2010 ) .
our conclusion also has implications for understanding grb prompt emission physics , in particular , the composition of the grb outflow .
the internal origin of gev emission in grb 080916c makes it essentially impossible to interpret the entire band spectrum with the photosphere model ( e.g. beloborodov 2010 ; lazzati & begelman 2010 ) .
the lack of photosphere emission then demands a poynting - flux - dominated outflow at least for this burst ( zhang & peer 2009 ; fan 2010 ) , and new models in the poynting flux dominated regime ( e.g. zhang & yan 2011 ) are called for .
|
recent observations of gamma - ray bursts ( grbs ) by the fermi large area telescope ( lat ) revealed a power law decay feature of the high energy emission ( above 100 mev ) , which led to the suggestion that it originates from a ( probably radiative ) external shock .
we analyze four grbs ( 080916c , 090510 , 090902b and 090926a ) jointly detected by fermi lat and gamma - ray burst monitor ( gbm ) , which have high quality lightcurves in both instrument energy bands . using the mev prompt emission ( gbm ) data
, we can record the energy output from the central engine as a function of time . assuming a constant radiative efficiency , we are able to track energy accumulation in the external shock using our internal / external shell model code . by solving for the early evolution of both an adiabatic and a radiative blastwave
, we calculate the high energy emission lightcurve in the lat band and compare it with the observed one for each burst .
the late time lat light curves after @xmath0 can be well fit by the model .
however , due to continuous energy injection into the blastwave during the prompt emission phase , the early external shock emission can not account for the observed gev flux level .
the high energy emission during the prompt phase ( before @xmath0 ) is most likely a superposition of a gradually enhancing external shock component and a dominant emission component that is of an internal origin .
[ firstpage ] gamma - rays : bursts fermi telescope .
|
at the lagrangian level , qcd has , in addition to @xmath3 chiral symmetry , an approximate @xmath2 symmetry , under which all left - handed quark fields are rotated by a common phase while the right - handed quark fields are rotated by an opposite phase .
it is well known that the @xmath2 symmetry is violated by the axial anomaly present at the quantum level and thus can not give rise to the goldstone boson which would occur when @xmath4 chiral symmetry is spontaneously broken .
the @xmath2 particle , known as @xmath5 in the @xmath6 case , acquires an additional mass through the quantum tunneling effects mediated by instantons @xcite , breaking up the mass degeneracy with pions and @xmath0 in the chiral limit when all quarks ( @xmath7 , @xmath8 and @xmath9 ) are massless .
the @xmath10 particle also acquires an additional mass through the mixing with @xmath11 .
it is believed that at high temperatures the instanton effects are suppressed due to the debye - type screening @xcite .
then one expects a practical restoration of @xmath2 at high temperatures .
if the restoration occurs at a temperature lower than the chiral phase transition temperature @xmath12 , there may be some interesting phenomenological implications in high - energy heavy - ion collisions , as suggested first by pisarski and wilczek @xcite and more recently by shuryak @xcite .
one of the consequences of @xmath2 restoration is the enhancement of @xmath0 particle production at small and intermediate transverse momenta due to the softening of its mass at high temperatures .
however , the final yield of the @xmath0 particles and their @xmath13 distributions both depend crucially on the chemical and thermal equilibrating processes involving the @xmath0 . in this paper , we shall examine the rates of various processes relevant for the thermal @xmath0 particle production , in particular , whether or not the @xmath0 can decouple early enough from the thermal system expected to be produced in relativistic heavy ion collisions .
we shall present a theoretical calculation of the thermal cross sections for the processes @xmath14 , @xmath15 and @xmath16 , essential to the thermal and chemical equilibration .
our calculations are based on models which explicitly incorporate the @xmath2 anomaly .
we also assume an exponential suppression of the @xmath2 anomaly due to the debye - type screening of the instanton effect @xcite , which leads to the temperature dependence of the @xmath0 and @xmath11 masses .
our results suggest that the chemical equilibrium breaks up for @xmath0 particles long before the thermal freeze - out .
we suggest a modest enhancement of thermal @xmath0 production as a signal for the relic of @xmath2 restoration .
this paper is organized as follows : in sec .
[ ii ] , we compute the mass spectrum of @xmath0 and @xmath11 using the di vecchia - veneziano model , which incorporates the @xmath2 anomaly and the @xmath0-@xmath11 mixing effect .
we obtain the low - energy theorems for various scattering amplitudes . in sec .
[ iii ] , we incorporate the @xmath17 and the @xmath18 resonances using the t hooft model and reevaluate the @xmath0 scattering cross sections . in sec .
[ iv ] , we study the thermal averaged cross sections responsible for maintaining thermal and chemical equilibria , and suggest that the chemical equilibrium between @xmath0 and @xmath1 breaks up considerably earlier than the @xmath0 thermal equilibrium .
we discuss two scenarios for the @xmath0 freeze - out and their corresponding signals for the @xmath0 production .
we briefly comment on the roles of @xmath11 and the qcd sphalerons in sec . [ v ] and sec .
[ vi ] respectively .
up to now , there has been no direct experimental measurement of the @xmath0 scattering cross sections ( or the scattering lengths ) .
one has to rely on theoretical models to calculate the interaction rates which are complicated by many uncertainties .
nevertheless , the scattering amplitudes at low energy can be more or less precisely predicted if the meson masses are soft , thanks to the soft - meson theorems which are based on the symmetry of the interactions and depend very little on the detailed dynamics .
the current - algebra predictions of these scattering amplitudes have been made very early by osborn @xcite based on @xmath19 , where the anomalous @xmath2 and the @xmath0 and @xmath11 mixing are not included . in the light of softening of @xmath0 and @xmath11 masses at high temperatures
, we argue that the symmetry can be extended to @xmath20 .
we shall rederive the low - energy amplitudes incorporating the anomalous @xmath2 using the nonlinear @xmath17-model that at the lowest order should give us the low - energy theorems .
the standard di vecchia - veneziano model @xcite , which incorporates the explicit @xmath2 anomaly , reads after integrating out the gluon field @xmath21 where @xmath22 , @xmath23 mev , @xmath24 and @xmath25 the last term in eq .
( [ 1 ] ) is the anomaly term which breaks @xmath2 explicitly .
it is easy to check that eq .
( [ 1 ] ) satisfies the anomalous ward identity which is crucial for determining the form of @xmath2 breaking @xcite . in eq .
( [ 1 ] ) , @xmath26 is related to the topological charge correlation function in pure yang - mills theory @xmath27\rangle_{\rm ym}\ ; , \label{3}\ ] ] where @xmath28 is the dual gluon field strength tensor and @xmath29 stands for the vacuum expectation value at zero temperature or the thermal average at finite temperature .
the integral @xmath26 is identically zero in perturbation theory ; it only receives nonperturbative contributions arising from the topologically nontrivial instanton configurations .
the calculation of @xmath26 at both zero and finite temperature is done by gross , pisarski and yaffe @xcite using a dilute gas approximation , and by dyakonov and petrov and by shuryak @xcite using an instanton liquid model . for our purpose , the phenomenological value of @xmath26 at @xmath30 can be fixed by the meson mass spectroscopy , while @xmath31 will be modeled by assuming an exponential suppression shown by pisarski and yaffe @xcite at high @xmath32 .
the quadratic terms for the octet @xmath33 and the singlet @xmath34 from the lagrangian reads @xmath35 \ ; . \label{4}\ ] ] clearly , there is a mixing between the octet @xmath33 and the singlet @xmath34 .
the physical @xmath10 and @xmath5 are defined by @xmath36 to diagonalize the quadratic terms with the mixing angle @xmath37 and the physical masses are @xmath38 the mixing angle @xmath39 as well as @xmath40 and @xmath41 depend on the instanton - induced quantity @xmath26 which is a function of temperature .
the precise form of @xmath42 at low temperature is not known .
nevertheless , if the @xmath2 breaking becomes soft at a temperature lower than the chiral phase transition temperature @xmath12 , one may model the suppression effect by an exponential dependence @xcite @xmath43 where @xmath44 mev , while keeping the masses of the pion and kaon approximately temperature independent , since they change very slowly with the temperature .
it is known that mixing angle @xmath39 , @xmath40 and @xmath41 at @xmath30 can not be simultaneously fit to their experimental values by a single parameter @xmath45 .
the best fit is to use the measured value of @xmath46 as an input to determine @xmath47 and use this @xmath45 to predict @xmath39 , @xmath40 and @xmath41 using eqs .
( [ 5 ] ) and ( [ 6 ] ) . at @xmath30 ,
the predicted values are @xmath48 , @xmath49 mev and @xmath50 mev , compared to the measured values @xmath51 from @xmath52 , @xmath53 mev and @xmath54 mev .
the temperature dependence of @xmath55 and @xmath56 is completely determined by a temperature - dependent @xmath42 given in eq .
( [ 7 ] ) . throughout this paper , we take @xmath57 mev in eq .
( [ 7 ] ) .
it should be emphasized that in relativistic heavy - ion collisions , the thermal system freezes out at about @xmath58 mev , when the collision time scale exceeds the size of the system mainly determined by the nuclear radius @xmath59 fm for central @xmath60 or @xmath61 collisions . below the freeze - out temperature @xmath62
, the finite - temperature calculation of @xmath42 does not make sense and the behavior of @xmath26 is determined by the nonequilibrium dynamics .
figure 1 schematically plots such a temperature dependence .
clearly , the @xmath0 becomes soft at high @xmath32 and eventually is degenerate with the pions .
the mass of @xmath11 also decreases .
however , it does not become degenerate with the pion because of the large strange - quark mass , as is seen from eq .
( [ 6b ] ) . from fig . 1 we see that the @xmath11 mass at high temperatures is still higher than the @xmath0 mass at zero temperature .
in fig . 1 we also plot the temperature dependence of the @xmath18 resonance mass which we will discuss in the following section . at temperatures higher than @xmath63 ,
the masses of these excitation modes will all rise again .
the interaction terms are obtained by expanding @xmath64 in eq .
( [ 1 ] ) .
in contrast to the pion field , there are no derivative couplings involving @xmath0 and @xmath11 .
we shall ignore the interactions of @xmath0 and @xmath11 with kaons since they are heavy compared with pions . to the lowest order , the quartic terms involving @xmath1 , @xmath0 and @xmath11 are @xmath65\ ; , \label{8}\end{aligned}\ ] ] where @xmath66 .
at very high @xmath32 , as @xmath67 and @xmath68 , we can see that the @xmath11 decouples from interactions with @xmath1 and @xmath0 .
the low - energy theorems on the two - body scattering amplitudes can be easily derived from eq .
( [ 8 ] ) : @xmath69 \ ; , \nonumber\\ { \cal a}(\eta\eta\leftrightarrow\pi^a\pi^a ) & = & { \cal a}(\pi^a\eta \leftrightarrow \pi^a\eta ) = \frac{1}{f_\pi^2}m_\pi^2\sin ^2\chi \ ; , \nonumber\\ { \cal a}(\eta ' \eta ' \leftrightarrow\pi^a\pi^a ) & = & { \cal a}(\pi^a\eta ' \leftrightarrow \pi^a\eta ' ) = \frac{1}{f_\pi^2}m_\pi^2\cos ^2\chi \ ; , \nonumber\\ { \cal a}(\eta\eta ' \leftrightarrow\pi^a\pi^a ) & = & { \cal a}(\pi^a\eta ' \leftrightarrow \pi^a\eta ) = \frac{1}{f_\pi^2}m_\pi^2\sin \chi\cos\chi \ ; , \nonumber\\ { \cal a}(\eta ' \eta ' \leftrightarrow\eta ' \eta ' ) & = & \frac{1}{f_\pi^2}[m_\pi^2\cos ^4\chi + ( 4m_k^2 - 2m_\pi^2)\sin ^4\chi ] \ ; , \nonumber\\ { \cal a}(\eta ' \eta ' \leftrightarrow\eta \eta ) & = & { \cal a}(\eta\eta ' \leftrightarrow \eta\eta ' = \frac{1}{f_\pi^2}(4m_k^2-m_\pi^2)\sin ^2\chi\cos^2\chi \ ; , \nonumber\\ { \cal a}(\eta \eta \leftrightarrow\eta \eta ' ) & = & \frac{1}{f_\pi^2}[m_\pi^2\sin^3\chi\cos \chi -(4m_k^2 - 2m_\pi^2)\sin\chi\cos^3\chi ] \ ; , \nonumber\\ { \cal a}(\eta \eta ' \leftrightarrow\eta ' \eta ' ) & = & \frac{1}{f_\pi^2}[m_\pi^2\sin\chi\cos^3 \chi -(4m_k^2 - 2m_\pi^2)\sin^3\chi\cos\chi ] \ ; . \label{9}\end{aligned}\ ] ] the results calculated by osborn @xcite based on the current algebra can be recovered by taking @xmath70 and using the gell - mann - okubo relation @xmath71 .
these low - energy theorems must be satisfied by any dynamical models , because they are solely based on the symmetry properties of the theory .
the amplitudes listed in eq . ( [ 9 ] ) grossly underestimate the strength of scatterings at higher energies , especially in the resonance regions .
however , the inclusion of resonances introduces many uncertainties , such as which resonances should be included and what are the couplings of these resonances to the mesons .
in addition , there is no guarantee that a naive lowest - order calculation will preserve the unitarity because of the strong interactions .
fortunately , the low - energy theorems provide us some guidelines as to how the amplitudes should approach their low - energy limits .
the linear @xmath17-model based on the chiral symmetry is known to satisfy the low - energy theorems , and at the same time to be able to incorporate the resonances . to further reduce the input parameters , we consider the @xmath17 and @xmath72 [ now called @xmath73 $ ] resonances , which , together with @xmath1 and the @xmath74 to be defined below , form a complete representation of @xmath75 . we shall concentrate on the @xmath0 particle , since there is no dramatic change of the @xmath11 mass with temperature , as shown in fig . 1
. we study the most relevant processes for the @xmath0 production : @xmath76 , @xmath77 , and @xmath78 . in this case
, @xmath20 reduces to @xmath75 except for the mixing effects which we have already calculated .
let us introduce the nonstrange mode @xmath79 and take @xmath80 to be heavy .
then @xmath74 is approximately a mass eigenstate , @xmath81 , whose mass is determined from eq .
( [ 4 ] ) to be @xmath82 . at zero temperature , @xmath83 mev .
we then define the ( 2,2 ) representation multiplet of @xmath75 as @xmath84 the most general @xmath75 invariant potential is @xmath85 and the mass term is @xmath86 where @xmath87 , @xmath88 are dimensionless constants .
the @xmath2-breaking term , consistent with the ward identity , is introduced by t hooft @xcite as @xmath89 and the coefficient in @xmath90 is chosen such that it gives the correct mass for @xmath74 .
the mass spectrum can be derived from eqs .
( [ 10 ] ) , ( [ 11 ] ) and ( [ 12 ] ) by making a shift @xmath91 : @xmath92 the decay widths are @xmath93 [ ( m_\delta^2-(m_{\rm ns}-m_\pi ) ^2]\}^{1/2 } \frac{(m_\delta^2-m_{\rm ns}^2)^2}{16\pi f_\pi^2m_\delta^3}\ ; .\end{aligned}\ ] ] at zero temperature , @xmath94 gev ( if @xmath95 mev ) and @xmath96 mev . in principle
, we should also take into account the temperature dependence of @xmath97 and @xmath98 below @xmath63 . here
, we assume the chiral phase transition is very rapid after which @xmath97 and @xmath98 have very slow temperature dependences .
furthermore , due to the large width of the @xmath17 , the slow temperature dependence of @xmath98 will not change our results significantly .
under such an assumption , the linear @xmath17-model predicts also some softening of the @xmath18 resonance as @xmath32 increases , because @xmath18 is the chiral partner of @xmath1 and acquires some mass from the @xmath2 anomaly .
the temperature dependence of @xmath99 is plotted in fig . 1 .
the interaction terms are @xmath100 the coupling constants @xmath87 and @xmath88 can be obtained from the mass relations of eq .
( [ 13 ] ) .
it is worth pointing out that the above model should not be used to estimate the pion - pion scattering amplitude , because it does not include the important vector resonances such as @xmath101 and @xmath102 .
however , since @xmath103 and @xmath104 scatterings can not go through @xmath105 channel , they do not directly affect the interaction rates for @xmath0 .
similarly , we have also neglected the @xmath0-@xmath101 interaction . to calculate the scattering amplitudes at the lowest order , we have to remove a pole singularity encountered when a resonance appears in the @xmath9-channel .
a naive introduction of breit - wigner resonance width will spoil the delicate cancellation between the contact interaction and the pole exchange at low energy , leading to the violation of the low - energy theorems .
we adopt a minimal prescription to save the low - energy limit developed by chanowitz and gaillard @xcite , making the following replacement @xmath106 the scattering amplitudes are calculated as follows : @xmath107 \ : , \nonumber\\ { \cal a}(\eta\eta\leftrightarrow \pi^a\pi^a ) & = & \sin^2\chi { \cal a}(\eta_{\rm ns}\eta_{\rm ns } \leftrightarrow \pi^a\pi^a)\nonumber\\ & = & \sin^2\chi\lambda_1(1-i\gamma_\sigma / m_\sigma ) \frac{s - m_\pi^2}{s - m_\sigma^2+im_\sigma\gamma_\sigma } \nonumber \\ & & + \sin^2\chi\lambda_2(1-i\gamma_\delta / m_\delta ) \left [ \frac{t - m_{\rm ns}^2}{t - m_\delta^2+im_\delta\gamma_\delta } + \frac{u - m_{\rm ns}^2}{u - m_\delta^2+im_\delta\gamma_\delta } \right ] \ : , \nonumber\\ { \cal a}(\eta\pi^a\leftrightarrow \eta\pi^a ) & = & \sin^2\chi { \cal a}(\eta_{\rm ns}\pi^a \leftrightarrow \eta_{\rm ns}\pi^a)\nonumber\\ & = & \sin^2\chi\lambda_1(1-i\gamma_\sigma / m_\sigma ) \frac{t - m_\pi^2}{t - m_\sigma^2+im_\sigma\gamma_\sigma } \nonumber \\ & & + \sin^2\chi\lambda_2(1-i\gamma_\delta / m_\delta ) \left [ \frac{s - m_{\rm ns}^2}{s - m_\delta^2+im_\delta\gamma_\delta } + \frac{u - m_{\rm ns}^2}{u - m_\delta^2+im_\delta\gamma_\delta } \right ] \ ; .\end{aligned}\ ] ] the cross sections for these processes are readily calculated by integrating out the scattering angle in @xmath7 and @xmath108 , most conveniently in the cm frame : @xmath109 where @xmath110 for ( non-)identical particles in the final state .
we are interested in the production of @xmath0 from a thermal source . to learn about the thermal history of the @xmath0 , one needs to calculate the thermal averaged cross sections for various reaction channels .
since we are only concerned with the qualitative picture , we assume throughout the rest of this paper boltzmann distribution functions for thermalized @xmath1 s and @xmath0 s and ignore the quantum bose - einstein enhancement .
the thermal averaged cross section for @xmath111 is @xmath112 where @xmath113[s-(m_i - m_j)^2]$ ] and @xmath114 is the reaction threshold .
the reactions @xmath115 and @xmath116 determine the collision time scale responsible for maintaining the thermal equilibrium while @xmath117 is responsible for the chemical equilibrium between @xmath1 s and @xmath0 s .
we define the time scales @xmath118 and @xmath119 as @xmath120 respectively , where @xmath121 and @xmath122 are the number densities for @xmath1 and @xmath0 , and the summation over different pion states is understood .
we have performed a numerical integration in eq .
( [ 19 ] ) and plotted @xmath118 and @xmath119 as functions of the temperature in figure 2 . in the calculation , we have explicitly taken into account the temperature dependence of @xmath123 , @xmath124 , @xmath125 and @xmath126 as calculated in sections ii and iii .
we take a typical value @xmath127 fm for the transverse freeze - out radius of the system .
we define the thermal and chemical freeze - out temperatures @xmath62 and @xmath128 respectively as @xmath129 and @xmath130 .
one finds from fig . 2 @xmath131 which are the temperatures at which the thermal and chemical equilibria start to break up , respectively .
it is worth noting that @xmath62 is comparable to the decoupling temperature of the thermal pions .
the result that @xmath128 is considerably higher than @xmath62 offers an interesting possibility to detect the suppression of the @xmath2 anomaly effect at high temperatures caused by the debye - type screening . at sufficiently high temperatures @xmath132
, the @xmath0 rescattering and the @xmath1-@xmath0 conversion are frequent so that the system possesses both thermal and chemical equilibria .
as the system expands and the temperature falls into the range @xmath133 , the @xmath1-@xmath0 conversion process becomes slow and effectively is turned off ; the system can no longer maintain the chemical equilibrium .
there is an approximate conservation of the total number of @xmath0 s since neither @xmath115 or @xmath116 can change the total @xmath0-number .
the number density of @xmath0 at the chemical break - up temperature @xmath134 is determined by the mass of @xmath0 at such a temperature @xmath135 : @xmath136 = \frac{1}{2\pi^2 } m_\eta ( t_{\rm ch})^2 t_{\rm ch}k_2\left [ \frac{m_\eta ( t_{\rm ch})}{t_{\rm ch}}\right]\ ; , \label{22}\ ] ] and the momentum distribution is just the boltzmann distribution with zero chemical potential .
however , this is not the final particle distribution , because the thermal collisions can still alter the momentum distribution .
nevertheless , the total number @xmath137 given by @xmath138 \label{23}\ ] ] is conserved at any time @xmath139 since @xmath76 is turned off . here
@xmath140 mev and @xmath141 is the proper time when the temperature of the system reaches @xmath128 .
as the system cools down to @xmath62 , the mass of @xmath0 should tend to @xmath142 mev , according to fig . 1 .
if the rate for increasing @xmath123 is comparable to the thermal collision rate , the @xmath0 particle adiabatically relaxes to @xmath143 . in this case , which we shall call scenario a , one expects a standard thermal distribution for @xmath0 at the freeze - out temperature @xmath62 with a mass @xmath143 .
the total number conservation requires @xmath0 to develop a chemical potential @xmath144 such that ( neglecting the transverse expansion ) @xmath145 = \tau_cn_\eta \left [ m_\eta ( t_{\rm ch } ) , t_{\rm ch}\right ] \ ; , \label{24}\ ] ] where @xmath146 is the freeze - out time when @xmath147 .
the momentum distribution function in the local comoving frame is @xmath148 the chemical potential @xmath149 is a function of temperature , whose value at freeze - out can be determined from eq .
( [ 24 ] ) once @xmath150 is known .
we assume that pions dominate the energy - momentum tensor ( in fact we explicitly checked the contribution from @xmath0 and found it negligible ) so that @xmath150 can be estimated by solving the ideal @xmath151 dimensional hydrodynamic equation @xmath152 where @xmath153 is the energy density and @xmath154 is the pressure , for massive pions .
we find that @xmath155 , given @xmath156 .
substituting the ratio back in eq .
( [ 24 ] ) , one finds @xmath157 .
we thus predict that if there is a partial @xmath2 restoration at high temperatures , the thermal @xmath0 production given by eq .
( [ 25 ] ) will be enhanced in this scenario due to both the finite chemical potential @xmath158 and a smaller @xmath0 mass @xmath159 mev at the thermal freeze - out temperature @xmath62 . to quantify such an enhancement
, we use eq .
( [ 25 ] ) to calculate the @xmath13 distribution of @xmath0 particle , employing the fireball model and taking into account the transverse flow effects as described in ref .
@xcite : @xmath160 where @xmath161 and @xmath162 ( with @xmath163 , @xmath164 ) is the transverse flow velocity profile @xcite . to reduce the possible normalization ambiguity
, we also calculate the @xmath13-distribution for pions at the same freeze - out temperature @xmath62 , taking into account only the dominant resonance decays , @xmath165 , and plot the ratio @xmath166 as a function of @xmath13 in figure 3 .
it should be noted that the thermal ratio is only relevant when @xmath13 is small . at very large @xmath13 ,
hard processes become important and the fireball model is no longer applicable . for comparison , we also plot the same ratio for a normal case in which the @xmath0 particles freeze out at the same temperature @xmath62 but with the zero - temperature mass @xmath167 mev .
another situation , which we shall call scenario b , is that the rate for increasing @xmath123 when @xmath133 is considerably smaller than the thermal collision rate . in this case , things get more complicated because the screening process is out of equilibrium .
the @xmath0 number conservation still holds , but the momentum distribution is quite different from that in scenario a.
roughly one may imagine that even though the temperature drops to @xmath62 after the chemical breakup , @xmath55 will still have the value @xmath135 , in close analogy to a `` quenching '' situation .
the number density at the thermal freeze - out temperature is then @xmath168 $ ] , and the chemical potential is determined by @xmath169 = \tau_cn_\eta \left [ m_\eta ( t_{\rm ch } ) , t_{\rm ch}\right ] \ ; , \label{29}\ ] ] yielding @xmath170 .
the momentum distribution function is @xmath171 which predicts larger @xmath0 enhancement at low @xmath13 than at high @xmath13 .
we also plot the ratio @xmath172 based on this scenario in fig . 3 . what happens after the thermal freeze - out
it is clear that there must exist some mechanism for the @xmath0 to relax from the ` temporary ' entity whose mass is either @xmath143 or @xmath135 to its true identity at zero temperature with @xmath173 mev .
a possible picture might be that the @xmath0 particles still feel a negative potential in the fireball .
the height of the potential barrier is determined by the mass difference @xmath174 . the @xmath0 particles with @xmath13 smaller than @xmath175
will be trapped in the potential well until the rarefaction wave reaches the center of the interaction volume .
such a picture has been suggested by shuryak @xcite and is similar to the mechanism of cold kaon production @xcite . at this stage ,
we do not attempt to address this nonequilibrium issue , but just to remark that our calculation here may have underestimated the enhancement effect at small @xmath13@xmath176 mev . both scenarios
a and b predict an enhancement of the thermal @xmath0 production in the light of a partial @xmath2 symmetry restoration .
it would be very interesting to test the idea experimentally by measuring the ratio @xmath177 , especially its @xmath13 dependence . although preliminary data from wa80 @xcite on @xmath177 ratio in both central and peripheral @xmath178 collisions at the cern sps energy , as indicated in fig .
3 , have shown such a trend of enhancement , one certainly needs better statistics in order to make a definite conclusion .
a related matter is the enhanced dilepton pair production via @xmath0 dalitz decay @xmath179 .
if the @xmath0 production is enhanced about 3 times , as we have predicted , the observed dilepton enhancement with the invariant mass below 500 mev at the cern sps @xcite may be partially accounted for .
there should be also some enhancement of the ratio @xmath180 , since the mass of @xmath11 also decreases as the temperature increases .
moreover , since the couplings of @xmath11 to @xmath0 and @xmath1 in our model become small and eventually goes to zero when @xmath2 is completely restored , @xmath11 might decouple from the system earlier than @xmath0 .
the decay @xmath181 can also enhance the @xmath0-production .
however , in our model , we postulate that the @xmath2 restoration occurs at a temperature below the chiral phase transition temperature .
therefore , the kaon mass @xmath182 is large and the @xmath11 does not become very soft . at @xmath134 , the @xmath11
mass @xmath56 is about 750 mev . even without the effect of chiral symmetry breaking
, the large strange - quark mass can give rise to a large mixing between @xmath0 and @xmath11 according to eq .
( [ 4 ] ) .
this mixing gives @xmath11 a mass @xmath183 even if @xmath2 is completely restored .
this mass is significantly larger than the @xmath0 mass at any temperature .
therefore , in the context of our model , the @xmath11 effects are only moderate .
the ratio @xmath180 should never exceed that of @xmath177 .
so far we have confined ourselves to the possible suppression of the instanton effects at finite temperature that causes the softening of the masses arising from the topological charge transitions . at very high temperatures
, it is known that such transitions can occur without going through the instanton configurations .
in fact , they are dominated by sphaleron - like transitions whose electroweak counterparts have been extensively studied in the literature @xcite .
it is pointed out by mclerran , mottola and shaposhnikov @xcite that the rate of a qcd sphaleron transition should be estimated in analogy to the electroweak theory for temperatures above the symmetry - restoration , which may not be quite suppressed . in the range of temperatures discussed in this paper ,
the rate of the qcd sphaleron transition may be unimportant . a rough estimate by giudice and shaposhnikov @xcite is @xmath184 where @xmath185 is the strength of the transition .
the characteristic time scale of the sphaleron transition is @xmath186 there is some evidence for @xmath185 to be @xmath187 from lattice calculations @xcite . unless @xmath185 is really big , greater than 10 , the sphalerons should be decoupled from the system in the hadronic phase , where the instanton effect is most dominant .
we wish to thank m. suzuki and v. koch for helpful discussions .
this work was supported by the director , office of energy research , office of high energy and nuclear physics , divisions of high energy physics and nuclear physics of the u.s .
department of energy under contract no.de-ac03-76sf00098 , and by the natural sciences and engineering research council of canada .
g. t hooft , phys .
rev . * d14 * , 3432 ( 1976 ) .
d. j. gross , r.d .
pisarski and l.g .
yaffe , rev .
phys . * 53 * , 43 ( 1981 ) .
r. d. pisarski and f. wilczek , phys . rev .
* d29 * , 338 ( 1984 ) .
e. shuryak , comments nucl .
* 21 * , 235 ( 1994 ) .
h. osborn , nucl .
b15 * , 501 ( 1970 ) .
p. di vecchia and g. veneziano , nucl .
b171 * , 253 ( 1980 ) .
h. chen , phys . rev . *
d44 * , 166 ( 1991 ) ; a. pich and e. de rafael , nucl . phys . *
b367 * , 313 ( 1991 ) .
s. aoki and t. hatsuda , phys
* d45 * , 2427 ( 1992 ) .
d. i. dyakonov and v.yu .
petrov , nucl . phys . *
b245 * , 259 ( 1984 ) ; _ ibid _ , * b272 * , 475 ( 1986 ) ; e. shuryak , _ ibid _ , * b302 * , 559 ( 1988 ) .
r. d. pisarski and l. g. yaffe , phys . lett .
* b97 * , 110 ( 1980 ) . h. kikuchi and t. akiba , phys
b200 * , 543 ( 1988 ) .
t. kunihiro , nucl .
b351 * , 593 ( 1991 ) .
g. t hooft , phys .
rep . * 142 * , 357 ( 1986 ) m. chanowitz and m. k. gaillard , nucl .
b261 * , 379 ( 1985 ) .
e. schnedermann , j. sollfrank and u. heinz , phys . rev . *
c48 * , 2462 ( 1993 ) ; e. schnederman and u. heinz , phys . rev . *
c50 * , 1675 ( 1994 ) .
a. lebedev _
et al . _ ,
wa80 collaboration , nucl .
phys . * a566 * , 355 ( 1994 ) .
i. tserruya , cern preprint , cern - ppe-95 - 52 ( 1995 ) ; g. agakichev _ et al .
_ , ceres collaboration , cern preprint , cern - ppe-95 - 026 ( 1995 ) . v. koch phys . lett . *
b351 * , 29 ( 1995 ) .
v. kuzmin , v. rubakov and m. shaposhnikov , phys . lett . *
b155 * , 36 ( 1985 ) ; p. arnold and l. mclerran , phys . rev . *
d36 * , 581 ( 1987 ) ; _ ibid _ , * d37 * , 1020 , ( 1988 ) . l. mclerran , e. mottola and m. shaposhnikov , phys . rev . *
d43 * , 2027 ( 1991 ) .
g. giudice and m. shaposhnikov , phys . lett . * b326 * , 118 ( 1994 ) ; t. askarrd , h. porter and m. shaposhnikov , nucl . phys . * b353 * , 346 ( 1991 ) . j. kapusta , d. kharzeev and l. mclerran , hep - ph/9507343 .
fig . 1 : : the temperature dependence of @xmath55 , @xmath56 , @xmath99 .
the parameter in the exponential suppression of the instanton effect is taken to be @xmath57 mev .
fig . 2 : : the characteristic time scales of the thermal and chemical equilibration for the @xmath0 particle .
fig . 3 : : the predicted ratio @xmath177 as a function of the transverse momentum @xmath13 in three scenarios as discussed in the text .
the preliminary data from wa80 @xcite are also indicated .
|
we calculate the thermally averaged rates for the @xmath0-@xmath1 conversion and @xmath0 scattering using the di vecchia - veneziano model and t hooft model , which incorporate explicitly the @xmath2 anomaly . assuming an exponential suppression of the @xmath2 anomaly
, we also take into account the partial restoration of @xmath2 symmetry at high temperatures .
we find that the chemical equilibrium between @xmath0 and @xmath1 breaks up considerably earlier than the thermal equilibrium .
two distinct scenarios for the @xmath0 freeze - out are discussed and the corresponding chemical potentials are calculated .
we predict an enhancement of the thermal @xmath0-production as a possible signal of the partial @xmath2 restoration in high - energy heavy - ion collisions .
|
the reaction @xmath1 near threshold was studied relatively long ago @xcite .
the authors of those papers claimed to have found an abnormal behavior of the production amplitude for this reaction near threshold that is not yet understood theoretically @xcite .
this conclusion was based on a comparison of the measured cross section with that for the production of a stable particle .
specifically , they found that the production cross section is proportional to @xmath2 instead of @xmath3 , as expected .
( @xmath3 denotes the momentum of the outgoing neutron in the center of momentum system . )
this behavior was interpreted as possible evidence for a resonance in the @xmath4 system not far above threshold @xcite . at the same time
there are no direct indications of the existence of such a resonance in the @xmath5channel .
recently a behavior similar to that of the cross section under discussion was also found in the reaction @xmath6 @xcite . the transition amplitude @xmath7 , while interesting in its own right , is also of great importance in other reactions .
theoretical analyses of the reactions @xmath8 @xcite , @xmath9 @xcite and @xmath10 @xcite , as well as @xmath0 production in proton nucleus collisions @xcite all rely on the @xmath7 transition amplitude , in which the pion enters as an exchanged particle , as the basic mechanism for the reactions studied .
it is thus of great theoretical interest to obtain direct , reliable experimental information on this reaction .
@xcite information the experiments cited above were all performed in an unusual kinematical situation : instead of measuring the momentum distribution of the final state for a fixed beam energy , the excitation function for a fixed neutron momentum versus initial energy was measured . in this work
we analyze the general expression for the production cross section of unstable particles in near threshold binary reactions . both situations the standard one , wherein the energy is fixed and the final state momentum is varied and that of the above experiments , wherein one of the final momenta is fixed while varying the energy , are compared .
we shall demonstrate that the dependence of the count rates on the outgoing center of mass momentum depends on how the analysis is done .
we conclude that the behavior of the @xmath11 amplitude is quite normal .
that means that the earlier interpretation of the experimental data @xcite is incorrect .
to explain the results of the cited papers , we need a smooth behavior of the averaged matrix element that must be a decreasing function of energy in the near threshold region .
we begin by discussing the production of a resonance using a monochromatic beam .
we then consider the integration of the so - obtained cross section over the beam energy , which is the procedure that was carried out experimentally in refs .
we close with a discussion of the formulae used in the cited papers to analyze the experimental data .
let us consider the case of a monochromatic beam of energy @xmath12 . in this case
the differential cross section @xmath13 for production of a stable particle is @xmath14 where @xmath15 ( @xmath16 ) and @xmath17 ( @xmath18 ) denote the reduced mass and the momentum of the initial state ( final state ) respectively .
this integral is proportional to @xmath19 after the integration is performed .
if we consider the cross section for the production of an omega meson or any other resonance with finite width @xmath20 , expression ( [ stabl ] ) must be convoluted with the spectral density @xmath21 . for simplicity
we use a breit - wigner form for the spectral density , namely @xmath22 here @xmath23 is the average mass of the unstable particle . in this case
the resonance production cross section is given by the expression : @xmath24 where @xmath25 and @xmath26 is the maximum momentum of the outgoing neutron for the reaction @xmath1 .
@xmath26 is determined by the masses of the lightest decay products of the unstable particle . note that , because of the experimental setup
, the authors of the papers [ 13 ] have not measured the total differential cross section for the production of an unstable particle as given by eq .
( 3 ) , but only a fraction of it , as the momentum of the outgoing neutron was constrained to lie in a small band around a given @xmath3 . in other words
, they have measured the following part of differential cross section : @xmath27 let us estimate the remaining integral for the case when the scattering amplitude @xmath28 is approximately constant in the interval @xmath29 , as one would expect close to the production threshold . in this case
we get @xmath30 where @xmath31 and @xmath32 . here
the limits are @xmath33 .
as will become clear below , the behavior of the integral depends on the parameter @xmath34 let us consider the case of small @xmath3 and @xmath29 such that the condition @xmath35 is satisfied . in this limit
the denominator under the integral is practically constant , so that we have @xmath36 where @xmath37 .
the dependence of this integral on energy looks like a bw - resonance with strength proportional to @xmath38 .
this was the dependence found in ref .
@xcite .
[ sec : integr ] in the experiments @xcite an additional integration over the beam energy ( still keeping @xmath3 fixed ) was performed in order to remove the width - dependence from eq .
( [ wide ] ) .
indeed , since the spectral density is normalized , integrating over the beam energy gives @xmath39 the above derivation shows specifically that @xmath40 where @xmath41 .
the right hand side of this equation is displayed in figure [ fig ] , based on the data of ref .
@xcite for the points with @xmath42 . as for the point at @xmath43
, we used the data from ref .
@xcite , averaged over an interval of @xmath44 .
note that the errors of @xmath45 displayed in the plot contain the uncertainty in @xmath3 as well .
figure 1 gives evidence for a matrix element @xmath46 that is practically constant , at least for the points @xmath47 .
the data above @xmath48 give evidence for a matrix element that decreases smoothly with @xmath3 , as would be expected in the usual effective range approximation .
we conclude therefore that the existing experimental data @xcite for the reaction @xmath49 give no indication of a growth of the matrix element for increasing @xmath3 in a wide interval of momenta @xmath3 above threshold .
note that the over all dependence of the matrix element on @xmath3 is contrary to the conclusions of refs .
[ 1 - 3 ] .
[ sec : chi ] we now investigate the second limiting case , @xmath50 to estimate the integral ( [ integr ] ) in this situation we must further distinguish separately two possibilities : \i ) the energy parameter @xmath51 is within the limits @xmath52 and @xmath53 of the integral ( [ integr ] ) . in this case @xmath54
\ii ) the energy parameter @xmath51 is not within the interval @xmath55 $ ] .
the integral @xmath56 is then strongly suppressed . in short ,
if condition ( [ more ] ) is satisfied we get the usual energy behavior for the differential cross section , namely a linear @xmath3dependence . the condition @xmath57 determines the critical value of @xmath3
thus , by measuring the count rates versus @xmath3 one may observe a transition from a @xmath38 behavior of the cross section at low @xmath3 to a linear dependence at high @xmath3 , even for a constant matrix element . in the case of the omega
this takes place at @xmath58 , if @xmath59 mev , as specified in ref .
@xcite . in ref .
@xcite the production of @xmath60 and @xmath61 was studied as well .
the authors report that here a behavior very different from what they found for @xmath0 production . using the above discussion
one can now easily understand this : for both mesons condition ( [ more ] ) was satisfied , since @xmath62 mev for the @xmath60 and @xmath63 mev for the @xmath61 .
[ sec : incor ] to complete our criticism of the analyses of refs .
@xcite , we compare our formulae to those given therein .
let us start by briefly repeating the arguments for a linear dependence of the @xmath0 production cross section on @xmath3 given in @xcite . instead of eq .
( 1 ) for the production of a stable particle , ref .
@xcite starts directly from the expression for the double differential cross section , @xmath64 in order to get the total production rates for producing final particles with a given @xmath3 , expression ( [ pru ] ) was integrated over the initial energy under the constraint @xmath65 . by employing energy conservation , i.e. using the condition @xmath66 ( [ pru ] )
can be formally integrated .
since the spectral density is normalized , this integration yields @xmath67 this procedure to obtain eq .
( [ binniecs ] ) from eq .
( [ pru ] ) looks formally correct , but it is not .
the reason for this is that in order to derive eq .
( [ pru ] ) the energy conserving @xmath68function in eq .
( 1 ) was evaluated .
therefore @xmath3 in eq . ( [ pru ] )
implicitly depends on @xmath12 and @xmath69 and thus must be treated as a dependent variable in any argument based on eq .
( [ pru ] ) .
therefore the use of the relation ( [ encon ] ) in this context is simply incorrect .
instead , the condition of allowing @xmath3 to vary only in a small interval translates into a condition on the ranges of integration of @xmath69 , given a fixed energy @xmath12 , namely @xmath70 with @xmath71 .
this formula actually agrees with our eq .
( 5 ) if we rewrite it in terms of an integration over @xmath72 .
therefore we conclude that in the theoretical analysis of refs .
@xcite the limits of integration were not properly treated , thereby leading to an inappropriate conclusion for the momentum dependence of the cross section .
it is this point that was overlooked in the earlier works . if we impose the limit @xmath73 on eq .
( [ invo2 ] ) and integrate over the energy we again find @xmath74 to clarify the situation we would like to add that the final result agrees with what one expects when taking the decay of the unstable particle into account explicitly . let us , for simplicity , assume a two particle decay , as illustrated in figure [ diagram ] . in this case the phase space is the 3 body phase space and we get @xmath75 where @xmath76 denotes the dressed @xmath0 propagator , @xmath77 is the decay amplitude , @xmath78 is the reduced mass of the decay particles and @xmath79 their relative momentum .
therefore eq .
( [ 3part ] ) agrees with ( [ invo ] ) when we identify @xmath80where we used unitarity for the second identity .
( [ specdef ] ) agrees with the standard definition of a spectral function .
to summarize , we demonstrated that the interpretation of the experimental results for the reaction @xmath1 given in refs .
@xcite is incorrect .
a proper treatment of the independent variables leads to an expression for the momentum dependence of the integrated cross section that is consistent with a constant matrix element near the production threshold .
the procedure of refs .
@xcite was also used in ref .
@xcite and thus our criticism applies to the conclusion of this paper as well .
however , we want to emphasize that we regard the method of integrating over the beam energy while keeping the final momentum fixed as useful way to examine the production of narrow resonances close to their production threshold .
this technique allows for a more direct access to the production amplitude and , simultaneously , to an increase in the counting rate .
we demonstrated that as the momentum @xmath3 grows , the formula for the cross section reduces to the standard one for the production of a stable particle , as expected .
the relevant parameter is @xmath81 .
the knowledge of the transition amplitude @xmath7 is an important input for several approaches investigating @xmath0 production in hadron
hadron collisions @xcite
. a better understanding of its energy dependence therefore will help us to get deeper insight in the strong interaction of vector mesons and nucleons and nuclei in the intermediate energy regime .
the authors are grateful to c.wilkin for useful discussion and information about the papers @xcite , and to k. brinkmann , o. krehl , n.n.nikolaev , f.plouin and a. sibirtsev for useful discussions .
thanks v.baru for help in numerical estimations .
we also wish to thank j. durso for editorial assistance .
acknowledges the hospitality of the institute fr theoretische kernphysik ikp , forschungzentrum jlich .
a.k is also thankful to the russian foundation for basic research ( grant 98 - 02 - 17618 ) for partial support .
c.h . is grateful for financial support by the cosy ffe
project no .
41324880 , department of energy grant de - fg03 - 97er41014 and by the humboldt foundation .
99 d.m.binnie et al.phys.rev.d8 , 9(1973)2789 .
j.keyne et al.phys.rev.d14,1 ( 1976)28 . h.karami et al .
nucl.phys.b154 ( 1979)503 .
c. wilkin , talk presented at the baryon98 conference and nucl - th/9810047 .
r.wurzinger et al .
phys.rev.c51 ( 1995)r443 .
kondratyuk , y.s .
golubeva , m. bscher , nucl - th/9808050 .
kondratyuk , yu.n .
uzikov , acta phys.polon.b27(1996)2977 .
a. sibirtsev , w. cassing , u. mosel , z.phys.a358(1997)357 .
|
we discuss the near threshold behavior of the @xmath0 production amplitude in the reaction @xmath1 .
in contrast to the results of earlier analyses we find that the averaged squared matrix element of the production amplitude must be a decreasing function of energy in order to describe the existing experimental data . fzj
ikp(th)199831 + nt@uw9940 + doe / er/4056163int99 + + + +
|
vibrational degrees of freedom in atomic nuclei can be described in terms of phonon excitations that arise from nuclear shape oscillations .
vibrations of nuclei with ellipsoidal symmetry can be of two types @xcite : @xmath4 vibrations which preserve axial symmetry and give rise to a band with @xmath5 , and @xmath0 vibrations which break axial symmetry and yield a @xmath6 band , where @xmath7 is the projection of the angular momentum on the axis of symmetry . at the experimental level ,
@xmath0 bands have been identified in many well - deformed nuclei ; in contrast , the identification of @xmath4 bands is still full of questions and difficulties .
this is mainly because , when the energy surface has a well - deformed minimum in @xmath4 but is rather flat in @xmath0 , the @xmath4-band increases in excitation energy and approaches the energy region where other degrees of freedom are important . in that case band mixing may occur and can give rise to non - pure structures with decay patterns difficult to identify as those of a @xmath4 band @xcite .
since single-@xmath0 excitations are very well established , it is natural to look for double-@xmath0 vibrations and to develop models that can deal with such multi - phonon excitations .
double-@xmath0 excitations correspond to @xmath8 and @xmath9 bands which are the anti - parallel and parallel combinations of single-@xmath0 phonons , respectively .
the experimental identification of two-@xmath0 states in deformed nuclei is difficult because their expected excitation energy is around the pairing gap and hence they can mix strongly with two - quasiparticle excitations .
however , recent experimental improvements in nuclear spectroscopy following coulomb excitation @xcite , inelastic neutron scattering @xcite , and thermal - neutron capture @xcite have made possible the study of highly excited low - spin states .
many states have been proposed as possible candidates of double-@xmath0 vibrations .
there is , however , some controversy about their interpretation . the author of reference @xcite claims that some of the presumed double-@xmath0 states can be interpreted as single hexadecapole - phonon excitations . in fact , to identify the band - head of a double-@xmath0 band
it is not sufficient to analyze just @xmath10 values ; data from single - nucleon transfer reactions , @xmath4-decay studies , and inelastic scattering experiments must be considered as well .
one of the key properties to disregard a band as a double-@xmath0 band is the fact that its members , in first order , can not be populated in single - nucleon transfer reactions .
many examples of @xmath9 states that are identified as double-@xmath0 excitations but are strongly populated in single - nucleon transfer reactions , can be found in the literature : @xmath11gd , @xmath12dy , @xmath13yb , @xmath14hf , and @xmath15os .
since the double - phonon character of the states in question is in doubt , they are not considered here .
however , some candidates seem to have a genuine double - phonon nature . such is the case with @xmath1dy and @xmath16er .
in particular , in reference @xcite a @xmath9 state in @xmath1dy at @xmath17 mev is found to exhibit all properties of a double-@xmath0 band . in references
@xcite the observation is reported of @xmath8 and @xmath9 double-@xmath0 states in @xmath2er , at energies of @xmath18 mev and @xmath19 mev , respectively .
finally , in @xmath3er a @xmath9 double-@xmath0 excitation is identified at an energy of @xmath20 mev @xcite .
one of the most striking features of the observed double-@xmath0 bands is their high anharmonicity , _
i.e. _ the ratio of double-@xmath0 over single-@xmath0 energy is different from @xmath21 and ranges from @xmath22 to @xmath23 .
this information is very important since it provides a stringent test of nuclear models .
the nuclei @xmath1dy and @xmath24er have been interpreted in the context of many different models such as the quasi - phonon model @xcite , the geometrical model @xcite , the multi - phonon model @xcite , the self - consistent collective - coordinate method @xcite , and the @xmath25-ibm @xcite , and it is now of interest to revisit these models in connection with anharmonic vibrational behavior . in this paper the simplest version of the interacting boson model ( ibm ) @xcite
is extended by adding to the usual hamiltonian higher - order interactions between the bosons with the purpose of creating a framework that accommodates the high anharmonicities observed in @xmath1dy and @xmath24er .
the structure of the paper is as follows .
first , the ibm is reviewed with special reference to its harmonic character . in section [ 3b ]
the inclusion of three - body terms in the hamiltonian is discussed .
the introduction of four - body terms is presented in section [ su3 - 4b ] and some analytic results are pointed out . in section [ 4b ] a detailed study of possible four - body terms is carried out and realistic calculations for @xmath1dy and @xmath24er are presented .
finally , in section [ conclu ] the conclusions of this work are made .
the ibm describes low - lying collective excitations in even even nuclei in terms of monopole ( @xmath26 ) and quadrupole ( @xmath27 ) bosons @xcite .
the boson number that corresponds to a given nucleus equals half the number of valence nucleons ( @xmath28 ) .
the rotationally invariant and number - conserving boson hamiltonian usually includes up to two - body interactions between the bosons although higher - order terms can be added in principle .
the most general two - body ibm hamiltonian can be written in a multipole expansion as @xmath29 where @xmath30 and @xmath31 are the @xmath26- and @xmath27-boson number operators , respectively , and @xmath32 the symbol @xmath33 represents the scalar product ; in this paper the scalar product of two operators with angular momentum @xmath34 is defined as @xmath35 where @xmath36 corresponds to the @xmath37 component of the operator @xmath38 . in the previous equations the
operator @xmath39 ( where @xmath0 refers to @xmath26 or @xmath27 ) is introduced so that the annihilation operator verifies the appropriate properties under spatial rotations .
it is not _ a priori _ clear to what extend @xmath4 and @xmath0 vibrations are anharmonic in the ibm even if one just considers the hamiltonian ( [ ham1 ] ) with up to two - body interactions .
a partial analysis of this problem was given in reference @xcite .
there , the authors find that the ibm in its simplest version is a harmonic model in the limit of infinite boson number @xmath40 and even for finite @xmath40 the model can not accommodate large anharmonicity if one considers up to two - body interactions ; only the interplay between one+two - body terms and higher - order interactions can induce , in principle , a sizeable anharmonicity in the double - phonon excitations .
the reason why one - body terms and two - body interactions can not create a large anharmonicity can be understood as follows .
if one considers a hamiltonian with one parameter that controls the ratio of the strength of the one - body energies and the two - body interactions , @xmath41 where @xmath42 ranges from @xmath43 to @xmath44 , one finds two ` phases ' separated by a critical value , @xmath45 : a first phase where the one - body term plays the main role ( @xmath46 ) and a second phase where the two - body interaction is the driving force ( @xmath47 ) .
the crucial point is that the separation between the two phases is very sharp @xcite and essentially no interplay between one- and two - body terms can be found . since , to a good approximation ,
the force is either one body or two body but not both , harmonic behavior can not be avoided .
the inclusion of high - order interactions in a system with a high boson number also leads to a harmonic description . only for finite boson number
the interplay between one+two - body terms and higher - order interactions can induce an anharmonicity in the double - phonon excitations that is comparable to the observed one .
these ideas will be used as a guideline in the following sections . to carry out a quantitative study of anharmonicities ,
it is convenient to define a ratio between single- and double - phonon excitation energies . because the experimental situation for @xmath4 excitations is not clear we concentrate on @xmath0 vibrations and define energy ratios for @xmath0 phonons only ( although similar definitions can be given for @xmath4 phonons ) : @xmath48 where @xmath49 and @xmath50 are the band heads of the @xmath8 and @xmath9 double-@xmath0 bands , respectively , and @xmath51 stands for excitation energy .
this particular definition removes any rotational influence .
let us consider in the following a hamiltonian that includes a quadrupole term , a rotational @xmath52 term , and three - body interactions between the @xmath27 bosons , @xmath53 where @xmath54 , @xmath55 .
five independent three - body @xmath27-boson interactions exist which have @xmath56 , 2 , 3 , 4 , and 6 .
interactions with the same @xmath57 but different @xmath58 are not independent but differ by a normalization factor only @xcite .
the combinations @xmath59 , ( 0,2 ) , ( 2,3 ) , ( 2,4 ) , and ( 4,6 ) are chosen here .
the hamiltonian ( [ ham ] ) is certainly not the most general one+two+three - body hamiltonian that can be considered .
notably , a vibrational term @xmath60 which dominates in spherical nuclei is omitted since it is thought of lesser importance in the deformed nuclei considered here . and , of all possible three - body interactions ( seventeen terms ) ,
only those between the @xmath27 bosons are retained here since they are the more efficient terms to produce anharmonicity @xcite . for the discussion of the anharmonicities of @xmath0 vibrations we study the behavior of the energy ratios ( [ ratios ] ) as a function of the ratio @xmath61 .
the identification of the states @xmath49 and @xmath50 is based on the @xmath10 values for decay into the single-@xmath0 states . in figure
[ fig-3b - ratios ] . ] the influence of the various three - body interactions is shown for a typical value of @xmath62 ( @xmath63 ) and for @xmath64 bosons .
it is seen that a @xmath0-vibrational anharmonic behavior is obtained which can be different for the @xmath8 and @xmath9 bands ( _ e.g. _ positive for the former while negative for the latter .
) care has been taken to plot results only up to values of @xmath65 that do not drastically alter the character of rotational spectrum ; beyond these values , the three - body interaction , being of highest order in the hamiltonian ( [ ham ] ) , becomes dominant .
also shown in figure [ fig-3b - ratios ] are the ratios @xmath66 as observed in @xmath2er @xcite , @xmath67 and @xmath68 .
figure [ fig - spec-3b ] shows the experimental spectrum of @xmath2er @xcite and compares it to the eigenspectrum of hamiltonian ( [ ham ] ) with an @xmath69 three - body interaction .
the parameters are @xmath70 kev , @xmath71 , @xmath72 kev , and @xmath73 kev is @xmath74 of that given in reference @xcite which has an error in its definition .
the results shown in that paper are correct after a simple rescaling of the parameter .
] , with boson number @xmath64 . with these values
the calculated excitation energies of the double-@xmath0 band heads are 1926 kev and 1972 kev for the @xmath8 and @xmath9 levels , respectively , leading to the ratios @xmath75 and @xmath76 , in excellent agreement with observation .
note , however , that although all @xmath0-band heads are well reproduced by the calculation , problems arise for the moments of inertia , in particular of the @xmath0 band . in next sections
we will come back on the moments of inertia to see that three - body hamiltonians provide a very poor description of them .
in the previous section and in reference @xcite a very good description of double - phonon excitation energy has been obtained , but at expense of spoiling the moments of inertia of ground and @xmath0 bands .
these drawbacks seem to be a general feature of three - body hamiltonians . in this and the following section it is shown that the drawbacks of three - body interactions can be overcome by going to the next order .
since @xmath0 anharmonicity has been observed exclusively in well - deformed nuclei , it is appropriate to consider the problem in the su(3 ) limit of the ibm which is suited to deal with nuclei in this mass region @xcite . therefore , in a first approach , the @xmath77 limit is used and later , in the next section , these results are used as a guidance in more realistic calculations .
let us consider the following hamiltonian : @xmath78 + b_1~\hat c_3[su(3)]+ b_2~\hat n \hat c_2[su(3 ) ] \nonumber\\ & + & c_1~\hat c_2[su(3)]^2 + c_2~\hat n \hat c_3[su(3)]+ c_3~\hat n^2 \hat c_2[su(3)],\end{aligned}\ ] ] where @xmath79 $ ] stands for the casimir operator of order @xmath80 of @xmath77 . note the inclusion of cubic terms for completeness in the @xmath77 analysis .
since only the spectrum of a single nucleus is of interest here , the number of bosons can be fixed in every case and the hamiltonian ( [ cas - ham ] ) can be simplified by combining terms into a single one , leaving a hamiltonian with three coefficients @xmath81 , @xmath82 , and @xmath83 , @xmath84 + b~\hat c_3[su(3)]+ c~\hat c_2[su(3)]^2,\end{aligned}\ ] ] with eigenvalues @xmath85 no quartic casimir operator exists for @xmath77 because the number of independent casimir operators equals the number of labels that characterize an irreducible representation .
the hamiltonian ( [ cas - ham2 ] ) has no rotational term @xmath52 since of primary interest , at this point , is the description of band - head energies of single- and double-@xmath0 excitations . the definition of the energy ratios ( [ ratios ] )
must now be adapted to incorporate the symmetry labeling of the states . in the @xmath77
limit the @xmath0 band belongs to the @xmath77 representation @xmath86 where @xmath40 is the number of bosons .
the double-@xmath0 band with @xmath87 is contained in the @xmath88 representation ; the double-@xmath0 band with @xmath5 is predominantly contained in the @xmath89 representation , although an important component is in @xmath88 @xcite . in this section the energy ratios ( [ ratios ] ) are thus defined as follows : @xmath90 because no rotational term is included , the ratios ( [ ratios - su3 ] ) can be compared directly with equations ( [ ratios ] ) . in the following the effect of the different terms in equation ( [ cas - ham2 ] ) on the degree of anharmonicity of the two - phonon excitation energy
is analyzed . * @xmath91 , @xmath92 , @xmath93 .
+ this corresponds to the simplest version of ibm ; the energy ratios become @xmath94 an almost pure harmonic @xmath0-vibrational spectrum is found since the energy ratios ( [ ratios - a ] ) are only slightly lower than @xmath21 .
this will be referred to as negative anharmonicity as opposed to the positive anharmonicity for energy ratios above @xmath21 .
* @xmath91 , @xmath95 , @xmath93 .
+ in this case the hamiltonian ( [ cas - ham2 ] ) is a combination of two- and three - body terms . for given values of @xmath81 and @xmath82 and for a high enough boson number ,
only the three - body part of the hamiltonian is dominant and a harmonic spectrum is recovered . for obtaining an anharmonic spectrum the values of the hamiltonian parameters
are very constrained once the number of bosons has been fixed , as the following analysis shows .
+ without lost of generality @xmath81 can be fixed to @xmath96 .
the energy ratios are then @xmath97 to keep the energy of the @xmath0 excitation positive , the value of @xmath82 has an upper limit @xmath98 the value @xmath99 leads to a divergence in @xmath100 and around this value anharmonic behavior is found . from equation ( [ ratios - b2 ] ) one observes that the behavior of the @xmath88 representation is completely harmonic and does not depend on @xmath82 . on the other hand , from equation ( [ ratios - b1 ] )
one sees that a wide range of anharmonic ratios is found for the @xmath89 representation . as an illustration , in figure [ fig - ratio-3b ] equation ( [ ratios - b1 ] )
is represented as a function of @xmath82 , for three values of @xmath40 ( @xmath101 , @xmath102 , and @xmath103 ) .
only positive values of @xmath82 are plotted because for the negative ones @xmath100 decreases smoothly to the asymptotic values @xmath104 , @xmath105 , and @xmath106 for @xmath107 , @xmath102 , and @xmath103 , respectively .
+ the conclusion is that a hamiltonian with @xmath93 does not agree with the experimental situation observed in the mass region of well - deformed nuclei , where the @xmath108 state is highly anharmonic .
* @xmath91 , @xmath92 , @xmath109 .
+ in this case the hamiltonian ( [ cas - ham2 ] ) is a combination of two- and four - body terms . as in the previous case , anharmonicity requires very constrained hamiltonian parameters once the number of bosons is fixed .
+ again , without loss of generality , we fix @xmath96 .
the energy ratios then read as follows : @xmath110 to keep the energy of the @xmath0 excitation positive , the value of @xmath83 has the upper limit @xmath111 the value @xmath112 produces a divergence in the two energy ratios and in its neighborhood highly anharmonic behavior is found . in figures [ fig - ratio-60 ] and [ fig - ratio-84 ] are plotted the ratios @xmath100 and @xmath113 , respectively . again , for negatives values of @xmath83 , @xmath100 goes asymptotically to the values @xmath114 , @xmath115 , and @xmath116 for @xmath107 , @xmath102 , and @xmath103 , respectively , while @xmath113 goes to @xmath117 , @xmath118 , and @xmath119 .
these negative anharmonicities have no phenomenological interest .
+ in this case the experimental situation can be nicely described . both @xmath120 and @xmath108 states
can be accommodated in a anharmonic description .
the conclusion is thus that a hamiltonian with @xmath121 $ ] and @xmath121 ^ 2 $ ] terms seems to be a good starting point to treat the @xmath0 anharmonicity in deformed nuclei .
the description presented here only provides band heads and , to recover a rotational structure , a @xmath52 term must be included .
the rotational structure is the same in every band because in the @xmath77 limit no mixing exists between rotational and vibrational degrees of freedom . in next subsection
these results are illustrated with a schematic calculation for energies and transition probabilities .
let us consider the case of @xmath2er which is , as already mentioned , one of particular interest because both double-@xmath0 excitations ( with @xmath5 and @xmath87 ) have been identified @xcite . to carry out the schematic calculation
, we use the hamiltonian ( [ cas - ham2 ] ) with @xmath92 .
the experimental values for the single and double-@xmath0 energy ratios are @xmath122 and @xmath123 and can be compared directly with the expressions ( [ ratios - c1]-[ratios - c2 ] ) leading two values for @xmath124 , namely @xmath125 and @xmath126 .
both solutions are fairly close and any value in between them will correctly describe the anharmonicity of the @xmath5 and @xmath87 bands . the value of @xmath81 and the strength of the rotational term are fixed from the excitation energies of the @xmath127 and @xmath128 levels . after these simple considerations one arrives at the following hamiltonian : @xmath129 + 9.296~10^{-3}~\hat c_2[su(3)]^2,\ ] ] where the coefficients are given in kev .
the theoretical and experimental spectra are compared in figure [ fig-166er - su3 ] and a very good agreement is obtained .
however , due to the simplicity of the calculation , @xmath0 and @xmath4 bands are degenerate in energy which is not the case experimentally . to complete the description ,
@xmath130 transition probabilities must be computed also .
the calculation of @xmath10 values in the @xmath77 limit , as in other situations where degeneracies occur , must be treated with care and an appropriate basis must be chosen for states with the same energy . a natural way to do
this work is to slightly lift the degeneracy of the @xmath77 hamiltonian .
the levels can be split using in the hamiltonian a value @xmath131 which is very close to its @xmath77 value . with this @xmath77 breaking the degeneracy
is lifted in a natural way because the @xmath4 band is pushed up in energy , as is observed .
one may expect that with this small change in @xmath62 the @xmath77 spectrum will keep its properties .
the @xmath130 transition operator is : @xmath132 the value of @xmath62 that best reproduces the data is @xmath133 . note that @xmath62 in the @xmath134 operator and in the hamiltonian is different .
the effective charge is fixed to reproduce @xmath135 : @xmath136 w.u .. in table [ table-166er - su3 ] theoretical and experimental transition rates involving the ground and the @xmath0 bands are compared .
this simple analysis suggests that a four - body operator of the type in ( [ ham - su3-sch ] ) provides a good description of both single- and double-@xmath0 bands . in the next section this schematic analysis
is extended to non-@xmath77 situations .
a general hamiltonian with all possible three- and four - body terms can , in principle , be constructed but the number of parameters is so high that a study , even a schematic one , of the effect of the different terms on energy spectra and electromagnetic transitions is impossible .
schematic ibm hamiltonians have been used for many years and , in particular , the quadrupole - quadrupole interaction has been very successful in describing a wide variety of nuclear spectra @xcite . on the other hand , in the previous section it was shown that an expansion in terms of casimir operators , which are mainly related to quadrupole operators , leads to a satisfactory description of ground , single- and double-@xmath0 bands .
it is worth noting that such an expansion in terms of casimir operators has been successfully used in molecular physics where spectroscopic data provide many anharmonic states @xcite .
this leads us to propose a hamiltonian as a quadrupole expansion that includes up to four - body terms .
an alternative hamiltonian can be based on an expansion in terms of pseudo - casimir operators , which we define here as operators that become a true casimir operator only for a particular choice of one structure parameter .
for example , the @xmath137 $ ] operator is related to the quadrupole operator through @xmath138 where @xmath139 .
if the value @xmath140 is changed to @xmath62 , a new operator @xmath137_\chi$ ] is obtained which we refer to as a pseudo - casimir operator .
it is a casimir operator only for @xmath141 . guided by the results of the previous section ,
two possible hamiltonians that include up to four - body interactions , can be proposed , one based on a quadrupole expansion and the other on a pseudo - casimir expansion : @xmath142_\chi+ b~\hat c_3[su(3)]_\chi+ c~\hat c_2[su(3)]^2_\chi,\end{aligned}\ ] ] where @xmath143 and @xmath144_\chi&= & 2\hat q(\chi)\cdot\hat q(\chi)+ { 3\over4}\hat l^2,\\ \hat c_3[su(3)]_\chi&= & -4\sqrt{35}(\hat q(\chi)\times\hat q(\chi)\times\hat q(\chi))^{(0 ) } -{9\over2}\sqrt{15}(\hat l\times\hat l\times\hat q(\chi))^{(0)}.\end{aligned}\ ] ] for simplicity and taking into account the analysis done in the previous section , in the following @xmath92 is taken in equations ( [ ham - q ] ) and ( [ ham - pc ] )
. there are three nuclei that have double-@xmath0 bands identified without ambiguity : @xmath1dy , @xmath2er , and @xmath3er @xcite . in this section the band heads of these nuclei are studied using the hamiltonians ( [ ham - q]-[ham - pc ] ) to get an improved description of the anharmonicity phenomenon .
the number of bosons for @xmath1dy and @xmath3er is @xmath145 while for @xmath2er it is @xmath64 . in the different calculations shown in this section the parameters of the hamiltonian
have been chosen as to reproduce as well as possible not only the heads of single- and double-@xmath0 bands but also the structure of the bands . because these calculations are schematic and do not try to be the best answer ,
the parameters of the hamiltonian , for simplicity , will not be given fully and only in the case of final spectra will be shown .
for comparison , the calculation with three - body terms ( see section [ 3b ] and reference @xcite ) is also included . in figures
[ fig-164dy - sch ] , [ fig-166er - sch ] , and [ fig-168er - sch ] the heads of single- and double - phonon bands are shown . in each figure six
panels are included : a ) experimental data , b ) calculation with hamiltonian ( [ ham - q ] ) and @xmath146 , c ) calculation with hamiltonian ( [ ham - q ] ) and @xmath71 , d ) calculation with hamiltonian ( [ ham - pc ] ) and @xmath146 , e ) calculation with hamiltonian ( [ ham - pc ] ) and @xmath71 , and f ) calculation with a three - body term @xmath147 . in each panel , from left to right are represented the ground state , the @xmath0 band head , the @xmath4 band head , the double-@xmath0 @xmath5 band head , and the double-@xmath0 @xmath87 band head . due to the controversy on the nature of the @xmath4 band ,
several candidates for the latter have been included .
for the same reason in the figures will be used the label `` @xmath4 '' . in @xmath1dy and @xmath3er no information on the double-@xmath0 @xmath5 band exists .
the value @xmath71 is chosen as an alternative to the @xmath77 value because it describes very well the @xmath130 transition probabilities in this mass region .
also , the same value of @xmath62 is taken in the hamiltonian and in the transition operator , in line with the consistent - q formalism ( cqf ) @xcite . in section [ realistic ] this
_ ansatz _ is used in the complete analysis of spectra and @xmath10 transitions for @xmath1dy , @xmath2er , and @xmath3er .
the most striking feature of figures [ fig-164dy - sch][fig-168er - sch ] is that in all calculations the position of the double-@xmath0 band heads and the degree of anharmonicity is well reproduced .
thus , the energies of the different band heads only are not sufficient to completely determine the hamiltonian
. nevertheless , not all possible terms are able to create sufficient anharmonicity in the double-@xmath0 bands .
for example , in the case of three - body terms , a phenomenological study of the most relevant type of term shows that only @xmath148 is able to produce the required anharmonicity ( see section [ 3b ] and reference @xcite ) . on the other hand , only a few four - body interactions have been explored here and it can not be excluded that other four - body terms can produce the appropriate degree of anharmonicity . in order to decide which
hamiltonian is more appropriate , a description should be attempted not only of band heads but also of the structure of the bands and of @xmath130 transition probabilities .
the study of the moments of inertia of the lowest bands is a very sensitive way to test the different calculations shown in figures [ fig-164dy - sch ] , [ fig-166er - sch ] , and [ fig-168er - sch ] .
particular attention will be paid to the dynamic moment of inertia which can be obtained from the relation between angular momentum and @xmath0-ray energy @xcite and can be approximated by , @xmath149 where @xmath150 is given dimensionless . in a plot of @xmath0-ray energy versus @xmath150 , @xmath151 will be the slope .
equation ( [ m - inertia ] ) can be used to study the structure of the rotational bands in comparison with experimental results . in figure [ fig - iner-3b ] the moments of inertia of the ground and @xmath0-bands in the nuclei @xmath1dy , @xmath2er , and @xmath3er are compared to those obtained with the hamiltonian of section [ 3b ] .
the parameters for @xmath1dy are @xmath152 kev , @xmath71 , @xmath153 kev , and @xmath154 kev , for @xmath2er are indicated in section [ 3b ] and for @xmath3er are @xmath155 kev , @xmath71 , @xmath72 kev , and @xmath156 kev .
the predicted moments of inertia disagree completely with the almost pure rotational structure observed experimentally .
so , although this simple description based on one single three - body term describes well the anharmonic position of the double - phonon band heads , it fails in the moments of inertia of ground and @xmath0-bands .
a more realistic description needs others three - body terms .
a recent analysis @xcite shows , however , that hamiltonians with _ two _ different three - body terms can not get the correct moment of inertia .
the results with the hamiltonian ( [ ham - q ] ) are shown in figure [ fig - iner - q ] .
two different calculations are shown which correspond to panels @xmath82 and @xmath83 , respectively , of figures [ fig-164dy - sch][fig-168er - sch ] . in this case
a better agreement is obtained but still some discrepancies remain , especially when a realistic value for @xmath62 is taken ( @xmath71 ) .
finally , in figure [ fig - iner - cas ] the results with the hamiltonian ( [ ham - pc ] ) are compared with the data .
again , two different calculations are shown which correspond to panels @xmath27 and @xmath157 , respectively , of figures [ fig-164dy - sch][fig-168er - sch ] . here , good agreement is found both for @xmath146 and @xmath71 .
the different results with the hamiltonians ( [ ham - q ] ) and ( [ ham - pc ] ) , can be understood qualitatively by analyzing the structure of @xmath158 .
with just two - body terms the hamiltonians ( [ ham - q ] ) and ( [ ham - pc ] ) are equivalent .
this is different when up to four - body terms are included because @xmath158 does not only contribute to @xmath137 ^ 2_\chi$ ] and @xmath52 but also to @xmath159 .
this can be clarified with the equation @xmath160 ^ 2_\chi-{3\over 8}\hat c_2[su(3)]^2_\chi
\hat l^2+{9\over 64}(\hat l^2)^2.\end{aligned}\ ] ] as a consequence , @xmath158 substantially modifies the rotational structure of a band even in the case of pure @xmath77 .
this is how can be qualitatively understood that _ a description in terms of pseudo - casimir operators is the most appropriate for dealing with anharmonic vibrations .
_ in the next section a complete analysis is given of the nuclei under study in the framework of cqf using the hamiltonian ( [ ham - pc ] ) .
the complete calculated spectra of @xmath1dy , @xmath2er , and @xmath3er and the most relevant @xmath130 transition probabilities are presented in this section .
they are compared with existing data . in each calculation
the same value of @xmath62 has been used both in the hamiltonian ( [ ham - pc ] ) and in the electromagnetic operator ( [ t - e2 ] ) . finally , the effective charge in the transition operator ( [ t - e2 ] ) , @xmath161 , was fixed for each nucleus to reproduce the @xmath135 value .
the experimental and calculated spectra for @xmath1dy , @xmath2er , and @xmath3er are shown in figures [ fig-164dy - fin ] , [ fig-166er - fin ] , and [ fig-168er - fin ] , respectively .
the parameters used in the calculations are listed in table [ table - ham ] .
the parameters that yield the best fit to the energy spectra are very similar in the three cases which is consistent with the analysis carried out in the preceding sections .
the overall description of the energies is satisfactory .
the calculated low - lying bands are in good agreement with their experimental counterparts ( see also figure [ fig - iner - cas ] ) while the double-@xmath0 band - head energies are close to the experimental values .
the calculated ratios ( [ ratios ] ) are : @xmath162 and @xmath163 for @xmath1dy , @xmath164 and @xmath165 for @xmath2er , and @xmath166 and @xmath167 for @xmath3er , to be compared with the experimental ones : @xmath168 for @xmath1dy , @xmath75 and @xmath76 for @xmath2er , and @xmath68 for @xmath3er . only for the @xmath120 vibration
the experimental and theoretical results are slightly different in the sense that this framework overestimates the anharmonic behavior for the @xmath120 band .
this can be corrected by increasing the value of @xmath169 in the hamiltonian which , however , will introduce one more parameter because in the electromagnetic operator a different value of @xmath62 must be used . for the calculation of @xmath130 transition
probabilities the same @xmath62 values as in the hamiltonian are adopted .
the effective charges are @xmath170 w.u .
, @xmath171 w.u . , and @xmath172 w.u . , for @xmath1dy , @xmath2er , and @xmath3er respectively . in tables [ table-164dy - fin ] , [ table-166er - fin ] , and [ table-168er - fin ] the observed @xmath10 values and ratios concerning @xmath0-vibrational states
are compared with the theoretical results .
in general , a good overall agreement is obtained in the three cases under study .
in this paper the problem of anharmonicity in the @xmath4 and @xmath0 vibrations of deformed nuclei was addressed in the context of the interacting boson model . the occurrence or not of anharmonicity
was shown to be related to the order of the interactions between the bosons and the conclusions of the analysis can be summarized as follows .
if the hamiltonian includes up to two - body interactions , no sizeable anharmonicity can be obtained and the observed behavior can not be obtained .
the origin of this behavior is related to the existence of a first - order phase transition between rotational and vibrational nuclei @xcite which excludes any interplay between one- and two - body terms , necessary to obtain anharmonic spectra . for
up to three - body interactions that preserve @xmath77 symmetry it can be _ shown _ that the observed anharmonicity can not be fully reproduced .
furthermore , extensive numerical calculations indicate that even a general ibm hamiltonian that includes up to three - body interactions has difficulty in reproducing all observed aspects of ground , single- and double-@xmath0 bands .
however , due to the large parameter space of three - body interactions which is difficult to search exhaustively , this can not be considered as a firm conclusion and alternative approaches ( _ e.g. _ mean - field ) should be tried to tackle the same problem .
finally , it was shown that a simple parametrization of the ibm hamiltonian that includes up to four - body interactions can account for all observed properties of the three deformed nuclei with firmly established double-@xmath0 vibrations .
in particular , the introduction of @xmath77 pseudo - casimir operators allows to describe the double - phonon states while keeping correct the properties of low - lying bands .
it should be emphasized that , in spite of its fourth - order character , the hamiltonian considered here is only slightly more complex than the usual ibm hamiltonian ( one more parameter ) and is a straightforward extension of the consistent - q formalism that was previously successfully applied to many nuclei . also clear from our study
is the need for more experimental information about the double - phonon vibrations in deformed nuclei beyond the three cases known at present : our analysis indeed has shown that this information represents a challenging test of any theoretical description of deformed nuclei .
we are grateful to c.e .
alonso , k. heyde and , f. iachello for valuable comments .
two of the authors ( pvi and jegr ) wish to thank the institute for nuclear theory , university of washington , where this work was initiated .
one of the authors ( jegr ) thanks the f.w.o . for financial support .
this work was supported in part by the spanish dgicyt under project number pb98 - 1111 .
.observed and calculated @xmath173(e2 ) values and ratios for @xmath2er in a schematic calculation using an @xmath77 hamiltonian .
the e2 operator ( [ t - e2 ] ) is used with @xmath174 w.u.and @xmath133 . [ cols="<,^,^ " , ]
|
double-@xmath0 vibrations in deformed nuclei are analyzed in the context of the interacting boson model .
a simple extension of the original version of the model towards higher - order interactions is required to explain the observed anharmonicities of nuclear vibrations .
the influence of three- and four - body interactions on the moments of inertia of ground- and @xmath0-bands , and on the relative position of single-@xmath0 and double-@xmath0 bands is studied in detail . as an example of a realistic calculation , spectra and transitions of the highly @xmath0-anharmonic nuclei @xmath1dy , @xmath2er , and @xmath3er
are interpreted in this approach . *
pacs numbers : 21.60 -n , 21.60 fw , 21.60 ev *
|
the appearance of cracks is an effective mechanism for a mechanical system under load to release elastic energy and to relax towards equilibrium .
it is therefore not surprising that aging processes in a broad class of materials can lead to the emergence of microcracks that weaken a specimen .
they do not necessarily lead to complete failure , but their presence alters the elastic properties of the system . for many practical applications
it is therefore highly desirable to develop simple , but still precise predictions for the resulting elastic properties of a medium that contains defects , cracks or other inhomogeneities .
cracked material is just one case in the widely dealt - with topic of physical properties of heterogeneous media @xcite .
the different physical properties to be described encompass conductive , transport and also elastic quantities @xcite . often , one starts from a coarse - grained picture and aims to find an effective description for the heterogeneous mixture .
much effort has been put into the calculation of effective elastic properties of composed media where the constituents have different elastic coefficents @xcite .
it turns out that the elastic properties of the system depend strongly on the positional and orientational distribution of the inclusions .
even different loading paths can lead to a different elastic response of the material under investigation @xcite .
therefore , special attention has to be paid to the underlying assumptions of the cavity distribution .
it has been shown that the hashin - shtrikman - typ bounds obtained for the bulk modulus of two - phase materials set important restrictions in terms of phase moduli and volume fractions @xcite , and improvement to these bounds have to involve considerations of statistical details of phase distributions .
the starting point for various _ effective medium theories _ is the effect of a single impurity or crack inside an otherwise homogeneous medium .
also in the following work we will implicitly employ the results of eshelby @xcite concerning the elastic fields around and inside a single ellipsoidal inhomogeneity in an infinitely extended , linearly elastic homogeneous solid .
in fact , this `` dilute '' limit for a single imperfection already gives an expression for the effective elastic constants for vanishingly low defect concentration .
higher concentrations of inhomogeneities can be treated in the framework of self - consistent or differential effective medium theories . generally speaking ,
these approaches make use of the idea , that a medium which already contains inclusions of another `` phase '' can be approximated as a homogeneous material with different material properties , to which then , step by step , additional inhomogeneities are added to reach a finite concentration of them .
the underlying assumption , that all effective properties depend only on the material constants of the pure phases and their volume fractions , is of course only an approximation , and the quality of the theoretical predictions can hardly be controlled .
a careful comparison to either experiments or numerical calculations of the effective properties is therefore highly recommended to judge the quality of the different homogenization methods .
the purpose of the present paper is manifold : first , it is intended as a numerical check for the analytical estimates for the effective elastic constants . obviously , all schemes mentioned above are approximative in nature , and it is one goal of this paper to shed light on the range of applicability of the theoretical models .
we use both finite difference and finite element methods for the numerical investigations , and the comparison to earlier results serves as benchmark for these approaches .
this will be done for the important case of spherical inclusions , since rigorous theoretical statements can be used to test the numerical methods .
this methodological confirmation is essential for the following tests of homogenization theories concerning the weakening of materials through cracks , which is the second main subject of this work .
as will be pointed out , the effect of percolation plays an important role here , and therefore deviations from differential homogenization theories , which predict an exponential weakening of the material , are noticeable already for moderate crack densities . in this context ,
the only situation where percolation does not play a role is that of parallel cracks .
the third important subject of this paper is the prediction of effective elastic constants in such a geometry , which surprisingly turn out to decay here according to a power law behavior , in contrast to exponential decays that could be expected from related situations @xcite .
it must be pointed out that this fully analytical prediction becomes accurate in the limit of _ high _ crack densities , and is therefore complementary to conventional theories .
the predictions are confirmed by the same numerical methods that have been justified before .
the first system under investigation is that of a two - dimensional isotropic solid in a plane strain situation that contains randomly placed circular holes which are allowed to overlap .
this system has already been investigated numerically by day et al .
we use this scenario to demonstrate the applicability of our numerical method to determine the effective elastic constants . for @xmath0
spherical holes of radius @xmath1 in the solid phase with area @xmath2 , the true void concentration @xmath3 is related to the void area ratio @xmath4 according to the relation @xmath5 which takes into account that the circles can overlap .
starting from the exact expression for a single inclusion @xcite , low - density expressions for the effective elastic constants can be derived in terms of the two - dimensional elastic moduli : @xmath6 with the elastic constants @xmath7 of the solid phase ; this result is attributed to numerous authors @xcite .
we use here the explicit annotation @xmath8 to emphasize that the elastic constants are those of a two - dimensional plane strain material , since some peculiarities in the behavior of the effective constants are purely attributed to the dimensionality of representation , as will be elucidated below .
the expressions for conversion between 2d and 3d are given in appendix [ conversion ] .
the truncation of the above series after the linear term already provides a low density prediction for the effective elastic constants . within this effective medium theory ,
the elastic modulus @xmath9 vanishes for @xmath10 ; however , the true percolation point is@xcite @xmath11 , and only then @xmath9 should become strictly zero .
this deviation already shows that the effective medium theory loses its predictive power for higher concentrations , underestimating the true stiffness of the material . using the above low - density expressions ( [ sphericalholes::eq1 ] ) and ( [ sphericalholes::eq2 ] )
, we can also derive another approximative model for the elastic constants in the framework of the differential medium theory ( see also appendix [ diffhom ] ) . according to equation ( [ diffhom::eq3 ] ) ,
we start from @xmath12 and obtain as solution @xmath13 apparently , this model predicts `` percolation '' for @xmath14 , i.e. if the solid phase disappears completely .
it is obvious that this model therefore must be invalid for high cavity concentrations as well , overestimating the elastic constants of the heterogeneous system .
we note that in both approximative theories the effective elastic modulus does not depend on the poisson ratio , a behavior that is known to hold exactly @xcite .
we use a straightforward finite difference method to solve the problem numerically . in a discretized rectangular system in the @xmath15 plane an `` order parameter '' @xmath16
is set to zero at the grid points which are covered by the circles of equal diameter , and @xmath17 in the remaining solid .
then the local elastic modulus is set to @xmath18 , and the elastic equilibrium conditions @xmath19 are solved by relaxation .
the system is strained and the average stress calculated , from which the effective elastic constants can be deduced as follows : for a system that is strained in @xmath20 direction and has periodic boundary conditions in @xmath21 direction , the average strain @xmath22 vanishes .
the average diagonal stress components in the system @xmath23 and @xmath24 are measured for this plane strain scenario .
then the effective elastic constants are determined through @xmath25 where the average strain @xmath26 is fixed through the boundary conditions .
we typically used systems sizes of @xmath27 grid points , with up to 1000 circles with a radius of 20 grid points .
further details on the elastic solver are presented in @xcite .
the dependence of the effective elastic modulus on the concentration as predicted by the theories , see eq . and
eq . , and as obtained by numerical simulations is shown in fig .
[ sphericalholes::fig2 ] .
the independence of @xmath28 on the poisson ratio is clearly visible also in the numerics , where we checked this explicitly for @xmath29 and @xmath30 ( corresponding to @xmath31 and @xmath32 respectively ) ; the latter case of an auxetic material is sometimes observed e.g. in foams @xcite , and is here only used as an extreme case to confirm the independence on the poisson ratio . in fact , we find that for the same random arrangement of circular holes the elastic constants match .
since we wanted to obtain a reasonable statistical averaging , we also performed repeated runs with different initializations .
as we increase the void concentration @xmath3 , one can clearly see that the scattering of the data points increases for higher concentrations , since larger clusters can form which can become comparable to the ( finite ) system size used in the simulations .
also , the relaxation time increases strongly with @xmath3 , thus results for higher concentrations are not shown here . in ref .
, day et al .
performed simulations based on an elastic spring network formulation for system setups analogous to ours .
the comparison of our numerical results to the simulation data of day et al .
are also included in fig .
[ sphericalholes::fig2 ] .
the results for the independent numerical approaches are in reasonable agreement .
in particular , all sets correctly reproduce the exactly known low density limit @xmath33 .
for higher concentrations , we obtain a higher effective elastic modulus than day et al . , and we believe that this is a consequence of the considerably larger systems that we used . .
the plot shows numerical data for different poisson ratios , as obtained with the present method , in comparison to numerical results obtained by day et al .
here we used different distributions of cracks , and evaluated the effective elastic modulus for the two different poisson ratios using exactly the same arrangement of cracks ; the independence of the poisson ratio is clearly visible . ] . for @xmath34 , both , the effective medium theory and the differential theory show fairly good agreement with the numerical results .
the predictions from the effective medium theory are shown only up to the percolation point @xmath10 . for negative poisson ratios ,
the differential theory coincides much better with the simulations . ]
similarly , the effective poisson ratio agrees well with the differential theory , as can be seen in fig .
[ fig_spherenueff ] , especially in the case of a negative poisson ratio . even at the highest densities that were simulated here
, we do not observe a noticeable deviation from this homogenization model .
finally , we briefly remark that the results depend on the dimension of representation .
conversion of the results for the differential homogenization theory gives according to eqs .
( [ conversion::eq2 ] ) @xmath35 / \big[(c\;(4\nu^{(3d)}-1)(c^2 - 3c + 3 ) - 3)^{2 } \nonumber \\ & & \times ( \nu^{(3d)}+1)\big ] , \\ { \nu_{\mathit{eff}}}^{(3d ) } & = & \frac{c\;(4\nu^{(3d ) } - 1 ) ( c^2 - 3c+3 ) - 3\nu^{(3d)}}{c\;(4\nu^{(3d)}-1)(c^2 - 3c+3 ) - 3}.\end{aligned}\ ] ] in particular , the effective three - dimensional elastic modulus does not have the property of being independent of the poisson ratio .
furthermore , for negative poisson ratios the effective elastic modulus can first increase if the material is `` weakened '' by spherical holes . a similar behavior was reported for cracks in ref .
, and here we see that this effect is rather generic and results mainly from the definition of the elastic constants . indeed , this counterintuitive behavior is obviously an artifact of the three - dimensional representation that is already contained in the low density expressions and not related to a specific homogenization scheme .
already for low concentrations we get @xmath36 which can start with a positive slope for negative poisson ratios .
in this section we investigate a random arrangement of cracks in a solid and compare the prediction for the effective elastic constants to numerical simulations . to that end , we use the same geometry as in ref . ,
where the normal vectors of the planar cracks are located in the @xmath15 plane , and they are infinitely extended in @xmath37 direction . therefore , the system becomes again effectively two - dimensional , and we restrict our investigations to a plane - strain scenario . in the @xmath15 plane ,
all cracks have the same length @xmath38 ; here we assume that the orientation is random and all angles @xmath39 appear with the same probability ; in the notation of ref .
this means for the orientational order parameter @xmath40 .
we introduce a crack density parameter @xmath41 where @xmath0 is the number of cracks per area @xmath2 in the @xmath15 plane .
the prediction for the effective ( three - dimensional ) elastic constants in the framework of the differential homogenization method is for plane strain according to @xcite @xmath42}{[\nu^{(3d ) } + ( 1-\nu^{(3d)})e^{\alpha}]^2 ( 1+\nu^{(3d)})}\\ { \nu_{\mathit{eff}}}^{(3d ) } & = \frac{\nu^{(3d)}}{\nu^{(3d ) } + ( 1-\nu^{(3d)})e^{\alpha}}\ > , \label{eq_nueffgiordano}\end{aligned}\ ] ] which predicts an exponential weakening of the material with the density parameter @xmath43 .
in particular , the effective medium is still isotropic , since there is no preferred orientation for the cracks , and therefore the elastic properties are still fully described by two elastic constants .
interestingly , the two - dimensional representation of the above result gives simply @xmath44 so both constants decay according to a simple exponential decay to zero .
notice in particular that the effective poisson two - dimensional ratio also tends to zero , in contrast to the spherical case discussed before , where it approaches @xmath45 .
we also mention that here the effective elastic modulus does not depend on the bare poisson ratio @xmath46 .
notice that the above conversion implies also that the effect of an increase of stiffness with the crack density for negative poisson ratios , that was discussed in ref .
, is indeed an artifact of the three - dimensional representation , similar to the spherical example discussed above .
we note that the limit @xmath47 is only reached for @xmath48 , which means that this theory does not predict percolation .
however , in reality percolation occurs for @xcite @xmath49 , and then a network of cracks penetrates the whole system , thus the true effective modulus vanishes .
therefore the differential homogenization method overestimates the true elastic modulus for higher crack densities . to check the quality of the above analytical predictions , we investigated the case of randomly oriented cracks also numerically for plane strain using finite difference relaxation methods , see figs .
[ fig_randomorientationeeffnu0p3 ] and [ fig_randomorientationeeffnum0p7 ] . for low crack densities ,
the numerical results agree with the prediction eq . , but for higher values they are indeed systematically lower due to prospective percolation .
, for several random distributions ( @xmath50 ) of cracks of equal length . ] , for several random distributions ( @xmath50 ) of cracks of equal length .
the initial stiffness increase predicted by the differential homogenization theory is clearly visible . for higher crack densities
, the theory overestimates the effective elastic modulus significantly . ] ) of cracks of equal length . ]
nevetheless , the analytical prediction from ref . can be considered as a very good approximation at least for crack densities @xmath51 .
we also see good agreement for the poisson ratio in this range of @xmath43 , see fig .
[ randomcrack::fig4 ] . for a negative bare poisson ratio , here @xmath32
, the numerical results seem to indicate that it approaches even a positive value instead of just decaying to zero .
from a more general point of view , all setups with random crack orientations have a finite percolation threshold , even if the probability distribution for the choice of the angle is not uniform .
the only exception is the case that all cracks are parallel ; then percolation does not occur .
thus only here a nontrivial asymptotic behavior exists for high crack densities .
it turns out that for this special case analytical predictions for the effective elastic constants can be made , which become accurate in the limit @xmath48 , and in this respect they differ fundamentally from conventional homogenization theories . , the average vertical distance between neighboring cracks @xmath52 . ] first , it should be noted that in this case the material that is pierced by cracks becomes anisotropic , and we therefore characterize its elastic properties by the tensor @xmath53 , with @xmath54 .
if we assume that in the @xmath15 plane all cracks are aligned in @xmath21 direction ( see fig .
[ parallel::fig1 ] ) , it is immediately clear that e.g. @xmath55 , since a pure stretching in @xmath21 direction does not open the cracks ; hence the strain tensor is homogeneous in the material and unaffected by the cracks . for low crack densities ,
the effective elastic constants were calculated in ref . , and in particular we get @xmath56 d^{-1 } e^{(3d)}\ ] ] with @xmath57 ( 1+\nu^{(3d)})\end{aligned}\ ] ] and @xmath58 for the parallel arrangement .
we start with looking at high crack densities , @xmath59 : two different lengthscales are important for a complete description of the problem at hand , the length @xmath38 of the cracks and the average vertical distance @xmath52 between them . for high crack densities @xmath43 , the vertical distance @xmath52 between neighboring cracks is smaller than the average crack length @xmath38 , and the relation between the two characteristic length scales can be given through @xmath43 only , so we obtain @xmath60 . if the cracked body is subjected to tensile loading perpendicular to the cracks , the solid regions between two cracks can be understood as a thin bent plate of a width proportional to @xmath38 and thickness @xmath52 .
the opening of the cracks is the displacement @xmath61 .
the stress of a thin bent plate scales as @xcite @xmath62 with this equation , it follows readily that the average stress and the opening @xmath61 have to scale like @xmath63 the total displacement is distributed among the opening of all cracks , which relax the material around them . since for this
loading all other average strain components are small @xcite , the average strain @xmath64 is simply given by @xmath65 plugging this into eq .
, we finally obtain for the case @xmath66 @xmath67 in other words , the relevant elastic constant @xmath68 decays by a power law , @xmath69 we note that this scaling behavior holds also for situations where the cracks can have unequal lengths , distributed around the mean value @xmath38 ; details of the distribution function can affect only the numerical prefactor of the above prediction in the limit @xmath48 .
in addition , we also performed simulations for regular arrays of cracks .
also , we checked numerically that the scaling behavior holds for random parallel arrangements of cracks ; the results can be seen in fig . [
fig_scalingparallelcracks ] . as a function of the crack density @xmath43 for a parallel arrangement of cracks in logarithmic representation
for a regular arrangement of cracks , the agreement of the numerical simulations with thin plate theory is excellent .
if the cracks are placed at random positions , they still exhibit the same power law scaling behavior . ]
this graph shows the results for the low density theory , the asymptotic behavior and numerical simulation data from both finite difference and finite element methods @xcite .
we used different arrangements of cracks to illustrate the scaling behavior : first , we took a regular arrangement of cracks , where we can rigorously calculate the effective elastic constants for @xmath48 ; this is shown in appendix [ regular ] . due to the spatial periodicity it is sufficient to consider a system with only a few cracks .
we clearly see that both finite difference and finite element calculations give the same result .
the finite element method is computationally more efficient than the simple relaxation solver ; however , the geometrical description is easier with finite differences , since e.g. intersections with boundaries ( or overlaps of cracks for the random orientation case , as discussed in the preceding section ) do not require a separate treatment . to get clear predictions for the scaling behavior as function of the crack density @xmath43
, we randomly place the cracks in the system and solve the elastic problem by finite element methods .
then we change the value of @xmath43 by rescaling the height of the system , which means that the arrangement of cracks is the same for all points on one curve .
the correct scaling behavior is demonstrated here for a relatively small system with only @xmath70 cracks . obviously , the specific results depend then on the configuration , and only for @xmath71 these discrepancies between different arrangements would disappear . however , the results show , that the scaling holds for each configuration ( shown here for two cases ) , and therefore it must be correct also for the true ensemble average in an infinitely large system . the results , in particular the finite difference data for small @xmath43 show the crossover between the low density prediction ( [ parallel::eq1 ] ) and the asymptotic behavior ( [ parallel::eq2 ] ) . for the latter ,
the numerical prefactor was chosen such that it matches the particular case of regular cracks ( @xmath72 ) , as explained in appendix [ regular ] .
we investigated numerically the effective elastic constants for isotropic plane strain media with spherical holes , randomly oriented and parallel cracks . in all cases
we find a good agreement with predictions from different homogenization theories , with a better performance of differential media theories .
the results show clear deviations from the approximative theories , which are strictly valid only for low inclusion densities , since they do not correctly account for effects which go beyond mean - field approximations . in particular
, all discussed models do not correctly take into account percolation , which should lead to a sharp drop of the effective elastic modulus .
the only case where percolation does not occur is that of parallel cracks . by scaling arguments we derived analytically the scaling behavior of effective elastic constants in the limit @xmath48 and obtain a power law decay with the crack density .
this new prediction was confirmed numerically using finite - difference and finite - element methods .
we note that this prediction is complementary to conventional homogenization theories , as it becomes accurate for _ increasing _ crack densities .
even though the effective elastic constants are already low in this regime , the obtained results are therefore of principal interest and raise the question whether explicit solutions for other situations with high inclusion density are also possible .
this work has been supported by the german - israeli foundation .
r. s. would like to acknowledge the financial support from the industrial sponsors of icams , thyssenkrupp steel ag , salzgitter mannesmann forschung gmbh , robert bosch gmbh , bayer materials science ag , bayer technology services gmbh , benteler ag and the state of north - rhine - westphalia .
as already mentioned above , the dimensionality can play a role for the effective elastic constants .
we can convert the elastic constants of a two - dimensional setup to an equivalent three - dimensional plane strain situation .
the defining equation is hooke s law , @xmath73\ ] ] which holds for both 3d and 2d ; the difference is that in the first case all indices run over @xmath74 , in the second only over @xmath75 . in a plane strain 3d configuration , @xmath76 , we have @xmath77 , whereas this stress component does not appear in 2d .
hence the conversion rules for the elastic constants are given by @xmath78 which follows directly from hooke s law ( [ conversion::eq1 ] ) .
let a system of dimensionless `` volume '' @xmath79 contain inclusions of a second phase , characterized by the initial concentration ( volume fraction ) @xmath80 , which in turn means that the concentration of the first phase is @xmath81 .
now , a volume @xmath82 of the second phase to the original volume @xmath79 is added , leading to a total volume of @xmath83 .
the total volume of phase two has increased to @xmath84 , resulting in a total volume fraction of @xmath85 the change of concentration of the second phase is therefore @xmath86 .
let @xmath87 denote a complete set of effective elastic constants ( or other quantities of interest ) . in the framework of the homogenization methods used here ,
this set should depend on the properties of the pure phases and the concentration , @xmath88 with a universal function @xmath89 and the obvious relation @xmath90 in the framework of the differential homogenization method the increase of the amount of the new phase from @xmath3 to @xmath91 is interpreted as the addition of the amount @xmath82 to the already homogenized medium with properties @xmath87 .
hence we obtain @xmath92 since in the second step the change of concentration is @xmath93 ; in the last step the relation ( [ diffhom::eq2 ] ) was used . here
it is important to note that after the differentiation first @xmath3 has to be set to zero , and only then @xmath94 to be inserted .
therefore , we immediately obtain the fundamental equation @xmath95 for slit - like cracks , the volume fraction is zero , and therefore the prefactor @xmath96 disappears and @xmath3 is replaced by the density parameter @xmath43 .
to make the preceding scaling arguments in section [ parallel ] more explicit , we discuss here a regular arrangement of cracks , as depicted in fig .
[ regular::fig1 ] and solve this problem exactly in the limit @xmath48 .
the idea is that the displacement , which is applied to the sample is mainly stored in the opening of the cracks , and the material in between is only slightly stretched .
the region between adjacent cracks behaves then as a bent plate ( see dark region in fig .
[ regular::fig1 ] ) , which is thin in the limit @xmath97 .
we note that for this regular arrangement the plate length @xmath98 appears here as additional parameter , which is related to the gap distance @xmath37 by @xmath99 ; again , @xmath38 is the crack length which is now assumed to be exactly the same for all cracks .
therefore , the additional dimensionless parameter @xmath100 remains in the final solution , whereas for an irregular arrangement of cracks it would be determined statistically ; finally , it enters only into the numerical prefactor of the effective elastic constants . for the given geometry ,
the area that is occupied by a single crack , @xmath101 , is @xmath102 .
therefore , the crack density is @xmath103 the bending of the thin plate is described by the equation @xmath104 , since the upper and lower surfaces are stress free @xcite .
each plate is displaced by @xmath105 , since the total displacement is equally distributed among all crack openings .
together with the symmetry conditions @xmath106 and the reference value @xmath107 , we obtain for the coefficients of the general solution @xmath108 the values @xmath109 and @xmath110 .
the force per unit length in @xmath37 direction that is required to bend the plate by the given amount is given by @xcite @xmath111 and thus the average stress in vertical direction @xmath112 from hooke s law for the effective medium , @xmath113 follows @xmath114 the bare elastic constant @xmath115 is related to the isotropic moduli by @xmath116 and hence get get asymptotically for @xmath48 @xmath117
|
we investigate the weakening of elastic materials through randomly distributed circles and cracks numerically and compare the results to predictions from homogenization theories .
we find a good agreement for the case of randomly oriented cracks of equal length in an isotropic plane - strain medium for lower crack densities ; for higher densities the material is weaker than predicted due to precursors of percolation . for a parallel alignment of cracks , where percolation does not occur ,
we analytically predict a power law decay of the effective elastic constants for high crack densities , and confirm this result numerically .
|
ic 443 ( g189.1 + 3.0 ) , a galactic supernova remnant ( snr ) at a distance of 1.5 kpc ( welsh & sallmen 2003 ) , is located near the gem ob1 association and a dense giant molecular cloud ( cornett et al . 1977 ) with oh maser emission ( claussen et al . 1997 ) .
these facts strongly suggest that the remnant originated from a collapse of a massive progenitor .
braun & strom ( 1986 ) proposed that the shock wave has expanded into the pre - existing wind - blown bubble shell .
this was confirmed by a kinematical study of the optical filaments ( meaburn et al .
a comprehensive x - ray study of ic 443 was first made with the _ einstein _ and _ heao - a2 _ satellites ( petre et al .
they estimated the snr age to be @xmath43000 yr .
recently , troja et al . ( 2008 ) derived the age of @xmath44000 yr from the morphologies of the shocked ejecta and interstellar medium ( ism ) revealed by _ xmm - newton_. thus , ic 443 is a middle - aged snr . using _ asca _ , kawasaki et al . ( 2002 ) found that the ionization degrees of si and s were significantly higher than those expected from the electron temperature of the bremsstrahlung continuum .
therefore , it was argued that the plasma is in overionization ( @xmath5 @xmath6 @xmath7 , where @xmath5 and @xmath7 are ionization and electron temperatures , respectively ) . on the other hand , troja et al .
( 2008 ) found that the plasma is in collisional ionization equilibrium ( cie ) , or the overionization is only marginal based on _ xmm - newton _ data .
these two controversial results hinge upon small differences , if any , between the estimated electron and ionization temperatures . in this letter
, we investigate whether the overionized plasma is really present or not , by utilizing the superior spectral capabilities for diffuse sources of x - ray imaging spectrometers ( xis : koyama et al .
2007 ) aboard the _ suzaku _ satellite ( mitsuda et al . 2007 ) .
if present , we will study the plasma condition quantitatively to discuss the possible origin of the overionization .
the northern part of ic 443 was observed with _ suzaku _ on 2007 march 6 ( observation i d = 501006010 ) .
three xiss located at the foci of the independent x - ray telescopes ( xrt : serlemitsos et al .
2007 ) were operating .
two of the xiss are front - illuminated ( fi ) ccds and the other is a back - illuminated ( bi ) ccd .
the xis was operated in the normal full - frame clocking mode with a spaced - row charge injection technique ( uchiyama et al .
2009 ) during the observation . for the data reduction and spectral analysis
, we used the headas software package of version 6.5 and xspec version 11.3.2 , respectively .
the xis data of revision 2.0 were employed , but reprocessed using the ` xispi ` software and the latest calibration database released on 2009 february 3 .
after the screening with the standard criteria , an effective exposure of @xmath442 ksec was obtained .
the yellow square and the white rectangle indicate the xis field of view and the region used in our spectral analysis , respectively .
the optical digitized sky survey image is overplotted in contour .
[ fig : image ] ] emission lines from specific elements are labeled in the panel .
[ fig : full ] ] figure [ fig : image ] shows the vignetting - corrected xis image in 1.753.0 kev , the energy band including the major lines of k - shell emissions from si and s. we extracted the spectrum from the brightest region , the rectangular with an angular size of @xmath8 ( see fig . [
fig : image ] ) .
this approximately corresponds to the `` center '' region of kawasaki et al .
the xis spectra of the two fis in the entire energy range were made by subtracting the non x - ray background ( nxb ) constructed with the ` xisnxbgen ` software .
the spectra were merged to improve the statistics , because the response functions are almost identical each other .
figure [ fig : full ] shows the resultant spectrum .
we can see several prominent lines of k@xmath1 emission from he- and h - like ions ( hereafter , he@xmath1 and ly@xmath1 ) . the centroids of the ly@xmath1 lines were measured with a gaussian line model , and compared with the canonical values of the astrophysical plasma emission database ( aped : smith et al .
the averaged center energy difference was + 4 ev .
therefore , we added a 4 ev offset in the fi spectrum .
the bi spectrum was made with the same procedure as the fis , but an offset of @xmath9 ev was added to correct the energy scale . in order to examine the ionization states of si and s , we hereafter focus on the spectrum in the energy range above 1.75 kev ( fig . [
fig : fit ] ) .
detailed studies including the lower energies will be reported in a separate paper .
, si - ly@xmath10 , s - ly@xmath1 , and ar - ly@xmath1 ) , and cxb , respectively .
the lower panel shows the residual from the best - fit model .
two hump - like features are clearly found around the energies of @xmath42.7 kev and @xmath43.5 kev .
( b ) same spectrum as ( a ) , but for a fit with rrc components of h - like mg , si , and s ( magenta lines ) .
the residuals seen in ( a ) are disappeared .
[ fig : fit],title="fig : " ] , si - ly@xmath10 , s - ly@xmath1 , and ar - ly@xmath1 ) , and cxb , respectively .
the lower panel shows the residual from the best - fit model .
two hump - like features are clearly found around the energies of @xmath42.7 kev and @xmath43.5 kev .
( b ) same spectrum as ( a ) , but for a fit with rrc components of h - like mg , si , and s ( magenta lines ) .
the residuals seen in ( a ) are disappeared .
[ fig : fit],title="fig : " ] we first fitted the spectrum with a model of a thin - thermal plasma in cie state ( a vapec model ) .
the abundances ( anders & grevesse 1989 ) of si , s , and ar were free parameters , while the ca abundance was tied to ar .
interstellar extinction was fixed to a hydrogen column density of @xmath11 = @xmath12 @xmath13 with the solar elemental abundances , following kawasaki et al .
( 2002 ) and troja et al .
the cosmic x - ray background ( cxb ) spectrum was approximated by a power - law model with photon index of @xmath14 = 1.412 and the surface brightness in the 210 kev band of @xmath15 erg @xmath13 s@xmath16 sr@xmath16 ( kushino et al .
2002 ) . since ic 443 is located in the anti - galactic center direction , contribution of the galactic ridge x - ray emission was ignored . in the initial fit
, we found a significant inconsistency between the fi and bi data around the energy of neutral si k - edge ( 1.84 kev ) .
this is due to the well - known calibration issue of the xis . since the calibration for the fi ccds is currently far better than the bi , we decided to ignore the energy band below 1.9 kev in the bi spectrum
this fit leaved further large residuals , in both spectra , at the energies of s and ar ly@xmath1 lines , and hence was rejected with the @xmath17/dof of 935/270 .
we , therefore , applied a model of one - temperature vapec ( cie plasma ) plus narrow gaussian lines at 2006 , 2623 , and 3323 ev , the ly@xmath1 energies of the si , s , and ar , hydrogenic ions , respectively .
this process is essentially the same as kawasaki et al .
ly@xmath10 lines of the same elements were also added , but they were not significant except for si ( 2377 ev ) .
the result is shown in figure [ fig : fit]a , with the best - fit cie plasma temperature of @xmath7 @xmath4 0.94 kev .
although the @xmath18/dof value was significantly reduced to 657/266 , the model was still unacceptable .
in fact , apparent hump - like residuals are found around @xmath42.7 kev and @xmath43.5 kev , in addition to the systematic model excess in the continuum at the energies above @xmath44.5 kev . we checked systematic error due to the cxb fluctuation by allowing the cxb intensity as a free parameter .
the fit , however , did not improve at all .
we also tried two - component vapec models with independent temperatures and abundances , but these were rejected with @xmath18/dof = 737/265 .
the results were essentially the same as figure [ fig : fit]a ; the hump - like residuals remained at @xmath42.7 kev and @xmath43.5 kev .
no additional cie component nor non - equilibrium ionization ( nei ) plasma model removed the hump - like residuals with significant improvement of the @xmath17 value .
llcc component & & value + cie ( vapec ) & & 0.61 ( 0.590.64 ) + & & 0.82 ( 0.780.85 ) + & & 1.7 ( 1.61.8 ) + & & 2.5 ( 2.22.8 ) + & & 6.4 ( 6.36.6 ) + + & & @xmath19 ( kev ) & flux + line & si ly@xmath1 & 2.006 & 2.8 ( 2.72.9 ) + & si ly@xmath10 & 2.377 & 0.21 ( 0.140.29 ) + & s ly@xmath1 & 2.623 & 0.84 ( 0.790.90 ) + & ar ly@xmath1 & 3.323 & 0.11 ( 0.0850.14 ) + rrc & h - like mg & 1.958 & 1.2 ( 1.01.5 ) + & h - like si & 2.666 & 2.2 ( 2.02.3 ) + & h - like s & 3.482 & 0.46 ( 0.410.51 ) + at the energies of the humps , no emission line candidate from an abundant element is found
. however , the energies are consistent with the k - shell binding potentials ( @xmath20 ) of the h - like si ( 2666 ev ) and s ( 3482 ev ) .
therefore , the humps are likely due to the free - bound transitions to the k - shell of the h - like si and s. a formula for the spectrum of radiative recombination continuum ( rrc ) is found in equation ( 21 ) of smith & brickhouse ( 2002 ) . when the electron temperature is much lower than the k - edge energy ( @xmath7 @xmath21 @xmath20 ) , this formula is approximated as ; @xmath22 thus , the width of the rrc structure depends on the electron temperature .
we added the rrc of equation [ eq : rrc ] for h - like si and s. the @xmath7 values of the rrc were linked to that of the vapec component .
then , the fit was dramatically improved to an acceptable @xmath18/dof of 290/264 .
although the hump - like residuals were completely removed by this fit , we further added the rrc model of h - like mg .
this step is reasonable because mg is more abundant than si and s in the solar abundance ratio of anders & grevesse ( 1989 ) and the k - edge energy of h - like mg ( @xmath20 = 1958 ev ) falls into the analyzed band .
the @xmath18/dof value was significantly reduced to 267/263 , which gives an @xmath23-test probability of @xmath24 .
the best - fit parameters and model are given in table [ tab : fit ] and figure [ fig : fit]b , respectively .
the spectrum was also fitted with the independent electron temperatures for the vapec and rrc components , but the values are consistent with each other within their statistical uncertainties .
lines as a function of ionization temperature ( @xmath5 ) , predicted by the plasma radiation code of masai ( 1994 ) for recombining plasma with electron temperature ( @xmath7 ) of 0.6 kev .
black and red solid lines represent si and s , respectively . for comparison ,
the same ratios for cie ( @xmath5 = @xmath7 ) plasma ( the apec model : smith 2001 ) are indicated by dashed lines .
the horizontal dotted lines represent the 90% upper and lower limits of the observed values .
[ fig : ratio ] ] we have found that the 1.756.0 kev spectrum can not be represented with cie nor nei plasma alone , but need the additional fluxes of the lyman lines and the h - like rrc of si and s ( and possibly mg ) .
this is the first detection of clear rrc emissions in an snr . in the following ,
we quantitatively discuss the implications of our spectral results . from the best - fit model in table [ tab : fit ] , the flux ratios of h - like rrc to he - like k@xmath1 ( he@xmath1 ) line ( @xmath25 ) are given to be 0.28 ( 0.240.30 ) and 0.18 ( 0.150.20 ) for si and s , respectively .
these are compared , in figure [ fig : ratio ] , with the modeled emissivity ratios by the plasma radiation code of masai ( 1994 ) for the electron temperature of 0.6 kev .
we find that the large observed ratios of @xmath25 are significantly above those in the cie case ( @xmath5 = @xmath7 ) , but can be reproduced in the overionization case ( @xmath5 @xmath6 @xmath7 ) .
the ionization temperatures of si and s are determined to be @xmath2 kev and @xmath3 kev , respectively .
this is , therefore , the firm evidence of the overionized ( recombining ) plasma . the overionization claim for ic 443 was first argued by kawasaki et al .
( 2002 ) . using
data , they derived @xmath5 to be @xmath26 kev from the s ly@xmath1/he@xmath1 flux ratio compared with the predicted emissivity ratio in the cie plasma code .
troja et al .
( 2008 ) adopted the same analysis procedure to the _ xmm - newton _ spectrum , and claimed that @xmath5 obtained from the line flux ratio was nearly same as the bremsstrahlung temperature ( @xmath7 ) .
however , since the rrc process is accompanied with electron captures to the excited levels , as given in equation [ eq : rate ] below , the resulting cascade decay to the ground state contributes the line emission . in a cie plasma , on the other hand , origins of the line emissions
are more dominated by the collisional excitation .
therefore , a @xmath5 determination done by comparing with a cie plasma code is not a proper method .
also , the previous works assumed that the continuum spectrum purely consists of bremsstrahlung emission , and determined @xmath7 to be @xmath2 kev .
this value has been reduced owing to the discovery of the strong rrc emissions .
it should be noted that the elemental abundances in table [ tab : fit ] are determined as the parameters of the 0.6 kev vapec ( cie ) component , and hence should be modified in the real case of the overionized plasma .
the intensity of the si - he@xmath1 line is given as @xmath27 , where @xmath28 and @xmath29 are , respectively , emissivity coefficient for electron temperature @xmath30 and fraction of he - like ion for ionization temperature @xmath31 .
the abundance , therefore , can be modified by the fraction ratio of he - like ions , @xmath32 . according to the ionization population calculations by mazzotta et al .
( 1998 ) , the ion fractions of he - like , h - like , and fully - ionized si ( @xmath33 , @xmath34 , @xmath35 ) are estimated to be ( 0.83 , 0.15 , 0.01 ) for @xmath5 = 0.6 kev and ( 0.37 , 0.43 , 0.19 ) for @xmath5 = 1.0 kev , respectively .
then , the modified @xmath36 is @xmath37 solar . for s , ( @xmath33 , @xmath34 , @xmath35 )
are ( 0.91 , 0.02 , 0 ) for @xmath5 = 0.6 kev and ( 0.59 , 0.32 , 0.06 ) for @xmath5 = 1.2 kev .
thus , we similarly modified @xmath38 to be @xmath39 solar . using the measured rrc flux
, we can independently determine the volume emission measure , vem = @xmath40 , from a following equation : @xmath41 where @xmath42 and @xmath43 are k - shell rrc rate coefficient for electron temperature @xmath30 and number density of element @xmath44 , respectively . according to badnell ( 2006 ) ,
we find total radiative recombination rate for fully - ionized si , at @xmath7 = 0.6 kev , is @xmath45 @xmath4 @xmath46 @xmath47 s@xmath16 . since the rate of the recombination into a level of principal quantum number @xmath48 is described as ; @xmath49 ( e.g. , nakayama & masai 2001 ) , we obtain @xmath50 = 0.65 , for @xmath7 = 0.6 kev and @xmath20 = 2.666 kev .
thus , the vem is calculated to be @xmath51 @xmath52 @xmath53 @xmath54 .
similarly , @xmath45 at 0.6 kev for fully - ionized s is derived to be @xmath55 @xmath47 s@xmath16 .
therefore , the rrc flux of s corresponds to the vem of @xmath56 @xmath57 @xmath58 @xmath54 .
these values are almost consistent with that of the vapec component ( table [ tab : fit ] ) .
the rectangular region in figure [ fig : image ] corresponds to @xmath59 pc@xmath60 at @xmath61 = 1.5 kpc . assuming the plasma depth of 3 pc , the emitting volume is estimated to be @xmath62 @xmath47 .
therefore , the vem of @xmath63 @xmath54 is converted to the uniform electron density of @xmath64 @xmath65 . as the mechanism to form the overionized plasma , kawasaki et al .
( 2002 ) proposed that the snr consists of a central hot ( @xmath7 @xmath4 1.0 kev ) region surrounded by a cool ( @xmath7 @xmath4 0.2 kev ) outer shell , and interpreted that the hot interior cooled down via thermal conduction to the cool exterior . under the several reasonable boundary conditions
, they estimated the cooling time from @xmath7 = 1.5 kev to 1.0 kev is about ( 310 ) @xmath66 @xmath67 yr , roughly the same as the snr age ( @xmath44000 yr : troja et al .
2008 ) . however , this scenario can not work on our new results .
according to equation ( 5 ) of kawasaki et al .
( 2002 ) , the conduction timescale is estimated to be @xmath68 if the similar boundary condition is assumed .
thus , cooling via conduction requires far longer time than the snr age .
photo - ionization is also unlikely because no strong ionizing source is found .
furthermore , the temperature of @xmath40.6 kev is significantly higher than that of a typical photo - ionization plasma ( @xmath690.1 kev : e.g. , kawashima & kitamoto 1996 ) . since the progenitor of ic 443 has been suggested to be a massive star with strong stellar wind activity ( braun & strom 1986 ; meaburn et al .
1990 ) , we propose another possibility that the rapid and drastic cooling is due to a rarefaction process , as discussed by itoh & masai ( 1989 ) .
if a supernova explodes in a dense circumstellar medium made in the progenitor s super giant phase , the gas is shock - heated to high temperature and significantly ionized at the initial phase of the snr evolution .
subsequent outbreak of the blast wave to a low - density ism caused drastic adiabatic expansion of the shocked gas and resultant rapid cooling of the electrons .
the lifetimes of the fully - stripped ions are roughly estimated to be @xmath70 = @xmath71 @xmath72 yr for si and @xmath73 yr for s , respectively .
note that these values are underestimated compared to the actual timescale for the plasma to reach cie , because the contribution of collisional ionization processes is ignored .
nevertheless , the estimated lifetimes are longer than the age of ic 443 .
the overionized plasma can , therefore , still survive at present . future observations with very high energy resolution like the _
astro - h _ mission will give firm evidence for the rrc structure not only on si and s but also on the other major elements .
this will provide more quantitative study on the peculiar snr ic 443 .
the authors deeply appreciate the referee , randall smith , for his constructive suggestions on revising manuscript .
we also acknowledge helpful discussions with kazuo makishima and aya bamba .
h. yamaguchi and j. s. hiraga are supported by the special postdoctoral researchers program in riken .
m. ozawa is a research fellow of japan society for the promotion of science ( jsps ) .
this work is partially supported by the grant - in - aid for the global coe program `` the next generation of physics , spun from universality and emergence '' , young scientists ( hy ) , and challenging exploratory research ( kk ) from the ministry of education , culture , sports , science and technology ( mext ) of japan .
|
we present the _ suzaku _
spectroscopic study of the galactic middle - aged supernova remnant ( snr ) ic 443 .
the x - ray spectrum in the 1.756.0 kev band is described by an optically - thin thermal plasma with the electron temperature of @xmath0 kev and several additional lyman lines .
we robustly detect , for the first time , strong radiative recombination continua ( rrc ) of h - like si and s around at 2.7 and 3.5 kev . the ionization temperatures of si and s determined from the intensity ratios of the rrc to he - like k@xmath1 line are @xmath2 kev and @xmath3 kev , respectively .
we thus find firm evidence for an extremely - overionized ( recombining ) plasma . as the origin of the overionization ,
a thermal conduction scenario argued in previous work is not favored in our new results .
we propose that the highly - ionized gas were made at the initial phase of the snr evolution in dense regions around a massive progenitor , and the low electron temperature is due to a rapid cooling by an adiabatic expansion .
|
with the development of observational astrophysics and cosmology , the investigations of galaxy rotation curves , gravitational lensing and large scale structures have provided strong evidences for the existence and importance of dark matter .
the abundance of dark matter has been measured with increasingly high precision , such as @xmath11 by the latest planck data @xcite ; however , since our knowledge of dark matter exclusively comes from the gravitational effects , the physical nature of dark - matter particles remain mysterious .
nowadays it becomes a common view that to account for the observed dark matter , one needs to go beyond the su@xmath12su@xmath13u@xmath14 minimal standard model .
there are mainly two leading classes of dark - matter candidates : axions that are non - thermally produced via quantum phase transitions in the early universe , and generic weakly interacting massive particles ( wimps ) @xcite that freeze out of thermal equilibrium from the very early cosmic plasma and leave a relic density matching the present - day universe . in this paper , we are interested in the latter class , i.e. dark matter created as thermal relics .
we aim to correct and complete the pioneering investigations in ref.@xcite for cold relics in @xmath15 gravity , and provide a comprehensive investigation of thermal relics as hot , warm and cold dark matter in @xmath0 gravity .
this paper is organized as follows .
[ gravitational framework of power - law fr gravity ] sets up the gravitational framework of @xmath0 gravity , while sec .
[ preparations thermal relics ] generalizes the time - temperature relation for cosmic expansion and derives the simplified boltzmann equation .
[ hot warm relic dark matter and light neutrinos ] studies hot / warm thermal relics , and shows the influences of @xmath7 and @xmath1 to the bound of light neutrino mass . sec .
[ cold relic dark matter ] investigates cold thermal relics by solving the simplified boltzmann equation , while sec .
[ example fourth generation massive neutrinos and lee - weinberg bound ] rederives the lee - weinberg bound on fourth - generation massive neutrinos , and examines the departure from electroweak energy scale .
finally , the gr limit of the whole theory is studied in sec
. 7 . throughout this paper , for the physical quantities involved in the calculations of thermal relics , we use the natural unit system of particle physics which sets @xmath16 and is related to le systme international dunits by @xmath17 . on the other hand , for the spacetime geometry , we adopt the conventions @xmath18 , @xmath19 and @xmath20 with the metric signature @xmath21 .
@xmath10 gravity is a direct generalization of gr and extends the hilbert - einstein action @xmath22 @xmath23 into @xmath24\ , , \ ] ] where @xmath25 denotes the ricci scalar of the spacetime , @xmath1 is some constant balancing the dimensions of the field equation , and @xmath26 is the matter lagrangian density .
also , @xmath27 refers to the planck mass , which is related to newton s constant @xmath28 by @xmath29 and takes the value @xmath30 .
variation of eq.([f(r ) action ] ) with respect to the inverse metric @xmath31 yields the field equation @xmath32 where @xmath33 , @xmath34 denotes the covariant dalembertian @xmath35 , and @xmath36 is the stress - energy - momentum tensor of the physical matter content .
this paper considers the spatially flat , homogeneous and isotropic universe , which , in the @xmath37 comoving coordinates along the cosmic hubble flow , is depicted by the friedmann - robertson - walker ( frw ) line element @xmath38 where @xmath39 denotes the cosmic scale factor .
assume a perfect - fluid material content @xmath40 @xmath41 $ ] , with @xmath42 and @xmath43 being the energy density and pressure , respectively .
then eq.([field eq generic fr ] ) under the flat frw metric yields the generalized friedmann equations @xmath44 @xmath45 where overdot denotes the derivative with respect to the comoving time , @xmath46 , and @xmath47 .
in addition , the equation of local energy - momentum conservation gives rise to the continuity equation , @xmath48 for the very early universe that is radiation - dominated , integration of eq.([continuity eq ] ) with the equation of state @xmath49 yields that the radiation density is related to the cosmic scale factor by @xmath50 in this paper , we will work with the specific power - law @xmath10
gravity @xmath51where @xmath52 . with @xmath53 and @xmath54 ,
the generalized first friedmann equation ( [ friedmann eqi generic fr ] ) yields @xmath55 [ concrete friedmann eqi generic fr ] ^=32 ^ 2 - 2m_^-2 , & & where @xmath56 refers to the cosmic hubble parameter .
moreover , the weak , strong and dominant energy conditions for classical matter fields require the energy density @xmath42 to be positive definite , and as a consequence , the positivity of the left hand side of eq.([concrete friedmann eqi generic fr ] ) limits @xmath7 to the domain @xmath57 note that the ricci scalar for the flat frw metric with @xmath58 reads @xmath59 so @xmath60 and @xmath61 is always well defined in this domain .
moreover , we will utilize two choices of @xmath1 to balance the dimensions in @xmath0 gravity : 1 .
@xmath62 [ sec@xmath63 .
this choice can best respect and preserve existent investigations in mathematical relativity for the @xmath10 class of modified gravity , which have been analyzed for @xmath64 without caring the physical dimensions .
@xmath65 [ 1/s ] , or @xmath66 where @xmath67 refers to planck length . the advantage of this choice is there is no need to employ extra parameters outside the mathematical expression @xmath64 .
for the very early universe , the radiation energy density @xmath42 attributes to all relativistic species , which are exponentially greater than those of the nonrelativistic particles , and therefore @xmath68 , where @xmath69 are the numbers of statistical degrees of freedom for relativistic bosons and fermions , respectively .
more concisely , normalizing the temperatures of all relativistic species with respect to photons temperature @xmath70 , one has the generalized stefan - boltzmann law @xmath71 where , in thermodynamic equilibrium , @xmath72 is the common temperature of all relativistic particles . to facilitate the discussion of thermal relics ,
introduce a dimensionless variable @xmath73 to relabel the time scale , where @xmath74 denotes the mass of dark - matter particles .
@xmath75 is a well defined variable since the temperature monotonically decreases after the big bang : reheatings due to pair annihilations at @xmath76 only slow down the decrement of @xmath72 rather than increase @xmath72 @xcite . substitute eq.([rho and g for radiation ] ) into eq.([concrete friedmann eqi generic fr ] ) , and it follows that the cosmic expansion rate is related to the radiation temperature by @xmath77 which can be compactified into @xmath78 as time elapses after the big bang , the space expands and the universe cools .
eq.([hubble evolution ] ) along with @xmath79 leads to @xmath80 and the time - temperature relation @xmath81 for dark - matter particles @xmath82 in the very early universe ( typically before the era of primordial nucleosynthesis ) , there are various types of interactions determining the @xmath82 thermal relics , such as elastic scattering between @xmath82 and standard - model particles , and self - annihilation @xmath83 . in this paper
, we are interested in @xmath82 initially in thermal equilibrium via the pair annihilation into ( and creation from ) standard - model particles @xmath84 , @xmath85 as the mean free path of @xmath82 increases along the cosmic expansion , the interaction rate @xmath86 of eq.([wimp pair annihilation creation ] ) gradually falls below the hubble expansion rate @xmath56 , and the abundance of @xmath82 freezes out .
the number density of @xmath82 satisfies the simplified boltzmann equation @xmath87\ , , \ ] ] where @xmath88 is the thermally averaged cross - section .
employ the following quantity to describe the evolution of @xmath82 at different temperature scales : @xmath89 where @xmath90 is the comoving entropy density @xmath91 , @xmath92 here we have applied @xmath93 and @xmath94 in @xmath90 for relativistic matter , and @xmath3 denotes the entropic number of statistic degrees of freedom . according to the continuity equation eq.([continuity eq ] ) and the thermodynamic identities @xmath95
one has @xmath96 so the comoving entropy density @xmath97 of a particle species is conserved when the comoving particle number density @xmath98 is conserved or the chemical potential @xmath99 is far smaller than the temperature .
thus , @xmath100 , @xmath101 , and the time derivative of @xmath102 becomes @xmath103 substitute the simplified boltzmann equation ( [ simplified boltzmann equation ] ) into eq.([dm boltzamann eq1 ] ) , and one obtains @xmath104 now rewrite @xmath105 into @xmath106 . since @xmath107^\beta\right\}^{1/4}t^{-\beta/2 } \;\ ; \propto t^{-\beta/2}\ , , \ ] ] thus @xmath108 , and @xmath109 , which recast eq.([dm boltzamann eq3 ] ) into @xmath110 defining the annihilation rate of @xmath82 as @xmath111 , then eq.([dm boltzamann eq4 ] ) can be rewritten into the form @xmath112 = -\frac{\gamma_\psiup}{h } \left[\left(\frac{y}{y_{\text{eq}}}\right)^2 - 1 \right]\ , , \ ] ] which will be very useful in calculating the freeze - out temperature of cold relics in sec .
[ cold relic dark matter ] .
having set up the modified cosmological dynamics and boltzmann equations in @xmath0 gravity , we will continue to investigate hot dark matter which is relativistic for the entire history of the universe until now , and warm dark matter which is relativistic at the time of decoupling but become nonrelativistic nowadays . in the relativistic regime @xmath113 or equivalently @xmath114 ,
the abundance of @xmath74 is given by @xmath115 where @xmath116 , @xmath117 for bosons and @xmath118 for fermions .
@xmath119 only implicitly depends on @xmath75 through the evolution of @xmath3 along the temperature scale .
then , the relic abundance is still given by @xmath119 at the time of freeze - out @xmath120 : @xmath121 at the present time with @xmath122 @xcite , the entropy density is @xmath123 where in the minimal standard model with three generations of light neutrinos ( @xmath124 ) , @xmath125 thus , the present - day number density and energy density of hot / warm relic @xmath82 can be found by @xmath126 @xmath127 which , for @xmath128 , correspond to the fractional energy density @xmath129 this actually stands for an attractive feature of the paradigm of thermal relics : the current abundance @xmath130 of relic dark matter ( hot , warm , or cold ) can be predicted by @xmath82 s microscopic properties like mass , annihilation cross - section , and statistical degrees of freedom . since hot / warm relics can at most reach the total dark matter density @xmath131 @xcite , @xmath130 has to satisfy @xmath132 , and it follows from eq.([hot warm omega h2 ] ) that @xmath74 is limited by the upper bound @xmath133 moreover , particles of warm dark matter become nonrelativistic at present time , which imposes a lower bound to @xmath74 , @xmath134 where we have applied @xmath135 due to @xmath136 .
eqs.([mass bound upper ] ) and ( [ mass bound lower ] ) lead to the mass bound for warm relics that @xmath137 light neutrinos are the most popular example of hot / warm dark matter @xcite .
one needs to figure out the temperature @xmath138 and thus @xmath139 when neutrinos freeze out from the cosmic plasma .
the decoupling occurs when the hubble expansion rate @xmath56 balances neutrinos interaction rate @xmath140 . for the cosmic expansion , it is convenient to write eq.([hubble evolution ] ) into @xmath141}\ , , \end{split}\ ] ] where @xmath142 refers to the value of temperature in the unit of mev , @xmath143}$ ]
, @xmath144 is the value of @xmath1 in the unit of [ 1/s ] , and numerically @xmath145 [ 1/s ] . on the other hand ,
the event of neutrino decoupling actually indicates the beginning of primordial nucleosynthesis , when neutrinos are in chemical and kinetic equilibrium with photons , nucleons and electrons via weak interactions and elastic scattering .
the interaction rate @xmath140 is @xcite @xmath146}\ , , \ ] ] where @xmath147 is fermi s constant in beta decay and generic weak interactions , and @xmath148 .
neutrinos decouple when @xmath149 , and according to eqs.([dm hubble evolution ii ] ) and ( [ dm neutrino interaction rate ] ) , the weak freeze - out temperature @xmath150 is the solution to @xmath151 figs .
[ dmfiglightneutinoi ] and [ dmfiglightneutinoii ] have shown the dependence of @xmath150 on @xmath7 for @xmath152 mev and @xmath153 mev , respectively .
[ dmfiglightneutinoii ] clearly illustrates that @xmath150 spreads from 1.3030 mev to over 1000 mev , which goes far beyond the scope of @xmath154 mev ; thus , as shown in table [ table 1 gs ] , @xmath3 varies and the mass bound @xmath6 in light of eq.([mass bound upper lower ] ) is both @xmath5dependent and @xmath155dependent . [
cols="^ " , ]
now let s consider cold dark matter which is already nonrelativistic at the time of decoupling . in the nonrelativistic regime @xmath156 or equivalently @xmath157 , the number density and entropy density
are given by @xmath158 so one obtains the equilibrium abundance of nonrelativistic @xmath82 particles @xmath159 thus , @xmath160 and @xmath161 are exponentially suppressed when the temperature drops below @xmath74 . moreover , since cold relics are nonrelativistic when freezing out , one can expand the thermally averaged cross - section by @xmath162 , where @xmath163 corresponds to the decay channel of @xmath164wave , @xmath165 to @xmath166wave , @xmath167 to @xmath168wave , and so forth ; recalling that @xmath169 in light of the boltzmann velocity distribution , thus the annihilation cross - section can be expanded by the variable @xmath75 into @xmath170 then the boltzmann equation ( [ dm boltzamann eq4 ] ) becomes @xmath171\ , , \ ] ] where @xmath172 though initially in equilibrium @xmath173 , the actual abundance @xmath102 gradually departures from the equilibrium value @xmath174 as the temperature decreases ; @xmath102 freezes out and escapes the exponential boltzmann suppression when the interaction rate @xmath175 equates the cosmic expansion rate @xmath56 .
transforming eq.([integrate y i ] ) into the form @xmath176 x^{\frac{2}{\beta}-3-n } = -\frac{\gamma_\psiup}{h } \left[\left(\frac{y}{y_{\text{eq}}}\right)^2 - 1 \right]\ , , \ ] ] and the coupling condition @xmath177 at the freeze - out temperature @xmath178 yields @xmath179 thus , it follows that @xmath180 after taking the logarithm of both side , eq.([cdm xf eq ] ) can be iteratively solved to obtain @xmath181\\ & + \left(\frac{2}{\beta}-\frac{3}{2}-n\right)\ln \left[\ln\bigg(\frac{0.2199}{\left(5.7509 \right)^{1/\beta } } \sqrt{\frac{\beta-1}{\beta } } \left(\!\sqrt{\frac{-5\beta^2 + 8\beta-2}{\beta-1}}\right)^{1/\beta } \frac{g_\psiup}{(\!\sqrt{g_*})^{1/\beta}}\,\varepsilon^{\frac{1}{\beta}-1 } \ , m^{3-\frac{2}{\beta}}\ , m_{\text{pl}}^{1/\beta}\left\langle\sigma v \right\rangle_0 \bigg ) \right]\nonumber\\ & + \left(\frac{2}{\beta}-\frac{3}{2}-n\right)\ln \bigg[\cdots \cdots \bigg]\,,\nonumber\end{aligned}\ ] ] where @xmath182 has been treated as a constant , as the time scale over which @xmath182 evolves is much greater than the time interval near @xmath120 . to work out the actual abundance @xmath102 before the decoupling of @xmath183 ,
employ a new quantity @xmath184 , and then eq.([integrate y i ] ) can be recast into @xmath185 in the high - temperature regime @xmath186 before @xmath183 freezes out , @xmath102 is very close to @xmath119 , so that @xmath187 and @xmath188 . with @xmath161 in eq.([y equilibrium i ] ) , eq.([integrate y ii-1 ] ) can be algebraically solved to obtain @xmath189 and consequently @xmath190 after the decoupling of @xmath183 particles , the actual number density @xmath160 becomes much bigger than the ideal equilibrium value @xmath192 .
one has @xmath193 , @xmath194 , and the differential equations ( [ integrate y i ] ) or ( [ integrate y ii-1 ] ) leads to @xmath195 which integrates to yield the freeze - out abundance @xmath196 that @xmath197 following @xmath191 , the number density and energy density of @xmath183 are directly are directly found to be @xmath198 @xmath199 which gives rise to the fractional energy density @xmath200 unlike eq.([hot warm omega h2 ] ) for hot / warm relics , the relic density @xmath130 for cold dark matter is not only much more sensitive to the temperature of cosmic plasma , but also relies on the annihilation cross - section .
an example of cold relics can be the hypothetical fourth generation massive neutrinos @xcite . for the dirac - type neutrinos
whose annihilations are dominated by @xmath164wave @xmath201 , the interaction cross - section reads @xmath202 where @xmath147 is fermi s constant in beta decay and generic weak interactions , and @xmath203 . then with @xmath204 and @xmath205 , the neutrinos decouple at @xmath206\\ & + \left(\frac{2}{\beta}-\frac{3}{2}-n\right)\ln \left[\ln\bigg(\frac{0.5983\times 10^{-10 } } { \left(44.5463 \right)^{1/\beta } } \sqrt{\frac{\beta-1}{\beta } } \left(\!\sqrt{\frac{-5\beta^2 + 8\beta-2}{\beta-1}}\right)^{1/\beta } \,\varepsilon^{\frac{1}{\beta}-1 } \ , m^{5-\frac{2}{\beta}}\ , m_{\text{pl}}^{1/\beta } \bigg ) \right]\nonumber\\ & + \left(\frac{2}{\beta}-\frac{3}{2}-n\right)\ln \bigg[\cdots \cdots \bigg]\,,\nonumber\end{aligned}\]]which , through eq.([cdm omegah2 ] ) , gives rise to the fractional energy density @xmath207with the same amount of anti - particles , we finally have @xmath208 .
thus the lee - weinberg bound @xcite for massive neutrinos are relaxed in @xmath0 gravity .
in this paper , we have comprehensively investigated the thermal relics as hot , warm and cold dark matter in @xmath0 gravity . when light neutrinos act as hot and warm neutrinos , the upper limit of neutrino mass @xmath6 relies on the value of @xmath7 and the choice of @xmath1 . for cold relics ,
we have derived the freeze - out temperature @xmath209 in eq.([cold relics xf ] ) , @xmath102 before the freeze - out in eq.([cdm y before xf ] ) , the freeze - out value @xmath191 in eq.([cdm y after xf ] ) , and the dark - matter fractional density @xmath130 in eq.([cdm omegah2 ] ) .
note that we focused on power - law @xmath10 gravity because unlike the approximated power - law ansatz @xmath210 ( @xmath211 ) for generic @xmath10 gravity , @xmath58 is an exact solution to @xmath0 gravity for the radiation - dominated universe ; for gr with @xmath9 , eq.([exact solution ] ) reduces to recover the behavior @xmath212 which respects @xmath213 .
when light neutrinos serve as hot / warm relics , the entropic number of statistical degrees of freedom @xmath3 at freeze - out and thus the predicted fractional energy density @xmath4 are @xmath5dependent , which relaxes the standard mass bound @xmath6 . for cold relics , by exactly solve the simplified boltzmann equation in both relativistic and nonrelativistic regimes , we show that the lee - weinberg bound for the mass of heavy neutrinos can be considerably relaxed , and the `` wimp miracle '' for weakly interacting massive particles ( wimps ) gradually becomes invalid when @xmath7 departs @xmath8 .
the whole framweork reduces to become that of gr in the limit @xmath9 .
|
we investigate the thermal relics as hot , warm and cold dark matter in @xmath0 gravity , where @xmath1 is a constant balancing the dimension of the field equation , and @xmath2 for the positivity of energy density and temperature .
if light neutrinos serve as hot / warm relics , the entropic number of statistical degrees of freedom @xmath3 at freeze - out and thus the predicted fractional energy density @xmath4 are @xmath5dependent , which relaxes the standard mass bound @xmath6 . for cold relics ,
by exactly solve the simplified boltzmann equation in both relativistic and nonrelativistic regimes , we show that the lee - weinberg bound for the mass of heavy neutrinos can be considerably relaxed , and the `` wimp miracle '' for weakly interacting massive particles ( wimps ) gradually invalidates as @xmath7 deviates from @xmath8 .
the whole framework reduces to become that of gr in the limit @xmath9 . + * pacs numbers * 26.35.+c , 95.35.+d , 04.50.kd + * key words * thermal relics , dark matter , @xmath10 gravity
|
the demand for bandwidth is rapidly increasing due to the explosive growth of network traffic .
networking technologies play an important role in bridging the gap between limited resources and the constantly increasing demand . in order to avoid a full mesh architecture ,
a switching device is required to build a realistic network . over the past few years , a lot of enabling technologies have emerged as candidates for achieving high performance switching .
basically , switches act like automated patch - panels , switching all the electrical or optical signals from one port to another .
traditionally , digital switching can be done in many ways .
for example , by allocating physical separated paths , switching can be done in the space domain .
a 2-d mems optical switch with precisely controlled micromirrors is essentially a space domain switch .
similarly , by associating the data from each port with a unique resource , switching can be performed in many other ways , such as in the time domain , the wavelength domain , and even a combination of these mechanisms .
on the other hand , quantum information science is a relatively new field of study .
quantum computers were first discussed in the early 1980 s @xcite,@xcite,@xcite . since then , a great deal of research has been focused on this topic .
remarkable progress has been made due to the discovery of secure key distribution @xcite , polynomial time prime factorization @xcite , and fast database search algorithm @xcite .
these results have recently made quantum information science the most rapidly expanding research field .
other applications , such as clock synchronization @xcite,@xcite , and quantum boolean circuit implementation @xcite have driven this field further into the phase of real - world applications . in this paper , we present a architecture and implementation algorithm such that digital data can be switched in the quantum domain . first we define the connection digraph which can be used to describe the behavior of a switch at a given time , then we show how a connection digraph can be implemented using elementary quantum gates .
the proposed mechanism supports unicasting as well as multicasting and is strict - sense non - blocking @xcite .
it can be applied to perform either circuit switching or packet switching . compared with a traditional space or time switch ,
the proposed switching mechanism is more scalable .
assuming an @xmath0 quantum switch , the space consumption grows linearly , _
i.e. _ @xmath1 , while the time complexity is @xmath2 for unicasting and @xmath3 for multicasting . based on these advantages ,
a high throughput switching device can be built simply by increasing the number of i / o ports .
in a two - state quantum system , each bit can be represented using a basis consisting of two eigenstates , denoted by @xmath4 and @xmath5 respectively .
these states can be either spin states of a particle ( @xmath4 for spin - up and @xmath5 for spin - down ) or energy levels in an atom ( @xmath4 for ground state and @xmath5 for excited state ) .
these two states can be used to simulate the classical binary logic .
a classical binary logic value must be either * on * ( 1 ) or * off * ( 0 ) , but not both at the same time .
however , a bit in a quantum system can be any linear combination of these two states , so we have the state @xmath6 of a bit as @xmath7 where @xmath8 , @xmath9 are complex numbers and @xmath10 . in column matrices , this is written as @xmath11 the state shown above exhibits an unique phenomenon in quantum mechanics called _
superposition_. when a particle is in such a superposed state , it has a part corresponding to @xmath4 and a part corresponding to @xmath5 , at the same time .
when you measure the particle , the system is projected to one of its basis ( _ i.e. _ either @xmath4 or @xmath12 ) .
the overall probability for each state is given by the absolute square of its amplitude .
taking the state @xmath6 in eq.([superposition ] ) as an example , the coefficient @xmath13 and @xmath14 represents the probability of obtaining @xmath4 and @xmath12 respectively .
obviously , the sum of @xmath13 and @xmath14 will be @xmath15 to satisfy the probability rule . to distinguish the above system from the classical binary logic , a bit in a quantum system
is referred to as a quantum bit , or _
qubit_. two or more qubits can also form a quantum system jointly .
a two - qubit system is spanned by the basis of the tensor product of their own spaces .
hence , the joint state of qubit a and qubit b is spanned by @xmath16 , @xmath17 , @xmath18 , and @xmath19 , _
i.e. _ @xmath20 where @xmath8 , @xmath9 , @xmath21 , @xmath22 are all complex numbers and @xmath23 . in matrix form , this is equivalent to @xmath24 the notations described above can be generalized to multiple - qubit systems .
for example , in a three - qubit system , the space is spanned by a basis consisting of eight elements ( @xmath25 , @xmath26 , , @xmath27 ) .
a quantum system can be manipulated in many different ways , called _
quantum gates_. a quantum gate can be represented in the form of a matrix operation .
for example , a quantum _ not _ ( * n * ) gate applied on a single qubit can be represented by multiplying a @xmath28 matrix @xmath29 which changes the state from @xmath5 to @xmath30 and from @xmath4 to @xmath5 , as @xmath31 the symbol of an * n * gate is shown in fig.[figure1](a ) .
note that the horizontal line connecting the input and the output is not a physical wire as in classical circuits , it represents a qubit under time evolution .
similarly , a two - bit gate can be represented by a @xmath32 matrix .
for example , a _ control - not _ ( * cn * ) gate is represented by @xmath33 the symbol of a * cn * gate is shown in fig.[figure1](b ) .
a * cn * gate consists of one _ control _
bit @xmath34 , which does not change its value , and a _ target _
bit @xmath35 , which changes its value only if @xmath36 . assuming the first bit is the control bit , the gate can be written as @xmath37 , where @xmath38 denotes exclusive - or . in matrix form , a * cn * gate changes the probability amplitudes of a quantum system as follows : @xmath39 further generalization of the quantum gates described above involves _ rotation _ and _ phase shift_. they control the phase difference and relative contributions of the eigenstates to the whole state .
for example , a general single bit operation can be represented using a matrix @xmath40 this matrix can also be used to control the change between any two probability amplitude components in a quantum system . note that , to satisfy the probability rule , all quantum gates @xmath41 in their matrix form are unitary , _
i.e. _ @xmath42 where @xmath43 is the conjugate transpose of @xmath41 .
just like * and * and * not * form a universal set for classical boolean circuits , one- and two - bit gates are sufficient to implement any unitary operation @xcite , @xcite .
a set of quantum gates which can be used to implement any unitary operation is called a universal set .
there are many universal sets of one- and two - bit gates .
a practical approach is to use general one - bit rotation gates as in eq.([onebit ] ) and the * cn * gate as a universal set .
an important property regarding a quantum boolean operation is that any quantum boolean logic can be represented using a _
permutation_. a permutation is a one - to - one and onto mapping from a finite order set onto itself .
a typical permutation @xmath44 is represented using the symbol @xmath45 this permutation changes @xmath46@xmath47@xmath48 , @xmath48@xmath47@xmath46 , @xmath49@xmath47@xmath50 , @xmath50@xmath47@xmath51 , and @xmath51@xmath47@xmath49 , with state @xmath52 remaining unchanged .
a permutation can also be expressed as disjoint _ cycles_. a cycle is basically an ordered list , which is represented as : @xmath53 the order of the elements describes the operation .
for example , in eq.([cycle - rep ] ) , the cycle takes @xmath54@xmath47@xmath55 , @xmath55@xmath47@xmath56 , , @xmath57@xmath47@xmath58 , and finally @xmath58@xmath47@xmath54 .
the number of elements in a cycle is called _
length_. a cycle of length @xmath15
is called a _ trivial _ cycle , which can be ignored as it does not change anything . a cycle of length @xmath59
is called a _
transposition_. using this notation , the same permutation @xmath44 shown in eq.([permutation ] ) can be written as @xmath60 as we can see , a simple quantum boolean gate like * cn * can be regarded as a permutation , because the probability amplitudes in the quantum state are manipulated in the same way . in other words ,
a quantum boolean logic gate can be expressed as a permutation , or cycles .
for example , a * cn * gate is indicated by @xmath61 , changing @xmath62@xmath47@xmath63 and @xmath63@xmath47@xmath62 , leaving all other states unchanged .
in addition to permute the probability amplitude of each eigenstate , a qubit can be permuted as a whole .
this is equivalent to reshuffling the quantum states for each of the qubits .
since a permutation can be decomposed into disjoint cycles , the implementation actually consists of executing cycles of various lengths in parallel .
because a cycle of length @xmath15 does not permute anything , no circuit is required for a trivial cycle . for a cycle of length @xmath59
, the transposition can be done by three * cn * gates , as shown in fig.[figure2](a ) .
+ the circuit is described as follows .
for a two - qubit system @xmath64 the circuit transforms @xmath65 , @xmath66 , @xmath67 , and @xmath68 .
this is equivalent to the permutation @xmath69 assuming the state of these two unentangled qubits are @xmath70 and @xmath71 , where @xmath72 and @xmath73 , the joint state @xmath74 is transformed to @xmath75 which does the transposition .
note that once we have this basic function , we can build a switching network in the same way as a classical space switch .
however , a more efficient implementation exists , as will be presented later in this paper . for a general @xmath76-qubit ( @xmath77 ) cycle @xmath78
, it can be done by @xmath79 layers of * cn * gates without ancillary qubits @xcite .
the quantum operations required to implement @xmath80 are shown below .
for an even @xmath76 ( @xmath81 , @xmath82 ) , we define the following non - overlapping qubit transpositions as : @xmath83 the cycle can be implemented using @xmath84 on the other hand , for an odd @xmath76 ( @xmath85 , @xmath86 ) , we define the following non - overlapping qubit transpositions as : @xmath87 note that if the subscript @xmath88 then @xmath89 is used to avoid ambiguity . in the same way , the cycle can be implemented using @xmath84 two examples of @xmath90 and @xmath91 are shown in fig.[figure2](b ) .
note that both @xmath92 and @xmath93 consist of disjoint transpositions and can be executed in parallel using @xmath94 layers of * cn * gates , as shown in fig.[figure2](a ) . as a result
, each cycle and the whole permutation can be performed using @xmath79 layers of * cn * gates .
this achieves the constant time complexity of a qubit permutation .
if auxiliary qubits are used , a cycle can be implemented using only @xmath95 layers of * cn * gates @xcite .
in addition to permutation , qubit replication ( * fanout * ) is also an important and non - trivial operation .
qubit replication takes one bit as input and gives two copies of the same bit value as output . in the classical world
, we can do this simply with a metallic contact , but it is well - known that quantum mechanics does not allow us to make an exact copy of an unknown qubit .
this is called the quantum _ non - cloning _ theorem @xcite . however , if the source qubit is in either @xmath30 or @xmath5 , the quantum state can be replicated exactly using a * cn * gate .
for example , if @xmath6 is in either @xmath4 or @xmath5 , replicating @xmath96 to the qubit @xmath97 can be done simply by applying a * cn * gate with @xmath96 as the control and @xmath98 as the target , _
i.e. _ @xmath99 . moreover , since both @xmath96 and @xmath100 can be used as the source qubits for further replication processes , the number of copies will increase exponentially , which allows @xmath80 copies of the same quantum state being replicated using only @xmath101 layers of * cn * gates , as shown in fig.[figure3 ] .
note that the * cn * gates which have non - overlapping control and target qubits can be executed in parallel and are grouped into one layer .
in classical digital communication , switching is needed in order not to build a fully - meshed transmission network .
generally , digital switching technologies fall under two broad categories : _ circuit switching _ or _
packet switching_. in this section , we briefly introduce these two switching paradigms and describe various implementations that can be employed to implement the switching function .
we also define the connection digraph which can be used to illustrate the switching operation at a given time . in circuit switching ,
a dedicated path or time slot is reserved for an end - to - end bandwidth demand .
the connection is established at the time of call set - up and released when the call is torn down .
the function of the switching module is to transfer a particular time slot in the input port to a time slot in the output port . assuming a ( time slot @xmath102 of port @xmath103 ) and b ( time slot @xmath104 of port @xmath105 ) are making two - way communication via a @xmath106 digital switch , as shown in fig.[figure4](a ) . for the connection from a to b , the switching module transfers the data @xmath34 from @xmath102 of @xmath103 to @xmath104 of @xmath105 .
similarly , for the connection from b to a , it transfers the data @xmath35 from @xmath104 of @xmath105 to @xmath102 of @xmath103 .
these operations complete the data exchange between a and b. packet switching is more sophisticated than circuit switching .
modern packet switching networks take packets that share the same transmission line as input .
a packet can have either a fixed or variable length with a limited maximum size .
when a packet arrives at a node , it is stored first and then forwarded to the desired node according to its header as shown in fig.[figure4](b ) .
for example , assume each of the packets in fig.[figure4](b ) has the destination port number as indicated in the header of the packet .
the switching module at time @xmath107 needs to switch the data from input port @xmath108 , @xmath103 , @xmath105 , and @xmath109 to output port @xmath105 , @xmath109 , @xmath103 , and @xmath108 respectively .
although significant differences such as _ data dependency _ and _ output contention _ exist between circuit switching and packet switching , they still have similarities . in both circuit switching and packet switching
, the control block needs to specify the switching configuration for each individual time slot , so the data in that particular time slot can be switched correctly .
the configuration describes how the i / o ports should be switched at a given time .
the actual switching operation depends on which switching technique is used .
there are many switching techniques used today .
some of the basic switching techniques are described in the following section . in the field of classical digital switching ,
various techniques have been used to switch the input data to the corresponding output port .
for example , data can be switched in the space domain , the time domain , or the wavelength domain , etc . if the data is switched in the space domain , _
i.e. _ space division switching , usually a physical path or a dedicated time slot is reserved to establish the connection .
for example , in the crossbar architecture , a rectangular array of cross - points serve as a simple space switching architecture .
every output port can be reached by every input port in a non - blocking way by closing a single cross - point .
a more sophisticated space division switch utilizes multiple stages of rectangular arrays is shown in fig.[figure5](a ) .
a connection is established by closing proper cross - points to select a path from the inlet to the outlet @xcite . a device that switches the data in the time domain is called a time division switch .
time division technology is widely used in modern digital communication . in a time division switch
, connections are established in a time - sharing manner , so a connection occupies the resources for only a short duration of time . for example , in fig.[figure5](b ) , the connection from the inlet @xmath103 to the outlet @xmath109 is established by closing switch @xmath110 and @xmath111 .
this process is executed for each of the connections in a cyclic way to achieve switching functionality . primarily owing to the low cost of semiconductor devices , the implementation of a time division switch is usually done by using digital memory .
data received over an incoming port is written into the memory , the switching is accomplished by reading out the individual bits in the desired time slot , which is equivalent to connecting the inlet to the outlet for data transfer . before we describe how digital switching can be done in the quantum domain , we define a _ connection digraph _ as follows : + * definition 1 : * given an @xmath112 switch , the connection digraph at time @xmath113 , @xmath114 , is a digraph such that 1 .
each @xmath115 represents an i / o port .
2 . @xmath116 if and only if a connection exists from the input port @xmath117 to the output port @xmath118 at time @xmath113 . in a connection digraph
, each node represents an
i / o port , a directed edge @xmath119 is used to describe a connection when the connection from input port @xmath117 to output port @xmath118 is active .
the digraph describes the connection status of the switch at a given time , and is called the connection digraph at time @xmath113 .
note that the directed edge @xmath119 denotes only a one - way data path . for a point - to - point two - way communication between @xmath117 and @xmath118
, both @xmath119 and @xmath120 have to be used . obviously , due to the connection set - up and torn - down processes , the connection digraph is a function of time .
depending on the status of the switch , the topology of a connection digraph varies . in a general digraph
, it is possible that a node has multiple predecessors and multiple successors .
however , when there is no output contention or the problem is solved elsewhere , each node will have at most one predecessor . as to the number of successors , it depends on the type of the connection . in a multicast connection ,
the source node has multiple successors , while in a unicast connection , only a single successor is possible . in the following sections
, we will discuss the connection digraph based on this model and show that any connection digraph actually consists of a set of basic topologies as disjoint sub - digraphs .
these basic topologies are defined as follows : + * definition 2 : * given a digraph @xmath121 with only one node , _
i.e. _ @xmath122 .
@xmath123 is called a _ null node _ if @xmath124 .
otherwise @xmath123 is called a _ loopback _ when @xmath125 .
+ in a connection digraph , a null node without predecessor and successor means there is neither input traffic coming from that port nor output traffic going to that port . for a port without incoming traffic , we assume the stuff bits are all @xmath126 s
. however , a single node with a directed edge to itself means the input traffic goes back to the same port .
this trivial cycle effectively denotes a loopback .
a loopback @xmath127 can be made from a null node @xmath128 simply by linking the null node to itself .
@xmath127 is called the extension loopback of @xmath128 , denoted by @xmath129 .
an example consists of null nodes and loopbacks is shown in fig.[figure6](a ) .
the numbers in the boxes represents the destination port numbers .
an x represents no input traffic .
its corresponding connection digraph is depicted in fig.[figure6](b )
. + * definition 3 : * given a connected digraph @xmath121 with @xmath76 ( @xmath130 ) nodes .
@xmath123 is called a _ queue _ if 1 .
there exists one and only one _ head _ @xmath131 , such that for each @xmath132 , @xmath133 .
there exists one and only one _ tail _
@xmath134 , such that for each @xmath132 , @xmath135 .
3 . for each @xmath136
, there exists one and only one @xmath137 , such that @xmath138 .
+ a queue can be represented as a linear array from the head @xmath139 to the tail @xmath140 , and is denoted as @xmath141 $ ] .
this notation means the connection at a given time includes @xmath142 , @xmath143 , , and @xmath144 .
note that there is no input traffic coming from @xmath140 and no output traffic going to @xmath139 .
an example of a queue connection is shown in fig.[figure7](a ) , with its connection digraph @xmath145 $ ] depicted in fig.[figure7](b ) .
each connection in a queue is apparently a unicast connection , because there is at most one outgoing arrow from each node .
connecting the tail to the head of a queue forms a _ cycle _ , which is defined as follows : + * definition 4 : * given a connected digraph @xmath121 with @xmath76 ( @xmath130 ) nodes , @xmath123 is called a cycle if 1 . for each @xmath132 , there exists one and only one @xmath137 , such that @xmath146 .
2 . for each @xmath132
, there exists one and only one @xmath147 , such that @xmath148 . + using the same notation ,
a cycle connection is represented as @xmath149 .
this means the connection at a given time includes @xmath150 , @xmath143 ,
, @xmath151 , and @xmath152 . in the case of a cycle ,
each port has its input as well as output .
as described earlier , the tail and head of a queue @xmath153 can be connected to form a cycle @xmath154 .
@xmath154 is called the extension cycle of @xmath153 , denoted by @xmath155 .
an example of a cycle connection is shown in fig.[figure8](a ) , with its connection digraph @xmath156 depicted in fig.[figure8](b ) . in order to describe a multicast connection ,
we define the following connection digraphs : + * definition 5 : * given a connected digraph @xmath121 with @xmath76 ( @xmath130 ) nodes , @xmath123 is called a _ tree _ if 1 .
there exists one and only one _ root _ @xmath157 , such that for each @xmath132 , @xmath158 .
2 . there exists a collection of nodes @xmath159 called leaves , such that for each @xmath160 and @xmath132 , @xmath161 .
3 . for each @xmath162
, there exists at least one @xmath137 , such that @xmath138 .
+ the nodes in a tree can be divided into three categories : root , internal nodes , and leaves . for the root
, the output data is directed to possibly multiple output ports , but no data goes to the root .
however , all leaves receive data without generating traffic .
all internal nodes have exactly one predecessor and at least one successor .
a tree can be represented as a concatenation of queues like @xmath163[v^1_h , \ldots v^1_t]\ldots[v^n_h , \ldots v^n_t]$ ] , with @xmath164 be the root and each of the @xmath165 ( @xmath166 ) be the tail of one of the previous queues .
an example of a tree connection is shown in fig.[figure9](a ) .
if there are multiple numbers in a box , they represent a multicast connection .
its corresponding connection digraph @xmath167[p1,p3][p1,p6,p4][p3,p5][p3,p7][p4,p0][p4,p2]$ ] is depicted in fig.[figure9](b ) .
note that a queue is a special case of trees , with each node having only one successor .
connecting any leaf to the root of a tree forms a _ forest _ , which is defined as follows : + * definition 6 : * given a connected digraph @xmath121 with @xmath76 nodes @xmath168 , @xmath123 is called a forest if 1 .
there is one and only one cycle @xmath169 exists as a sub - digraph of @xmath123 .
2 . let @xmath170}. @xmath171 contains the cycle @xmath154 and a collection of disjointed null nodes , queues , and/or trees .
each @xmath137 is either one of the null nodes , the head of a queue , or the root of a tree in @xmath171 .
+ a forest basically contains one and only one cycle @xmath169 as a sub - digraph , with some of its nodes linked to either a null node , the head of a queue , or the root of a tree . following this structure ,
a forest can be represented by @xmath172 , where @xmath173 , @xmath174 , @xmath175 , be either a null node , a queue , or a tree .
a forest can be extended from a tree by connecting any leaf to the root .
a forest @xmath176 formed by connecting the leaf @xmath177 with the root of @xmath178 is called the extension forest of @xmath178 , denoted by @xmath179 .
an example of a forest connection is shown in fig.[figure10](a ) , with its connection digraph @xmath180[p3,p5][p3,p7],[p4][p4,p2][p4,p6]\}$ ] depicted in fig.[figure10](b ) .
since each node in a unicast connection has at most one successor , a unicast connection digraph only consists of disjoint null nodes , loopbacks , queues , and/or cycles as sub - digraphs .
however , a multicast connection switches the data from one node to multiple successors , so a multicast connection digraph consists of disjoint null nodes , loopbacks , queues , cycles , trees , and/or forests as sub - digraphs . based on these results ,
we describe the architecture of quantum switching and show how it can be used to implement a connection digraph in the next section .
the proposed architecture for building a digital _ quantum switch _ is depicted in fig.[figure11 ] . to switch classical digital data in the quantum domain
, first we have to convert the classical data into qubits .
for example , in a quantum switch with optical i / o ports , an optical to quantum converter ( o / q ) is used to convert optical input into qubits . in an o / q , @xmath126 is converted into @xmath4 and @xmath15 is converted into @xmath5 .
this can be done by exciting an electron using a light pulse of a certain frequency .
all qubits are then permuted ( _ i.e. _ switched ) by the unitary operations under the supervision of the control subsystem .
after the permutation , all qubits are converted back into their optical form by a quantum to optical converter ( q / o ) .
this can be done by measuring the qubits to recover the original classical information . in this section ,
we show how a connection digraph can be implemented using * cn * gates .
first we describe the connection digraph transformation guideline , then we demonstrate how this guideline can be used to implement a connection digraph .
both unicasting and multicasting will be covered in detail .
as described earlier , due to the nature of the connection , unicasting and multicasting have different connection digraphs .
the digraph of a unicast connection has a collection of disjointed null nodes , loopbacks , queues , and/or cycles as sub - digraphs .
however , in the digraph of a multicast connection , sub - digraphs like trees and forests are possible . as a matter of fact , these topologies are inter - related .
this is shown in fig.[figure12 ] and summarized as follows : 1 .
a null node can be regarded as a special case of a queue , denoted by the arrow s1 .
a queue can be regarded as a special case of a tree , denoted by the arrow s2 .
a loopback can be regarded as a special case of a cycle , denoted by the arrow s3 .
a cycle can be regarded as a special case of a forest , denoted by the arrow s4 .
of course , the binary relation `` _ is a special case of _ '' is transitive , so a null node and a loopback are special cases of tree and forest respectively .
fig.[figure12 ] also shows the binary relation `` _ can be extended to _ '' as follows : 1 .
a null node @xmath128 can be extended to a loopback @xmath181 , denoted by the process e1 .
a queue @xmath153 can be extended to a cycle @xmath182 , denoted by the process e2 .
a tree @xmath178 can be extended to a forest @xmath183 , denoted by the process e3 .
note that the process of extension only transfers the incoming data from an idle inlet ( all @xmath126 s ) to an outlet which has no outgoing traffic , this does not change the switching function .
the first step of our guideline for implementing a connection digraph is to transform each disjointed sub - digraph into loopbacks and/or cycles .
since no circuit is needed to implement a loopback and only @xmath79 layers of * cn * gates are sufficient to implement a cycle , the switching can be done efficiently . some of these transformations are straightforward .
for example , following e1 , a null node @xmath128 can be extended to a loopback @xmath181 . also ,
following e2 , a queue @xmath153 can be extended to a cycle @xmath182 . however , for a tree or a forest , `` _ cycle extraction _ '' and `` _ link recovery _ '' have to be used . the process of cycle extraction and link recovery are described as follows .
* cycle extraction * : a forest basically contains one and only one cycle @xmath169 as a sub - digraph with a subset of @xmath184 linked to either a null node , the head of a queue , or the root of a tree
. the process of cycle extraction detaches all the null nodes , queues , and trees from the cycle by cutting all the edges in @xmath185 , as shown in fig.[figure13](a ) .
this will transform a forest into one cycle ( arrow t1 in fig.[figure12 ] ) and a collection of null nodes , queues , and/or trees ( arrow t2 , t3 , and t4 respectively ) .
each of the null nodes and queues can further be transformed into loopbacks and cycles via process e1 and e2 .
if there are still any trees in the remaining digraph , extensions can be made again to transform the trees into forests ( process e3 ) and the procedure of cycle extraction can be applied recursively ( arrow t1 , t2 , t3 , and t4 ) until no trees are left .
this procedure eventually transforms a forest into loopbacks and/or cycles , so that the permutation can be implemented using @xmath79 layers of * cn * gates in parallel . * link recovery * :
after each cycle has been implemented , the links that had been cut must be recovered .
that is , for each @xmath186 , if @xmath187 but @xmath188 , @xmath137 must be replicated to @xmath147 , as shown in fig.[figure13](b ) .
since there will be at most @xmath189 such @xmath190 s in a multicast connection digraph , in the worst case the replication can be done by @xmath191 layers of * cn * gate .
this completes the implementation of a forest . for a tree
, it can be extended to a forest via process e3 and then follow the algorithm to do further reduction in the same way . following the guideline described above , in this section
we show how a unicast connection digraph can be implemented with a time complexity of @xmath2 and a space complexity of @xmath1 .
a typical unicast connection status at a given time is shown by the solid arrows in fig.[figure14](a ) .
the switching module needs to perform two connection sub - digraphs : @xmath192.\end{aligned}\ ] ] these can be done by first extending @xmath153 to @xmath193 , as shown by the dash link in fig.[figure14](a ) , and then implement @xmath154 and @xmath194 using @xmath79 layers of * cn * gates .
as described previously , the sub - digraph @xmath195 can be done by first applying @xmath196 and then @xmath197 the transposition @xmath198 is done by @xmath199 as shown by block b in fig.[figure14](b ) . in the same way ,
@xmath200 , @xmath201 , @xmath202 are done by blocks c , e , f respectively .
similarly , the implementation of @xmath193 can be done by first applying @xmath203 and then @xmath204 . these are implemented as blocks a and d in fig.[figure14](b ) .
note that , independent of the switch size @xmath76 , the whole circuit can be completed in @xmath79 layers of * cn * gates over @xmath76 qubits .
this achieves a time complexity of @xmath2 and a space complexity of @xmath1 . in classical packet switching ,
the input packets are usually buffered in the memory , multicasting can be easily achieved by reading the packet once and writing the same packet to multiple destinations . if the switching is done in the quantum domain , multicasting can be done by replicating the input qubit to multiple destination qubits .
a typical multicasting configuration is shown in fig.[figure15](a ) . in this example
, the switching module needs to perform the following connection digraph : @xmath205[q_1,q_4][q_1,q_3][q_3,q_5,q_2][q_3,q_6,q_7].\ ] ] following the guideline , each of the steps is shown below : 1 .
the tree @xmath178 can be extended to a forest by linking any leaf , say @xmath206 , to @xmath207 .
the cycle extraction procedure is then performed to cut @xmath208 and @xmath209 down .
the result is shown in fig.[figure15](b ) .
the extension and cycle extraction processes are recursively applied to @xmath210 $ ] until no tree is left , as shown in fig.[figure15](c ) .
3 . each of the disjointed sub - digraphs can be implemented in parallel .
the sub - digraph @xmath211 can be done by first applying @xmath212 and then @xmath213 , while @xmath214 can be implemented directly , as shown by blocks a , b , d , e , and c in fig.[figure16 ] .
4 . each of the disconnected edges has to be recovered , so @xmath215 needs to be replicated to @xmath216 , and @xmath217 needs to be replicated to @xmath218 , as shown in fig.[figure15](d ) .
these can be done by blocks f and g in fig.[figure16 ] .
in general , the total number of layers for implementing a multicast connection digraph is @xmath219 , where @xmath220 is the maximum number of @xmath221 ( @xmath222 ) that are to be recovered . in the worst case , when one inlet is broadcast to all other @xmath223 outlets , the whole connection digraph can be done in @xmath3 layers of * cn * gates over @xmath76 qubits
this results in a time complexity of @xmath3 and a space complexity of @xmath1 .
the advantages of performing digital switching in the quantum domain are summarized as follows .
first , switching in the quantum domain is _ strict - sense non - blocking_. a switch is called strict - sense non - blocking if the network can always connect each idle inlet to an arbitrary idle outlet independent of the current network permutation @xcite .
note that switching in the space domain is not always non - blocking .
sometimes , the required data path can not be established even if the output port is available .
it has been shown that for an @xmath112 network in fig.[figure5](a ) ) to be non - blocking , there must be at least @xmath224 modules in the middle stage @xcite .
however , switching in the quantum domain is actually a unitary transformation , which is always possible .
this results in the fact that a quantum switch is non - blocking in the strict sense .
second , it takes only @xmath76 qubits to build a quantum switch , the space complexity is @xmath1 in terms of the number of qubits .
the problem of space complexity is an important issue in the classical space switching . to make a classical space switch non - blocking , a certain number of modules in the middle stage have to be used to allocate a physical path for each connection , so the number of cross - points increases with the size of the switch .
for example , with optimal grouping , the minimum number of cross - points for the switch shown in fig.[figure5](a ) is @xmath225 , where @xmath76 is the total number of inlets / outlets @xcite .
however , an @xmath112 quantum switch uses only @xmath76 qubits as the basis to perform the switching , which is a reasonable resource consumption .
third , quantum switching is scalable in terms of time complexity . in a classical time switch ,
usually the bottleneck is the speed of the switching device . because when the throughput increases , the time duration for switching a particular bit of data decreases .
for example , in a memory switch with throughput @xmath226 , the memory speed must be at least @xmath227 to allow one read and one write operation to be performed .
however , in the quantum switching , the time complexity is not sensitive to the throughput .
a high throughput quantum switch can be achieved simply by increasing the number of i / o ports , which only induces a reasonable amount ( @xmath1 ) of space consumption . however , even in the worst case scenario , the throughput gain still outweights the time penalty in a classical time domain switch ( @xmath1 v.s .
@xmath3 ) .
networks are rapidly growing due to increased number of users and rising demands for bandwidth - intensive services . to support such a huge traffic volume , a wide range of different technologies
are being proposed as the core of a high performance switch . in this paper , an architecture of digital quantum switching
is presented .
the proposed mechanism allows digital data to be switched using a series of quantum operations .
the procedures of how to implement unicast and multicast connections are discussed in detail . in terms of the blocking rate , this architecture is strict - sense non - blocking . from a complexity point of view , the space complexity grows only linearly with the number of i / o ports , and the time complexity is constant for unicasting and logarithmic for multicasting .
this architecture is suitable for deploying high throughput switching devices so that high bandwidth demand can be met .
99 p. benioff , `` the computer as a physical system : a microscopic quantum mechanical hamiltonian model of computers as represented by turing machines , '' _ j. stat .
22(5 ) , pp . 563 - 591 , 1980 . c. bennett and g. brassard , ``
quantum cryptography : public key distribution and coin tossing , '' in _ proc . of ieee international conference on computers systems and
signal processing _
, 1984 , pp .
175 - 179 .
a. barenco , `` a universal two - bit gate for quantum computation , '' _ proc .
a _ , vol .
679 - 683 , 1995 . c. moore and m. nilsson .
( 1998 , aug . ) .
parallel quantum computation and quantum codes .
[ online ] .
available : http://www.arxive.org / quant - ph/9808027/.
|
in this paper , we present an architecture and implementation algorithm such that digital data can be switched in the quantum domain . first we define the connection digraph which can be used to describe the behavior of a switch at a given time , then we show how a connection digraph can be implemented using elementary quantum gates .
the proposed mechanism supports unicasting as well as multicasting , and is strict - sense non - blocking .
it can be applied to perform either circuit switching or packet switching .
compared with a traditional space or time domain switch , the proposed switching mechanism is more scalable . assuming an @xmath0 quantum switch
, the space consumption grows linearly , _
i.e. _ @xmath1 , while the time complexity is @xmath2 for unicasting , and @xmath3 for multicasting . based on these advantages ,
a high throughput switching device can be built simply by increasing the number of i / o ports .
|
symmetry - protected topological ( spt ) phases have attracted enormous research interest recently .
@xcite among the interesting models exhibiting spt order , a remarkable example is the spin-1 chain .
the generic spin-1 bilinear - biquadratic ( blbq ) model can be written down as @xmath5 , \label{eq - bbq}\ ] ] where the coupling @xmath6 sets the energy scale , and @xmath7 is a tunable parameter .
the phase diagram of the spin-1 blbq model with respect to various @xmath7 s is well known ( except for a subtlety in the thin region near @xmath8).@xcite when @xmath9 , the system is in the haldane phase .
@xcite although this phase has been intensively studied , it has been realized to be an spt phase only very recently.@xcite at @xmath10 , an exactly solvable point within the haldane phase , the ground state is termed aklt state,@xcite which can be exactly expressed as a matrix product state ( mps ) with bond dimension @xmath11 .
the haldane phase has a nonzero spin gap , called haldane gap , which can be interpreted in terms of spinon confinement . @xcite
no local order parameter can be found to distinguish the haldane phase from a trivial gapped phase , nevertheless , there exists a nonlocal string order parameter ( sop ) , @xcite @xmath12 , \label{eq - sop}\ ] ] where @xmath13 .
this string order parameter characterizes the topological order in the haldane phase .
further studies show that the string order parameter @xmath14 can be transformed to two ordinary ferromagnetic order parameters through a nonlocal unitary transformation @xmath15 .
therefore , a nonzero string order parameter actually reveals a hidden @xmath16 symmetry breaking .
@xcite .
( b ) two - leg spin ladder model with @xmath17 and @xmath18 for couplings along chain and rung directions , respectively .
( c ) three - leg spin tube model , @xmath17 is the coupling along the leg .
each isosceles triangle contains two kinds of couplings , @xmath18 for the two equal sides and @xmath19 for the third .
( d ) the su(2)-invariant matrix product state describing ground state of spin-1 chain ( a ) , ladder ( b ) or tube ( c ) .
@xmath20 represents a multiplet with quantum number @xmath21 . for the spin-1 model ,
each local space is a @xmath22 triplet .
the input bond multiplets on both open ends can be controlled , and three common choices are shown in ( d ) .
( e ) shows the phase diagram of the spin-1 blbq chain,@xcite h , c , fm , and d stand for haldane , critical , ferro - magnetic , and dimerized phases , respectively .
there is a narrow region near @xmath23 with possible spin nematic order , whose existence still remains debatable . ] the haldane phase is protected by several global symmetries . according to the valence bond solid ( vbs ) picture ,
the gapped haldane phase only possesses short - range entanglement , hence it is not an intrinsic topological phase.@xcite its nontrivial topological properties are protected by parity symmetry , time reversal symmetry , and @xmath16 rotational symmetry around the @xmath24 and @xmath25 axes.@xcite the haldane phase can not be adiabatically connected to the trivial one as long as one of the above symmetries is preserved along the path ; instead , a quantum phase transition ( qpt ) must occur along the way .
as is well known , the landau paradigm classifies the various symmetry - breaking phases according to symmetry groups.@xcite nevertheless , the existence of a qpt between spt phases and trivial ones shows that gapped phases without symmetry breaking in one dimension ( and also in higher dimensions ) can still be distinct and classified by the group cohomology of symmetries.@xcite to be specific , we consider the gapped phases in so(3 ) heisenberg chains , which can be generally classified by different projective representations of the rotational so(3 ) group , i.e. , the corresponding group cohomology @xmath26 .
( even ) representations of su(2 ) are linear representations of so(3 ) ; half - integer ( odd ) representation , which involve an additional minus sign after @xmath27 rotations ( owing to the su(2 ) double covering , so(3 ) = su(2)/@xmath28 ) are projective representations of so(3 ) . based on this observation , the classification theory states
: there are two distinct gapped phases in spin-1 chains corresponding to two different kinds of representations of so(3 ) , linear and projective .
they correspond to the trivial phase and the haldane phase , respectively .
@xcite it has recently been discovered that these two phases also differ strikingly in the structure of their entanglement spectra .
the entanglement spectrum consists of the eigenvalues of the entanglement hamiltonian @xmath29 , where @xmath30 is the reduced density matrix of a subsystem.@xcite the entanglement spectrum of the bulk has an intimate relationship with the real excitation spectrum on the boundary.@xcite closely related with the group cohomology classification , an interesting feature has been found : for the spin-1 chain , the entanglement spectrum is found to show at least two - fold degeneracy for the haldane phase , while it is generally non - degenerate for the trivial phase.@xcite the occurrence of the two - fold degeneracy can be used to numerically identify the haldane phase . actually , this degeneracy in the entanglement spectrum is a signature of the appearance of half - integer - spin multiplets in the mps geometric bond , which support projective representations of the so(3 ) group .
take the aklt point as an intuitive example : according to the construction of the aklt state , each local spin-1 is decomposed into two spin-1/2 ancillas .
the aklt state can be exactly expressed as mps with bond dimension 2 , hence only one @xmath31 doublet appears on each of its geometric bonds , and the entanglement spectrum is two - fold degenerate .
for other generic states in the haldane phase , the multiplets on the geometric bonds are generally @xmath32=half - integer , which leads to at least two - fold degeneracy and supports projective representations .
this key feature can be used to differentiate the spt phase from the trivial one , the latter instead has integer bond multiplets that support linear representations . therefore , if we could _ directly identify the virtual spins _ on the geometric bonds of the mps , it would be straightforward to see whether the representation is projective or linear , and thus to identify the spt or trivial phase .
one powerful numerical method for solving 1d quantum spin models is the density matrix renormalization group ( dmrg).@xcite in order to further improve its efficiency and stability , abelian and non - abelian symmetries have been implemented in the dmrg algorithm.@xcite in particular , the su(2 ) dmrg technique enables us to identify the spin of the multiplets on the virtual bonds .
note , though , that if open boundary conditions are adopted for su(2 ) dmrg , because only integer - spin sectors are allowed by adding spin-1 s together , the renormalized bases on the virtual bonds are automatically integer multiplets , i.e. , linear representations of so(3 ) .
this would imply the absence of two - fold degeneracy within each multiplet ( every multiplet contains odd number of individual states ) even in haldane phase , which seems paradoxical . to solve this problem
, we propose a protocol algorithm in this paper which automatically determines the proper bond representations .
in addition , by defining and calculating the integer and half - integer entanglement entropies , we elucidate why this protocol algorithm works , and obtain a simple criterion for identifying the spt phase .
we test these ideas in three spin-1 lattice models , and show that they succeed in telling the spt phase from the trivial one .
the paper is organized as follows . in sec .
ii , the mps and related dmrg algorithms are briefly introduced . in secs .
iii - v , we show that the entanglement entropies can be used to identify the spt phase , by studying three examples including the single spin-1 chain , 2-leg ladder and 3-leg tube models . in particular , in the spin-1 tube model
, the transition between the spt and trivial phases is verified to be a continuous qpt .
sec vi offers a summary .
are put on the end bonds , and the converged spectra are obtained after several dmrg sweeps .
@xmath33 ( asterisks ) is in the haldane phase , with half - integer - spin bond multiplets ; @xmath34 ( circles ) is in the trivial phase , with integer - spin bond multiplets .
every data point represents a multiplet ( not as usual a single state within a multiplet ) .
therefore , a multiplet with symmetry label @xmath32 corresponds to @xmath35 degenerate states .
( b ) multiplet spectrum of integer and half - integer representations for the haldane phase calculated for @xmath33 , and using @xmath31 ( asterisks ) or @xmath36 ( circles ) respectively , on the end bonds .
the half - integer spectrum has been shifted by @xmath37 , in order to reveal the one - to - one correspondence between each multiplet in the half - integer spectrum and a pair of degenerate multiplets in the integer spectrum . ]
the variational mps ground state of 1d heisenberg systems with hamiltonians like eq .
( [ eq - bbq ] ) can be written in an su(2)-invariant form .
corresponding bond spaces are factorized into two parts,@xcite @xmath38 where @xmath39 ( and @xmath40 , @xmath41 ) are composite multiplet indices .
@xmath42 specifies the symmetry sector , @xmath43 distinguishes different multiplets with the same @xmath42 , and @xmath44 ( @xmath45 , @xmath46 ) labels the individual states within a given multiplet in symmetry sector @xmath42 ( @xmath47 , @xmath48 ) .
the @xmath49-tensors can be regarded as physical tensors which combine the input multiplets @xmath50 with the local space @xmath51 , and transform ( and possibly truncate ) them into the output multiplets @xmath52 ; the @xmath53-tensors are the clebsch - gordan coefficients ( cgc ) which take care of the underlying mathematical symmetry structure .
the tensor product of physical tensor @xmath49 ( reduced multiplet space ) and its related mathematical tensor @xmath53 ( cgc space ) has been called the qspace,@xcite which is a generic representation used in practice to describe all symmetry - related tensors .
@xcite the qspace is a very useful concept not only for mps wavefunctions , but also for calculating the matrix elements of irreducible tensor operators , which can be treated in the same framework according to the wigner - eckart theorem . by implementing the qspace in our dmrg code , we need to determine only the physical @xmath49-tensors variationally as in plain dmrg , while the underlying cgc space ( @xmath53-tensors ) are fully determined by symmetry .
the @xmath49-tensors manipulate multiplets @xmath50 only on the reduced multiplet level , which leads to a large gain in numerical efficiency . in this work , by adopting the su(2)-invariant mps , we are able to keep track of the quantum numbers @xmath32 of the bond multiplets , and hence to distinguish the spt phase and the trivial phase straightforwardly . given an su(2)-invariant mps , it is natural to consider its _ multiplet entanglement spectrum _ , defined of multiplets , rather than individual states . to be explicit
, we note that any su(2)-invariant mps can be written in the following form : @xmath54 |q_1^z ... q_l^z \rangle .
\notag \\ \label{eq - su2-mps}\end{aligned}\ ] ] the trace includes all the quantum labels @xmath55 , while @xmath56 all equal 1 in the present spin-1 case .
( [ eq - su2-mps ] ) is an su(2)-invariant version of eq .
( 4 ) in ref . .
the difference is that the conventional mps matrix @xmath49 is represented in the factorized form of a direct product , i.e. , @xmath57 . in eq .
[ eq - su2-mps ] above , we have assumed the canonical mps forms in both the reduced multiplet space and the cgc space .
notice that since the @xmath53-matrices store cgc s , they automatically fulfill the left- and right - canonical conditions .
therefore , the diagonal @xmath58-matrices are identity matrices , and nontrivial diagonal matrices @xmath59 exist only on the multiplet level .
their eigenvalues @xmath60 determine the multiplet entanglement spectrum defined as @xmath61 where @xmath62 is the reduced - density - matrix eigenvalue corresponding to each multiplet . in order to illustrate the above concepts ,
let us now consider the spin-1 blbq model on a single chain [ see fig .
[ fig - sketch ] ( a ) for the lattice geometry and ( d ) for corresponding mps ] .
we use generalized boundary conditions on both ends of the mps , in that the left ( right ) input bases of @xmath63 ( @xmath64 ) can be specified as desired .
the most natural choice in dmrg is to take the input basis to be a singlet @xmath36 , as usually done for open boundary conditions . in that case , however , the spin quantum number @xmath32 of the virtual bond multiplets would automatically be integer , as only integer @xmath32 results when adding two integer spins together . for this reason , su(2 )
dmrg calculations with conventional open boundary conditions will never yield the half - integer bond ( projective ) representation of the so(3 ) symmetry , but always a trivial " state without the expected ( at least ) two - fold degeneracy in each bond multiplet expected for the haldane phase . on the other hand
, the boundary can also be set up by taking both end bonds to be @xmath31 doublets , instead of singlets @xmath36.@xcite since then only half - integer multiplets appear in the virtual bonds , this always yields an spt " state possessing doubly degenerate entanglement spectrum . in particular , for the spin-1 chain of hamiltonian eq .
( [ eq - bbq ] ) , this choice of boundary condition produces an spt " state for any @xmath65 $ ] .
however , this seemingly contradicts the well - established fact that haldane phase is confined to @xmath66 . in order to resolve this apparent paradox
, we here also study a more general situation , where we input the direct sum @xmath67 on the two boundary bonds .
this gives rise to the possibility of both integer and half - integer multiplets on the bonds , and allows us to do actually parallel dmrg calculations in two independent symmetry sections , i.e. , integer and half - integer bond spaces .
we thus adopt the following _ protocol algorithm for determining the bond representations _ : we input both integer and half - integer multiplets on the boundary virtual bonds , and perform several dmrg sweeps back and forth . in the presence of state space truncation along the bonds , depending on the hamiltonian parameters , the system will eventually converge to the half - integer projective representation or the integer linear representation of so(3 ) , thus telling the spt phase from a trivial one .
two typical multiplet entanglement spectra " selected through dmrg sweeps , and calculated using @xmath67 boundary states , are shown in fig .
[ fig - ent - spec ] ( a ) .
here each data point represents a multiplet , in contrast to the traditional state entanglement spectrum , where each data point corresponds to an individual state .
@xmath33 corresponds to the conventional heisenberg model , whose ground state belongs to the haldane phase .
the converged multiplet spectrum obtained is shown using asterisks : all points in the spectrum correspond to half - integer quantum numbers @xmath32 , and each asterisk with quantum number @xmath32 represents @xmath35 ( an even number ) degenerate u(1 ) states , as expected for an spt phase .
on the other hand , the system with @xmath34 is in the dimerized phase , a trivial gapped phase .
its su(2 ) multiplet spectrum is plotted using open circles in fig .
[ fig - ent - spec ] ( a ) .
in contrast to the @xmath33 case , the circles are all located at integer @xmath32 , as expected for a trivial ( non - spt ) phase . in the protocol algorithm , where @xmath68 is used as auxiliary boundary state
, dmrg allows the correct " bond representation to be found , as long as the system is not very close to the phase transition point . in the following , in order to compare the multiplet spectra between the integer and half - integer representations , we now change strategy and enforce the representation by specifying one of the two boundary state types on both ends of the chain , i.e. , @xmath36 ( @xmath69 ) for integer ( half - integer ) representation . in fig .
[ fig - ent - spec ] ( b ) , we choose @xmath33 ( corresponding for the haldane phase ) , and compare the multiplet entanglement spectra @xmath70 ( circle ) and @xmath71 ( asterisks ) , which are obtained by enforcing either integer or half - integer representations , respectively .
the integer - spin multiplet spectrum evidently displays a two - fold degeneracy , whereas the half - integer - spin multiplet spectrum does not .
instead , we observe a one - to - one correspondence between each multiplet in @xmath72 and a pair of degenerate multiplets in @xmath73 .
the shift value @xmath37 is chosen because the two representations have different numbers of states with nonzero weights in their reduced density matrices .
the nonzero individual states in the integer representation are twice as many as those in the half - integer one .
this different behavior of the degeneracies in the integer and half - integer multiplet entanglement spectra can be understood as follows : in the presence of space inversion symmetry , time reversal symmetry , or some @xmath16 rotational symmetry , etc . , which protects the haldane phase , it has been proven that @xmath74 has an even degeneracy of at least 2 .
@xcite therefore in the haldane phase , either @xmath75 or @xmath76 should have even degeneracy .
for the half - integer bond representation , the @xmath77 s are half - integer and therefore the @xmath76 s are identity matrices with an even number of diagonal elements , implying that an even degeneracy appears in the cgc space ; thus the @xmath75 in the reduced multiplet space is not necessarily two - fold degenerate , which explains the absence of degeneracies in the multiplet spectrum @xmath78 ( asterisks ) . on the other hand , for integer bond representations , the @xmath76 s are identity matrices of odd - number rank , therefore an even degeneracy must instead appear on the multiplet level , which explains the two - fold degeneracy obtained in @xmath79 ( circles ) .
this difference between integer and half - integer representations has an important consequence in the entanglement entropy , which will be discussed in the next section . to summarize , the lesson learnt from fig .
[ fig - ent - spec ] is as follows . in fig .
[ fig - ent - spec ] ( a ) we showed that , if mixed boundary @xmath68 is adopted , dmrg sweep can select the half - integer - spin representation in the haldane phase and integer - spin representation in the trivial phase .
[ fig - ent - spec ] ( b ) illustrates that if one studies the haldane phase using auxiliary spin @xmath36 or @xmath31 on the external bond , respectively , then the general requirement of having an entanglement spectrum of even degeneracy is satisfied by having the multiplet spectrum being degenerate or non - degenerate for the case of integer - spin or half - integer - spin representation , respectively . and @xmath1 , of the spin-1 blbq model , for @xmath80 ( dash - dotted lines ) and @xmath81 ( solid lines ) .
results for different system sizes coincide for @xmath7-values far from the critical point at @xmath82 ( vertical dash - dotted line ) , but differ in the intermediate region between the two vertical dashed lines .
@xmath0 and @xmath1 cross at a
pseudo - transition " point @xmath83 , which moves towards the critical point as the system size is increased ( shown in panel ( d ) , the extrapolated point is very close to the true critical one ) . in the above calculations ,
400 multiplets ( about 1600 individual states ) have been retained , the truncation errors are of the order @xmath84 around critical point , and are negligible ( @xmath85 or less ) for the rest parameters .
the entanglement entropies are evaluated at the center of the chain .
panel ( a ) also shows the entanglement entropy obtained by the itebd algorithm@xcite ( asterisks ) , which favors the minimally entangled states , and always follows the lower entanglement entropies .
( b ) the entanglement gap @xmath86 , which equals @xmath87 in the spt phase and the trivial phase respectively .
the dashed vertical lines mark the intermediate region , where @xmath88 is not a constant owing to finite size effects .
panel ( c ) shows the string order parameter ( sop @xmath89 ) of eq .
( [ eq - sop ] ) , obtained by itebd calculations , which retain up to 200 states . ]
during the dmrg sweeps in the protocol using @xmath68 as boundary , as long as the doublet @xmath31 is not physically coupled to the bulk ( the coupling strength between the auxiliary boundary spin-1/2 and the spin-1 chain can be set to be very weak or even turned off ) , the integer and half - integer symmetry sectors have exactly the same ground - state energy .
therefore , the energy is irrelevant in selecting the symmetry sector in the protocol algorithm . instead , since the two - site update scheme of dmrg is adopted during the sweeps , the truncation and hence the entanglement entropy is important in selecting the symmetry sector .
in order to uncover this mechanism underlying the protocol algorithm , we now study the bipartite entanglement entropies in the integer and half - integer symmetry sectors , respectively , by enforcing different boundary states .
the entanglement entropies are defined as @xmath90 , \label{eq - ent - ent}\ ] ] where @xmath91 or @xmath92 for half - integer or integer representations , and the difference @xmath93 will be called the entanglement gap " . in eq .
( [ eq - ent - ent ] ) , @xmath94 is the reduced density matrix on the multiplet level .
it is block - diagonal , with blocks @xmath95 labeled by @xmath42 and matrix elements @xmath96 .
@xmath97 is an identity matrix , with matrix elements @xmath98 whose trace thus equals the inner dimension of each multiplet .. consequently , @xmath99 $ ] refers to the trace over both , the multiplet index @xmath43 as well as as the internal multiplet space @xmath44 of a given symmetry sector @xmath42 .
note that the logarithm to base 2 ( @xmath100 ) is adopted in evaluating the entanglement entropy . @xmath94 and @xmath97 are readily obtained from dmrg simulations .
we note the su(2 ) multiplet language used to formulate eq .
( 6 ) for the von neumann entropy can easily be applied to also calculate the renyi entropy .
very recently , the latter has been employed to study the local differential convertibility and thereby probe the spt phase.@xcite though we here focus only on the von neumann entropy , our analysis can be be generalized straightforwardly to the renyi entropy . in fig .
[ fig - chain ] , @xmath0 and @xmath1 of the spin-1 blbq chain [ eq . ( [ eq - bbq ] ) ] are plotted in ( a ) , and the entanglement gap @xmath88 is shown in ( b ) . for the haldane phase ( @xmath101 in fig .
[ fig - chain ] ) we find @xmath102 , thus the half - integer bond representation has lower entanglement than the integer one , although the ground - state energies in both representations are the same . on the other hand , for the dimerized phase ( @xmath103 in fig .
[ fig - chain ] ) we find @xmath104 .
this observation explains why the protocol using @xmath105 on end bonds employed for fig .
[ fig - ent - spec ] ( a ) succeeded in selecting the correct " bond representations : dmrg always favors lower entanglement , and the representation ( integer or half - integer ) with higher entanglement would be discarded by truncations during the sweeps .
another interesting observation is that the entanglement gap @xmath106 is found to be a constant + 1 ( @xmath3 ) in the spt ( trivial ) phase .
it is rather robust and almost independent of different hamiltonian parameters and system sizes , except for the intermediate region near the critical point , where the finite - size effects become significant .
this region is marked by vertical dashed lines in fig .
[ fig - chain ] .
the entanglement curves cross within this region , and the crossing point moves to the true critical point @xmath107 , the exactly soluble takhtajan - bubujian point , @xcite with increasing system sizes .
the value @xmath108 actually originates from the different topology of the su(2 ) and so(3 ) groups , and hence can be regarded as a topological invariant in each phase . in order to understand this ,
let us again consider the exactly solvable aklt model with @xmath10 .
the reduced tensor at the multiplet level is @xmath109 , a simple tensor with bond dimension 1 , i.e. , a scalar number .
the corresponding cgc tensor is @xmath110 , which combines a spin doublet with a triplet into an output spin doublet .
the corresponding reduced density matrix of half - infinite aklt chain is a @xmath111 diagonal matrix , @xmath112 , fully encoded in the cgc space only , and resulting in an entanglement entropy @xmath113=1 $ ] .
however , for the integer bond representation , we instead have a 2 @xmath114 2 diagonal matrix @xmath115 in the reduced multiplet space .
the two degenerate multiplets contain 4 degenerate states in total , and the full reduced density matrix is a @xmath116 diagonal matrix with all elements @xmath117 .
the entanglement entropy is @xmath118=2 $ ] , which is larger than corresponding @xmath1 and the gap @xmath119 .
next , we consider a generic state in the spt phase away from the special aklt point . as
shown in fig .
[ fig - ent - spec ] , there exists a one - to - one correspondence between one @xmath120 multiplet in the half - integer sector and one pair of degenerate multiplets with @xmath21 and @xmath121 in the integer sector ( @xmath122 ) .
for the latter , the degeneracy on the multiplet level can not be trivially lifted owing to the protection of the symmetry .
consequently , this multiplet degeneracy enhances the entanglement entropies and opens an entanglement gap of @xmath119 , as shown in fig .
[ fig - chain ] . on the other hand , adopting integer virtual bonds would preferably lower the entropy by 1 for the trivial dimer phase . as shown in fig .
[ fig - chain ] , in this case entanglement gap is @xmath123 .
[ fig - chain](c ) presents the nonlocal string order parameter obtained by itebd calculations ; it is nonzero in the haldane phase and vanishes in the trivial phase .
the comparison of our entanglement entropy results with the sop data validates that @xmath88 can be used to distinguish spt phase from the trivial one .
lastly , we remark that the results in fig .
[ fig - chain ] were obtained by evaluating finite - size systems . when the system is close to the critical point , the entanglement entropies @xmath0 and @xmath1 are shown to cross each other . in figs .
[ fig - chain](a ) , the lower values of these two entropy curves can be regarded as giving the true " entanglement entropies .
the combined curve shows a sharp peak , which is missed when considering either @xmath0 or @xmath1 alone . in fig .
[ fig - chain](a ) , the results obtained by itebd are also shown , which always favor the low entanglement curves .
the itebd data coincide with the su(2 ) dmrg results ( except for the region near the critical point ) , which validates our arguments above .
the crossing point of the @xmath0 and @xmath1 curves ( as the peak of the low entanglement curve ) can be viewed as a
pseudo - transition " point .
as the system size increases , the pseudo - transition point approaches the true critical point @xmath107 [ see fig .
[ fig - chain](d ) ] . in the thermodynamic limit ,
the gap @xmath88 are supposed to show a jump between 1 and @xmath3 just at the critical point , and the peaks of the entanglement entropies are expected to diverge .
and @xmath1 for the spin-1 tube model . the critical point estimated from their crossing point is @xmath124 .
( b ) entanglement gap @xmath106 .
@xmath125 when @xmath126 , identifying the existence of an spt phase , and @xmath123 when @xmath126 , corresponding to a trivial phase .
the system size is @xmath127 , 400 multiplets are reserved , which lead to maximum truncation error of @xmath128 ( at the critical point ) . ] , of the ground state of the spin-1 three - leg tube heisenberg model , in the vicinity of @xmath129 . ( a ) and ( b )
show the entanglement entropy between boundary block of length x and the rest of the system , for several different system sizes and @xmath130-values , on ( a ) a linear scale and ( b ) a log(sin ) scale on the horizontal axis .
curves are vertically offset by 1 unit for clarity .
here we show the data on one of the three sublattices in tube model , which contains entanglement entropies cut at the @xmath131-th bond [ @xmath132 ; the other two curves give the same fitting results and are not present here .
the conformal central charge is determined as @xmath133 , ( c ) shows how the fitted @xmath134 s vary with @xmath130 , for three fixed system sizes .
( d ) and ( e ) show , respectively , the maximal @xmath134-values and corresponding @xmath130-values obtained for 5 different @xmath135-values .
the system size ranges from @xmath136 to @xmath137 ( @xmath135 is the total site number ) , and up to 450 bond multiplets are reserved in the calculations . a half - integer bond representation was adopted in the calculations ; the fittings of integer - representation entanglement entropies lead to the same conclusion . ] of a spin-1 tube versus the coupling ratio @xmath130 .
the system size varies from 60@xmath1143 to 120@xmath1143 . for the largest size @xmath138 , 500 su(2 ) multiplets ( @xmath139 equivalent u(1 ) states )
are retained in the calculations , truncation errors are less than @xmath140 .
the inset shows the first - order derivatives of energies @xmath141 , which are substantially converged with different system sizes , and are shown clearly to be continuous through the critical point .
( b ) the second - order derivative @xmath142 , which shows a diverging peak at @xmath143 .
the inset in ( b ) shows @xmath144 in the vicinity of critical point on a log - log scale .
the data points fall into two linear lines ( except for the points very close to the critical point @xmath129 , owing to the finite - size effects near the critical point ) , which implies algebraic divergence .
the dashed lines in the inset are fits to the form @xmath145 , with @xmath146 and @xmath147 , approaching critical point from left and right sides , respectively . ] in this section , we study the spt phase in a spin-1 tube model . this model has been studied by charrier et .
al . in ref . .
following their conventions , schematically depicted in fig .
[ fig - sketch ] ( c ) , the hamiltonian is given by : @xmath148 @xmath149 and @xmath150 are the intra- and inter - chain coupling terms , respectively . in ref .
, the authors found a haldane phase existing for @xmath151 , with @xmath152 and @xmath153 , where each triangle contains an effective spin-1 .
for @xmath154 , they found a trivial disordered phase , with each isosceles triangle carries an effective spin-0 , leading to a spin-0 chain ( note that the combined product space 1 @xmath155 1 @xmath155 1 allows for exactly one spin-0 singlet ) . at the critical point @xmath129 ,
the system undergoes a quantum phase transition between the haldane and the trivial phase . for @xmath156 , there are still phase transitions separating two phases , but at different @xmath129 ; if @xmath157 , no phase transition occurs because the trivial phase no longer exists .
@xcite next , we revisit this model using su(2 ) dmrg calculations , and study it by evaluating the entanglement entropies @xmath0 and @xmath1 . in fig .
[ fig - tube ] , the entropies @xmath0 and @xmath1 intersect at @xmath158 . for @xmath159 , @xmath160 ,
the half - integer representation has lower entanglement .
the dominating bond multiplets are doublets ( @xmath32=1/2 ) , and the system is in an spt ( haldane ) phase . in contrast , for @xmath161 , @xmath162 , the ground state favor integer bond representations .
the energy results show that the energy per triangle is uniform along the leg direction , without any translational symmetry breaking .
the leading bond multiplet in the entanglement spectrum is found to be a singlet ( @xmath32=0 ) , and the system is in a trivial disordered phase .
in addition , we remark that the proper definition of a sop in this spin-1 tube has been discussed by the authors in ref . .
the spt phase that we have here identified by entanglement entropy , indeed also possesses a nonzero sop .
compared with the spin-1 blbq model , finite - size effects are much less significant in the spin tube model . for
a system size of @xmath163 , the values at which the peaks of integer and half - integer entropies occur lie quite close together . by combining @xmath0 of @xmath164 and @xmath1 of @xmath126
, we can see a very sharp peak in the joint low entanglement curve , which suggests a second - order quantum phase transition .
next , we address the order of the phase transition in more details by checking the criticality at @xmath129 .
the block entanglement entropy of size @xmath24 can be fitted with the following form : @xmath165 + \rm{const .
} , \label{eq - cc}\ ] ] where @xmath135 is the total number of sites .
@xmath166 for the tube of length @xmath167 .
this is the cardy - calabrese formula @xcite with open boundary condition , showing that the block entanglement entropy has a logarithmic correction to the entanglement area law at the critical point.@xcite @xmath134 is the conformal central charge , which characterizes the criticality .
the fitting results are shown in fig .
[ fig - fitc ] , which strongly suggests that the transition point is critical or very close to some gapless point ( quasi - critical ) .
the central charge obtained from the fits is @xmath168 . by the dmrg ordering of sites into one linear sequence ,
the 3-leg tube has three different sublattices ( and hence three kinds of bonds ) , two of which are equivalent .
therefore , when cutting the systems in different ways , we can get three block entanglement entropy curves , one of which is shown in fig .
[ fig - fitc ] .
the fittings of the other two curves lead to the same results .
[ fig - fitc](a ) and ( b ) show fits for 5 different system sizes and @xmath130-values .
[ fig - fitc](c ) shows that the @xmath134-values obtained from each fit exhibit , for given tube with total site number @xmath135 , a clear maximum as function of @xmath130 .
this maximal value ( located at @xmath169 ) can be regarded as the best estimation of @xmath134 .
note , the system is most close to critical at @xmath169 , and away from critical when @xmath170 and @xmath171 ; cardy - calabrese formula ( eq . [ eq - cc ] ) gradually loses its legitimacy in the latter case , and fitted value of @xmath134 is reduced away from @xmath169 . collecting these maximal points , in figs .
[ fig - fitc](d ) and ( e ) we plot , respectively , how the fitted @xmath134 s and estimated transition points @xmath169 s vary with different system sizes ( from @xmath172 to @xmath137 ) . the fitted @xmath134 s ( estimated transition points from entanglement ) tend towards 3 ( critical point estimated from energy derivatives ) when @xmath135 is increased .
moreover , in the fits , we follow the same strategy as in refs . and fit the central charge in the central region of the chain .
typically we omit 10 to 20 sites ( depending on the total system sizes ) from both ends , and take @xmath134 to be the limiting value obtained when increasing the omitted site number .
the ground - state energy curves and their derivatives with respect to @xmath130 are presented in fig .
[ fig - eng - curv ] .
the energy per site is defined as @xmath173 , where @xmath174 is the total energy and @xmath135 is the number of sites .
the first - order derivatives of energies do not show any discontinuities at the transition point , but the second - order derivatives have very sharp peaks at @xmath129 . in the inset of fig .
[ fig - eng - curv ] ( b ) , we also plot @xmath175 on a log - log scale .
the observed power law behavior implies the algebraic divergence of @xmath175 approaching @xmath143 , i.e. , @xmath176 .
the exponent @xmath177 has two different values , depending from which side @xmath129 is approached . both , though , are less than 1 , which implies that @xmath178 maintains a smooth behavior at @xmath129 .
therefore , the results of entanglement entropies , block entropy fittings , along with the energy derivatives , all support the conclusion that there is a continuous phase transition at @xmath129 .
this contradicts the conclusion in ref . , where the transition is argued to be of weakly first - order .
in order to thoroughly clarify the transition order , more detailed studies of the correlation functions and excitation gaps are needed , which we leave as future studies .
the parameters could also be tuned ( say , take @xmath179 different from @xmath180 studied above ) and investigate the nature of the phase transition ; or introduce some other parameters in the hamiltonian ( say bilinear - biquadratic parameter @xmath7 ) and inspect the transition along some other paths in the parameter space .
we have done some preliminary calculations along these lines ( not shown in this paper ) , which reinforce the conclusion of a second - order phase transition .
200 multiplets are retained in the calculations , and the maximum truncation errors @xmath181 .
( a ) @xmath0 and @xmath1 represent integer and half - integer entropies , respectively .
@xmath182 or @xmath183 means that @xmath184 or @xmath185 dominates in the multiplet spectrum , respectively .
the dashed line is a guide for the eye . ] lastly , let us consider the spin-1 two - leg ladder model , @xmath186 there are two kinds of couplings in this model [ see fig .
[ fig - sketch ] ( b ) ] , @xmath17 along the chain direction and @xmath18 on the rungs . in fig .
[ fig - ladder ] , the entropies @xmath0 and @xmath1 are plotted .
two versions of @xmath0 are shown , @xmath182 and @xmath183 , both obtained with integer bond representations , but with different leading ( lowest ) multiplets in the entanglement spectrum : @xmath36 for @xmath182 and @xmath185 for @xmath183 .
the latter can be obtained by attaching auxilliary spin-1 s on both ends in our su(2 ) dmrg . for @xmath187 , @xmath188 in fig .
[ fig - ladder ] , thus the ground state favors integer - spin representation , verifying the triviality of the ground state .
indeed , for the limiting case @xmath189 , the ground state is a simple direct product of rung singlets . on the other side , for @xmath190
the system is in the same phase as the spin-2 antiferromagnetic heisenberg chain ( reached in the limiting case @xmath191 ) .
[ fig - ladder ] shows that the ground states in this region also favor integer representations .
however , the lowest multiplet in the entanglement spectrum is the spin triplet @xmath22 , rather than the singlet @xmath184 , consistent with the results of ref . .
the two low - entanglement curves from the @xmath192 and @xmath193 symmetry sectors together form a smooth line in fig .
[ fig - ladder ] ( a ) ( indicated by a dashed line ) , which represents the true " entanglement entropy of the system .
no sign of criticality can be seen from the entanglement entropies , and it is hence believed that only one disordered phase exists in the spin-1 heisenberg ladder model .
our observation is in agreement with the conclusion in ref .
, that the model does not undergo any phase transition from @xmath190 to @xmath187 .
the fact that there does not exist an spt phase in the spin-1 heisenberg ladder model studied above can be ascribed to the triviality of the standard @xmath194 aklt states , which can be adiabatically connected to the topologically trivial state without any phase transition .
@xcite the triviality of the standard @xmath194 aklt state can be also be intuitively understood as follows : it has _ two _ valence bonds ( corresponding to two virtual spin-1/2 ) living on each geometric bond , since these two virtual spin-1/2 couple to either spin 0 or 1 , the total spin forms integer - spin representations of so(3 ) on the geometric bond , leading to a conclusion of a topologically trivial phase .
this argument also applies to the spin-1 heisenberg ladder studied above ( especially when @xmath190 ) .
the bond states are more complicated for the general two - leg heisenberg ladder model , nevertheless , they form integer representations of so(3 ) and the corresponding groundstate belongs to a trivial phase .
we have proposed a novel way to identify spt phases in one dimension by evaluating entanglement entropies . with su(2 ) dmrg method , we can keep track of the bond multiplets , and readily tell half - integer - spin projective representation from integer - spin ones by checking the multiplet entanglement spectrum introduced in this paper . in addition , we have shown that auxiliary boundary spins attached on both ends of the chain can be used to control the bond representations ; this significantly changes the entanglement entropies in the bulk , depending on the topological properties of the phase . in the spt phase
, we showed that a two - fold degeneracy for the overall entanglement spectrum appears either in the reduced multiplet space or in the cgc space , depending on whether the integer or half - integer bond representations are adopted , respectively . in the latter case ,
the two - fold degeneracy occurs in cgc space , which reduces the entanglement entropy @xmath1 relative to @xmath0 ( entanglement gap @xmath125 ) , providing a practical criterion for identifying spt phases .
the existence of an entanglement entropy gap also allows us to automatically select the
correct " representation ( integer or half - integer ) through dmrg sweeps , which always favor low entanglement representation .
the entanglement gap closes at the critical point , which can be used to detect the quantum phase transitions .
several 1d and quasi-1d systems have been studied in this work ; the spt phase in the spin-1 chain and the spin-1 tube model are successfully identified by evaluating the entanglement entropies . for the spin-1 tube model ,
the numerical results indicate that the phase transition between the spt phase and the trivial phase is a continuous one .
the fact that the two - leg spin-1 heisenberg ladder has no spt phase for any @xmath18 is also validated by our entropy results .
wl would like to thank h .- h .
tu and t. quella for helpful discussions on symmetry - protected topological order and the projective representations of symmetry groups .
wl was also indebted to shou - shu gong for useful discussions on the numerical results and dmrg techniques .
this work was supported by the dfg through sfb - tr12 , sfb631 , the nim cluster of excellence , and also we4819/1 - 1 ( aw ) .
gu and x.g .
wen , phys .
b * 80 * , 155131 ( 2009 ) .
x. chen , z.c .
gu , and x.g .
wen , phys .
b * 83 * , 035107 ( 2011 ) ; x. chen , z.c .
gu , and x.g .
wen , phys .
b * 84 * , 235128 ( 2011 ) ; x. chen , z.c .
liu , x.g .
wen , phys .
b * 87 * , 155114 ( 2013 ) .
x. chen , z.c .
liu , and x.g .
wen , science * 338 * , 1604 ( 2012 ) .
f. pollmann , a.m. turner , e. berg , and m. oshikawa , phys .
b * 81 * , 064439 ( 2010 ) .
f. pollmann , e. berg , a.m. turner , and m. oshikawa , phys .
b * 85 * , 075125 ( 2012 ) .
f. pollmann and a.m. turner , phys .
b * 86 * , 125441 ( 2012 ) . in traditional dmrg , the boundary @xmath31 doublet can be realized by attaching two spin-1/2 on both ends of the chain , as the authors did in the paper s.r .
white and d.a .
huse , phys .
b * 48 * , 3844 ( 1993 ) . however , in the framework of su(2 ) mps here , we can directly manipulate the end bond spaces of the chain . c. holzhey , f. larsen , f. wilczek , nucl .
b , * 424 * , 443 ( 1994 ) .
g. vidal , j.i .
latorre , e. rico , and a. kitaev , phys .
lett . * 90 * , 227902 ( 2003 ) p. calabrese , j. cardy , j. phys .
a * 42 * , 504005 ( 2009 ) .
t. tonegawa _
et al_. , j. phys .
. jpn . * 80 * , 043001 ( 2011 ) ; k. okamoto _ et al_. , j. phys .
conf . ser . *
302 * , 012014 ( 2011 ) ; k. okamoto _
et al_. , j. phys .
. ser . * 320 * , 012018 ( 2011 ) ; y - c .
tzeng , phys .
b * 86 * , 024403 ( 2012 ) ; j.a .
kjall , m.p .
zaletel , r.s.k .
mong , j.h .
bardarson , f. pollmann , phys .
b * 87 * , 235106 ( 2013 ) .
|
according to the classification using projective representations of the so(3 ) group , there exist two topologically distinct gapped phases in spin-1 chains . the symmetry - protected topological ( spt ) phase possesses half - integer projective representations of the so(3 ) group , while the trivial phase possesses integer linear representations . in the present work ,
we implement non - abelian symmetries in the density matrix renormalization group ( dmrg ) method , allowing us to keep track of ( and also control ) the virtual bond representations , and to readily distinguish the spt phase from the trivial one by evaluating the multiplet entanglement spectrum .
in particular , using the entropies @xmath0 ( @xmath1 ) of integer ( half - integer ) representations , we can define an entanglement gap @xmath2 , which equals 1 in the spt phase , and @xmath3 in the trivial phase . as application of our proposal , we study the spin-1 models on various 1d and quasi-1d lattices , including the bilinear - biquadratic model on the single chain , and the heisenberg model on a two - leg ladder and a three - leg tube . among others , we confirm the existence of an spt phase in the spin-1 tube model , and reveal that the phase transition between the spt and the trivial phase is a continuous one .
the transition point is found to be critical , with conformal central charge @xmath4 determined by fits to the block entanglement entropy .
|
understanding the chemical make - up of stars like the sun is fundamental to our understanding of star formation and stellar evolution .
@xcite undertook a classical study of the chemical evolution of the galactic disc , deriving abundances for a sample of 189 nearby field f and g dwarfs .
they found a number of interesting relationships in the data , for example , that the strongest age dependent abundance comes from ba , which they attribute to the efficient s - element synthesis in low - mass agb stars that enrich the interstellar medium long after star formation .
they confirmed that metal - poor stars ( [ fe / h]@xmath3 - 0.4dex ) are relatively overabundant in @xmath4-elements .
since the [ @xmath4/fe ] for these stars shows a gradient that decreases with increasing galactocentric distance of the orbits , they show that star formation was probably more vigorous and started first in the inner parts of the galactic disc .
since this work , various other samples have been studied , a number of which were performed by the exoplanet community ( e.g. * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
these works have concentrated on studying the abundance distributions of exoplanet host stars compared with non - exoplanet hosts and a number of interesting trends have been found . along with the well established dependence of giant planet detection probability on host star metallicity @xcite
these abundance analyses indicate that various other atomic abundances are likely enhanced in exoplanet hosts compared to non - exoplanet hosts ( e.g. si & ni ) , at least for the gas giant population .
these over - abundances are explained in the framework of the core accretion scenario of planet formation @xcite where the more disc material present , the higher the probability of planet formation . in general
, the relative yields of different elements in the atmospheres of stars change with time , due to the processes of nucleosynthesis in the galaxy .
these processes are still poorly known , yet performing high quality analyses of homogeneous stellar samples can shed light on the fine underlying processes that are occurring , following the formation of planetary systems .
in recent years , series of large - scale abundance analyses have been published @xcite , mainly driven by the availability of high resolution spectral data from observational campaigns that are dedicated to searching for planets and the proliferation of software that allows us to automatically process the spectra of hundreds of stars in an automatic , or semi - automatic , manner .
the analysis we provide here is used to determine the chemical abundances in the atmospheres of metal rich stars , along with their relative behaviour versus iron .
it is worth noting that our sample is one of the generally homogeneous samples ( e.g. * ? ? ?
* ) in comparison to similar studies , which cover up to a thousand stars , but obtained using different instruments with different resolutions and pipeline reduction techniques ( e.g. * ? ? ?
on the other hand , some homogeneous studies cover rather smaller number of stars ( e.g. * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) or with lower resolution ( e.g. * ? ? ?
in this work we combined both large enough sample and high resolution to provide us with a good statistical sample to investigate .
the layout of the manuscript is as follows , in section [ _ observations ] we discuss the observational data , in section [ _ abundance_method ] we describe the method used to derive the abundances , and , our selection criteria of the examined lines , the influence of the microturbulence and rotation .
section [ _ results ] contains the description of our results , and in section [ _ comparison ] we compare these results to the previously published works .
finally , in section [ _ discussion ] we discuss the implication of our results in the field , and summarise our findings in section [ _ summary ] .
the calan - hertfordshire extrasolar planet search ( cheps ) program @xcite is monitoring a sample of the metal rich stars in the southern hemisphere , to improve the statistics for planets orbiting such stars , along with searching for short period planets that have a high probability to transit their host star .
all stars in the sample were initially selected from the _ hipparcos _ catalogue to have @xmath5-band magnitudes in the range 7.5 to 9.5 .
this range ensures that the sample is not overlapping with existing planet search programs , since most solar - like stars with magnitudes below 7.5 are already being examined as part of other planet search programs , and the upper limit is set to ensure that the stars are bright enough to allow the best follow - up to search for secondary eclipses and transmission spectroscopy , essentially bridging the gap between the long - term precision radial velocity programs , and the fainter samples that photometric transit surveys are generally biased towards .
more specifically , the cheps sample primarily contains objects that were drawn from a larger southern sample observed using the eso - feros spectrograph ( see * ? ? ?
* ; * ? ? ?
the secondary selection of these targets was focused on the inactive ( log@xmath6 - 4.5dex ) and metal rich ( [ fe / h]@xmath00.1dex ) subset of this sample @xcite to ensure the most radial velocity stable targets , and to make use of the known increase in the fraction of planet - host stars with increasing metallicity , mentioned above .
we note that , recent work has shown that the fraction of metal rich stars hosting low - mass planets may not follow the metallicity trend observed in the gas giant population @xcite .
our subset of targets have been followed - up using the harps at la silla in chile . in our analysis , we use high s / n ( @xmath7100 ) and high - resolution spectra ( r@xmath8120,000 ) of predominantly single stars from the cheps sample that have been well characterised photometrically based on _ hipparcos _ data @xcite . the general properties of the sample are shown in fig .
[ _ figure_general_properties ] .
@xcite carried out the photometric determinations of the surface gravity parameter for our stars ( shown with open circles ) , and we compare these values to our spectroscopically measured ( filled circles ) .
we work within a narrow range between 5200 to 6200k , with the majority of our stars being metal rich and on the main sequence ( @xmath74.1 ) , although a few are slightly metal deficient ( [ fe / h]@xmath7 - 0.2 ) .
most of the stars in our sample belong to the thin disc galactic population ; see the toomre diagram at the bottom right of fig .
[ _ figure_general_properties ] .
we used a modified numerical scheme developed by @xcite that allows to determine the atomic abundances , rotational velocities , microturbulences and surface gravities in the stellar atmospheres from high resolution and high s / n spectra , all in the framework of iterative approach , i.e. at each new step , the model atmospheres and synthetic spectra were recomputed for the metallicities and gravities derived before in our procedure , and for the fixed effective temperatures that were set photometrically .
the method is based on the minimisation of differences in the profiles of computed and observed lines .
we used the two independent procedures for the final model parameters .
firstly , we required there to be no dependence of the fe abundance on the line strengths to obtain .
secondly , we required the agreement between the abundances of fe and fe obtained for the previously found to determine . for our starting point we used the photometrically determined metallicities and
( see * ? ? ?
* ) , then we ran a few iterations to determine [ fe / h ] , , [ fe / h ] and .
the final step was the determination of abundances and for the model atmosphere parameters found in the procedure before .
all synthetic spectra were computed by the wita6 program @xcite using 1d local thermodynamic equilibrium ( lte ) model atmospheres computed with sam12 code @xcite combined with the minimization routine abel8 @xcite .
the lines to be fitted in the stellar spectra were compiled using the solar spectrum by @xcite and atomic line data from vald-2 and vald-3 @xcite .
in our analysis we made a few basic assumptions : the level of activity is low in these atmospheres , in other words , the continuum in the optical range forms in the photosphere of the star .
both the micro- and macroturbulent velocity fields are similar to the solar case . for simplicity
we adopt that @xmath9 and do not change with the depth in these atmospheres .
the convection zone is similar to the solar convection zone , even in the case of super metal rich stars .
the minimum of the atmospheric temperature is located above the line formation region .
spots on the surface of these stars are not sufficiently numerous to contribute to the formation of absorption lines or continuum , in terms of the emitted fluxes .
naturally , if any of these assumptions is not valid for the physical state of the studied stars , we will obtain a spread of abundances and other determined parameters , even in the case of ideally determined and .
the effective temperatures were estimated using the photometric methods ( see * ? ? ?
* ) , exploiting the existing large photometric databases , along with the latest relationships between stellar broad - band colours and their photospheric effective temperatures .
namely , we used the johnson @xmath5-band photometry that was taken from the _ hipparcos _ database @xcite as our optical anchor point , and combined this with near - infrared photometry from the 2mass database @xcite , in particular the @xmath10-band magnitude that gives a sufficiently large wavelength baseline to sample the shape of the spectral energy distribution .
we then used the relationships provided in @xcite to calculate accurate effective temperatures and place the stars on an hr - diagram to calculate the photometric values using the y2 evolutionary models @xcite .
we choose to use these photometric for the computations , rather than update them spectroscopically .
we believe this constrains uncertainties which may come from the quality of fits to the spectra and atomic line data , and these may also affect the results of our determination of abundances , microturbulent velocities , and surface gravities .
we provide our analysis for the pre - selected list of spectral lines extracted from vald-2 and vald-3 @xcite . to create the list of reliable lines for each element , we computed the solar synthetic spectra and convolved them to get the effective resolving power of r@xmath870,000 , which is the effective resolution due to macroturbulent velocity field in the solar atmosphere .
the theoretical spectra were fitted to the selected observed lines using the minimization routine abel8 . at this stage , we excluded severely blended lines , too weak lines with residual fluxes lower than 20 per cent and strong lines with @xmath110.7 to minimize the distortion of the results due to noise and possible nlte effects in the cores of the strong lines @xcite .
we computed our synthetic spectra using the damping constants provided by vald @xcite . for the absorption lines without damping constants we computed them using the unsold formulas @xcite .
the fitting part of each line profile was adopted manually to reduce the uncertainties introduced by wing blending . in some cases
, very close blends of the lines of the same element were fitted together , resulting in a single spectral range to fit multiple lines . the final list of fitting ranges for the neutral and ionized atoms available on cds .
the microturbulent velocity is an important parameter in 1d line - profile fitting techniques as there is an evident trend with abundances that causes uncertainties in their measurements .
the dependence of fe and fe abundances on microturbulent velocity for hd 102196 is shown in fig .
[ _ figure_method_microturbulence ] . here ,
fe and fe abundances differ by up to @xmath10.1dex at @xmath121.0but the same at 1.4 .
we carried out a set of synthetic spectra fits to fe and fe lines independently , using the grid of adopted microturbulent velocities in range from 0.0 to 2.6 , with a step of 0.2 .
in fact , we followed the procedure of @xcite , but here we investigated the dependence of @xmath13 , where @xmath14 and @xmath15 are the abundance and the central intensity of the corresponding iron absorption line computed for the grid of , the @xmath16 is known to show the dependence on .
therefore , we determine our at @xmath16=0 . our procedure is based on fitting to the line profiles , excluding the shallow parts of the wings , which could be more affected with blending by the weak lines of other elements .
furthermore , the wings of lines are more affected by the pressure broadening than the microturbulent velocities .
accounting both these factors should improve the determination of ( see fig .
[ _ figure_microturbulence_comparison ] ) .
the profiles of absorption lines in the stellar spectra affected by the rotation of a star , depending on the magnitude of rotation .
the rotational profile , in our case , determined by the formula of @xcite , is of a different shape compared to the instrumental broadening and macroturbulence . in our analysis ,
each line of the synthetic spectra was convolved with a profile of a different , and was fitted to the observed spectra until the best result was found .
despite we used only fe lines for the determination of , the procedure was also applied for every other line of all our elements to get a better fit to the observed line profiles .
we found that the majority of our stars are slow rotators , with @xmath34(fig . [ _ figure_rotation ] ) .
likely , a few stars with larger rotational velocity present the cases where sin@xmath171 , rather than large @xmath18 itself , since these stars are of lower activities , with only a few fast rotating active stars left over . on the other hand ,
correlates with , older stars rotate slower , in agreement with the theory , and vice versa , which is in line with their ages given by the preliminary cheps selection .
we can see an uprising trend of for the stars of lower , which represent a younger population .
for the elements other than fe , was used rather as the adjusting parameter . nevertheless , it agreed for the properly fitted lines of different elements within an uncertainty of less than 1 .
we can claim a clear measurement for the values of @xmath72 .
uniform results for the iron lines give credibility to our results .
we carried out a differential analysis with respect to the sun as a star , since our line lists were selected based on a comparison of the synthetic spectra to the observed solar spectrum by @xcite .
this allowed us to minimize possible blending effects , at least in the solar case .
we performed a quantitative analysis of the solar abundances to verify our procedure .
computations were done for the initial model atmosphere with //[fe
/ h]=5777/4.44/-4.40 , computed using sam12 @xcite . for the iron ions , we obtained n(fe)[email protected] and n(fe)[email protected] , which again highlights the known problem of the measured difference between fe and fe abundances obtained in the framework of the classical approach ( see * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
the abundances for the other elements ( see the first column of table [ _ table_solar ] ) are in agreement with those in the literature ( e.g. * ? ? ?
* ; * ? ? ?
we also determined the microturbulence and projected rotational velocity for the sun as a star , measuring = [email protected] = [email protected] , which also agree with known results .
the solar abundances determined for the different atomic line data and line lists are discussed in section [ _ comparison ] .
the complete table of abundances available on cds . in our analysis
, we used the parameter @xmath19 to describe the average slope of the distribution of abundances for a given element relative to iron .
we computed @xmath19 using a standard least squares approach to approximate the dependence of [ x / fe ] versus [ fe / h ] by a linear function . by definition , @xmath19 characterises , to first - order , the relative changes of the yield of elements with respect to the iron . in order to make a useful comparison with other works we needed to translate onto a common scale
we found the easiest way to achieve this was to adopt the scale of @xcite rather than in terms of the solar abundances determined by our procedure . in some papers ,
abundances are provided in relation to derived solar scales in order to compensate for the differences in procedures ( e.g. * ? ? ?
* ; * ? ? ?
other authors adopt @xcite or its later derivatives ( e.g. * ? ? ?
* ; * ? ? ?
* ) , though they did not consider their derived solar abundances .
in order to compare the different samples studied by different authors we converted abundances for the various comparison samples to the scale of @xcite ( fig .
[ _ figure_samples ] ) .
this also allowed us to assess the general accuracy level of stellar abundance determinations from the combined spread of abundances . in fig .
[ _ figure_overiron ] we show the dependence of abundances relative to iron with their corresponding error bars .
these uncertainties depend on the number of lines for each element , quality of fit , the scatter of abundances determined from the fits to different lines , the local continuum level , and the atomic line data taken from vald .
the uncertainties for [ fe / h ] are not plotted to make the plots easier to read .
the average [ fe / h ] uncertainty is @xmath10.02dex . below we discuss the specific results for the selected elements : _ fe , z = 26_. differences up to 0.1dex between fe and fe abundances are due to the iterative nature of our computations .
previously determined abundances can change after adoption of the refined model atmosphere .
in other words , it reflects the limit of accuracy implied by the model atmosphere , atomic line data , and line list .
_ na , z = 11_. the distribution of the na abundance versus iron is shown in fig .
[ _ figure_samples ] .
the computed @xmath19 for the metal rich stars in our sample and those of @xcite and @xcite show a well defined positive slope ( table [ _ table_abundance_slopes ] ) .
the abundance distribution of na is shifted toward larger abundances by 0.10.2dex compared to fe .
the general trend is similar to those obtained in all comparison works with higher abundances toward higher metallicities ( fig .
[ _ figure_samples ] ) .
_ mg , z = 12_. only up to 7 lines of mg i were used in the analysis .
we found similar result to that of sodium , with a notable over - abundance for our sample , and a mean abundance of [ mg / fe][email protected] . the shift is larger than the formal accuracy of our abundance determination procedure , but
may depend on the adopted continuum level and line list .
we note that @xmath20 for the stars in our sample .
however , it can be seen in fig .
[ _ figure_samples ] that this could be due to border effects , as this trend is rather marginal .
@xcite show the same order of over - abundance , but also a certain downward trend with metallicity , and for the sample of @xcite we see that their mg distribution is in agreement with [ fe / h ] . at the same time , @xcite show the results similar to our .
al , z = 13_. in our work aluminium show a definite positive slope . to some degree our findings similar to those of @xcite and @xcite . but
different to @xcite and @xcite who show no definite dependence on metallicity , and higher mean [ al / fe ] for their samples . _
si , z = 14_. the results for silicon agree well with most of the studies in comparison , showing the average [ si / fe ] over - abundance to be on the order of 0.1dex and no noticeable trend with metallicity . on the other hand ,
@xcite show no excess of silicon .
it is interesting to note that for the same lines in the spectra , our results for vald-2 show 0.1dex lower abundances than for vald-3 , highlighting the importance of reliable atomic line data .
_ ca , z = 20_. calcium is another element for which we see a high level of agreement between the different authors @xcite . in all these works
there is a clear negative trend with metallicity ( table [ _ table_abundance_slopes ] ) . on the whole , [ ca / fe]@xmath70 , but in the super metal rich domain , we note calcium deficiency .
however , various mean abundances in different works ( see fig .
[ _ figure_samples ] ) reflect that this deficiency is within the accuracy limit of the modern methods of abundance determination .
_ ti , z = 22_. in our sample ti follows the iron abundance and shows no dependence with metallicity . on the other hand ,
all comparison works show a larger spread of stars toward higher [ ti / fe ] abundances and a weak negative trend . at the same time , @xcite show no metallicity dependence for ti . for our stars ,
the ti lines show a large scatter , which makes it difficult to come to any firm conclusions in regards to the metallicity trends .
similar to si and fe , the lines of ionized titanium give rise to a 0.1dex higher abundance when the line parameters are taken from vald-3 .
_ cr , z = 24_. chromium is one of the few elements with reliable results that show no significant scatter between the different lines and stars in our sample . we can observe a very weak negative trend with iron .
the same results are shown by @xcite .
@xcite and @xcite show higher abundances for their stars and a weak positive trend , which could be an indication of the sample bias effect .
once again , the same chromium lines give different abundances using the different versions of vald , but unlike si , ti , and fe , we see a 0.05dex lower mean abundance for the vald-3 data .
_ mn , z = 25_. manganese abundances show a clear positive gradient with metallicity , much like zn , though with a larger scatter despite the relatively large number of lines used in the analysis .
manganese exhibits a higher abundance than iron , on average , by 0.3dex .
a positive trend was also found in all comparison works . as for most of the other elements , the abundance measured by @xcite is slightly higher than that of @xcite . and
@xcite show the results similar to our ( fig .
[ _ figure_samples ] ) . _
ni , z = 28_. [ ni / fe ] also shows an upward trend with iron of the same order in all the samples in comparison .
all the authors show different average over - abundances for ni : up to 0.05dex for @xcite and @xcite , 0.1dex for @xcite and our work , and 0.15dex for @xcite and @xcite .
_ cu , z = 29_. the distribution of cu with [ fe / h ] exhibits a large scatter ( fig .
[ _ figure_overiron ] ) , mainly affected by the quality of the line list and number of lines . formally speaking , within the range of metallicities [ fe / h]=-0.1+0.2 , there are two groups of stars , with mean [ cu / fe ] abundances of @xmath8 + 0.1 and -0.05dex , respectively , however with uncertainties at the level of @xmath10.4dex . the lower average abundances and defined positive trend were found by @xcite . the same trend could be observed for the @xmath80.0dex group of stars in our sample .
_ zn , z = 30_. for zn we found the same significant differences in the abundance distribution with iron that were found for manganese .
unfortunately , we used only one line of zn in our analysis , so the visible slope of [ zn / fe ] versus [ fe / h ] should be considered as a preliminary result at this point .
more work should be done to confirm this result using a wider set of zn atomic lines . a large scatter and similar abundance
are also shown by @xcite and @xcite .
our analysis shows that the different atomic line data can have a strong impact on the average abundances measured for similar samples of stars ( e.g. see silicon in this work ) . on the other hand ,
observed trends change rather marginally .
comparisons to our work and those in the literature highlights the similar tendencies for the na , si , ca , cr , mn , and ni ; for the elements with a small spread like si , ca , cr , ni we see various average abundances between the different samples , and similar trends with iron in all comparison works . and for the mg , al , and ti we see weak trends with iron in all comparison works due to large spread of abundances . in the framework of our work , we investigated the dependence of our abundance results on the effective temperature and surface gravity .
the results of these tests are shown in figs .
[ _ figure_temperature_dependence ] and [ _ figure_gravity_dependence ] in appendix .
we see the evidence for a weak trend of abundances versus in fig .
[ _ figure_temperature_dependence ] .
these rather indistinct trends may be explained by uncertainties in the choice of the photometric of the metal rich stars of earlier spectral classes , as well as limitations of the current model atmospheres and atomic line data . in each case
we obtained the trends of abundances versus for the elements with rather low ionization potentials , i.e. mg , al and si , with the pearson product - moment correlation @xmath21()@xmath80.40.5 ( table [ _ table_abundance_slopes ] ) .
the same we also observe for mn and zn .
however , because our data does nt show a sufficient level of homoscedasticity , it is hard to make definite conclusions in these regards .
we note the non - linear behaviour of the [ x / h ] versus dependence for practically all our elements in the stars with close to 6000k .
this apparent phantom gap is poorly constrained due to low number of stars in this temperature range .
what is more important , we do not see any evident trends of the abundances versus the adopted surface gravities ( table [ _ table_abundance_slopes ] ) . in some sense
, this provides us with the evidence that our adopted procedure performs in the way we had previously envisaged it would .
however , sensitivity of the spectroscopic and to the quality of fits and line data ( these are discussed in section [ _ comparison ] ) emphasizes the importance of at least one independent variable .
there have been a number of extensive works to determine abundances of main sequence dwarfs .
they follow similar approaches to the analysis of the observed high - resolution stellar spectra : lte , 1d model atmospheres .
we tabulate these in table [ _ table_abundance_slopes_procedures ] .
to compare our results to other authors we note that our sample primarily consists of metal rich stars ; the samples of some other authors consist of a larger total number of targets , and they maintain a broader metallicity range .
most of our objects are single stars , where our spectroscopic data have been observed solely using harps , using a photometric pre - selection , and thus our sample is homogeneous . the most of comparison works were done using the equivalent width analysis , we used the synthetic profile fitting . in comparison to @xcite and @xcite , who utilized synthetic spectra fitting for the broad spectral ranges and fixed microturbulent velocity , we adopted line - by - line fitting and microturbulence as a free parameter in addition to the other differences .
as noted in subsection [ _ abundances ] , some authors used their own solar abundances for the [ x / fe ] distribution analysis , whereas we use the @xcite abundances in our study . in our work
, we separate the relative and absolute solar scales .
the first case is defined by the reference solar abundances to which the absolute values are translated . and
the second is dependent on the adopted line lists and gf - values .
the latter we identify as the differences in the atomic line data or line lists , but not in the adopted solar scales .
|
we report results from the high resolution spectral analysis of the 107 metal rich ( mostly [ fe / h]@xmath07.67dex ) target stars from the calan - hertfordshire extrasolar planet search program observed with harps . using our procedure of finding the best fit to the absorption line profiles in the observed spectra
, we measure the abundances of na , mg , al , si , ca , ti , cr , mn , fe , ni , cu , and zn , and we then compare them with known results from different authors .
most of our abundances agree with these works at the level of @xmath10.05dex or better for the stars we have in common .
however , we do find systematic differences that make direct inferences difficult .
our analysis suggests that the selection of line lists and atomic line data along with the adopted continuum level influence these differences the most . at the same time
, we confirm the positive trends of abundances versus metallicity for na , mn , ni , and to a lesser degree , al .
a slight negative trend is observed for ca , whereas si and cr tend to follow iron .
our analysis allows us to determine the positively skewed normal distribution of projected rotational velocities with a maximum peaking at 3 .
finally , we obtained a gaussian distribution of microturbulent velocities that has a maximum at 1.2and a full width at half maximum @xmath2=0.35 , indicating that metal rich dwarfs and subgiants in our sample have a very restricted range in microturbulent velocity .
[ firstpage ] stars : abundances fundamental parameters
late - type solar type
|
a commonly used approach for the simulation of multi - phase fluid dynamics is the free energy lattice boltzmann method introduced by swift _
et al . _
this constitutes a so - called mesoscale method because it numerically solves the continuum equations of fluid dynamics by exploiting the underlying microscopic structure of these equations , without resorting to a description of the fluid in terms of molecular dynamics .
one obstacle to simulating some systems is that discretisation errors lead to unphysical flows near interfaces .
these so - called spurious velocities are present in multi - phase lattice boltzmann methods and in the other diffuse interface methods .
an illustration of these spurious velocities is given in fig .
[ fig1](a ) , which shows the flow profile around a liquid drop coexisting with a surrounding gas phase .
the simulation is left until the long time steady state behaviour is reached . from a physical point of view all velocities should go to zero .
what is observed , however , is that spurious flows persist indefinitely .
a number of papers have dealt with this problem .
wagner @xcite analysed the case of binary fluids , and identified that one way to eradicate spurious velocities was to remove non - ideal terms from the pressure tensor and introduce these as a body force of the form @xmath1 .
however , because this is no longer written in terms of the divergence of a pressure tensor ( note that in general @xmath2 can always be rewritten as @xmath3 ) then momentum is no longer conserved .
furthermore , wagner pointed out that this method is numerically unstable unless some additional viscosity is artificially added to the system .
lee and fisher @xcite use another forcing method for a different implementation of the lattice boltzmann algorithm .
again , they eliminate spurious velocities at the expense of sacrificing momentum conservation . an additional difficulty with using forcing methods ( including
the one we present later ) is that in order to update each lattice site the algorithm requires information from @xmath4 lattice sites away , rather than just @xmath5 for the standard method .
this makes boundary conditions more complicated and slows down parallel computations , since more information needs to be passed between processors .
seta and okui @xcite used a lattice boltzmann scheme proposed by inamuro _
@xcite , and considered calculating the derivatives in the pressure tensor using a more accurate fourth - order scheme ( as opposed to the usual second order accurate method ) . as will be shown later , however , they do not choose an optimum equilibrium distribution and hence their improvement in the spurious velocities is limited .
in this paper , we analyse the free energy lattice boltzmann scheme for a liquid - gas system proposed by swift _ et al .
_ @xcite and show that by making a careful choice of the equilibrium distribution ( and also finding the best way to calculate derivatives ) the magnitude of spurious velocities can be significantly reduced .
furthermore , we present a second numerical scheme which moves gradient terms in the equilibrium distribution into a body force .
this leads to a further reduction in spurious velocities whilst preserving momentum conservation .
the results of this analysis are equally applicable to other multiphase systems ( _ e.g. _ binary fluids ) when the free energy lattice boltzmann method is used to solve their equations of motion .
the pressure tensor for a liquid - gas system using a landau free energy is given by @xmath6 where @xmath7 is the fluid density and @xmath8 is a parameter related to the surface tension .
we choose the bulk pressure @xmath9 to be that of a van der waals fluid , @xmath10 this leads to liquid - gas phase separation below a critical temperature .
the analysis which follows is performed for a d2q9 lattice boltzmann scheme ( in section [ 3d ] the results are summarised for the d3q19 model ) which uses a square lattice of side @xmath11 , time - step @xmath12 , and has @xmath13 velocity vectors , @xmath14 , where @xmath15 , @xmath16 , @xmath17 , @xmath18 , and @xmath19 . the parameter @xmath20 is a lattice velocity .
a particle distribution function @xmath21 gives the mass density of particles travelling from lattice site @xmath22 , at time @xmath23 , in a direction @xmath14 .
the physical variables are related to this distribution function by @xmath24 where @xmath7 is the mass density and @xmath25 is the velocity of the fluid .
the time evolution equation for the particle distribution function , using the standard bgk approximation , is given by @xmath26 + f_i , \label{latbolt}\end{aligned}\ ] ] where @xmath27 is a relaxation parameter related to the viscosity , and @xmath28 is an equilibrium distribution .
it has been shown previously that this reduces to the navier - stokes equation provided the moments of @xmath29 and @xmath30 are chosen suitably @xcite ( see appendix [ app1 ] ) .
the final @xmath30 term is responsible for introducing a body force .
this is not present in the standard formulation of the free energy lattice boltzmann algorithm and so for now we set it to zero . in section
[ force ] , however , we discuss how this term can be usefully implemented to help reduce spurious velocities further .
the equilibrium distribution can be written as @xmath31 \times \nonumber\\ & & \quad \left ( \rho u_\alpha u_\beta + \lambda \left [ u_\alpha \partial_\beta \rho + u_\beta \partial_\alpha \rho + \delta_{\alpha \beta } u_\gamma \partial_\gamma \rho \right ] \right ) \big ) \nonumber\\ & & \quad + \tfrac{1}{c^2 } \big ( \,\ , w_{i}^p p_0 - w_{i}^t \rho \nabla^2 \rho + w_i^{xx } \kappa \partial_x \rho \partial_x \rho \nonumber\\ & & \quad + \,\ , w_i^{yy } \kappa \partial_y \rho \partial_y \rho + w_i^{xy } \kappa \partial_x \rho \partial_y \rho \big ) , \label{equilibrium}\end{aligned}\ ] ] for @xmath32 , where @xmath33 , @xmath34 , and summation over repeated indices is assumed .
the @xmath35 stationary value is chosen to conserve mass : @xmath36 the top two lines on the right hand side of eq .
correspond to a standard expansion of the maxwell boltzmann distribution in discretised space @xcite , and a correction term involving @xmath37 ( see eq .
( [ viscous ] ) ) which ensures galilean invariance @xcite .
these terms are not important from the point of view of spurious velocities because they each contain the fluid velocity @xmath38 to some power , which is expected to be zero in equilibrium .
the last two lines in eq .
give the pressure tensor contribution to the equilibrium distribution .
this has been written in its most general form involving the free parameter weights @xmath39 , @xmath40 , @xmath41 , @xmath42 , and @xmath43 .
through the course of this paper optimum values for these parameters will be obtained .
the derivatives in the equilibrium distribution are explicitly calculated within the algorithm using finite difference schemes .
for instance , one simple choice for calculating the @xmath44 derivative of @xmath7 is given by @xmath45 .
\label{drdx}\end{aligned}\ ] ] the bar above the partial derivative denotes that this is a discrete operator . by taylor
expanding the right hand side we find that @xmath46 the discrete operator is correct up to second order but there are higher order terms which are responsible for generating the spurious flows .
a useful representation of finite difference operators is to denote them by stencils .
for instance eq .
( [ drdx ] ) can be rewritten @xmath47_\rho .
\label{simple}\end{aligned}\ ] ] the central entry in the matrix represents the point at which the derivative is being made and the surrounding 8 entries correspond to the neighbouring lattice points surrounding this .
this , however , is not the only choice for calculating the @xmath44 derivative .
the most general stencil using only @xmath13 lattice nodes can be written @xmath48 \nonumber\\ & = & \partial_x + \tfrac{1}{6 } { \delta x } ^2 \partial^3_x + 2b { \delta x}^2 \partial_y^2 \partial_x + \dots \label{xderiv}\end{aligned}\ ] ] where @xmath49 is a free parameter which can be used to determine the third order term and @xmath50 is defined by @xmath51 .
similarly , the laplacian operator can be represented by @xmath52_\rho \nonumber\\ & = & \nabla^2 + \tfrac{{\delta x}^2}{12 } \left(\partial_x^4 + \partial_y^4 \right ) + d { \delta x}^2 \partial_x^2 \partial_y^2 + \dots \label{laplace}\end{aligned}\ ] ] where @xmath53 . in equilibrium
the navier - stokes equation reduces to @xmath54 in terms of the lattice boltzmann algorithm , the partial derivative operator acting on the pressure tensor in eq .
( [ main ] ) is implemented as a result of the choice of equilibrium distribution and the streaming and colliding operations .
when @xmath55 ( in section [ taune ] we discuss the more general case ) the lattice boltzmann equation ( [ latbolt ] ) reduces to @xmath56 we consider the idealised case when at some time @xmath23 the system is at rest , _
i.e. _ @xmath57 , and the density distribution is chosen such that the continuous operator equation ( [ main ] ) is solved exactly .
we ask the question what happens when the continuous operators are replaced by their discrete counterparts . in this case
( [ main ] ) will no longer be exactly satisfied and instead there will be some spurious force @xmath58 on the left hand side .
( [ equilibrium ] ) , this force can be expressed in terms of stencils of the various terms in the equilibrium distribution : @xmath59_{p_0 } \!\!\!\!\!\ ! - \left [ \begin{array}{ccc } -w_{5\text{-}8}^t & 0 & w_{5\text{-}8}^t\\ -w_{1\text{-}4}^t & 0 & w_{1\text{-}4}^t\\ -w_{5\text{-}8}^t & 0 & w_{5\text{-}8}^t \end{array } \right]_{\kappa \rho \nabla^2 \rho } \right .
\nonumber\\ & & \hspace{0.6 cm } + \left [ \begin{array}{ccc } -w_{5\text{-}8}^{xx } & 0 & w_{5\text{-}8}^{xx}\\ -w_{1\text{-}2}^{xx } & 0 & w_{1\text{-}2}^{xx}\\ -w_{5\text{-}8}^{xx } & 0 & w_{5\text{-}8}^{xx } \end{array } \right]_{m_{xx } } \!\!\!\!\!\!\!\ ! + \left [ \begin{array}{ccc } -w_{5\text{-}8}^{yy } & 0 & w_{5\text{-}8}^{yy}\\ -w_{1\text{-}2}^{yy } & 0 & w_{1\text{-}2}^{yy}\\ -w_{5\text{-}8}^{yy } & 0 & w_{5\text{-}8}^{yy } \end{array } \right]_{m_{yy } } \nonumber\\ & & \hspace{0.6 cm } \left
. + \left [ \begin{array}{ccc } -w_{7\text{-}8}^{xy } & 0 & w_{5\text{-}6}^{xy } \\ -w_{1\text{-}4}^{xy } & 0 & w_{1\text{-}4}^{xy } \\ -w_{5\text{-}6}^{xy } & 0 & w_{7\text{-}8}^{xy } \end{array } \right]_{m_{xy } } \right ) , \label{expand2}\end{aligned}\ ] ] in , for example , the @xmath44 direction . here
, we define @xmath60 . in writing this
we have made use of the symmetry properties of the system to immediately reduce the number of free parameters in the model .
for instance , the bulk pressure @xmath9 does not have a preferred direction ( _ i.e. _ it acts the same in the @xmath44 and @xmath61 directions ) and hence we expect that @xmath62 , which we denote by @xmath63 , and @xmath64 .
other terms do have a preferred direction .
for example , @xmath65 is less restricted and has the constraints @xmath66 , @xmath67 and @xmath68 . because the equilibrium should be invariant under simultaneous interchange of @xmath44 and @xmath61 and switching the velocities @xmath69
, then we expect that @xmath70 , @xmath71 , and @xmath72 .
to first order , @xmath73 should agree with eq .
( [ main ] ) , which in the @xmath44 direction is given by @xmath74 by comparing eq .
( [ fi ] ) with eq .
( [ expand2 ] ) further restrictions are possible .
for instance , by using eq .
( [ xderiv ] ) , the @xmath9 stencil becomes @xmath75 to second order provided that @xmath76 .
similarly , @xmath77 , @xmath78 , @xmath79 , @xmath80 , @xmath81 , and @xmath82 .
these constraints are also necessary to obtain the correct moments of the equilibrium distribution in eq .
( [ pcons ] ) . given all these conditions the spurious force can be rewritten as @xmath83_{p_0 } \!\!\!\!\!\ ! \right .
\nonumber\\ & & \hspace{0.6 cm } - \left [ \begin{array}{ccc } -w_{5\text{-}8}^t & 0 & w_{5\text{-}8}^t\\ -\left(\tfrac{1}{2 } - 2w_{5\text{-}8}^t\right ) & 0 & \left(\tfrac{1}{2 } - 2w_{5\text{-}8}^t\right)\\ -w_{5\text{-}8}^t & 0 & w_{5\text{-}8}^t \end{array } \right]_{\kappa \rho \nabla^2 \rho } \nonumber\\ & & \hspace{0.6 cm } + \left [ \begin{array}{ccc } -w_{5\text{-}8}^{xx } & 0 & w_{5\text{-}8}^{xx}\\ -\left(\tfrac{1}{4 } - 2w_{5\text{-}8}^{xx}\right ) & 0 & \left(\tfrac{1}{4 } - 2w_{5\text{-}8}^{xx}\right)\\ -w_{5\text{-}8}^{xx } & 0 & w_{5\text{-}8}^{xx }
\end{array } \right]_{m_{xx } } \!\!\!\!\!\!\!\ !
\nonumber\\ & & \hspace{0.6 cm } + \left [ \begin{array}{ccc } -w_{5\text{-}8}^{xx } & 0 & w_{5\text{-}8}^{xx}\\ -\left(-\tfrac{1}{4 } - 2w_{5\text{-}8}^{xx}\right ) & 0 & \left(-\tfrac{1}{4 } - 2w_{5\text{-}8}^{xx}\right)\\ -w_{5\text{-}8}^{xx } & 0 & w_{5\text{-}8}^{xx } \end{array } \right]_{m_{yy } } \nonumber\\ & & \hspace{0.6 cm } \left .
+ \frac{1}{4 } \left [ \begin{array}{ccc } 1 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & -1 \end{array } \right]_{m_{xy } } \right ) .
\label{expand}\end{aligned}\ ] ] there remains only three independent parameters in this expression , @xmath84 , @xmath85 , and @xmath86 . in the follow section
we choose these unknowns in order to minimise the spurious velocity contribution .
in this section , we explicitly calculate the spurious force per unit volume @xmath58 ( see eq .
( [ expand ] ) ) for the case of a liquid drop of radius @xmath87 .
if we take the origin to lie at the centre of the drop , then the density @xmath7 is solely a function of distance from that origin @xmath88 .
taylor expanding the @xmath9 stencil ( see eq .
( [ xderiv ] ) ) we find the contribution to the force from this term is given by @xmath89 transforming from cartesian into polar coordinates is achieved using the relations @xmath90 by sequentially substituting these operators and performing derivatives , eq .
( [ fxp ] ) can be rewritten @xmath91 where we define @xmath92 . by symmetry
, the @xmath61 component can be obtained by interchanging the @xmath44 and @xmath61 labels in this expression .
the force can be decomposed into two terms ; a term parallel and a term perpendicular to the interface .
the perpendicular contribution results in a small deviation in the laplace pressure difference across the interface . the parallel term can not be corrected for in this way and thus it is responsible for inducing spurious flows . a parallel unit vector is given by @xmath93 and , therefore , this tangential contribution can be calculated using @xmath94 this is zero provided that @xmath95 . an analysis of other terms in eq .
( [ expand ] ) can be performed in a similar way .
for instance , the force contribution from the laplacian term is given by @xmath96 .
\end{aligned}\ ] ] by repeating the process that was used to derive eq .
( [ fp ] ) , we find that this contribution vanishes provided that @xmath95 and @xmath97 . when transformed into polar coordinates , the tangential force from the @xmath98 terms in eq .
( [ expand ] ) is given by @xmath99\nonumber\\ & & \hspace{-0.7cm}+ \left(x^3 y - x y^3 \right ) \left ( \tfrac{1}{6 } - 2b \right ) { \delta x}^2 ( d_r \rho ) ( d_r^3 \rho ) \big ] .
\label{lines}\end{aligned}\ ] ] the last two lines on the right hand side become zero when @xmath100 . generally , it is not possible to make the first two lines simultaneously zero .
however , it turns out that the first term dominates over the second , and so the best choice is @xmath101 .
the reason for this is that the width of the interface is much smaller than the radius of curvature .
the density @xmath7 is approximately constant in the bulk regions but varies sharply in the interface .
if we denote the width of the interface to be @xmath102 then the largest value for a derivative can be typically obtained using @xmath103 . since the operator @xmath104 appears one more time on the first line than the second , and it contains an extra factor of @xmath105 or @xmath106
, then we expect the ratio in the magnitude of the first two lines to be approximately @xmath107 .
in fact , a detailed analysis explicitly calculating the two functions based on a hyperbolic tangent interface profile reveals that their maxima differ by a factor @xmath108 .
thus provided @xmath109 the second line will be negligible compared to the first . now that we have obtained a unique choice for the equilibrium , it is interesting to note that , to the best of our knowledge , none of the previously proposed free energy lattice boltzmann schemes make this optimum choice .
for example , inamuro _ et al . _
@xcite choose @xmath110 and desplat _ et al . _
@xcite choose @xmath111 .
to test the predictions made in the previous section , we perform simulations on a grid of size @xmath112 .
parameters used were @xmath113 , @xmath114 , and @xmath115 , leading to liquid - gas phase separation with densities @xmath116 and @xmath117 .
the interfacial tension was set using @xmath118 , giving an interface width of approximately @xmath119 lattice sites .
a drop of radius @xmath120 was initialised at the centre of the system and simulations were run for @xmath121 time - steps to allow steady state conditions to be reached .
figure [ fig1](a ) shows the flow profile around the drop for a typical set of parameters .
we clearly observe eight vortices in the gas phase surrounding the curved interface of the drop .
[ fig1](b ) shows the dramatic reduction in the spurious flow when the best parameter choice is used . to verify that the we have , indeed ,
obtained an optimum choice of parameters , we show that the spurious velocity is minimised for each of the parameters separately .
the effect of changing one parameter in isolation was found numerically by fixing all other degrees of freedom and scanning the chosen parameter s value over some range .
this scanning procedure was performed sufficiently slowly to be quasi - static .
figure [ fig2 ] shows the results . the spurious velocity on the @xmath61-axis is defined to be the maximum velocity magnitude in the system .
the solid curve in fig . [ fig2](a ) shows how this velocity varies with @xmath84 .
it clearly reaches a minimum very close to that predicted theoretically ( @xmath122 ) .
the spurious velocity never reaches exactly zero because our analysis only considered terms up to @xmath123 in the taylor series expansion for the stencils ( eqs .
( [ xderiv ] ) and ( [ laplace ] ) ) . in reality , higher order terms also induce spurious velocities but these terms will be @xmath124 smaller , and so have much less effect provided the interface width is reasonably large . the other curves in fig .
[ fig2 ] show minima which correspond well with the values @xmath125 , @xmath126 , @xmath100 , and @xmath97 predicted in section [ unique ] . note that when the simplest choice for calculating the derivatives is used ( see eq .
( [ simple ] ) ) , the spurious velocities are @xmath127 times larger than for the optimum choice ( this corresponds to @xmath128 in fig .
[ fig2](b ) ) .
to obtain eq .
( [ expand ] ) we assumed that @xmath55 and so the lattice boltzmann equation reduced to @xmath130 .
the more general case can be calculated under steady state conditions by sequentially substituting eq .
( [ latbolt ] ) back into the @xmath131 term on the right hand side .
this gives @xmath132.\end{aligned}\ ] ] therefore , @xmath133 can be expressed in terms of the equilibrium distributions along lines of points radiating out following the velocity vector directions .
the magnitude of these contributions decrease by a factor @xmath134 for each step away .
the stencils in eq .
( [ expand ] ) are no longer finite in size .
for instance , the inner @xmath135 region of the @xmath9 stencil now looks like @xmath136_{p_0}.\end{aligned}\ ] ] converting this into continuous operators gives @xmath137 \!+\ !
\dots \right ) p_0,\end{aligned}\ ] ] where the sum @xmath138 is @xmath139 since @xmath138 is simply a numerical factor multiplying all the @xmath140 terms then it will also pre - multiply the spurious force expressions in eqs .
( [ fp ] ) and ( [ lines ] ) .
such a change does not alter the optimum choice of equilibrium when @xmath129 . the solid line in fig .
[ fig3 ] shows how the numerically calculated spurious velocities depend on @xmath27 using the optimum choice for all other parameters .
the function @xmath138 passes through zero when @xmath141 .
this condition was calculated by swift _
_ @xcite using a different method .
it does not correspond exactly with the minimum of the curve because higher order spurious velocities become important in this region .
as @xmath27 is increased the spurious velocities rapidly increase in magnitude .
these are principally generated by the small term on the second line of eq .
( [ lines ] ) being multiplied by the very large numerical factor @xmath138 , which grows as @xmath142 .
rather than incorporate the problematic @xmath98 terms into the equilibrium distribution , it is also possible to put them into a body force . the term @xmath30 in eq .
( [ latbolt ] ) is given by @xmath143 \left ( u_\alpha g_\beta \!+ \!u_\beta g_\alpha \right)\right ) , \nonumber\\\end{aligned}\ ] ] where @xmath144 is a body force that now appears on the right hand side of the navier - stokes equation ( [ nsfinal2 ] ) .
in this new forcing scheme , the @xmath145 , @xmath146 , and @xmath147 terms are removed from the equilibrium distribution ( [ equilibrium ] ) and replaced by @xmath148_{\tfrac { m_{xx}-m_{yy}}{2 } } \nonumber\\ & & - \frac{1}{\delta x } \left [ \begin{array}{ccc } f & e & f \\ 0 & 0 & 0 \\ -f & -e & -f \\ \end{array } \right]_{m_{xy}},\end{aligned}\ ] ] in the @xmath44 direction .
@xmath149 may be obtained by interchanging the labels @xmath44 and @xmath61 and transposing the stencils .
such a procedure leaves the continuum navier - stokes equation unchanged .
this method has the advantage of allowing extra degrees of freedom in choosing the stencils as compared to the standard lattice boltzmann .
in particular , we can choose to have a symmetry between the derivatives in the @xmath44 and @xmath61 directions ( _ i.e. _ the @xmath61 stencil can be obtained by transposing the @xmath44 ) . by comparison , eq .
( [ expand ] ) clearly can not have this property .
this improvement in the isotropy of the governing equation helps to reduce spurious velocities further . the dotted line in fig .
[ fig2](b ) shows numerical results of how the spurious velocities change as a function of the stencil parameter @xmath150 .
the minimum of this curve lies at @xmath151 , corresponding to a standard choice . by comparing the magnitude of the spurious velocity at this point with the minima from the other curves ,
we conclude that the forcing method leads to a further @xmath152 fold reduction , giving a typical value of @xmath153 .
another advantage of using forcing is shown in fig .
[ fig3 ] . as @xmath27
is increased the spurious velocities normally become non - negligible due to the large numerical factor @xmath138 in eq .
( [ s ] ) multiplying the otherwise small contribution from the second line in eq .
( [ lines ] ) . in the forcing method
this term goes to zero allowing for accurate simulation of more viscous systems . in general , the disadvantages of using forcing methods are that they make boundary conditions more complicated and , if being run on a parallel computer , require more information to be passed between computer micro - processors .
this is because the standard two dimensional lattice boltzmann method only requires information from the surrounding 8 points to update each lattice site , whereas the forcing method requires information from 24 points .
a number of different lattice boltzmann schemes have been proposed for simulating 3d systems using @xmath154 , @xmath155 or @xmath156 lattice velocities . in this paper
we find that @xmath155 lattice vectors are necessary to ensure the reduction in spurious velocities .
one way to define the velocity vectors in this model is the following : @xmath157 lie along the nearest neighbour directions @xmath158 , \nonumber\end{aligned}\ ] ] and @xmath159 are in the 12 square diagonal directions @xmath160\!. \nonumber\end{aligned}\ ] ] analogous to the definitions for the gradient and laplacian stencils given in eqns .
( [ xderiv ] ) and ( [ laplace ] ) , we define @xmath161 , \nonumber\\ \bar{\nabla}^2\!\ ! & = & \!\ !
\frac{1}{\delta x^2 } \!\ !
\left [ \ !
\left ( \begin{array}{ccc } 0 & d & 0 \\ d & c & d \\ 0 & d & 0 \\ \end{array } \right ) \!\!,\!\ ! \left ( \begin{array}{ccc } d & c & d \\ c & e & c \\ d & c & d \\ \end{array } \right ) \!\!,\!\ !
\left ( \begin{array}{ccc } 0 & d & 0 \\ d & c & d \\ 0 & d & 0 \\ \end{array } \right ) \
! \right ] , \nonumber\\ \label{laplacian3d}\end{aligned}\ ] ] where @xmath162 , @xmath163 , @xmath164 and the left , middle , and right matrices show slices of the stencil when @xmath165,@xmath166 , and @xmath167 , respectively . in three dimensions ,
additional terms containing @xmath168 , @xmath169 , and @xmath170 appear in the equilibrium distribution ( [ equilibrium ] ) . by using the same procedure as in section [ unique ] , this distribution can be uniquely defined .
one additional complication in the three dimensional case is that there is no longer a single vector defining a tangent to the surface of a drop .
instead we use three vectors @xmath171 , @xmath172 , and @xmath173 and require that the spurious force parallel to each one of these is zero . for instance , the previous expression in eq . ( [ lines ] ) becomes @xmath174.\nonumber\\\end{aligned}\ ] ] for the first line to be zero then @xmath175 .
the second line , therefore , is zero only when @xmath176 .
the last two lines vanish when @xmath100 .
a summary of all parameters obtained using this procedure is given below : @xmath177 note that for a system one lattice unit wide this equilibrium reduces to the 2d result .
in this paper we analysed the spurious velocities from two different methods : a standard lattice boltzmann scheme and a new forcing method .
firstly , we calculated the spurious forces which originate when the continuous operators in the navier - stokes equation are replaced by stencils ( in other word the contribution from the next order in the taylor series expansion of the stencils ) .
secondly , we identify that spurious velocities result from the component of these spurious forces acting parallel to the interface .
finally , we find that by making a suitable choice of the equilibrium distribution and stencils we were able to set these parallel forces to zero ( up to fourth order in the derivatives ) . in 2d ,
the best choice of stencils for calculating the derivatives and the laplacian are : @xmath178 , % \quad \bar{\nabla}^2 = \tfrac{1}{6 { \delta x}^2 } \left [ \begin{array}{ccc } 1 & 4 & 1 \\ 4 & -20 & 4 \\ 1 & 4 & 1 \\ \end{array } \right ] .
\label{best}\end{aligned}\ ] ] using the standard lattice boltzmann model the equilibrium is given by eq .
( [ equilibrium ] ) , where the optimum choice of parameters is @xmath179 , @xmath180 , @xmath181 , @xmath182 , @xmath183 , @xmath80 , and @xmath184 . in 3d
the corresponding results are summarised in eqns .
( [ laplacian3d ] ) and ( [ 3dres ] ) .
one way to improve spurious velocities further is to remove the @xmath60 terms from the equilibrium distribution and implement them as a body force .
this force is then explicitly calculated by taking derivatives of @xmath98 using the stencil in eq .
( [ best ] ) ( or eq . ( [ laplacian3d ] ) in the 3d case ) .
the additional symmetry in the resulting equations leads to a further reduction in spurious velocity size .
the authors would like to thank prof .
yeomans for her help in writing this paper .
to conserve mass and momentum the first two constraints on the equilibrium distribution must be @xmath185 the higher order moments of @xmath29 are chosen such that the resulting continuum equations describe the dynamics of a non - ideal fluid .
a suitable choice is @xmath186 where @xmath187 will become the shear and bulk kinematic viscosities , respectively .
the speed of sound is given by @xmath188 , where @xmath9 is the fluid pressure ( [ p0 ] ) .
the term involving @xmath37 on the right hand side of eq .
( [ pcons ] ) is necessary to ensure galilean invariance @xcite .
for an ideal gas with @xmath189 it is zero , but in the more general case it must be included . by applying the chapman - enskog expansion to the lattice boltzmann equation ( [ latbolt ] ) @xcite
, we obtain the continuity equation for the total density @xmath191 and the navier - stokes equation for the fluid momentum @xmath192 where the fluid velocity is defined by @xmath193 note that this definition differs slightly from the lattice fluid velocity ( [ rho ] ) in the case when the body force is non - zero .
it is @xmath194 , and not @xmath25 , which is used to calculate the spurious velocities in section [ force ] .
swift , e. orlandini , w.r .
osborn and j.m .
yeomans , phys .
e * 54 * 5041 ( 1996 ) .
a.j . wagner , int . j. mod . phys .
b * 17 * 193 ( 2003 ) .
t. lee and p. f. fischer , phys .
e * 74 * , 046709 ( 2006 ) .
t. seta and k. okui , j. fluid sci .
* 2 * 139 ( 2006 ) .
t. inamuro , n. konishi , f. ogino , comp .
* 129 * 32 ( 2000 ) .
holdych , d. rocas , j.g .
georgiadis , and r.o .
buckius , int . j. mod
c * 9 * 1393 ( 1998 ) .
luo , phys .
e * 62 * 4982 ( 2000 ) .
desplat , i pagpnabarraga , and p. bladon , comp .
. comm . * 134 * 273 ( 2001 ) . c. m. pooley , _
. thesis _ , oxford university ( 2003 ) .
|
spurious velocities are unphysical currents that appear close to curved interfaces in diffuse interface methods .
we analyse the causes of these spurious velocities in the free energy lattice boltzmann algorithm . by making a suitable choice of the equilibrium distribution , and by finding the best way to numerically calculate derivatives
, we show that these velocities may be decreased by an order of magnitude compared to previous models .
furthermore , we propose a momentum conserving forcing method that reduces spurious velocities by another factor of @xmath0 . in three dimensions
we find that 19 velocity vectors is the minimum number necessary .
|
the weekly vla monitoring of sagittarius a * , the @xmath5 supermassive black hole at the center of our galaxy , has accumulated over 20 years of variability data at @xmath2 , @xmath6 , @xmath1 , @xmath7 , and @xmath4 cm wavelengths . the sampling within this period has been somewhat irregular ( zhao , bower , and goss 2001 ) .
nonetheless , the power spectral density ( psd ) reveals a clear peak near @xmath8 hz , with a progressively smaller significance at longer wavelengths .
this frequency corresponds to a periodic modulation of approximately @xmath9 days ; the actual best - fit period extracted from the combined data sets is @xmath10 days .
this result is at once intriguing and unsettling .
the fact that a @xmath11-minute keplerian period has been seen in sagittarius a * s infrared emission ( genzel et al . 2003 ) makes this cyclic modulation easier to accept . yet the implied radio period ( @xmath12 days ) contrasts sharply with the dynamical time scale ( @xmath13 minutes ) associated with motion in the inner disk ( see melia and falcke 2001 for a recent review ) .
perhaps the periodic radio signal is a false detection due to a combination of a random process and the irregular sampling pattern .
however monte carlo tests with data created from various sources of noise using this same sampling do nt seem to bear this out .
regardless of the type of noise used in the simulations including white noise , gaussian noise around a mean , and a poisson distribution of flares the probability of false detection due to any such random process appears to be less than @xmath14 .
( however , one should keep in mind that @xmath15or red noise could in principle mimic such a periodic signal if the random fluctuations have an appropriate _ scale _ ; g. bower , private communication . )
perhaps also supporting the view that this period may be real is the observation that both the absolute ( @xmath16 ) and fractional ( @xmath17 amplitudes of the pulsed component increase toward shorter wavelengths , yet the period appears to be independent of wavelength .
the @xmath10-day cycle evident primarily at @xmath2 and @xmath6 cm may be a valuable tool for probing sagittarius a * s inner workings should it truly have something to do with the source .
high - resolution vla observations have already ruled out the possibility that such a period might be produced by an orbiting emitting object ( bower and backer 1998 ) .
the @xmath10-day orbit of a companion to sagittarius a * would have a radius @xmath18 a.u .
, corresponding to an angular separation of @xmath19 milliarcseconds at @xmath20 kpc .
a compact @xmath21-jy source separated from sagittarius a * by this amount would have easily been detected with the vlba at wavelengths shorter than @xmath1 cm .
the unlikelihood of sagittarius a * having an orbiting companion is further supported by its observed lack of proper motion , which precludes any possible association with rapidly moving components .
in addition , a stellar origin for such a source would fall well short of the power required to account for the measured radio emission . all in all , the evidence seems to favor an interpretation in which the @xmath10-day periodic variations , if real , are intrinsic to sagittarius a * itself .
the characteristics of this 106-day cycle constrain the nature of its origin rather tightly .
first , the observed period is , as we have said , independent of wavelength .
the emission in sagittarius a * at different frequencies is produced on different spatial scales ( see , e.g. , melia , jokipii , and narayanan 1992 ) , so the period should be induced by a single process .
otherwise , we would expect to see different periods at different frequencies .
what is required is something that can cause correlated fluctuations across a broad range of wavelengths .
second , the period is four orders of magnitude longer than the dynamical time scale in the inner disk surrounding sagittarius a*. could it be produced on a much larger spatial scale ?
sagittarius a * s 2-cm emission is produced within a region no bigger than @xmath22 a.u .
( krichbaum et al . 1999 ) , for which the corresponding dynamical time scale is calculated to be about one and a half days ; so the answer is apparently no .
higher frequency emission is produced within still smaller regions , associated with even smaller time scales .
we may ask then , whether this modulation could be produced by a corrugation wave in an accretion disk , which is used to account for the quasi - periodic oscillations ( qpos ) seen in low - mass x - ray binaries .
these waves have periods that are much longer than the corresponding dynamical time scale ( kato 1990 ) , but they depend on radius and thus may not be able to account for the first feature described above . moreover , sagittarius a * s light curves show quite stable periodic fluctuations , rather than the uncorrelated segments constituting qpos .
the evidence is pointing to a single process evolving in a relatively confined region , certainly no bigger than 100 schwarzschild radii ( @xmath23 cm for a @xmath24 black hole ) . in an earlier paper ( liu and melia 2002 )
, we noted that the gravitational acceleration in a kerr metric acquires a dependence on poloidal angle ( relative to the black hole s spin axis ) , so that matter orbiting above or below the equatorial plane experiences a restoring force toward the equator .
this results in the precession of its angular momentum vector about the black hole s axis of rotation .
the physical conditions in sagittarius a * s keplerian region lead to strong coupling between neighboring rings , and the @xmath25 compact disk therefore torques more or less as a rigid body . under an appropriate set of circumstances
( see liu and melia 2002 ) , the precession period can exceed a hundred days , and the long - term radio modulation in sagittarius a * may be closely related to its short - term x - ray and infrared variability after all via the dynamical properties of the disk in a rotating spacetime .
however , whereas this earlier treatment established the viability of a spin - induced disk precession in accounting for the 106-day modulation , it left open the question of how exactly this periodicity is manifested .
the purpose of this paper is to demonstrate that the observed fluctuations , frequency - dependent amplitudes , and periodicity can in fact be produced as a result of partial occultation of a nonthermal halo surrounding sagittarius a * by the pivoting disk .
as we shall see , the properties inferred for this source based on its spectrum and polarization characteristics ( see liu and melia 2001 ) , produce a surprisingly accurate fit to the observed radio lightcurve , its modulated amplitude , and the frequency - dependent signal strength . in 2 we describe the method used for this analysis , including the source geometry and particle properties .
we summarize our results in 3 , and present our conclusions in 4 .
the level of polarization seen in sagittarius a * at mm / sub - mm wavelengths approaches @xmath26 ( aitken et al . 2000 ) .
however , this object reveals a lack ( @xmath27 ) of linear polarization below @xmath28 ghz , though some circular polarization ( @xmath29 ) has been detected ( bower et al .
1999 ; bower et al .
2001 ) . these prominent spectral and polarimetric differences ( melia , bromley , & liu 2001 ) between the cm and the mm / sub - mm bands suggest two different emission components in sgr a*. as we have already noted , higher frequencies correspond to smaller spatial scales ( see also narayan et al .
1995 ) , so the mm / sub - mm radiation is likely produced in the vicinity of the black hole .
earlier work ( e.g. , melia 1992 , 1994 ; coker & melia 1997 ) has indicated that sgr a * is accreting from the stellar winds surrounding the black hole and that the infalling gas circularizes at a radius of @xmath30 .
recent work on sagittarius a * s emissivity ( melia , liu , & coker 2001 ; bromley et al .
2001 ) has demonstrated that the inner @xmath31 of the resultant keplerian structure can not only account for the mm / sub - mm properties via thermal synchrotron emission , but it may also produce sgr a * s x - ray spectrum in the quiescent state ( baganoff et al . 2001 ) via comptonization of the mm / sub - mm photons . on the other hand , the cm radio emission appears to be produced by non - thermal synchrotron emission ( liu & melia 2001 ) .
the integrated @xmath32 cm luminosity of sgr a * is comparable to the power extracted from its spin energy via a blandford - znajek type of electromagnetic process if @xmath33 ( liu and melia 2002 ) .
the 106-day modulation could therefore presumably arise when the precessing disk periodically shadows the non - thermal particles flooding the region surrounding the black hole as they escape from their creation site near the event horizon , essentially forming an expanding halo of relativistic particles . some observational evidence for this has recently been provided by vlba closure amplitude imaging techniques at 7 mm ( bower et al .
2004 ) , which point to an intrinsic source size of @xmath34 for sagittarius a * at this wavelength .
we will therefore adopt the basic model displayed in figure 1 .
the compact disk ( with outer radius @xmath35 ) that produces the mm / sub - mm emission is opaque to radiation longward of @xmath36 cm , which we assume originates from the surrounding , semi - transparent halo with radius @xmath37 . to accurately determine the disk shadowing effect
, we integrate the non - thermal synchrotron emissivity @xmath38 ( erg @xmath39 s@xmath40 ster@xmath40 hz@xmath40 ) along each given line - of - sight ( see figure 1 ) , including the effects of opacity , such that @xmath41 in this expression , @xmath42 is the optical depth in terms of the absorption coefficient @xmath43 @xmath44@xmath45 .
the flux is then calculated by integrating @xmath46 over all solid angles pertaining to the source : @xmath47 where @xmath48 is the angle between the line of sight to the black hole ( at the center of this geometric configuration ) and the emitting area associated with @xmath49 .
numerically , we calculate the flux on a grid ( see figure 2 ) , assuming @xmath50 to be constant over each mesh element , so that the total flux may be written @xmath51 where @xmath52 is the distance to the galactic center , @xmath53 is the side length of each mesh element , and @xmath54 for @xmath55 .
we assume that within the halo the emitting particles form a power - law distribution @xmath56 with @xmath57 .
thus , @xmath58 where @xmath59 is the magnetic field strength , @xmath60 is the pitch angle ( the angle between the electron s velocity and @xmath61 ) , @xmath62 , and @xmath63 is the modified bessel function of the second kind . similarly , @xmath64 equations ( [ j ] ) and ( [ alpha ] ) then become @xmath65 where @xmath66 equations ( [ j_nu ] ) and ( [ alpha_nu ] ) are used in equations ( [ inu ] ) through ( [ flux2 ] ) to calculate the total flux . for the calculations described here ,
the disk precession is handled analytically , with an orientation prescribed as a function of time , based on our previous work ( liu and melia 2002 ) .
a full sph simulation is currently underway , and a more realistic time - dependent geometric profile will be published elsewhere ( rockefeller , fryer , and melia 2005 ) . a key concern is whether the disk succumbs to the bardeen - pettersen effect , in which the radially - dependent precession frequency can lead to dissipation between neighboring rings , ultimately forcing the inner @xmath67 of a disk inclined to the black hole s spin axis to eventually settle into the equatorial plane .
it is commonly thought that an accretion disk surrounding a spinning black hole must be warped , with an overall angular momentum vector possibly misaligned relative to the spin axis , but with its inner portion fully flattened at the equator . but detailed hydrodynamical simulations are now showing that this effect , although fully realized under a majority of physical conditions , can be negated in cases where other couplings between neighboring rings in the disk are strong .
an example of this occurs when the disk plasma has a small mach number ( typically @xmath68 ) , for which large pressure gradients can then distribute the radially - dependent precession torque and force the disk to rotate as a rigid body ( see , e.g. , nelson & papaloizou 2000 ) . in the case of sgr a * , we estimate from the spectrum and polarization properties of its disk that the mach number in this system is @xmath69 .
not surprisingly then , the latest sph simulations confirm the earlier semi - analytic results that the small disk in sgr a * is probably precessing as a rigid body . in our calculation , we therefore evolve the disk orientation analytically in time , assuming rigid body precession , and recalculate the flux at @xmath2 , @xmath6 , and @xmath1 cm for each step .
as we can see from equations ( [ j_nu ] ) and ( [ alpha_nu ] ) , the model contains five basic parameters : @xmath70 , @xmath59 , @xmath37 , @xmath35 and @xmath71 .
here @xmath71 is the angle between the angular momentum vector of the disc and the spin axis of the black hole ( see fig .
[ fig:1 ] ) . the other two angles @xmath72 and @xmath73 change in every time step .
the pitch angle @xmath60 depends on distance @xmath74 in equation ( [ inu ] ) . for relativistic electron energies , we can approximate @xmath75 , where @xmath76 is the angle between the magnetic field vector and the line of sight . for the sake of simplicity
, @xmath70 is taken to be a power - law function @xmath77 ( with @xmath78 ) , in terms of @xmath79 , the distance from the center of the halo .
the magnetic field @xmath59 is assumed constant throughout the halo . consequently , there are effectively six overall parameters in this model : @xmath80 , @xmath59 , and @xmath81 , @xmath37 , @xmath35 and @xmath71 . throughout this work the accretion disk
is considered to be optically thick to all radiation onward of @xmath82 cm , so any ray of light intercepting the disk is immediately stopped at that point . since the distance to the galactic center @xmath52 is very large compared to the radius of the halo we may also assume parallel lines of sight .
we have explored models with @xmath83 , @xmath84 for the halo , and a tilting angle @xmath71 between @xmath85 and @xmath86 of @xmath87 ( or @xmath88 measured from the xy - plane ) . based on earlier spectral fitting calculations , we know that typical values for @xmath80 are of order @xmath89 erg@xmath90 . for a given power - law dependence of @xmath70 , this then also fixes the density throughout the emitting region . also based on earlier spectral fitting calculations ( see liu and melia 2001 ) , we infer a magnetic field intensity of order @xmath82 g. we point out , however , that the specific choice of the function @xmath91 has little effect on the shape of the light curve ( as we shall see in figs . 3 to 8 below ) , though the calculated flux does change in response to the changing column depth through the emitting medium .
consequently , we have chosen the model assumptions to be as simple as possible . a uniform electron distribution with @xmath92 , though unrealistic , allows a first critical analysis of the shadowing effect .
a good fit to the folded light curves ( shown in figs .
3 and 4 ) is possible with either a uniform electron distribution , or a power law . for the sake of specificity , we here focus on the results produced with a uniform halo ( figs . 3 , 4 , and 5 ) , and one in which @xmath93 ( figs .
6 , 7 , and 8) .
we have also found that the light curve produced as a result of shadowing ( dark solid curve in these figures ) is only weakly dependent on the absolute value of @xmath94 and @xmath59 , which instead affect primarily the absolute flux level .
the shadowing effect is therefore relatively independent of the details of the underlying model ; it appears to be a rather robust phenomenon . in figure [ fig:3 ] ,
we show the calculated light curve in comparison with the data at @xmath2 cm .
this fit was produced with @xmath95 erg@xmath96 , @xmath97 , @xmath98 and @xmath99 .
every data point on the solid line represents the flux ( in jy ) obtained for a certain position of the accretion disk , which precesses around the ( fixed ) spin vector of the black hole . in every time step between time
@xmath100 days ( which also corresponds to @xmath101 ) and @xmath10 days ( which corresponds to an angle of @xmath102 of @xmath85 with respect to the starting point ) the precessing angle is increased . at the maximum ( @xmath103 days ) , the accretion disk is seen edge - on and consequently , since the disk is here assumed to have no thickness , the integration over the synchrotron emissivity extends over the full extent of the halo . in order to illustrate this , for a tilting angle of @xmath104 degrees , i.e. , @xmath85 parallel to the @xmath105-axis in figure [ fig:1 ]
, the curve produced by our simulation would be constant at the value of the maximum in figure [ fig:3 ] , since the accretion disk would be seen edge - on in every time step .
it should be emphasized that in figure [ fig:3 ] , the mean flux has not been adjusted vertically , but rather is set by the parameter values suggested by earlier spectral studies . one of the principal results of our simulation is that the parameters that yield this correct flux , together with a reasonable tilting angle of @xmath106 , yield both the correct amplitude of the observed modulation , and its variation in time . in figure 3 , we also show the corresponding light curve for @xmath6 cm , using the same parameters as those for @xmath2 cm .
notice that here the shadowing is less effective , which is due to two factors : first , the halo becomes progressively more opaque with increasing wavelength , which means that a greater fraction of the observed intensity arises from the medium in front of the disk ; second , the emitting volume contributing to the overall flux increases , rendering the fraction of the halo occulted by the disk less significant with increasing wavelength . the optical depth through the halo at @xmath2 cm is @xmath107 , so most of the photons pass through the medium unaffected .
however , @xmath108 goes as @xmath109 ( see eq .
[ alpha_nu ] ) ; at @xmath6 cm , it is @xmath110 .
we note that both the shape of the light curve and the amplitude of modulation fit the data just as well at @xmath6 cm as they do at @xmath2 cm , for the _ same _ system parameters . by 3.6 cm
, @xmath108 has increased to values @xmath111 and , not surprisingly , most of the modulation due to the precessing disk , which lies below the photosphere at this wavelength , has disappeared ( see fig .
[ fig:5 ] ) .
this trend is reflected in the data as well , though we note that our calculated flux is a factor 2 lower than that observed .
we attribute this to the fact that for simplicity we have assumed a fixed halo size . in reality
, the fact that @xmath108 is increasing with increasing wavelength also means that the emitting volume contributing to the overall flux must itself increase .
it would be a simple matter to reproduce the correct flux at @xmath1 cm by making @xmath37 wavelength dependent , but this would violate our goal of keeping this model as simple as possible . in principle , a full numerical simulation of the disk precession and particle acceleration and escape should yield the correct behavior of quantities such as @xmath70 with radius , which would then make this point moot .
figure 4 demonstrates the same effects as those discussed in figure 3 , except now for a halo with uniform magnetic field , but with a density @xmath93 , with @xmath112 erg@xmath96 and with @xmath113 .
the fact that a reasonable fit to the data may be made with such disparate halo geometries affirms the robustness of this model .
however , changing the parameter values does produce quantitative variations in the modulation amplitude , and its dependence on wavelength .
figure [ fig:5 ] shows the sensitivity of our simulation to changes in @xmath94 .
we see that the shape of the curve ( at @xmath2 cm ) is not affected .
this is hardly surprising , since @xmath114 for all these densities .
however , since @xmath115 , the overall flux does change with increasing @xmath70 . the consequence of changing the magnetic field is shown in figure [ fig:5 ] . as was the case for @xmath70 , the shape of the curve does not depend on this value
also , since the dependence of @xmath116 and @xmath38 on @xmath59 is only to the power @xmath117 , the impact on the average flux is also less pronounced .
the impact of parameter variations on the light curve at other frequencies is similar to that for @xmath2 cm , so we wo nt show these here .
changes in the disk s outer radius produce the modified modulations shown in figure [ fig:5 ] .
all the curves coincide at the peak since the edge - on disk does not shadow any part of the halo .
also , the amplitude decreases as the disk becomes smaller .
the picture described here works rather well in accounting for ( 1 ) the absolute flux , ( 2 ) the observed modulation , and ( 3 ) the amplitude of the observed peaks in the @xmath2 and @xmath6 cm light curves of sagittarius a*. however , future work should include a more realistic time - dependent disk profile , the halo , and a more detailed exploration of possible magnetic field configurations .
for example , in this paper , we have restricted our attention exclusively to isotropic halos surrounding sgr a * , though we have allowed for a possible radial variation in its physical conditions , such as the particle number density and the magnetic field .
clearly , though , one might expect some impact on the results should the halo not be spherical , but rather toroidal , or cylindrical .
but as we have already indicated earlier , the halo is optically thin to radio emission , and the dominant effect is the shadowing of this emitter by the disk .
our brief survey of the various possible geometries has indicated that the _ shape _ of the modulated light curve is only weakly dependent on the halo s structure .
nonetheless , the absolute flux level _ is _ indeed sensitive to the halo s geometry .
some of the additional work necessary to fully explore the range of possibilities ( e.g. , with sph simulations ) is currently underway and will be reported elsewhere . finally , we point out here something that was overlooked in liu and melia ( 2002 ) .
this is the fact that if the disk precession is responsible for the modulation , then the observed period should actually correspond to half of the precession period because of symmetry .
this has some impact on the inferred value of the black hole spin @xmath118 .
in addition , it may be possible to see two periods , the primary one associated with half the precession period , and a second at roughly twice the first , corresponding to the full period .
the latter would arise if the disk were not perfectly symmetric , so that the tilt in the first half of a cycle is not exactly the same as the tilt in the second .
this is a prediction of the model presented in this paper and could be included in future work .
interestingly , there may already be some evidence for a second period , roughly twice as long as the first , in data acquired since 2000 ( zhao , private communication ) .
to close , it is still not entirely clear whether this @xmath0-day period in sagittarius a * s emission is in fact due to an intrinsic modulation .
however , we have now shown that a spin - induced disk precession for a slowly rotating black hole can account for this long period and , in addition , that the partial occultation of a @xmath119-cm emitting halo by the radio - opaque precessing disk can also produce the correct amplitude and time - dependent light curves seen at @xmath2 and @xmath6 cm .
the halo becomes optically thick longward of @xmath120 cm , so the shadowing by the disk can no longer produce a modulation at these wavelengths . future work
observationally should focus more carefully on the question of whether red noise fluctuations could mimic the effect of a long periodic modulation in this source .
theoretically , future work will determine the temporal profile of the precessing disk more accurately , and will provide us with a better grasp of the halo s internal structure .
m.p . would like to thank the german fulbright commission for support far beyond the financial aspect of academics , which made this work possible .
furthermore he likes to thank steward observatory for kind hospitality and casey meakin and philipp strack for helpful discussions .
this work was supported by nasa grant nag5 - 9205 and nsf grant 0402502 at the university of arizona .
|
there is some evidence , though yet unconfirmed , that sagittarius a*the supermassive black hole at the galactic center emits its radio waves modulated with a @xmath0-day period .
what is intriguing about this apparent quasi - periodicity is that , though the amplitude of the modulation increases with decreasing wavelength ( from @xmath1 to @xmath2 cm ) , the quasi - period itself does not seem to depend on the frequency of the radiation .
it is difficult to imagine how a binary companion , were that the cause of this modulation , could have escaped detection until now . instead
, it has been suggested that the spin - induced precession of a disk surrounding a slowly rotating black hole could have the right period to account for this behavior . in this paper
, we examine how sagittarius a * s light curve could be modulated by this mechanism .
we demonstrate that the partial occultation of a nonthermal halo by a compact , radio - opaque disk does indeed produce the observed frequency - dependent amplitude .
this appears to be in line with other observational arguments suggesting that sagittarius a * s mm / sub - mm spectrum is produced by a @xmath3 schwarzschild - radius disk , whereas its cm - waves originate from a nonthermal particle distribution in a halo extending out to over @xmath4 schwarzschild radii .
interestingly , this model suggests that the observed period corresponds to half the precession period and that a non - axisymmetric disk could produce a second period roughly twice as long as the first .
|
the cosmic microwave background ( cmb ) is now a well known probe of the early universe .
the temperature fluctuations in the cmb , especially the so - called acoustic peaks in the angular power spectrum of cmb anisotropies , capture the physics of primordial photon - baryon fluid undergoing oscillations in the potential wells of the dark matter @xcite . the associated physics involving the evolution of a single photon - baryon fluid under compton scattering and gravity
are both simple and linear , and many aspects of it have been discussed in the literature since the early 1970s @xcite . the gravitational redshift contribution at large angular scales @xcite and the photon - diffusion damping at small angular scales @xcite complete this description . by now , there are at least five independent detections of the first , and possibly the second and the third , acoustic peak in the anisotropy power spectrum @xcite .
we summarize these results in figure [ fig : cl ] . given the variety of experiments that are either collecting data or reducing data that were recently collected , more detections that extend to higher peaks are soon expected .
the nasa s map mission is expected to provide a significant detection of the acoustic peak structure out to a multipole of @xmath0 1000 and , in the long term , the esa s planck surveyor , will extend this to a multipole of @xmath0 2000 with better frequency coverage and polarization sensitivity . a discussion of these recent results and implications for cosmology are presented in the contribution to these proceedings by a. melchiorri .
an additional important aspect with respect to cmb is that in transit to us , photons are affected by the large scale structure along the way .
the modifications come from effects related to gravity , such as through frequency shifts associated with the integrated sachs - wolfe effect @xcite , and through effects related to scattering such as the sunyave - zeldovich ( sz ) effect @xcite .
we refer the reader to ref .
@xcite for a discussion of these contributions and their importance for understanding the large scale structure via precision cmb measurements .
the detection of coherent oscillatory features in the cmb temperature anisotropy power spectrum suggests that if our physical description of them is correct , then , the large scale structure at low redshifts should also contain a signature of oscillations .
these troughs and peaks are associated with baryons which under went acoustic oscillations with photons as a single photon - baryon fluid prior to recombination at small wavenumbers , between the radiation and matter oscillatory peaks , due to a velocity term that source the growing mode of matter perturbations .
also , radiation oscillations project the sound horizon at the recombination while baryon oscillations project the sound horizon at the end of the compton - drag epoch . in principle
, these two epochs can be different , but for all purposes , the horizons are similar for currently favorable @xmath1cdm cosmological models .
we refer the interested reader to discussions in @xcite , and references therein , for further details . ] . unlike oscillations in the angular power spectrum of cmb anisotropies ,
oscillations due to baryons in the matter power spectrum of the large scale structure are highly suppressed due to the low baryon content .
they are also easily erased at late times due to the non - linear gravitational evolution of density perturbations @xcite . compared to amplitudes of oscillations in the radiation ,
which vary by at least a factor of 2 , baryon oscillations have amplitudes which are significantly lower with variations at the level of @xmath0 8% for currently favored @xmath1cdm cosmologies .
these oscillations have effective widths in fourier space of order @xmath2 h mpc@xmath3 .
as we discuss , these issues put several strong observational constraints for a reliable detection of these oscillations .
the verification on the presence of baryon oscillations in the large scale structure , however , is not impossible .
several attempts have already been made with three - dimensional redshift surveys such as the 2df galaxy redshift survey and information from measurements related to angular clustering @xcite , though there is still no significant evidence for the presence of baryon oscillations .
the current three - dimensional surveys lack the required volume , with sizes of order @xmath4 h@xmath3 mpc in all three dimensions , to reliably resolve the oscillations while three dimensional power spectra obtained through an inversion of the angular correlation function should not contain oscillatory features due to the nature of the inversion and the widths of window functions involved , given the expected widths of oscillatory features @xcite . while ongoing redshift surveys such as the sloan digital sky survey ( sdss ; @xcite ) allow a strong possibility to detect these oscillations , in future , one can also use projected clustering in fourier space , such as the angular power spectrum similar to that of cmb anisotropies , to detect baryon oscillations .
this is useful given that one does not require precise redshift information but rather spatial information , and estimates of redshift for binning purposes , from wide - field surveys .
in addition to galaxy survey data , a number of additional observational efforts are under way or planned to image the large - scale structure of the universe out to redshift of a few .
these wide - field surveys typically cover tens to thousands of square degrees on the sky and include the weak gravitational lensing shear observations with instruments such as the snap and the large aperture synoptic survey telescope , dedicated small angular scale cmb telescopes that will map the sz effect through instruments such as the south pole telescope , and the wide - field x - ray imaging survey using the dark universe explorer telescope ( duet ) mission .
the latter was recently proposed to nasa as a medium explorer mission .
in addition to their primary science goals , these surveys are expected to produce catalogs of dark matter halos , which in the case of lensing and sz surveys , such catalogs are expected to be essentially mass selected @xcite . in the case of x - ray , using cluster temperature data , one can also construct a mass selected catalog of galaxy clusters .
lensing and other optical surveys are particularly promising in that they will provide photometric redshifts on the member galaxies of a given halo ; this will render accurate determination of the halo redshift .
halo number counts as a function of redshift is a well - known cosmological test @xcite .
we can also consider the additional information supplied by the angular clustering of halos , in particular , a possibility to use them for a detection of the baryon oscillations , and to use clustering information for a new cosmological test .
it is well - known that a feature in the angular power spectrum of known physical scale and originating from a known redshift can be used to measure the angular diameter distance between us and this redshift ; this has most notably been applied to the case of cmb anisotropy power spectrum to determine the distance to redshift @xmath5 . here
, one uses the peak location of the first acoustic peak which projects on the multipolar space as @xmath6 , where @xmath7 is the sound horizon at the recombination and the @xmath8 is the associated angular diameter distance in comoving coordinates .
given the location , and a physical model for @xmath7 , one can constrain cosmology using the variation of @xmath8 as a function of cosmological parameters .
we can apply a similar argument to estimate cosmology using the clustering related to the large scale structure . in figure
[ fig : pk ] , we show the three - dimensional power spectrum and recent measurements from the 2df galaxy redshift survey , following the analysis by @xcite . in the case of the large scale structure ,
there are two prominent features in the linear power spectrum associated with the clustering of matter .
these are the horizon at the matter - radiation equality @xmath9 which controls the overall shape of the power spectrum , including a turn over , and the sound horizon at the end of the compton drag epoch , @xmath10 ; the latter controls the peak location of oscillations due to baryons in the power spectrum .
we denote these features in figure [ fig : pk ] .
similar to the projection of the sound horizon at the recombination on to the cmb anisotropy power spectrum , the angular , or multipole , locations of features in the large scale power spectrum shift in redshift as @xmath11 . in @xcite ,
this allowed us to propose the following test : measure angular power spectrum of projected clustering , @xmath12 , in several redshift bins and , using the fact that @xmath13 scales with @xmath14 , constrain the angular diameter distance as a function of redshift . unlike the case with cmb
, we can use tracers over a wide range in redshift and measure the distance as a function of redshift , or redshift bin .
when combined with cmb observations , note that the absolute physical scale of @xmath15 can be directly calibrated as cmb anisotropies provide estimates of parameters such as @xmath16 and @xmath17 accurately . when combined with the large scale structure , one can break degeneracies , such as the one between @xmath18 and @xmath19 , or given sufficient prior information on @xmath19 to constrain information related to @xmath18 and @xmath20 . to understand the projected clustering , consider the angular power spectrum of a tracer field of large scale structure in a redshift bin .
this is simply the projection of the the tracer density power spectrum @xmath21 where @xmath22 is the distribution of tracers in a given redshift bin normalized so that @xmath23 , @xmath24 is the hubble parameter . for comparison , in the bottom panel of figure [ fig : pk ] , we show the projected angular power spectrum of matter at the last scattering surface and the angular power spectrum of cmb anisotropies .
note that the acoustic oscillations associated with photons always show as peaks in the anisotropy power spectrum .
there is an overall shift in oscillation phases in the projected angular power spectrum of matter at the last scattering surface when compared to those of cmb anisotropy power spectrum .
if we have a tracer of the matter clustering near recombination , say , just after the compton - drag epoch , we should be able to make a comparison similar to the one suggested in the bottom panel of figure [ fig : pk ] . for the purpose of this discussion
, we assume a tracer of the large scale structure that correspond to mass selected halos , or galaxy clusters . if these halos trace the linear density field , @xmath25 where @xmath26 is the mass - averaged halo bias parameter , @xmath27 is the present day matter power spectrum computed in linear theory , and @xmath28 is the linear growth function @xmath29 .
a scale - independent halo bias is commonly assumed in the so - called `` halo model '' @xcite and should be valid at least in the linear regime .
equation ( [ eqn : cl ] ) then becomes @xmath30 where the function @xmath31 , associated with redshift bin @xmath32 , contains information on the halo bias , the growth function , the power spectrum normalization , and terms involved in the 3-d to 2-d projection ( such as a @xmath33 term ; see , eq .
( [ eqn : cl ] ) ) .
note that @xmath22 comes directly from the observations of the number counts as a function of redshift . in figure
[ fig : halo ] , we illustrate the proposed test .
the two curves show the halo power spectra in two redshift bins : @xmath34 and @xmath35 .
the angular power spectrum corresponding to the higher redshift bin is shifted to the right in accordance to the ratio of angular diameter distances @xmath36 .
much of this shift simply reflects the hubble law , @xmath37 .
since the physical size of the two features the overall shape of the spectrum and the baryon oscillations can be calibrated from the cmb , these measurements can in principle be used to determine the hubble constant independently of the distance ladder . in figure
[ fig : lssosc ] , we remove the smooth curves and focus only on oscillations .
it is clear from this figure that peak locations of oscillations shift in multipole space as a function of the redshift , or more appropriately , the comoving distance .
the ability to measure oscillations crucially depend on both the cosmic variance and the shot - noise associated with the tracer number counts .
we illustrate this in figure [ fig : nm ] . in the top panel
we show the associated cosmic variance errors for an all - sky survey that image two - dimensional clustering at @xmath38 .
the vertical lines correspond to errors resulting from the finite number of halos . for a reliable detection of the first three baryon oscillations
, one needs a survey with a tracer surface density of @xmath39 sr@xmath3 and this surface density corresponds to a mass limit of @xmath40 m@xmath41 at this redshift .
the cumulative signal - to - noise ratios are illustrated in figure [ fig : sn ] .
these are the values to distinguish oscillations , which we define for this purpose as any deviation about the mean prediction for clustering in the absence of any baryons .
note that in addition to the horizontal shift due to the change in angular diameter distance , the power spectra in figure [ fig : halo ] are shifted vertically due to the change in @xmath42 ( eq . [ eqn : fz ] ) . by ignoring the information contained in @xmath42 ,
the proposed geometric test in @xcite is robust against uncertainties in the mass selection , mass function and linear bias . of course , if these uncertainties are pinned down independently , both @xmath42 and the halo abundance in @xmath22 will help measure the growth rate of structure . in @xcite , we discussed in detail cosmological information that can be gained from such a test using halos that will be detected in planned sz and lensing surveys .
we can perform a similar test with cluster catalogs that are expected to be produced with upcoming wide - field x - ray surveys such as the one planned with the proposed duet mission .
the duet mission plans to map the @xmath43-steradians of the sdss and the sz deep fields to be observed with the south pole telescope .
we show expected errors and improvements in cosmological parameters by combining the duet clustering information with cmb data in figure [ fig : duet ] .
the duet mission allows a detection of baryon oscillations in the projected angular power spectra of galaxy clusters at the few sigma level .
this , by itself , is enough to break significant degeneracies associated with cosmological parameter measurements from cmb data .
since the combined test measures cosmology through the angular diameter distance at low redshifts , from clustered halos , and at the last scattering , from the cmb acoustic peak , the test allows one to probe not only the curvature but also relative abundances of various energy densities and properties of the dark energy .
the bottom panel of figure [ fig : duet ] shows errors on @xmath44 , the ratio of pressure to density of the dark energy component , and @xmath16 , from map and planck , and improvements on these errors when these cmb data sets are combined with clustering information from the duet survey .
there is at least a factor of 5 improvement in error associated with @xmath44 .
such an improvement clearly demonstrate the need for combined studies involving large scale structure and cmb .
we strongly encourage combined approaches such as the one proposed with duet and cmb data both to understand astrophysics , mainly the presence of baryon oscillations , and cosmology , such as properties of the dark energy .
we thank zoltan haiman , wayne hu and dragan huterer for collaborative work discussed in this article .
the author is grateful to the duet science team for providing him an opportunity to contribute aspects related to a combined cmb and large scale structure data analysis .
the author acknowledges support from the sherman fairchild foundation and the department of energy and apologizes , before hand , for any missed references .
|
the advent of high signal - to - noise cosmic microwave background ( cmb ) anisotropy experiments now allow detailed studies on the statistics related to temperature fluctuations .
the existence of acoustic oscillations in the anisotropy power spectrum is now established with the clear detection of the first , and to a lesser confidence the second and the third , peak . beyond the acoustic peak structure associated with cmb photon temperature fluctuations , we study the possibility for an observational detection of oscillations in the large scale structure ( lss ) matter power spectrum due to baryons . we also suggest a new cosmological test using the angular power spectrum of dark matter halos , or clusters of galaxies detected via wide - field surveys of the large scale structure .
the _
standard rulers _ of the proposed test involve overall shape of the matter power spectrum and baryon oscillation peaks in projection .
the test allows a measurement of the angular diameter distance as a function of redshift , similar to the distance to the last scattering surface from the first acoustic peak in the temperature anisotropy power spectrum .
the simultaneous detection of oscillations in both photons and baryons will provide a strong , and a necessary , confirmation of our understanding related to the physics during the recombination era .
the proposed studies can be carried out with a combined analysis of cmb data from missions such as the map and the large scale structure data from missions such as the duet .
|
an important support for graph exploration is interactive visualization , which can help to quickly identify the main components of a graph , its outliers , the most important edges and communities of related nodes .
interaction - enabled visualization allows to pick detailed and contextualized information on demand , interact with nodes and edges and determine topology aware arrangements for clearer inspection . however , up - to - date applications have produced graphs on the order of hundred thousand nodes and possibly million edges ( referenced from here on as large graphs ) .
large graphs can be found in numerous real - life settings : web graphs ( web pages , pointing to others with hypertext links ) @xcite , computer communication graphs ( ip addresses sending packets to other ip addresses ) , recommendation systems @xcite , who - trusts - whom networks @xcite , bipartite graphs of web - logs of who visits what page ; blogs and similar . at this magnitude ,
efficient graph visualization becomes prohibitive because of the excessive processing power requirements that prevent interaction . besides that , hundred - thousand - node drawings result in unintelligible cluttered images that do not aid the user s cognition . to face these challenges we present a system that explores two new ideas to address scalability in large graph visualization .
the first idea establishes a hierarchical partitioned arrangement from a graph in order to allow multi - resolution visualization .
the second idea utilizes an innovative algorithm to extract a subgraph of interest based on an initial set of target nodes .
our system uses either or both of these ideas to process large graphs bypassing the aforementioned limitations of massive graph drawing . the proposed interface permits to navigate through the levels of a graph hierarchy and also to mine subgraphs information for targeted graph exploration .
the remaining of this paper is structured as follows .
section [ dataset ] introduces the dblp dataset that will be used along this work .
section [ hierarchical ] describes our multi - resolution visualization idea and section [ subgraph ] illustrates our subgraph extraction algorithm .
section [ conclusions ] concludes the work .
throughout this text we employ the dblp dataset to illustrate the functionalities of our system .
this dataset originates from the digital bibliography & library project ( or dblp ) .
dblp is a publicly available database of publication data that embraces authors ( also co - authors ) from the computer science community and their published works .
its content is periodically updated and detailed information from dblp can be achieved at _ http://dblp.uni - trier.de/_. the version of dblp dataset that we use defines a graph with @xmath0 nodes and @xmath1 edges , where each node represents an author of a publication and each edge denotes a co - authoring relationship between two authors .
our first idea to deal with massive graphs is the use of a commu - nities - within - communities structured visualization . in the next sections we overview the steps to come up with such proposal at the same time that we describe its features for visualization and interaction . for this work
, initially we need to recursively and hierarchically partition a given graph .
we adopted the methodology named _ k - way _ partitioning ( however any partitioning methodology fits our system ) .
that is , given a graph @xmath2 with @xmath3 , we want to have @xmath4 subsets @xmath5 of @xmath6 , such that @xmath7 for @xmath8 , @xmath9 and @xmath10 .
also , the partitioning must minimize the number of edges of @xmath11 whose incident vertices belong to different subsets .
this partitioning methodology is implemented by metis , whose details are found in the work by karypis and kumar @xcite and in related works . hence , given a graph , we perform a sequence of recursive partitionings to achieve a hierarchy of communities - within - communities . at each recursion , each partition is submitted to a new partitioning cycle that will create another set of partitions .
this process repeats until we get the desired granularity for the partitions ( communities ) . for each new set of partitions ,
a new subtree is embedded in an r - tree like structure . at each new level of the tree
, the tree nodes ( communities ) just created will have the formerly partitioned tree node as their parent .
we call this structure g - tree ( named after graph - tree ) , which is the data structure that supports our system , illustrated in figure [ r - tree ] .
the references for the graph nodes properly said are at the bottom level of the tree .
the entire structure is stored in a single file and the nodes are transferred to main memory only when necessary . to demonstrate our methodology , we recursively partition dblp dataset into @xmath12 hierarchy levels each with @xmath12 partitions .
the dataset , thus , is broken into @xmath13 , or @xmath14 , communities with an average of @xmath15 nodes per community .
the communities reflect the connectivity ( number of edges ) among their members according to metis partitioning algorithm .
we propose an innovative interactive presentation for large graphs . for this purpose , our system promotes the navigation across the levels of the tree that represents the hierarchical partitioning of a large graph .
as the user interacts with the visualization , the system keeps track of the connectivity among communities of nodes at different levels of the partitioned graph . when the user changes the focus position on the tree structure , the system works on demand to calculate and present contextual information .
when we display a graph as communities - within - communities , we have new representations for graph drawing , as illustrated in figure [ nodesandedges ] . besides conventional nodes and edges that appear only at the bottom level of the tree ( leaf nodes ) , we also have community nodes , that comprehend a number of sub communities and nodes , and we have connectivity edges , that represent the number of edges between community nodes .
these connectivity edges represent the number of edges between nodes from the original graph , but that are in different communities .
the storage and management of this information is out of the scope of this demonstration paper .
these features are illustrated in figure [ dblpcomplete ] , which presents a sequence of interactive actions taken by the user when navigating in dblp dataset . in figure [ dblpcomplete](a ) , it is possible to see dblp partitioned into @xmath12 communities in its first hierarchy level , and other @xmath16 , or @xmath17 communities in its second hierarchy level . at this point , @xmath18 communities
are highly connected to every other community and also highly connected among their @xmath12 sub communities .
the other @xmath19 first - level communities are relatively isolated from the other @xmath18 and totally isolated among their sub communities .
one can conclude that the @xmath18 highly connected communities hold long term active and collaborating authors , while the other @xmath19 hold casual , less productive authors who seldom interact with each other . in figure [ dblpcomplete](b )
we focus on community _ s034 _ and verify that its sub communities are isolated from each other
. a deeper focus in community _
_ in figure [ dblpcomplete](c ) shows that among its sub communities ( highlighted ) , only two of these sub communities present an edge .
our system allows to inspect this specific outlier edge to reveal that authors `` d. b. miller '' and `` r. g. stockton '' define this co - authoring relation for their unique dblp publication dated from @xmath20 .
it is also possible to execute a label query to locate a specific author within the hierarchy , as for example author jiawei han in figure [ dblpcomplete](d ) . in figure [ dblpcomplete](e )
we go to its subgraph community and verify other important nodes surrounding this author .
in figure [ dblpcomplete](f ) we interact with the graph to discover author ke wang , which is another very active author who has worked for years with author jiawei han .
main communities and its @xmath17 sub communities .
( b ) contextualization of community _ s034_. ( c ) closer look and complete expansion of community _ s034_. ( d ) we locate author jiawei han .
( e ) subgraph community of author jiawei han .
( f ) interaction with the subgraph reveals co - author ke wang as one of the main contributors to jiawei han . ]
the exploration of communities of nodes instead of all the nodes at a time , the way we are doing , allows the perception of the relationships among communities of nodes . this way it is possible to trace the distribution of edges among communities , their connectivity degree and their scope of connectivity .
it is also possible to pick outlier edges for suspicious connections between communities .
the user can focus at different communities of nodes according to his / her interest and browse the levels of the hierarchy in order to identify interesting connections or to inspect specific graph nodes . at the bottom level of the tree ,
the user can access a subgraph that is part of the larger graph being analyzed . to do so
, the system brings the correspondent graph nodes from disk to memory and draws them inside the region attributed to its parent community ( tree node ) .
then this area of the visualization scene becomes a regular area for graph drawing .
for this subgraph , besides basic interaction ( zoom , pan and details on demand ) the user can also ask for the calculation of metrical features corresponding to this subgraph only .
our system supports the following calculations : degree distribution , number of hops , number of weak components , number of strong components and page rank calculation for the nodes .
gmine also offers pop up node information , edge expansion and edition of nodes and edges .
the presentation of the node communities together with the edges that connect them may cause sensory overload .
this is due to the fact that every community can potentially be connected to every other community .
this problem is aggravated if the graph has many hierarchy levels exhibited simultaneously when communities are expanded to show their content . to cope with this aspect of our multi - resolution graph visualization ,
we propose to display a small , but carefully chosen set of communities .
we refer to this method as the `` tomahawk '' principle , because the chosen nodes remind of a tomahawk ax when shown on g - tree method , illustrated in figure [ tomahawk ] .
that is , in order to limit the number of items presented at a time , we make use of g - tree structure to determine a well - established context every time in response to user interaction .
thus , as the user chooses a community node to focus on , we traverse the tree in order to gather the desired node of interest , its sons and its siblings .
then we plot only these items inside the minimum node that bears this contextualization , see figure [ dblpcomplete](b ) .
we argue that the tomahawk principle can provide a minimum contextualization to the user by presenting nodes above , beneath and by the side of a node of interest .
our second idea to deal with massive graphs is the use of a novel algorithm for connection subgraph extraction .
our algorithm , which is not to be detailed in this demonstration work , aims to maximize what we call `` goodness score '' of the nodes within a subgraph . to reach this goal , an independent random walk with restart
is simulated for each source node , and the goodness score of a node is computed by the steady - meeting probability that the random particles will finally meet each other at the given node .
then , a dynamic programming is used to discover important paths iteratively .
the proposed algorithm can deal with multi - source queries , while the existing one @xcite is restricted to pairwise source queries . a typical scenario to apply connection subgraph extraction is `` given an initial set of interesting individuals , find a small number of individuals from a large social network that can best capture the relationship among the individuals of the initial set '' . for large graphs ,
extracting a small ( say , with tens of nodes ) yet representative connection subgraph brings feasibility to large graph visual exploration . also , due to the multi - faced nature of many real life relationships , connection subgraphs provide a better way to describe such kind of relationships if compared to single path descriptions . for ( limited static ) demonstration , a connection subgraph with @xmath21 nodes extracted from the whole dblp dataset
is plotted in figure [ connsubgraph ] . the initial query set in figure [ connsubgraph ]
is composed of three authors from the database community : `` philip s. yu '' , `` flip korn '' and `` minos n. garofalakis '' . in figure
[ connsubgraph ] , instead of a thousand nodes graph , one can concentrate on a subgraph of interest extracted from the original graph .
the magnitude of the subgraph is thousand fold smaller than the original dataset and the subgraph being visualized is directly related to the interconnection defined by our initial set of target nodes . on the visualization ,
if the user moves the mouse over a node , gmine pops up more information about that node - in the example , one can see prof . h. v. jagadish data and his edges highlighted .
jagadish has direct connection with flip korn , and 1-step - away connections with dr .
philip yu and dr .
minos garofalakis . in our system ,
subgraph extraction can be utilized alone or combined to communities - within - communities visualization .
alone , one can extract a subgraph of interest from a given large graph .
combined , ( see figure [ dblppartial ] ) , it can be used to generate a subgraph to be hierarchically partitioned for visualization or , alternatively , it can be used to generate a subgraph from an existing graph partition .
figure [ dblppartial ] illustrates the combination of subgraph extraction and com - munities - within - communities visualization .
figure , [ dblppartial](a ) displays a @xmath22 nodes subgraph extracted from the dblp dataset . in figure [ dblppartial](b ) it possible to see this subgraph partitioned into @xmath18 main communities . in figures [ dblppartial](c ) and
[ dblppartial](d ) we go deeper into the hierarchy to analyze the connectivity between communities and , finally , the very nodes of the graph .
nodes subgraph extracted from dblp dataset .
( b ) the same graph presented as three partitions .
( c ) one level down the hierarchy and we have three other communities inside the community highlighted in ( b ) .
( d ) zoom in the community highlighted in ( c ) and another level down the hierarchy .
we reach the very nodes of the graph.,scaledwidth=46.0% ]
we have demonstrated a system that supports the visualization of large graphs in an interactive environment . in our tool
the user can navigate through the graph structure in a hierarchical fashion , having different perspectives of the graph arrangement , varying from multiple resolution levels to detailed inspection of specific graph nodes .
the system also supports an innovative subgraph extraction algorithm that can speed up large graph exploration by concentrating on a targeted subset of the graph .
the benefits of our ideas come from its compartmented graph management that promotes scalability while keeping visual comprehension .
the scalability is due to the fact that smaller parts of the graph are processed one at a time instead of the whole graph at every cycle .
visual comprehension derives from limited visual data presentation in contrast to cluttered visualizations generated when large graphs are entirely drawn . due to
space limitations it is not possible to show all the gmine functionalities
. therefore , for a better exposition , we have gmine available online at _ ` http://www.cs.cmu.edu/~junio/gmine ` _ , where the software , datasets and videos can be downloaded . for vldb demonstration session
, we plan to let the interested vldb participants interact directly with the system , possibly checking for their name , their connection - subgraphs with their colleagues , and zooming in and out their corresponding communities .
this work has been supported by fapesp ( so paulo state research foundation ) , cnpq ( brazilian national research foundation ) , capes ( brazilian committee for graduate studies ) , national science foundation , ( pita ) pennsylvania infrastructure technology alliance and donations from intel , ntt and hewlett - packard .
any opinions , findings and conclusions or recommendations expressed here are those of the author(s ) and do not necessarily reflect the views of the funding parties .
|
several graph visualization tools exist . however , they are not able to handle large graphs , and/or they do not allow interaction .
we are interested on large graphs , with hundreds of thousands of nodes .
such graphs bring two challenges : the first one is that any straightforward interactive manipulation will be prohibitively slow .
the second one is sensory overload : even if we could plot and replot the graph quickly , the user would be overwhelmed with the vast volume of information because the screen would be too cluttered as nodes and edges overlap each other .
gmine system addresses both these issues , by using summarization and multi - resolution .
gmine offers multi - resolution graph exploration by partitioning a given graph into a hierarchy of com - munities - within - communities and storing it into a novel r - tree - like structure which we name _ g - tree_. gmine offers summarization by implementing an innovative subgraph extraction algorithm and then visualizing its output .
|
the fractional quantum hall ( fqh ) states have long fascinated the condensed - matter community due to their remarkable transport properties and the exotic nature of their quasiparticle excitations .
it is in the context of fqh states that the notion of topological order in gapped two - dimensional states first arose .
@xcite recently there has been enhanced interest in fqh states with _ non - abelian statistics _
, @xcite due to the possibility of implementing quantum computation schemes topologically protected from decoherence .
@xcite the unusual features of fqh states have been notoriously difficult to characterize using traditional condensed - matter concepts such as local order parameters and @xmath1-point correlation functions . in a separate development , recent years have seen growing understanding that entanglement measures borrowed from the discipline of quantum information can be useful in probing global features of quantum many - particle states . @xcite it is thus natural to ask what features of fqh states can be characterized by entanglement measures . in a recent short report,@xcite
three of the present authors have shown that one such entanglement measure , the _
bipartite entanglement entropy _ , indeed elucidates the subtle correlations and topological order in the simplest fqh states , those in the so - called laughlin sequence . the bipartite entanglement entropy is defined by partitioning the system under question into two blocks @xmath2 and @xmath3 , and using the reduced density matrix of one part ( e.g. , @xmath4 obtained by tracing over @xmath3 degrees of freedom ) to calculate the von neumann entropy @xmath5 $ ] . in ref .
, numerical calculations of the entanglement entropy between two spatial regions allowed us to extract from the laughlin wavefunctions the so - called _ topological entanglement entropy _
, a concept introduced in ref . .
[ for brevity we write _ topological entropy _ where no confusion can arise . ] in addition , we provided results on the entanglement entropy between subsets of the particles making up the state .
we showed that such _ particle entanglements _ are bounded by expressions that manifest the exclusion statistics in the laughlin states . in this article , we present a systematic discussion of bipartite entanglement entropies for fqh states , elaborating on our results in ref . .
in addition to the abelian laughlin ( l ) states , we consider a series of non - abelian fqh states : the moore - read ( mr ) ( or pfaffian ) states .
@xcite in planar geometry , the respective wavefunctions are given by @xmath6 with @xmath7 denoting the antisymmetric pfaffian symbol .
we shall here consider these same states in spherical geometry , so as to eliminate boundary effects . for both series of states ,
we derive upper bounds @xmath8 for particle entanglement entropies .
a marked difference between the @xmath9 laughlin state and the @xmath10 moore - read states is that in the latter the leading correlations have a 3-body nature , whereas those in the laughlin states are 2-body effects .
this difference is nicely manifested in the leading terms of a @xmath11 expansion of the upper bounds @xmath8 , which are given by ( see section iii below for details ) @xmath12 & & m=2\ { \rm moore - read}\ { \rm state : } \nonumber \\ & & \qquad s_a^{\rm bound } = s^f_a - \frac{3}{4n^2 } n_a(n_a-1)(n_a-2 ) + \ldots\end{aligned}\ ] ] another marked difference between the abelian and non - abelian states is in the value for the topological entropy @xmath0 . comparing the laughlin and moore - read states at the same filling fraction @xmath13 we have @xmath14 the difference being due to the non - abelian nature of the moore - read states ( see section iv for details ) . in this article
we extract values for @xmath0 directly from wavefunctions for a limited number of particles in spherical geometry ( up to @xmath15 for the @xmath9 laughlin state and up to @xmath16 for the @xmath10 moore - read state ) , finding values that are consistent with the expected result .
these results illustrate how the entanglement entropy can be used in diagnosing the topological order for a fqh state that is only known in the form of wavefunctions for a limited number of particles , as is often the case in numerical studies .
in section [ sec_partition - choices ] we give a general discussion of possible bipartite entanglement measures in itinerant many body systems , carefully distinguishing between _ particle _ and _ spatial _ partitioning schemes . in section [ sec_pcle - partition ]
we present analytical and numerical results for particle entanglement in the laughlin and moore - read fqh states . in this , the eigenvalue distribution of the reduced density matrix @xmath17 plays a central role . in subsection [ sec_pcle - rdm - n - corrln - fn ]
we relate the eigenvalue distribution for @xmath18 to the two - particle correlation function @xmath19 . in section [ sec_spatial - n - gamma ]
we discuss spatial partitioning , paying particular attention to the numerical procedure followed in extracting the topological entropy @xmath0 . while most results in this article are for fermionic fqh states ( meaning @xmath20 odd in the laughlin sequence and @xmath20 even for the moore - read states ) , we briefly comment on bosonic states in subsection [ sec_bosonic ] .
the entanglement entropy , being a _ bipartite _ measure of entanglement , depends on the particular partitions being considered .
obviously , a many - particle system can be partitioned in many ways . rather than asking
which partition is the `` correct '' one , we find it more useful to ask what information one can extract from various kinds of partitioning .
accordingly , we have partitioned both the spatial degrees of freedom , and the particles themselves .
we find that both schemes are useful , for revealing distinct features of the many - particle state .
the kinds of partitioning one is able to study depend on the available degrees of freedom .
our calculations are all performed for fqh states in a spherical geometry .
@xcite in this representation the fermions are placed on a sphere containing a magnetic monopole .
the magnetic orbitals of the relevant landau level are then represented as angular momentum orbitals ; the total angular momentum is half the number of flux quanta , @xmath21 .
the @xmath22 orbitals are labeled either @xmath23 to @xmath24 or @xmath25 to @xmath26 . for @xmath27 particles at fractional filling @xmath13
, one finds the interesting fqh states for @xmath28 , where @xmath29 is a finite - size shift .
the laughlin states appear at @xmath30 while for the moore - read states @xmath31 .
the `` filling '' acquires the usual meaning @xmath32 only in the thermodynamic limit .
the orbitals are each localized around a `` circle of latitude '' on the sphere , with the @xmath23 orbital localized near one `` pole . '' since the fqh wavefunctions on a sphere
are obtained in terms of orbital occupancies , one can either partition orbitals or partition particles . because of the spatial arrangement of the orbitals , partitioning orbitals is in fact equivalent to partitioning spatial regions .
the difference between spatial and particle partitioning has not been stressed in the literature because the most common systems studied ( in the context of entanglement entropies in many - particle states ) are spin models , for which there is no such distinction . in the cases of itinerant particles where there is a difference , the common default scheme has been spatial partitioning .
in particular , conformal field theory results on entanglement scaling @xcite and the distinction between gapless and gapped states observed in entanglement scaling @xcite actually pertain to the blocking of _ space _ rather than the particles or spins themselves .
the definition of the topological entropy for two - dimensional topologically ordered states is also based on the entanglement entropy between spatial blocks .
@xcite in previous work , @xcite three of the present authors studied the entanglement entropy between subsets of particles making up a laughlin fqh state .
we presented upper bounds and gave an interpretation in terms of exclusion statistics . in this paper
we extend these results to the moore - read states , where the exclusion effects are more intricate .
we refer to refs . for other studies of particle entanglement properties . for orbital or spatial partitioning ,
we define block @xmath2 to be the first @xmath33 orbitals , extending spatially from one pole of the sphere out to some latitude . in the thermodynamic limit , this is equivalent to choosing a disk - shaped block @xmath2 within an infinite planar system . in this limit , since each orbital @xmath34 is associated with a wavefunction of the form @xmath35 in usual complex coordinate language , a disk with @xmath33 orbitals has radius @xmath36 .
the spatial arrangement of the orbitals constrains us to either disc or ring - shaped spatial regions as the @xmath2 partition .
since the orbital indexes do not give us access to the full two - dimensional degrees of freedom , it is not possible to experiment with the various kinds of topologically nontrivial partitions suggested by preskill and kitaev , @xcite levin and wen , @xcite and furukawa and misguich .
@xcite however , as we have reported previously , @xcite the spherical geometry is sufficient to probe the topological entropy of fqh wavefunctions . comparing particle and spatial entanglement entropies
, we remark that the effect of correlations is opposite between the two cases . for particle
partitioning , the maximal entropy is realized for uncorrelated fermions ; correlations tend to lower @xmath37 from the fermion bound @xmath38 .
spatial entanglement , on the other hand , is entirely due to correlations .
the point is illustrated by considering the @xmath39 laughlin state , where the fermions are uncorrelated . here
the particle entanglement entropy equals @xmath38 , while the spatial entanglement entropy vanishes .
in this section we provide close upper bounds to the entropy of entanglement between @xmath40 particles of the state and the remaining @xmath41 particles .
we also discuss the @xmath40-particle reduced density matrices @xmath17 that arise in this context . for fqh states on a sphere ,
the @xmath40-particle reduced density matrices @xmath17 commute with the total angular momentum operators @xmath42 and @xmath43 of the selected @xmath40 particles .
this implies that the eigenvalues of @xmath17 are organized in a multiplet structure of the corresponding @xmath44 algebra : an eigenvalue for total angular momentum @xmath45 will be @xmath46-fold degenerate . for @xmath47 fermions , each having angular momentum @xmath48 ,
the 2-particle states have total angular momenta @xmath49 , @xmath50 , @xmath51 , @xmath52 , for @xmath48 integer ( half - integer ) , giving a total number of @xmath53 states . a naive upper bound to the entanglement entropy
is thus @xmath54 .\ ] ] inspecting the explicit structure of the fermionic laughlin states with @xmath55 , one finds that the eigenvalues corresponding to 2-particle states with @xmath49 , @xmath50 , @xmath51 , @xmath56 all vanish .
the reason is that the correlations in the laughlin states are such that particles can not come too close together .
for example , if a first fermion occupies the @xmath23 orbital , localized near the north pole , the laughlin wavefunction has zero amplitude for finding a second fermion in orbitals @xmath57 , @xmath58 , @xmath51 , @xmath59 .
the highest possible value of the angular momentum of the two fermions combined is thus @xmath60 .
the remaining number of non - zero eigenvalues is @xmath61 , leading to an improved bound on the entropy @xmath62 @xmath63\ ] ] with @xmath64 as before . for @xmath65 , the multiplet structures are more complicated and we need to resort to a different method for finding a non - trivial upper bound to the particle entropy . in the next subsection
we give a general derivation for both the laughlin and the moore - read series of fermionic fqh states . for @xmath27 fermionic particles , @xmath40 particles in the @xmath2 block , and the total number of orbitals given by @xmath66 ,
fermionic statistics lead to an obvious upper limit @xmath67 to the entropy @xmath37 @xmath68 in the fqh states the correlations are such that the particles avoid each other and the entropy is further reduced . to obtain a handle on this , one may reason as follows .
the model fqh states in the laughlin and moore - read series can be characterized as zero - energy eigenstates of a hamiltonian penalizing pairs and/or triplets of particles coming to the same position . after tracing out the coordinates for the @xmath3 set ,
the dependence on those in the @xmath2 set is such that one still has a zero - energy eigenstate .
however , the number of orbitals available to the @xmath2 particles is larger than what is needed to make the model fqh state in the @xmath2 sector , and one instead has a certain number of quasi - holes on top of the @xmath2 set model state .
the total ground state degeneracy for this situation has been studied in the literature : see ref . for the laughlin and moore - read states and ref . for the read - rezayi and ardonne - schoutens series of non - abelian fqh states . for
the laughlin states the details are as follows .
the @xmath27-particle laughlin state is realized on a total of @xmath69 landau orbitals , corresponding to @xmath70 flux quanta .
the laughlin state for @xmath40 particles would need @xmath71 flux quanta ; we thus have an excess flux of @xmath72 . with the laughlin gauge argument this corresponds to the presence of @xmath73 quasi - holes over the groundstate .
according to ref .
each of the quasi - holes has a number of @xmath74 effective orbitals to choose from , with bosonic counting rules ( meaning that two or more quasi - holes can be in the same effective orbital ) .
this gives a number of quasi - hole states equal to @xmath75 leading to the following upper bound to the entropy @xmath37 @xmath76 we remark that this expression has a clear interpretation in terms of exclusion statistics : the counting factor in eq .
( [ laughlinbound ] ) gives the number of ways @xmath40 particles can be placed in @xmath69 orbitals , in such a way that a particle placed in a given orbital @xmath34 excludes particles from orbitals @xmath77 with @xmath78 . in a @xmath11 expansion
we find ( assuming @xmath79 ) @xmath80 & & \frac{1}{n } \frac{m-1}{m } n_a ( n_a-1 ) \nonumber \\[2 mm ] & & + \frac{1}{n^2 } \frac{m-1}{2 m^2 } n_a ( n_a-1 ) [ 2 m + ( n_a-1)(m+n_a-4 ) ] \nonumber \\[2 mm ] & & + \mathcal{o}(1/n^3 ) \label{laughlin1overn}\end{aligned}\ ] ] the particle entropy reaches a maximum for @xmath81 . for this case our eq .
( [ laughlinbound ] ) gives , in the limit of large @xmath27 , @xmath82/2 \ .\ ] ] this bound is sharper than a bound recently presented in ref .
, which gives a larger coefficient for the linear - in-@xmath27 behavior . for the fermionic moore - read states at @xmath13 , with @xmath83
, we can reason in a similar way , with now @xmath84 .
as for the laughlin states we have an excess flux of @xmath85 but now the number of quasi - holes is twice this number due to the fact that the fundamental quasi - holes correspond to half a flux quantum .
thus , @xmath86 .
we now take from ref .
the following result for the total quasi - hole degeneracy @xmath87 this gives us an upper bound @xmath88 as before .
putting @xmath10 , one easily checks that @xmath88 coincides with @xmath89 for @xmath47 . in a @xmath11 expansion ,
the leading deviation from @xmath38 is a 3-body term at order @xmath90 , @xmath91 this result nicely illustrates the fact that the leading correlations in the @xmath10 moore - read state have a 3-body character : the wave - function vanishes if at least three particles come to the same position .
for @xmath92 the leading correlations do have a 2-body character , as for the laughlin states , @xmath93 inspecting the particle entanglement at @xmath81 and for @xmath27 large , our bound implies that for the @xmath10 moore - read state @xmath94 this bound is reduced from the fermi bound @xmath95 , but it is larger than the bound for the @xmath10 ( bosonic ) laughlin state , which has asymptotic form @xmath96 .
this indicates that , at equal filling @xmath97 , the particles in a moore - read state are more entangled than those in a laughlin state .
the quasi - hole counting rules for the order-@xmath98 clustered spin - polarized ( read - rezayi ) and spin - singlet ( ardonne - schoutens ) states are all known in the literature . @xcite they can be used to generalize the upper bounds on particle entanglement entropy given in this subsection to these more intricate non - abelian fqh states .
we briefly comment on the case of bosonic fqh states .
the realization that a rapidly rotating bose gas may eventually enter a regime of bosonic quantum hall states motivates the theoretical study of the effects of bosonic statistics .
we consider bosonic laughlin states at filling fraction @xmath99 with @xmath83 .
the naive upper bound to the the entropy associated to placing @xmath40 bosons in @xmath69 orbitals is @xmath100 the expression for @xmath88 remains unchanged , giving the following leading correction in a @xmath11 expansion @xmath101 for a bosonic moore - read state , with filling fraction @xmath13 with @xmath102 ,
the leading @xmath11 correction becomes @xmath103 in the case @xmath39 the leading correlations have 3-body character , leading to the vanishing of the leading @xmath11 correction . in deriving the upper bound @xmath8 we relied on the fact that a certain number of eigenvalues of the reduced density matrix vanish .
the bounds would be exact if all non - zero eigenvalues were equal , but since they are not the bounds overestimate the actual values for the entropies .
[ fig_2-pcle_eigen ] plots the eigenvalues for the @xmath47-particle reduced density matrix for @xmath104 particles on a sphere in the @xmath9 laughlin state , for which the single particle angular momentum is @xmath105 .
the horizontal axis represents the degeneracy @xmath106 of the eigenvalues , in descending order .
the eigenvalue at @xmath107 , with degeneracy 47 , vanishes ; the non - zero eigenvalues show some scatter around an asymptotic value . due to this scatter
the entropy @xmath108 is somewhat lower than the upper bound @xmath109 .
( color online ) eigenvalues for the 2-particle reduced density matrix , plotted against their multiplicities , for @xmath104 particles in the @xmath9 laughlin state . ] an important difference between the @xmath9 laughlin and the @xmath10 moore - read states is the absence of vanishing eigenvalues for the 2-particle reduced density matrix . the eigenvalue distribution shown in fig .
[ fig_2-pcle_pf_eigen ] illustrates this point .
( color online ) eigenvalues for the 2-particle reduced density matrix , plotted against their multiplicities , for @xmath110 particles in the @xmath10 moore - read state . ] in the @xmath10 moore - read state , there are vanishing eigenvalues in the reduced density matrix of @xmath111 particles .
the number of nonzero eigenvalues predicted by eq .
( [ mrbound ] ) agrees with numerical results .
for example , for @xmath112 and @xmath15 particles there are 770 nonvanishing eigenvalues , in agreement with eq .
( [ mrbound ] ) . in figs .
[ fig_pcle_lghln_23 ] , [ fig_pcle_pf_23 ] we compare numerically computed particle entanglement entropies with the bounds derived above .
( color online ) entanglement entropy for @xmath47 and @xmath112 particles for the @xmath9 laughlin state .
dots are numerical exact values , the dotted line represents @xmath38 and the solid curve is the bound @xmath8 . ]
( color online ) entanglement entropy for @xmath47 and @xmath112 particles for the @xmath10 moore - read state .
dots are numerical exact values , the dotted line represents @xmath38 and the solid curve is the bound @xmath8 . ] it is interesting to consider in some detail the deviation between the bounds @xmath8 and the actual entropies computed numerically . as mentioned above , this deviation arises from the fact that the non - zero eigenvalues of the reduced density matrices are not all equal . to estimate the effect on @xmath113 $ ] of the spread in the non - zero eigenvalues , we do a rough modeling of the eigenvalue distribution ( fig.[fig_2-pcle_eigen ] ) of @xmath18 for the @xmath9 laughlin state . for this case
the number @xmath114 of non - zero eigenvalues is @xmath115 . if these nonzero eigenvalues were all equal ( to @xmath116 ) , the entanglement entropy would have the maximum value @xmath117 , which is the predicted upper bound ( [ laughlinbound ] ) .
we now take into account the deviations from @xmath116 , guided by fig .
[ fig_2-pcle_eigen ] , with the following toy distribution : we take @xmath118 out of @xmath114 of the eigenvalues to be equal to @xmath119 , with @xmath120 , while the rest of the eigenvalues are at value @xmath121 , @xmath122 , such that the sum of eigenvalues is unity . assuming @xmath123 leads to @xmath124 guided by the eigenvalue distributions in fig .
[ fig_2-pcle_eigen ] , we assume that @xmath125 is of order @xmath11 ; for concreteness we put @xmath126 equal to the multiplicity of the largest eigenvalue , which is @xmath127 . taking @xmath128 between 1.2 and 1.5 ( as observed for the largest available laughlin wavefunctions )
gives a @xmath11 correction in the entropy with coefficient in the range 0.03 0.14 .
fitting the difference @xmath129 to a form @xmath130 gives a coefficient @xmath131 .
the vanishing eigenvalues account for @xmath132 , see eq .
( [ laughlin1overn ] ) , and we see that the remaining difference @xmath133 is consistent with the @xmath11 correction due to the spread in the non - zero eigenvalues .
we made similar estimates for a the @xmath10 moore - read state with up to @xmath110 particles , where the eigenvalues are all non - zero and @xmath8 agrees with @xmath38 . in this case the deviation between data and bound show a @xmath11 dependence with a coefficient of about 0.14 .
these considerations are of some general interest , as they make the point that a @xmath11 expansion of particle entanglement entropies are indicative of correlations in a many - body state . in the concrete case studied here ,
the sizeable value of the leading @xmath11 correction in the laughlin state indicates strong 2-body correlations , while the small value for the moore - read state indicates the absence of such correlations .
since the @xmath40-particle reduced density matrices @xmath17 are obtained by integrating out all but @xmath40 of the particles , one expects these matrices to be related to the @xmath40-particle correlation functions . in this subsection
we study this relation for the case @xmath47 .
in particular , the eigenvalue distributions of @xmath18 in figs .
[ fig_2-pcle_eigen ] , [ fig_2-pcle_pf_eigen ] , although discrete , are reminiscent of the well - known two - particle correlation functions @xmath19 for laughlin and moore - read states .
we will show that the eigenvalue distributions are in fact very closely related to the correlation functions the eigenvalue distribution function is a kind of discretized version of @xmath19 .
the two - particle correlation function @xmath19 is conventionally defined as @xmath134 where @xmath27 is the number of particles and @xmath1 is a density , which is chosen such that @xmath135 .
we express the 2-particle reduced density matrix @xmath136 on a sphere in a basis of polar spherical coordinates .
because of the rotational symmetry it should be a function of the angular distance @xmath137 between the two particles . in appendix [ eigenvalues_correlations ] we show that @xmath138 can be written in the form @xmath139 in this expression , @xmath140 is the eigenvalue with multiplicity @xmath141 , corresponding to the total angular momentum of the two particles equal to @xmath34 .
the functions @xmath142 are explicitly given in eq .
( [ rtheta ] ) .
( color online ) two - particle correlation as a function of distance in units of the magnetic length , for the @xmath9 laughlin state with @xmath143 particles and for the @xmath10 moore - read state with @xmath110 . ] since the distance @xmath144 between two particles is simply equal to @xmath145 , with @xmath146 the radius of the sphere , the 2-particle correlation function @xmath19 is directly proportional to @xmath138 . through eq .
( [ rho2_theta ] ) it is expressed as a transfrom from @xmath34 space to @xmath137 space , with basis functions @xmath142 . in fig .
[ fig_g2_lghln_pf ] , we show some curves for @xmath138 ; they agree with known results . as illustrated in the inset to fig .
[ fig_g2_lghln_pf ] , the basis functions @xmath142 have a peak structure , with the position of the peak depending on the total angular momentum @xmath34 .
large values of @xmath34 correspond to small angular distances and vice versa .
this is easy to understand from the following classical picture .
when two particles with angular momenta @xmath48 have total angular momentum @xmath147 , the corresponding vectors @xmath148 and @xmath149 should point a the same direction .
the angular momentum vector points at the position of the particle on a sphere , therefore two particle should be close to each other . on the other hand , if total angular momentum is zero then the angular momentum vectors should point into opposite directions .
this means that particles are placed at the opposite sides of the sphere .
the fact that the @xmath142 are localized functions , peaked at @xmath137 values monotonically decreasing with @xmath34 , indicates that @xmath19 curve is simply a continuous form of the @xmath140 versus descending-@xmath150 curves of figs .
[ fig_2-pcle_eigen ] and [ fig_2-pcle_pf_eigen ] .
the similarity between the discrete @xmath140 and the continuous @xmath19 curves is not accidental . at small distances @xmath151 .
for a laughlin state the lowest value of @xmath152 is @xmath20 , thus @xmath153 .
this behavior is a direct consequence of the vanishing of eigenvalues with the largest multiplicities . for the @xmath10 moore - read state
the lowest value of @xmath152 is @xmath154 because there are no vanishing eigenvalues .
therefore at small distances @xmath155 . of course , our observations on the 2-particle correlations agree with known results ; our main point has been to stress the intimate relation with the eigenvalue distribution of the 2-particle reduced density matrices .
we now turn to dividing the landau level orbitals into two blocks and calculating the entropy of entanglement between them . in a previous publication , @xcite we used this scheme to extract the topological entropy of the laughlin state . here
, we mainly focus on the ( @xmath10 ) moore - read state . after reviewing the topological entropy @xmath0 and the total quantum dimension @xmath156 , especially in the context of the moore - read state ( [ subsec_gamma - review ] ) , we detail some issues with taking the thermodynamic limit necessary for extracting @xmath0 from numerical data ( [ subsec_extrapoln - issues ] ) , and then present our numerical results ( [ subsec_numerical - gamma ] ) . we also present observations on the spectral structure of the reduced density matrices ( [ subsec_orbital - rdm - spectrum ] ) . for spatial partitioning of many - particle states ,
the general rule ( `` area law '' ) is that the entanglement entropy scales as the size of the boundary between the @xmath2 and @xmath3 blocks . @xcite
subtle information about the nature of the many - particle state can be provided by the presence or absence of logarithmic corrections , values of coefficients , or subleading terms in this basic relationship . for topologically ordered states in two dimensions ,
the following theorem has been presented recently @xcite concerning the scaling of entanglement entropy between spatial partitions . if @xmath48 is the length of the boundary between the two blocks , the entanglement entropy scales as @xmath157 .
as usual the scaling law applies to situations where @xmath2 is large and the total system is infinite .
the subleading term @xmath0 is called the _
topological entanglement entropy_. the striking result of refs .
has been that it can be expressed as the logarithm of a quantity @xmath156 known as the _ total quantum dimension _ of the topological field theory describing the topological order of the state .
the total quantum dimension is given by @xmath158 where the @xmath159 s are the quantum dimensions of the individual sectors making up the topological field theory .
these quantum dimensions are set by fusion rules of the fundamental anyons in the field theory .
the topological field theory for a @xmath13 laughlin state has a fundamental anyon ( of fractional charge @xmath160 ) , which generates @xmath20 abelian sectors . the quantum dimension @xmath159 is unity in all sectors so that @xmath161 for the @xmath13 laughlin state . for @xmath9
this gives @xmath162 . for states with non - abelian quasiparticles ,
the situation is more interesting because some anyon sectors contribute @xmath163 .
details for some examples have been provided in refs . .
in particular , for the @xmath10 moore - read state , there are six sectors ( two each of quasiparticles denoted by @xmath164 , @xmath165 , @xmath166 ) which contribute @xmath167 , @xmath168 , @xmath169 , leading to @xmath170 and @xmath171 .
the non - abelian nature shows up in the fact that @xmath0 is larger than @xmath172 , six being the degeneracy of the @xmath10 moore - read state on the torus . + a compact general expression for the total quantum dimension for a read - rezayi state with order-@xmath98 clustering and at filling fraction @xmath173 is ( see also ref . ) @xmath174= \frac{\sqrt{(k+2)(km+2)}}{2 \sin(\pi /(k+2 ) ) } \ . \label{scriptdrr}\ ] ] it includes the laughlin states ( @xmath175 , @xmath176 ) and the moore - read states ( @xmath177 , @xmath178 ) as special cases . for a general spin - singlet non - abelian fqh state with order @xmath98 clustering and filling fraction @xmath179 ,
the result is @xmath180= \frac{(k+3)\sqrt{(2km+3)}}{16 \cos(\pi/(k+3 ) ) \sin^3(\pi /(k+3 ) ) } \ , \label{scriptdas}\ ] ] giving @xmath181 for the paired spin - singlet state ( @xmath177 , @xmath182 ) at @xmath183 .
in general , the @xmath184-dependence of these expressions for total quantum dimensions is linked to the ground state degeneracy in torus geometry .
denoting the latter by @xmath185 $ ] we have the relation @xmath186= \mathcal{d}[k,0 ] \sqrt{\frac{\#[k , m]}{\#[k,0 ] } } \ , \ ] ] the conformal field theories underlying the states at @xmath187 are of wess - zumino - witten type ( @xmath188 for the rr states and @xmath189 for the as series ) and the quantities @xmath190 $ ] can be expressed in the modular @xmath29-matrix for these wzw models .
first of all , we note that , since our degrees of freedom are ordered essentially one - dimensionally , we can not use one of the two - dimensional schemes proposed previously @xcite in which an appropriate addition / subtraction of the entanglement entropies of several regions cancels the boundary parts of the entropy ( @xmath191 ) leaving the subleading term @xmath0 . with the orbital degrees of freedom on a sphere , we can choose only regions corresponding to disks , concentric rings , and combinations thereof .
any combination of entropies of disk- and ring - like regions that cancels out the boundary terms also unfortunately cancels out the @xmath0 term .
we are thus led to using directly the scaling law , @xmath192 .
our choice of block @xmath2 as the first @xmath33 orbitals , extending spatially from one pole out to some latitude , corresponds to a disk - shaped block only in the thermodynamic limit .
the block area is proportional to the square of @xmath193 while its boundary is proportional to @xmath194 ; these are equivalent only in the same @xmath195 limit . one way to numerically access
the thermodynamic limit is to take the entanglement entropy of @xmath33 orbitals with the rest , for accessible wavefunctions of various sizes @xmath27 , and then take the @xmath195 limit .
the @xmath196 versus @xmath193 points thus obtained should then follow a linear curve at large @xmath33 , whose vertical intercept gives the topological entropy .
results following this procedure were provided for the @xmath197 laughlin state in our earlier paper ; @xcite here we will focus on the moore - read state .
the extrapolation of @xmath198 values to the thermodynamic limit is a tricky issue .
we therefore discuss the extrapolation in some detail here , providing some general results .
we are interested in the function @xmath199 , where @xmath200 .
we have access to @xmath201 at several integer values of @xmath27 , and would like to estimate @xmath202 . for each dataset that we have access to ( each @xmath33 ; both laughlin and moore - read ) , we note the following : the @xmath201 versus @xmath203 values form a _ monotonic _ curve and this curve gets flatter ( slope magnitude decreases ) with decreasing @xmath204 .
two examples can be seen in the inset to fig .
[ fig_extrapolatn ] . in other words ,
the first and second derivatives of the @xmath199 function have the same sign and neither derivative changes sign . motivated by the above observations , we provide the following result . assuming only that the signs of the first two derivatives of the @xmath199 function are the same and that the signs remain unchanged until @xmath205
, we have : 1 .
the value @xmath206 corresponding to the smallest value @xmath207 of the available @xmath203 is a strict lower ( upper ) bound for @xmath202 if the @xmath208 is negative ( positive ) .
the intercept found by connecting the @xmath199 corresponding to the smallest two @xmath203 values ( @xmath207 , @xmath209 ) , namely @xmath210 is a strict upper ( lower ) bound if @xmath208 is negative ( positive ) and @xmath211 is positive ( negative ) .
the limits @xmath212 and @xmath213 thus obtained give us conservative bounds for the required entanglement entropies in the thermodynamic limit , @xmath196 . to obtain a sharper extrapolation
, one can use various polynomial extrapolations and take the average , as done in our earlier work .
@xcite here , we improve the extrapolation by using the extrapolation algorithm of bulirsh and stoer ( bst algorithm ) , based on rational polynomial fraction approximations .
@xcite .[table_bst-1 ] extrapolation using the bst algorithm , using three different @xmath214 values for the same initial dataset ( first column ) , corresponding to @xmath215 values for @xmath216 12,14,16 , 18 . in the last case
, @xmath214 has been tuned to ensure that the sets obtained after first and second iterations converge to the same value , i.e. , the lowest elements of the second and third columns are the same . [
cols="<,<,<,<",options="header " , ] in tables [ table_mr_entropies ] and [ table_laughlin_entropies ] we list orbital - partitioning entanglement entropies calculated using the numerical wavefunctions . for each wavefunction
( each colmun ) , the entanglement entropies are only listed up to their maximum value , typeset in bold , because the values after this are determined by the symmetry @xmath217 . the extrapolation procedure of sec .
[ subsec_extrapoln - issues ] involves extrapolating each row of numbers to @xmath195 .
note that , in each column ( for a particular @xmath27 ) , the omitted part after the midpoint ( in bold ) is a _ decreasing _ function of @xmath33 and thus does not give useful information about the thermodynamic limit of @xmath218 . in the extrapolation ,
it is therefore important to avoid values from these parts of the table .
we therefore restrict ourselves to values which in tables [ table_mr_entropies ] and [ table_laughlin_entropies ] are to the right of ( i.e. , above ) the diagonal line through the midpoint numbers typeset in bold .
kitaev , annals phys . * 303 * , 2 ( 2003 ) ; m. freedman , m. larsen , and z. wang , commun .
227 , 605 ( 2002 ) ; n. e. bonesteel , l. hormozi , g. zikos , s.h .
simon , phys .
95 * , 140503 ( 2005 ) ; s. das sarma , m. freedman , c. nayak , phys .
lett . * 94 * , 166802 ( 2005 ) .
j. i. latorre , e. rico , and g. vidal , quantum information and computation , * 4 * , 48 ( 2004 ) ; v.e .
korepin , phys .
. lett . * 92 * , 096402 ( 2004 ) ; v. popkov and m. salerno , phys .
a * 71 * , 012301 ( 2005 ) .
m. srednicki , phys .
71 * , 666 ( 1993 ) ; m. m. wolf , f. verstraete , m. b. hastings , j. i. cirac , arxiv : 0704.3906 ; m. b. plenio , j. eisert , j. dreiig , m. cramer , phys .
lett . * 94 * , 060503 ( 2005 ) .
|
we present a detailed analysis of bipartite entanglement entropies in fractional quantum hall ( fqh ) states , considering both abelian ( laughlin ) and non - abelian ( moore - read ) states .
we derive upper bounds for the entanglement between two subsets of the particles making up the state .
we also consider the entanglement between spatial regions supporting a fqh state .
using the latter , we show how the so - called topological entanglement entropy @xmath0 of a fqh state can be extracted from wavefunctions for a limited number of particles .
|
by @xmath6 we denote the unit disk in the complex plane @xmath7 .
its boundary is the unit circle @xmath8 .
let @xmath9 and @xmath10 be subdomains of the complex plane @xmath11 , and @xmath12 be a function that has both partial derivatives at a point @xmath13 . by @xmath14
we denote the matrix @xmath15 for the matrix @xmath16 we define @xmath17 and @xmath18 where @xmath19 a mapping @xmath20 , between metric spaces @xmath21 and @xmath22 is said to be @xmath23_lipschitz _ and @xmath24_bi - lipschitz _ , for some constants
@xmath25 , if @xmath26 and @xmath27 respectively .
we define @xmath28 we say that a function @xmath29 is acl ( absolutely continuous on lines ) in the region @xmath9 , if for every closed rectangle @xmath30 with sides parallel to the @xmath31 and @xmath32-axes , @xmath33 is absolutely continuous on a.e . horizontal and a.e .
vertical line in @xmath34 .
such a function has of course , partial derivatives @xmath35 , @xmath36 a.e . in @xmath9 .
a sense - preserving homeomorphism @xmath37 where @xmath9 and @xmath10 are subdomains of the complex plane @xmath38 is said to be @xmath39-quasiconformal ( @xmath39-q.c ) , @xmath40 , if @xmath0 is acl in @xmath9 in the sense that the real and imaginary part are acl in d , and @xmath41 ( cf .
@xcite , pp .
notice that the condition ( [ defqc ] ) can be written as @xmath42 and @xmath43 is the complex dilatation of @xmath0 .
sometimes instead of @xmath39 quasiconformal we write @xmath44 quasiconformal . a homeomorphism @xmath45 is called @xmath46 quasisymmetric if for all @xmath31 and @xmath47 @xmath48 and @xmath49 .
we easy can modify the previous definition for self - homeomorphisms of the unit circle .
it is well known that , every quasisymmetric function has quasiconformal extension to the half - plane .
we want to point out two most important extensions : _ beurling- ahlfors extension _
@xcite , and the _ barycentric extension _ of douady and earle @xcite ( see also @xcite ) .
let @xmath10 be a starlike jordan domain with respect to the origin .
let @xmath50 and let @xmath51 be a homeomorphism .
the _ radial extension _ of a homeomorphism is defined by @xmath52 and it defines a homeomorphism of the unit disk onto @xmath10 .
radial extension maps piecewise - linearly , but not smoothly .
note this important and simple fact , if @xmath10 is not starlike w.r .
to 0 , then the radial extension is not a mapping between @xmath6 and @xmath10 .
one of primary aims of this paper is to describe all homeomorphisms , whose radial extensions are quasiconformal .
we will show that the extension is quasiconformal if and only if it is bi - lipschitz .
it is well known that every bi - lipschitz is quasiconformal .
the converse is not true .
however , if the mapping is quasiconformal , then it is hlder continuous under some conditions on the boundaries ( see @xcite and @xcite ) . for connection between these two concepts ( bi - lipschitz mappings and quasiconformal mappings )
we also refer to the paper @xcite .
we say that a mapping @xmath51 is _ polar parametrization _ , if @xmath53 .
thus @xmath54 , for some positive continuous function @xmath55 , such that @xmath56 . for
a given homeomorphism @xmath51 define * @xmath57\to \gamma$ ] , @xmath58 * @xmath59 , @xmath60 .
take @xmath61 . _
the spherical and the chordal distance _ between points @xmath62 and @xmath0 are defined by @xmath63 notice that @xmath64 . for
a given function @xmath4 define the following four constants : * @xmath65 , * @xmath66 , * @xmath67 and * @xmath68 .
in this paper we will compare these constants .
we will show that , if @xmath69 , then @xmath70 ( theorem [ cir ] ) .
the condition @xmath69 is essential , see example [ she ] .
however , if @xmath4 is a polar parametrization of a starlike jordan curve w.r . 0 , then we will show the following interesting fact @xmath71 ( theorem [ ara ] ) ; in addition we will show that , for polar parametrizations of a curve that is not a circle centered at origin @xmath72 ( theorem [ rrr ] ) . in the last section
, we will show that , the radial extension is quasiconformal if and only if it is bi - lipschitz ( theorem [ star ] ) .
finally we provide two explicit examples .
in this section we will derive some auxiliary results .
further we will consider the case @xmath69 .
since @xmath73 and @xmath74 it follows that @xmath75 recall the following fundamental result of rademacher : ( ( * ? ? ?
* theorem 6.15 ) ) . every lipschitz function in an open of @xmath76 is differentiable almost everywhere .
[ dri ] if @xmath77 is a @xmath78 lipschitz mapping , such that @xmath79 for some @xmath80 and @xmath81 and every @xmath31 , then there exist a sequence of @xmath82 @xmath78 lipschitz functions @xmath83 such that @xmath84 converges uniformly to @xmath85 , and @xmath86 .
this result is well - known .
we refer to @xcite . by rademacher theorem , lemma [ dri ] and mean value theorem , for a lipschitz mappings @xmath87 and @xmath0
, we have the following simple facts @xmath88 and @xmath89 if @xmath90 , then @xmath91 thus @xmath92 therefore @xmath93 moreover @xmath94 thus @xmath95 on the other hand @xmath96 thus @xmath97 hence @xmath98 thus @xmath99 and @xmath100 which implies that @xmath101 now we have the following theorem : [ cir ] if @xmath102 is a lipschitz surjective mapping , then @xmath103 here and in the sequel by @xmath104 we mean the @xmath105 norm of @xmath106 . if @xmath87 is a mapping of the unit circle onto itself , then @xmath107 for some increasing bijective function @xmath108\to [ 0,2\pi]$ ] .
moreover , @xmath109 thus @xmath110 and @xmath111 from and it follows the theorem . from the previous theorem
we infer the following corollaries : [ qo]@xcite let @xmath112
. then @xmath87 is @xmath113lipschitz continuous with respect to spherical distance if and only if @xmath87 is @xmath113lipschitz continuous with respect to chordal distance .
let @xmath114 , where @xmath115 .
then @xmath0 is @xmath113lipschitz if and only if @xmath4 is @xmath113lipschitz .
the question arises , can we replace the unit circle by some other starlike jordan curve @xmath2 in the previous statements .
the following example shows that , in general we do not have that @xmath71 .
[ she ] let @xmath116 and @xmath117
. then @xmath118
let @xmath2 be a starlike jordan curve w.r . to the origin .
let @xmath119 be the polar parametrization of @xmath2 . in this section we will prove the following intrigue results .
* for all polar parametrization of starlike curves holds @xmath71 ( theorem [ ara ] ) . * for a polar parametrization @xmath87 we have @xmath120 if and only if @xmath121 for some @xmath122 ( theorem [ rrr ] ) . as we said before , @xmath123 as @xmath124 for all @xmath125 , one expect that , for some ( or all ) polar parametrizations @xmath87 we should have @xmath126 .
however we have [ ara ] let @xmath127\to \mathbf r_+$ ] be a continuous positive function with @xmath56 .
. then @xmath129 and consequently @xmath71 .
the condition that @xmath55 is a continuous positive function means that the jordan curve @xmath130 is starlike . on the other hand only starlike curves w.r . to the origin
have this representation .
we need the following simple lemma : for @xmath131 and @xmath132 $ ] we have @xmath133 denote by @xmath87 the periodic extension of @xmath87 in @xmath134 .
we will show that , for @xmath125 @xmath135 let @xmath136 .
then either @xmath137 or @xmath138 .
put @xmath139 , @xmath140 , in the first case , and @xmath139 , @xmath141 in the second case , and @xmath142 .
then @xmath143 for simplicity , denote @xmath144 by @xmath145 , @xmath146 by @xmath147 and @xmath148 by @xmath87 .
let @xmath149 be a triangle with vertexes @xmath150 , @xmath151 and @xmath152 ( see figure 1 ) .
assume without loos of generality that [ poincare ] @xmath153 let @xmath154 and @xmath155 be a segment with endpoints @xmath156 and @xmath157 $ ] , such that latexmath:[$|lp|=\varepsilon @xmath155 .
then @xmath159 moreover the angle @xmath160 between @xmath155 and @xmath161 is given by @xmath162\}}{p|(1-\varepsilon)p+\varepsilon e^{it } q|}.\ ] ] let @xmath31 and @xmath32 be the lengths of chords of the unit circle that correspond to the angles @xmath147 and @xmath160 , respectively . then @xmath163 and @xmath164 thus @xmath165 let @xmath166 and @xmath167 .
as @xmath168 , then @xmath169 .
thus @xmath170 .
let @xmath171 be the point of @xmath172 $ ] that belong to the half - line @xmath173 .
since @xmath174 , we have the following simple geometric fact @xmath175 on the other hand @xmath176 having in mind the fact that @xmath177 it follows the desired conclusion . to show that @xmath71 we only need to point out that @xmath178 together with the assumptions of theorem [ ara ] assume that @xmath55 is a smooth function
. then @xmath179 by theorem [ ara ] we obtain the following theorem : [ rrr ] let @xmath2 be a lipschitz starlike curve w.r .
the the origin , parameterized by polar coordinates @xmath180 .
let @xmath181 be its extension between the unit disk @xmath6 and the jordan domain @xmath182 .
if @xmath183 , then @xmath2 is a circle with the center at origin .
assume first that @xmath55 is a smooth lipschitz function .
let @xmath58 .
proceeding as in we have @xmath184 and @xmath185 so @xmath186 let @xmath187 then is equivalent to @xmath188 by theorem [ ara ] , @xmath189 .
it follows from that , if @xmath190 , then @xmath191 therefore there exists @xmath192 $ ] such that @xmath193 from @xmath194 we obtain that @xmath195 .
thus @xmath196 this implies that @xmath197 and therefore @xmath198 for some complex constant @xmath80 .
the general case follows from lemma [ dri ] .
let @xmath50 be a smooth starlike jordan curve w.r . to the origin in @xmath11 .
we will recall some properties of @xmath2 .
let @xmath199 be the polar parametrization of @xmath2 .
the tangent @xmath200 of @xmath2 at @xmath201 is defined by @xmath202 following the notations in @xcite , the angle @xmath203 between @xmath204 and the positive oriented tangent at @xmath204 is defined by @xmath205 hence @xmath206 consequently @xmath207 observe that for smooth starlike jordan curve @xmath2 , we have @xmath208 put @xmath209 let @xmath210 be a continuous locally injective function from the unit circle @xmath8 onto the star - like jordan curve @xmath2 . then @xmath211 is a parametrization of @xmath2 which represents @xmath106 . if @xmath106 is a orientation preserving then @xmath212 obviously is monotone increasing .
suppose that @xmath106 is differentiable .
since @xmath213 , we deduce that @xmath214 .
hence @xmath215 from and we obtain @xmath216 from we obtain @xmath233 and consequently @xmath234 by making use of , we obtain ( i ) . further by we have @xmath235 since @xmath236 is follows that @xmath237 furthermore , since @xmath238=[0,2\pi]$ ] , it follows that @xmath239 .
let @xmath240 . for @xmath241 and @xmath242 , we have @xmath243 and @xmath244 respectively .
on the other hand @xmath245 i.e. @xmath246 thus @xmath87 is @xmath226quasiconformal where @xmath227 this concludes @xmath225 .
moreover the previous proof shows that @xmath247 . to show @xmath248 , and the last assertion of the theorem we do as follows .
if @xmath87 is @xmath226quasiconformal , then @xmath87 is differentiable almost everywhere .
assume that @xmath249 .
then from , we obtain that @xmath250 thus @xmath251 by using again , we obtain that @xmath252 then we infer that @xmath253 since the right hand side of the last inequality is increasing in @xmath254 $ ] , by making use of it follows that @xmath255 this finishes the proof of @xmath228 and @xmath230 .
the question arises , which homeomorphism of the unit circle onto a smooth starlike jordan curve @xmath2 induces a radial quasiconformal mapping with the smallest constant of quasiconformality .
it follows from theorem [ star ] that , if there exists a @xmath39-quasiconformal radial mapping between the unit disk and a smooth starlike domain , then @xmath256 . to motivate the previous question ,
recall the teichmller problem . for a given @xmath257-quasisymmetric selfmapping of the unit circle or ( equivalently ) of the real line , find an extension with minimal constant of quasiconformality .
this problem is related to unique extremality . for this topic
we refer to the paper @xcite .
let @xmath259 and let @xmath260 be a radial mapping of the unit disk onto the interior of the ellipse @xmath261 then for @xmath262 we have @xmath263 if @xmath264
we have @xmath265 on the other hand by theorem [ rrr ] @xmath266 moreover @xmath267 is a @xmath39 quasiconformal mapping where @xmath268 to show , we begin by @xmath269 the minimum is @xmath270 and is achieved for @xmath271 , @xmath272 , @xmath273 or @xmath274 and the maximum for @xmath275 where @xmath276 .
the maximum is equal to @xmath277 the larger solution of the equation @xmath278 is given by .
let @xmath279 then @xmath87 is a polar parametrization of the unit square @xmath280 moreover , since @xmath281 , and @xmath282 , by theorem [ star ] , we have @xmath283 the constant of quasiconformality of @xmath0 is @xmath284 1 l. ahlfors : _ lectures on quasiconformal mappings , _ van nostrand mathematical studies , d. van nostrand 1966 .
a. beurling , l. ahlfors : _ the boundary correspondence under quasiconformal mappings .
_ acta math .
* 96 * ( 1956 ) , 125142 . c. j. bishop : _ bilipschitz approximations of quasiconformal maps .
fenn . math .
* 27 * ( 2002 ) , no .
1 , 97108 .
d. kalaj , _ on harmonic diffeomorphisms of the unit disc onto a convex domain .
_ complex var .
theory appl .
48 ( 2003 ) , no .
2 , 175187 .
d. kalaj , m. pavlovi : _ on quasiconformal self - mappings of the unit disk satisfying the poisson s equation _ , to appear in transactions of ams .
a. fletcher , v. markovi , _ quasiconformal maps and teichmller theory .
_ oxford graduate texts in mathematics , 11 .
oxford university press , oxford , 2007 .
viii+189 pp .
|
in this paper we discus the radial extension @xmath0 of a bi - lipschitz parameterization @xmath1 of a starlike jordan curve @xmath2 w.r . to origin .
we show that , if parameterization is bi - lipschitz , then the extension is bi - lipschitz and consequently quasiconformal . if @xmath2 is the unit circle , then @xmath3 . if @xmath2 is not a circle centered at origin , and @xmath4 is a polar parametrization of @xmath2 , then we show that @xmath5 .
subjclassprimary 26a16 , secondary 30c62 , 51f99 ifundefinedsubjclassname@2000 xpxpsubjclassname@2000
|
infrared luminous ( @xmath6 l@xmath7 ) and ultraluminous galaxies ( ulirgs ) ( @xmath8 l@xmath7 ) are believed to be major mergers of massive , gas - rich disk galaxies .
they are spectacular systems , related to super - starbursts and massive gas flows to their central regions .
their molecular gas distribution is often found in rotating nuclear disks , toruses or bars only a few hundred pc in extent .
these ulirgs have been suggested as precursors of qsos ( e.g. norman and scoville 1988 ) .
still , galaxies collide that do not have the necessary properties to become ultraluminous objects .
the importance of these intermediate , often minor i.e. unequal - mass mergers , to the general evolution of galaxies is not well understood .
both major and minor mergers undergo bursts of star formation , but the activity seems to take place on a larger linear scale in the minor mergers . whether the starburst processes and triggering mechanisms remain the same for the compact and large scale bursts is unclear .
it appears however , that some of the extended bursts can reach the same high sfes as the ulirgs .
probing the properties and distribution of the molecular ism enables modelling of the triggering and evolution of the starburst . for this purpose , it is necessary to go beyond and use other ( fainter ) molecular tracers such as the isotopomer and high dipole moment molecules such as hcn , hnc and cn .
these lines are powerful diagnostic tools , in particular in combination with other extinction - free tracers like ir and radio . the minor merger , and young shell galaxy , ngc 4194 ( `` the medusa '' ) has an extended region ( 2 kpc ) of intense star formation ( e.g. prestwich 1994 ) which is responsible for most of the fir luminosity ( @xmath9 l@xmath7 at @xmath10=39 mpc ) .
the sfe , @xmath11 , is high , 40 @xmath12 , _ similar to the sfe of the ulirg arp 220_. the global is large , @xmath13 , indicating a highly excited or disrupted ism ( aalto & httemeister 2000 , ah ) . despite the high sfe , emission from the high density tracer molecule hcn remains undetected in ngc 4194 .
we present a preliminary analysis of the and distribution obtained at high resolution with ovro .
we offer an explanation for the apparent lack of dense gas in the medusa and suggest `` cartoon models '' of a high pressure ( ulirg ) and a comparatively low pressure ( medusa ) molecular ism .
since both high kinetic temperatures and large turbulent line widths will decrease the 10 optical depth ( @xmath14 ) , a map of the @xmath0 10 intensity ratio ( ) can be used to identify regions of extreme or unusual physical conditions in the molecular gas .
large ratios indicate low to moderate @xmath14 and aalto ( 1995 ) established some general diagnostics of the cloud conditions and environment based on global values of : small ratios , @xmath15 are an indication of a normal galactic disk population of clouds dominated by cool giant molecular clouds ( gmcs ) ; intermediate ratios @xmath16 are associated with the inner regions of normal starburst galaxies ; the extreme values @xmath17 originate in turbulent , _ high pressure _ gas in the centers of luminous mergers . in the most luminous mergers , the gas surface density implied by the galactic conversion factor from luminosity to m(h@xmath18 )
is well over 10@xmath19 pc@xmath20two orders of magnitude higher than in typical milky way gmcs .
this led aalto ( 1995 ) to suggest that large values of are related to the large gas surface densities in compact nuclear starbursts .
large surface densities require high pressures in hydrostatic equilibrium ; as the @xmath21 line ratios in these objects indicate _ low _
( @xmath22 @xmath5 ) densities of the emitting gas - component it must be supported by large turbulent line widths ( @xmath23 ) .
thus , @xmath14 can be reduced to moderate ( @xmath24 ) values , resulting in large .
however , the global of the medusa is high , but the gas surface density is more than an order of magnitude lower than that of the ulirgs .
therefore , the pressure will be considerably lower and the ism is likely much less turbulent ( even though it is still significantly higher than the average pressure for the galactic disk ) .
thus , for the medusa , the high may well be caused by elevated kinetic temperatures rather than large line widths .
there is indeed a strong correlation between fir colour temperature and large values of : galaxies with @xmath17 all have @xmath25 flux ratios @xmath26 ( e.g. aalto 1991 , 1995 ) indicating high average dust temperatures .
thus , large values of are also related to the effect of the starburst itself , heating dust and gas to high temperatures .
figure 1 shows the emission overlayed on an r - band image of ngc 4194 .
the distribution is surprisingly extended , @xmath27 kpc , for an advanced merger even if the bulk ( 60 - 70 % ) of the emission emerges from the 2 kpc starburst region .
the morphology is complex , tracing large - scale dust lanes , one which curves along the north - eastern edge of the main body and continues into the tidal tail , and one which is crossing the central region along the minor axis .
the gas mass ( for a standard co to h@xmath18 conversion factor ) is @xmath28 m@xmath7 ( ah ) . there is substantial variation in across ngc 4194 : from quiescent values of 7 in the the eastern dust lane 5 kpc from the center , to high values , @xmath120 , in the starburst and central dust lane .
this clearly demonstrates that there is a strong connection between and gas environment . in figure 2
we show the and contours overlayed on an archival hst wfpc2 image of the inner 2 kpc of ngc 4194 . the and morphologies differ substantially showing significant variation in also within the central region .
peaks west of , associated with a peculiar dust feature . elevated values of are found in two regions : a ) througout the extended starburst region and b ) in the curved part of the central dust lane .
the global is dominated by the starburst region where is likely elevated because of large kinetic temperatures ( see next section ) . in region
b ) gas streaming may cause local effects of large line widths ( see also httemeister and aalto , this volume ) .
.line ratios in high starbursts [ cols="<,^,^",options="header " , ] in the table we list typical line ratios for compact high pressure and extended low pressure starbursts .
both types may have large values of @xmath29 , but their average ism properties differ on several accounts . in figure 3
we show cartoons of the two ism types , `` raisin roll '' and `` fried eggs '' . for the former ( ulirg ) scenario ,
the emission is emerging from low density @xmath30 @xmath5 diffuse gas of large linewidths and filling factor , while the hcn 1 - 0 emission is coming from dense ( @xmath4 @xmath5 ) , embedded clouds ( but see e.g. aalto 1995 for a discussion on mid - ir pumping of hcn ) .
the bulk of the molecular mass is in the dense gas , but the high value of can primarily be attributed to the large linewidths of the diffuse molecular gas ( which can remain molecular because of the extreme pressure ) .
galaxies which may be characterised by this ism scenario include arp 220 , ngc 6240 and mrk 231 .
it may seem surprising that the emitting gas is somewhat denser , @xmath31 @xmath5 , in the lower pressure scenario ( and with a smaller volume filling factor ) .
this may be because ( in contrast to the `` raisin roll '' ism ) the intercloud medium is likely atomic ( or ionized ) since the destructive forces of the newborn stars are not balanced by a high ambient pressure .
thus , the emitting gas is more `` cloudy '' in the lower pressure model , but the relatively faint hcn emission indicates that the mass fraction of _ high _ density gas , @xmath32 @xmath5 , is considerably lower .
the emission may , however , be filamentary and not always in bound clouds .
the large value of is likely caused by elevated gas temperatures ( @xmath33 k ) since the gas is dense enough for to be thermalized ( @xmath34 ) .
the clouds may be pdrs ( photon dominated regions ) if the impact of the starburst is strong enough .
galaxies which may be characterised by this ism scenario include the medusa ( ngc 4194 ) , ugc 2866 and ngc 1614 .
because of its high dipole moment , hcn 1 - 0 emission is often used as a tracer of high density ( @xmath32 @xmath5 ) gas and the @xmath35 line intensity ratio as a measure of the mass fraction of dense gas .
bright hcn ( low @xmath35 ratios ) in ulirgs has been used to argue for a starburst origin of their luminosity ( e.g. solomon , downes and radford 1992 ) because of a global correlation between @xmath36 and @xmath37 .
it is therefore interesting that the sfe of the medusa rivals that of the ulirg arp 220 even though the hcn emission is faint towards ngc 4194 .
how can there be an efficient transformation of gas into stars when there is only little dense gas present ?
the answer may lie in the time the molecular gas spends in a dense phase which is related to the average gas pressure and dynamics .
the medusa starburst takes place in an environment of reduced shear because it occurs inside the region of solid body rotation .
thus , gravitational instability may dominate over tidal shear with a resulting increase in the sfe .
the rate of star formation ( sfr ) is indeed very high , 40 m@xmath7 yr@xmath38 , which is close to the maximum sfr per kpc@xmath39 found by lehnert & heckman ( 1996 ) .
since the hydrostatic pressure is much lower than in a ulirg , gas is not maintained at high densities , but passes through a dense phase quickly on its way to becoming stars .
the time spent in a dense phase is short and will not be an observed signature of the ism .
the rate of star formation has been found to be dependent on large scale dynamics ( e.g. kenney , carlstrom and young 1993 ) .
perhaps , for a threshold level of the gas surface density , the dynamical environment of the clouds is also important in regulating the _ efficiency _ of star formation , even more so than the relative amount of high density gas
. for the compact burst in the deep potential well of a massive major merger , the gas surface density ( and matter density ) is high and thus also the ambient pressure leading to large _ average _ gas densities .
still , the dynamical and radiative environment may be unfavourable to star formation . within 200 pc of the deep potential
well of a differentially rotating galaxy with rotational speed 250 , the clouds must have average densities of @xmath32 @xmath5 just to be stable against tidal shearing .
lower density gas will only exist as diffuse unbound clouds .
in addition , the feedback mechanisms of the starburst itself may help regulate the sfe , and these mechanisms may be effective in the densely packed , high gas surface density central regions of ulirgs .
thus , in some circumstances , the gas will remain at substantial average density without forming stars . in this context , the @xmath36 - @xmath37 correlation is , at least partially , caused by high pressure gas being more centrally concentrated than low pressure gas , and that fir emission in galaxies tend to emerge from the inner regions at least the 60 @xmath40 m emission .
the fir emission may indeed originate in starburst activity , but also from an agn or from an evolved starburst where densely packed stars heat a fragmented ism .
a possible caveat in the notion that the sfe is similar in the medusa and arp 220 is the possibility that the luminosity is tracing molecular mass differently in a compact and extended starburst .
furthermore , the ulirgs deviate from the limiting sfr per kpc@xmath39 found by lehnert and heckman .
their maximum star formation rates are in the range 100 - 300 m@xmath7 yr@xmath38 instead of 20 - 40 and from a smaller area than less powerful galaxies .
dust opacity and hidden agns may explain some of the discrepancy in the limiting sfrs , but we can not exclude that the underlying star formation mechanisms are different .
|
high resolution observations of and 10 in the medusa ( ngc 4194 ) minor merger show the @xmath0 10 intensity ratio ( ) increasing from normal values ( 5 - 10 ) in the outer parts of the galaxy to high ( @xmath1 20 ) values in the central , extended starburst region .
ratios @xmath2 are otherwise typical of more luminous mergers .
the medusa @xmath3 ratio rivals that of ultraluminous galaxies ( ulirgs ) , despite the comparatively modest luminosity , indicating an exceptionally high star formation efficiency ( sfe ) .
we present models of the high pressure ism in a ulirg and the relatively low pressure ism of the medusa .
we discuss how these models may explain large in both types of distributions .
since the hcn emission is faint towards the medusa , we suggest that the sfe is not primarily controlled by the mass fraction of dense ( @xmath4@xmath5 ) gas , but is probably strongly dependent on dynamics .
the bright hcn emission towards ulirgs is not necessarily evidence that the ir emission there is always powered by starbursts . -1s^-1 # 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in
|
the study of matter - energy density distribution in a quantum gravitational system ( qgs ) is of interest in connection with the problem of the mechanism of nucleation of the expanding universe from the initial cosmological singularity point , as it is asserted by the standard cosmological model .
this problem , as well as the question about the prehistory of the universe before its nucleation , can find the solution in quantum theory which considers matter and gravitation as quantum fields .
the method of constraint system quantization @xcite can be taken as a basis of quantum theory of gravity suitable for the investigation of cosmological and other quantum gravitational systems @xcite . in this theory
, the state vector ( wave function ) satisfies the set of wave equations which describes the time evolution of a quantum system in a generalized space of quantum fields .
the probabilistic interpretation of the state vector of qgs implies its normalizability . in the simplest case of the maximally symmetric geometry with the robertson - walker metric
, the geometric properties of the system are determined by a single variable , namely the cosmic scale factor @xmath0 .
we will consider the homogeneous isotropic qgs formed by matter in the form of a uniform scalar field @xmath1 .
this field can be interpreted as a surrogate of all possible real physical fields of matter averaged with respect to spin , space and other degrees of freedom .
in addition , it will be accepted that qgs is filled with a perfect fluid in the form of a relativistic matter ( further referred as radiation ) which defines a material reference frame enabling us to introduce the time variable ( recognize the instants of time ) @xcite .
it is convenient to formulate quantum theory in terms of dimensionless variables and parameters , in which length is measured in modified units of planck length @xmath2 , proper time is expressed in planck time units @xmath3 , and mass - energy is taken in planck mass units @xmath4 .
the planck density @xmath5 is used as a unit of energy density and pressure .
the scalar field is taken in @xmath6 . here
@xmath7 is newton s gravitational constant .
then the basic equations of the qgs model can be reduced to the following simple set of two differential equations in partial derivatives for the state vector @xmath8 @xcite @xmath9 @xmath10 where @xmath11 is a conformal time expressed in radians . in general relativity ,
the cosmic scale factor @xmath0 describes the overall expansion or contraction of the cosmological system , being the function of proper time @xmath12 which is connected with the conformal time @xmath11 by the equation @xmath13 in eqs .
( [ 1 ] ) and ( [ 2 ] ) , the quantities @xmath0 , @xmath1 and @xmath11 are independent variables of the state vector @xmath14 .
the parameter @xmath15 is a real constant which is determined by the energy density of a perfect fluid @xmath16 taken in the form @xmath17 in natural physical units , @xmath15 has the dimension of [ energy @xmath18 length ] , @xmath19 = [ \hbar c]$ ] .
the coefficient @xmath20 in eq .
( [ 1 ] ) is caused by the choice of the parameter @xmath11 as the conformal time variable .
the operator @xmath21 in eq .
( [ 2 ] ) is the hamiltonian of the field @xmath1 .
this hamiltonian is defined in a curved space - time and therefore it depends on a scale factor @xmath0 as a parameter , @xmath22 . if the potential term of the uniform scalar field @xmath1 is described by the scalar function @xmath23 , then @xmath24 and @xmath25 where @xmath26 and @xmath27 are the operators of the energy density and pressure .
the @xmath28 can be interpreted as the lagrangian of the scalar field .
the variable @xmath1 is defined on the interval : @xmath29 . in eq .
( [ 2 ] ) , we single out the curvature constant @xmath30 , @xmath31 for for spatially closed , flat , and open qgs .
the derivation of eqs .
( [ 1 ] ) and ( [ 2 ] ) @xcite does not depend on the numerical value of @xmath30 .
the variables @xmath0 and @xmath1 satisfy the commutation relations @xmath32 = i , \qquad [ \phi ,- i \partial_{\phi } ] = i.\ ] ] all other commutators vanish .
the equations ( [ 1 ] ) and ( [ 2 ] ) can be rewritten as one time equation of schrdinger type in the space of two variables @xmath0 and @xmath1 with the time - independent operator with the dimension of [ energy @xmath18 length ] instead of a hamiltonian .
it is convenient to pass from @xmath33-representation of the wave function @xmath8 to a representation in which a continuous variable @xmath1 is replaced by a discrete or continuous set of values of quantum number @xmath34 which characterizes the states of the matter field in a comoving volume @xmath35 . with that end in view
, we introduce the complete set of orthonormalized state vectors of the scalar field @xmath36 in a representation of a rescaled variable @xmath37 , in which the hamiltonian @xmath21 is diagonalized , @xmath38 after averaging of @xmath21 with respect to the field @xmath39 , we transform from the scalar field to the new effective matter in discrete and/or continuous @xmath34th state with the proper mass - energy @xmath40 .
the energy density @xmath41 and pressure @xmath42 of such an averaged matter are @xmath43 introducing the equation of state parameter @xmath44 we find that the averaged matter is a barotropic fluid with the equation of state @xmath45 the explicit form of the field @xmath39 and the mass - energy @xmath46 are different for the different functions @xmath23 . if , for example , @xmath47 , where @xmath48 is the coupling constant , @xmath49 takes arbitrary non - negative values , and the summation with respect to @xmath49 is not assumed , then @xcite @xmath50 where @xmath51 is an eigenvalue of the equation @xmath52 in a particular case of the @xmath53-model , the parameter @xmath54 and the barotropic fluid becomes an aggregate of separate macroscopic bodies ( dust ) with the mass - energy @xmath55 , which does not depend on @xmath0 , and @xmath56 , where @xmath57 .
the equation ( [ 12 ] ) for @xmath58 is the equation for quantum oscillator .
the mass @xmath59 can be interpreted as a sum of masses of separate excitation quanta of the spatially coherent oscillations of the field @xmath39 about the equilibrium state @xmath60 .
the quantum number @xmath34 is the number of these excitation quanta with the mass @xmath61 . taking the mass of proton @xmath62 gev ( @xmath63 in planck mass units ) as such a mass , one obtains @xmath64 gev ( @xmath65 ) when the number of protons @xmath66 .
such a mass of a dust leads to the actual density of matter in the observed part of our universe @xcite .
the review of the properties of the barotropic fluid in the @xmath67-models with @xmath68 is given in refs .
using the completeness and orthonormality of the state vectors @xmath36 , the state vector @xmath8 of qgs in @xmath69-representation can be represented in the form of the superposition of all possible @xmath34th states of the barotropic fluid @xmath70 where @xmath71 satisfies the differential equations @xmath72 @xmath73 the general solution of this set has the form @xmath74 where @xmath75 @xmath76 the wave functions @xmath77 in eq .
( [ 16 ] ) satisfy the equation @xmath78 where @xmath79 and @xmath80 for a discrete @xmath81th state of radiation with the density ( [ 4 ] ) , and @xmath82 for a continuous @xmath15th state of radiation . in both cases ,
the qgs is considered as a material point moving in the potential function @xmath83 formed by the barotropic fluid in @xmath34th state .
the parameter @xmath84 in eqs .
( [ 17 ] ) and ( [ 18 ] ) is an arbitrary constant taken as a time reference point .
the equation ( [ 19 ] ) determines the stationary quantum state of qgs at some fixed instant of time @xmath84 , the choice of which is arbitrary , @xmath85 . in probabilistic interpretation of quantum theory , the coefficient @xmath86 gives the probability @xmath87 to find the qgs in @xmath81th state of the discrete spectrum of radiation and @xmath34th state of the barotropic fluid at the instant of time @xmath84 . the same interpretation can be given for the coefficient @xmath88 in the case of continuous state of radiation .
the conditions of normalization and orthogonality can be imposed on the wave functions @xmath77 , @xmath89 then the state vector @xmath8 appears be normalized to unity , @xmath90 under the condition that the probability summed over all possible quantum states of radiation and barotropic fluid equals to unity , i.e. the qgs with the state vector @xmath8 exists .
according to eq . ( [ 5 ] ) , the equation ( [ 2 ] ) is invariant under the inversion @xmath91 . taking into account that the robertson - walker line element contains only even powers of @xmath0 and also that the sign of @xmath0 has not been fixed when deriving eqs .
( [ 1 ] ) and ( [ 2 ] ) , the equation ( [ 19 ] ) can be generalized by extending to the domain of negative values of @xmath0 , so that @xmath92 ( cf . refs .
@xcite ) . in order to clarify the physical meaning of the solutions of eq .
( [ 19 ] ) in the domain @xmath93 , let us integrate eq .
( [ 3 ] ) , @xmath94 it gives @xmath95 from eqs .
( [ 13])-([18 ] ) , it follows that the dependence of the state vector @xmath8 on time @xmath96 is determined by the exponential multiplier @xmath97 , where the inessential multiplier @xmath20 is omitted and the natural condition @xmath98 is imposed .
let this exponential describes the wave expanding from @xmath99 in the direction @xmath100 and passing through the point @xmath101 , where @xmath102 according to eq .
( [ 24 ] ) .
an illustration is given in fig .
[ fig:1 ] .
.,scaledwidth=70.0% ] then the scale factor @xmath103 $ ] corresponds to the values @xmath104 $ ] , and the scale factor @xmath105 corresponds to the values @xmath106 . as a result for the arrow of time from @xmath107 to @xmath108 , the state vector @xmath8 describes the qgs contracting on the semiaxis @xmath93 , since @xmath109 decreases , and the qgs expanding on the semiaxis @xmath110 , because @xmath109 increases . the instant of time @xmath101 can be interpreted as the instant of nucleation of the quantum system expanding in time from the point @xmath111 , although any nucleation `` from nothing '' does not occur physically .
what happens at the instant @xmath101 is that the regime of the preceding contraction of the system changes into the subsequent expansion .
the equation ( [ 19 ] ) describes the stationary states of qgs for a given value of constant @xmath15 .
the state vector ( [ 13 ] ) contains all information about the system as a whole : the cross - section @xmath112 determines the quantum state of qgs at the time instant @xmath12 , when such a value of the scale factor holds . if one applies the above - described scenario to our universe at the planck epoch , interpreting the passage through the point @xmath111 at @xmath101 as the nucleation of expanding universe with @xmath110 at @xmath113 , then the answer to the question `` what was with the quantum system before the instant of nucleation of the universe of our ( expanding ) type ? ''
can be given : there has existed another universe with the same mass - energy @xmath114 and wave function @xmath77 characterized by the same quantum numbers for matter and radiation as the nucleated universe , however that universe has been contracting up to the state with @xmath111 , which not necessarily will be singular ( see sect . 5 below ) .
let us consider qgs in which the barotropic fluid ( matter ) and radiation are in some definite quantum states .
in such a quantum system , the intensity distribution of matter - energy as a function of @xmath0 is given by the expression @xmath115 where indices of the states of matter ( @xmath34 ) and radiation ( @xmath81 ) are omitted .
the wave function @xmath77 is the solution of eq .
( [ 19 ] ) complemented with the appropriate boundary conditions which determine , for example , the behaviour of @xmath77 in the asymptotic domain of large values of @xmath109 .
the intensity summed over all possible values of @xmath0 gives the mean mass - energy of matter in qgs in the state @xmath77 normalized to unity , @xmath116 we note that @xmath117 for the @xmath53-model . as an example , we will calculate the intensity ( [ 25 ] ) in the exactly solvable models of spatially closed and flat qgs filled with dust and radiation . in the model of spatially
closed qgs formed by dust whose mass does not depend on @xmath0 , @xmath118 , the equation ( [ 19 ] ) is reduced to the equation for an oscillator by substitution of the variable @xmath119 , @xmath120 where an eigenvalue @xmath121 . changing from variable @xmath0 to @xmath122
restores inversion invariance of eq .
( [ 19 ] ) violated in the case when matter is represented by dust .
the variable @xmath122 describes the deviation of @xmath0 from its equilibrium value at the point @xmath123 .
this variable lies in the interval @xmath124 , if @xmath0 takes only positive values and zero . for @xmath125 , the interval of change of @xmath122 in the normalization integral for the function @xmath126
can be extended to the whole semiaxis of the negative values of @xmath122 .
the error arising here is @xmath127 , where @xmath128 @xcite .
then the solution of eq .
( [ 27 ] ) , decreasing at @xmath129 and normalized to unity , gives the intensity distribution of dust matter in the form @xmath130 where @xmath131 is the hermite polynomial and the variable @xmath0 takes any values in the interval @xmath132 .
the constant @xmath15 is quantized in accordance with the condition @xmath133 it takes a sequence of discrete positive values for @xmath134 and discrete negative values for @xmath135 . in the latter case , radiation as a perfect fluid is characterized by the negative energy density ( [ 4 ] ) and pressure @xmath136 , i.e. a perfect fluid acquires the properties of the antigravitating matter at small quantum numbers @xmath81 and large masses @xmath137 .
the equation ( [ 28 ] ) determines the intensity distribution of dust matter in qgs both in the regime of its contraction , when @xmath93 , and in the regime of expansion , when @xmath110 .
the quantities @xmath81 and @xmath137 in eqs .
( [ 28 ] ) and ( [ 29 ] ) are free parameters . in fig .
[ fig:2 ] , it is shown the intensity distribution for the parameters @xmath138 and @xmath139 ( bold line ) , when @xmath140 , and @xmath141 ( thin line ) , when @xmath142 . in the case of the positive energy density of radiation , the intensity evolves so that the first maximum of @xmath143 is reached in the domain of contraction , straight before the boundary point @xmath111 , where the regime of contraction changes into expansion .
it is as if qgs accumulates the energy just before the beginning of expansion .
the intensity at the point @xmath111 is found to be finite ( @xmath144 in fig .
[ fig:2 ] ) . in the case
@xmath142 , the negative pressure of radiation pushes out the first maximum into the domain of expansion near the point @xmath111 . in both cases , the intensity oscillates between maximum values and zero in the domain of expansion .
the value @xmath123 corresponds to the smallest maximum and @xmath145 . for @xmath146 ,
the intensity decreases exponentially . ) for the cases @xmath140 ( bold line ) and @xmath142 ( thin line).,scaledwidth=70.0% ] for @xmath147 , the intensity distribution oscillates according to the law @xmath148 averaging with respect to oscillations gives the intensity which does not depend on @xmath0 , @xmath149 the general behaviour of the intensity ( [ 28 ] ) reproduces a position probability density of a harmonic oscillator with respect to the variable @xmath0 renormalized by the mass @xmath137 .
for the qgs under consideration , the equation ( [ 27 ] ) is exact .
its application to the cosmological problem leads to nontrivial conclusions about the evolution of the intensity distribution of matter in the qgs with all possible parameters @xmath81 and @xmath137 close to the planck epoch .
the prediction about the exponential decreasing of the intensity distribution of matter for @xmath146 is made in the model of spatially closed qgs .
now let us consider another exactly solvable model , namely that of flat space . in the model of spatially flat qgs formed by dust with the constant mass @xmath137 , the equation ( [ 19 ] ) is reduced to @xmath150 by introducing the variable @xmath151 its general solution is a linear superposition of airy functions @xmath152 and @xmath153 .
we will look for the solution which has a form of the outgoing wave at @xmath154 and satisfies the boundary condition @xmath155 . by normalizing this solution to delta - function as in eq .
( [ 21 ] ) , we get the following expression for the intensity distribution of matter ( [ 25 ] ) @xmath156 in fig .
[ fig:3 ] , it is depicted the intensity @xmath157 ( [ 34 ] ) as a function of @xmath0 for the same parameters as in fig .
[ fig:2 ] for @xmath140 . ) for @xmath139 , @xmath158.,scaledwidth=70.0% ] as in the case of spatially closed qgs , the intensity ( [ 34 ] ) increases exponentially with contraction of the system , reaching a maximum , passing through the point @xmath111 with the finite value ( @xmath159 ) and then oscillating . from the asymptotic expression for @xmath160
, it follows that the intensity averaged over the oscillations decreases with @xmath0 according to the law @xmath161 if one assumes that during the quantum epoch the intensity ( [ 35 ] ) decreases in time as @xmath162 like the energy density in general relativity , then the scale factor should increase in time according to a power law @xmath163 .
such a growth of @xmath0 corresponds to an inflationary model , in which it is supposed that the scale factor increases more slowly than in the exponential regime @xcite .
the intensities ( [ 28 ] ) and ( [ 34 ] ) obtained from exact solutions of eq .
( [ 19 ] ) allows us to draw analogy with known phenomena which are described by the equations of quantum mechanics and optics .
we will consider some of these analogies in detail .
the eigenfunction of the ground state of an oscillator ( [ 27 ] ) @xmath164 has the form of the normalized minimum packet whose center of gravity is displaced in the positive @xmath0 direction by an amount @xmath165 @xcite .
we assume that such a state corresponds to the time instant @xmath101 , @xmath166 .
if the equivalent classical system evolves in time @xmath12 with a power - law scale factor , @xmath167 , where @xmath49 and @xmath168 are some positive constants , then the time phase in eq .
( [ 17 ] ) at @xmath98 takes the form @xmath169 \omega \tau,\ ] ] where the frequency @xmath170 depends on @xmath0 . from the requirement @xmath171
, it follows the restriction : @xmath110 and @xmath172 . under these conditions we have @xmath173 .
taking into account only the sum with respect to @xmath81 in eq .
( [ 16 ] ) ( continuos spectrum is absent ) , using the representation ( [ 17 ] ) with @xmath98 , and the explicit form of the function ( [ 36 ] ) , as well as the representation of the phase ( [ 37 ] ) , one can calculate the wave function @xmath174 . as a result
, the intensity distribution @xmath175 of matter with the mass @xmath137 in wave packet moving in time appears to be equal @xmath176^{2 } \right\}.\ ] ] if the mass of matter ( dust ) goes to zero , then the wave functions tends to the eigenfuction corresponding to the lowest energy of radiation @xmath177 at @xmath178 , while the intensity does not depend on time and goes to zero as @xmath179 at @xmath180 .
if the mass @xmath181 , then the condition @xmath182 is ensured by including the stationary states with the quantum numbers @xmath183 in the packet . in the case @xmath147 , the main contribution to the packet is made by the eigenfunction with @xmath184 , and the intensity is described by the expression ( [ 39 ] ) . in fig .
[ fig:4 ] , it is shown the intensity @xmath185 ( [ 39 ] ) for the parameters @xmath139 , and @xmath186 in eq .
( [ 38 ] ) . the mods with @xmath187 and @xmath188 make the most important contribution to @xmath185 .
matter is distributed in @xmath0 and @xmath12 in the form of periodic structures like petals or stretched bubbles and displaced to their edges .
these structures are limited by the value @xmath189 with respect to @xmath0 ( as in fig .
[ fig:2 ] ) , and their number increases with time . for the spatially flat qgs formed by dust and filled with radiation , described by eq .
( [ 19 ] ) with @xmath190 and @xmath191 , an analogy with the motion of a particle in a uniform external field along the coordinate @xmath0 with the energy @xmath15 under the action of the force @xmath192 is obvious @xcite .
therefore we consider a more interesting analogy with the distribution of the light intensity in the neighbourhood of the point where its ray is tangent to the caustic @xcite . from eq .
( [ 34 ] ) , it follows the asymptotic expression for @xmath157 at @xmath193 , which it is convenient to rewrite in the form @xmath194 where we denote @xmath195 the introduced quantities have a clear physical meaning .
the amplitude @xmath196 is the intensity far from the caustic which would be obtained from geometrical optics neglecting diffraction effects , @xmath197 is the distance from the point of observation along the normal to the caustic which takes positive values for points on the normal in the direction of its center of curvature , @xmath198 is the energy of the ray of light , @xmath199 is the radius of curvature of the caustic at the point of observation .
the equation ( [ 40 ] ) describes the intensity for the ray of light at large negative values of @xmath197 . in the radiation - dominated epoch , @xmath200 ,
the radius of curvature is @xmath201 .
it means that the wave surface is practically flat . in the matter - dominated epoch , @xmath202
, we have @xmath203 . for @xmath204 ,
the wave surface is practically spherically symmetric , and its center of the caustic coincides with the focus .
far from the focus , the averaged intensity decreases as @xmath205 with account of diffraction effects ( see eq .
( [ 35 ] ) ) .
diffraction in the qgs can be explained by scattering of electromagnetic waves of radiation on massive dust particles playing the role of opaque bodies ( screens ) .
the observed diffraction picture is similar to that depicted in fig .
[ fig:3 ] .
|
in the framework of the method of constraint system quantization , a quantum gravitational system ( qgs ) with the maximally symmetric geometry is studied .
the state vector of the qgs satisfies the set of wave equations which describes the time evolution of a quantum system in the space of quantum fields .
it is shown that this state vector can be normalized to unity .
the generalization of the wave equations to the domain of negative values of the cosmic scale factor is made .
for the arrow of time from past to future , the state vector describes the qgs contracting for the negative values of the scale factor and expanding for its positive values .
the intensity distributions of matter are calculated for two exactly solvable models of spatially closed and flat qgss formed by dust and radiation .
the analogies with the motion in time of minimum wave packet for spatially closed qgs and with the phenomenon of diffraction in optics for flat qgs are drawn .
pacs numbers : 98.80.qc , 98.80.cq , 95.35.+d , 95.36.+x
|
the introduction of a symmetry ( i.e. collineation ) is most conveniently studied if the lie derivative of the field equations is taken with respect to the symmetry vector .
more specially , the lie derivative of the ricci and independently the energy momentum tensor can be computed by the symmetry .
so , the field equations can be obtained as lie derivatives along the symmetry vector of the dynamical variables .
previous works on rcvs have been undertaken by several authors .
oliver and davis , who gave necessary and sufficient conditions for a matter space - time to admit an rcv , @xmath2 , with @xmath3 where @xmath4 is the dynamic four - velocity @xcite .
tsamparlis and mason have considered ricci collineation vectors ( rcvs ) in fluid space - times ( perfect , imperfect and anisotropic ) @xcite .
duggal have also considered timelike ricci inheritance vector in perfect fluid space - times @xcite .
et al . _ have discussed space times with conformal killing vectors @xcite .
the study of string fluid has aroused considerable interest as they are believed to give rise to density perturbations leading to the formation of galaxies @xcite .
the existence of a large scale network of strings in the early universe is not contradiction with present day observations of the universe @xcite .
also , the present of strings in the early universe can be explained using grand unified theories ( guts ) @xcite .
thus , it is interesting to study the symmetry features of string fluid .
recently , work on symmetries of the string has been taken yavuz and yilmaz , and yilmaz _ et al .
_ who have considered inheriting conformal and special conformal killing vectors , and also curvature inheritance symmetry in the string cosmology ( string cloud and string fluids ) , respectively @xcite .
yilmaz has also considered timelike and spacelike ricci collineations in the string cloud @xcite .
et al . _ have studied conformal collineation in the string cosmology @xcite .
the theory of spacelike congruences in general relativity was first formulated by greenberg , who also considered applications to the vortex congruence in a rotational fluid @xcite .
the theory has been developed and further applications have been considered by mason and tsamparlis , who also considered spacelike conformal killing vectors and spacelike congruences @xcite . a space - time admits a ricci collineation vector ( rcv ) @xmath5 if @xmath6 where @xmath7 is the ricci tensor and @xmath8 denotes lie derivative along @xmath5 .
a conservation law , valid for any rcv , was established by collinson @xcite . if @xmath9 is an rcv
, then it can be verified that @xmath10 and if einstein s field equations @xmath11 are satisfied , then @xmath12_{;a}=0.\ ] ] equation ( [ eq4 ] ) plays an important role as one of necessary and sufficient conditions for a space - time to admit an rcv , @xmath9 .
it is important to establish a connection between timelike or spacelike rcvs and material curves in the string fluid . in this paper
, we will investigate the properties of rcvs , @xmath2 , parallel to the string fluid unit four - velocity vector @xmath13 : @xmath14 and spacelike rcvs , @xmath15 , orthogonal to @xmath13 : @xmath16 we will express the necessary and sufficient conditions for string fluid space - time to admit a timelike rcv parallel to @xmath13 and a spacelike rcv parallel to @xmath17 in terms of the kinematic quantities of the timelike congruence of world - lines generated by @xmath13 and the expansion , shear , and rotation of the spacelike congruence generated by @xmath17 , respectively .
the energy - momentum tensor for a string fluid can be written as @xcite @xmath18 where @xmath19 is string density and @xmath20 is `` string tension '' and also `` pressure '' .
the unit timelike vector @xmath13 describes the fluid four - velocity and the unit spacelike vector @xmath17 represents a direction of anisotropy , i.e. , the string s directions .
also , note that @xmath21 the paper may be outlined as follows . in section 2 ,
necessary and sufficient conditions for string fluid space - time to admit a timelike rcv parallel to @xmath0 are derived . in section 3 ,
the basic aspects of the theory of spacelike congruences required in this paper are reviewed briefly .
also , necessary and sufficient conditions for a string fluid space - time to admit a spacelike rcv parallel to @xmath1 are given . finally , concluding remarks are made in section 4 .
if einstein s field equation ( [ eq3 ] ) are satisfied , then string fluid with energy - momentum tensor ( [ eq5 ] ) admits an rcv , @xmath2 , if and only if @xmath22 -2\sigma_{c(a}\gamma^{c}_{b ) } -2\omega_{c(a}\gamma^{c}_{b)},\ ] ] @xmath23=0,\ ] ] @xmath24 where @xmath25 is the rate - of expansion , @xmath26 is the rate - of - shear tensor , @xmath27 is the vorticity tensor of the timelike congruence generated by @xmath13 , @xmath28 and @xmath29 .
* proof : * from the definition of the lie derivative it follows that @xmath30 which , using einstein s field equation ( [ eq3 ] ) for string fluid , may be rewritten as @xmath31 suppose first that @xmath32 is an rcv .
then ( [ eq1 ] ) holds and the right - hand side of ( [ eq11 ] ) vanishes . by contracting ( [ eq11 ] ) in turn with @xmath33 , and @xmath34 and by using the expansion @xmath35
we obtain , respectively , @xmath36 = 0,\label{eq14}\\ 2\dot\rho_{s}-\dot q+\frac{2}{3}(2\rho_{s}-q)\theta+2\gamma^{ab}\sigma_{ab}=0,\label{eq15}\end{aligned}\ ] ] and equation ( [ eq7 ] ) . the energy momentum conservation equation will also be required . for string fluid , the momentum conservation equation , which follows from einstein s field equations ,
is @xmath37 ( i ) : : condition ( [ eq7 ] ) was derived directly in the decomposition of ( [ eq11 ] ) .
( ii ) : : in order to determine condition ( [ eq8 ] ) , we first obtain an expression for @xmath38 by eliminating @xmath39 and @xmath40 from ( [ eq13 ] ) . substituting from ( [ eq16 ] ) for @xmath39 into ( [ eq15 ] ) gives @xmath41 and using ( [ eq16 ] ) for @xmath39 and ( [ eq17 ] ) for @xmath40 , equation ( [ eq13 ] ) becomes @xmath42 condition ( [ eq8 ] ) is derived immediately from ( [ eq14 ] ) and ( [ eq18 ] ) .
( iii ) : : in order to derive condition ( [ eq9 ] ) , we observe that ( [ eq13 ] ) may be written as @xmath43 if ( [ eq18 ] ) is used to replace one of the terms @xmath38 in ( [ eq19 ] ) , then ( [ eq19 ] ) becomes @xmath44 from which ( [ eq9 ] ) follows directly .
conditions ( [ eq7])- ( [ eq9 ] ) are therefore necessary conditions if @xmath45 is an rcv .
conversely , suppose that conditions ( [ eq7])- ( [ eq9 ] ) are satisfied . then if ( [ eq7 ] ) for @xmath46 is substituted into ( [ eq11 ] ) , and ( [ eq12 ] ) is used to expand @xmath47 and @xmath48 , ( [ eq11 ] ) becomes @xmath49h_{ab}\bigg{\}}.\
] ] now , @xmath39 is given by the energy conservation equation ( [ eq16 ] ) . in order to obtain an expression for @xmath40
, we first observe that ( [ eq9 ] ) can be expanded as @xmath50 but , by contracting ( [ eq8 ] ) with @xmath13 , ( [ eq18 ] ) is again obtained and by eliminating @xmath38 from ( [ eq22 ] ) , equation ( [ eq17 ] ) for @xmath40 is again derived . by using ( [ eq16 ] ) for @xmath39 and ( [ eq17 ] ) for @xmath40 it is easily verified that the coefficients of @xmath51 and @xmath52 in ( [ eq21 ] ) vanish and therefore @xmath45 is an rcv .
before we discuss the calculation some general results can be presented for convenience on spacelike congruences that will be used in this work .
let @xmath53 where @xmath1 is a unit spacelike vector ( @xmath54 ) normal to the four velocity vector @xmath0 .
the screen projection operator @xmath55 projects normally to both @xmath0 and @xmath1 .
some properties of this tensor are @xmath56 the @xmath57 can be decomposed with respect to @xmath0 and @xmath1 as follows : @xmath58,\ ] ] where @xmath59 and @xmath60 .
we decompose @xmath61 into its irreducible parts @xmath62 where @xmath63 is the traceless part of @xmath61 , @xmath64 is the trace of @xmath61 , and @xmath65 is the rotation of @xmath61 .
we have the relations : @xmath66},\nonumber \\\label{eq26 } \stackrel{*}\theta & = & p^{ab}n_{a;b}.\end{aligned}\ ] ] it is easy to show that in equation ( [ eq23 ] ) the @xmath0 term in parenthesis can be written in a very useful form as follows : @xmath67 where the vector @xmath68 is the greenberg vector . using ( [ eq27 ] ) ,
equation ( [ eq23 ] ) becomes @xmath69 the vector @xmath70 is of fundamental importance in the theory of spacelike congruences .
geometrically the condition @xmath71 means that the congruences @xmath0 and @xmath1 are two surface forming .
kinematically it means that the field @xmath1 is `` frozen in '' along the observers @xmath0 .
having mentioned a few basic facts on the spacelike congruences we return to the computation of the lie derivative of the ricci tensor using the field equations . if einstein s field equation ( [ eq3 ] ) are satisfied , then string fluid with energy - momentum tensor ( [ eq5 ] ) admits an rcv , @xmath72 if and only if @xmath73=0,\label{eq31}\\ & & q\stackrel{*}\theta = 0,\label{eq32}\\ & & \left(\xi q n^{a}\right)_{;a}=0.\label{eq33}\end{aligned}\ ] ] * proof : * from the definition of the lie derivative it follows that @xmath74\ ] ] which , using einstein s field equation ( [ eq3 ] ) for string fluid , may be rewritten as @xmath75 + 2\rho_{s } n_{(a;b)}\bigg{\}}.\end{aligned}\ ] ] suppose first that @xmath76 is an rcv
. then equation ( [ eq1 ] ) is satisfied .
the right - hand side of equation ( [ eq35 ] ) is therefore zero and by contracting it in turn with @xmath77 , @xmath78 , @xmath79 , @xmath80 , @xmath81 , @xmath82 , and @xmath83 the following seven equations are derived : @xmath84=0 , \label{eq37}\\ \rho_{s}p^{b}_{a}\dot n_{b}-(\rho_{s}+q ) p^{b}_{a}\stackrel{*}u_{b}+q p^{b}_{a}n^{t}u_{t;b}=0 , \label{eq38}\\ \stackrel{*}q + 2q(\ln\xi)^{*}=0 , \label{eq39}\\ q p^{b}_{a}\left[\stackrel{*}n_{b}+(\ln\xi)_{,b}\right]=0 , \label{eq40}\\ \stackrel{*}\rho_{s}+\rho_{s}\stackrel{*}\theta = 0 , \label{eq41}\\ \rho_{s } s_{ab}=0.\label{eq42}\end{aligned}\ ] ] the energy momentum conservation equation will also be required . for string fluid ,
the momentum conservation equation , which follows from einstein s field equations , is @xmath85 ( i ) : : condition ( [ eq29 ] ) is derived from ( [ eq38 ] ) .
we have @xmath86}+\stackrel{*}u_{b}=-2\omega_{bt}n^t-(n_t \dot u^{t})u_{b}+\stackrel{*}u_{b},\ ] ] and by substituting from ( [ eq44 ] ) into ( [ eq38 ] ) , ( [ eq29 ] ) follows directly .
( ii ) : : condition ( [ eq30 ] ) is given by equation ( [ eq42 ] ) .
( iii ) : : to derive condition ( [ eq31 ] ) we first expand ( [ eq40 ] ) and use ( [ eq37 ] ) ; this gives @xmath87=0.\ ] ] if we subtract ( [ eq39 ] ) from ( [ eq36 ] ) , then we have @xmath88 if we substitute equation ( [ eq46 ] ) into equation ( [ eq45 ] ) , then we have condition ( [ eq31 ] ) . ( iv ) : : to derive condition ( [ eq32 ] ) , we substitute equation ( [ eq43 ] ) into ( [ eq41 ] ) , then we have condition ( [ eq32 ] ) .
( v ) : : consider the final condition ( [ eq33 ] ) . from ( [ eq26 ] )
, we have @xmath89 substitute ( [ eq47 ] ) into ( [ eq46 ] ) ; this gives @xmath90 if one of the terms @xmath91 into equation ( [ eq39 ] ) is replaced by ( [ eq48 ] ) and used condition ( [ eq32 ] ) , then ( [ eq39 ] ) may be written as @xmath92 from which ( [ eq33 ] ) follows directly .
hence , if @xmath72 is an rcv then conditions ( [ eq29])- ( [ eq33 ] ) are satisfied .
conversely , suppose that ( [ eq29])- ( [ eq33 ] ) are satisfied and einstein s field equations hold . using ( [ eq28 ] ) for @xmath93 , ( [ eq30 ] ) and ( [ eq31 ] ) for @xmath94 equation ( [ eq35 ] ) becomes @xmath95 - 2\rho_{s } n_{(a}u_{b)}\bigg{\}}.\end{aligned}\ ] ] further , by using ( [ eq44 ] ) for @xmath96 and ( [ eq29 ] ) for @xmath97 and by replacing @xmath64 by @xmath98 with the aid of ( [ eq26 ] ) , ( [ eq50 ] ) reduces to @xmath99 now , @xmath100 is given equation ( [ eq43 ] ) . to obtain @xmath101 in terms of @xmath98 we use the remaining condition ( [ eq33 ] ) , which may be expanded as @xmath102 but if ( [ eq31 ] ) is contracted with @xmath1 , equation ( [ eq46 ] ) is again derived .
therefore ( [ eq52 ] ) becomes @xmath103 it easily verified with the aid of ( [ eq43 ] ) , ( [ eq53 ] ) , and condition ( [ eq32 ] ) that the right - hand side of ( [ eq51 ] ) vanishes and therefore @xmath72 is an rcv .
in the case of timelike ricci collineation vectors parallel to @xmath0 , we have the following results : ( a ) : : in this case , it is easily verified that condition ( [ eq9 ] ) is the conservation law ( [ eq4 ] ) with @xmath104 .
( b ) : : condition ( [ eq8 ] ) may be rewritten alternatively either @xmath105 or @xmath106 .
if , @xmath105 the energy - momentum tensor reduces @xmath107 which is pure string . (
a ) : : in this case , it is easily verified that ( [ eq33 ] ) is the conservation law ( [ eq4 ] ) for the string fluid with @xmath108 .
( b ) : : from equation ( [ eq30 ] ) , we have @xmath109 ( c ) : : when @xmath110 , equation ( [ eq29 ] ) reduces to @xmath111 and hence either @xmath112 or @xmath113 . when @xmath113 , the integral curves @xmath1 are material curves and string fluid form two surface . when @xmath112 , strings disappear .
( d ) : : when @xmath114 , equation ( [ eq29 ] ) reduces to @xmath115 ( i ) : : if @xmath113 , then equation ( [ eq56 ] ) reduces to @xmath116 and hence if @xmath117 then @xmath118 and since @xmath119 we find by contracting ( [ eq57 ] ) with @xmath120 that @xmath121\omega^{a}.\ ] ] since both @xmath122 and @xmath123 it follows that @xmath124 .
( ii ) : : if the integral curves of @xmath1 are material curves in the fluid then @xmath125 .
hence , since @xmath117 , from ( [ eq46 ] ) @xmath118 and @xmath124 .
( iii ) : : if @xmath124 and @xmath117 then from equation ( [ eq56 ] ) , @xmath125 and the integral curves of @xmath1 are material curves .
|
we study the consequences of the existence of timelike and spacelike ricci collineation vectors ( rcvs ) for string fluid in the context of general relativity . necessary and sufficient
conditions are derived for a space - time with string fluid to admit a timelike rcv , parallel to @xmath0 , and a spacelike rcv , parallel to @xmath1 . in these cases ,
some results obtained are discussed .
|
weakly interacting massive particles , or wimps , have become paradigmatic in the construction of models for the particle nature of dark matter .
particles with mass in the hundreds of gev to few tev range , and interacting via standard model weak interactions , can naturally have a thermal relic density in a range that includes the observed cosmological density of the mysterious dark matter .
this derives from a thermal history where wimps were once in thermal equilibrium with the high - density and high - temperature primordial plasma
a condition dependent upon the pair - annihilation rate @xmath0 being much larger than the hubble expansion rate @xmath1 ; as the temperature dropped below a fraction of the mass of the wimp , the equilibrium number density decayed exponentially with temperature , as dictated by its maxwell - boltzmann equilibrium distribution .
shortly thereafter , the precipitous decline of @xmath2 brought it below @xmath1 , causing the freeze - out of the wimps .
the resulting number density today is then a function of a combination of effective couplings and masses such that , for wimps , one obtains a relic abundance parametrically close to the observed dark matter density .
what described in the previous paragraph pertains to the so - called _ chemical _ decoupling of wimps : the wimp number density @xmath3 ceases to follow the equilibrium distribution once the pair - annihilation and pair - creation rates go out of equilibrium ( i.e. they occur less frequently than once per hubble time ) .
after chemical decoupling , wimps do not entirely forget about the surrounding thermal environment : elastic scattering processes where a wimp scatters off of , for example , a light lepton @xmath4 ( @xmath5 ) keep wimps in _ kinetic _ equilibrium .
wimps continue to trace the thermal background kinetically , and structures can not start to form via gravitational collapse .
when the rate for elastic scattering processes also falls out of equilibrium , structures eventually start forming , and a small - scale cutoff is imprinted in the power - spectrum of density fluctuations in the universe .
this cutoff scale also defines the size of the smallest possible dark matter halos ( `` protohalos '' ) , some of which might survive and populate the late universe , with potentially important implications @xcite .
kinetic decoupling of wimps was first discussed in ref .
@xcite for heavy neutrinos as dark matter candidates , and for supersymmetric neutralinos , first in ref .
@xcite some time later .
it was subsequently argued in ref .
@xcite that the typical kinetic freeze - out temperature could be as low as a kev , a value that would yield a cutoff scale on the same order of the mass of dwarf galaxies the smallest observed dark matter halos .
this would have been a profound result , potentially impacting our understanding of the mismatch between the predicted and observed number of small - scale dark matter halos in cold dark matter cosmology @xcite .
unfortunately , ref .
@xcite pointed out important kinematic effects that were neglected in @xcite , leading the latter analysis to vastly overestimating the cross sections relevant for kinetic decoupling .
the kinetic decoupling temperature calculated in @xcite pointed , instead , to the mev to gev range , with a resulting cutoff scale significantly smaller than dwarf galaxies halos , and on the order of the sun s mass or small fractions of it .
a number of more recent studies addressed the question of calculating the kinetic decoupling temperature with increasingly finer detail , see e.g. ref .
@xcite , including wimp models beyond supersymmetric neutralinos @xcite as well as addressing the question of how to connect the kinetic decoupling temperature to the scale at which the matter power spectrum is effectively cut off @xcite .
these studies were paralleled by a series of n - body simulations that targeted the nature and fate of the smallest dark matter halos , starting with ref .
@xcite and continuing in @xcite .
further analyses studied the question of whether the smallest - scale halos would survive tidal stripping and stellar encounters , and whether they would then be potentially hovering around in today s galaxies @xcite , with potentially important implications for indirect @xcite as well as for direct @xcite dark matter detection .
other work also targeted the direct detection of these primordial halos ( alternately named protohalos , mini - halos or micro - halos : the latter two names allude to the size of the halos , which strongly depends on the particle physics model , see e.g. @xcite ) via gravitational lensing ( e.g. @xcite ) . in the present analysis we point out that there might be orthogonal handles to pinpoint the size of primordial dark matter halos , and thus of the effective cutoff scale of structure in the universe .
our main observation is that the same class of processes entering kinetic decoupling namely , elastic scattering off of light fermions also enters the cross section for direct dark matter detection , which is determined by elastic scattering off of quarks inside nucleons .
additionally , in a situation of capture - annihilation equilibrium , the rate of high - energy neutrinos expected from the capture and annihilation of wimps in celestial bodies such as the earth or the sun also depends on the same scattering cross section .
we therefore ask , in the present study , whether the mass of dark matter protohalos correlates with quantities that could be measured by experiments on earth , be it via direct detection or with neutrino telescopes . here
, we take a model - dependent view , and focus on two specific and well - defined wimp scenarios : the lightest neutralino of the minimal supersymmetric extension to the standard model ( mssm ) @xcite , and the lightest kaluza - klein ( kk ) excitation of universal extra dimensions ( see ref .
* for a review ) ( we will take a model - independent look at the same problem , based on an effective field - theoretic setup , in a forthcoming study @xcite ) . for these two paradigmatic wimp setups ,
we study correlations between the dark matter cutoff scale and rates for direct and indirect dark matter detection .
we find that strong correlations exist for some quantities , and not for others .
we discuss approximations and analytical formulae that help understanding our detailed numerical results , and we conclude that `` earthly probes '' of the size of the smallest dark matter halos are , in principle and with the model assumptions we detail here , possible . the ensuing study is articulated as follows : we review in sec .
[ sec : halosize ] the calculation of both the kinetic decoupling temperature and the cutoff scale as a function of this temperature , along with potential ways to directly measure the size of the smallest collapsed dark matter structures ; sec . [
sec : susy ] and [ sec : ued ] discuss in detail , respectively , the case of supersymmetric neutralinos and of kk dark matter ; finally , we discuss our results and draw our conclusions in sec .
[ sec : concl ] .
the most thorough and comprehensive method to calculate the temperature of kinetic decoupling @xmath6 is a numerical one described in ref .
this treatment begins with the boltzmann equation in a flat friedmann - robertson - walker spacetime : @xmath7.\ ] ] here @xmath8 is the wimp phase - space density , @xmath9 and @xmath10 are the wimp energy and comoving momenta respectively , and @xmath1 is the hubble parameter . @xmath11 $ ] is the collision term , and to find @xmath6 , the necessary @xmath11 $ ] is that of the scattering of a massive wimp off of a standard model ( sm ) particle that is in thermal equilibrium with the plasma in the early universe . in @xcite , to lowest order in @xmath12 and the sm particle momentum , this collision term is shown to be of the form @xmath13 = c(t ) m_\chi^2 \left[m_\chi t \nabla^2_{\bf p } + { \bf p } \cdot \nabla_{\bf p } + 3 \right ] f ( { \bf p}).\ ] ] @xmath14 is an expression which contains the scattering amplitude of the wimps off all possible sm scattering partners .
we can define a temperature parameter @xmath15 before kinetic decoupling , wimps are in thermal equilibrium with the heat bath , and therefore @xmath16 . after kinetic decoupling , the rate of wimp scattering off sm particles drops below the level which is needed to keep them in thermal equilibrium , and so the wimps cool down due to hubble expansion , with @xmath17 .
the transition between these two asymptotic behaviors is rapid and corresponds to the temperature of kinetic decoupling , @xmath6 . to find when this change occurs , eq .
[ eq : boltzmann ] is multiplied by @xmath18 and integrated over @xmath10 .
using integration by parts , this can be shown to give an equation describing the evolution of @xmath19 with the temperature of the universe : @xmath20 the author of @xcite has developed a routine which interfaces with the darksusy code @xcite and numerically solves this equation . by equating the limiting behavior of @xmath19 in the two regimes described above , when @xmath16 and when @xmath21 , @xmath6 is found .
it is important to note that often before kinetic decoupling the universe passes through the the qcd phase transition at @xmath22 , during which the number of relativistic degrees of freedom decreases substantially and scattering interactions with quarks are suppressed . to deal with this
, the code only considers scattering off quarks when @xmath23 , and after this point all scattering is assumed to be with leptons .
it has also been shown in ref .
@xcite that an analytic solution for @xmath6 can be found . for this to be done
, certain simplifying assumptions need to be made ; namely , sm scattering partners are relativistic , variations in the universe equation of state are ignored ( i.e. @xmath24 , the number of relativistic degrees of freedom , is assumed to be constant ) , and the scattering amplitude is of the form @xmath25 where @xmath26 is the energy of the sm scattering partner . with these assumptions , the solution to eq .
[ eq : tchi ] is @xmath27 \ \gamma [ -(n+2)^{-1},z ] \right\}_{z = ( a / n+2)(t / m_\chi)^{n+2}},\ ] ] where n is the power of the leading term in @xmath28 and a is a term that contains the leading coefficient of the scattering amplitude and @xmath24 . in the limit @xmath29
, this equation becomes @xmath30 \frac{t^2}{m_\chi}.\ ] ] @xmath6 occurs when the above limiting behavior matches the high temperature behavior @xmath16 .
therefore @xmath31 \right)^{-1}.\ ] ] in our work we will use both the numerical code and the analytic approximation of eq . [
eq : tkd_ana ] to find @xmath32 . in the period before kinetic decoupling , wimps behave as a fluid coupled to the cosmic heat bath via scattering interactions with standard model particles .
this coupling leads to bulk and shear viscosity in the wimp fluid which damps out the primordial structure in the fluid @xcite .
the amount of this damping has been shown to be given by a damping term of the form @xcite @xmath33.\ ] ] in the equation above , @xmath34 is the conformal time and @xmath35 a function which quantifies the amount of fluctuation in the wimp density over an isotropic state , so @xmath36 is the initial primordial value of the density perturbation function .
the characteristic damping scale @xmath37 is given by : @xmath38 after kinetic decoupling , there are no longer interactions between the wimps and the cosmic heat bath , so damping no longer occurs due to viscosity in the fluid .
however , in this epoch free streaming effects are found to significantly damp out density perturbations . in this process
, the dark matter particles propagate from regions of high density to low density , smoothing out inhomogeneities . by considering the collisionless boltzmann equation and using the results of the viscosity calculation as initial conditions ,
a characteristic scale for this damping process can be found , and this comoving scale approaches a constant value after matter - radiation equality @xcite : @xmath39 where @xmath40 is the scale factor at matter radiation equality .
the damping term for this free streaming @xmath41 is of a similar form as @xmath42 , and to find the total damping term , the two are multiplied together , i.e. @xmath43 : @xmath44 \exp \left[- \left ( \dfrac{k}{k_\mathrm{fs } } \right)^2 - \left ( \dfrac{k}{k_\mathrm{d } } \right)^2 \right].\ ] ] comparing @xmath45 and @xmath37 , one finds that @xmath46 : it is therefore @xmath45 that determines where the exponential cutoff in the mass spectrum is . therefore , to find the mass of the smallest protohalo allowed by these processes , one just calculates the mass of wimps contained in a sphere of radius @xmath47 , i.e. @xcite : @xmath48 in later papers @xcite , it was shown how an additional damping scale is set by acoustic oscillations in the cosmic heat bath itself .
the calculation of this damping scale is independent of the other two , @xmath37 and @xmath45 , that we previously presented in equations [ eq : d ] and [ eq : fs ] respectively .
these oscillations , which are remnants of the inflationary epoch , couple to modes of oscillation in the wimp fluid with @xmath49 values large enough that they enter the horizon before kinetic decoupling .
these modes in the wimp fluid then oscillate with the acoustic modes in the heat bath and are damped out , while modes with @xmath49 values that correspond to a distance larger than the horizon size at kinetic decoupling do not experience such a damping and grow logarithmically .
the damping scale for this process is just the size of the horizon at kinetic decoupling ( @xmath50 ) , and therefore the cutoff mass for this process is the mass of wimps enclosed by the horizon at the kinetic decoupling time @xcite : @xmath51 depending on the parameters of the wimp model , either @xmath52 or @xmath53 can be larger , so to find a cutoff mass both are calculated and the larger one is used , i.e. @xmath54 $ ] .
a comprehensive review on the detection of sub - solar - mass dark matter halos is given in ref .
some of these detection methods might provide a more or less direct way to infer the value @xmath55 , even if most of the studies mentioned here do not claim to probe cutoff masses as small as @xmath55 .
dense , nearby protohalos could host enough dark matter pair - annihilation to be visible as gamma - ray sources , as envisioned in a number of studies , e.g. @xcite .
small - scale subhalos could also contribute to the local cosmic - ray electron - positron population , potentially producing a sizable amount to be relevant for the reported anomalies in the abundances of these cosmic rays at 10 - 100 gev energies @xcite .
the possibility that gamma - ray data would be able to determine the proper motion of protohalos ( not necessarily only the smallest protohalos , however ) was first entertained in ref .
@xcite , but it was shown in @xcite that the diffuse gamma ray background makes this idea unfeasible in practice . rather than aiming to resolve individual substructures , ref .
@xcite considered the anisotropy in the diffuse gamma ray emission , arguing that it could be possible to use a statistical analysis to measure the substructure mass function . recently , direct observational constraints from the fermi lat collaboration were reported in ref .
@xcite in the form of a search , leading to a null result , for unassociated gamma - ray sources with spectra that could be conducive to particle dark matter annihilation .
if the earth were to pass through a dark matter clump , this would lead to an enhanced _ direct _ detection rate ( which scales directly with the local dark matter density ) , although in @xcite it was shown that the presence of substructure in the milky way halo is expected , on average , to _ reduce _ the direct detection rate relative to the rate with a smooth halo and no substructure . a long duration direct detection experiment might in principle detect variations in the rate due to intervening substrcture , as envisioned in @xcite .
it has been noted in @xcite that substructure should affect pulsar timing measurements , with @xmath55 having an effect on the amount of the frequency shift .
it has also been discussed in ref .
@xcite that when there is a time - variable compact source that that is multiply imaged by strong gravitational lensing , small perturbations in the gravitational potential , such as those caused by protohalos , can lead to variations in the images which could be used to make statements about the size of the protohalos .
nanolensing from sub - solar - size dark matter halos was discussed in ref .
@xcite , together with the possibility of detecting events with much shorter durations and smaller amplitudes than the microlensing events due to stars with future surveys .
note that none of these studies would directly provide a probe of the size of the small - scale cutoff in the matter power spectrum .
we first consider correlations between direct detection rates and protohalo size for mssm neutralino dark matter . to calculate the dark matter direct detection rates we use the routines in the numerical package darksusy ( see ref .
@xcite ; for an in - depth description of the direct detection calculation see also ref .
@xcite ) , while @xmath32 is calculated numerically as described in section [ sec : tkd ] .
we define our mssm models by 9 parameters given at the weak scale : @xmath56 , @xmath57 , @xmath58 , @xmath59 , @xmath60 , @xmath61 , @xmath62 , @xmath63 and @xmath64 .
this is the same parameterization as the `` mssm-7 '' described in @xcite ( to which we refer the reader for further details ) , with the change that we let @xmath57 , @xmath58 and @xmath59 vary freely , while in the mssm-7 the two parameters @xmath57 and @xmath59 are related to @xmath58 through gut - scale gaugino mass universality relations .
the parameters @xmath56 , @xmath57 , @xmath58 , @xmath59 , @xmath60 and @xmath62 are scanned over logarithmically in the range of 50 gev to 5 tev , with @xmath58 and @xmath56 allowed to take positive or negative values .
@xmath61 is scanned logarithmically over the range 2 to 50 , while @xmath63 and @xmath64 are scanned over linearly in the range of -5 to 5 .
all of the models we present in are checked against the accelerator and other particle physics constraints contained in the most recent version of darksusy , 5.0.5 .
they are also checked to see if they satisfy the @xmath65 bounds on the relic density from the most recent seven year release of wmap data , in which @xmath66 is constrained to the values @xmath67 @xcite . the relic density for each model
is calculated with coannihilations using the routines in darksusy @xcite .
current ( solid line ) and future ( dashed line ) sensitivities from direct detection experiments are also included on many of the the plots . for plots with spin independent scattering cross sections , we present the current sensitivity of the xenon100 experiment from @xcite and the projected sensitivity of the xenon1 t experiment found at @xcite . for spin dependent plots , we present the current sensitivity of the 4 kg coupp detector @xcite and the expected sensitivity of the future 60 kg coupp detector @xcite .
the elastic spin - independent neutralino - nucleon cross section depends , at the microscopic ( quark ) level , and at tree - level in perturbation theory , on two sets of diagrams : ( i ) higgs exchange ( including , in absence of cp violation , the two cp - even higgses ) and ( ii ) squark exchange @xcite .
elastic spin - dependent ( axial ) interactions are also mediated by squark exchange , as well as by @xmath68 exchange .
processes relevant to elastic scattering of neutralinos off of light leptons and quarks depend on all scattering processes .
since kinetic decoupling typically occurs at low temperatures , where heavier fermions no longer participate in the thermal bath , yukawa - suppressed higgs - exchange processes are generically subdominant with respect to @xmath68 and squark / slepton exchange .
this consideration leads us to anticipate that the correlation between kinetic decoupling temperature @xmath6 and the spin - independent elastic neutralino - proton cross section @xmath69 be _ weaker _ than the correlation with spin - dependent processes , @xmath70 .
the theoretical anticipation is in fact confirmed by the results of the extensive scan over mssm parameters we carried out , shown in fig .
[ fig : susy ] .
the upper panels correlate the scalar and axial cross sections with @xmath6 , while the lower panels with @xmath71 . as expected , although a general trend is present , we do not find a tight correlation for scalar interactions ( panels to the left ) , while a correlation is definitely present for axial interactions ( panels to the right , especially for large and potentially experimentally interesting values of the cross section ) .
the scatter in the correlation between @xmath6 and @xmath70 is within a factor 2 down to @xmath72 , and grows for smaller values of the cross section .
the correlation has a very small spread at values of the cross section currently probed by the most sensitive detectors ( for the most recent results from coupp , probing cross sections as small as few @xmath73 see @xcite ) .
we investigate and discuss the origin of this scatter in the following subsections .
note that @xmath72 ( for a 100 gev wimp ) corresponds approximately to the projected reach of a large ( 1 cubic meter ) dmtpc detector with 50 kev threshold operating for one year @xcite .
this corresponds to a jump of about three orders of magnitude over the current detector performance ( this does not include indirect limits from neutrino telescopes ) @xcite .
there are two main sources for the scatter in the correlation found between @xmath6 and @xmath70 : the neutralino mass and the squark mass scale .
we discuss these effects both analytically and numerically in this section . in the limit of heavy squark masses , kinetic decoupling and elastic neutralino - nucleon axial scattering
are only mediated by @xmath68-exchange , and should thus be tightly correlated .
however , at a fixed value of @xmath70 , corresponding to a fixed value of the neutralino-@xmath68 coupling , @xmath6 inherits a dependence on the neutralino mass scale beyond that produced by the dependence of @xmath70 on @xmath74 , resulting in a scatter in the values of @xmath6 for a given value of @xmath70 .
we illustrate this effect ( for both @xmath6 and @xmath71 ) in the left panels of fig .
[ fig : zmediator ] . for this scan
, we use the set of parameters as before , with the changes that the pseudoscalar higgs boson mass @xmath60 is set to 1000 gev , the trilinear couplings @xmath75 , and , most importantly , @xmath62 is set to the high value of 10 tev .
the color - coded dots show models with neutralino masses in the @xmath76 gev ( red ) , 100 gev @xmath77 500 gev ( orange ) and @xmath78 gev ( black ) .
the dependence of @xmath6 in the limit of heavy squarks can be understood analytically when we calculate both the neutralino proton scattering cross section and @xmath32 using just the z exchange tree level diagram , ignoring contributions from slepton and squark exchange diagrams .
the cross section of scattering a neutralino off a proton when z exchange is the only diagram is given by ( following , e.g. @xcite ) : @xmath79 to analytically approximate the kinetic decoupling temperature , we use the prescription described in section [ sec : tkd ] , which gives us @xmath80 where @xmath81 here @xmath8 is all possible sm scattering partners and @xmath82 is the number of degrees of freedom for the sm partners .
combining these expressions , we find @xmath83 the @xmath84 behavior can be seen in the left hand side plots in figure [ fig : zmediator ] .
to illustrate the validity of this approximation , we plot this expression as the black line in figure [ fig : zmediator ] , with the sum over fermions including all sm leptons except the quarks , and @xmath85 , a value which corresponds to @xmath86 .
@xmath74 is set to 100 gev , and from the upper left hand plot , we see that the analytic approximation follows the numerical result well for low @xmath32 , with the validity of the approximation becoming less valid at high @xmath32 because of that fact that before the qcd phase transition there is quark scattering which we do not consider and @xmath24 varies significantly from the value we chose . as squark and slepton masses
are lowered , the relative contribution from squark , but especially slepton exchange in the kinetic decoupling process increases . as a result , for sufficiently low squark / slepton masses ( which we assume to be at the same scale ) the correlation between @xmath6 and @xmath70 is lost .
namely , we expect @xmath6 to be driven by the sfermion mass , while @xmath70 is tuned by the @xmath68-neutralino coupling and is relatively insensitive to the squark mass scale .
we illustrate this numerically , and we assess the importance of this factor in a robust determination of @xmath6 from a measurement of @xmath70 , in the right panels of fig .
[ fig : zmediator ] . the color - coding shows models where we fix the sfermion mass scale to 10 tev ( red ) , 5 tev ( yellow ) , 1 tev ( blue ) and 500 gev ( green ) .
the figure shows that deviations from the expected correlation arise at @xmath87 for 5 tev sfermions , and at @xmath88 for 1 tev sfermions . for sub - tev sfermions
the correlation can become weaker , although given the negative results on squark and slepton searches from the lhc @xcite we expect little effect from light sfermion contributions if @xmath70 is close to the range that could be probed by next generation experiments , @xmath89 .
we also consider the situation in which there is no @xmath68-exchange in scattering processes , corresponding to the limit of a purely bino - like neutralino . to obtain models with such composition
, we use a modified version of the original parameter set where we enforce @xmath90 , @xmath91 , and @xmath75 .
this leaves us with the parameters @xmath57 , @xmath59 and @xmath92 , which are scanned logarithmically over the range 50 gev to 5 tev , and @xmath61 , which is scanned logarithmically over the range 2 to 50 . a plot of @xmath32 and @xmath55 versus @xmath93 for these models is shown in fig .
[ fig : bino ] .
as binolike neutralinos are very weakly interacting , only about 100 of the @xmath94 models in the plot do not produce a relic density greater than the wmap-7 5@xmath95 range .
for such a class of models , we see a correlation between @xmath32 , @xmath55 , and @xmath93 , with , just as before , an additional dependence on the neutralino mass .
unlike in the high @xmath92 case , there is a scatter of large neutralino mass models down into the smaller mass bands .
we have identified these models as having a small splitting between @xmath74 and @xmath92 . in @xcite , it was shown that for binolike neutralinos , @xmath32 is of the form : @xmath96 since in the mssm-7 parameterization the squark mass scale is the same as the slepton mass scale , when @xmath97 , @xmath32 is driven to a much smaller value than when the splitting is large , as can be appreciated from eq .
( [ eq : bino ] ) .
when @xmath68-exchange is the only process relevant for the interaction , the mass splitting does not enter into @xmath32 , and we see no similar effect . for these models ,
the highest possible spin dependent cross section is significantly smaller than that of the models from the full parameter space scan as well as that for models where sfermion exchange diagrams are suppressed . as such ,
these are not models that the current generation of direct detection experiments would explore : this additional source of scatter in the correlation between protohalo size and scattering cross section is thus not worrisome , at a practical level .
furthermore , for general sets of models where one has both @xmath68 and sfermion exchange scattering diagrams , these results show that when there is a relatively large scattering cross section , the sfermion exchange contribution to that cross section is subdominant to the contribution from @xmath68 exchange : for supersymmetric models that might be detectable with current or future generation detectors it is thus valid to approximate the scattering cross section as being due solely to @xmath68 exchange .
spin - dependent neutralino - nucleon interactions drive , quantitatively more than spin - independent interactions , the capture rate of neutralinos in the sun .
if the neutralino pair - annihilation rate is large enough so that capture and pair - annihilation in the sun are in equilibrium ( which is typically the case across the mssm parameter space @xcite ) , the actual rate of neutralino annihilation is governed by the capture rate .
we thus expect a correlation of the rate of high - energy neutrinos resulting from neutralino annihilation inside the sun and @xmath6
. note that effects such as the detector energy threshold are expected to impact the correlation and to potentially disrupt it .
we investigate in fig .
[ fig : nuflux ] the correlation between the flux of neutrinos from the sun integrated above two representative energy threshold , namely 10 gev ( upper panels ) and 100 gev ( lower panels ) .
models in the two panels on the right assume heavy squark masses , while those on the left scan over the general mssm parameter space as before . to calculate the neutrino flux ,
the routines from darksusy described in @xcite are used .
we note that a rather tight correlation exists between the neutrino flux from the sun , @xmath98 , integrated above 10 gev and @xmath71 , as long as @xmath99 the most recent icecube results looking for neutrinos from solar wimp annihilation through the @xmath100 channel claim a sensitivity to a total muon flux from this annihilation of about @xmath101 for a 1 tev wimp @xcite . using the routines from darksusy
, we find that this muon flux corresponds to a range of incoming total neutrino fluxes from about @xmath102 for mssm models when the neutrino threshold energy is 10 gev .
larger energy thresholds tend to loose the desired correlation : a very significant dependence on the neutralino mass is present in the flux above 100 gev , as , for example , almost no neutrinos with those energies are produced for neutralinos with masses below 200 gev or so .
therefore , a model with a given @xmath71 can well have a vanishing neutrino flux if the neutralino mass is small enough !
interestingly , with the deployment of the deepcore detector @xcite the effective energy threshold for neutrino detection of the icecube system has been significantly lowered .
again , for fluxes large enough to be above potentially detectable levels , we find that a tight correlation with @xmath71 is present .
the correlation we find is expected to further improve should plans to deploy an additional , even more thickly instrumented section of the detector , pingu , come to fruition @xcite .
the universal extra dimensions ( ued ) framework offers an interesting setup for a wimp model alternative to supersymmetry @xcite ( see also ref . @xcite for a review ) . the lightest kaluza - klein ( kk ) @xmath103 excitation , usually the first kk mode of the hypercharge gauge boson @xmath104 , is stable by virtue of the so - called kk parity @xcite . the lightest kk particle , or lkp , makes for a phenomenologically viable wimp dark matter candidate . the particle properties of the lkp interchangeably in what follows . ] depend , for the minimal version of the ued scenario that we will consider here @xcite , upon three parameters : the effective cut - off scale @xmath105 , the inverse compactification radius @xmath106 , and the value of the standard model higgs mass .
the latter quantity is especially crucial for the calculation of the spin - independent lkp - nucleon scattering cross section .
@xmath106 , instead , sets the mass scale of the kk levels , including the mass of the lkp , while @xmath105 feeds in the details of the particle spectrum . here
, we consider the range @xmath107 , @xmath108 for our scans .
we also scan over the values of the higgs mass allowed by current collider constraints @xcite , with a maximum higgs mass of 600 gev .
we then find the relic density for all models using the results of @xcite and check these against wmap relic density constraints .
as for neutralinos , the same diagrams contributing to elastic @xmath104-nucleon scattering contribute to the process of kinetic decoupling .
in particular , spin - dependent scattering depends upon kk - quark exchange , while spin - independent scattering is primarily driven by processes mediated by higgs exchange . as a result ,
the general expectation for ued is not dissimilar to what we awaited in the context of supersymmetry : a tight correlation between spin - dependent elastic processes and kinetic decoupling , and a looser correlation for the scalar cross section .
figures [ fig : ued ] and [ fig : uedsi ] accurately confirm these expectation . in fig .
[ fig : ued ] we show , for 5000 minimal ued models , the correlation between the elastic spin - dependent cross section and the kinetic decoupling temperature ( top panel ) and cut - off mass scale ( lower panel ) .
all of these models have the same higgs boson mass , 125 gev , which does not effect the calculation of @xmath32 or @xmath55 , but does have a large effect on the relic density . here
@xmath32 is once again calculated using the numerical method described in section [ sec : tkd ] , while the spin dependent cross section is calculated using the approximate formula from ref .
@xcite : @xmath109 in the formula above , @xmath110 is the mass splitting between the right handed quarks and the lkp , @xmath111 , with all of the kk quarks taken to have the same mass for simplicity . an analytic approximation for @xmath32
was already found in @xcite , which is @xmath112 with @xmath113 .
putting together equations ( [ eq : uedsdcs ] ) and ( [ eq : tkdued ] ) , we get an analytic approximation relating @xmath32 and @xmath114 : @xmath115 there is no dependence on dark matter mass as there was in the susy case , rather the kinetic decoupling temperature just goes like @xmath116 , leading to the strong correlation displayed in figure [ fig : ued ] .
the validity of this approximation is shown by plotting it as the black line in figure [ fig : ued ] , with @xmath117 and @xmath118 .
for the relatively low values of @xmath32 we find for ued models , @xmath55 is always set by the acoustic oscillation cutoff of equation ( [ eq : mao ] ) . in plotting the analytic approximation for @xmath55
, we use @xmath119 , which corresponds to @xmath120 . in the spin - independent case , to find the cross section we use the approximation from @xcite : @xmath121 ^ 2.\ ] ] the results of using this formula are shown in fig .
[ fig : uedsi ] , where were are now scanning over a full range of higgs masses . here
the scattering cross section at a given value of @xmath106 depends sensitively on the value of the standard model higgs mass @xmath122 , as shown in the top panel to the left . in the figure
, the green points correspond to values allowed before the most recent lhc results on searches for the standard model higgs @xcite , which are now ruled out ( the region without points corresponds to values of @xmath122 already ruled out by searches with the tevatron and lep @xcite ) .
lhc results have therefore severely constrained the prediction for the scalar @xmath104-nucleon scattering cross section to a range of about an order of magnitude around @xmath123 , along with significantly tightening the correlation between @xmath106 and this cross section .
this , in turn , implies a correlation of the latter cross section with the cutoff scale , illustrated in the lower - left panel . in the future , more accurate measurements of @xmath122 will yield an increasingly tighter correlation between the spin - independent cross section and the kinetic decoupling temperature and cutoff scales , as we show in the right panels
there , we show with different colors models corresponding to the ranges @xmath124 ( cyan ) , @xmath125 ( brown ) and @xmath126 ( red ) .
we also calculate the expected analytic form for the correlation between the quantities shown in the right panels . as higgs exchange dominates over the kk quark exchange processes in the calculation of the spin independent scattering cross section , we approximate eq .
( [ eq : uedsi ] ) as @xmath127 combining equations ( [ eq : tkdued ] ) and ( [ eq : uedapprox ] ) , we find @xmath128 this approximation for @xmath32 and the corresponding result for @xmath55 are plotted in the right hand side of fig .
[ fig : uedsi ] , for @xmath129 and @xmath130 .
the analytic approximation underestimates @xmath32 slightly , which is due to ignoring the kk quark exchange processes in the expression we use for @xmath131 .
however , the behavior of the numerical and analytic results is the same , with @xmath132 and with a scatter occurring due to the varying mass splittings .
finally , we note that we estimated the neutrino flux from the sun in the case of lkp dark matter , with results that mirror the same behavior and correlation as we found in the case of supersymmetric models , illustrated in fig .
[ fig : nuflux ] .
it is shown in ref .
@xcite that the event rate in neutrino telescopes correlates strongly with @xmath133 . we have shown that for both the spin dependent and independent cases in ued there is also a correlation between @xmath133 and the scattering cross section .
this is shown analytically in eq .
( [ eq : uedsdcs ] ) for the spin dependent case and numerically in the upper left hand panel of fig .
[ fig : uedsi ] for the spin independent case .
therefore , there is a resulting correlation between the neutrino flux and @xmath93 or @xmath131 .
the larger mass of lkp with respect to the possibly light neutralino case produces a slightly smaller spread as the one shown in fig .
[ fig : nuflux ] , especially with a larger neutrino energy threshold .
we addressed the possibility of establishing the small - scale cutoff of the cosmological matter power spectrum in a variety of particle dark matter models via dark matter direct and indirect detection experiments .
we argued , and showed with analytical calculations and numerical results , that the kinetic decoupling temperature , which sets the cutoff scale , correlates tightly with the spin - dependent elastic scattering cross section of wimps off of nucleons .
there also is a generically tight correlation with the flux of high - energy neutrinos from the sun - whose intensity depends on the capture rate of wimps in the sun , in turn set by the same axial scattering cross section .
a weaker correlation is found in the case of scalar wimp - nucleon interaction .
control over the spectrum and properties of the higgs sector will dramatically improve this latter correlation , especially in the case of minimal universal extra dimensions .
all correlations we found are tightest when the detection rates are largest . in summary , we find that `` earthly '' probes of the small - scale cutoff to the cold dark matter power spectrum are possible in the foreseeable future .
multiple measurements , for example of both scalar and axial scattering cross sections and of a flux of neutrinos from the sun , would help pinpoint a concordance dark matter model and , in view of the results reported here , its cosmological bearings on structure formation .
we gratefully acknowledge the assistance of torsten bringmann , who provided the code for numerical calculation of the decoupling temperature .
jmc is supported by the nsf graduate research fellowship under grant no .
( dge-0809125 ) .
sp is partly supported by an outstanding junior investigator award from the us department of energy and by contract de - fg02 - 04er41268 , and by nsf grant phy-0757911 .
j. e. gunn , b. w. lee , i. lerche , d. n. schramm and g. steigman , astrophys .
j. * 223 * , 1015 ( 1978 ) . c. schmid , d. j. schwarz and p. widerin , phys .
d * 59 * , 043517 ( 1999 ) [ astro - ph/9807257 ] . c. boehm , p. fayet and r. schaeffer , phys .
b * 518 * , 8 ( 2001 ) [ astro - ph/0012504 ] .
l. e. strigari , j. s. bullock , m. kaplinghat , j. diemand , m. kuhlen and p. madau , astrophys .
j. * 669 * , 676 ( 2007 ) [ arxiv:0704.1817 [ astro - ph ] ] .
x. -l .
chen , m. kamionkowski and x. -m .
zhang , phys .
d * 64 * , 021302 ( 2001 ) [ astro - ph/0103452 ] .
a. m. green , s. hofmann , d. j. schwarz , mon . not .
soc . * 353 * , l23 ( 2004 ) .
[ astro - ph/0309621 ] .
t. bringmann and s. hofmann , jcap * 0407 * , 016 ( 2007 ) [ hep - ph/0612238 ] . j. diemand , m. kuhlen and p. madau , astrophys .
j. * 649 * , 1 ( 2006 ) [ astro - ph/0603250 ] . h. -s .
zhao , j. taylor , j. silk and d. hooper , astro - ph/0502049 .
b. moore , j. diemand , j. stadel and t. r. quinn , astro - ph/0502213 .
t. goerdt , o. y. gnedin , b. moore , j. diemand and j. stadel , mon . not .
soc . * 375 * , 191 ( 2007 ) [ astro - ph/0608495 ] .
j. diemand , m. kuhlen and p. madau , astrophys . j. * 657 * , 262 ( 2007 ) [ astro - ph/0611370 ] .
j. diemand , m. kuhlen , p. madau , m. zemp , b. moore , d. potter and j. stadel , nature * 454 * , 735 ( 2008 ) [ arxiv:0805.1244 [ astro - ph ] ] ; m. kuhlen , n. weiner , j. diemand , p. madau , b. moore , d. potter , j. stadel and m. zemp , jcap * 1002 * , 030 ( 2010 ) [ arxiv:0912.2358 [ astro-ph.ga ] ] .
l. a. moustakas , k. abazajian , a. benson , a. s. bolton , j. s. bullock , j. chen , e. cheng and d. coe _ et al .
_ , arxiv:0902.3219 [ astro-ph.co ] .
s. baghram , n. afshordi and k. m. zurek , phys .
d * 84 * , 043511 ( 2011 ) [ arxiv:1101.5487 [ astro-ph.co ] ] .
g. jungman , m. kamionkowski and k. griest , phys .
rept . * 267 * , 195 ( 1996 ) [ hep - ph/9506380 ] .
j. m. cornell and s. profumo , in preparation .
p. gondolo , j. edsjo , p. ullio , l. bergstrom , m. schelke and e. a. baltz , jcap * 0407 * , 008 ( 2004 ) [ astro - ph/0406204 ] . s. hofmann , d. j. schwarz , h. stoecker , phys . rev .
* d64 * , 083507 ( 2001 ) .
[ astro - ph/0104173 ] .
s. m. koushiappas , new j. phys . * 11 * , 105012 ( 2009 ) [ arxiv:0905.1998 [ astro-ph.co ] ] .
a. tasitsiomi and a. v. olinto , phys
. rev .
d * 66 * , 083006 ( 2002 ) [ astro - ph/0206040 ] .
s. m. koushiappas , a. r. zentner and t. p. walker , phys .
d * 69 * , 043501 ( 2004 ) [ astro - ph/0309464 ] .
e. a. baltz , j. e. taylor and l. l. wai , astro - ph/0610731 .
l. pieri , g. bertone and e. branchini , mon . not .
* 384 * , 1627 ( 2008 ) [ arxiv:0706.2101 [ astro - ph ] ] .
m. kuhlen , j. diemand and p. madau , arxiv:0805.4416 [ astro - ph ] .
t. ishiyama , j. makino and t. ebisuzaki , arxiv:1006.3392 [ astro-ph.co ] . b. anderson , m. kuhlen , j. diemand , r. p. johnson and p. madau , astrophys .
j. * 718 * , 899 ( 2010 ) [ arxiv:1006.1628 [ astro-ph.he ] ] .
p. brun , t. delahaye , j. diemand , s. profumo and p. salati , phys .
d * 80 * , 035023 ( 2009 ) [ arxiv:0904.0812 [ astro-ph.he ] ] .
t. delahaye , p. brun , j. diemand , s. profumo and p. salati , j. phys .
ser . * 203 * , 012050 ( 2010 ) .
s. m. koushiappas , phys .
lett . * 97 * , 191301 ( 2006 ) [ astro - ph/0606208 ] .
s. i .
ando , m. kamionkowski , s. k. lee and s. m. koushiappas , phys .
d * 78 * , 101301 ( 2008 ) [ arxiv:0809.0886 [ astro - ph ] ] .
s. k. lee , s. i .
ando and m. kamionkowski , jcap * 0907 * , 007 ( 2009 ) [ arxiv:0810.1284 [ astro - ph ] ] .
m. ackermann _ et al .
_ [ the fermi lat collaboration ] , astrophys .
j. * 747 * , 121 ( 2012 ) [ arxiv:1201.2691 [ astro-ph.he ] ] . m. kamionkowski and s. m. koushiappas , phys .
d * 77 * , 103509 ( 2008 ) [ arxiv:0801.3269 [ astro - ph ] ] . j. chen and s. m. koushiappas , astrophys .
j. * 724 * , 400 ( 2010 ) [ arxiv:1008.2385 [ astro-ph.co ] ] .
l. bergstrom and p. gondolo , astropart .
* 5 * , 263 ( 1996 ) [ hep - ph/9510252 ] .
e. komatsu _ et al .
_ [ wmap collaboration ] , astrophys . j. suppl . *
192 * , 18 ( 2011 ) arxiv:1001.4538 [ astro-ph.co ] ] e. aprile _ et al . _
[ xenon100 collaboration ] , phys .
lett . * 107 * , 131302 ( 2011 ) [ arxiv:1104.2549 [ astro-ph.co ] ] .
see e.g. v. khachatryan _ et al .
_ [ cms collaboration ] , phys .
b * 698 * , 196 ( 2011 ) [ arxiv:1101.1628 [ hep - ex ] ] ; g. aad _ et al .
_ [ atlas collaboration ] , phys .
* 106 * , 131802 ( 2011 ) [ arxiv:1102.2357 [ hep - ex ] ] .
f. halzen and d. hooper , new j. phys .
* 11 * , 105019 ( 2009 ) [ arxiv:0910.4513 [ astro-ph.he ] ] .
l. bergstrom , j. edsjo and p. gondolo , phys .
d * 58 * , 103519 ( 1998 ) [ hep - ph/9806293 ] . c. d. l. heros , pos idm * 2010 * , 064 ( 2011 ) [ arxiv:1012.0184 [ astro-ph.he ] ] . c. wiebusch and f. t. i. collaboration , arxiv:0907.2263 [ astro-ph.im ] .
d. j. koskinen , mod .
a * 26 * , 2899 ( 2011 ) .
g. servant and t. m. p. tait , nucl .
b * 650 * , 391 ( 2003 ) [ hep - ph/0206071 ] ; g. servant and t. m. p. tait , new j. phys .
* 4 * , 99 ( 2002 ) [ hep - ph/0209262 ] .
d. hooper and s. profumo , phys .
* 453 * , 29 ( 2007 ) [ hep - ph/0701197 ] .
t. appelquist , h. -c .
cheng and b. a. dobrescu , phys .
d * 64 * , 035002 ( 2001 ) [ hep - ph/0012100 ] .
f. gianotti ( atlas collaboration ) , talk at cern public seminar , dec . 13 , 2011 ; atlas collaboration , atlas - conf-2011 - 163 ( 2011 ) ; g. tonelli ( cms collaboration ) ,
talk at cern public seminar , dec .
13 , 2011 ; atlas and cms collaborations , atlas - conf-2011 - 157 , cms pas hig-11 - 023 ( 2011 ) .
[ tevnph ( tevatron new phenomina and higgs working group ) and cdf and d0 collaborations ] , arxiv:1107.5518 [ hep - ex ] .
r. barate _
[ lep working group for higgs boson searches and aleph and delphi and l3 and opal collaborations ] , phys .
b * 565 * , 61 ( 2003 ) [ hep - ex/0306033 ] .
m. kakizaki , s. matsumoto and m. senami , phys . rev .
d * 74 * , 023504 ( 2006 ) [ hep - ph/0605280 ] .
|
dark matter kinetic decoupling involves elastic scattering of dark matter off of leptons and quarks in the early universe , the same process relevant for direct detection and for the capture rate of dark matter in celestial bodies ; the resulting size of the smallest dark matter collapsed structures should thus correlate with quantities connected with direct detection rates and with the flux of high - energy neutrinos from dark matter annihilation in the sun or in the earth . in this paper
we address this general question in the context of two widely studied and paradigmatic weakly - interacting particle dark matter models : the lightest neutralino of the minimal supersymmetric extension of the standard model , and the lightest kaluza - klein particle of universal extra dimensions ( ued ) .
we argue and show that while the scalar neutralino - nucleon cross section correlates poorly with the kinetic decoupling temperature , the spin - dependent cross section exhibits a strong correlation in a wide range of models . in ued models
the correlation is present for both cross sections , and is extraordinarily tight for the spin - dependent case .
a strong correlation is also found , for both models , for the flux of neutrinos from the sun , especially for fluxes large enough to be at potentially detectable levels .
we provide analytic guidance and formulae that illustrate our findings .
|
in astronomical survey projects , it is common practice to produce a _ catalog _ of detected and measured sources , using only data from the survey itself .
this approach has the benefit that the survey can be thought of as an independent experiment , but the disadvantage that it ignores the huge amount of information we already have about astronomical sources measured in previous surveys covering the same part of the sky .
while new surveys typically bring some new capability that previous surveys lacked ( eg , depth , resolution , or wavelength coverage ) , it is seldom the case that a new survey surpasses all previous data in all regards ; there is usually some complementary information that could be of value .
this approach of compiling `` independent '' catalogs has two shortcomings , in particular when it comes to comparing the new survey with existing surveys .
first , when the new survey has lower resolution , there will be some nearby sets of sources that are blended together ( detected and measured as a single source ) in the new catalog , but resolved in existing data .
second , when the new survey has lower sensitivity ( at least to some types of sources ) , sources known from previous surveys will not be detected in the new survey and will not appear in its catalog . when investigators attempt to cross - match the new catalog with existing catalogs ( usually via astrometric cross - matching ) , the first problem ( blended sources ) typically results in either failed matches ( because the blended source has a different centroid ) , or very strange inferred properties ( for example , bizarre colors because the new survey matches the sum of a set of sources to a single source in the existing survey ; or unexpected non - matching sources ) .
the second problem ( non - detections ) means that fewer sources are available to cross - match ; a catalog cross - match is limited by the weaknesses of both catalogs .
in contrast , in this paper we perform `` forced photometry '' of a new survey ( wise ) given a great deal of knowledge from an existing survey ( sdss ) . while wise has comparable _ depth _ to sdss for many sources , its resolution is significantly lower .
we therefore get significant benefit from using sdss detections to decide _ where to look _ in the wise data .
the wide - field infrared survey explorer ( wise ; @xcite ) measured the full sky in four mid - infrared bands centered on 3.4 @xmath0 m , 4.6 @xmath0 m , 12 @xmath0 m , and 22 @xmath0 m , known as w1 through w4 . during its primary mission , it scanned the full sky in all four bands .
after its solid hydrogen cryogen ran out and w4 became unusable , it continued another half - sky scan in w1 , w2 , and w3 . during the `` neowise post - cryo '' continuation ( @xcite ) , it continued to scan another half - sky in w1 and w2 . over 99% of the sky
has 11 or more exposures in w3 and w4 , and 23 or more exposures in w1 and w2 .
median coverage is 33 exposures in w1 and w2 , 24 in w3 , and 16 in w4 . in december 2013 , wise was reactivated and is expected to complete several more full scans of the sky in w1 and w2 ( @xcite ) .
the wise team have made a series of high - quality data releases , the most recent of which is the allwise data release . ] the allwise release includes a source catalog of nearly 750 million sources , a database of photometry in the individual frames at each source position , and `` atlas images '' : coadded matched - filtered images .
the allwise atlas images were intentionally convolved by the point - spread function ( psf ) , making it challenging to use them for forced photometry .
instead , we use the `` unwise '' coadds from @xcite , which preserve the resolution of the original wise images . the sloan digital sky survey ( sdss ; @xcite ) imaged over @xmath1 square degrees of sky in five bands ( @xmath2 , @xmath3 , @xmath4 , @xmath5 , @xmath6 ) , detecting and measuring over @xmath7 million sources .
we use the imaging catalogs from sdss - iii data release 10 ( @xcite ) .
these catalogs contain the outputs of the _ photo _ pipeline ( @xcite ) , and include star / galaxy separation and galaxy shape measurements using either exponential , de vaucouleurs , or composite ( sum of exponential and de vaucouleurs ) profiles .
the combination of data from sdss and wise has proven to be very powerful for a variety of studies .
@xcite give a survey of the properties of extragalactic sources , showing that sdss
wise colors and morphology can be used to select type-2 dust - obscured quasars and ultra - luminous infrared galaxies at redshift @xmath8 .
the sdss - iii boss survey ( @xcite ) includes quasars targeted using sdss color cuts and wise detection in the w1 , w2 , and w3 bands , to select @xmath9 quasars .
the work described here was motivated by the need to select targets for the sdss - iii sequels and sdss - iv eboss programs .
myers _ etal . _ ( in prep . )
describe the use of our results to select quasars , while prakash _ etal .
_ ( in prep . )
describe the selection of luminous red galaxy ( lrg ) targets .
the lrg targets are fairly bright in wise , so a catalog match produces satisfactory results .
however , due to the lower resolution of the wise images , nearby sources that are resolved in sdss may be blended in wise , resulting either in missed astrometric matches ( because the wise centroid is shifted ) , or incorrect colors ( because the wise catalog source includes flux from multiple sdss sources ) .
using our results improves this situation , since we photometer a consistent set of sources . for the quasar targets , the often few - sigma flux measurements we make are of considerable utility . in the redshift range of interest , the quasar and stellar loci
are significantly separated in sdss
wise colors , so even a noisy measurement of the wise flux can effectively eliminate stellar contamination . in similar work , the `` extreme deconvolution '' quasar target selection and redshift - estimation method ( xdqsoz ; @xcite ) makes effective use of forced photometry of galex uv ( @xcite ) and ukidss near - ir ( @xcite ) images , based on sdss source positions .
while often low - signal - to - noise , these measurements nevertheless can be very effective in eliminating degeneracies in quasar classification and redshift determination .
indeed , the xdqsoz method has been extended to incorporate the measurements we present here by dipompeo _ etal . _ ( in prep ) .
we use _ the tractor _ code ( lang _ etal .
_ , in prep . ) in `` forced photometry '' mode . in general ,
_ the tractor _ optimizes or samples from a full generative model that includes parameters of the image calibration and all the parameters of the sources in the images ( positions , shapes , and fluxes ) . in forced photometry mode , the image calibration parameters are frozen ( held fixed ) , as are all properties of the sources except for their fluxes in the bands of interest . in this case , the photometry task becomes linear : we know what each source should look like in the wise images , and we must compute the linear sum of the sources that best matches the observed image .
the image calibration parameters include the astrometric calibration , described by a world coordinate system ( wcs ) ; the photometric calibration , described by a zeropoint ; a point - spread function model ; a noise model ( per - pixel error estimates ) ; and a `` sky '' or background model .
we are photometering the `` unwise '' coadds , which are tiles of roughly @xmath10 in extent .
the tiles use a gnomonic projection ( tangent plane ; wcs code `` tan '' ) , are sky - subtracted , and have a photometric zeropoint of @xmath11 in the vega system . in turn , these coadds use the `` level 1b '' calibrated individual exposures from the wise all - sky data release .
we use the wise psf models from the wise all - sky release , ] averaged over the focal plane and approximated by a mixture of three isotropic concentric gaussian components .
we have been impressed by the quality of the wise psf models from @xcite , ] but have opted to use the wise team s models here for consistency .
[ fig : psf ] shows the allwise psf models and our gaussian approximations .
[ tab : psf ] lists the parameters of our psf models . ,
scaledwidth=80.0% ] c@[email protected]@.l@[email protected]@.l@[email protected]@.l@[email protected]@.l 1 & @xmath12 & @xmath13 & @xmath12 & @xmath14 & @xmath12 & @xmath15 & @xmath12 & @xmath16 & @xmath12 & @xmath17 & @xmath12 & @xmath18 & @xmath19 & @xmath20 & @xmath21 & @xmath22 + 2 & @xmath12 & @xmath23 & @xmath24 & @xmath25 & @xmath12 & @xmath26 & @xmath21 & @xmath27 & @xmath12 & @xmath28 & @xmath24 & @xmath29 & @xmath12 & @xmath30 & @xmath21 & @xmath31 + 3 & @xmath12 & @xmath32 & @xmath33 & @xmath34 & @xmath12 & @xmath35 & @xmath36 & @xmath37 & @xmath12 & @xmath38 & @xmath39 & @xmath40 & @xmath12 & @xmath41 & @xmath33 & @xmath42 [ cols="^,^,^ " , ]
forced photometry is one approach for applying information learned in one survey to data gathered in a second survey .
it is rigid , in the sense that we _ only _ photometer the images at locations containing a source in the input catalog . as such , forced photometry is most useful when the survey providing the catalog of sources to photometer has at least the depth and resolution of the images being photometered .
in addition , forced photometry demands that the images being photometered are well calibrated . while it is possible to fit for the psf model , astrometric solution and sky level using _ the tractor _ , this incurs additional computational cost and is not necessarily the most efficient approach for recalibrating images .
we are fortunate that the wise team have produced superbly calibrated images so that image recalibration has been unnecessary
. a more holistic approach than forced photometry would be to do _
simultaneous fitting_. we could , for instance , fit all parameters of the sources ( galaxy shapes and positions as well as fluxes ) , and include both the sdss and wise images in the fitting .
this would extract additional information from both surveys , yielding stronger constraints on the source properties .
it would also allow fitting for proper motions and parallaxes of nearby stars .
in addition , we could detect sources that are below the individual survey detection thresholds but are significant when the surveys are combined .
this approach would , however , be significantly more computationally expensive : to start , sdss images have roughly 50 times more pixels per area than wise .
further , this would require non - linear optimization , in contrast to the much cheaper linear optimization required by forced photometry . since we do not expect the wise images ( with their lower resolution ) to have much constraining power on the positions or shapes of galaxies
, an alternative approach would be first to re - fit the sdss catalog to the sdss images using _ the tractor _ , and then repeat our forced photometry with that improved catalog . for moving sources , we could search for regions of the wise images that are poorly fit by the sdss catalog and allow the sources in these regions to shift their positions slightly . this post - processing approach would allow us to improve upon the forced photometry results without increasing the computational cost excessively . in this work ,
we have used coadds of the wise imaging , rather than the individual frames . while in general it would be preferable to photometer the individual frames ( at least in principle ) , the wise images have a stable and approximately isotropic psf with little variation over the focal plane , so little information is lost in the coadding process . the computational time and
memory requirements scale roughly with the number of pixels being fit , so photometering the individual frames would have cost roughly 30 times more
. forced photometry assumes that the profile of a galaxy is the same between bands .
this does not describe the ( typically small ) color gradients across galaxies , although the wise images lack the resolution to inform any such gradients in the infrared colors .
our forced photometry of the wise fluxes is equivalent to a weighted - aperture flux that has the property of being well - defined and consistently applied to all objects .
therefore , any biases in the inferred infrared fluxes would be consistent between galaxies that have the same intrinsic properties . in this paper
, we used the sdss @xmath4-band galaxy shape measurements as the galaxy profiles for forced photometry .
one might expect the @xmath6-band shapes to be closer to the wise shapes , but the sdss @xmath6-band images are generally of significantly lower signal - to - noise . since we use the same galaxy profiles as used in the @xmath4-band `` cmodelmag '' measurements , our measurements can be used consistently with those mags .
when the `` fracdev '' devaucoulers - to - total fraction is zero or one , our measuments are also consistent with the sdss `` modelmag '' measurements for all bands .
it is a pleasure to thank adam myers , john moustakas and abhishek prakash for early testing and feedback .
dwh was partially supported by the nsf ( grant iis-1124794 ) , nasa ( grant nnx12ai50 g ) and the moore
sloan data science environment at nyu .
this publication makes use of data products from the wide - field infrared survey explorer , which is a joint project of the university of california , los angeles , and the jet propulsion laboratory / california institute of technology , and neowise , which is a project of the jet propulsion laboratory / california institute of technology .
wise and neowise are funded by the national aeronautics and space administration .
this publication makes use of data from the sloan digital sky survey iii .
funding for sdss - iii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , and the u.s .
department of energy office of science .
the sdss - iii web site is http://www.sdss3.org/[_http://www.sdss3.org/_ ] .
sdss - iii is managed by the astrophysical research consortium for the participating institutions of the sdss - iii collaboration including the university of arizona , the brazilian participation group , brookhaven national laboratory , carnegie mellon university , university of florida , the french participation group , the german participation group , harvard university , the instituto de astrofisica de canarias , the michigan state / notre dame / jina participation group , johns hopkins university , lawrence berkeley national laboratory , max planck institute for astrophysics , max planck institute for extraterrestrial physics , new mexico state university , new york university , ohio state university , pennsylvania state university , university of portsmouth , princeton university , the spanish participation group , university of tokyo , university of utah , vanderbilt university , university of virginia , university of washington , and yale university .
this research used resources of the national energy research scientific computing center ( nersc ) , which is supported by the office of science of the u.s .
department of energy under contract no .
de - ac02 - 05ch11231 .
this research has made use of nasa s astrophysics data system .
our output files are row - by - row parallel to the sdss _ photoobj _ files , and are named _
photowiseforced_. for example , the sdss _ photoobj _ file containing objects observed in run 1000 , camera column 1 , field 100 , in data reduction version 301 , is found in the file ] ` photoobj/301/1000/1/photoobj-001000 - 1 - 0100.fits ` and our results are found in the file ` 301/1000/1/photowiseforced-001000 - 1 - 0100.fits ` where both of these files contains 368 rows , describing row - by - row the same objects . `
has_wise_phot ` : : ( boolean ) true for sdss sources that were photometered in wise .
the object must be primary in sdss for this to be set .
when this column is false , all other columns have value zero . ` ra ` , ` dec ` : : ( floats ) j2000.0 coordinates from sdss . ` objid ` : : ( string ) object identifier from sdss . ` treated_as_pointsource ` : : ( boolean ) the sdss source is a galaxy ( ` objc_type ` = = 3 ) but was treated as a point source for the purposes of forced photometry .
if you want an optical / wise color , it would be best to use the sdss psf mags , not the model mags , for these objects . `
pointsource ` : : ( boolean ) the sdss source is a point source ( ` objc_type = = 6 ` ) . ` coadd_id ` : : ( string ) the unwise coadd tile name , for example `` 3570p605 '' ` x ` , ` y ` : : ( float ) zero - indexed pixel coordinates of the source on the unwise image tile ( 2048 @xmath43 2048 pixels ) .
` w1_nanomaggies ` : : ( float ) wise flux measurement for this object .
note that these are in the native wise photometric system : vega , not ab .
a source with magnitude 22.5 in the vega system would have a ` w1_nanomaggies ` flux of 1 . ] ` w1_nanomaggies_ivar ` : : ( float ) wise formal error as inverse - variance .
note that this formal error does not include error due to poisson variations from the source .
as such , it is most appropriate for faint objects . `
w1_mag ` , ` w1_mag_err ` : : ( floats ) vega magnitude and formal error in the forced photometry .
these are simple conversions from the `` nanomaggies '' columns above . `
w1_prochi2 ` , ` w1_pronpix `
: : ( floats ) profile - weighted chi - squared and number - of - pixels values .
`` profile - weighted '' means that these are weighted according to the profile of the source in the wise images ( eg , weighted by the point - spread function for point sources ; weighted by the galaxy profile convolved by the point - spread function for galaxies ) .
the column ` w1_prochi2 ` is supposed to measure the quality of fit at the location of the source .
note that ` w1_pronpix ` effectively counts the fraction of the source that was inside the image , and should be close to unity for all sources . `
w1_proflux ` : : ( float )
profile - weighted , the amount of flux contributed by other nearby sources .
this will be zero for isolated sources , but can be larger that ` w1_nanomaggies ` if this source is blended with a brighter source . ` w1_profracflux ` : : ( float ) equal to ` w1_proflux ` divided by ` w1_nanomaggies ` ; the amount of flux at the location of this source that is due to other sources , relative to the flux of this source . `
w1_npix ` : : ( integer ) the number of pixels included in the fit . `
w1_pronexp ` : : ( float ) the number of wise exposures included in the unwise coadd at the location of this source .
this is a proxy for the depth ( which is also reflected in the formal error ) .
|
we present photometry of images from the wide - field infrared survey explorer ( wise ; @xcite ) of over 400 million sources detected by the sloan digital sky survey ( sdss ; @xcite ) .
we use a `` forced photometry '' technique , using measured sdss source positions , star galaxy separation and galaxy profiles to define the sources whose fluxes are to be measured in the wise images .
we perform photometry with _ the tractor _ image modeling code , working on our `` unwise '' coaddds and taking account of the wise point - spread function and a noise model .
the result is a measurement of the flux of each sdss source in each wise band .
many sources have little flux in the wise bands , so often the measurements we report are consistent with zero .
however , for many sources we get three- or four - sigma measurements ; these sources would not be reported by the wise pipeline and will not appear in the wise catalog , yet they can be highly informative for some scientific questions . in addition , these small - signal measurements can be used in _ stacking _ analyses at catalog level .
the forced photometry approach has the advantage that we measure a _
consistent _ set of sources between sdss and wise , taking advantage of the resolution and depth of the sdss images to interpret the wise images ; objects that are resolved in sdss but blended together in wise still have accurate measurements in our photometry .
our results , and the code used to produce them , are publicly available at http://unwise.me[_http://unwise.me_ ] .
mcwilliams center for cosmology , department of physics , carnegie mellon university , 5000 forbes ave , pittsburgh , pa , 15213 , usa department of physics & astronomy , university of waterloo , 200 university avenue west , waterloo , on , n2l 3g1 , canada to whom correspondence should be addressed : [email protected] center for cosmology and particle physics , department of physics , new york university , 4 washington place , new york , ny , 10003 , usa max - planck - institut fr astronomie , knigstuhl 17 , d-69117 , heidelberg , germany lawrence berkeley national laboratory , 1 cyclotron road , berkeley , ca , 94720 , usa
|
four hundred years ago , when johannes kepler and others observed the `` new star '' of 1604 , those observing the event had no concept of what it was that they were observing . today we know that supernovae are exploding stars and that they even come in different varieties .
the type ia sne , which have become so important for cosmology because they are standard candles , arise from the incineration of white dwarf stars .
the other main class , core collapse sne , come from more massive stars and produce the sub - classes called type ib , type ic , and type ii .
of the six historical sne in our galaxy that have occurred over the last millennium ( including cas a for which the sn itself apparently escaped detection ) , only kepler s sn has remained uncertain as to the type of the star that exploded . today
of course , what we observe is the expanding young supernova remnant ( snr ) that resulted from the explosion . by studying this snr across the electromagnetic spectrum
, modern astronomers are still trying to discern a clear picture of the precursor star of this event .
in doing so , they are hindered by the lack of an accurate distance to the object , which makes the derivation of even basic properties like the diameter or mean expansion velocity uncertain . in this paper
, i will review the current observational status of this enigmatic object and point out some of the apparent inconsistencies in existing interpretations .
kepler s snr is located at galactic coordinates @xmath0 and @xmath1 ( i.e. , nearly directly toward the galactic center from the sun but significantly out of the plane ) .
the full extent of the remnant is most visible in radio and x - ray regimes , where a circle of diameter 200 encompasses the entire shell except for two `` ears '' of emission on the east and west sides ( see figure 1 ) .
unfortunately , the distance to kepler s snr is only poorly constrained by observations to date .
most recent literature cites the study by reynoso & goss ( 1999 ) and adopts a distance near 5 kpc .
however , a careful reading of this paper shows that many authors misquote or misunderstand reynoso & goss s result .
these authors use the h i kinematics with a galactic rotation model to place a rather inaccurate `` lower limit '' of 4.8 @xmath2 1.4 kpc on the distance , and independently place an `` upper limit '' of 6.4 kpc based on the proposed association of the snr with an h i cloud .
kinematic distances are inherently uncertain along the line toward the galactic center .
table 1 shows how some basic parameters for the snr depend critically on the assumed distance . in particular , note that the larger distances imply a very large distance off the galactic plane and , when combined with the observed current shock velocity , apparently require a very significant deceleration , which is not consistent with the absence of a well - defined reverse shock in the x - ray data . in figure 1 , i show 6 cm vla radio ( delaney et al .
2002 ) and 0.2 - 10 kev _ chandra _ x - ray observations ( hwang et al . 2000 ) of kepler s snr .
the overall similarity is striking , showing a roughly spherical thick shell of emission brightest in the north .
the apparent band of emission cutting across the middle from nw to se is largely an illusion , caused by projection effects from material on the front and back sides of the shell ( see optical section below ) .
however , this apparent similarity may be deceiving .
flat and steep radio spectrum deconvolutions look quite different from the total intensity maps , and soft ( 0.3 - 1.4 kev ) and hard ( 4 - 6 kev ) x - ray bands , either from _ chandra _ or _ xmm - newton _
( cassam - chena et al .
2004 ) also show different structures .
( in particular , the harder x - rays form a distinct outer rim likely associated with the primary shock wave . )
also , when one looks at the dynamics of the snr , discrepancies are seen .
both radio and x - ray observations extend over a long enough baseline that expansion of the snr has been measured .
hughes ( 1999 ) finds an x - ray expansion rate of r @xmath3 , which is nearly free expansion .
this is almost twice that found in the radio ( r @xmath4 , dickel et al .
1988 ; delaney et al . 2002 ) .
the reason for this discrepancy is not understood .
optical observations ( blair , long , & vancura 1991 ) are brightest in the nw , with the northern cap and isolated central patches of emission also visible ( see figure 2 ) .
the optical data indicate substantial and variable foreground extinction , with e(b - v ) = 1.0 @xmath2 0.2 , consistent with and x - ray determined n(h ) = 5.0 @xmath5 .
bandiera & van den bergh ( 1991 ) performed a careful study of the space velocity of the object , finding a value of 278 @xmath6 toward the nw , which is away from the galactic plane . assuming this motion is due to the precursor star ,
this would account for the large angular distance off the galactic plane and the observed morphology in all bands , showing the brightest emission in the n and nw .
the optical emission comes from two components : radiative shocks into dense , knotty structures ( presumably circumstellar mass loss ) , and smoother filamentary emission visible only in h@xmath7 from so - called nonradiative shocks ( e.g. blair et al .
1991 ; sollerman et al .
the radiative knots show enhanced [ n ii]/h@xmath7 , indicative of probable enrichment from the precursor .
interpretation of the broad line components in the nonradiative shocks ( e.g. chevalier , kirshner , & raymond 1980 ) indicate a current shock velocity of about 1750 @xmath2 250 @xmath6 .
relatively little information is available in the infrared .
the snr is small enough in angular size that _ iras _ data are not useful .
however , douvion et al .
( 2001 ) observed kepler s snr with thr _ infrared space observatory s _ isocam ( see figure 3 ) .
the morphology of the @xmath812 @xmath9 m emission is very similar to the optical , and 6 - 16 @xmath9 m spectra are well - fitted by shock - heated dust models with t@xmath10 = 95 - 145 k , n@xmath11 of several thousand ( similar to optical [ s ii ] densities ) , and t@xmath12 of several hundred thousand k. this makes it very likely that the _ iso _ emission arises primarily from dust heated directly by the primary shock front .
scuba sub - mm observations at 450 @xmath9 m and 850 @xmath9 m by morgan et al .
( 2003 ) were interpretted as indicating a large mass ( @xmath13 of cold ( t = 17 k ) dust in the remnant , but this result has been called into question ( dwek 2004 ) . upcoming _
spitzer space telescope _
observations may resolve this issue .
the literature is extremely confusing on the issue of the sn type .
baade ( 1943 ) reconstructed the historical light curve and claimed a type ia designation , a result that is still quoted in many recent papers .
however , doggett & branch ( 1985 ) showed consistency with a type ii - l light curve , and schaefer ( 1996 ) has also called the historical curve and type ia designation into question .
a decade ago , the preponderance of evidence seemed to point toward a core - collapse event , and much of this evidence is still relevant .
a large distance off the galactic plane might be suggestive of a white dwarf precursor , but high space motion away from the plane and n - rich circumstellar material points to a massive runaway star from an earlier sn as the precursor for kepler s snr .
borkowski et al .
( 1992 , 1994 ) developed a massive star model that was consistent with observations available at that time .
however , x - ray analyses in particular , from exosat ( smith et al . 1989 ; decourchelle & ballet 1994 ; rothenflug et al .
1994 ) , asca ( kinugasa & tsunemi 1999 ) , _ chandra _ ( hwang et al . 2000 ) , and now _ xmm - newton _
( cassam - chena et al .
2004 ) have alternately claimed better fits to type ia or core collapse models .
the modelling of _ xmm - newton _
data by cassam - chena et al .
( 2004 ) is the most careful and accurate to date .
although these authors do not claim to have determined the sn type , their determination of si and fe abundances similar to type ia models and their lack of detection of overabundances of o , ne , ar , s as seen in core collapse objects such as cas a ( hughes et al . 2000 ) , is a strong indicator of a type ia event .
i also note that , even with its exquisite sensitivity and resolving power , the _ chandra _ observation has failed to detect any hint of a stellar remnant ( quite in contrast to the situation with cas a ! ) .
while the issue may not be closed , the pendulum has swung back toward a white dwarf precursor star .
this is not an entirely comfortable situation .
if the type ia designation is correct , then the closest example of our cosmological standard candle has some very peculiar properties !
at least one significant near term advance is in the offing .
recent optical _ hst _ acs images have been obtained ( see figure 4 ) that not only show the bright optical filaments in exquisite detail , but will permit the proper motion of key filaments to be measured . with a refined estimate of the shock velocity from x - ray and optical data , it should soon be possible to measure the distance to kepler s snr with relative precision .
johannes kepler was an intriguing personality ( ferguson 2002 ) .
he came from extremely humble beginnings , he possessed a fierce and staunch religious faith that was entwined with his world view but at odds with his contemporary culture , and he was a visionary scientist . while understanding the new star of 1604 was a sidelight for him ,
i think he would be secretly pleased to know that his name has been attached to this equally intriguing supernova remnant .
baade , w. 1943 , apj , 97 , 119 bandiera , r. , & van den bergh , s. 1991 , apj , 374 , 186 blair , w. p. , long , k. s. , & vancura , o. 1991 , apj , 366 , 484 borkowski , k. j. , sarazin , c. l. , & blondin , j. 1992 , apj , 400 , 222 borkowski , k. j. , sarazin , c. l. , & blondin , j. 1994 , apj , 429 , 710 cassam - chena , g. decourchelle , a. , ballet , j. , hwang , u. , hughes , j. , & petre , r. 2004 , a&a , 414 , 545 chevalier , r. a. , kirshner , r. p. , & raymond , j. c. 1980 , apj , 235 , 186 decourchelle , a. , & ballet , j. 1994 , a&a , 287 , 206 delaney , t. , koralesky , b. , rudnick , l. & dickel , j. r. 2002 , apj , 580 , 914 dickel , j. r. , sault , r. arendt , r. g. , maysui , y. , & korista . k. t. 1988 , apj , 330 , 254 doggett , j. b. , & branch , d. 1985 , aj , 90 , 2303 douvion , t. , lagage , p. o. , cesarsky , c. j. , & dwek , e. 2001 , a&a , 373 , 281 dwek , e. 2004 , apj , 607 , 848 ferguson , k. 2002 , `` tycho and kepler , '' ( new york : walker & company ) hughes , j. p. 1999
, apj , 527 , 309 hughes , j. p. , rakowski , c. , burrows , d. n. , & slane , p. o. 2000 , apj , 528 , 109 hwang , u. , holt , s. s. , petre , r. , szymkowiak , a. e. , borkowski , k. j. 2000 , baas , 32 , 1236 kinugasa , k. , & tsunemi , h. 1999 , pasj , 51 , 239 morgan , h. l. , dunne , l. , eales , s. a. , ivison , r. j. , & edmunds , m. g. 2003 , apj , 597 , l33 reynoso , e. m. , & goss , w. m. 1999 , aj , 118 , 926 rothenflug , r. , magne , b. , chieze , j. p. , & ballet , j. 1994 , a&a , 291 , 271 schaefer , b. e. 1996 , apj , 459 , 438 smith , a. , peacock , a. , arnaud , m. , ballet , j. , rothenflug , r. , & rocchia , r. 1989 , apj , 347 , 925 sollerman , j. , ghavamian , p. , lundqvist , p. , & smith , r. c. 2003 , a&a , 407 , 249
|
october 2004 marks the 400th anniversary of the sighting of sn 1604 , now marked by the presence of an expanding nebulosity known as kepler s supernova remnant . of the small number of remnants of historical supernovae ,
kepler s remnant remains the most enigmatic .
the supernova type , and hence the type of star that exploded , is still a matter of debate , and even the distance to the remnant is uncertain by more than a factor of two . as new and improved multiwavength observations become available , and as the time baseline of observations gets longer , kepler s supernova remnant is slowly revealing its secrets .
i review recent and current observations of kepler s supernova remnant and what they indicate about this intriguing object . to appear in 1604 - 2004 : supernovae as cosmological lighthouses , " a conference held june 2004 in padua , italy ; to be published as an asp conference proceedings .
|
a considerable body of evidence from diverse sources leads to the conclusion that star - forming disc galaxies such as the milky way accrete @xmath0 of gas each year ( e.g. * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* and references therein ) , and have built up their observed discs gradually over the last @xmath1 ( e.g. * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
what remains unclear is from what reservoir this gas is drawn , and how it enters the thin gas disc within which stars are formed .
a central question is the temperature of the reservoir : is this low enough ( @xmath2 ) for the reservoir to contain largely neutral gas , or comparable to the virial temperature , @xmath3 , of the gravitationally bound groups within which milky - way type galaxies currently reside ? in recent years there has been much enthusiasm for so - called cold - mode accretion of gas that has failed to be shock heated to the virial temperature @xcite .
cosmological simulations suggest that cold - mode accretion is the dominant process at redshifts @xmath4 , but gradually becomes less important . a powerful argument against its currently being
the dominant process is the persistent failure of 21-cm surveys to identify significant bodies of intergalactic hi in the nearby universe @xcite .
in particular the galaxy s `` high - velocity '' hi clouds , which as late as 1999 were argued to be distant and massive @xcite , are now known to be at distances @xmath5 and have masses @xmath6 too small for these clouds to be a cosmologically significant reservoir of gas @xcite .
@xcite already inferred from the presence of interstellar absorption in the spectra of high - latitude stars that the galactic disc must be embedded in pervasive medium of temperature @xmath7 .
the empirical case for such `` coronal gas '' was greatly strengthened by the copernicus and fuse missions , which detected ions such as ovion high - latitude sight lines to distant uv sources , in particular sight lines that pass close to high - latitude hi clouds @xcite .
the highly ionised gas detected must be collisionally ionised and is most readily interpreted as material at the interface between coronal gas with @xmath8 and clouds of cooler , partly neutral hi .
cosmology strongly suggests that galaxies should be embedded in virial - temperature coronae .
first , standard cosmology predicts that only a minority of baryons are contained in stars and cool interstellar gas ( e.g. * ? ? ?
* ; * ? ? ?
* ) . in rich clusters of galaxies the `` missing '' baryons
are directly detected through their x - ray emission @xcite . in lower - density environments , such as the local group , it is thought that the surface - brightness of x - ray emission is too low to be detected by current instrumentation @xcite .
second , three lines of argument indicate that star - formation is an inefficient process in which as much gas is ejected from a star - bursting system as is converted into stars : ( i ) we actually see winds blowing off star - forming discs @xcite ; ( ii ) the spectra of several quasars show blue - shifted absorption - line systems indicative of massive winds flowing away from the star - bursting host galaxy @xcite ; ( iii ) in clusters of galaxies of order half the metals synthesised by stars are in the intergalactic medium @xcite .
our premise in this paper is that most of the baryons originally associated with the milky way s dark matter comprise a corona of gas at the virial temperature , and that the gas that sustains star formation in the disc is drawn from this corona .
the question is how gas makes the transition from a pressure supported corona to the centrifugally supported disc . for more than thirty years
x - ray astronomers have studied the virial - temperature coronae of rich groups and clusters of galaxies .
most of these coronae have central cooling times that are significantly shorter than their ages , and it is natural to ask whether we can understand the milky - way s corona by extrapolating results for cluster `` cooling flows '' to lower masses .
it seems , however that the dynamics of these systems is qualitatively different from the dynamics of the milky way s corona because the central galaxies of rich clusters do not have massive stellar discs .
the extent to which gas in rich clusters is cooling ( as opposed to radiating ) is controversial , but it is now widely accepted that radiative losses by the inner corona are largely offset by mechanical feedback from the central black hole @xcite .
it is important to understand the origin of this qualitative difference in the dynamics of the coronae of star - forming galaxies of the `` blue cloud '' and `` green valley '' in colour - luminosity space @xcite and that of the coronae of massive galaxies within the `` red sequence '' .
sensitive hi observations of star - forming disc galaxies reveal that these galaxies keep @xmath9 to @xmath10 per cent of their hi a kiloparsec or more above or below their disc plane ( * ? ? ?
* ; * ? ? ?
* and references therein ) .
most of this gas is thought to have been driven out of the disc by supernova - powered bubbles @xcite and must consist of clouds that are moving on essentially ballistic trajectories because if it were in hydrostatic equilibrium , its vertical density profile would be very much steeper than that observed by virtue of its low temperature @xcite .
the gas is expected to return to the disc within @xmath11 , so the phenomenon of extraplanar hi is evidence for galactic fountains @xcite .
hi observations of extraplanar gas in the milky way are hard to interpret on account of our location within the plane , but the leiden argentina
bonn ( lab ) survey @xcite , which mapped local hi emission with high sensitivity , is consistent with the milky way having a distribution of extraplanar hi that is similar to the distributions of hi studied around external nearby galaxies .
the extraplanar gas of nearby galaxies has two key properties : ( i ) the mean rotation speed of the gas declines quite rapidly with distance from the plane ; ( ii ) the gas shows net motion towards the galaxy s symmetry axis .
fraternali & binney ( 2006 ; hereafter fb06 ) and fraternali & binney ( 2008 ; hereafter fb08 ) fitted models of a galactic fountain to observations of ngc891 and ngc2403 and showed that these models can only account for the observed rotation rates and inward motion of the extraplanar gas if the observed hi clouds accrete gas such that the mass of a cloud exponentiates on a timescale @xmath12 .
such an accretion rate simultaneously accounts for data from the two rather different galaxies modelled and yields rates of accretion onto the discs , @xmath13 and @xmath14 that happen to be similar to the rates at which star - formation consumes gas in these galaxies .
the accreted gas is required to have significantly smaller specific angular momentum about the galaxy s symmetry axis than thin - disc gas .
fb08 suggested that the gas swept up by the fountain s clouds comes not from the corona but from cool streams embedded within it .
they were driven to this conclusion by the shortness of the time it takes coronal gas to flow past a cloud this time is very much shorter than the cooling time of coronal gas . in this paper we argue that notwithstanding the disparity between the cooling and flow times of the coronal gas , the interaction between a cold cloud and coronal gas leads to cooling of the coronal gas and accretion of it onto the star - forming disc . in section [ sec : theory ]
we give analytical arguments why cloud - corona interaction must lead to cooling of coronal gas rather than evaporation of hi . in section [ sec : simuls ] we present hydrodynamical simulations of the flow past a cloud , which confirm the analytic arguments . in section
[ sec : discuss ] we discuss the implications of these simulations both for further observations and the theory of galaxy formation .
section [ sec : conclude ] sums up .
[ fig : ctime ]
in model 2 of @xcite , the density of the corona decreases with radius as @xmath15 and at @xmath16 has electron density @xmath17 .
the corona is assumed to be isothermal with @xmath18 .
figure [ fig : ctime ] shows the cooling time of gas with metallicities @xmath19}=-1 $ ] and @xmath20 that is in pressure equilibrium with plasma of this temperature and the mean density of this corona between 8 and @xmath21 . at @xmath22 , the cooling time is @xmath23 .
once the temperature has fallen to @xmath24 , the cooling time has dropped by nearly forty to @xmath25 , which is less than the dynamical time at the solar galactocentric radius , @xmath26 . by the time the temperature has dropped by a further factor of two to @xmath27 ,
the cooling time has fallen by more than a further order of magnitude and is a mere @xmath28 .
thus although the cooling time of ambient gas in the lower corona is long , any diminution in temperature will dramatically shorten the cooling time . in the model of @xcite
the radiating plasma flows inwards at a rate @xmath29 .
the work done by compression offsets radiative losses , so the plasma temperature remains @xmath30 .
the mass of an hi halo is rather ill - defined : in an external galaxy it is not clear at what value of @xmath31 we should place the boundary between disc and halo hi , and in the galaxy there is a similar ambiguity in the line - of - sight velocity that divides halo from disc hi .
if in ngc 891 the disc - halo boundary is conservatively place at @xmath32 , the mass of the hihalo is @xmath33 ( fb06 ) .
@xcite conclude that @xmath34 per cent of the galaxy s hi is halo gas , and that the total himass within @xmath21 ( the region within which most star formation occurs ) is @xmath35 , so the mass of the hi halo within @xmath21 is @xmath36 . the hi halo is largely confined to the region @xmath37 , so we compare the mass of the hi halo at @xmath38 to the mass of coronal gas in the cylindrical annulus @xmath39 , @xmath37 .
the coronal mass is @xmath40 , so within this volume there is over twice as much hi as coronal gas , although the coronal gas will occupy nearly all the space . the clouds that make up the hi halo typically take @xmath41 to travel from their launch point in the plane through the halo and back to the plane .
consequently , in one gyr @xmath42 of hi is passed through the @xmath43 of gas in the lower corona .
hence the temperature of the coronal gas would be halved if just 4.5 per cent of the gas that passes through the corona in a gyr were to mix with the coronal gas .
this drop in temperature would bring the cooling time of the coronal gas down to @xmath44 , the local orbital time .
the hi clouds of the halo plough through the corona at speeds @xmath45 that are less than but comparable to the sound speed of the coronal gas . consequently the flow around these clouds is likely to be in a high reynolds number regime , and each cloud must be decelerated by ram pressure of order @xmath46 , where @xmath47 is the coronal density . in these circumstances
the cloud loses its momentum on a timescale equal to @xmath48 times the time @xmath49 for the cloud to move its own length ( fb08 ) .
clouds have diameters of a few tens of parsecs and travel a few kpc , while @xmath50 .
hence clouds must surrender a significant fraction of their momentum to coronal gas @xcite .
above we showed that in a gyr @xmath51 of hi passes through the @xmath52 of the lower corona .
clearly the coronal gas can not absorb a significant fraction of the momentum of more than 20 times its mass of hi ( fb08 ) .
moreover , if the hi disc were losing angular momentum to the corona at the rate this calculation implies , it would be contracting on a gyr timescale .
the natural resolution of these problems is that the disc _ accretes _ most of the coronal gas that hi clouds encounter .
then the momentum lost by hi clouds would be returned to the disc , and it would not build up in the corona .
as was mentioned in section 1 , coronal gas in dark - matter halos more massive than those of spiral galaxies has been extensively studied for four decades .
cooling within these halos shows no tendency to produce a cold stellar disc
the coldest gas is near the centre , where the cooling time is shortest , and whatever gas cools out of the corona feeds the central black hole rather than forming a star - forming disc . the argument of the last subsection may help us to understand why in less massive dark halos cooling coronal gas flows into the disc rather than onto a central black hole : so long as the halo has a star - forming disc , that disc sustains its star formation by reaching up and grabbing coronal gas .
discs are disrupted by major mergers , which occur rather frequently .
analytic arguments and hydrodynamical simulations of cosmological clustering suggest that when the disc of a relatively low - mass galaxy is disrupted by a major merger , it will quickly re - form from filaments of cold inflowing gas ( e.g. * ? ? ?
thereafter it will sustain itself by grabbing coronal gas . when a more massive galaxy experiences a major merger , the disrupted star - forming disc is less likely to re - form .
if it does not form from cold inflowing gas , it will not form from coronal gas , because in the absence of a star - forming disc , catastrophically cooling coronal gas will feed the central black hole and reheat the corona .
thus star - formation permanently ceases if there is insufficient cool gas to re - form a star - forming disc after a merger . in this picture
it is likely that cooling at the centre of the corona of a disc galaxy leads to episodic reheating of the central corona , just as in a classical cooling flow .
that is , we suggest that the corona of a star - forming disc galaxy accretes onto central black hole _ as well as _ onto the disc . for this to be a viable proposal , the central reheating associated with accretion by the black hole
must not undermine the ability of the star - forming disc to grab coronal gas from the part of the corona that lies above it
. this condition could be satisfied if the agn outburst were sufficiently small and sufficiently directed perpendicular to the galactic plane .
this is a topic for a later paper , however . as a cloud moves through the ambient coronal gas , turbulence in the boundary layer at the interface of the hot and cold fluids
must cause gas to be stripped from the cloud at some rate . to calculate this rate from ab - initio physics is a daunting task because one would have to consider plasma instabilities in addition to hydrodynamical ones such as the kelvin - helmholtz instability , and turbulent energy will be cascading to very small scales . in view of these difficulties ,
the obvious way forward is to parametrise the problem by hypothesising that mass is stripped from the a cloud of mass @xmath53 at a rate @xmath54 .
a turbulent wake of stripped gas will run back through the corona from the moving cloud . in this wake turbulence will mix the stripped gas with ambient gas . a key quantity is the cooling time of the plasma that results from this mixing .
we estimate this under the assumption that mixing occurs so quickly that radiative losses during mixing can be neglected .
let @xmath55 be a short length of the wake , which has cross - sectional area @xmath56 and originally contained a mass @xmath57 of coronal gas .
in the time @xmath58 that it took the cloud to pass through the length , a mass @xmath59 of gas was stripped from it .
after the cold gas has mixed and come into thermal equilibrium with the coronal gas , the temperature of the resulting fluid is @xmath60 where @xmath61 and @xmath62 are the coronal and cloud temperatures , respectively .
the natural unit for the cross - sectional area of the wake is the characteristic cross - section @xmath63 of the cloud .
we write @xmath64 , where @xmath65 .
thus @xmath66 we simplify this expression under the assumption that the cloud is in approximate pressure equilibrium with the corona , so @xmath67 , and have @xmath68 since @xmath69 is the fraction of the cloud s mass that is stripped in the time taken for the cloud to travel its own length , and @xmath65 , the second term in the numerator of equation ( [ eq : trat ] ) must be small compared to unity and may be neglected .
the second term in the denominator is larger by a factor @xmath70 , so @xmath71 may be significantly lower than @xmath61 . on account of the steepness of the cooling - time curve plotted in fig .
[ fig : ctime ] , this result implies that the cooling time in the wake may fall below the ambient cooling time by a factor of several . from this back - of - envelope calculation
we draw the following conclusions * material in the wake will become hi on the timescale that it takes the parent hi cloud to fly its trajectory if the mass - loss rate @xmath72 exceeds the critical value @xmath73 for @xmath74 this condition is that in the time taken to travel its own length the cloud lose at least a fraction @xmath75 of its mass . *
if the mass - loss rate of clouds falls below this critical value , what mass is stripped from the cloud will be integrated into the corona .
this will lower the cooling time of the ambient corona , but not lead to prompt accretion of the wake onto the disc .
this drop in the temperature of the corona near the disc will lower the critical mass - loss rate required for subsequent wakes to cool promptly . * our estimate of @xmath76
was obtained under the assumption that we can neglect cooling during mixing .
although the mixing process is likely to be fast , the cooling rate is extremely large at temperatures that lie within a factor 30 of the cloud s temperature .
hence it is likely that a more exact calculation would produce a lower estimate of @xmath76 .
we have also neglected compression of the coronal gas as it flows around the cloud .
however , the effect of compression on the cooling time of ambient gas is unclear because , while the cooling time decreases with increasing density at constant temperature , it increases with temperature , and compression will be associated with a ( largely adiabatic ) rise in @xmath77 . *
the effective value of @xmath78 is uncertain . on one hand ,
as the simulations will show , the cloud flattens in its direction of motion , making @xmath79 . on the other hand ,
the distribution of cloud material is likely to be concentrated in a network of thin sheets .
the effective value of @xmath78 for an individual sheet could be small .
* dimensional analysis indicates that @xmath72 will lie near @xmath76 : the rate of stripping is essentially determined by the rate at which coronal gas hits the leading surface of the cloud and strips a comparable mass from the cloud .
quantitatively , @xmath80 , where @xmath81 .
when we eliminate @xmath82 and @xmath47 in favour of @xmath53 and @xmath61 we find that the characteristic mass - loss parameter is @xmath83 * if we take the mass - loss rate to be given by the previous item , we find that the mass of a cloud that will completely mix with the corona after travelling distance @xmath84 is @xmath85 equivalently , the distance travelled prior to destruction is predicted to be @xmath86 times the size of the cloud .
numerical simulations of a cloud of gas at @xmath87 moving through coronal gas at an initial speed @xmath88 will illustrate these points and give insight into both typical mass - loss rates and the critical mass - loss rate required for prompt cooling of the wake .
the simulations are idealised in that they neglect gravitational acceleration and the variation of the coronal density along the cloud s trajectory .
however , the galactocentric radius of the clouds varies by @xmath89 per cent , so the density variation is not extreme .
the range of relevant velocities depends on the extent to which the corona corotates with the disc , which is currently unclear .
the velocity we have chosen to explore is at the low end of the relevant range because it is of order the velocity at which clouds have to be ejected from the disc in order to reach heights of a few kiloparsecs .
larger velocities would clearly lead to more rapidly interaction with the corona than that explored here .
it is important to be clear about what can and can not be learnt from the simulations .
first any simulation incorporates limited physics and limited spatial resolution . in the real world
the fluid is a magnetised , almost perfectly conducting , largely collisionless plasma
. the shear flow at the cloud - corona interface will draw out and strengthen the field lines originally in the plasma .
this drawing out will make the velocity distribution of particles anisotropic , which will excite plasma instabilities .
these instabilities will both heat and mix the plasma . by contrast the tendency of the field lines to follow stream lines will strongly inhibit mixing . none of this complex physics is included in our simulations . instead , in the simulations
ablation and mixing are driven by the kelvin - helmholtz instability , which gives rise to ripples in the fluid interface .
these ripples develop into vortices , which shed smaller vortices , which themselves shed vortices . in a numerical simulation numerical viscosity
prematurely truncates this hierarchy of vortices on a scale of a few times the grid spacing @xmath90 . on scales finer than @xmath90 , the fluid
is represented as perfectly mixed , whereas it is in reality a roughly fractal foam of high- and low - density regions .
the condition for the simulations to be a reliable guide to the large - scale structure of the flow is that the large - scale dynamics of this foam is equivalent to the dynamics of the locally homogeneous fluid that is actually represented .
this is a reasonable proposition , but we have to expect that the values of the flow s macroscopic parameters , such as ambient pressure and cooling rate , at which a simulation of given resolution most closely approximates reality are likely to vary with resolution .
that is , we anticipate a phenomenon analogous to renormalisation in quantum field theory , where the appropriate value of the bare electron mass , for example , is a function of the largest wavenumber summed over .
the simulations do not include thermal conduction . on the smallest scales conduction must play an important role in homogenising a mixture of cloud and coronal gas , and in the simulations this role is effectively covered by numerical mixing and diffusion .
consequently , our failure to model thermal conduction is most worrying on intermediate and larger scales . in a magnetised plasma heat
is largely conducted along field lines .
the field lines in upstream ambient gas are inevitably disconnected from field lines in the cloud .
any connection between these sets of field lines must occur in the turbulent wake .
this fact is likely to severely limit the effectiveness of thermal conduction .
the physical properties of the clouds and the corona considered here are such that the clouds would be stable against conductively - driven evaporation if they were stationary ( see * ? ? ?
* ) , so it is very unlikely that conduction plays an important role in the energetics of our problem . in principle conduction lowers the rate at which a moving cloud is ablated , by damping the kelvin - helmholtz instability @xcite , but this effect will be unimportant if conduction is magnetically suppressed to values small compared to the spitzer or saturated values . in light of these remarks , the aims of the simulations are as follows * to estimate the mass - loss rate @xmath72 for comparison with @xmath91 .
it is advantageous to do this in the absence of radiative cooling for then ( a ) the calculations are slightly faster , and ( b ) one can identify the cloud with gas at @xmath92 since any ablated material will be heated to and remain at higher temperatures .
we will show that the mass - loss rate is reasonably independent of @xmath90 . * to show that for any given metallicity of the gas , there is a critical ambient pressure @xmath93 above which the mass of cool gas increases with time through condensation in the wake and below which the wake tends to evaporate .
we shall find that although our values of @xmath93 vary with both metallicity and @xmath90 , they lie within the range of values that occur in practical cases . hence it is plausible that the true value of @xmath94 lies below the actual ambient pressures , so real wakes give rise to condensation and accretion .
@xmath95 $ ] & @xmath96 $ ] & @xmath97 $ ] & grid size + + z_0 & 25 & no met .
& @xmath98 & @xmath99 & @xmath100 & @xmath101 & c , m , f + z_1 & 25 & -3.0 & @xmath98 & @xmath99 & @xmath100 & @xmath101 & c , m , f + z_2 & 25 & -2.0 & @xmath98 & @xmath99 & @xmath100 & @xmath101 & c , m , f + z_3 & 25 & -1.5 & @xmath98 & @xmath99 & @xmath100 & @xmath101 & c , m , f + z_4 & 25 & -1.0 & @xmath98 & @xmath99 & @xmath100 & @xmath101 & c , m , f + z_5 & 25 & -0.5 & @xmath98 & @xmath99 & @xmath100 & @xmath101 & m , f + z_6 & 25 & 0.0 & @xmath98 & @xmath102 & @xmath103 & @xmath104 & m + + t_4_z_3 & 25 & -1.5 & @xmath98 & @xmath105 & @xmath100 & @xmath106 & m + t_4_z_4 & 50 & -1.0 & @xmath98 & @xmath105 & @xmath100 & @xmath106 & m + t_3_z_4 & 50 & -1.0 & @xmath98 & @xmath105 & @xmath107 & @xmath108 & m + the parameters of the simulations are listed in table [ tab : simuls ] . in all simulations
the initial cloud velocity was @xmath109 and the initial radius was @xmath110 .
the temperature of the corona is restricted to a narrow range around @xmath111 by the requirement that the corona be bound to the galaxy and yet be extensive enough to contain a cosmologically significant mass ( * ? ? ?
* e.g. ) , so in all simulations we made the corona s temperature @xmath111 . in simulations of the low - pressure sequence the total particle density of the corona was @xmath112 ( except in the z_6 simulation in which it was reduced to @xmath113 ) , implying @xmath114 , while in the high - pressure simulations it was twice as great .
these values are both lower than the density @xmath17 at @xmath115 in model 2 of @xcite , and may be compared with the total particle density @xmath116 at @xmath16 above the plane adopted by @xcite . in all but two simulations
the initial cloud temperature was @xmath117 ; in the last two ( high - pressure ) simulations the cloud temperature was lowered to @xmath118 since at the higher temperature ( and therefore lower density contrast ) the cloud was totally disrupted by @xmath119 .
the cloud mass ranged from @xmath104 to @xmath120 depending on the pressure of the corona and the temperature of the cloud .
the jeans mass of the standard cloud ( @xmath121 pc , @xmath122 k ) is @xmath123 so our neglect of self gravity is amply justified .
the calculations were performed on two - dimensional , cartesian grids of three sizes : @xmath124 ( c ) , @xmath125 or @xmath126 for simulations run to @xmath119 ( m ) , and @xmath127 ( f ) .
ghost cells are used to ensure that the pressure gradient vanishes at the grid boundaries .
every configuration was simulated twice , once with cooling on and once with cooling off .
when radiative cooling is permitted , it follows the prescription of @xcite .
the metallicity of the cloud is always the same as that of the ambient medium , and is varied from zero up to solar .
we used the eulerian code echo , which is flux - conserving and uses high - order shock - capturing schemes ; a detailed description and tests of it can be found in @xcite . since one of the dimensions perpendicular to the cloud s velocity has been suppressed , we are in effect simulating flow around an infinite cylindrical cloud that is moving perpendicular to its long axis .
the cylinder initially has a circular cross section of radius @xmath82 . from the simulations we obtain quantities per unit length of the cylinder .
we relate these to the corresponding quantities for an initially spherical cloud of radius @xmath82 by multiplying the cylindrical results by the length @xmath128 within which the mass of the cylinder equals the mass of the spherical cloud .
we first study the simulated flows in the absence of radiative cooling , in order to assess the importance of the limited resolution of the simulations .
figure [ fig : tplot ] shows the temperature distribution on the grid after @xmath129 and @xmath119 with radiative cooling turned off .
the cloud has been flattened into a pancake by ram pressure from the medium it is moving into .
the shear flow over the leading face of the pancake is causing vortices to be shed from the pancake s edges that are analogous to the vortices shed by an aeroplane wing . in the highly turbulent wake behind the cloud
, the temperature fluctuates around @xmath130 depending on the fraction of the gas in each cell that comes from the cloud rather than the corona .
figure [ fig : nocoolablate ] quantifies the speed at which ablation reduces the cloud s mass by plotting the mass of gas below @xmath24 versus time for simulations of three resolutions .
we see that over @xmath129 @xmath131 of gas ( @xmath132 per cent of the cloud s mass ) are heated to above @xmath24 , regardless of the grid resolution ; it seems that even the coarsest grid has sufficient resolution to model satisfactorily the stripping of gas from the leading edge of the cloud . in the absence of radiative cooling ,
any gas that is stripped from the cloud will eventually be heated to above @xmath24 as it mixes with coronal gas .
the details of the smallest vortices involved in the mixing process are resolution - dependent in the sense that at higher resolutions some gas remains cold for slightly longer before numerical mixing on the grid scale eliminates it .
the cloud s mass - loss rate is resolution - independent because all stripped gas will be heated within a couple of large - scale eddy - turnover times , regardless of resolution . the mass - loss rate increases gently with time because it depends on the area of the cloud s leading face , which increases with time as the cloud is squashed into a thinner and thinner pancake , a process that is apparent in fig .
[ fig : tplot ] . the dotted straight lines in fig .
[ fig : nocoolablate ] show the critical mass - loss rate defined by equation ( [ eq : defsac ] ) for @xmath133 and @xmath134 .
we see that , as predicted above on dimensional grounds , the measured mass - loss rate lies near @xmath76 .
actually the simulations must underestimate @xmath135 because mass is lost from the cloud s edges , which have total length @xmath136 in cylindrical geometry and @xmath137 in the spherical case .
thus we expect the numerical rates to be @xmath138 of the true rate and in three dimensions @xmath78 would lie close to unity .
these results enable us to estimate the minimum size that a cloud must have if it is to survive a typical passage through the corona . from fig .
10 of fb06 we have that trajectories last @xmath41 , and in this time a cloud travelling at @xmath139 will move @xmath140 . setting @xmath141 and @xmath142 in equation ( [ eq : mofl ] )
we find that clouds with masses less than @xmath143 will completely mix with the corona before they can return to the plane .
it is important to know how the mass - loss rate scales with cloud mass and thus cloud radius . since mass is lost from the leading surface of the cloud , we would expect the absolute mass - loss rate to scale as @xmath144 and therefore the specific mass - loss rate @xmath145 .
we have verified this dependence by simulating the evolution of a cloud of the standard density but radius doubled to @xmath146 . when radiative cooling is switched on , the strength of the dependence of the cooling rate upon @xmath77 that is apparent in fig .
[ fig : ctime ] , substantially increases the difficulty of the simulations and the uncertainties surrounding their results because now the structure of the turbulent wake is crucial .
indeed , evaporation is favoured over cooling by more effective dispersal of stripped material through a large volume of the corona .
the higher the resolution delivered by the code , the greater the dynamic range of the hierarchy of vortices , and the more effective is the dispersal of stripped material , with the consequence that increased resolution favours evaporation over condensation .
each panel of fig .
[ fig : cool-0 ] is the analogue of fig .
[ fig : nocoolablate ] but with radiative cooling turned on ; in the three panels the metallicity increases from zero ( top panel ) through @xmath147}}=-1 $ ] ( middle panel ) to @xmath20 ( bottom panel ) .
the simulations with [ fe / h ] @xmath148 -1 do not differ significantly from that at zero metallicity . in all three panels the mass of cool gas starts by increasing rather than decreasing , and the rate of increase naturally increases with metallicity .
when @xmath147}}=-0.5 $ ] , the increase continues throughout the @xmath129 simulated , but at lower metallicities the mass of cool gas eventually starts to decrease . as anticipated
, the results are unfortunately dependent on numerical resolution : in general there is a tendency for the mass of cold gas to decrease as the resolution increases , although the medium and high - resolution simulations give rather similar results .
figure [ fig : cool - hp ] shows the effect of doubling the particle density of the corona to @xmath149 : even at metallicity @xmath147}}=-1 $ ] the mass of cool gas now increases throughout the @xmath119 simulated , and in fact in the latter half of the simulation cold gas accumulates at an accelerating rate .
this result should be contrasted with that shown by the middle panel of fig .
[ fig : cool-0 ] , which shows that at the same metallicity but half the density , the mass of cool gas starts to decrease after @xmath150 .
if we extrapolate the behaviour shown in fig .
[ fig : cool - hp ] to the whole milky way halo , assuming as its total mass the value given in sect .
2.1 , we obtain a global accretion rate of @xmath151 .
figure [ fig : cooltplot ] shows the temperature distribution at @xmath152 in a simulation with cooling of plasma with @xmath147}}=-0.5 $ ] and ambient particle density @xmath112 . comparing this figure with the upper panel of fig .
[ fig : tplot ] we see that cooling makes the wake longer and less laterally extended .
we have also run simulations with lower values for the coronal gas density and found that accretion is still present for @xmath153 provided that the metallicity is about solar .
figure [ fig : x - vx ] shows for the high - pressure simulation t_3_z_4 the distribution of cool gas along the cloud s direction of travel ( top panel ) and its distribution in velocity along the same direction ( bottom panel ) at the start of the simulation ( black curve ) and after @xmath119 when radiative cooling is ( blue curve ) or is not ( red curve ) included . from the upper panel
we clearly see the effectiveness of cooling in enhancing the mass of cool gas .
we also see that @xmath154 percent of the cold gas lies more that @xmath155 behind the cloud . in the lower panel around @xmath109
we clearly detect the deceleration of the main body of the cloud , but more striking is the width of the velocity range over which small amounts of cool gas are distributed .
a few times @xmath156 is accelerated to higher velocities than the cloud s .
a slightly larger mass of gas is decelerated to negative velocities .
this velocity distribution implies that gas circulates around vorticies at speeds that are comparable to the speed of the cloud s forward motion .
.[fig : vwidth ] figure [ fig : vwidth ] shows the distribution of cold gas in @xmath157 , the direction of travel , and in @xmath158 for the high - pressure simulation t_3_z_4 with and without cooling .
turning on cooling greatly broadens the extent in both @xmath157 and @xmath158 of low - density cool gas . in both simulations
, the region of highest gas density runs from @xmath159 at the cloud s location down to @xmath160 @xmath161 behind the cloud .
thus hydrodynamics leads to a steep velocity gradient , @xmath162 in the part of the wake that will dominate 21-cm emission .
@xcite assume that a gradient in the measured line - of - sight velocity along complex a , @xmath163 , is entirely produced by the galaxy s gravitational field .
in light of our simulations this assumption should be treated with some caution .
the lower panel of fig .
[ fig : vwidth ] shows that the velocity width of the wake increases with distance from the cloud for @xmath164 from the cloud as a result of the lower envelope ( velocities in the opposite direction to the cloud s motion ) moving downward towards @xmath165 , while the upper boundary remains at @xmath166 .
this is further evidence that turbulent eddies impart peculiar velocities comparable to the cloud s velocity . just behind the cloud material
has to flow faster than the cloud in order to flow into the space vacated as the cloud moves on .
further back similar vortices carry cold gas away from the cloud with similar velocities . in the region @xmath167
to @xmath168 behind the cloud , the turbulence damps quite rapidly . from these simulations we conclude the following .
* there is clear evidence that the simulations limited resolution is adequate for determining the rate at which gas is stripped from a cloud , although the two - dimensional nature of the simulations suggests that they will underestimate the stripping rate by a factor @xmath132 . *
the stripping rate tends to increase with time as a result of the cloud being flattened by ram pressure . *
the stripping rate is best determined from simulations in which cooling is turned off , and it is found to lie close to the rate expected on dimensional grounds . in
[ sec : theory ] we saw that this rate is very similar to the critical rate at which cooling takes over from evaporation . * whether stripped gas is evaporated or leads to the condensation of coronal gas depends on the structure of the turbulent wake .
consequently the finite spatial resolution of the simulations makes it impossible to determine with confidence the combinations of pressure and metallicity which divide evaporation from condensation .
however the simulations confirm analytic arguments , which imply that for a given cloud there is a critical path in the space spanned by the coronal pressure and metallicity which divides situations in which , at high pressure or metallicity , coronal gas condenses in the wake and those in which , at lower pressure or metallicity , the gas stripped from the cloud is evaporated by the corona . both the simulations and
analytic arguments suggest that the parameters of the galactic corona lie close to this critical path .
* the neutral gas that trails the cloud is strongly influenced by hydrodynamics forces and can not be considered to be on an orbit .
there is a large velocity gradient along its high - density ridge in velocity - position space . within @xmath169 from the cloud
the velocity width of the stream is of order of the cloud s velocity .
the picture developed here of the connection between the galactic fountain and accretion onto the star - forming disc differs materially from that proposed by fb08 : in that paper hi clouds grew in mass as they moved through the corona ; here each cloud loses mass , but the mass of cold gas in a cloud and its wake taken together increases with time .
a natural question is whether the present picture predicts a similar overall picture of 21-cm emission to that proposed by fb08 .
if it does , it will fit the data reasonably well .
fb08 fitted 21-cm data cubes for ngc891 and ngc2403 .
individual clouds are for the most part not resolved in these data , so the observed emission is due to many superposed clouds . in the fb08 model ,
each cloud places a blob of emission in the cube , centred on its sky - position and line - of - sight velocity , and smeared by the angular and velocity resolution of the survey .
if the picture developed here is correct , the emission of each cloud is extended in velocity and is elongated on the sky .
very sensitive 21-cm data would be required to see the full extent of the trail of an individual cloud the antenna temperature will drop by two orders of magnitude within @xmath161 and @xmath170 of the cloud s position and velocity . however , the integrated emission from many individually undetectable trails must contribute to the hi `` beards '' of nearby galaxies . in a forthcoming study
we will modify the pseudo - data cubes of fb08 to include the extended emission from the trails of clouds .
we suspect that these improved data cubes will have similar observable properties to those of fb08 because the velocity centroid of each cloud s total emission will be unchanged .
clearly it is to surveys of our own galaxy that we must turn for evidence that clouds have hi - rich wakes .
maps of the 21-cm emission of individual high - velocity clouds generally display a tadpole - like structure :
the cloud is elongated and the point of highest surface brightness lies towards one end ( e.g. * ? ? ?
* ; * ? ? ?
* ) . in the scenario proposed here ,
tadpole - like structures can be explained by the interaction between the cloud and the ambient ( coronal ) medium .
this interaction leads to the formation of trailing material behind the cloud . even in our galaxy
most emission from extraplanar gas is unresolved in the sense that along any direction in the lab survey , emission is detected over a wide band in velocity around 0 .
however , a scan through the data cube , one heliocentric velocity at a time , reveals numerous elongated structures along which there is a systematic trend in velocity
. these could well be the wakes of relatively massive clouds .
clouds with masses well below the threshold for detection of their 21-cm emission can be detected through the absorption lines to which they give rise in the ultraviolet spectra of background sources .
absorption - line studies suggest that large hi complexes are associated with numerous small hiclouds @xcite
. could these small clouds be the knots of cold gas visible in the wake of fig .
[ fig : cooltplot ] ? if this interpretation is correct , the small clouds would be found only on one side of the large complex , and they would have a mass spectrum that was restricted to masses very much smaller than that of the complex . in the simulations the metallicity of the cloud is the same as that of the corona , whereas real - world clouds will be more metal - rich than the corona by a factor up to 10 . whether gas condenses or evaporates depends on the cooling rate of gas that is roughly a 5050 mixture of gas stripped from the cloud and coronal gas , so the most realistic simulations are those in which the universal metallicity is about half that of real clouds ; that is the simulations with @xmath147}}\sim-0.5 $ ] .
we must obviously ask how valid a guide two - dimensional simulations will be to the real , three - dimensional problem . reducing the dimensionality of the problem
reduces the number of high - wavenumber modes relative to the immediately driven low - wavenumber modes , thus making the power - spectrum of turbulence less steep .
this argument suggests that turbulent mixing will be more effective in two dimensions than in three , with the consequence that our simulations have a tendency to over - estimate the pressure or metallicity of the transition from evaporation to condensation .
however , this conclusion must be considered very tentative at this time .
the ablation of clouds that move through a low - density medium has been discussed by , among others , @xcite , @xcite , @xcite and @xcite .
in addition to studying the ablation of pressure - bounded clouds as here , some of these authors have considered also gravitationally bound clouds and included thermal conduction in addition to radiative cooling .
@xcite find that thermal conduction stabilises a moving cloud by reducing the magnitude of the velocity gradient in the boundary layer where the cloud meets the ambient medium ; a smaller velocity gradient leads to slower growth of the kelvin - helmholtz instability .
most previous simulations assume the flow to be axisymmetric around the velocity of the cloud s motion . in such simulations material
becomes trapped on the assumed symmetry axis , where the radial velocity must vanish by symmetry .
@xcite also simulated clouds with the symmetry assumed here and found that the main results were independent of the adopted geometry .
@xcite found ablation to proceed faster than we do and interpreted their ablation timescale as the inverse of the kelvin - helmholtz growth rate , which is larger than @xmath91 by @xmath171 .
the difference between their ablation rate and ours probably arises from their use of a different criterion for identifying cloud gas : they took this to be the mass within an appropriate sphere , whereas we have defined cloud gas to be cool gas , regardless of where it resides . in other respects our results are in accordance with earlier findings .
the simulations most comparable to ours are those of @xcite , who simulated the ablation of cool clouds that fall towards the disc through the corona .
consequently , their characteristic coronal density , @xmath172 , was a factor 2 to 4 lower than ours .
their simulations used a three - dimensional grid , so their grid spacing was coarser and they could not provide evidence of numerical convergence .
their clouds , which had similar initial masses to ours , fragmented in a similar fashion . in most simulations
the total mass of hiin the computational volume declined with time , although in some simulations an upturn in the hi mass is evident at late times as the cloud approaches the plane .
our numerical results are entirely consistent with theirs , although our physical motivation is different and , crucially , our parameter regime extends to the higher coronal densities expected near the plane .
recent attempts to model extraplanar hi include those of @xcite and @xcite .
@xcite investigated the equilibria of differentially rotating distributions of gas in flattened gravitational potentials .
they showed that by making the specific entropy and angular momentum of the gas vary appropriately within the meridional plane , the kinematics of the gas can be made consistent with the data for hi around ngc891 .
they noted that dynamical equilibrium required the gas to be too hot to be neutral , so the gas within their model could not be the observed as hi itself , but suggested it might be coronal gas within which hi clouds were embedded as almost stationary structures ( see also * ? ? ? * ) .
@xcite studied the settling of hot gas within the gravitational potentials of spherical galaxy - sized dark halos .
the gas was initially spinning with a rotation velocity that was independent of radius ( so specific angular momentum @xmath173 ) . a prompt cooling catastrophe caused cold gas to accumulate in a centrifugally supported disc , and clouds of cold gas
were subsequently seen to be falling through hotter gas towards this disc .
this simulation is interesting but one has to worry that the cool clouds are numerical artifacts .
it is well known that standard smooth - particle hydrodynamics artificially stabilises contact discontinuities @xcite .
moreover , observed high - velocity clouds all have masses that lie below the resolution limit of the simulations of kaufmann et al .
, so the clouds they see have no counterparts in reality . finally , @xcite showed that unless the specific entropy profile of coronal gas is unexpectedly flat , a combination of buoyancy and thermal conductivity ( which was not included in the kaufmann et al .
simulations ) suppresses thermal instability .
there is abundant evidence that in galaxies like ours , star - formation powers a fountain that each gigayear carries @xmath174 of hi to heights in excess of @xmath155 above the plane .
several lines of argument strongly suggest that galaxies like ours are surrounded by gas at the virial temperature coronae .
the density of the corona , which must vary with position , is very uncertain , especially in the region above the star - forming disc .
however , there are indications that this density is such that the local coronal cooling time is of order a gigayear .
at least half of the baryons in the universe are believed to reside in coronae and their extensions to intergalactic space .
models of the chemistry and stellar content of the galactic disc require the disc to accrete @xmath175 of low - metallicity gas .
the corona is the only reservoir of baryons that is capable of sustaining an infall rate of this order for a hubble time .
therefore there is a strong prima - facie case that star formation in the disc is sustained by cooling of coronal gas .
the dynamical interaction of hi clouds of the fountain with coronal gas is inevitable . for any plausible coronal density
, the ram pressure arising from motion through the corona leads to non - negligible loss of momentum by fountain clouds .
if this momentum were retained by the corona rather than returned to the disc , the corona would rapidly become rotation - dominated .
we have not pursued this possibility because we think it is more likely that coronal gas that absorbs momentum from fountain clouds is shortly thereafter accreted by the disc .
we suggest that the absorption proceeds as follows : ( i ) coronal gas strips gas from the leading edge of the cloud as a result of kelvin - helmholtz instability ; ( ii ) in the turbulent wake of the cloud , the stripped gas mixes with a comparable mass of coronal gas ; ( iii ) as a result of this mixing the cooling time becomes shorter than the cloud s flight time and coronal and stripped gas together form knots of hi that trail behind the cloud and fall onto the disc within a dynamical time .
this scenario is suggested by a combination of analytic and observational arguments , and supported by hydrodynamical simulations .
analytic arguments imply that for a given coronal pressure and metallicity there is a critical rate of mass loss by a cloud , @xmath76 , such that at lower mass - loss rates , stripped gas will be evaporated by the corona , and the total mass of hi will decrease during a cloud s flight .
by contrast , when the mass - loss rate exceeds @xmath76 , stripped gas will lead to condensation of coronal gas , so the mass of hi increases over time .
dimensional arguments suggest that the actual mass - loss rate must lie close to @xmath76 .
we have used grid - based hydrodynamical simulations of the flight of a cloud to check the analytic arguments . any hydrodynamical simulation
is severely limited by its finite spatial resolution .
however , we present evidence from simulations in which radiative cooling has been switched off that our simulations have sufficient resolution to provide reliable estimates of the mass - loss rate .
this rate is such that clouds with masses @xmath176 ( eq . [ eq : mcrit ] ) will be totally disrupted before they return to the disc .
simulations that include radiative cooling confirm the existence of a critical mass - loss rate @xmath76 that depends on coronal pressure and metallicity in the expected manner . on account of the restricted resolution of our simulations
, we can not give a definitive value for @xmath76 within the local corona .
however , we can argue that if the critical mass - loss rate lay above the actual mass - loss rates , the coronal density and metallicity would rise secularly at the expense of the star - forming disc . as a consequence , the @xmath76 would decrease until it fell below the actual mass - loss rate .
that is , the conditions at the base of the corona have a tendency to adjust until they lead to accretion by the disc .
although the present model differs materially from that of fb08 in that we envisage the masses of clouds decreasing rather than increasing over time , it seems likely that when the model is used to simulate data cubes for the galaxies studied by fb08 , similar agreement with the data will be achieved .
this is because fb08 correctly give the dependence on time of both the mass of hi associated with a given cloud , and its velocity centroid .
however , a revision of the fb08 models is required to test this conjecture .
the simulations make detailed predictions for what should be seen in studies of the high - latitude hi distribution in the galaxy .
trails behind clouds should show large gradients in mean velocity that are dominated by hydrodynamical rather than gravitational forces .
high - sensitivity data should reveal small quantities of gas distributed around the mean velocity by of order the cloud s velocity .
the trail should be studded by knots of cold gas .
the idea that star - forming discs reach up into the surrounding corona and grab the relatively pristine gas required to sustain their star formation for a hubble time , makes a good deal of sense cosmologically by explaining how discs can remain star - forming as long as they are not disrupted by a major merger . if after a major merger there are significant streams of cold gas , this gas can seed a new star - forming disc , but in the absence of cold seed - gas , star formation ceases because coronal gas can only condense onto a disc that is already star - forming .
@xcite argued that thermal evaporation by coronal gas of filaments of cold gas determines whether a central cusp reforms when an early - type galaxy experiences a major merger .
similarly , when a late - type galaxy experiences a major merger , thermal evaporation of filaments of cold gas can prevent the gas disc reforming . as a halo proceeds up the clustering hierarchy , the effectiveness of thermal evaporation increases , and at some point a gas disc is prevented from forming after a major merger . following this event
, there is a dramatic reduction in the rate at which coronal gas cools , and the density and x - ray luminosity of the corona increase rapidly .
the higher they get , the lower the chance that at the next major merger a cool - gas filament will reach the galactic centre and renew the stellar cusp .
thus there is a direct connection between the process discussed here and the stark contrast in the detectability with x - rays of the coronae of star - forming and early - type galaxies @xcite .
the progressive reduction with cosmic time in the typical halo mass of galaxies making the transition from the blue cloud to the red sequence ( `` downsizing '' ) , follows from the decrease in the abundance of cold infalling gas with both cosmic time and halo mass - scale that is expected on analytic grounds and observed in ab - initio cosmological simulations . it would be interesting to add this idea to semi - analytic models of the evolution of the galaxy population
we thank an anonymous referee for his / her valuable comments .
most of the numerical simulations were performed using the bcx system at cineca , bologna , with cpu time assigned under the inaf - cineca agreement 2008 - 2010 .
fm gratefully acknowledges support from the marco polo program , university of bologna .
|
it is argued that galaxies like ours sustain their star formation by transferring gas from an extensive corona to the star - forming disc . the transfer is effected by the galactic fountain cool clouds that are shot up from the plane to kiloparsec heights above the plane .
the kelvin - helmholtz instability strips gas from these clouds .
if the pressure and the the metallicity of the corona are high enough , the stripped gas causes a similar mass of coronal gas to condense in the cloud s wake .
hydrodynamical simulations of cloud - corona interaction are presented .
these confirm the existence of a critical ablation rate above which the corona is condensed , and imply that for the likely parameters of the galactic corona this rate lies near the actual ablation rate of clouds . in external galaxies trails of hi behind individual clouds will not be detectable , although the integrated emission from all such trails should be significant .
parts of the trails of the clouds that make up the galaxy s fountain should be observable and may account for features in targeted 21-cm observations of individual high - velocity clouds and surveys of galactic hi emission .
taken in conjunction with the known decline in the availability of cold infall with increasing cosmic time and halo mass , the proposed mechanism offers a promising explanation of the division of galaxies between the blue cloud to the red sequence in the colour - luminosity plane .
[ firstpage ] hydrodynamics turbulence
ism : kinematics and dynamics galaxy : kinematics and dynamics galaxy : structure galaxies : formation intergalactic medium cooling flows
|
the static electromagnetic properties of deuterium provide interesting information on the dynamics at work within the nucleus .
the fact that deuterium s charge is one teaches us little other than the validity of charge conservation in the nuclear system , but that its magnetic moment @xmath12 and that it has of a non - zero quadrupole moment are facts which played an important role in establishing that non - central components of the @xmath2 potential are at work within the deuterium nucleus .
the most accurate value for the deuteron quadrupole moment comes from a molecular physics experiment @xcite .
it is : @xmath13 meanwhile the best determination of @xmath14 comes from the spectroscopy of the deuterium atom .
it is @xcite : @xmath15 but elastic electron scattering from deuterium provided the one - photon - exchange approximation is valid probes m1 and e2 responses for ( virtual ) photons that have a finite three - momentum @xmath16 .
the relevant form factors are related to breit - frame matrix elements of the two - nucleon four - current @xmath17 via @xmath18 these , together with the charge form factor , @xmath19 : @xmath20 provide a complete set of invariant functions for the description of the deuterium four - current that interacts with the electron s current in this approximation . in eqs .
( [ eq : gm])([eq : gq ] ) we have labeled the deuteron states by the projection of the deuteron spin along the direction of the three - vector @xmath21 , and @xmath22 , with @xmath23 since we are in the breit frame .
we can then calculate the deuteron structure functions : @xmath24 in terms of @xmath25 and @xmath26 the one - photon - exchange interaction yields a lab .
frame differential cross section for electron - deuteron scattering @xmath27 , \label{eq : dcs}\ ] ] here @xmath28 is the electron scattering angle , and @xmath29 is the ( one - photon - exchange ) cross section for electron scattering from a point particle of charge @xmath30 and mass @xmath31 .
the form factors defined in eqs .
( [ eq : gm])([eq : gc ] ) are related to the static moments of the nucleus by : @xmath32 with @xmath33 the nucleon mass . for recent reviews of experimental and theoretical work on elastic electron - deuteron scattering
see refs .
@xcite . from eq .
( [ eq : dcs ] ) it is already clear that measurements of the differential cross section alone can not yield uncorrelated information on all three form factors . to measure @xmath8 , @xmath34 , and @xmath19 in a model - independent way one must obtain data with polarized deuterium targets or polarized electron beams .
a new set of measurements of polarization observables in electron - deuteron scattering will soon be available from the data set obtained at the bates large - acceptance spectrometer toroid ( blast ) .
there polarized electrons of energy 850 mev circulated in the bates ring and were scattered from an internal target containing polarized deuterium .
the significant amount of beam on target ( 3 million coulombs since late 2003 ) , and high degree of beam and target polarization achieved at blast , means that we anticipate data on electron - deuteron polarization observables that is more precise than that derived from any previous measurement .
the electron beam circulating in the bates ring was roughly 70% polarized , and the deuterium target employed could operate in both a vector - polarized and tensor - polarized mode .
this gives access to all of the elastic electron scattering deuterium structure functions .
prominent among these are @xmath35 , the vector analyzing power , and @xmath36 , the tensor analyzing power .
they are related to the form factors defined above by @xcite @xmath37 where the ratios @xmath38 and @xmath39 are : @xmath40 and @xmath41 .
therefore measurements of @xmath35 and @xmath36 should facilitate the extraction of the ratios @xmath42 ( which at the @xmath11 s we will consider here mainly affects @xmath35 ) and @xmath43 ( which mainly affects @xmath36 ) . in this paper
we provide predictions for these ratios which are based on chiral effective theory ( @xmath0et ) .
this approach ( for reviews see refs .
@xcite ) is based on the use of a chiral expansion for the physics of the two - nucleon system . in the formulation suggested by weinberg @xcite , the @xmath0et treatment of the @xmath2 system
is based on a systematic chiral and momentum expansion for the two - nucleon - irreducible kernels of the processes of interest . in particular ,
wave functions are computed using an @xmath2 potential expanded up to a given order in the small parameter : @xmath44 where @xmath45 is the momentum of the nucleons and @xmath46 is the breakdown scale of the theory . for electron - deuteron scattering the other two - nucleon - irreducible kernel that must be calculated
is the deuteron current operator @xmath17 .
we also expand this object as : @xmath47 where the operator @xmath48 contains @xmath49 powers of the small parameter @xmath50 , which now includes the momentum transfer to the nucleus , @xmath51 , as one of the small scales in the numerator .
for chiral effective theories without an explicit delta degree of freedom @xmath46 will in general be @xmath52 , but in electron - deuteron elastic scattering the @xmath53 intermediate state is not allowed and so @xmath46 will be larger , @xmath54 . the @xmath2 potential has now been computed up to @xmath55 @xcite , @xmath56 @xcite and @xmath57 @xcite . in this paper
we will employ wave functions computed using the next - to - leading order [ nlo=@xmath55 ] , next - to - next - to - leading order [ nnlo=@xmath56 ] and n@xmath60lo potentials [ @xmath57 ] developed in ref .
these potentials are regularized in two different ways : first , spectral - function regularization ( sfr ) at a scale @xmath62 @xcite , is applied to the two - pion contributions .
then , after the sfr potential @xmath63 is obtained in a particular @xmath2 partial wave , it is multiplied by a regulator function @xmath64 , so that the lippmann - schwinger equation can be straightforwardly solved : @xmath65 there have been questions raised as to the consistency of the wave functions computed in this way @xcite .
partly because of these questions we will , for comparison , also present results for electron - deuteron matrix elements using the form ( [ eq : sum ] ) for the current operator , and wave functions derived from the nijm93 @xcite or cd - bonn @xcite potentials , as well as potentials with one - pion exchange at long range and a square well and surface delta function of radius @xmath66 @xcite .
we stress that such calculations are not chirally consistent . however , common features of deuteron observables that can be identified within calculations that use these different types of wave functions
chiral effective theory , potential models , and one - pion - exchange tails should be independent of the details of physics at ranges @xmath67 in deuterium , and so should not be sensitive to any subtleties pertaining to the renormalization of the @xmath0et . the operators @xmath48 and the coefficients @xmath68 in eq .
( [ eq : sum ] ) are constructed according to the counting rules and lagrangian of heavy - baryon chiral perturbation theory ( hb@xmath0pt ) , which is reviewed in ref .
@xcite . here
the results we will present for @xmath19 and @xmath8 include all contributions to @xmath17 up to chiral order @xmath69 .
this is the next - to - next - to - leading order ( nnlo ) for these quantities .
calculations of electron - deuteron scattering with the nnlo @xmath0et operator were already considered in ref .
@xcite , which improved upon results with the @xmath70 operator in ref .
@xcite and the @xmath71 results of ref .
however , as was already observed in ref .
@xcite and is reiterated below , calculation of the quadrupole combination of matrix elements at nnlo does not reproduce the experimental value of @xmath72 to the accuracy one would expect at that order .
we identify the cause of this as short - distance two - body contributions to @xmath73 of natural size ( i.e. with @xmath74 ) through which quadrupole photons induce an @xmath75 transition in the @xmath76 deuteron state @xcite .
we use the operator induced by these short - distance contributions to renormalize the deuteron quadrupole moment , and hence the form factor @xmath8 .
we also provide results for @xmath34 up to nlo .
not surprisingly , @xmath34 at nlo proves more sensitive to short - distance physics than does the renormalized @xmath8 . throughout this work
we will use the factorization of nucleon structure employed in ref .
@xcite in order to include the effects of finite nucleon size in the calculation .
there it was shown that the chiral expansion for the ratios : @xmath77 with @xmath78 and @xmath79 the isoscalar single - nucleon electric and magnetic form factors , is better behaved than the chiral expansion for @xmath19 , @xmath8 , and @xmath34 themselves . the ratios ( [ eq : ratios ] )
allow us to focus on the ability of the chiral expansion to describe deuteron structure , and we will employ the @xmath0et results for the ratios in our efforts to predict blast s results for polarized electron scattering from a deuterium target .
we note that , up to the order we work to here , our predictions for @xmath1 are independent of the manner in which we include nucleon structure in the calculation . our invoking the factorization of nucleon structure in the electron - deuteron matrix elements plays no role in our predictions for @xmath1 .
the chiral perturbation theory for this calculation was laid out in refs .
@xcite , and so here we merely summarize the pertinent features of the chiral expansion for the deuteron currents in section [ sec - kernel ] .
however , in doing so we find that we must address the issue of how to calculate the corrections to this ratio that have coefficients which are fixed by low - energy lorentz invariance .
we deal with this problem in section [ sec - oneoverm ] , by recalling results of friar , adam _ et al .
_ , and arenhvel _ et al .
_ , which show that such corrections can be calculated unambiguously , as long as they are included consistently in both the @xmath2 potential and the current operator . then , in section [ sec - other ] we discuss effects in the @xmath73 operator beyond @xmath80 , and explain how we will estimate their impact on @xmath19 and @xmath8 . in particular ,
we write down an operator that represents the effects of physics at mass scales above 1 gev on @xmath8 , and can repair the discrepancy between the experimental value of @xmath72 and our predictions for @xmath81 .
we also discuss how to estimate the remaining uncertainty in our results .
then , in section [ sec - j0results ] we present results for @xmath19 , @xmath8 , and the ratio @xmath1 .
we show that the shape of @xmath8 can be predicted in a model - independent way for @xmath82 gev@xmath5 , but the uncertainty in the ratio @xmath1 is sizeable at @xmath83 gev@xmath5 .
finally , in section [ sec - jplusresults ] we present results for @xmath10 , and in section [ sec - conclusion ] we summarize and provide an outlook .
we now discuss the charge and current operators in turn .
such a decomposition is , of course , not lorentz invariant , so here we make this specification in the breit frame .
the vertex from @xmath84 which represents an @xmath85 photon coupling to a point nucleon gives the leading - order ( lo ) contribution to @xmath73 as depicted in fig .
[ fig - twobodycharge](a ) . at @xmath86
this is corrected by insertions in @xmath87 that generate the nucleon s isoscalar charge radius .
this gives a result for @xmath73 through @xmath70 : @xmath88 with @xmath89 the `` relativistic '' corrections to the single - nucleon charge operator .
these contributions have fixed coefficients that are determined by the requirements of poincar invariance .
since these coefficients scale as @xmath90 this particular set of @xmath86 contributions are generally smaller than one would estimate given the formula ( [ eq : p ] ) for the parameter @xmath50 .
these `` relativistic '' corrections can be calculated by writing down a @xmath73 operator that , when inserted between deuteron wave functions calculated in the two - nucleon center - of - mass frame , yields results for the matrix elements that are lorentz covariant up to the order to which we work . to do this we employ the formalism of adam and arenhvel , as described in ref .
@xcite .
meanwhile the only contribution at @xmath91 , or nnlo , comes from the tree - level two - body graph shown in fig .
[ fig - twobodycharge](b ) . in hb@xmath0pt
the relevant single - nucleon photo - pion vertex arises as a consequence of the foldy - wouthuysen transformation which generates a term in @xmath92 .
straightforward application of the feynman rules for the relevant pieces of the hb@xmath0pt lagrangian gives the result for this piece of the deuteron current @xcite : @xmath93 , \label{eq : j02b}\ ] ] where @xmath94 and @xmath95 are the ( breit - frame ) relative momenta of the two nucleons in the initial and final - state respectively ( see ref .
@xcite for a much earlier derivation ) .
thus we now have a result for the deuteron s charge operator which can be summarized as : @xmath96 \delta^{(3)}(p ' - p - q/2 ) + \langle { \bf p}'|j_0^{(3)}({\bf q})|{\bf p } \rangle + o(ep^4 ) .
\label{eq : pure}\ ] ] however , it was shown in ref .
@xcite that the parameterization ( [ eq : structure ] ) of the nucleon s isoscalar charge distribution breaks down at @xmath97 mev . in order to avoid this difficulty we observe that the result ( [ eq : pure ] ) may be recast , up to the order to which we work , as : @xmath98 g_e^{(s)}(q^2 ) + o(ep^4 ) , \label{eq : factor}\ ] ] with @xmath99 the complete one - loop hb@xmath0pt result for the nucleon s isoscalar electric form factor @xcite
this means that we can write a result that is independent of @xmath0pt s difficulties in describing nucleon structure if we focus on the ratio of @xmath100 to @xmath99 .
we will then use experimental data , in particular the parameterization of mergell _ et al . _
@xcite , for @xmath99 in all computations we present below .
this use of experimental data for the single - nucleon matrix element that appears in eq .
( [ eq : factor ] ) allows us to focus on how well the @xmath0et is describing deuteron structure , since it removes the nucleon - structure issues from the computation of the deuterium matrix element . our technique to achieve this is rigorous , up to the chiral order to which we work here .
the ratio @xmath1 can also be computed independent of nucleon - structure issues , as is made clear by a brief examination of eq .
( [ eq : factor ] ) , together with the definitions ( [ eq : gc ] ) and ( [ eq : gq ] ) . the counting for the isoscalar three - vector current @xmath101
was already considered in detail by park and collaborators @xcite .
@xmath101 begins at @xmath102 , but at @xmath80 there are finite - size and relativistic corrections , which are suppressed by two powers of @xmath103 .
this is the highest order we will calculate @xmath34 to here , and at this order , using factorization we have : @xmath104 g_m^{(s)}(q^2 ) \delta^{(3)}(p ' - p - q/2).\ ] ] with @xmath105 is the momentum of the struck nucleon , and @xmath106 is the isoscalar magnetic moment of the nucleon , whose value we take to be @xmath107 . at @xmath108 [ nnlo ]
two kinds of magnetic two - body current enter the calculation .
one is short - ranged , and one is of pion range @xcite .
each of them has an undetermined coefficient . in principle
those coefficients should be fit to data ( e.g. the deuteron magnetic moment , which is not exactly reproduced by the current @xmath101 and the wave functions employed here ) and the low-@xmath11 shape of the form factor .
in this section we discuss the constraints imposed by poincar invariance or the low - energy manifestations thereof on the breit - frame isoscalar @xmath2 charge operator .
recall that in hb@xmath0pt @xmath109 arises from a piece of @xmath110 that has a fixed coefficient obtained via a foldy - wouthuysen transformation .
therefore this piece of the charge operator is a low - energy consequence of lorentz covariance of @xmath111 .
as such the contribution ( [ eq : j02b ] ) should be computed in a manner consistent with that used to derive the @xmath90 corrections to the one - pion exchange part of the @xmath2 potential .
those @xmath90 corrections can be obtained from the chiral lagrangian specifically from the @xmath3 pieces in @xmath112 and the @xmath90 pieces in @xmath113 .
but the relevant operators involve the energy of the individual nucleons , and so it is not immediately obvious how to convert them to contributions to an energy - independent quantum - mechanical potential . in fact , in the 1970s and 1980s many techniques were developed by which quantum - mechanical operators could be obtained from a relativistic quantum field theory @xcite . in all of these techniques
there was freedom in choosing whether ( and if so , which ) nucleon lines to put on shell , as well as freedom in how to include meson retardation . as we shall see , the choices made with respect to these two issues have an impact on the form of the operators ( both @xmath114 and @xmath73 ) that are obtained .
ultimately though , as long as operators and potentials are derived in a consistent way , the different choices are related by unitary transformations that leave matrix elements unaffected @xcite .
that unitary transformation is labeled by two parameters : @xmath115 , which parameterizes the energy , @xmath116 , of the exchanged pion via @xmath117 where @xmath118 ( @xmath45 ) is the length of the relative three - momentum vector of the @xmath2 system after ( before ) the meson exchange ; and @xmath119 , which denotes a choice for the change in the nucleons energy after absorption of the pion @xcite : @xmath120 note that in quantum mechanics energy is not conserved at each vertex , and so @xmath121 need not necessarily hold .
indeed , it turns out that since we are in the @xmath2 c.m
. frame the difference @xmath122 is the same for both nucleons once an energy shell is chosen .
the full expression for the @xmath90 corrections to the one - pion - exchange potential in the case of arbitrary @xmath119 and @xmath115 can be found in ref .
the main result for our purposes here is that if @xmath123 then the potential takes the form : @xmath124 this is the one - pion - exchange potential used in the n@xmath60lo computation of ref .
( corrections to `` leading '' two - pion exchange diagrams that are suppressed by @xmath3 are also included there , but are associated with pieces of the @xmath73 which are of higher order than we work to here . ) the computation of ref .
@xcite employed the form for @xmath125 that corresponds to @xmath126 .
all other potentials we discuss here ( including the nnlo and nlo ones used in ref .
@xcite ) employed the non - relativistic form of ope , i.e. the result ( [ eq : relope ] ) , but without the additional factor in the round brackets .
meanwhile , all of the potentials we have used neglect retardation , which means they have set @xmath127 . consistent reduction of the contributions to the deuteron charge operator then leads to a more general result for diagram fig .
[ fig - twobodycharge](b ) than that given in eq .
( [ eq : j02b ] ) @xcite : @xmath128 ^ 2 } + ( 1 \leftrightarrow 2)\right].\nonumber\\ \label{eq : j02bmutilde}\end{aligned}\ ] ] in eq .
( [ eq : j02b ] ) we obtained the result for the @xmath80 piece of the charge operator that corresponds to @xmath126 and @xmath129 , because the field - theoretic manipulations used to arrive at eq .
( [ eq : j02b ] ) assume that the fields represent physical particles , i. e. they are on - shell . the result ( [ eq : j02bmutilde ] )
may be obtained from eq .
( [ eq : j02b ] ) by applying a unitary transformation @xcite : @xmath130 where the form of @xmath131 can be found in the original papers .
the same unitary transformation generates consistent @xmath90 corrections to the one - pion - exchange part of the @xmath2 potential : @xmath132 including the form ( [ eq : relope ] ) if the choice @xmath123 , @xmath133 is adopted .
this is not consistent with the choice made in obtaining eq .
( [ eq : j02b ] ) in ref .
@xcite because the @xmath2 potential of ref .
@xcite was computed using the okubo formalism developed in ref .
@xcite . in ref .
@xcite this issue of the choice made for @xmath119 and @xmath115 does not arise until the n@xmath60lo potential is derived , because in that paper , and in the earlier refs .
@xcite , epelbaum _ et al .
_ chose to count @xmath134 .
in doing this they were adhering to weinberg s original argument as to why it is the two - nucleon - irreducible kernel and not the @xmath2 amplitude itself which admits a chiral expansion .
@xmath2 intermediate states introduce factors of @xmath33 in the amplitude for loop graphs , and if @xmath135 then the @xmath136th iterate of the one - pion - exchange potential is the same order as one - pion exchange itself @xcite . however , in ref . @xcite the need to iterate the one - pion - exchange potential to all orders was established without any reference to counting @xmath134 , being justified instead by the singular , and attractive , nature of the @xmath2 force ( see also ref .
therefore , while it is true that corrections to the @xmath2 potential which are suppressed by powers of @xmath3 are often smaller than , e.g. those arising from excitation of the delta(1232 ) , in discussing electron - deuteron scattering we will consider a regime in which @xmath137 can be sizeable .
therefore we follow the original hb@xmath0pt counting and take @xmath138 .
as we shall see , this counting is supported by the fact that the contribution ( [ eq : j02b ] ) plays a significant , but not excessive , role in the deuteron charge and quadrupole form factors .
if we count @xmath138 the dilemma presented by the inconsistency between @xmath114 and @xmath73 arises already at @xmath80 . a way out of this dilemma is provided by eqs .
( [ eq : j0unitary ] ) and ( [ eq : vopeunitary ] ) .
they guarantee that we will obtain the same result ( up to @xmath139 corrections ) for deuteron matrix elements @xmath140 , regardless of what choices for @xmath119 and @xmath115 we make when constructing the operators @xmath114 and @xmath73 from the chiral effective field theory , provided that we consistently include the @xmath141 pieces of the potential @xmath114 and the @xmath91 pieces of the operator @xmath73 . therefore in order to be consistent with the calculation of the @xmath90 corrections to @xmath142 in ref .
@xcite we must adopt the choice @xmath123 in the formula ( [ eq : j02bmutilde ] ) for @xmath143 . if we do this , and also make sure to calculate one - pion exchange according to the formula ( [ eq : relope ] ) , then our results for matrix elements of the deuteron charge operator will incorporate the low - energy consequences of lorentz invariance , up to corrections of @xmath139 ( higher order than we consider here ) .
note that the cd - bonn potential is a different case , since the use of a pseudoscalar @xmath144 coupling means that there we have @xmath126 .
therefore in that case , and only in that case , we have used the expression ( [ eq : j02b ] ) for the first part of @xmath143 , with no modification by the factor of @xmath145 that must be present if @xmath146 is constructed with @xmath123 .
this still leaves us with the issue of how the @xmath147 corrections in the one - pion exchange potential ( [ eq : relope ] ) and the @xmath147 corrections to the nucleon kinetic energy operator are to be accounted for in the calculations using the nnlo and nlo wave functions of ref .
@xcite ( or included in calculations with the nijm93 wave function of ref .
@xcite or the wave functions of ref .
the original calculations of these wave functions did not include such effects , but since we count @xmath138 here , we need to include them in order to have a consistent calculation of @xmath19 and @xmath8 to @xmath80 .
starting from the kamada - glckle transformation @xcite , we show in appendix [ ap - p2overm2corrns ] the major part of these effects can be included by making changes to the short - distance pieces of the @xmath2 potential , and using a slightly modified wave function @xmath148 in the computation of @xmath19 , @xmath8 , and @xmath34 .
that wave function is related to the original non - relativistic wave function @xmath149 obtained in ref .
@xcite by @xcite : @xmath150 the solution of the non - relativistic schrdinger equation for the wave function @xmath149 , followed by the use of the formula ( [ eq : wfreln2 ] ) to obtain the solution of the relativistic schrdinger equation , is the method by which we incorporate @xmath90 effects for the nlo and nnlo wave functions of ref .
@xcite , the wave function of ref .
@xcite , and the wave functions of ref .
the effects of using @xmath148 , rather than the wave function , @xmath149 , to calculate electron - deuteron observables increase with photon momentum transfer @xmath151 , but are small over the entire range for which the @xmath0et predictions can be trusted . at @xmath152 mev for the nijm93
wave function they change @xmath19 by 6.3% , @xmath8 by 1.2% , and @xmath34 by 2.0% .
( the effect on @xmath19 is proportionately larger because 700 mev is quite close to the form factor minimum . )
so far we have obtained the deuteron two - body charge operator up to @xmath91 , or next - to - next - to - leading order .
this is the order up to which the calculation we present here is fully systematic . in this section
we discuss the role of contributions that are nominally higher order , and identify one particular mechanism that apparently could generate significant effects at @xmath153 .
we are particularly interested in this operator because `` the @xmath72 problem '' that is present in all modern potential models ( see , e.g ref .
@xcite ) persists in the @xmath0et .
the problem is that all such calculations under - predict the value ( [ eq : qdexpt ] ) for @xmath72 by about 23% when they use a charge operator that includes all effects up to nnlo in the @xmath0et .
the remaining discrepancy is large compared to the expected @xmath154 size of higher - order effects .
it is also large compared to other discrepancies between theory and experiment in the @xmath155s@xmath156-@xmath155d@xmath156 channel of the @xmath2 system .
and the situation is actually worse than this , because at @xmath157one order higher than we are considering here there are two - meson - exchange contributions to the deuteron charge operator .
one might hope that these provide the missing strength in the e2 response of deuterium at @xmath153 .
these diagrams are presently being calculated for finite @xmath11 , and will be incorporated in a future computation of the charge and quadrupole form factors @xcite .
however , it is already known that they do not give a sizeable contribution to the deuteron quadrupole moment .
et al . _
@xcite computed their effect on @xmath72 using the av18 wave function @xcite , and found : @xmath158 therefore these effects will _ not _ repair the discrepancy between the calculated @xmath81 and the experimental @xmath72 . at the next order , @xmath159 , there are additional two - pion - exchange contributions to the deuteron charge .
however , short - distance two - body currents that contribute to @xmath160 and @xmath72 are also present , and are depicted in fig .
[ fig - twobodycharge](c ) . even though it is suppressed by five powers of @xmath50 relative to the lo result
, the latter operator can have a noticeable impact on the quadrupole moment of deuterium , since the numerical value of @xmath72 corresponds to a distance that is small on the typical scale of deuteron physics @xmath161 fm .
the operator is @xcite : @xmath162 and is designed to be diagonal in two - body spin and isospin and contribute only for @xmath76 and @xmath163 states . in the case of deuterium it represents an e2 photon inducing a @xmath155s@xmath164s@xmath156 transition .
such a transition is possible because the photon interacts with the total spin of the two - nucleon system through the two - body operator ( [ eq : e2sd ] ) .
the two - nucleon operator ( [ eq : e2sd ] ) will be induced when high - momentum modes in the @xmath2 system are integrated out to obtain the low - momentum effective theory .
it could also be induced when heavy mesons which can couple to a quadrupole photon in the requisite way are integrated out of the @xmath0et .
this heavy - meson origin for the operator leads us to anticipate that with a scale @xmath165 of about 1.2 gev the coupling @xmath166 will be of order 1 . in particular , if we used resonance saturation in the @xmath2 system @xcite to estimate the size of this operator the first mesonic current that would contribute to the operator would be the @xmath167 current @xcite .
we now give arguments which demonstrate that physics at roughly this scale could indeed remedy the discrepancy between the experimental @xmath72 and the result found from the mechanisms already discussed .
let us take the accepted number from `` high - quality '' potential models @xmath168 fm@xmath5 ( see , e.g. ref .
calculations with @xmath0pt wave functions obtain similar , or even slightly smaller , numbers @xcite .
then , we adopt @xmath170 fm@xmath5 as an estimate for the nlo and nnlo corrections ( which come mainly from the two - body operator @xmath143 ) .
this leaves us with a remaining discrepancy between theory and experiment of 0.008 fm@xmath5 , or about 3% . inserting the operator ( [ eq : e2sd ] ) between deuteron wave functions we obtain its contribution to @xmath72 as : @xmath171 where @xmath172 is the deuteron wave function at the origin . while @xmath172 is not an observable , neither is the dimensionless number @xmath166 : it is a wave - function - regularization dependent coefficient .
since only s - waves contribute at @xmath173 if we assume @xmath174 gev we can constrain the combination of @xmath172 and @xmath166 to be : @xmath175 ^ 2 \approx 6.5~{\rm fm}^{-3},\ ] ] where @xmath176 is the @xmath155s@xmath156 radial wave function .
therefore the operator ( [ eq : e2sd ] ) can be associated with physics at scales of about 1 gev and still remedy the discrepancy between the theoretical value of @xmath72 ( including the meson - exchange contribution to the charge ( [ eq : j02bmutilde ] ) ) and the experimental value ( [ eq : qdexpt ] ) .
this higher - order effect has an importance greater than one would anticipate in weinberg s counting ( [ eq : sum ] ) not because it is unnaturally large , but because , from the point of view of that counting , @xmath72 is unnaturally small @xcite .
this is hardly a surprise , since , at both leading and next - to - leading order , @xmath72 is generated by one - body operators that connect the deuteron s - state to the deuteron d - state .
such effects are suppressed by the ratio @xmath177 @xcite .
in contrast , the operator ( [ eq : e2sd ] ) is not @xmath178-suppressed and so its contribution to @xmath72 is significantly larger than the naive estimate of @xmath179% leads us to anticipate . in this context it is worth noting that such an estimate for the short - distance effects in @xmath180 is completely validated by calculation @xcite .
the leading - order piece of @xmath160 is of the expected size @xmath181 , and two - body contributions , beginning with effects from @xmath143 and continuing through the two - pion - exchange mechanisms of @xmath182 and the c0-photon short - distance operator of @xmath183 , give contributions of the expected size , approximately @xmath184% .
while this is reassuring , the ( relatively ) large impact of @xmath183 on @xmath81 means that we must ask how well we know this operator .
its value at @xmath153 can be fixed by the requirement that it repair the discrepancy between theory and experiment for @xmath72 .
but at this stage we know nothing about its @xmath11 dependence .
for the purposes of this paper we will assume that this operator arises from heavy - meson exchange , and so model its @xmath11 dependence by : @xmath185 the uncertainty in the operator is now encoded as uncertainty in the scale @xmath46 of its momentum variation .
we anticipate @xmath186 gev , because there are no meson resonances below 1.2 gev which , when integrated out of the theory , will yield this operator .
the only danger with this reasoning is that two - pion - range mechanisms that occur at @xmath159 may ultimately prove to be responsible for the @xmath72 discrepancy .
this possibility is under investigation @xcite .
however , evaluation of relevant processes in models which calculate , e.g. the role of @xmath187 components in deuterium , suggest that the dominant two - pion - exchange contributions to @xmath183 are not large enough to remedy the @xmath72 discrepancy @xcite . as for an upper bound on the value @xmath46 ; the effects of the operator ( [ eq : j05uncert ] )
persist to higher @xmath11 as the scale of its momentum variation is increased . at the @xmath11 s considered here ,
the impact of this operator on observables when we choose @xmath188 gev is within 30% of what one would obtain at @xmath189 , so we will vary @xmath46 between 1.2 and 2 gev in order to assess the theoretical uncertainty of our calculation .
we will see below that even with this range of variation our ignorance as to the precise value of @xmath46 ( or the precise function of @xmath11 that modulates the current @xmath183 ) is the dominant contribution to our theoretical uncertainty in the ratio @xmath1 .
our goal in introducing the @xmath11-dependence ( [ eq : j05uncert ] ) into our calculation is to assess the potential impact on our @xmath0et calculation from physics that is not explicitly included in it .
the @xmath11-dependence of @xmath143 will be modified by these sorts of effects , as well as by higher - order loop contributions that can be calculated in the @xmath0et . however ,
once such higher - order calculations are carried out the @xmath11-dependence of @xmath143 can presumably be constrained by input from electro - production in the single - nucleon sector .
therefore here we take the operator @xmath143 as given . when we quote ranges for its impact on observables those ranges arise from the fact that @xmath190 varies when evaluated with different deuteron wave functions .
in this section we present results for the matrix elements of the deuteron charge operator : @xmath19 and @xmath8 .
.[table - evgenywfs ] values of the sfr cutoff @xmath62 and the lippmann - schwinger equation cutoff @xmath46 that are employed in the different wave functions of ref .
@xcite that are used to compute deuteron form factors in this work .
the wave functions are in groups of five : first those generated with the nlo @xmath0et potential , then nnlo , then n@xmath60lo . [ cols="^,^,^,^",options="header " , ]
lastly we focus on the region where the @xmath0et is reliable : @xmath191 mev .
an enhanced view of this region is shown in fig . [ fig - gcgqdetail ] . for each choice of @xmath2 potential two curves
are shown : the upper one of the pair corresponds to choosing @xmath192 gev when evaluating the operator @xmath183 and the lower one corresponds to choosing @xmath188 gev .
a conservative estimate for the impact of short - distance physics which is not well - constrained in this @xmath0et calculation is given by combining the uncertainties from @xmath193 variation and the uncertainty coming from lack of knowledge about the momentum dependence of @xmath183 .
the black bars then represent the range of possible values that the @xmath0et predicts for @xmath1 at the kinematics where there will be data from blast .
these ranges are reproduced in table [ table - blastpreds ] .
the error is about @xmath194% at the lowest blast point and increases with @xmath11 , as it should .
note that we have not included the n@xmath60lo @xmath0et wave function , or the nlo @xmath0et wave function , in generating these predictions .
as already discussed , the predictions for @xmath19 and @xmath8 with the n@xmath60lo wave function deviate already from the extant data at quite low @xmath11 , while the nlo wave function is , in the @xmath0et sense , less accurate than the operator being used here .
the predictions obtained with these wave functions are , however , within 2@xmath195 , if the theoretical error bars we have obtained are taken to have the usual one - standard - deviation interpretation .
blast will also measure the ratio @xmath10 .
predictions for that observable are provided in fig .
[ fig - gcovergm ] .
we do not show any experimental data in fig .
[ fig - gcovergm ] because , as far as we can glean from the literature , all previous data on @xmath34 comes from different data sets to that used for the extraction of @xmath8 and @xmath19 @xcite .
therefore in general data on @xmath19 and @xmath34 are not at the same @xmath11 and have different systematic errors .
the blast data set will be pioneering in this regard . .
calculations shown are for extremal nlo ( long dashed ) , extremal nnlo ( dot - dashed ) , extremal n@xmath60lo ( solid green ) ( solid red ) , cd - bonn ( solid blue ) and @xmath196 fm square well + one - pion exchange ( short dashed ) wave functions.,width=415 ] the calculations displayed in fig . [ fig - gcovergm ] are accurate to relative order @xmath103 , although the @xmath19 used here is actually computed up to relative order @xmath197 .
once again the shape of the low - momentum part of the ratio is fairly wave - function independent , but the value at @xmath153 changes as we move through the different wave functions used for the computation of fig .
[ fig - gcovergm ] , due to short - distance contributions to @xmath14 being different for different wave functions .
however , even without renormalization there is a robust prediction for the ratio out to @xmath198 gev@xmath5 .
the robust prediction is that @xmath10 is approximately flat
. this would be exactly the case in the absence of relativistic , meson - exchange , and nucleon - structure contributions to the operator , and if @xmath199 .
the relativistic corrections at @xmath55 are negligible at @xmath200 gev@xmath5 , and the meson - exchange piece of the charge operator is higher order than we are attempting to calculate the ratio @xmath10 to . as far as the operator is concerned this leaves only the nucleon - structure effects , which depend on the ratio : @xmath201 if a strict chiral expansion is used for the form factors then this ratio depends ( again , at this order ) on the difference of the isoscalar magnetic and charge radii , and amounts to a @xmath202% effect at @xmath203 gev@xmath5 .
even though taking the ratio @xmath10 does not ( as it did in the case of the ratio of the previous section ) eliminate nucleon - structure effects , it does reduce their size .
meanwhile , the effects of @xmath204 grow with @xmath11 , and so at @xmath205 gev@xmath5 it is thus not particularly surprising that @xmath42 is , to quite a good approximation , flat .
given the extent of the variation in the prediction for the ratio beyond @xmath206 mev it is difficult to believe that the nlo predictions for the ratio shown here are reliable beyond that point .
this situation could improve once the nnlo pieces of the operator @xmath101 were computed , but short - distance pieces of @xmath207 appear already at that order .
therefore it is a prediction of the chiral expansion that this ratio will be more sensitive to short - distance physics than is @xmath1 .
the position of the minimum in @xmath34 is known to be very sensitive to such short - distance physics @xcite . in this context
it is worth noting that the minimum for the n@xmath60lo wave functions is already at @xmath208 mev much lower than in any of the calculations of refs . @xcite and indeed , much lower than the experimental data allows the position of the minimum to be @xcite .
we have used the @xmath0et isoscalar charge operator in the nucleon - nucleon sector computed to nnlo ( including a consistent treatment of the @xmath3 pieces of the charge ) , and the wave functions of ref .
@xcite , to obtain the form - factor ratios @xmath209 , @xmath210 , @xmath1 . these ratios test @xmath0et s ability to describe deuteron structure . we confirm and extend the finding of ref .
@xcite , that the nlo and nnlo @xmath0et wave functions , combined with the nnlo @xmath73 , yield results for these ratios that agree within the experimental uncertainties with the extractions of ref .
@xcite for @xmath4 gev@xmath5 .
in contrast , the n@xmath60lo wave function of ref .
@xcite , when used in conjunction with the n@xmath5lo charge operator , produces form factors that depart from the data at @xmath211 gev@xmath5 . in light of the upcoming release of data on @xmath36 from blast we examined the ratio @xmath1 in detail .
we found variation in the @xmath0et predictions for this ratio at @xmath153 , and also found that even allowing for this variation@xmath0et is in disagreement with the experimental value for this quantity .
this phenomenon the `` @xmath72 problem''is familiar in high - precision @xmath2 potential models with the modern value of the @xmath212nn coupling constant . in @xmath0et
its solution arises naturally through a higher - order two - body current that couples exclusively to quadrupole photons .
we added this operator to our analysis , and showed that when we do so the @xmath0et predictions for @xmath1 ( with the nnlo wave functions ) fall within a narrow band out to @xmath213 gev@xmath5 ( see fig .
[ fig - gcgqdetail ] ) .
we also performed the calculation with the same charge operator and potential - model wave functions that have a one - pion - exchange tail identical to that of the @xmath0et potential @xcite .
we found that these wave functions make the band obtained at nnlo in @xmath0et about a factor of two wider .
we conservatively adopt the full width of that band as representative of the theoretical uncertainty in our calculation . meanwhile , the @xmath0et predictions for @xmath10 , which will also be measured at blast , are not reliable to as high a @xmath11 . in saying this ,
it should , in fairness , be pointed out that @xmath101 has not been computed to as high an order as @xmath73 , and it could therefore be that @xmath10 can also be well described to @xmath214 gev@xmath5 in @xmath0et once @xmath108 corrections to @xmath215 are included .
this is a topic for future work .
another topic for future work is the inclusion of the @xmath108 pieces of @xmath73 that were already computed in ref .
@xcite at @xmath153 in the finite-@xmath151 calculation @xcite .
in addition , the operators and the @xmath0et wave functions used in this paper to make predictions for the blast data can be further tested by comparing their predictions with experimental results for deuteron electro - disintegration although this will require the computation of the isovector pieces of the operators .
irrespective of such future efforts , one thing is already clear from fig .
[ fig - gcgqdetail ] .
when the theoretical predictions for @xmath1 are renormalized in the manner we advocate here , the theoretical uncertainty in @xmath1 for @xmath216 gev@xmath5 is less than the uncertainty in the data .
this makes the low-@xmath11 data from blast all the more crucial , since it will provide an important test of @xmath0et s ability to organize contributions to deuteron observables , and its ability to use that organization to provide estimates of the theoretical uncertainty in a given calculation .
i thank michael kohl and chi zhang for a number of conversations regarding the blast data , and in particular for stimulating questions regarding the theoretical uncertainty that arises from the @xmath11-dependence of the two - body current that renormalizes @xmath72 .
i am also grateful to richard milner for inviting me to a blast workshop in january 2005 where a number of the results in this paper were presented in preliminary form .
thanks also to evgeny epelbaum for supplying me with the wave functions of ref .
@xcite , and for his comments on both my results and this manuscript .
i am also grateful to lucas platter for his careful reading of the manuscript and assistance with the spelling of dutch names . this work was supported by the us department of energy under grant de - fg02 - 93er40756 .
one way to include relativistic corrections to the nucleon kinetic energy operator in the @xmath0et is to solve the `` relativistic schrdinger equation '' @xcite @xmath217 \psi_l^{sj}(p ) + \int \frac{dp ' { p'}^2}{(2 \pi)^3 } \ , v_{ll'}^{sj } ( p , p ' ) \psi_{l'}^{sj } ( p')=e \psi_l^{sj}(p ) , \label{eq : relse}\ ] ] with the ( partial - wave decomposed ) potential @xmath63 the one that is obtained from the @xmath0et using the counting rules explained in section [ sec - intro ] . to facilitate computation of the @xmath147 ( and beyond ) corrections to the nucleon kinetic energy operator
we adopt the kamada - glckle transformation @xcite and define a new relative momentum @xmath218 such that @xmath219 the inverse transformation is then : @xmath220 as shown by kamada and glckle , any `` relativistic '' schrdinger equation that employs the kinetic energy operator with the relativistic form on the right - hand side of eq .
( [ eq : kg ] ) can be recast as a non - relativistic schrdinger equation , i.e. @xcite @xmath221 where the potential @xmath222 is obtained from @xmath223 via : @xmath224 and @xmath225 here the factor @xmath226 is introduced to ensure that , as long as @xmath148 is normalized to one , @xmath149 is also normalized to one .
calculation of the jacobian associated with the transformation ( [ eq : kg ] ) yields : @xmath227 we can also work this procedure in reverse , i.e. interpret solutions of the non - relativistic schrdinger equation as solutions to a dynamical equation with a relativistic kinetic energy operator and a relativistic potential .
( something similar is done to obtain the @xmath2 c.m .
frame hamiltonian in approaches to electron - deuteron scattering based on relativistic hamiltonian dynamics , see , for example , ref .
@xcite . ) . here
we will make this interpretation for the deuteron wave functions obtained with the nlo and nnlo chiral potentials .
suppose that @xmath228 is the chiral potential computed ( at nlo or nnlo ) _ without _ @xmath147 corrections , i.e. @xmath229 where @xmath230 is the ( partial - wave decomposed ) sum of @xmath2-irreducible diagrams that is computed at nlo and nnlo using spectral - function regularization in ref .
@xcite and @xmath231 is the `` lippmann - schwinger equation regulator '' used there .
a potential @xmath114 that includes @xmath147 corrections and is associated with the same sum of feynman diagrams can be constructed by inverting eq .
( [ eq : tildev ] ) .
we find that this potential , which when inserted in the relativistic schrdinger equation ( [ eq : relse ] ) will be phase - equivalent to @xmath228 of eq .
( [ eq : tildevchiral ] ) , is : @xmath232 we have not bothered to distinguish between @xmath218 and @xmath45 in the functions @xmath64 and @xmath226 , because we now define a new regulator function @xmath233 to obtain : @xmath234 up to terms that are of order @xmath235 . since we count @xmath33 as the same order as the cutoff scale @xmath46 absorbing the functions @xmath226 into the definition of the regulator in this way makes physical sense .
apart from these short - distance effects the only difference between the potentials @xmath114 and @xmath228 is the use of the `` stretched '' momenta in @xmath114 .
if we wish we can also incorporate the factor @xmath236 that distinguishes eq .
( [ eq : relope ] ) from the non - relativistic treatment of one - pion exchange used at nlo and nnlo in ref .
@xcite . to do this we redefine the regulator again , writing @xmath237 where , up to the order to which we work : @xmath238 if one ignores the regulator @xmath239 , then the factors in front of the one - pion - exchange part of the potential in eq .
( [ eq : vfinal ] ) agree with eq .
( [ eq : relope ] ) and the expression for the two - pion - exchange pieces has been modified only at n@xmath60lo and beyond . as for the short - distance pieces of the potential
, the changes in the way the chiral @xmath2 potential is regularized ( @xmath240 ) mean that the @xmath2 lec @xmath241 will have to be re - defined : its value is shifted by a term of @xmath242 .
however , @xmath241 is fitted to data , so no change in observables should result from doing this . therefore we have shown that , via modifications of the ( unobservable ) short - distance part of the potential that do not affect the low - energy observables , we can absorb much of the `` relativistic effects '' that would enter the nlo and nnlo chiral potentials were we to count @xmath243 . the only such effect that can not be absorbed by a redefinition of the regulator used when solving the lippmann - schwinger equation with the potential @xmath114 is the different arguments at which @xmath230 is evaluated in eq .
( [ eq : vfinal ] ) as compared to eq .
( [ eq : tildevchiral ] ) .
this difference is an nlo effect for the part of @xmath114 coming from one - pion exchange , where it represents the interplay of the relativistic kinetic energy operator and the ( leading - order ) one - pion - exchange potential .
this interplay is not included in our calculations , but all other relativistic effects are , and as reported in sec .
[ sec - oneoverm ] , they prove to be small . the main difference between interpreting the expression ( [ eq : tildevchiral ] ) as a non - relativistic potential and interpreting it as resulting from a relativistic calculation after application of the kamada - glckle transformation therefore resides in the momentum arguments that enter the schrdinger equation .
in the first case the wave function that is the solution of that equation will be a function of @xmath218 . in the second case
it will be a function of @xmath244 .
the relationship between the two wave functions is given by eq .
( [ eq : wfreln ] ) . this can be inverted to yield eq .
( [ eq : wfreln2 ] ) , which we rewrite here : @xmath245 thus , if we are provided with the wave - function @xmath149 that is obtained by solving the non - relativistic schrdinger equation with the potential @xmath228 , then we may derive from that the wave function @xmath148 , that is the solution of the relativistic schrdinger equation ( [ eq : relse ] ) with a phase - equivalent relativistic potential @xmath114 ( [ eq : vfinal ] ) , with @xmath114 constructed to include all the corrections to one - pion - exchange that must be present in the nlo and nnlo potentials if we count @xmath138 .
this procedure is not necessary in the case of the n@xmath60lo potential . in that case
the @xmath90 corrections which are connected with the current ( [ eq : j02bmutilde ] ) were explicitly computed in ref .
@xcite .
p. f. bedaque and u. van kolck , ann .
nucl . part .
52 * , 339 ( 2002 ) .
d. r. phillips , j. phys .
g * 31 * , s1263 ( 2005 ) , and references therein .
e. epelbaum , prog .
phys . *
57 * , 654 ( 2006 ) .
s. weinberg , phys .
b. * 251 * , 288 ( 1990 ) .
s. r. beane , p. f. bedaque , m. j. savage and u. van kolck , nucl .
a * 700 * , 377 ( 2002 ) .
d. eiras and j. soto , eur .
j. a * 17 * , 89 ( 2003 ) .
a. nogga , r. g. e. timmermans and u. van kolck , phys .
c * 72 * , 054006 ( 2005 ) .
m. pavon valderrama and e. ruiz arriola , phys .
c * 72 * , 054002 ( 2005 ) .
m. c. birse , phys .
c * 74 * , 014003 ( 2006 ) .
v. g. j. stoks , r. a. m. klomp , c. p. f. terheggen and j. j. de swart , phys . rev .
c * 49 * , 2950 ( 1994 ) .
r. machleidt , phys .
c * 63 * , 024001 ( 2001 ) .
d. r. phillips and t. d. cohen , nucl .
a668 * , 45 ( 2000 ) .
v. bernard , h. w. fearing , t. r. hemmert and u. g. meiner , nucl .
a * 635 * , 121 ( 1998 ) [ erratum - ibid .
a * 642 * , 563 ( 1998 ) ] .
s. platchkov _ et al .
_ , nucl .
a * 510 * , 740 ( 1990 ) .
mergell , u. g. meiner and d. drechsel , nucl .
phys . a * 596 * , 367 ( 1996 ) .
park , d .-
min , and m. rho , nucl .
a596 * , 515 ( 1996 ) .
j. l. forest , phys .
c * 61 * , 034007 ( 2000 ) .
e. epelbaum , w. glckle and u. g. meiner , nucl .
a * 637 * , 107 ( 1998 ) .
h. kamada and w. glckle , phys .
* 80 * , 2547 ( 1998 ) . y .-
sung , t .- s .
park , n. kaiser , and d. r. phillips , in progress .
j. j. de swart , c. p. f. terheggen and v. g. j. stoks , arxiv : nucl - th/9509032 .
w. p. sitarski , p. g. blunden and e. l. lomon , phys .
c * 36 * , 2479 ( 1987 ) ; p.
g. blunden , w. r. greenberg and e. l. lomon , phys .
c * 40 * , 1541 ( 1989 ) .
a. amghar , n. aissat and b. desplanques , eur .
j. a * 1 * , 85 ( 1998 ) .
d. abbott _
et al . _ ,
j. * a7 * , 421 ( 2000 ) .
d. r. phillips , s. j. wallace and n. k. devine , phys .
c * 72 * , 014006 ( 2005 ) .
r. schiavilla and i. sick , phys .
c * 64 * , 041002 ( 2001 ) .
p. l. chung , w. n. polyzou , f. coester and b. d. keister , phys .
c * 37 * , 2000 ( 1988 ) .
|
we use chiral effective theory ( @xmath0et ) to predict the deuteron form factor ratio @xmath1 as well as ratios of deuteron to nucleon form factors .
these ratios are calculated to next - to - next - to - leading order . at this order
the chiral expansion for the @xmath2 isoscalar charge operator ( including consistently calculated @xmath3 corrections ) is a parameter - free prediction of the effective theory .
use of this operator in conjunction with nlo and nnlo @xmath0et wave functions produces results that are consistent with extant experimental data for @xmath4 gev@xmath5 .
these @xmath0et wave functions predict a deuteron quadrupole moment @xmath6@xmath7with the variation arising from short - distance contributions to this quantity .
the variation is of the same size as the discrepancy between the theoretical result and the experimental value .
this motivates the renormalization of @xmath8 via a two - nucleon operator that couples to quadrupole photons .
after that renormalization we obtain a robust prediction for the shape of @xmath1 at @xmath9 .
this allows us to make precise , model - independent predictions for the values of this ratio that will be measured at the lower end of the kinematic range explored at blast .
we also present results for the ratio @xmath10
. plus 2pt minus 2pt = -1 cm * chiral effective theory predictions for deuteron form factor ratios at low @xmath11 * + + + + pacs nos .
: 12.39.fe , 25.30.bf , 21.45.+v = -1 cm
|
progress in material technology , especially in nanofabrication , ultrathin film manufacturing , ultraclean and high vacuum systems , _ etc . , _ requires better understanding of boundary scattering in physical processes .
the boundary effects should be an integral part of any study of quantum wires , wells , and films .
boundary scattering is especially important for transport in ultrathin and/or clean systems in which the particle mean free path is comparable to the system size .
below we consider the effect of random surface roughness on quantum transport in quantized quasi-@xmath5 systems such as , for example , ultrathin metal films .
the main issue is to find how sensitive is the transport along such film to the statistical properties of random surface inhomogeneities ( thickness fluctuations ) .
an important by - product of our systematic comparison of different classes of random surface inhomogeneities is the prediction of a new type of size effect in quantized films .
this effect manifests itself as large oscillations of conductivity @xmath0 as a function of the film thickness @xmath1 .
in contrast to the usual quantum size effect ( qse ) , the peaks can be observed only at relatively large values of @xmath6 .
the distance between the peaks is large and is roughly proportional to @xmath7 .
the observation of this qse opens a new experimental method of identification of the type of surface roughness .
the choice of quasi-@xmath5 systems is explained by a desire to avoid divergence of surface fluctuations and strong localization effects which are inherent to @xmath8 systems and make a systematic quantitative study of the effect of surface inhomogeneities on transport virtually impossible . in contrast to @xmath8 systems , the randomly fluctuating @xmath5 surfaces are practically stable while the localization length in systems with weak surface roughness is exponentially large .
( in general , the _ transport _ problems are more interesting in systems with weak rather than with strong roughness .
transport in systems with strong roughness is trivial : each wall collision completely dephases the particles and the mean free path can not exceed the distance between the walls ) .
the prevalent way to characterize the random surface roughness and/or thickness fluctuations is to use the correlation function of surface inhomogeneities @xmath9 where the vector @xmath10 gives the @xmath5 coordinates along the surface , @xmath11 describes the deviation of the position of the surface in the point with @xmath5 coordinates @xmath10 from its average position , @xmath12 , and @xmath13 is the averaging area . here it is assumed that the correlation properties of the surface do not depend on direction
. two main characteristics of the surface correlation functions @xmath14 are the average amplitude ( height ) @xmath15 and correlation radius ( size ) @xmath2 of surface inhomogeneities .
any transport theory for systems with rough boundaries should provide the explicit dependence of the particle mean free path ( or the conductivity along the walls ) on the correlator of surface inhomogeneities @xmath16 . without bulk scattering ,
the conductivity @xmath0 is determined by the relation between three length scales : particle wavelength , @xmath17 ; the width of the channel , @xmath1 ; and the correlation radius of inhomogeneities , @xmath2 .
if the roughness is weak , the fourth length parameter , @xmath15 , enters conductivity as a coefficient , @xmath18 note , that this @xmath5 conductivity differs by a length unit from the usual @xmath19 conductivity and , as a result , has a dimensionality of conductance .
the form of the surface correlator @xmath20 can vary from surface to surface .
most of the theoretical calculations assume that this correlator is gaussian .
the numerical simulations , on the other hand , often rely on various generators for random rough surfaces without paying much attention to the correlation function of the generated inhomogeneities .
both approaches are not satisfactory since the experiments on surface scattering and diffraction patterns show that real surfaces exhibit surface correlators @xmath21 of various forms@xcite .
even one and the same film can exhibit various correlation properties on different stages of growth . as a result ,
the behavior of the functions @xmath22 in eq .
@xmath23 , which reflects the correlation properties of inhomogeneities , can vary from surface to surface even when the main correlation parameters @xmath15 and @xmath2 remain the same .
the correlation functions @xmath21 are characterized by different long - range behavior that can be reliably identified in various surface diffraction and scattering experiments .
what we would like to know is how sensitive is the _ particle _ _ transport _ to the form of the surface correlator .
in contrast to surface diffraction and scattering data with angular and/or wavelength scanning , the transport coefficients are integral parameters that include angular and wavelength averaging .
this leaves the question of how sensitive is the conductivity to the shape of the surface correlator wide open .
in addition , we are asking a question whether it is possible to identify the type of surface inhomogeneities from transport experiments in ultrathin films or multilayer systems without beforehand information on the form of the surface correlator .
the interrelated question is , of course , to what extent one should pay attention to the details of the correlator of surface inhomogeneities in analytical or numerical transport calculations for particles with large mean free paths .
the former issue has already been raised in refs.@xcite for a small set of surface correlators on the basis of the born approximation for wall scattering .
below we present a systematic study which is based on a more general transport formalism and involves a variety of classes of surface correlators .
in short , we want to compare functions @xmath24 in eq.@xmath25 calculated for various types of the correlation functions @xmath20 in a wide range of parameters .
we start from degenerate ballistic fermions in quantized metal films .
the choice is not arbitrary : transport in such systems involves the minimal degree of averaging ( integration ) and can be the most sensitive to the long - range properties of the surface correlators @xmath26 .
quantum size effect ( qse ) in metal films is a subject of intensive experimental study .
recent qse experiments with quantized metal films include conductivity @xcite , spectroscopy @xcite , susceptibility @xcite , and stm @xcite measurements .
one of the signature features of qse in metals is a pronounced saw - like dependence of conductivity on , for example , film thickness , @xmath27 .
this dependence was predicted for both bulk @xcite and surface @xcite scattering .
experimentally , qse in conductivity was studied for metals in refs.@xcite ( for earlier results see references therein ) .
however , experiments on qse in metals have to overcome a difficulty which one does not encounter in semiconductors .
the period of the qse oscillations in the dependence @xmath27 is usually small , almost atomic , @xmath28 ( below , except for final results , @xmath29 ) . for this reason
typical experimental object are lead or semimetal films such as bismuth .
below we predict a new type of qse with large - period oscillations of @xmath30 at relatively large values of @xmath31 that could lead to observation of qse in a wider group of metals .
large - period qse oscillations have already been observed ( see the second ref .
@xcite ) ; however , sketchy experimental details do not allow one to identify reliably this observation as the new type of qse predicted below .
our results can also resolve the long - standing controversy on the influence of the structure of the nanoscale film on its resistivity @xcite .
recently , we developed a transparent semi - analytical formalism for transport in systems with rough boundaries that allows simple uniform calculations in a wide range of parameters and for various types of roughness with and without bulk scattering @xcite .
this formalism unites approaches by tesanovic _
et al _ @xcite , fishman and calecki @xcite , kawabata @xcite , meyerovich and s. stepaniants @xcite , and makarov _
et al_@xcite ( for a brief comparison between different theoretical approaches see refs.@xcite ) .
below we apply this formalism with an explicit purpose of studying the dependence of the transport coefficients on the shape of the correlation function of surface inhomogeneities .
the well - defined limits of applicability of our approach to metal and semiconductor films are discussed in detail in refs.@xcite . since the @xmath5 mobility of particles is described by essentially the same equations as the exponent in the expression for the localization length in films , our study provides the dependence of the localization length on the type of the correlation function of random surface inhomogeneities .
the paper has the following structure . in the next section
we introduce various types of surface correlation functions .
section iii briefly describes the transport equation used for conductivity ( mobility ) calculations in qse conditions .
the results of transport calculations for different types of correlators are given in section iv .
conclusions and experimental implications are discussed in section v. appendix a contains useful analytical expressions for the power density spectral functions of inhomogeneities responsible for the behavior of scattering probabilities for different types of correlators .
appendix b deals with the positions of new type of qse peaks .
we consider an infinite @xmath5 channel ( or film ) of the average thickness @xmath1 with random rough boundaries @xmath32 ( the walls are assumed hard with infinite potential ) .
the inhomogeneities are small , @xmath33 , and random with zero average , @xmath34 .
their correlation function @xmath35 and its fourier image @xmath36 , which is often called the power spectral density function or power spectrum , are defined as @xmath37 where @xmath38 and @xmath39 are the @xmath5 vectors . in homogeneous systems ,
the correlation function depends only on the distance between points @xmath40 and not on coordinates themselves .
the correlation functions @xmath41 and @xmath42 describe intrawall correlations of inhomogeneities , and @xmath43 are the interwall correlations . usually , but not always , the inhomogeneities on different walls are not correlated with each other , @xmath44 .
thus , everywhere , except for section i ve , it is assumed that @xmath44 . to avoid parameter clutter
, we also assume that the correlation parameters are the same on both walls , @xmath45 .
then the effective correlator contains @xmath46 with @xmath20 given by equations below .
surface inhomogeneities exhibit a variety of types of the correlation functions @xcite . to have a meaningful comparison , we consider the correlation functions that involve only two characteristic parameters , namely , the amplitude ( average height ) @xmath15 and the correlation radius ( average size ) @xmath2 of surface inhomogeneities .
the most commonly used in theoretical applications is the gaussian correlation function , @xmath47 including its limit for small correlation radius @xmath48 , _
i.e. , _ the @xmath49-type correlations , @xmath50 sometimes , a better fit to experimental data on surface scattering is provided by the use of the exponential correlation function @xmath51 or by the even more long - range , power - law correlators @xmath52 with different values of the parameter @xmath53 .
the most commonly used are the staras function with @xmath54 and the correlator with @xmath55 which has the exponential power spectrum @xmath56 , @xmath57 the use of the lorentzian correlator , which differs from the definition @xmath58 at @xmath59 by the factor @xmath53 in the numerator@xmath60 @xmath61 deserves a special comment .
this correlator is often considered as
unphysical .
its fourier image @xmath62 contains function @xmath63 that diverges logarithmically at long wavelengths @xmath64 .
the issue to what extent the correlators are physical and can be reproduced experimentally is irrelevant in our context . for us ,
the fact that the lorentzian correlator is sometimes used in calculations is sufficient enough to consider this correlator in the paper . to deal with the divergency
, one can truncate the lorentzian correlator at large distances ( the common practice is to make a cut - off at the distances about 0.1 of the system length @xcite ) .
another option is to use the generalized power - law correlator @xmath65 with small @xmath53 instead of the lorentzian @xmath66 . in order not to introduce additional parameters
, we use the untruncated equation @xmath62 .
even though the divergence of @xmath67 does not lead to any singularities in transport coefficients , the transport coefficients for lorentzian surfaces ( see below ) often behave qualitatively different from systems with other types of random inhomogeneities , even from the systems @xmath68 with small @xmath53 .
( sometimes , the divergence of the power spectrum @xmath56 is associated with the fractal nature of the surface @xcite ; to what extent our transport formalism can be used for films with fractal surfaces is an open question ) .
the last class of correlation functions covers the power - law correlators in momentum space , @xmath69 the correlators from this group include the lorentzian in momentum space @xmath70 that was observed in ref.@xcite ( see also ref_._@xcite ) and the exponential correlator @xmath71 at @xmath72 .
the constants in all these correlators are chosen in such a way that the value of @xmath73 is the same .
this provides a reasonable basis of comparison for transport coefficients in films with all these different types of random surfaces .
indeed , the scattering cross - section for @xmath64 does not depend on the details of short- and mid - range structure of surface inhomogeneities .
therefore , at fermi momenta @xmath74 ( more precisely , at @xmath75 ) , the transport coefficients should be the same for all random surfaces .
( the only exception is the lorentzian @xmath62 for which @xmath76 diverges at small @xmath77 ) . in what follows
we compare the transport properties of the films @xmath78 - @xmath79 in various ranges of the film thickness @xmath1 , correlation radius @xmath2 , and the particle wavelength @xmath80 ( or the @xmath5 particle density @xmath81 ) .
qse is caused by quantization of motion in the direction perpendicular to the film , @xmath82 , and leads to a split of the energy spectrum @xmath83 into a set of minibands , @xmath84 .
for simplicity , we consider spherical fermi surfaces @xmath85 , @xmath86 , \
q_{j}\equiv q_{fj}=\left [ 2m\epsilon _ { f}-\left ( \pi j / l\right ) ^{2}\right ] ^{1/2 } , \label{t1}\ ] ] where @xmath87 is the fermi momentum for the miniband @xmath88
. one can introduce the overall fermi momentum as @xmath89 the relationship between this fermi momentum @xmath90 and the @xmath5 density of fermions @xmath91 in quantized films is somewhat cumbersome @xcite : @xmath92 , \label{cc1}\ ] ] where @xmath93 is the number of the occupied minibands , @xmath94 \label{ee12}\ ] ] if the density of fermions is the same as in the bulk , then @xmath95 where @xmath96 is the usual bulk density . even in this case , the number of the occupied minibands @xmath93 , according to eqs . @xmath97,@xmath98is a complicated function of @xmath99 .
asymptotically , at large @xmath93 @xmath100 . \label{eee12}\ ] ] according to refs .
@xcite , scattering by random surface inhomogeneities results in intra- and interband transitions @xmath101 with transition probabilities @xmath102 that are expressed explicitly via the surface correlation function @xmath103 : @xmath104 \left ( \frac{\pi j}{l}\right ) ^{2}\left ( \frac{\pi j^{\prime } } { l}\right ) ^{2}. \label{rrr3}\ ] ] the generalization to other , more complicated energy spectra is straightforward @xcite .
the transport equation for the distribution functions @xmath105 @xmath106 \delta \left ( \epsilon _ { j{\bf q}}-\epsilon
_ { j^{\prime } { \bf q}^{\prime } } \right ) \frac{d^{2}q^{\prime } } { \left ( 2\pi \right ) ^{2 } } , \label{aa3}\ ] ] reduces , after standard transformations , to a set of linear equations @xmath107 \nonumber\end{aligned}\ ] ] where @xmath108 is the first angular harmonic of the distribution function @xmath109 at @xmath110 , and @xmath111 are the zeroth and first harmonics of @xmath112 over the angle @xmath113 . for some of the correlation function from the previous section the angular harmonics can be calculated analytically ( see appendix a ) . for others ,
this calculation is performed numerically .
the solution of eqs . @xmath114 provides the conductivity of the film : @xmath115 equations @xmath114 have simple analytical solution when the matrix @xmath116 can be approximated by a diagonal matrix , @xmath117@xmath118 this happens when the matrix @xmath119 is almost or exactly diagonal , @xmath120 and @xmath121 then the conductivity @xmath122 is equal to @xmath123 such a diagonalization occurs in three physical situations .
the simplest one is the one when only one miniband is occupied and @xmath124 the second case is the case of systems with large correlation length @xmath125 . in such systems
the intraband scattering is much stronger than the interband one and the off - diagonal matrix elements @xmath126 are small in comparison with the diagonal ones ( see appendix a )
. then both matrices @xmath127 are almost diagonal , @xmath128 and the expression for the conductivity @xmath129 reads @xmath130 such diagonalization of the matrices @xmath127 @xmath131 at @xmath125 can often be an oversimplification ( see section iv ) .
the third situation with diagonal @xmath116 is the case of small @xmath132 . in this limit ,
the correlation function is a constant with the zero first harmonic , @xmath133 according to eq.@xmath134 @xmath135 and @xmath136 note , that all our surface correlators @xmath20 are introduced in such a way that in the longwave limit @xmath137 they are , except for the lorentzian @xmath138 , equal to each other , @xmath139 this means that in this limit the conductivities @xmath140 are the same irrespective of the shape of the correlator , @xmath141 ( _ cf .
_ ref.@xcite ) . in all other situations
eqs.@xmath114 are not diagonal and should be solved numerically .
the results for conductivity ( mobility ) also provide the exponent in the expression for the localization length @xmath142 that describes localization caused by particle scattering by random wall inhomogeneities @xcite : @xmath143 \label{loc1}\ ] ] where @xmath144 is the mean free path and the diffusion coefficient @xmath145 is proportional to the conductivity @xmath0 .
as it is mentioned in introduction , the @xmath5 conductivity @xmath0 of the film has the dimensionality of conductance and is described by a dimensionless function @xmath146 in eq .
@xmath147 this function , in turn , depends on the relation between three length scales - particle fermi wavelength @xmath148 the width of the channel @xmath6 , and the correlation radius of the surface inhomogeneities @xmath2 .
the fourth length parameter , @xmath15 , is perturbative and enters conductivity as a coefficient , @xmath149 note , that we consider only the contribution from surface roughness and disregard bulk scattering . as a result ,
the conductivity @xmath150 diverges in the limit of vanishing inhomogeneities @xmath151 or @xmath152 .
the proper account of bulk scattering @xcite eliminates this divergence .
the dimensionless function @xmath153 depends only on the ratio of these three lengths . of three ratios @xmath154 , @xmath155 , and @xmath156 only two
are independent , @xmath157 . which two of these ratios should be used as independent dimensionless variables depends on whether one wants to display the dependence of @xmath0 on @xmath158 , @xmath1 , or @xmath2 .
the study of the dependence of the conductivity on film thickness , @xmath27 , should be performed at constant @xmath158 and @xmath2 .
this means that @xmath159 is best displayed by the function @xmath160 @xmath161 for various values of @xmath162 .
plots of the function @xmath163 at constant values of @xmath164@xmath165 reflect the dependence @xmath166 .
similarly , plots of the function @xmath167 at constant @xmath168@xmath169 characterizes the dependence of conductivity on density of particles @xmath81 or the fermi momentum @xmath90 .
below we compare these dimensionless functions , @xmath170 @xmath171 and @xmath167 for various types of correlation functions in wide ranges of parameters .
needless to say , the results at @xmath172 should coincide for all types of correlators except , maybe , for the lorentzian , since , by design , all the correlation functions are the same in this limit [ see eq .
@xmath173 .
curves in all figures below are labeled in a uniform way by the type of surface correlator used in calculations .
curves @xmath174 correspond to gaussian inhomogeneities @xmath175 , curves @xmath1 describe the surfaces with lorentzian correlations @xmath62 , curves @xmath176 , @xmath177 , and @xmath178 give the results for the correlators @xmath65 with @xmath179 , and curves @xmath180 , @xmath181 , and @xmath182 correspond to eq . @xmath183 with @xmath184 .
note , that correlator @xmath177 has the exponential power spectrum @xmath185 and that correlator @xmath181 is actually the exponential correlator @xmath71 .
figures 1 - 2 for the function @xmath186 , eq .
@xmath187 , show the dependence of the conductivity @xmath30 for two different values of @xmath188 , @xmath189 , for various types of the correlation functions .
the labeling of the curves @xmath190 is explained in the end of previous subsection .
the main feature of the curves , namely , their saw - like character , is well known .
the sharp drops occur when the number of the occupied minibands , eq@xmath191 @xmath192 changes by 1 , _
i.e. , _ in the points @xmath193 with integer @xmath194 the only unexpected feature is a wrong
periodicity of the initial part of the gaussian curve @xmath174 at small values of @xmath195 for @xmath196 ( see insert in figure 2 ) . this feature will be explained later .
the lorentzian curve @xmath1 is different from others : at @xmath196 the curve has already lost its qse structure .
= 3.4 in at these , relatively small values of @xmath197 the curves for all types of correlators have roughly the same shape though the exact values of the conductivity are different .
( the curves @xmath177 and @xmath181 are indistinguishable in both figures 1 and 2 , and curves @xmath174 and @xmath178 are indistinguishable in figure 1 ) . to underscore this point , in figures 3 and 4 we plotted instead of the curves @xmath198 the normalized curves @xmath199 with the normalization coefficients ensuring that the values of the normalized conductivity are equal to 1 at the highest values of @xmath195 in the plot .
strikingly , for @xmath200 ( figure 3 ) all the normalized curves with these 8 correlation functions lie within one bold line and are _ all _ indistinguishable with this resolution . for larger @xmath201 ,
the difference is still insignificant : all the curves are compressed between curves @xmath174 and @xmath1 . the only anomaly is the loss of qse structure by curve @xmath1 .
= 3.4 in = 3.4 in the main conclusion here is that the _ shape _ of the dependence @xmath30 at constant @xmath2 and @xmath90 is not sensitive to and can not provide any information on the type of the correlator at not very large values of @xmath188 . since @xmath15 is unknown and enters the conductivity as a coefficient , the absolute values of @xmath202 can not serve as a clue either : experimental data on @xmath30 at moderate @xmath188 can be fitted by any type of the correlator by a choice of @xmath15 . in this case , it is impossible to make any conclusion on the type of correlation function from transport measurements and it does not matter what correlator to use in theoretical calculations . meaningful analysis requires some _ _ beforehand _ _ information on the correlation parameters .
the only correlator that can be identified is the lorentzian ; however , this type of correlations is the least probable and might be unphysical .
the situation changes dramatically at higher @xmath162 as is shown in figures 5 [ function @xmath203 and 6 [ normalized function @xmath204 for the same 8 correlators ( the labeling of the curves is explained in the end of sec .
= 3.4 in we anticipated one feature , namely , the decrease in the amplitude of saw teeth with increasing @xmath201 and even the disappearance of such teeth for the gaussian correlator .
the sharp drops in conductivity in the points where the number of the occupied minibands @xmath93 increases by 1 is explained by opening of @xmath93 _ new _ scattering channels associated with interband transitions in and from this newly opened miniband . without the interband transitions , the increase of @xmath93 by 1 results not in a sharp drop in @xmath0 , but in an insignificant kink on the curve @xmath27 as it is shown in the third of refs.@xcite .
the interband transitions are described by the off - diagonal components of the matrix of transition probabilities @xmath205 . with increasing @xmath188
, these off - diagonal ( interband ) transition probabilities go to zero though with different rate for different types of the correlation function .
the rate of decrease of the interband transition probabilities as a function of @xmath188 for different correlation functions is discussed in the appendix .
this rate is a good predictor for observing the saw - like shape of @xmath27 .
the fastest decrease happens in the case of the gaussian correlator ; thus the curve for the gaussian correlator should be the smoothest and should exhibit the smallest traces of the saw teeth . therefore , the visibility of the saw teeth on the experimental curve can be a clue to the form of the correlation function .
= 3.4 in what is completely unexpected is the appearance of a new type of oscillation structure on @xmath27 in a limited range of @xmath195 for the gaussian and power - law correlators ( curves @xmath174 and @xmath206 in figures 5,6 ) .
it looks as if there is a transition between two distinct regimes with several sharp oscillations in the transition range .
the effect looks even more striking in figure 6 for the normalized curves which , in contrast to since figure 5 , is plotted in a linear scale .
this new type of qse requires an explanation .
these new oscillations are not related to abrupt changes in the number of occupied minibands @xmath207 : the oscillations are less sharp , have a much larger period , and , most important , appear only in a limited range of @xmath195 where the number of occupied minibands @xmath93 is already large .
these new oscillations are observed for the correlators for which the interband transitions are the smallest and the saw - like structure is suppressed , namely for the gaussian and power - law correlation functions . the power spectrum for these correlators
@xmath56 goes to zero exponentially at large @xmath77 .
then one would expect that the off - diagonal ( interband ) transition probabilities are exponentially small in comparison with intraband scattering and that the conductivity can be well described by the
approximation @xmath208 that does not have an oscillation feature .
this turns out not to be the case .
the oscillations are indeed related to off - diagonal ( interband ) scattering probabilities @xmath126 . a qualitative explanation of the effectand an estimate of the peak positions are the following .
scattering by surface inhomogeneities changes the tangential momentum by @xmath209 . according to the momentum conservation law ,
this scattering can cause the interband transition @xmath210 only when @xmath211 .
if the miniband index @xmath88 is relatively small and @xmath212 , then @xmath213 .
the energy conservation requires that @xmath214 .
the combination of these conservation laws defines the peak positions @xmath215 , which correspond to the opening of robust interband transitions @xmath210 and which are given by equations @xmath216 . in dimensionless variables ,
this is equivalent to @xmath217 accordingly , with increasing film thickness @xmath1 the transition channel opens first for the electrons in the lowest miniband @xmath218 with @xmath219 .
note , that these are the grazing electrons which are responsible for the dominant contribution to the conductivity .
thus , the conductivity drops almost by half at the film thickness @xmath220 where @xmath221 becomes comparable to @xmath222 and the effective cross - section doubles . at higher value of @xmath1 , @xmath223
, a new channel @xmath224 opens for the electrons from the next miniband @xmath225 with @xmath226 and the conductivity drops again , and so on .
the only difference is that the contribution of the electrons from the higher minibands falls rapidly with an increase in the band index @xmath227 and the drops in conductivity @xmath27 , which are associated with the opening of new scattering channels for electrons from these minibands , become smaller and smaller . the number of the visible peaks on the curve @xmath27 and their relative heights give a good visual estimate of the number of important minibands and of their relative contribution to the conductivity .
with further increase in the film thickness , when @xmath1 becomes large , @xmath228 , the change of momentum @xmath229 is sufficient to excite _ all _ interband transitions and the ordinary qse with the saw teeth at the points @xmath230 is restored . the above explanation works for the films with the exponential decay of the power spectrum of inhomogeneities in which the size of inhomogeneities @xmath2is well - defined . in the films with a non - exponential power spectrum of inhomogeneities , _
i.e. _ , with a more uniform distribution of inhomogeneities over the sizes @xmath2 in momentum space , this new size effect can not be observed because the particles from all minibands can always find the inhomogeneities of the right size that ensure the interband transitions irrespective of what is the separation between the walls .
= 3.4 in more accurate explanation is the following .
the off - diagonal elements @xmath231are functions of @xmath232 and rapidly decrease with increasing @xmath233 ( see appendix a ) . in general , the off - diagonal @xmath234 is large at large @xmath2 ( or @xmath201 ) while the diagonal elements @xmath235 .
however , for large @xmath195 ( large @xmath93 ) some of the elements @xmath234 with _ small _ @xmath236 which are close to the main diagonal , could become small even for large @xmath201 : @xmath237 ( @xmath88 changes from @xmath238 to int@xmath239 )
. then at large @xmath195 the transitions @xmath210 can become noticeable and eqs .
@xmath240 become coupled .
this coupling changes the solution of transport equation and , therefore , conductivity . according to eqs . @xmath114
the coupling between the minibands @xmath88 and @xmath241 becomes noticeable , @xmath242 , when @xmath243 at fixed @xmath201 , eq .
@xmath244 can be considered as the equation for the values of @xmath245 at which one can observe the opening of transitions @xmath210 .
the opening of such transition channels is accompanied by drops in conductivity . since for the gaussian and power - law
correlators the interband transition probabilities @xmath126 depend exponentially on parameters @xmath246 , these drops in conductivity are sharp and deep as illustrated in figures 5,6
. solutions @xmath247 of eqs .
@xmath248 are discussed in appendix b. at @xmath249 , @xmath221 is the first of transition probabilities to acquire the
normal order of magnitude . at @xmath250 ,
@xmath251 becomes noticeable , then @xmath252 , _ etc .
_ the amplitudes of the drops rapidly decrease with increasing @xmath88 . in the end ,
when several interband channels with @xmath253 are open , @xmath27 becomes smooth , but with a much lower slope than in its initial part .
the growth of transition probabilities for transitions @xmath254 does not result in new oscillations in @xmath27 . in the points @xmath255 where @xmath256 becomes large , @xmath257 , the states @xmath88 and @xmath258
are already strongly coupled via @xmath259 and @xmath260 . according to appendix b , eq.([apb4 ] ) , the positions of the drops for films with gaussian surface inhomogeneities are similar to eq .
( [ eq35 ] ) : @xmath261 ^{-1/4}. \label{w3}\ ] ] the values @xmath262 agree well with the positions of the conductivity drops on curve 1 of figures 5 and 6 .
for the surface with the power - law correlations of inhomogeneities @xmath263 the solution of eq.@xmath244 with logarithmic accuracy [ appendix b , eq.([apb9 ] ) ] again resembles eq .
( [ eq35 ] ) : @xmath264 \right\ } ^{\mu /2 + 1/4}\right ] .
\nonumber\end{aligned}\ ] ] this expression is barely sensitive to @xmath53 .
this almost complete independence of the peak positions from @xmath53 can be clearly seen in figure 6 .
the difference between this new type of size effect and the usual saw - like qse is dramatic .
the saw - like drops in conductivity for usual qse occur in the points @xmath265 with integer @xmath266 and are direct consequence of quantization of momentum in thin films .
the interband transitions are not germane to the existence or positions of this qse and are responsible only for the amplitude of the conductivity oscillations .
the drops in conductivity are equidistant with the period @xmath267 along the @xmath195 axis , _
i.e , _ are equidistant as a function of the film thickness .
in contrast to this , the new qse oscillations in figures 5,6 are not related directly to the quantization of momentum and are a consequence of the exponential opening of interband transitions between minibands with small quantum numbers at certain values of the film thickness .
the transitions in and out of higher minibands remain suppressed .
( in some sense , the effect resembles magnetic breakthrough between separated parts of the fermi surface in high magnetic fields ) .
the peaks are roughly equidistant if plotted against @xmath268 ; weak deviation from periodicity is due to a logarithmic terms in eqs . @xmath269 .
the period of the new qse is much larger than for the usual qse .
the large period of oscillations can open the way to direct observation of qse in transport measurements in metal films in which usual qse has atomic period and can hardly be observed .
there is a strong possibility that the conductivity oscillations reported in the last of refs .
@xcite are actually this new type of qse . the initial part of the curves @xmath174 , @xmath206 in figures 5,6 for @xmath30 is described analytically by eq.@xmath270 with appropriate values of @xmath271 from appendix a. this curve is close to the power law @xmath272 ( small @xmath273 depends on @xmath201 ) and to experimental data of the third ref .
@xcite . after the region of new qse oscillations ,
the curves are again smooth , but with a much smaller tangent .
we do not have an analytical description for this regime .
the numerical approximation can be done equally well by either @xmath274 with small @xmath275 ( @xmath275 also depends on @xmath201 ) or a quadratic expression @xmath276 .
this behavior explains the experimental data @xcite and the last ref .
@xcite . as a result ,
the power - law dependence of @xmath202 is qualitatively different for ultrathin and more thicker films .
this type of behavior is different from the earlier studied behavior of @xmath159 at small @xmath277 .
@xcite .
= 3.4 in the initiation of this new type of oscillations with a large period can be seen on the initial part of curve @xmath174 in figure 2 for @xmath278 with growing @xmath279 these new oscillations get overtaken by the standard qse .
the transition from standard to new qse is illustrated in figure 7 that contains normalized
curves @xmath174 for the gaussian inhomogeneities , @xmath280 , for @xmath281 .
it is clear from these curves how usual qse is replaced by new oscillations with increasing @xmath201 .
the transitional
curve for @xmath282 is especially interesting : it shows new qse at smaller @xmath195 and a restoration of standard qse at higher @xmath195 .
this restoration occurs when a noticeable number of interband transitions become open at higher @xmath195 .
it seems that such restoration does not happen on curves @xmath283 .
this impression is wrong .
such restoration indeed occurs for curves @xmath284 , but at values of @xmath195 that are much larger than those in the figure . at very large @xmath201 ,
all curves @xmath186 consist of four parts : rapid increase at small @xmath195 , region of new qse oscillations , smooth monotonic part , and the region of relatively smooth standard qse oscillations at the largest values of @xmath195 . with increasing @xmath201 , the amplitude of new qse oscillations and the length of region separating new and old qse increase rapidly .
the dependence of the conductivity on the correlation radius of surface inhomogeneities , @xmath166 , is best illustrated by the function @xmath285 , eq .
since the number of the occupied minibands @xmath93 does not depend on the correlation radius of inhomogeneities , the curves @xmath163 at constant @xmath195 do not exhibit the saw - like structure .
instead , the two main features are the presence of the minimum in @xmath163 and the step - like structure that corresponds to the oscillations in figures 5,6 . = 3.4 in the scattering of fermions by surface inhomogeneities
is most effective at @xmath287 , _
i.e. , _ at @xmath288 .
this leads to a minimum of the conductivity @xmath166 at such values of @xmath289 . at @xmath290
the particle wavelength is much larger than the size of surface inhomogeneities and the scattering is almost specular and does not contribute to the formation of the mean free path . in the opposite limit @xmath291 the walls
are flat on the particle length scale and surface scattering also does not limit the effective mean free path .
therefore , at @xmath292 the conductivity @xmath166 for non - divergent correlators is infinite in both limits @xmath293 and @xmath294 with a minimum around @xmath288 .
the curves @xmath295 close to this minimum are plotted for different correlators in figure 8 ( @xmath296 ; the labeling of the curves is explained in the end of sec .
it is important that the position of the minimum , its width , and even the order of magnitude of the function @xmath297 in the minimum are roughly the same for all types of surface correlators .
this is , probably , the most universal feature of the system .
the only correlator that does not display a well - defined minimum is @xmath298 with @xmath299 ( the lorentzian in momentum space ; curve @xmath180 ) .
this feature is related to the logarithmic divergence of this correlator in real space .
this feature is especially interesting because the surfaces with such inhomogeneities were observed in experiment @xcite .
the drops in @xmath27 at large @xmath245 , which are analyzed in the previous section ( figures 5,6 ) , correspond to the points @xmath300 on the curves @xmath163 .
the positions of these points @xmath300 are implicitly determined by eq . @xmath301 and @xmath302 for the gaussian and power - law correlations provided that @xmath303 .
these values of @xmath289 are far away to the right from the minimum in the curves @xmath304 and can not be presented in the same figures .
the feature that corresponds to the oscillations from the previous section is clearly seen as a set of steps in figure 9 for the same value of @xmath195 as in figure 8 , @xmath296 on curves @xmath174 and @xmath177 for gaussian and power - law inhomogeneities . for the surfaces with the gaussian inhomogeneities ,
the first interband transition @xmath221 becomes visible for @xmath296 at @xmath305 , the next one at @xmath306 , and so on . at these values of @xmath289 one
can see well - pronounced steps on the curve @xmath174 in figure 9 . the same feature , though barely discernible , is also observed for the power - law correlator @xmath177 . for comparison , curves @xmath307 @xmath308 and @xmath181 do not exhibit any anomalies .
interestingly , the curve for the lorentzian inhomogeneities is the only one that decreases with increasing @xmath289 after the initial increase at small @xmath289 ( figure 8) .
how is this feature related to the peculiarities of the lorentzian that have been discussed in section ii is unclear .
the curve @xmath180 remains essentially flat .
= 3.4 in the dependence of the conductivity @xmath0 on the density of fermions @xmath81 or their fermi momentum @xmath90 is best displayed by the function @xmath309 at constant @xmath310 , see eq .
this dependence @xmath312 is similar to @xmath30 .
the function @xmath312 exhibits a clear saw - like structure of usual qse at not very high @xmath289 for _ all _ correlators . with increasing @xmath289 , the saw teeth disappear first for the gaussian correlator @xmath174 , then for the power - law correlators @xmath206 , but persist for the power - law correlators in momentum space @xmath313 . instead ,
at large @xmath289 the functions @xmath314 for gaussian and power - law inhomogeneities exhibit new type of qse oscillations similar to that for @xmath186 in sec .
the positions of these oscillations can be found from eqs.@xmath315 after substitution @xmath303 .
= 3.4 in this effect is illustrated in figure 10 ( the labeling of the curves is explained in the end of sec .
the figure presents functions @xmath316 , eq.@xmath317 , for the gaussian ( curve @xmath174 ) and power - law ( @xmath318 ; curve @xmath177 ) correlators , and for the correlator with a power - law power spectrum ( @xmath319 ; curve @xmath181 ) . to compensate for different orders of magnitude of the data for these correlators , the functions are normalized by their values at @xmath320 , @xmath321 .
curve @xmath181 exhibits a saw - like behavior typical to the usual qse with period @xmath267 along the @xmath195-axis .
curves @xmath174 and @xmath177 exhibit new qse oscillations with a much larger period .
surprisingly , the possibility of interwall correlation of surface inhomogeneities gives an interesting insight into usual and new qse and provides an additional proof for our explanation of qse oscillations reported above .
the study of the effect of interwall correlation of inhomogeneities has been initiated in ref.@xcite for gaussian correlations .
below we supplement those results for other types of surface correlators with an emphasis on new qse . to decrease the number of parameters , we assume that , as in ref.@xcite , the correlation functions of inhomogeneities on both walls @xmath322 and @xmath42 are given by the same function , @xmath323 . the structure of the interwall correlator of inhomogeneities @xmath324 is assumed to be the same as for the intrawall correlations with the same correlation radius @xmath2 .
however , the amplitude @xmath325 of the interwall correlations is different from the intrawall ones , @xmath326 to compare the effect of such interwall correlations for different classes of the function @xmath20 , we calculate the relative change of conductivity @xmath0 ( _ i.e. , _ functions @xmath327 ) caused by introduction of such correlations @xmath328 where @xmath329 and @xmath330 are the functions @xmath331 calculated with and without interwall correlations .
an additional benefit is that the functions @xmath332 for all types of correlators are automatically normalized thus eliminating a difference by orders of magnitude between the functions @xmath331 for different types of correlation functions .
= 3.4 in in the presence of such interwall correlations , the transition probabilities @xmath333 @xmath334 become proportional , in accordance with @xcite , to @xmath335 \zeta \left ( \left| { \bf % q}_{j}-{\bf q}_{j^{\prime } } ^{\prime } \right|
\right ) . \label{i2}\ ] ] the most interesting effects of the interwall correlations are related to the oscillating structure of the term with @xmath325 in eq .
@xmath336 . if the interband transition probabilities @xmath337 are large , _ i.e _
, if @xmath338 is not small for @xmath339 , then the contribution of the term with @xmath325 in @xmath340 has a different sign for different @xmath205 depending on whether @xmath341 is even or odd .
this should result in an oscillating structure of the function @xmath342 @xmath343 as a function of the number of occupied minibands @xmath93 , _
i.e. , _ as a function of film thickness @xmath1 ( the existence of such oscillations was first reported in ref.@xcite for gaussian inhomogeneities ) .
the period of such oscillations should be equal to that for standard qse and their amplitude should decrease rapidly with increasing @xmath1 . since our explanation of the standard qse ties it to large interband transitions , the oscillation nature of the function @xmath342 @xmath343 should exist in the same range of parameters as the standard qse . in accordance with sec .
ivb , these oscillations should be noticeable for the function @xmath344 at small @xmath201 for _ all _ types of surface correlators .
this is illustrated in figure 11 ( @xmath200 ) for the correlators @xmath174 , @xmath1 , @xmath181 , @xmath177 .
the figure is plotted for @xmath345 .
the similarity of the functions @xmath346 is striking , but not surprising .
the flat part of all curves at small @xmath195 is explained below . at higher values of @xmath201 , the interband transitions
( off - diagonal @xmath126 @xmath347 ) become more and more suppressed .
when the interband transitions become negligible , the only non - zero scattering probabilities are diagonal @xmath348 that are proportional to @xmath349 \zeta \left ( \left| { \bf q}_{j}-{\bf q}_{j}^{\prime } \right| \right ) $ ] eq.@xmath336 .
since _ all _ @xmath350 are scaled by the same factor @xmath351and the conductivity is inversely proportional to @xmath271 , the function @xmath352 in the absence of the interband transition becomes a constant , @xmath353 if @xmath354 , the value of this constant is @xmath355 .
eq.@xmath356 also describes the initial part of all curves @xmath357 for all values of @xmath201 at small @xmath195 when only the first miniband is occupied , @xmath358 .
this explains all curves in figure 11 have identical flat pat at small @xmath195 .
= 3.4 in figure 12 illustrates @xmath359 at @xmath360 and @xmath354 for several correlators . at this value of @xmath201 , the exponential correlator @xmath181 @xmath71 exhibits , according to the results and explanation of sec .
ivb , the usual qse .
therefore , the function @xmath361 for this correlator should have an oscillation structure ; this is clearly seen in figure 12 .
the gaussian and power - law correlators @xmath174 and @xmath177 , according to sec .
ivb , ensure the absence of interband transitions at small and moderate @xmath195 where the function @xmath362 in figure 12 . our explanation for the new type of qse in sec .
ivb is an abrupt sequential appearance of noticeable interband transitions @xmath221 , @xmath224 , @xmath252 , _ etc .
_ at certain values of @xmath363 .
since for the term with @xmath325 in eq.@xmath340 is negative for all transitions @xmath364 , one should observe spikes in conductivity and , therefore , in the function @xmath365 , at @xmath366 . in some sense , figure 12 provides the best illustration for our explanation of new qse .
figure 12 also provides an insight into anomalous behavior of conductivity for lorentzian correlation of inhomogeneities @xmath367 , curve @xmath1 . at @xmath368 ,
the interband transitions are suppressed and @xmath369 . at higher @xmath195 ,
the interband transitions become more noticeable and start increasing , but very slowly .
why does the curve remain smooth when a sufficient number of transitions is already visible , is still a puzzle .
a possible explanation is that oscillations should appear only at very large @xmath93 ( or @xmath195 ) when their amplitude should be vanishingly small .
in summary , we compared the behavior of conductivity for various types of surface correlators in a wide range of parameters .
the following conclusions can be important when analyzing the experimental data or discussing theoretical predictions .
* the rough shapes of the curves of the transport coefficients are similar at small and moderate @xmath2 for _ all _ types of correlators though the orders of magnitude of the transport coefficients and more fine details of the curves can be different . to make any definite conclusions from the rough shapes of the experimental curves
, one should have at least some idea of the type of the correlation function of surface inhomogeneities and/or the value of the correlation radius @xmath2 and the amplitude of inhomogeneities @xmath15 .
since @xmath15 plays the role of a scaling parameter , getting the values of parameters of surface inhomogeneities from experimental data on transport without any additional information on the correlation of inhomogeneities could result in mistakes by orders of magnitude . in the same way , the use of the wrong correlator in theoretical calculations could result in absolutely wrong predictions without evoking any warning signals from comparison of the rough shapes of experimental and theoretical curves . *
the most universal feature is the shape of the curves and order of magnitude of @xmath166 near the minimum at @xmath370 .
this minimum allows experimental evaluation of the correlation length of surface inhomogeneities @xmath2 without any assumptions about the type of the correlation function . *
the shape of the curves @xmath27 , @xmath371 , and @xmath166 becomes very sensitive to the type of surface correlator at large correlation radius of inhomogeneities @xmath2 .
experimentally , this is important for better quality films ( see , for example , in ref .
@xcite ) in which stm and other usual methods are not well - suited for the study of the long - range behavior of the thickness fluctuations . here
transport measurements can be used as a good alternative for identification and analysis of the thickness fluctuations . *
the underlying reason is very high sensitivity of coupling between quantum well states with _ low _ quantum numbers to film thickness and the long - range behavior of the thickness fluctuations .
this phenomenon is quite general and should lead to observable effects not only in metal films , but for other types of quantum wells such as semiconductor films or quantum wave guides @xcite . * the persistence of the saw - like dependence of the transport coefficients on the thickness of the film , fermi momentum , or the density of fermions should signal the long range nature of the surface correlations in momentum space @xmath56 .
the observation of the saw - like structure for @xmath372 is a distinct signature of the power - law decay of the power spectral density function @xmath56 though , by itself , is insufficient to make conclusions about the index in this power law .
the easy suppression of the saw - like behavior points at the exponential decay of the power spectral density .
the rate of this suppression is significantly different for simple exponential and gaussian decays of @xmath373 .
* thickness fluctuations with gaussian correlations and correlations with exponential power spectrum lead to a new type of qse in @xmath202 , @xmath312 , and @xmath166 for surface inhomogeneities of a relatively large size @xmath2 .
this new qse produces large oscillations in @xmath27 and @xmath371 and steps in dependence @xmath166 .
the spacing between these new qse anomalies provides important direct information on the correlation parameters of inhomogeneities .
the peaks are almost equidistant if plotted against @xmath374 . * in contrast to the usual saw - like qse , the new qse oscillations are not related directly to the quantization of momentum and are a consequence of the exponential opening of interband transitions between minibands with small quantum numbers at certain values of the film thickness . in some sense , the effect is reminiscent of magnetic breakthrough that describes the opening of transitions between disconnected parts of the fermi surface .
* large period of new qse oscillations opens the way to direct observation of qse in conductivity of quantized metal films and may be responsible for experimental data in the second ref .
an additional experimental signature should be the appearance of these new qse oscillations only at relatively large values of the thickness of quantized metal films . *
the gaussian correlation of inhomogeneities affects particle transport in a unique way .
first , the values of the transport coefficient are , except for the smallest correlation radii , larger than for other , slower correlators by orders of magnitude .
this is explained by this correlator having the shortest tails resulting in the least effective scattering .
second , this type of correlation does not exhibit a saw - like dependence of the transport coefficients on the system parameters except for small correlation radii @xmath2 .
third , this type of correlation of the surface inhomogeneities leads to the above - mentioned new type of large - scale oscillations of the transport coefficients .
the combination of these features can make the gaussian correlator readily identifiable in transport experiments . *
the lorentzian correlation of inhomogeneities in configuration space is also readily identifiable by several abnormal features .
the combination of these features could be another manifestation of an
unphysical nature of this correlator .
if possible , this correlator should be avoided in theoretical and computational models . a power - law correlator @xmath375 with small index @xmath53 can serve as a good replacement in the calculations . *
the results explain the observed difference in power - law regimes of the thickness dependence of the conductivity @xmath27 between ultrathin and more thicker films . *
the relative contribution of the interwall correlation of surface inhomogeneities strongly depends on the type of qse . for usual qse ,
the contribution of the interwall correlations is a rapidly decaying oscillation function of the film thickness .
for qse of the new type , this contribution is constant in a wide range of small and moderate thicknesses , and becomes an oscillating function with a big period in a limited range of large thicknesses .
the work is supported by nsf grant dmr-0077266 .
various correlation functions from section ii allow different degrees of analytical calculations of the scattering probabilities .
the angular harmonics of the correlation function @xmath376 in the transport equation @xmath377 are defined as @xmath378 where @xmath379 is the angle between the @xmath5 vectors @xmath380 and @xmath381 .
the harmonics for the gaussian correlator @xmath175 are @xmath382 e^{-\left ( q - q^{\prime } \right ) ^{2}/2 } , \label{ap1 } \\
\zeta ^{\left ( 1\right ) } \left ( q_{j},q_{j^{\prime } } \right ) & = & 4\pi \ell ^{2}r^{2}\left [ e^{-qq^{\prime } } i_{1}\left ( qq^{\prime } \right ) \right ] e^{-\left ( q - q^{\prime } \right ) ^{2}/2 } \nonumber\end{aligned}\ ] ] where @xmath383 , @xmath384 .
note , that in refs.@xcite we used equivalent expressions with hypergeometric functions instead of modified bessel functions .
expressions in square brackets in eqs .
@xmath385 are smooth functions of @xmath386 and @xmath387 .
the exponential coefficients , @xmath388 $ ] , on the other hand , are rapidly going to zero for large @xmath132 if @xmath389 .
this explains why the off - diagonal scattering probabilities @xmath126 are much smaller than the diagonal ones at large @xmath132 .
such a drastic difference between interband and intraband scattering probabilities is a unique feature of the gaussian correlator .
the physical consequences are discussed in section iv . for the exponential correlator @xmath71 the harmonics are @xmath390 \sqrt{1+\left ( q+q^{\prime } \right ) ^{2 } } } \label{ap2 } \\
\zeta ^{\left ( 1\right ) } \left ( q_{j},q_{j^{\prime } } \right ) & = & \frac{4\ell ^{2}r^{2}}{qq^{\prime } } \frac{\left ( 1+q^{2}+q^{\prime 2}\right ) e\left ( \omega \right ) -\left ( 1+\left ( q - q^{\prime } \right ) ^{2}\right ) k\left ( \omega \right ) } { \left [ 1+\left ( q - q^{\prime } \right ) ^{2}\right ] \sqrt{% 1+\left ( q+q^{\prime } \right ) ^{2 } } } , \nonumber \\ \omega & = & 2\sqrt{qq^{\prime } /\left [ 1+\left ( q+q^{\prime } \right ) ^{2}% \right ] } , \nonumber\end{aligned}\ ] ] where @xmath391 and @xmath392 are complete elliptic integrals .
here the diagonal and off - diagonal transition probabilities ( probabilities of the intraband and interband scattering ) differ mainly by the terms @xmath393 in denominator that are insignificant in comparison with the exponential factors for the gaussian correlator above .
the physical consequences are discussed in section iv .
the power - law @xmath65 correlation functions correspond to @xmath394 ^{\mu } \,d\phi \nonumber \\ \zeta ^{(1 ) } & = & 4\ell ^{2}r^{2\,}\sum_{m=0}^{\infty } \left ( \mu + m\right ) \frac{k_{\mu + m}\left ( q_{\max } \right ) } { q_{\max } ^{\mu } } \frac{i_{\mu + m}\left ( q_{\min } \right ) } { q_{\min } ^{\mu } } \nonumber \\ & & \times \int_{0}^{2\pi } c_{m}^{\mu } ( \cos \phi ) \left [ q^{2}+q^{\prime 2}-2qq^{\prime } \cos \phi \right ] ^{\mu }
\cos \phi \,d\phi \nonumber\end{aligned}\ ] ] where @xmath395 are the ultraspherical ( gegenbauer ) polynomials , and @xmath396 and @xmath397 .
the off - diagonal transition probabilities disappear exponentially at large @xmath398 , approximately as @xmath399 , _
i.e. , _ much slower than for the gaussian correlator @xmath400 but faster than for the correlator @xmath401 .
the integrals in eqs.@xmath65 can be simplified for the lorentzian correlator@xmath402 @xmath403 note , that the function @xmath404 diverges logarithmically at @xmath405 .
this divergence is discussed in sections ii and iv .
the expressions for the harmonics @xmath406 can also be simplified for the staras correlator , @xmath407 when @xmath408 /\sin \phi $ ] , @xmath409 , \\ \int_{0}^{2\pi } c_{m}^{1}\left ( \cos \phi \right ) \cos \phi \,d\phi & = & \left [ 0,\text { } m=2k\text { ; } 2\pi , \text { } m=2k+1\right ] , \end{aligned}\ ] ] and the harmonics @xmath406 reduce to the rapidly converging sums of the bessel functions with alternating coefficients .
for all other power - law correlators with different values of @xmath53 the integration should be performed numerically .
the last group of correlators involves power - law behavior in momentum space , eq .
this group includes the lorentzian in momentum space @xmath299 that was observed in ref.@xcite and the exponential correlator @xmath410 at @xmath411 . in general , the angular harmonics are @xmath412 ^{\left ( 1+\lambda \right ) /2}}p_{\lambda } \left ( \omega \right ) , \label{ap5 } \\ \zeta ^{\left ( 1\right ) } & = & \frac{4\pi \ell ^{2}r^{2}/\lambda } { \left [ 1+\left ( q^{2}-q^{\prime 2}\right ) ^{2}+2\left ( q^{2}+q^{\prime 2}\right ) % \right ] ^{\left ( 1+\lambda \right ) /2}}p_{\lambda } ^{1}\left ( \omega \right ) , \nonumber \\
& = & \left ( 1+q^{2}+q^{\prime 2}\right ) /\sqrt{1+\left ( q^{2}-q^{\prime 2}\right ) ^{2}+2\left ( q^{2}+q^{\prime 2}\right ) } \nonumber\end{aligned}\ ] ] where @xmath413 are the associated legendre functions of the first kind .
note , that the argument @xmath414 of the legendre functions in our expressions is larger than @xmath238 .
one should be cautious when doing calculations with expressions @xmath415 : some of the handbooks ( and software packages , _
e.g. , mathematica _ ) do not use the same normalization for legendre polynomials and legendre functions , _
i.e. , _ for functions @xmath416 with integer and non - integer @xmath417 . in the case of the lorentzian in momentum space , @xmath418 @xmath419 the harmonics @xmath420 note ,
that this correlator diverges in real space at @xmath421 .
the peak positions are determined by the condition that the absolute value of the diagonal and the first off - diagonal matrix elements in transport equation ( [ ee6 ] ) become comparable : @xmath422 rewriting this condition via transition probabilities @xmath423 we get @xmath424 + \sum_{j^{\prime } \neq j}w_{j , j^{\prime } } ^{\left ( 0\right ) } \left ( x , z\right ) \sim w_{j , j+1}^{\left ( 1\right ) } \left ( x , z\right ) , \label{apb1}\ ] ] where @xmath425 are the zeroth and first harmonics of @xmath426 over the angle @xmath427 that can be expressed explicitly via the surface correlation functions [ see eq .
( [ rrr3 ] ) and appendix a ] . for large @xmath428 ,
the off - diagonal scattering probabilities @xmath126 are exponentially suppressed for gaussian and power - law inhomogeneities , eqs .
( [ ap1 ] ) , ( [ ap3 ] ) : @xmath429 . with a logarithmic accuracy
, the condition ( [ apb1 ] ) corresponds to the equation @xmath430 where @xmath432 .
when @xmath433 , we can put @xmath434 in the denominator .
the exponent should be evaluated more carefully : @xmath435 then eq . ( [ apb3 ] ) yields the following values of the peak positions : since @xmath303 , these peak positions @xmath247 can be also used to get the peak positions for the conductivity at fixed @xmath289 , @xmath437 as a solution of the following algebraic equation : @xmath438 ^{1/2}}% .
\label{apb5}\ ] ] similar but more cumbersome calculations , can be performed for the power - law correlators ( [ ee2 ] ) .
for example , if @xmath55 , eq .
( [ apb2 ] ) reads @xmath439 where we introduced @xmath440for large @xmath441 an asymptotic estimate for the integral in the left - side is @xmath442 .
rough asymptotic estimate for the integral in the right - side of the equation is @xmath443 in order to estimate this integral , we can substitute @xmath444 by @xmath445 then @xmath446 this leads to the following estimate for peak positions @xmath447 where @xmath448 is the root of the transcendental equation @xmath449 the last equation can be solved by iterations : @xmath450 , ... \;\end{aligned}\ ] ] finally , with a logarithmic accuracy , the solution of eq . ( [ apb6 ] ) for the positions of peaks becomes @xmath451 } } .
\label{apb8}\ ] ] similar asymptotic estimates for the power - law correlators with arbitrary @xmath452 yield m. jalohowski , h. hoffman , and e. bauer , phys.rev.lett . * 76 * , 4227 ( 1996 ) ; * 51 * , 7231 ( 1995 ) ; l. a. kuzik , yu .
e. petrov , f. a. pudonin , and v. a. yakovlev , sov.phys .
- jetp * 78 * , 114 ( 1994 ) ; g. m. mikhailov , i. v. malikov , and a. v. chernykh , jetp lett . *
66 * , 725 ( 1997 ) j. j. paggel , t. miller , and t. c. chang , science , * 283 * , 1709 ( 1999 ) ; d. a. evans , m. alonso , r. cimino , and k. horn , phys.rev.lett . * 70 * , 3483 ( 1993 ) ; j. e. ortega , f. j. himpsel , g. j. mankey , and r. f. willis , phys.rev .
b*47 * , 1540 ( 1993 ) m. jalohowski , e. bauer , h. knoppe , and g. lilienkamp , phys.rev .
b * 45 * , 13607 ( 1992 ) ; m. jalohowski , h. hoffman , and e. bauer , phys.rev .
b * 51 * , 7231 ( 1995 ) ; h. sakaki , t. noda , k. hirakawa , m. tanaka , and t. matsusue , appl .
lett . * 51 * , 1934 ( 1987 ) ; l .- w .
tu , g. k. wong , and j. b. ketterson , appl .
. lett . * 55 * , 1327 ( 1989 ) j. henz , h. von knel , m. ospelt , and p. wachter , surf .
sci . * 189/190 * , 1055(1987 ) ; j. y. duboz , p. a. badoz , e. rosencher , j. henz , m. ospelt , h. von knel , , and a. briggs , appl .
. lett . * 53 * , 788 ( 1988 ) j. j. paggel , t. miller , and t. c. chang , phys.rev.lett .
* 81 * , 5632 ( 1998 ) ; f. patthey and w .- d .
schneider , phys.rev .
b * 50 * , 17560 ( 1994 ) ; m. schmid _
et al . _ ,
lett . * 76 * , 2298 ( 1996 ) ( 1988 )
|
the effect of random surface roughness on quantum size effect in thin films is discussed .
the conductivity of quantized metal films is analyzed for different types of experimentally identified correlation functions of surface inhomogeneities including the gaussian , exponential , power - law correlators , and the correlators with a power law decay of the power density spectral function .
the dependence of the conductivity @xmath0 on the film thickness @xmath1 , correlation radius of inhomogeneities @xmath2 , and the fermion density is investigated . the goal is to help in extracting surface parameters from transport measurements and to determine the importance of the choice of the proper surface correlator for transport theory .
a new type of size effect is predicted for quantized films with large correlation radius of random surface corrugation .
the effect exists for inhomogeneities with gaussian and exponential power spectrum ; if the decay of power spectrum is slow , the films exhibit usual quantum size effect .
the conductivity @xmath3 exhibits well - pronounced oscillations as a function of the channel width @xmath1 or the density of fermions , and large steps as a function of the correlation radius @xmath4 these oscillations and steps are explained and their positions identified .
this phenomenon , which is reminiscent of magnetic breakthrough , can allow direct observation of the quantum size effect in conductivity of nano - scale metal films .
the only region with a nearly universal behavior of transport is the region in which particle wavelength is close to the correlation radius of surface inhomogeneities .
|
shock fronts or shock waves occur when disturbances in a medium propagate faster than the local speed of sound .
shocks are characterized by nearly discontinuous changes in the characteristics of the medium , like pressure , temperature , energy etc .
the width of the discontinuity is quite infinitesimal in comparison to the actual system , and therefore it is assumed to be a single propagating front .
the relativistic magneto - hydrodynamic ( mhd ) theory of shock waves is being applied to problems of cosmology and relativistic heavy - ion collisions . in the field of cosmology
the shock waves are usually collision - less mhd shocks and are thought to be mostly responsible for the acceleration of particles in a variety of astrophysical objects ranging from active galactic nuclei ( agn ) to gamma ray bursts ( grb ) ( known as first - order fermi acceleration or diffusive shock acceleration ) .
this phenomenon occurs when the charged particles in the plasma interact with the magnetic fields in the shock layers , and are repeatedly transported back and forth across the shock and thereby gaining energy .
the generation of such collision - less shocks have been extensively studied in the literature @xcite .
the commonly observed earth bow shock indicate that the velocity anisotropic distributions ( vad ) of the particles play an important role in the shock formation .
observation with the new generation of air cherenkov tev @xmath0-ray telescopes such as hess , magic and veritas @xcite have established the fact that agns , microquasars and pulsar wind nebulae are powerful sources of high - energy photon radiation .
they generate relativistic jets of particles , which collide with the surrounding intergalactic and interstellar medium to give rise to high energy tev @xmath0 radiation .
this is the currently accepted hypothesis .
however , how the collision - less shock is generated is still not well understood .
shock waves are usually created under rapid compression of matter . however , a similar discontinuity in matter can also be generated if the system suddenly expands and there is a phase transition ( pt ) @xcite .
taub was the earliest one to study the relativistic hydrodynamic shocks , by writing the conservation equation across the shock boundary also called rankine - hugoniot equation .
de hoffmann and teller were first to study the magnetized shock @xcite .
this led the way for the study of mhd shocks for the following years @xcite .
lichnerowicz extended the analysis to the relativistic case @xcite , where he treated the matter to be an ideal fluid with an infinite conductivity .
the more recent important studies of mhd shocks @xcite used relativistic equation of state to obtain the solutions from the conservation condition .
the conservation condition of mhd shocks in a gyrotropic fluid has also been extensively studied @xcite .
furthermore , several studies of mhd shocks , the first - order fermi acceleration and its connection with the astrophysical observation have also been performed @xcite .
in the above mentioned astrophysical scenarios , the shock wave propagates with a velocity less than the velocity of light .
that is , the normal vector of the surface of the discontinuity , is space - like .
however , it may not always be the case . in some situations there
may be a fast pt ( first order ) where the normal vector to the surface of the discontinuity can be time - like @xcite .
a system undergoing rapid and homogeneous rarefaction , bubbles at different spatial points are formed which are causally unconnected to each other .
for such cases the phase boundary separating the two phases of matter becomes time - like .
if the thickness of the time - like surface depends on the rate of formation of the bubbles and its growth , and if it is sufficiently thin , we can assume that the pt happens along an approximate structureless time - like surface .
an example of such type of pt is the hadronization of a supercooled quark - gluon plasma ( qgp ) in a heavy - ion collision .
as the qgp fireball expands , it cools , and well below critical temperature the qgp hadronizes .
the time - like shock hadronization in connection with heavy - ion collision has been studied extensively @xcite .
the inflationary universe model can be thought to be a cosmological example .
the main aim of this work is to study both the space - like and time - like shock in a relativistic mhd formalism . in our specific calculation
, we assume that the shock converts hadronic matter to quark matter , and so the shock has on one side hadronic matter and on the other side quark matter .
we treat the shock as a generalised oblique shock both in the de hoffmann teller frame and normal incidence frame .
such shock studies can be important both to the study of shocks in astrophysics and in heavy - ion collisions . in the astrophysical scenario
such shock and pt can occur in neutron stars which have huge inner magnetic fields . on the other hand , in the field of heavy - ion collisions such scenario can also occur when extremely large magnetic field are generated by colliding particles . in the next section
we write the conservation equations for the time - like and space - like shock front in the de hoffmann frame . in section
iii we give the transformation equation from the fluid frame to the normal incidence frame . in section
iv we present our results . in the final section ( section v )
we discuss our results and their applicability .
we denote the surface of discontinuity as @xmath1 and the normal to the surface as @xmath2 .
the normalization condition is given as @xmath3 the energy momentum tensor of the system reads @xmath4 where , @xmath5 is the enthalpy ( @xmath6 ) , @xmath7 being the energy density and @xmath8 being the pressure . @xmath9
is the 4-velocity of the fluid , and is normalized such that @xmath10 and @xmath0 is the lorentz factor .
@xmath11 is the metric tensor and is @xmath12 using the flat space - time convention .
the conservation conditions are nothing but the energy - momentum and baryon number conservation laws , across the discontinuity of the shock surface .
we denote @xmath13 as the initial state ahead of the shock front and @xmath14 as the final state behind the shock .
the general derivation of the shock wave can be found in @xcite and also in many subsequent literature @xcite .
one of the other important constraints of the transition is the thermodynamic stability condition , which requires non - decreasing entropy @xcite @xmath15 where , @xmath16 and @xmath17 are the entropy densities before and after the shock .
the generalised conservation conditions can be written as @xmath18 the relativistic conservation conditions for the space - like ( sl ) and time - like ( tl ) shocks are derived from the above generalised equations . closely following ref .
@xcite , the equation reads \a .
space - like @xmath19 solving the equations we get the solution of the downstream and upstream velocities , which are @xmath20 \b .
time - like @xmath21 the corresponding downstream and upstream velocities are @xmath22 comparing the matter velocities for the tl and sl shocks , we arrive at the equation @xmath23 where , @xmath24 stands for tl shocks and @xmath25 for sl shocks , respectively .
next , we address the conservation equation for the mhd shocks . in this case
, the energy - momentum tensor has both matter and magnetic contributions .
we assume an ideal infinitely conducting fluid , therefore the electric field vanishes .
the total energy momentum tensor is given by @xmath26 the @xmath27 represents the matter part of the tensor and @xmath28 the magnetic part .
the magnetic part of the tensor is defined as @xmath29 where , @xmath30 is the magnetic field vector .
the shock conservation condition or the relativistic rankine - hugoniot condition is very difficult to solve for the mhd shocks . to make it more manageable we go to the de hoffmann teller frame ( ht ) @xcite . the ht frame is the shock rest frame , where there is no @xmath31 drift electric fields .
therefore , for the subluminal flows the ht frame is an obvious choice where the conservation conditions are reduced to simple forms .
the system of equations are then transformed from the ht frame to normal incidence ( ni ) frame , with a boost .
the ni frame is a frame where the incident velocity is normal to the shock front .
[ fig1 ] gives the pictorial description of the two frames .
we then have a system of simple simultaneous equations in which the imaginary terms in the superluminal shocks are absent . here
we should mention that our approach of this frame transformation is similar to the one used by ballard & heavens and summerlin & baring @xcite . in this work
we adopt the following conventions : the ht frame quantities do not have any subscript , whereas the ni frame quantities are labelled by a subscript @xmath25 .
the angle between the magnetic field and the shock normal in the ht frame is denoted by @xmath32 and that in the ni frame by @xmath33 .
correspondingly , the angle between flow velocities and the shock normal is denoted by @xmath34 for the ht frame and by @xmath35 in the ni frame . along with the system of conservation equations we have also a set of equations of state ( eos ) describing the matter phases in the upstream and downstream .
we also assume that in the ht frame the fluid flows along the magnetic lines and there is no @xmath31 electric fields .
the four matter jump conditions are given by the conservation of baryon number , momentum ( 2 components ) and energy density across the front .
the electromagnetic jump condition is given by the equation @xmath36 the last term arises trivially because @xmath37 holds everywhere .
thus we define the oblique shock conservation condition .
first we write the conservation condition in the ht frame . in this frame the magnetic field and the matter velocities are aligned .
let us assume that the @xmath38-direction defines the normal to the shock plane .
the magnetic field is constant and lies in the @xmath39 plane .
therefore the velocities in the @xmath38 and @xmath40 direction are given by @xmath41 and @xmath42 , respectively .
similarly the magnetic fields in the @xmath38 and @xmath40 direction are given by @xmath43 and @xmath44 .
the lorentz factor is defined as @xmath45 with that the conservation conditions follow .
space - like @xmath46 \b .
time - like @xmath47 for the ht frame we also have @xmath48 with the assumption of infinite conductivity , the electric field is zero .
the maxwell equation @xmath49 , defines that there are no monopoles and so we have @xmath50 0.2 in cc + however , this equations are valid only for subluminal flows . for broader applicability ( including superluminal flows ) , we now transform the equations to the ni frame .
as done by summelin & baring and kirk & heavens @xcite , we arrive from the local fluid frame to the ni frame by a boost of @xmath51 along the @xmath38 direction and from the ni frame to the ht frame by a boost of @xmath52 in the @xmath40 direction .
the shock planes of the ht and ni frames are coincident . in the ni frame ,
as the upstream velocity is normal incident to the shock front the @xmath40 component of the velocity is zero ( @xmath53 ) .
the transformation of the velocities can be written as @xmath54 the two parameters that connect the quantities of the ht frame with the ni frame are defined as @xmath55 the angles of the ni and the ht frame are connected by the relation @xmath56 the relationship between the magnetic field component in the two frames are given by @xmath57 0.2 in cc .
@xmath58,@xmath59 , @xmath60 , @xmath61 and @xmath62 are shown for two different values of magnetic field ( @xmath63 g and @xmath64 g ) .
the incident velocity is @xmath65 and the incident magnetic angle @xmath66 is @xmath67 . in fig .
2a , the full lines are for the @xmath38 component of the velocity and the dotted for the @xmath40 component . in fig 2c ,
the full curves are used for downstream magnetic angle and the dotted curves are for flow velocities .
we follow this convention throughout the paper . ] .
@xmath58,@xmath59 , @xmath60 , @xmath61 and @xmath62 are shown for two different values of magnetic field ( @xmath63 g and @xmath64 g ) .
the incident velocity is @xmath65 and the incident magnetic angle @xmath66 is @xmath67 . in fig .
2a , the full lines are for the @xmath38 component of the velocity and the dotted for the @xmath40 component . in fig 2c ,
the full curves are used for downstream magnetic angle and the dotted curves are for flow velocities .
we follow this convention throughout the paper . ]
@xmath58,@xmath59 , @xmath60 , @xmath61 and @xmath62 are shown for two different values of magnetic field ( @xmath63 g and @xmath64 g ) .
the incident velocity is @xmath65 and the incident magnetic angle @xmath66 is @xmath67 . in fig .
2a , the full lines are for the @xmath38 component of the velocity and the dotted for the @xmath40 component . in fig 2c ,
the full curves are used for downstream magnetic angle and the dotted curves are for flow velocities .
we follow this convention throughout the paper . ]
+ now we need the relations transform brings the field angles from the fluid frame to the ni frame .
these equations are given by @xmath68 where , @xmath69 and @xmath70 respectively .
the downstream flow angle is the same for fluid and ni frame ( @xmath71 ) .
we solve the four conservation conditions in the ht frame .
subsequently , we transform the solution to the ni frame and obtain the unknown solutions in this frame .
the input to the equations are @xmath72 and @xmath73 .
the solution of the set of equations defines @xmath74 .
we also have both the upstream and downstream eos that relate @xmath8 and @xmath7 in the two phases .
the importance of this methodology lies in the fact that the equation now has thermodynamic quantities relating the upstream and downstream components and the transformation velocities to get to the ni frame from the fluid frame .
this method is useful for the fact that it removes all the imaginary and unphysical quantities in the superluminal shock and there is a smooth mathematical transition of the shock from the subluminal to the superluminal regime .
0.2 in cc , @xmath59 , @xmath60 , @xmath61 and @xmath62 are plotted against density @xmath75 for two different upstream magnetic angles ( @xmath76 and @xmath77 ) .
the incident velocity is @xmath65 and the magnetic field strength is @xmath63 g .
this figure shows curves for sl shocks . ] , @xmath59 , @xmath60 , @xmath61 and @xmath62 are plotted against density @xmath75 for two different upstream magnetic angles ( @xmath76 and @xmath77 ) .
the incident velocity is @xmath65 and the magnetic field strength is @xmath63 g .
this figure shows curves for sl shocks . ] + , @xmath59 , @xmath60 , @xmath61 and @xmath62 are plotted against density @xmath75 for two different upstream magnetic angles ( @xmath76 and @xmath77 ) .
the incident velocity is @xmath65 and the magnetic field strength is @xmath63 g .
this figure shows curves for sl shocks . ]
+ in this work we have assumed constant magnetic field at all densities .
concentrating on the scenario of a hadron - to - quark pt , we have used a hadronic nonlinear walecka model eos @xcite to describe the upstream quantities and mit quark bag model eos @xcite to describe the downstream quantities , respectively .
the two eos are standard choices and their detailed description can be found in previous literature .
we assume that the shock front induces a pt from normal hadronic mater to quark matter .
the input quantities are the upstream pressure , baryon number density , the magnetic field and the angle between the magnetic field and the shock normal .
the effect of the magnetic field is quite negligible if the field strength is less than @xmath78 g . if it is larger it affects both downstream velocities and the downstream magnetic and flow angles .
first we analyse the sl shock . for a fixed upstream velocity ( @xmath65 in units of @xmath79 ) and a fixed magnetic angle ( @xmath80 ) , as we increase the magnetic field , the downstream @xmath38 component of the velocity decreases slightly ( usually at low densities , fig .
[ fig2 ] ) .
however , the @xmath40 component of the downstream velocity increases with increasing magnetic field , but the sign is negative , which means that the angle between the downstream flow velocity and the shock normal is negative .
the positive and negative directions / angles convention are chosen with respect to the angle between the incident velocity and the incident magnetic field ( shown in fig .
[ fig1 ] ) .
we define a quantity @xmath81 , which is the velocity compression ratio . at low density
@xmath60 increases with increase in magnetic field .
the magnetic field acts as an extra pressure term which enhances the compression of the medium .
next we show the downstream magnetic and flow angles . as can be seen in fig .
[ fig2]b , @xmath60 is greater than @xmath82 ( other that at very low densities ) .
this shows that we have velocity compression due to the pt . at higher densities @xmath60 saturates and
is quite independent of the magnetic field strength .
as the @xmath40 component of the velocity is negative the flow angle is also negative , that is , the downstream velocity vector points downwards with respect to the shock normal .
the flow angle becomes more negative with increasing magnetic field , which points to the fact that the velocity anisotropy in the downstream flow is directly proportional to the magnetic field .
the downstream magnetic angle is also slightly negative and becomes more negative with increasing magnetic field strength . in the plots ( fig .
[ fig2 ] ) we have also shown shocks occurring at different densities
. however , we should not confuse them to be some kink of temporal evolution . as the density increases the flow velocity decreases ( both @xmath38 and @xmath40 component ) .
this is because as matter gets denser , it generates a much higher damping effect for the velocities .
the parameter @xmath60 increases with density , because for the same value of incident velocity the downstream velocity reduces with increasing density .
the flow angle first increases because the @xmath38 component of the downstream velocity decreases much faster than the @xmath40 component , but after some density ( @xmath83 times saturation density ) it decreases , because from there on @xmath40 component decreases much faster than the @xmath38 component . with increasing density the magnetic angle tends to move from negative to positive values .
0.2 in cc , @xmath59 , @xmath60 , @xmath61 and @xmath62 are plotted against density @xmath75 for two different incident velocities ( @xmath84 and @xmath85 ) .
the incident magnetic angle is @xmath67 and the magnetic field strength is @xmath63 g .
] , @xmath59 , @xmath60 , @xmath61 and @xmath62 are plotted against density @xmath75 for two different incident velocities ( @xmath84 and @xmath85 ) .
the incident magnetic angle is @xmath67 and the magnetic field strength is @xmath63 g . ] + , @xmath59 , @xmath60 , @xmath61 and @xmath62 are plotted against density @xmath75 for two different incident velocities ( @xmath84 and @xmath85 ) .
the incident magnetic angle is @xmath67 and the magnetic field strength is @xmath63 g . ] + in fig .
[ fig3 ] , we plot curves for a fixed magnetic field ( @xmath63 g ) and for a fixed upstream velocity ( @xmath65 ) with different incident magnetic field inclination ( @xmath66 ) . for smaller incident angle ,
the @xmath38 component starts with a high value ( @xmath86 ) and then decreases with density , saturating at higher densities . for larger incident angle ,
the behaviour is quite opposite , it starts with some low value ( @xmath87 ) and increases with density .
it attends a peak ( @xmath88 ) and then decreases and saturates at higher densities .
the saturation value is quite similar for a different incident magnetic angle . as , the @xmath38 component of the velocity shows such a behaviour
, the @xmath60 also reflects its behaviour . for larger incident angle
it first decreases and then increases with density .
however , the value is always greater than @xmath82 .
the @xmath40 component of the velocity increases ( becomes more negative ) with increase in incident angle .
the incident magnetic angle adds to the anisotropy of the downstream velocity .
this can be seen from the flow angle , which increases with increase in upstream magnetic angle .
the downstream magnetic angle ( @xmath61 ) is directly proportional to the upstream magnetic angle ( @xmath89 ) .
as the density increases the downstream magnetic angle increases , however , the flow angle decreases . now keeping the magnetic field ( @xmath63 g ) and incident angle fixed ( @xmath67 ) ( fig .
[ fig4 ] ) , if we increase the incident velocity the downstream @xmath38 component of the velocity increases and saturates at larger densities .
the @xmath40 component of the downstream velocity for small incident velocity is small and does not decrease much .
however , for large incident velocity , initially @xmath40 is large , but decreases very fast and saturates at a value very close to @xmath90 .
@xmath60 for smaller incident velocity is slightly larger than for larger incident velocity .
@xmath60 increases very fast initially and saturates at higher densities . as the incident velocity increases the anisotropy in the downstream velocity decreases
this can be seen by the fact that the flow angle decreases with increasing upstream velocity . the magnetic angle ( which is positive )
increases with the increase in upstream velocity .
overall , from the above set of curves , we find that the anisotropy in the downstream velocity is enhanced by the magnetic field and also to some extent by the incident magnetic angle .
the downstream velocity vector always points downward .
higher incident velocity tends to lessen this effect . with the increase in density
the anisotropy is also reduced .
@xmath60 is usually greater than @xmath82 .
0.2 in cc , @xmath59 , @xmath60 , @xmath61 and @xmath62 are shown as function of density @xmath75 , for two different incident magnetic angles ( @xmath76 and @xmath77 ) .
the incident velocity is @xmath65 . ] , @xmath59 , @xmath60 , @xmath61 and @xmath62 are shown as function of density @xmath75 , for two different incident magnetic angles ( @xmath76 and @xmath77 ) .
the incident velocity is @xmath65 . ]
+ , @xmath59 , @xmath60 , @xmath61 and @xmath62 are shown as function of density @xmath75 , for two different incident magnetic angles ( @xmath76 and @xmath77 ) .
the incident velocity is @xmath65 . ]
+ for the tl shocks , the magnetic fields have no contribution to the shock velocities ( fig . [ fig5 ] and fig . [ fig6 ] ) .
the upstream and downstream magnetic angles remain the same , before and after the shock .
the magnetic field has no effect on the downstream flow velocities or on the flow angle .
the anisotropy in the downstream flow velocities is caused by the boost that brings the quantities from the fluid frame to the ni frame .
the @xmath38 component of the downstream velocity decreases with incident magnetic angle , whereas the @xmath40 component decreases ( fig .
[ fig5 ] ) . for the tl shocks both the @xmath38 and @xmath40 component of the downstream velocities
are positive .
therefore , an increase in the incident magnetic angle makes the downstream velocity more anisotropic .
the tl shock differs from the sl shock by the fact that the @xmath40 component of the velocity is positive and so is the flow angle .
the downstream velocity vector points upwards .
this is the case because the anisotropy is generated by the boost and not by the magnetic field .
the @xmath60 parameter also differs , it is always less than @xmath82 and decreases with density .
this shows that there is no velocity compression but rarefaction in velocity due to the pt .
as the magnetic field has no effect on the tl shock , the magnetic angle remains the same before and after the shock . the flow angle ( positive for tl shocks )
increases with incident magnetic angle as the anisotropy of the downstream velocity increases .
the @xmath38 component of the downstream velocity is directly proportional to the upstream velocity ( fig .
[ fig6 ] ) , and has a small effect on the @xmath40 component .
the velocity component @xmath38 in the downstream increases with increase in incident velocity whereas , the @xmath40 component decreases .
@xmath60 increases with increase in incident velocity .
the flow angle decreases with increase in incident velocity , and so the anisotropy in the downstream velocity is reduced .
the above discussion highlights that the system does not feel the magnetic effect , as if the fluid is non magnetic .
thus , the magnetic field present in a neutron star will not affect a tl shock , and so the pt would also remain unaffected . even in the case of a heavy - ion collision
the magnetic field would have no effect on a tl shock produced by the transition of a qgp to hadrons during the expansion of the fireball .
this is due to the fact that the electromagnetic field , which we assumed , that one can neglect the tl component ( electric ) compared to the sl component ( magnetic ) .
0.2 in cc , @xmath59 , @xmath60 , @xmath61 and @xmath62 versus density @xmath75 for tl shocks are plotted for two different incident velocities ( @xmath84and @xmath85 ) .
the magnetic angle is @xmath67 . as the magnetic angle is unaffected by the shock , in fig [ fig6]c the black straight line marks the downstream magnetic angle , which is the same for two incident velocity .
] , @xmath59 , @xmath60 , @xmath61 and @xmath62 versus density @xmath75 for tl shocks are plotted for two different incident velocities ( @xmath84and @xmath85 ) .
the magnetic angle is @xmath67 . as the magnetic angle is unaffected by the shock , in fig [ fig6]c the black straight line marks the downstream magnetic angle , which is the same for two incident velocity .
] + , @xmath59 , @xmath60 , @xmath61 and @xmath62 versus density @xmath75 for tl shocks are plotted for two different incident velocities ( @xmath84and @xmath85 ) .
the magnetic angle is @xmath67 . as the magnetic angle is unaffected by the shock , in fig [ fig6]c the black straight line marks the downstream magnetic angle , which is the same for two incident velocity .
summarising our work , we have derived the general rankine - hugoniot condition for a mhd oblique shock .
first we have derived the equations for the subluminal shock in the ht frame and from there we have derived a more general shock ( sub and superluminal shock ) in the ni frame . to go from the fluid frame to the ni frame we have used a set of boosts along the coordinate axes .
this is very useful as it removes all the imaginary and unphysical quantities present in the superluminal shock . in the ht frame
we have written the four matter conservation equations and two electromagnetic conservation equations ( maxwell equations ) .
there is also a set of equations which transforms the quantities from the fluid to the ni frame .
we have formulated the equations for both the sl and tl shock and the sl conservation condition matches well with previous works @xcite . along with these equations
we also have introduced a set of standard eos describing matter in the upstream and downstream phases .
we assume that the shock wave brings about a pt from hadronic to quark matter
. the important results of this work are relevant both for astrophysics and heavy - ion collision .
the input quantities are the upstream variables and we are solving the conservation equation to obtain the downstream variables .
the magnetic effects are relevant once the field strength is greater than @xmath78 g .
the general behaviour for sl shocks , the anisotropy in the downstream velocity is enhanced by the magnetic field and also to some extent by the incident magnetic angle .
higher incident velocity and higher baryon density tends to reduce this effect .
the tl shock differs from sl shocks , as the magnetic field has effectively no effect on the former .
the slight anisotropy in the downstream flow velocities is caused by the boosting that brings the quantities from the fluid frame to the ni frame .
however , the cause may lie in our assumption of small electric fields . in the sl shocks
there is compression in the velocity ratio whereas in the tl shocks there is rarefaction in the velocity ratio due to pt . in this work
we consider a special case of pt from hadronic to quark matter .
this is a phenomenon that might take place inside a neutron stars ( ns ) , where the star is ultimately converted to a quark star ( qs ) .
some perturbation ( like spin - down @xcite ) may induce such a pt brought about by a shock wave , originating at the centre .
if such is the case , then a detailed study of shock waves inside the star is needed .
the first step towards it is the study of mhd shock waves in those environments .
here we study such a scenario where a mhd shock wave ( both sl and tl ) brings about a pt from hadronic to quark matter .
we find that the sl and tl mhd shock waves are quite different from one another .
the downstream flow velocity vectors for sl shocks points in opposite direction ( i.e. the flow angle has opposite signs ) in comparison to the tl shocks .
the velocity compression ratio is also typically different . and
the most significant difference comes from the magnetic effect . for the sl shocks all the downstream components depends strongly on the magnetic field and its angle with respect to the shock
whereas , tl shocks are unaffected by magnetic field . the difference in the downstream behaviour of matter variables for sl and
tl shocks could have some observational consequences for ns .
the observed pulses from the ns are the jets of particles emitted by the ns and are directly in our line of sight .
the jets of particles interacts with the ns environment surrounding it . also , it was suggested @xcite that gamma ray bursts ( grb ) can be a consequence of pt in ns .
therefore , if the pt is brought about by a shock wave , its signature may differ for sl and tl shocks and could be strongly dependent on the magnetic field involved .
it is suggested that the particle acceleration and production of high energy ( tev ) @xmath0-rays from blazars and agns depends on the nature of the shock environment , the shock speed and the magnetic field present there .
the spectral indices which are measured depends strongly on these parameters , and provide deep insight regarding blazars and grb s .
this may be also true for shock waves in ns , where the downstream flow variables for sl and tl shocks are quite different .
for more significant observational inferences , we need more detailed calculation of the particle emission , their acceleration and their interaction with the ns environment . in heavy - ion collision ,
the pt is quite the opposite .
there we have hadronization of particles from an expanding and subsequently cooling qgp fireball .
the pt can also be thought to be initiated by a shock wave .
although , in our work we analyse the opposite pt , the results can provide a insight even for heavy - ion collision , for the fact that the magnetic field which are produced in heavy - ion collisions would have no effect on the tl shock whereas it would significantly affect the sl shocks .
we should mention that the transformation equations from the fluid to the ni frame are similar to ballard & heavens and summerlin & baring ( @xcite ) .
we have solved the rankine - hugoniot condition for the mhd fluid with the assumption that there is a pt from hadronic to quark matter .
there is no temporal evolution . to look into the temporal evolution or how shock waves bring about a pt in neutron stars we have to study and solve the corresponding equations of motion ( the continuity and euler equation )
however , before tackling such complex problems we can guess the initial conditions from this work .
therefore , these results can serve as the starting point of such calculations which we hope to report upon in the future work .
colburn , d. s. , & sonett , c. p. , space sci .
* 5 * , 439 ( 1966 ) kukel , w. b. , & brown , s. c. , phys . today
* 20 * , 88 ( 1967 ) hinton , j. a. , & hoffmann , w. , ara&a * 47 * , 523 ( 2009 ) taub , a. h. , phys . rev .
bf 74 , 328 ( 1948 ) de hoffmann , f. , & teller , e. , phys .
rev . * 80 * , 692 ( 1950 ) landau , l. d. , & lifshitz e. m. , _ fluid mechanics _ ( oxford , pergamon press , 1959 ) cabannes , h. , _ theoretical magnetofluiddynamics _ ( new york , academic , 1970 ) lichnerowicz , a. , _ relativistic hydrodynamics and magnetohydrodynamics _
( new york , benjamin , 1967 ) majorana , a. , & anile , a. m. , phys .
fluid * 30 * , 3045 ( 1987 ) ballard , k. r. , & heavens , a. f. , mnras * 251 * , 438 ( 1991 ) gerbig , d. , & schlickeiser , r. , astrophys .
j. * 733 * , 32 ( 2011 ) kirk , j. g. , & heavens , a. f. , mnras * 239 * , 995 ( 1989 ) summerlin , e. j. , & baring , m. g. , astrophys .
j. * 745 * , 63 ( 2012 ) csernai , l. p. , zh . eksp .
fiz . * 92 * , 379 ( 1987 ) gorenstein , m. i. , miller h. g. , quick , r. m. , & ritchie , r. a. , phys .
b * 340 * , 109 ( 1994 ) glendenning , n. k. , & matsui , t. , phys . lett .
b * 141 * , 419 ( 1984 ) csernai , l. p. , & gong , m. , phys .
d * 37 * , 11 ( 1988 ) rosenhauer , a. , maruhn , j. a. , greiner , w. , & csernai , l. p. , z. phys .
a * 326 * , 213 ( 1987 ) gyulassy , m. , kajantie , k. , kurki - suonio , h. , & mclerran , l. , nucl .
b * 237 * , 477 ( 1984 ) walecka , j. d. , astrophys . j. * 83 * , 491 ( 1974 ) alcock , c. , farhi , e. , & olinto , a. , astrophys .
j. * 310 * , 261 ( 1986 ) glendenning , n. k. , phys .
d * 46 * , 1274 ( 1992 ) bombaci , i. , & datta , b. , astrophys .
j * 530 * , l69 ( 2000 ) berezhiani , z. , bombaci , i. , drago , a. , frontera , f. , & lavagno , a. , astrophys .
j * 586 * , 1250 ( 2003 ) mallick , r. , bhattacharyya , a. , ghosh , s. k. , & raha , s. , int .
e * 22 * , 135008 ( 2013 )
|
shock waves constitute discontinuities in matter which are relevant in studying the plasma behaviour in astrophysical scenarios and in heavy - ion collision
. they can produce conical emission in relativistic collisions and are also thought to be the mechanism behind the acceleration of energetic particles in active galactic nuclei and gamma ray bursts .
the shocks are mostly hydrodynamic shocks . in a magnetic background
they become magnetohydrodynamic ( mhd ) shocks . for that reason we study the space - like and time - like shock discontinuity in a magnetic plasma .
the shocks induce a phase transition in the plasma , here assuming a transition from hadron to quarks .
the mhd conservation conditions are derived across the shock .
the conservation conditions are solved for downstream velocities and flow angles for given upstream variables .
the shock conditions are solved at different baryon densities . for the space - like shocks the anisotropy in the downstream velocity arises due to the magnetic field .
the downstream velocity vector always points downward with respect to the shock normal . with the increase in density
the anisotropy is somewhat reduced .
the magnetic field has effectively no effect on time - like shocks .
the slight anisotropy in the downstream flow velocities is caused by the boosting that brings the quantities from the fluid frame to normal incidence ( ni ) frame .
|
low energy observables sensitive to cp violation in @xmath9 transitions constitute excellent probes of possible new cp violating phases in extensions of the standard model ( sm ) .
indeed , as cp violation in such processes is predicted to be tiny in the sm , evidence for sizable cp violation in @xmath9 transitions would be a clear hint for the presence of new physics ( np ) .
examples of such low energy probes are observables that are sensitive to the @xmath0 mixing phase , as the semi - leptonic asymmetry @xmath10 in decays of @xmath0 mesons to `` wrong sign leptons '' or the time dependent cp asymmetries in the @xmath2 and @xmath3 decays . in the context of generic two higgs doublet models with minimal flavor violation ( mfv ) , where the ckm matrix is the only source of flavor violation , a large @xmath0 mixing phase
can be realized if additional cp violating phases are allowed @xcite .
simultaneously , these models can also address tensions in fits of the ckm matrix that seem to indicate a sizable np contribution to the @xmath1 mixing phase @xcite .
possible relations between a non - standard @xmath0 mixing phase and the baryon asymmetry of the universe in these models have been studied in @xcite . in the context of the minimal supersymmetric standard model ( mssm ) on the other hand ,
a mfv soft sector is not sufficient to generate sizable np phases in meson mixing @xcite due to the strong experimental bound on the @xmath5 branching ratio .
non sm - like @xmath0 and @xmath1 mixing phases in the mssm require new sources of flavor violation in addition to the ckm matrix ( see e.g. @xcite for studies of such frameworks ) .
supersymmetric models with mfv _ can _ generate large @xmath11 mixing phases _ if _ they allow for a strongly reduced muon yukawa coupling such that the br@xmath12 constraint can be avoided . such a situation can be realized for example in the so - called uplifted susy higgs region @xcite , even though this framework is strongly constrained by other @xmath11 physics observables and @xmath13 @xcite .
as studied in detail in @xcite , non - negligible corrections to cp violating observables in meson mixing can also be generated if the mssm with mfv is extended by the two leading higher dimensional operators in the higgs sector with complex coefficients @xcite .
still , the stringent bounds on br@xmath12 only allow for a @xmath0 mixing phase of @xmath14 in specific regions of parameter space of this model .
in this work we analyze an extension of the mssm , introducing higher dimensional operators not exclusively in the higgs sector , but also considering dimension 5 operators that induce non - holomorphic higgs - fermion couplings already at the tree level @xcite .
such operators are a possible source of flavor and cp violation . assuming that these operators obey the minimal flavor violation ansatz @xcite
, we explore to which extent the considered framework allows for non - standard cp violation in @xmath11 mixing without being in conflict with the bounds on the br@xmath15 . for a study of a similar framework , that however does not consider cp violation in @xmath11 mixing , see @xcite .
the @xmath16 mixing amplitude @xmath17 consists on an dispersive part , @xmath18 , and an absorptive part , @xmath19 .
the absorptive part is dominated by tree level sm contributions and therefore hardly affected in many np models . throughout this work
we will assume that @xmath19 has no significant np contributions .
the dispersive part @xmath18 on the other hand is highly sensitive to new heavy degrees of freedom .
the effects of np in @xmath18 can be parametrized by @xmath20 the main impact of the parameters @xmath21 and @xmath22 is on the mass differences @xmath23 and @xmath24 , respectively .
the np phases @xmath25 and @xmath26 affect observables that are sensitive to cp violation in b meson mixing , like the semileptonic asymmetries @xmath27 and @xmath28 as well as the time - dependent cp asymmetries in @xmath29 , @xmath2 and @xmath3 . in order to constrain the np parameters through measurements of these observables , knowledge of the respective sm contributions is required . as several of the observables ,
in particular @xmath23 and @xmath30 the time dependent cp asymmetry in @xmath29 are a key ingredient in the determination of the unitarity triangle ( ut ) , a simultaneous fit of the ckm parameters and the np parameters as performed in @xcite is the most consistent approach . here
we focus mainly on the impact of the recent improvements on the determination of the @xmath0 mixing phase at cdf and d0 @xcite and in particular at lhcb @xcite .
as these measurements have no significant effect on the determination of the ckm parameters and the np parameters other than @xmath26 , we consider a simplified approach and take the ckm parameters as well as @xmath21 and @xmath22 from the generic np fit in @xcite , and fit only the np parameters @xmath26 and @xmath25 .
we expect this approach to give a good estimate of the allowed region of parameter space compatible with the present experimental data on b meson mixing .
in particular we will use @xmath31 = ( 27.2^{+1.1}_{-3.1})^\circ \\ \beta_s & = & \textnormal{arg}\left [ ( v_{tb } v_{ts}^*)/(v_{cb } v_{cs}^*)\right ] = ( -1.3\pm 0.1)^\circ \nonumber\end{aligned}\ ] ] and the following @xmath32 bounds @xmath33 we now give expressions for the observables that are sensitive to cp violation in b meson mixing . for the semileptonic asymmetries
we obtain @xcite @xmath34 where the uncertainties on the numerical coefficients are at the level of @xmath35 . for the time
dependent cp asymmetries one has @xmath36 these expressions hold under the usual assumption that the @xmath29 , @xmath2 and @xmath3 decays are dominated by the sm tree level amplitudes . using ( [ eq : beta ] ) the corresponding sm predictions are @xmath37 the d0 collaboration measured the like - sign dimuon charge asymmetry that is predicted to be composed out of the semi - leptonic asymmetries in the @xmath1 and @xmath0 decays @xcite @xmath38 the corresponding sm prediction @xmath39 @xcite is roughly a factor 25 below the central value in ( [ eq : aslb_d0 ] ) and differs from it by @xmath40 .
the value ( [ eq : aslb_d0 ] ) updates an earlier d0 study @xcite that found a @xmath41 evidence for an anomalous like - sign dimuon charge asymmetry . a separate extraction of the semileptonic asymmetries results in @xcite @xmath42 the results in eqs . ( [ eq : aslb_d0 ] ) , ( [ eq : asld_d0 ] ) and ( [ eq : asls_d0 ] ) hint towards large negative values for the np phases @xmath25 and , in particular , @xmath26 .
interestingly enough , there is a ( 2 - 3)@xmath43 tension between the sm prediction of @xmath30 ( [ eq : spsiks ] ) and its experimental value @xcite @xmath44 that is largely driven by the @xmath45 measurements that prefer a large value of @xmath46 .
this tension points in the same direction for the np phase @xmath25 as the data on the like sign dimuon charge asymmetry .
a small preference for a negative np phase in @xmath0 mixing was also observed in cdf and d0 data on the time dependent cp asymmetry in @xmath2 that give @ 95% c.l .
@xcite @xmath47 combining the results from @xcite with @xcite , global fits to the data found a @xmath0 mixing phase @xmath48 @xcite . and @xmath25
at the 1 and 2 @xmath43 level , taking into account the measurements of the time - dependent cp asymmetries in @xmath2 at cdf , d0 and lhcb @xcite , in @xmath3 at lhcb @xcite and in @xmath29 at the b factories @xcite .
the measurement of the like - sign dimuon charge asymmetry at d0 @xcite is included in the black dotted contours but _ not _ in the red solid contours . , scaledwidth=47.0% ] recently however , lhcb presented results on the time dependent cp asymmetries in @xmath2 and @xmath3 that are consistent with the tiny sm prediction and that strongly restrict the possible values for a np phase in @xmath0 mixing @xcite @xmath49 in fig .
[ fig : phi_fit ] we show the result of a simple fit of the np phases @xmath25 and @xmath26 to the combined lhcb result on the time - dependent cp asymmetries in @xmath2 and @xmath3 @xcite , the results on the time - dependent cp asymmetry in @xmath2 from cdf and d0 @xcite as well as the measurement of @xmath30 at the b factories @xcite , using the values for @xmath50 and @xmath51 , @xmath21 and @xmath22 from above .
the allowed region is mainly determined by the measurements of @xmath52 , @xmath53 at lhcb and @xmath30 at the b factories , while the measurements of @xmath52 at cdf and d0 lead to a small shift of the central value of @xmath26 towards a small negative value .
we stress that the very large value of the like - sign dimuon charge asymmetry observed by d0 can not be explained given the current data on the time dependent cp asymmetries in @xmath2 and @xmath3 . using our fit results
we find a central value of @xmath54 and a @xmath32 range of @xmath55 .
this differs from the measured value ( [ eq : aslb_d0 ] ) by @xmath56 .
including the like - sign dimuon charge asymmetry directly into the fit leads only to a small shift towards slightly larger negative @xmath26 values . in the following we focus on the fit that does not include the @xmath57 measurement . due to the small discrepancy between the experimental determination of @xmath30 and its sm prediction coming from
the ut fits @xcite , the np phase in @xmath1 mixing shows preference towards a negative value @xmath58 that is roughly 2@xmath43 below 0 . while also for the np phase in @xmath0 mixing we find a slight preference for a small negative value , @xmath26 is perfectly consistent with zero @xmath59 presently , this still leaves room for np contributions , but the bound ( [ eq : phi_s ] ) will improve significantly in the near future with more data from lhcb .
the cdf collaboration reported a small excess in @xmath60 candidates @xcite , leading to @xmath61 no excess has been observed by lhcb and cms that report the following bounds @xcite @xmath62 combining the bounds from lhcb and cms one finds @xcite @xmath63 that is only a factor of 3.5 above the sm prediction @xcite @xmath64 the current bounds on the @xmath65 branching ratio @xcite are still a factor 40 - 50 above the sm expectation and therefore @xmath65 is much less constraining than @xmath5 in models with mfv . given the strong bound on br@xmath12 , possible neutral higgs contributions to b mixing in the mssm with mfv are strongly constrained @xcite .
also in the bmssm model considered in @xcite non - standard b mixing phases are rather restricted ( @xmath14 ) and can only be generated in particular corners of parameter space . in the following
we present an extension of the mssm that respects the mfv principle but allows nonetheless for sizable np phases in b mixing without being in conflict with the br@xmath12 constraint .
at the tree level , the mssm is a 2 higgs doublet model of type ii and the couplings of the neutral higgs bosons to fermions are flavor conserving . at the loop level on the other hand ,
non - holomorphic higgs couplings are generated and have important consequences .
loop induced couplings of down - type quarks and charged leptons to the up - type higgs can lead to large threshold corrections to the corresponding masses @xcite and modify significantly ckm matrix elements @xcite as well as charged higgs couplings to quarks @xcite .
finally , they also generate flavor changing neutral higgs couplings that can have a profound impact on flavor phenomenology @xcite .
all these effects become relevant for large values of @xmath66 that can compensate for the 1-loop suppression .
we now consider possible extensions of the mssm with new degrees of freedom at a scale @xmath67 several tev .
as long as the susy breaking scale @xmath68 of the new degrees of freedom is small compared to m and as long as m is sufficiently larger than the scale of the mssm degrees of freedom , one can describe the effects of the beyond the mssm ( bmssm ) physics by higher dimensional operators suppressed by @xmath69 @xcite .
an analysis up to order @xmath69 captures the physics of several mssm uv extensions while the effective description of others need to include @xmath70 effects @xcite . in this work
we restrict ourselves to the @xmath69 level .
we consider both the leading higher dimensional superpotential operators that involve only higgs fields @xcite @xmath71 and in particular also @xmath69 suppressed khler potential operators that induce non - holomorphic higgs - fermion couplings already at the tree level @xcite ) and ( [ eq : kahler ] ) are discussed e.g. in @xcite and briefly presented in the appendix . ]
@xmath72 in the above expressions , @xmath73 is an auxiliary dimensionless spurion that develops a susy breaking f - term @xmath74 the phenomenological consequences of the operators in ( [ eq : super ] ) have been thoroughly studied in the literature @xcite . in particular
, they can significantly enhance the tree level mass of the lightest higgs boson of the mssm and lead to sizable mass splittings between the two heavy neutral higgs bosons and also the charged higgs boson .
possible phases of the coefficients @xmath75 and @xmath76 lead also to scalar - pseudoscalar mixing . their impact in the context of higgs and
flavor phenomenology has been analyzed in @xcite .
the khler potential operators ( [ eq : kahler ] ) modify the interactions of higgs bosons , ( s)quarks and ( s)leptons of the mssm at the @xmath69 level @xcite .
the supersymmetric part of ( [ eq : kahler ] ) for example leads to corrections of the holomorphic mssm yukawa couplings @xmath77 after susy breaking also non - holomorphic higgs - quark couplings are generated @xmath78 where we now made flavor indices explicit .
these terms can lead to flavor changing neutral higgs vertices and correspondingly to tree level contributions to fcnc processes like @xmath11 mixing and @xmath60 . in the following
we will focus on them .
the mfv hypothesis as formulated in @xcite amounts to the assumption that the @xmath79 quark flavor symmetry of the gauge sector is broken by only two spurions @xmath80 and @xmath81 that transform as @xmath82 and @xmath83 respectively .
correspondingly , the couplings @xmath84 and @xmath85 can be expanded in powers of these spurions . to keep notation simple and concise
, we conveniently choose @xmath86 to be these spurions .
any other linear combination of @xmath84 and @xmath85 leads to equivalent results . for the non - holomorphic higgs couplings @xmath85 to the down quarks one then has @xmath87 where for simplicity we dropped the prime on the corrected yukawa couplings .
an analogous expression holds for the up quark coupling which is however not relevant for the following discussion .
the coefficients @xmath88 are generically of o(1 ) and complex . for later convenience we define @xmath89 @xmath90 the couplings in ( [ eq : non_holomorphic ] ) modify the relation between the down quark masses @xmath91 and the corresponding yukawa couplings @xmath92 @xmath93 with the higgs vev @xmath94 gev and we only show the leading term in a @xmath66 expansion .
similar to the quark masses , also the ckm matrix receives @xmath66 enhanced corrections .
the relations between the affected elements of the bare ckm matrix @xmath95 and the physical ckm matrix @xmath96 read ( @xmath97 ) @xmath98 finally we also give explicit expressions for the corrected flavor changing couplings of right handed down quarks with the higgs bosons .
the leading @xmath66 enhanced terms read ( @xmath99 ) @xmath100 @xmath101~. \label{eq : xbi}\end{aligned}\ ] ] the flavor changing @xmath102 couplings @xmath103 are generated by the @xmath104 and @xmath105 terms and they are proportional to @xmath106 .
the flavor changing @xmath107 couplings @xmath108 on the other hand are generated by the @xmath104 and @xmath109 terms and suppressed by light quark masses .
the expression ( [ eq : xbi ] ) generalizes the results given in @xcite and , to the best of our knowledge , has not been presented in the literature . in the mssm ,
the @xmath110 factors can only be loop induced .
gluino - down squark loops generate for example @xmath111 , while @xmath112 is generated by chargino - stop loops . due to the loop suppression ,
the @xmath110 factors are generically of o(0.01 ) in the mssm . correspondingly , the corrections in ( [ eq : masses ] ) , ( [ eq : ckm ] ) , ( [ eq : higgs_couplings ] ) and ( [ eq : xbi ] ) become important only for large values of the ratio of the two higgs vevs @xmath113 . on the other hand , in generic 2 higgs doublet models with mfv as analyzed in @xcite , where the @xmath110 are free parameters , moderate values of @xmath114 are sufficient to generate o(1 ) effects .
the same is true in the supersymmetric framework considered here as long as the bmssm scale that controls the size of the @xmath110 factors is not too high , i.e. @xmath115 .
observables that are highly sensitive to a non - standard higgs sector are observables in meson mixing as well as the branching ratios of the rare decays @xmath116 that receive tree level contributions from flavor changing neutral higgs exchange .
charged higgs effects are relevant in the @xmath117 decay as well as in the @xmath45 , @xmath118 and @xmath119 decays .
the first one is modified only at the loop level while the others receive contributions from tree level charged higgs exchange .
however these tree level decays turn out to give only mild constraints in regions of parameter space with non - standard b meson mixing phases and we do not discuss them in detail here although they are included in our numerical analysis . for a recent study of @xmath45 in the context of multi higgs doublet models with mfv see @xcite .
the flavor changing neutral higgs couplings in ( [ eq : fcnc_lagrangian ] ) give rise to tree level contributions to @xmath0 mixing mediated by neutral higgs exchange .
these contributions can be described by the following effective hamiltonian @xmath120 with the wilson coefficients @xmath121 analogous contributions to @xmath1 mixing can be obtained through the replacements @xmath122 . or @xmath123 in the considered framework and therefore negligibly small . ] in writing ( [ eq : wilson2 ] ) and ( [ eq : wilson4 ] ) , we assume the decoupling limit @xmath124 and treat the effect of the operators in ( [ eq : super ] ) in a mass insertion approximation . in our numerical analysis
instead we work with higgs mass eigenstates that we derive from the full higgs potential including mssm 2-loop corrections @xcite .
the wilson coefficient @xmath125 is proportional to @xmath126 .
consequently it can only lead to sizable effects in @xmath0 mixing while its impact on @xmath1 mixing is rather restricted . on the other hand , @xmath127 is proportional to @xmath128 and therefore leads to np contributions of the same size and phase both in @xmath0 and @xmath1 mixing .
we stress that @xmath125 is complex only if higher orders of the bottom yukawa are considered in the expansion ( [ eq : non_holomorphic ] ) ( see also @xcite ) . indeed ,
one easily checks that switching off @xmath129 , @xmath130 and @xmath131 leads to a real @xmath125 .
the wilson coefficient @xmath127 however is also highly sensitive to the phases of @xmath132 , @xmath133 , @xmath76 and @xmath75 . from ( [ eq : wilson2 ] ) it is clear that @xmath127 is only relevant in presence of the higher dimensional operators ( [ eq : super ] ) in the superpotential and for small higgs masses not far above the electroweak scale .
the @xmath5 decay receives tree level contributions from flavor changing neutral higgs exchange .
one finds @xmath134 @xmath135 with the sm loop function given by @xmath136 .
the above expression assumes again the decoupling limit @xmath124 . for small higgs masses , corrections at the @xmath69 level
become important and are included in our numerical analysis . in writing ( [ eq :
a ] ) we also assume that the non - holomorphic lepton - higgs coupling @xmath137 is proportional to the lepton yukawa @xmath138 as the np contribution ( [ eq : a ] ) to the @xmath5 amplitude grows with @xmath139 , the large @xmath66 regime of the mssm is strongly constrained by the experimental bound ( [ eq : bsmm_lhc ] ) .
however , as already stressed above , the non - holomorphic tree level higgs - fermion couplings allow to generate np contributions to b mixing already for moderate values of @xmath66 , where the bound from br@xmath15 is considerably relaxed as long as the muon yukawa coupling is not largely enhanced by the @xmath66 resummation factors .
to obtain the @xmath16 mixing amplitudes , we use 2-loop renormalization group running for the wilson coefficients @xcite and the hadronic matrix elements from @xcite .
we check compatibility of the model with various constraints .
\(i ) vacuum stability : the higher dimensional operators in the superpotential can lead to a second minimum in the higgs potential . requiring that the electroweak minimum is stable , gives a lower bound on the charged higgs mass for given values of the @xmath140 term and the susy breaking scale @xmath68 @xcite .
\(ii ) electroweak precision observables can constrain regions of parameter space where the dimension 5 superpotential operators lead to a very heavy sm like higgs boson or a large splitting between the heavy higgs bosons .
we implement the s and t parameter following @xcite . also the @xmath141 coupling can be modified significantly by higgs loops @xcite
. however the @xmath141 constraint can be avoided if the higgs - top coupling is suppressed by non - holomorphic corrections and therefore we do not include it in the numerical analysis .
\(iii ) electric dipole moments ( edms ) can be induced both by phases of the higher dimensional operators in the khler potential and the superpotential .
experimentally accessible edms , like the edms of thallium , mercury or the neutron can be generated by the electron and quark edms and chromo edms ( cedms ) , by cp violating 4 fermion operators @xcite as well as by the weinberg 3 gluon operator @xcite .
the fermion ( c)edms are generated at the 1-loop level by sparticle loops that are sensitive to the phase of the higgs vev @xcite as well as to possible complex @xmath69 corrections to the sfermion mass matrices that are induced by modified higgs - sfermion couplings after electroweak symmetry breaking .
the latter corrections can however always be avoided if the parameters @xmath142 and @xmath143 are real . at the 2-loop level ,
barr - zee diagrams contribute to the ( c)edms @xcite .
they are directly sensitive to both the phases in the non - holomorphic higgs couplings and the scalar - pseudoscalar mixing in the higgs sector .
the cp violating 4 fermion operators are induced by neutral higgs exchange at tree level @xcite and , analogously to the 2-loop barr - zee contributions , they are sensitive to both the phases in the non - holomorphic higgs couplings and the scalar - pseudoscalar mixing in the higgs sector . finally , contributions to the weinberg 3 gluon operator can be induced by 2-loop diagrams that are sensitive to the scalar - pseudoscalar mixing in the higgs sector and in particular also to the phases in the non - holomorphic higgs couplings @xcite .
these contributions can be sizable , but they can be avoided to a large extent if the higgs - top couplings are suppressed by non - holomorphic corrections . keeping also in mind the large uncertainties in estimating the contribution of the 3 gluon operator to the neutron edm , we do not include it in our numerical analysis . following this approach , we find that the most important contributions are typically 2-loop barr - zee contributions to the mercury edm and 1-loop higgsino - wino - slepton contributions to the thallium edm in the regions of parameter space that we consider below .
\(iv ) constraints from direct higgs searches at lep , tevatron and lhc are implemented using ` higgsbounds ` @xcite as well as the latest updates from atlas and cms @xcite .
generically direct sm higgs searches do not lead to strong constraints , as the lightest higgs boson is usually in the range @xmath144 gev in the regions of parameter space considered below .
the two heavier higgs bosons can be much heavier and , due to the moderate values of @xmath66 , susy higgs searches are also not constraining possible large effects in b mixing .
\(v ) flavor observables : the main constraints come from @xmath23 and @xmath24 as well as the branching ratios of the decays @xmath117 and @xmath5 .
we also implement the constraints from @xmath145 , @xmath146 and @xmath147 . in fig .
[ fig : ma_tanb ] we show in two representative scenarios the possible values of the np phase in @xmath0 mixing in the @xmath148 - @xmath66 plane together with the above mentioned constraints . the plot on the left of fig .
[ fig : ma_tanb ] shows scenario i , where we chose a common sfermion mass of @xmath149 tev and trilinear couplings @xmath150 , @xmath151 .. higher order terms in the expansion can in principle induce additional flavor changing effects at the loop level . in the case of neutral meson mixing and @xmath60 however
, the loop level effects induced by the higher order tems will be subdominant compared to the dominant tree level contributions from the modified higgs sector . among the considered flavor observables , only the loop induced @xmath117 decay can be noticeably affected by possible higher order terms in the trilinear parameters .
considering such higher order terms would therefore add more flexibility in controlling the @xmath117 constraint .
as this constraint turns out to have only a small impact on our analysis , the higher order terms would not change any of our conclusions .
] we assume the absence of the higher dimensional operators ( [ eq : super ] ) in the superpotential and set @xmath152 correspondingly , np contribution to the @xmath11 mixing amplitudes are generated through the wilson coefficient @xmath125 and therefore effects are much larger in @xmath0 than in @xmath1 mixing .
we observe that the @xmath0 mixing phase can easily reach the @xmath32 bound given in ( [ eq : phi_s ] ) , @xmath153 , even for moderate values of @xmath154 and very large higgs masses of @xmath155 tev . in this region of parameter space , higgs contributions to the @xmath60 decay and to @xmath117 are well under control .
the strongest edm constraints in this scenario come from the mercury edm but due to the large higgs masses they turn out to be easily fulfilled .
as the higher dimensional operators in the superpotential are not present , vacuum stability bounds as well as electroweak precision constraints are always fulfilled .
the plot on the right of fig .
[ fig : ma_tanb ] shows scenario ii , where we allow both for higher dimensional operators in the superpotential and khler potential , but consider new sources of cp violation only in the superpotential .
we chose third generation squark soft masses of @xmath156 gev , all remaining sfermion masses @xmath157 and trilinear couplings @xmath158 tev , @xmath151 .
in addition we set @xmath159 in this setup the np phases in @xmath0 and @xmath1 mixing are induced by the wilson coefficient @xmath127 .
therefore they are of comparable size in both cases and typically well within the @xmath32 ranges of ( [ eq : phi_d ] ) and ( [ eq : phi_s ] ) .
we observe that sizable values for the @xmath0 mixing phase are possible even for small @xmath160 but require a rather light higgs spectrum , which agrees with our expectation that @xmath127 can be important only for small higgs masses . because of the small values of @xmath66 , both constraints from @xmath5 and from the edms ( that mainly come from the thallium edm ) are well under control in the considered scenario .
concerning the bound from br@xmath161 we remark that for the light higgs masses , there are sizable charged higgs loop contributions to the @xmath162 amplitude that are further enhanced by higher order @xmath66 resummation factors .
these contributions can be partly canceled either by @xmath163 corrections to the couplings of the charged higgs to the right - handed top quark , or by chargino - stop loops as long as stops are rather light , below 1 tev .
in contrast to scenario i , a very important constraint is now coming from @xmath23 . also bounds from vacuum stability start to be important .
contrary to the framework discussed in @xcite however , vacuum stability bounds can be compatible with a large @xmath0 mixing phase without the need of additional physics that stabilizes the electroweak vacuum .
term is required to generate sizable flavor changing neutral higgs couplings at the loop level . in the scenario considered here however , small values for @xmath140 are possible that considerably soften the vacuum stability bounds . ] in fig .
[ fig : asl ] we present the results of a parameter scan of the model .
the left plot shows the correlation between the semi - leptonic asymmetries @xmath27 and @xmath27 , while the right plot shows the correlation between the time dependent cp asymmetries @xmath30 and @xmath52 .
blue ( dark gray ) points correspond to a scenario where new sources of cp violation are arising entirely from the modified yukawa couplings and the higher dimensional operators in the superpotential ( [ eq : super ] ) are switched off completely .
we consider mass scales as in scenario i above and allow @xmath164 as well as epsilon parameters @xmath165 with @xmath142 and @xmath143 real and arbitrary phases for the remaining @xmath88 .
as expected , in such a setup the np effects in @xmath0 mixing are much larger than in @xmath1 mixing and the lhcb bounds on @xmath166 exclude a sizable np phase in @xmath1 mixing .
green ( light gray ) points correspond to a scenario where also the higher dimensional operators in the superpotential ( [ eq : super ] ) are considered .
we fix mass scales as in scenario ii above and allow the parameters @xmath167 with arbitrary phases and real epsilon parameters @xmath168 as well as @xmath164 . in this setup , the np contributions to cp violation in @xmath1 and @xmath0 mixing are comparable .
consequently , a sizable np phase in @xmath1 mixing can be compatible with the lhcb constraints on the np phase in @xmath0 mixing . even though new cp phases come entirely from the superpotential operators , we stress that the presence of the khler potential operators is crucial .
they allow for low @xmath66 values and therefore the constraint from @xmath60 can be avoided .
we conclude that in order to generate sizable corrections to cp violation in @xmath1 mixing that are in agreement with the lhcb data on cp violation in @xmath0 mixing , both the higher dimensional operators in the superpotential ( that modify the higgs spectrum ) as well as the higher dimensional operators in the khler potential ( that lead to non - holomorphic higgs - fermion couplings at tree level ) are required .
recent results from lhcb on the time - dependent cp asymmetries in @xmath2 and @xmath3 significantly restrict the allowed values for the @xmath0 mixing phase . combining these results with measurements of cdf and d0 of the time - dependent cp asymmetry in @xmath2 as well as the measurements of the time - dependent cp asymmetry in @xmath29 at the b factories ,
we find the following 2@xmath43 ranges for possible np phases in @xmath0 and @xmath1 mixing : @xmath169 and @xmath170 .
the preference for a negative np phase in @xmath1 mixing is driven by tensions in fits of the unitarity triangle , while the np phase in @xmath0 mixing is perfectly consistent with 0 . under the assumptions that neither the absorptive part of the @xmath0 mixing amplitude nor the sm tree - level @xmath2 decay amplitude are significantly affected by np , the anomalous like sign dimuon charge asymmetry observed by d0
can not be explained given the above bounds . and
@xmath3 and the d0 measurement of the like sign dimuon charge asymmetry with np effects in @xmath171 has been analyzed very recently in @xcite . ] in view of these results we studied the possible impact of higher dimensional operators in the mssm on b physics .
we considered dimension 5 operators both in the superpotential and in the khler potential assuming that they are generated at a scale of @xmath172 tev .
the 1/m suppressed operators in the superpotential have important impact on the higgs spectrum .
they can significantly enhance the tree level mass of the lightest higgs boson and lead to a mass splitting between the two heavy neutral higgs bosons . with complex coefficients
they also lead to scalar - pseudoscalar mixing .
the 1/m operators in the khler potential can induce non - holomorphic higgs couplings and consequently flavor changing neutral higgs couplings at the tree level .
assuming that the khler potential operators follow the minimal flavor violation ansatz , we find a flavor phenomenology that resembles to a large extent the 2 higgs doublet model with mfv discussed in @xcite .
in particular large values for @xmath66 are not required to generate sizable corrections to b meson mixing from neutral higgs exchange and therefore the strong constraint from br@xmath12 can be significantly relaxed .
we find that both superpotential and khler potential operators are required to generate non - standard effects in the @xmath1 mixing phase that are in agreement with the current bounds on the @xmath0 mixing phase from lhcb .
the corresponding region of parameter space is characterized by rather small values of @xmath173 as well as low masses for the heavy higgs bosons @xmath174 gev .
we stress that in the class of models discussed in this work , a non - standard @xmath1 mixing phase does imply also non - standard effects in the @xmath0 mixing phase at a level that can be tested in the near future by lhcb .
[ [ acknowledgments ] ] acknowledgments : + + + + + + + + + + + + + + + + we thank michael trott for useful comments .
fermilab is operated by fermi research alliance , llc under contract no .
de - ac02 - 07ch11359 with the united states department of energy .
here we briefly present uv completions that lead to the higher dimensional operators in ( [ eq : super ] ) and ( [ eq : kahler ] ) .
the dimension 5 superpotential operator in ( [ eq : super ] ) can be generated by integrating out a heavy singlet s with the following superpotential interactions @xcite @xmath175 additional gauge interactions that are broken at a high scale @xmath176 can be effectively described by dimension 6 operators @xcite . one way to generate the khler potential operators in ( [ eq : kahler ] ) is to introduce two heavy @xmath177 doublets @xmath178 , @xmath179 with hypercharge @xmath180 and @xmath181 that couple to fermions and mix with the mssm higgs doublets @xmath182 , @xmath183 @xcite .
neglecting for simplicity gauge interactions , their khler potential and superpotential read @xmath184 integrating out the heavy higgs doublets generates the supersymmetric term in ( [ eq : kahler ] ) and susy breaking can be incorporated with the auxiliary spurion @xmath73 .
in addition to ( [ eq : kahler ] ) , integrating out the heavy higgses from ( [ eq : uv ] ) also generates @xmath69 suppressed terms in the superpotential @xmath185 that in turn generate dimension 5 fermion sfermion interactions .
such interactions are however not relevant for the topics discussed in the present work .
a. j. buras , m. v. carlucci , s. gori and g. isidori , jhep * 1010 * , 009 ( 2010 ) .
a. j. buras , g. isidori and p. paradisi , phys
. lett .
b * 694 * , 402 ( 2011 ) .
m. trott , m. b. wise , jhep * 1011 * , 157 ( 2010 ) . for studies of related frameworks see a. pich , p. tuzon , phys .
rev . * d80 * , 091702 ( 2009 ) ; m. jung , a. pich , p. tuzon , jhep * 1011 * , 003 ( 2010 ) ; phys .
rev . * d83 * , 074011 ( 2011 ) .
a. lenz _ et al .
_ , phys .
d * 83 * , 036004 ( 2011 ) ; a. lenz and u. nierste , arxiv:1102.4274 [ hep - ph ] .
e. lunghi and a. soni , phys .
b * 666 * , 162 ( 2008 ) ; a. j. buras and d. guadagnoli , phys .
d * 78 * , 033005 ( 2008 ) ; e. lunghi and a. soni , phys .
b * 697 * , 323 ( 2011 ) .
j. m. cline , k. kainulainen , m. trott , arxiv:1107.3559 [ hep - ph ] .
t. liu , m. j. ramsey - musolf , j. shu , arxiv:1109.4145 [ hep - ph ] .
w. altmannshofer , a. j. buras and p. paradisi , phys .
b * 669 * , 239 ( 2008 ) .
w. altmannshofer , a. j. buras , s. gori , p. paradisi and d. m. straub , nucl .
b * 830 * , 17 ( 2010 ) .
see e.g. p. ko and j. h. park , phys .
d * 80 * , 035019 ( 2009 ) ; g. f. giudice , m. nardecchia and a. romanino , nucl .
b * 813 * , 156 ( 2009 ) ; j. k. parry , phys .
b * 694 * , 363 ( 2011 ) ; s. f. king , jhep * 1009 * , 114 ( 2010 ) ; j. girrbach , s. jager , m. knopf , w. martens , u. nierste , c. scherrer and s. wiesenfeldt , jhep * 1106 * , 044 ( 2011 ) [ erratum - ibid . * 1107 * , 001 ( 2011 ) ] ; r. barbieri , g. isidori , j. jones - perez , p. lodone , d. m. straub , eur .
j. * c71 * , 1725 ( 2011 ) .
a. crivellin , l. hofer , u. nierste , d. scherer , phys .
* d84 * , 035030 ( 2011 ) . b. a. dobrescu and p. j. fox , eur .
j. c * 70 * , 263 ( 2010 ) ; b. a. dobrescu , p. j. fox and a. martin , phys .
* 105 * , 041801 ( 2010 ) .
w. altmannshofer and d. m. straub , jhep * 1009 * , 078 ( 2010 ) .
w. altmannshofer , m. carena , s.
gori and a. de la puente , arxiv:1107.3814 [ hep - ph ] .
m. dine , n. seiberg and s. thomas , phys .
d * 76 * , 095004 ( 2007 ) .
i. antoniadis , e. dudas , d. m. ghilencea and p. tziveloglou , nucl .
b * 808 * , 155 ( 2009 ) .
r. s. chivukula and h. georgi , phys .
b * 188 * , 99 ( 1987 ) .
a. j. buras , p. gambino , m. gorbahn , s. jager and l. silvestrini , phys .
b * 500 * , 161 ( 2001 ) .
g. dambrosio , g. f. giudice , g. isidori and a. strumia , nucl .
b * 645 * , 155 ( 2002 ) .
k. j. bae , phys .
d * 82 * , 055004 ( 2010 ) .
t. aaltonen _ et al . _
[ cdf collaboration ] , phys .
* 100 * , 161802 ( 2008 ) ; v. m. abazov _ et al . _
[ d0 collaboration ] , phys .
* 101 * , 241801 ( 2008 ) .
cdf collaboration , cdf public note 10206 ; d0 collaboration , d0 note 6098-conf .
a. lenz and u. nierste , jhep * 0706 * , 072 ( 2007 ) .
v. m. abazov _ et al . _
[ d0 collaboration ] , phys .
rev . * d84 * , 052007 ( 2011 ) .
v. m. abazov _ et al . _
[ d0 collaboration ] , phys .
d * 82 * , 032001 ( 2010 ) .
d. asner _
[ heavy flavor averaging group ] , arxiv:1010.1589 [ hep - ex ] .
z. ligeti , m. papucci , g. perez and j. zupan , phys . rev
. lett . * 105 * , 131601 ( 2010 )
. t. aaltonen _ et al .
_ [ cdf collaboration ] , arxiv:1107.2304 [ hep - ex ] .
lhcb collaboration , lhcb - conf-2011 - 037 ; s. chatrchyan _ et al .
_ [ cms collaboration ] , arxiv:1107.5834 [ hep - ex ] .
lhcb collaboration , lhcb - conf-2011 - 047 ; cms collaboration , cms - pas - bph-11 - 019 .
a. j. buras , phys .
b * 566 * , 115 ( 2003 ) .
a. j. buras , p. h. chankowski , j. rosiek and l. slawianowska , nucl .
b * 659 * , 3 ( 2003 ) .
m. s. carena , a. menon , r. noriega - papaqui , a. szynkman and c. e. m. wagner , phys .
d * 74 * , 015009 ( 2006 ) .
r. hempfling , phys .
d * 49 * , 6168 ( 1994 ) ; l. j. hall , r. rattazzi and u. sarid , phys .
d * 50 * , 7048 ( 1994 ) ; m. s. carena , m. olechowski , s. pokorski and c. e. m. wagner , nucl .
b * 426 * , 269 ( 1994 ) .
t. blazek , s. raby and s. pokorski , phys .
d * 52 * , 4151 ( 1995 ) .
m. s. carena , d. garcia , u. nierste and c. e. m. wagner , nucl .
b * 577 * , 88 ( 2000 ) ; phys .
b * 499 * , 141 ( 2001 ) . c. hamzaoui , m. pospelov and m. toharia , phys .
d * 59 * , 095005 ( 1999 ) ; s. r. choudhury and n. gaur , phys .
b * 451 * , 86 ( 1999 ) ; k. s. babu and c. f. kolda , phys .
* 84 * , 228 ( 2000 ) ; a. dedes , h. k. dreiner and u. nierste , phys .
lett . * 87 * , 251804 ( 2001 ) ; g. isidori and a. retico , jhep * 0111 * , 001 ( 2001 ) .
a. brignole , j. a. casas , j. r. espinosa and i. navarro , nucl .
b * 666 * , 105 ( 2003 ) .
m. carena , k. kong , e. ponton and j. zurita , phys .
d * 81 * , 015001 ( 2010 ) .
i. antoniadis , e. dudas , d. m. ghilencea and p. tziveloglou , nucl .
b * 831 * , 133 ( 2010 ) ; nucl .
b * 848 * , 1 ( 2011 ) .
see e.g. i. antoniadis , e. dudas and d. m. ghilencea , jhep * 0803 * , 045 ( 2008 ) ; i. antoniadis , e. dudas , d. m. ghilencea and p. tziveloglou , nucl .
b * 808 * , 155 ( 2009 ) ; n. bernal , k. blum , y. nir and m. losada , jhep * 0908 * , 053 ( 2009 ) ; n.
bernal and a. goudelis , jcap * 1003 * , 007 ( 2010 ) ; m. carena , e. ponton and j. zurita , phys . rev .
d * 82 * , 055025 ( 2010 ) .
k. blum , c. delaunay , m. losada , y. nir and s. tulin , jhep * 1005 * ( 2010 ) 101 . l. hofer , u. nierste and d. scherer , jhep * 0910 * , 081 ( 2009 ) . g. blankenburg and g. isidori , arxiv:1107.1216 [ hep - ph ] . m. s. carena , j. r. espinosa , m. quiros and c. e. m. wagner , phys . lett .
b * 355 * , 209 ( 1995 ) ; a. pilaftsis and c. e. m. wagner , nucl .
b * 553 * , 3 ( 1999 ) .
a. l. kagan , g. perez , t. volansky and j. zupan , phys .
d * 80 * , 076002 ( 2009 ) .
m. ciuchini , e. franco , v. lubicz , g. martinelli , i. scimemi and l. silvestrini , nucl .
b * 523 * , 501 ( 1998 ) ; a. j. buras , m. misiak and j. urban , nucl .
b * 586 * , 397 ( 2000 ) .
d. becirevic , v. gimenez , g. martinelli , m. papinutto and j. reyes , jhep * 0204 * , 025 ( 2002 ) k. blum , c. delaunay and y. hochberg , phys .
d * 80 * , 075004 ( 2009 ) g. degrassi , p. slavich , phys . rev . * d81 * , 075001 ( 2010 )
. s. m. barr , phys .
lett . * 68 * , 1822 - 1825 ( 1992 ) .
s. weinberg , phys .
* 63 * , 2333 ( 1989 ) .
s. m. barr , a. zee , phys . rev .
lett . *
65 * , 21 - 24 ( 1990 ) .
a. pilaftsis , nucl .
b * 644 * ( 2002 ) 263 .
d. a. demir , o. lebedev , k. a. olive , m. pospelov and a. ritz , nucl .
b * 680 * ( 2004 ) 339 .
g. boyd , a. k. gupta , s. p. trivedi , m. b. wise , phys .
* b241 * , 584 ( 1990 ) ; e. braaten , c. -s .
li , t. -c .
yuan , phys .
* 64 * , 1709 ( 1990 ) .
p. bechtle , o. brein , s. heinemeyer , g. weiglein and k. e. williams , comput .
phys . commun .
* 181 * , 138 ( 2010 ) ; comput .
commun . * 182 * , 2605 ( 2011 ) .
cms collaboration , cms - pas - hig-11 - 022 ; atlas collaboration , atlas - conf-2011 - 135 .
|
we study a minimal flavor violating extension of the mssm , where higher dimensional operators in the khler potential induce tree level non - holomorphic higgs couplings that are controlled by the scale of the physics beyond the mssm and analyze their possible impact on cp violation in @xmath0 and @xmath1 mixing .
we consider results on the time dependent cp asymmetries in @xmath2 and @xmath3 from lhcb , in @xmath2 from cdf and d0 and in @xmath4 from the b factories as well as the measurement of an anomalous like - sign dimuon charge asymmetry at d0 .
taking into account the stringent bounds on the branching ratio of the rare @xmath5 decay , we investigate to which extent the framework allows to address the observed @xmath6 discrepancies in fits of the unitarity triangle .
we find that a non - standard @xmath1 mixing phase , that is in agreement with the current bounds on cp violation in @xmath0 mixing , requires the presence of higher dimensional operators both in the khler potential and the superpotential .
the corresponding region of parameter space is characterized by small @xmath7 , a light higgs spectrum with masses below @xmath8 gev and will be probed by future measurements at lhcb .
|
a major shift in the field of thermodynamics in the last century was from idealized equilibrium processes to natural irreversible processes [ 1 - 4].chemical reactions continue to play a pivotal role in this development and provide significant motivation in studying the non - equilibrium thermodynamic properties of systems _ in vitro _ as well as _ in vivo _ [ 5 - 10 ] . since a closed system always tends to thermodynamic equilibrium ( te ) , a natural generalization in the theory of irreversible thermodynamics has been achieved via the concept of a steady state @xcite . in this regard , the quantity of primary importance is the entropy production rate ( epr ) @xcite .
the epr vanishes for a closed system in the long - time limit that reaches a true te . on the other hand , epr is positive definite for a steady state that can emerge in an _ open _ system .
the easiest way to model such a system in the context of chemical reactions is to assume that concentrations of some of the reacting species are held fixed @xcite . under this condition , aptly known as the chemiostatic condition @xcite , epr tends to a non - zero constant , reflecting a steady dissipation rate ( sdr ) to sustain the system away from equilibrium @xcite .
the corresponding steady state is denoted as the non - equilibrium steady state ( ness ) @xcite .
this concept has been extensively used in analyzing single - molecule kinetic experiments @xcite .
the ness also includes the te as a special case when detailed balance ( db ) is obeyed @xcite , thus providing a very general framework .
recently , an important progress was made in the theory and characterization of ness , considering a master equation formalism @xcite .
these studies have established that the classification of ness requires _ not only _ the steady distribution ( as in te ) but _ also _ the stationary fluxes or probability currents .
this approach enables one to identify _ all possible _ combinations of transition rates that ultimately lead the system to the _ same _ ness .
however , these nesss in general have different values of the epr , and hence the sdr .
this proposition prompts one to check ( i ) how states with the same epr at ness can be generated and ( ii ) whether there exist ways to distinguish these states . here
, we shall address both the issues by considering an enzyme - catalyzed reaction under chemiostatic condition . expressing the epr as a function of experimentally measurable reaction rate , we emphasize also
that , the quantity that identifies the various nesss having the same epr is linked with the enzyme efficiency , a useful measure that is expressible in terms of enzyme kinetic constants .
the basic scheme of enzyme catalysis within the michaelis - menten ( mm ) framework with reversible product formation step is shown in fig.[fig1 ] . under chemiostatic condition ,
@xmath0 $ ] and @xmath1 $ ] are kept constant by continuous injection and withdrawal , respectively .
this is the simplest model to mimic an open reaction system . unlike the usual case of full enzyme recovery with total conversion of substrate into product in a closed system , here both the concentrations of free enzyme e and the enzyme - substrate complex
es reach a steady value . also , instead of the rate of product formation , the progress of reaction is characterized by the rate of evolution of @xmath2 $ ] ( or @xmath3 $ ] ) .
we define the pseudo - first - order rate constants as @xmath4 $ ] and @xmath5.$ ] concentration of e is denoted by @xmath6 and that of es is given by @xmath7 we have then @xmath8 here @xmath9 is a constant that stands for the total enzyme concentration
. then the rate of the reaction , @xmath10 , is written as @xmath11 where @xmath12 with the initial condition , @xmath13 , the time - dependent solution is given as @xmath14 the steady state enzyme concentration corresponds to the long - time limit of eq.([ct ] ) : @xmath15 at any steady state , we thus note @xmath16 the fluxes of the reaction system are defined pairwise as @xcite @xmath17 @xmath18 from eq.([const ] ) , eq.([remm ] ) , eq.([j1 ] ) and eq.([j2 ] ) , one gets @xmath19 at the steady state , eq.([rateflx ] ) leads to @xmath20 an ness is characterized by a non - zero flux , @xmath21 . at te ,
the fluxes vanish for both the reactions .
one may note , then the system satisfies db . the conjugate forces of the fluxes given in eqs ( [ j1])-([j2 ] ) are defined as @xcite @xmath22 @xmath23 corresponding to the scheme depicted in fig.[fig1 ] , the epr is then given by @xcite @xmath24 we set here ( and henceforth ) the boltzmann constant @xmath25 . in the present case ,
the steady value of epr becomes @xmath26 therefore , unless the substrate and the product take part in equilibrium , the reaction system reaches an ness with a sdr equal to @xmath27 .
the problem is now transparent .
if the rate constants become different , the steady concentrations will also differ .
but , one can adjust them in such a way that @xmath27 remains the same . in these situations
, one needs an additional parameter to distinguish these states . to proceed ,
we define a small deviation in @xmath6 around ness as @xmath28 it then follows from eq.([const ] ) that @xmath29 from eq.([remm ] ) and eq.([delmm1 ] ) , the reaction rate becomes @xmath30 now , putting eqs ( [ j1])-([f2 ] ) and eqs ( [ delmm1])-([vnsmm ] ) in eq.([eprmm ] ) and taking only the first terms of the logarithmic parts , we obtain the epr close to ness as @xmath31 here @xmath32 @xmath33 @xmath34 as @xmath10 vanishes at any steady state , the sdr at ness is given by @xmath35 however , at te , @xmath36 one may check that here db holds : @xmath37 inspection of eq.([eprmm1 ] ) reveals that , near ness , @xmath38 varies _ linearly _ with @xmath10 with a slope @xmath39 .
thus , while @xmath40 distinguishes an ness from a true te , @xmath39 plays the same role in identifying systems with the _ same _ sdr but having _ different _ time profiles .
in this section , we consider various situations where the reaction system reaches ness with the same sdr . focusing on eq.([x3 ] ) , the different cases that keep @xmath40 invariant are discussed next .
case a : any parent choice of rate constants .
case b : only @xmath41 and @xmath42 are exchanged .
case c : only @xmath43 and @xmath44 are exchanged .
case d : both @xmath45 and @xmath46 are exchanged .
case e : both @xmath47 and @xmath48 are exchanged .
case f : both @xmath49 and @xmath50 are exchanged .
case g : @xmath41 changed to @xmath51 , @xmath43 changed to @xmath52 , @xmath42 changed to @xmath53 and @xmath44 changed to @xmath54 , such that @xmath55 it can be easily verified that cases d and e possess not only identical @xmath40 but also the same @xmath39 and @xmath56 .
this is true for cases a and f as well .
so , we do not consider cases e and f any further . a simple explanation of the equivalence is given in fig.[fig2 ] schematically , based on reflection symmetry . to explore the characteristics of various cases given above
, we take the rate constants from the single molecule experimental study of english _ et al . _
@xcite on the _ escherichia coli _ @xmath57-galactosidase enzyme .
they are as follows : @xmath58 e07 @xmath59 e04 @xmath60 e02 @xmath61 we clarify that , in their study @xcite , @xmath42 had actually been shown to be a fluctuating quantity with a distribution .
however , only an _ average _ value of @xmath42 will suffice our purpose .
the constant substrate concentration is set at @xmath0=1.0 $ ] e02 @xmath62 and thus , @xmath4 = 5.0 $ ] e03 @xmath63 .
we choose @xmath64 e-05 @xmath63 to make the reaction scheme almost identical to the conventional mm kinetics . here
@xmath65 @xmath66 with magnitudes given above represents the parent choice of rate constants , _
i.e. _ , case a. the value of the constant @xmath67 e01 , in case g. the time - evolution of epr @xmath38 , determined using both the exact ( eq.([eprmm ] ) ) and the approximate ( eq.([eprmm1 ] ) ) expressions , are shown in fig.[fig3 ] , for the various cases .
the concentrations @xmath68 are made dimensionless by scaling with respect to the total enzyme concentration @xmath9 .
this ensures that @xmath38 has the unit of @xmath63 . from the figure ,
it is evident that eq.([eprmm1 ] ) nicely approximates the behavior near ness .
specifically , the curves of exact and approximate cases merge quite well for any @xmath69 e-04 s. the evolution of reaction rate @xmath10 is shown in fig.[fig4 ] for all the distinct cases .
the curves are displayed over a time - span where eq.([eprmm1 ] ) is valid , as mentioned above .
this gives us a quantitative understanding of the magnitude of @xmath10 up to which the _ close to _ ness approximation , and hence eq.([eprmm1 ] ) , is valid .
we note the variation of @xmath38 as a function of @xmath10 in all the relevant cases in fig.[fig5 ] .
both the exact ( fig.[fig5](a ) ) as well as the approximate results ( fig.[fig5](b ) ) are shown .
two features are interesting .
first , in all the situations , the system reaches an ness with identical @xmath70 e03 @xmath63 .
secondly , the quantity that distinguishes one case from the other is the slope @xmath39 of @xmath38 vs. @xmath10 curve near the ness
. this slope can be positive as well as negative .
one may like to next investigate the role of the rate constants in governing the overall dissipation in various cases . specifically
, we like to enquire if the efficiency of the enzyme has anything to do with the total dissipation . in this context
, it may be recalled that , the conventional mm kinetics requires the rate constant @xmath44 to be negligible compared with the others .
so , the enzyme kinetic constants , like the mm constant @xmath71 and catalytic efficiency @xmath72 , are meaningful in the limit @xmath73 .
our choice of parent rate constants ensures that in case a , the system follows mm kinetics .
case b , which leaves @xmath44 unchanged and case g , which changes @xmath44 to @xmath54 ( with @xmath74 e01 ) , can also be included within the mm scheme .
but , cases c to f , which exchange @xmath44 with any one of the other bigger rate constants , can not follow the usual mm kinetics . therefore , we focus on cases a , b and g in finding any possible connection between the kinetic constants of the enzyme and the total dissipation .
while the sdr @xmath27 is the same for all of them , the time - integrated epr , giving the total entropy production , is different .
we define it as @xmath75 the upper limit @xmath76 is fixed at such a time when all the systems reach ness .
in the present set of cases , we find that setting @xmath77 e-03 s is satisfactory .
the values of @xmath78 and @xmath79 ( determined by integrating @xmath38 from eq.([eprmm ] ) ) are listed in table [ tab1 ] , along with the slope @xmath39 [ see eq.([eprmm1 ] ) ] .
it is clear from the data that , in going from case a to case g , @xmath80 gradually increases , whereas @xmath81 falls .
both these features indicate that the enzyme becomes _ less efficient_. more interesting is to note that the corresponding @xmath79 values also exhibit a decreasing trend from case a to case g. thus , we can say that , with identical sdr , the more efficient enzyme ( bigger @xmath81 and smaller @xmath80 ) involves higher _ total _ dissipation .
this can be rationalized by the fact that , higher efficiency corresponds to a _
faster _ conversion of substrate into product .
this implies an increased irreversibility in the process .
consequently , a higher entropy production is noted .
.values of the quantities @xmath39 , @xmath80 , @xmath81 and @xmath79 for cases a , b and g. [ cols="<,^,^,^,^",options="header " , ] [ tab1 ] before ending this section , we mention briefly the fate of the different situations when db , eq.([db ] ) , gets satisfied
. in this scenario , whatever be the values of the rate constants , the final epr is trivially zero as the reaction system reaches te [ see eq.([epreq ] ) ] .
for the same reason , @xmath39 also becomes zero [ see eq.([y3 ] ) ] .
however , it follows from eq.([eprmm1 ] ) that , epr varies _ quadratically _ with @xmath10 near te .
then , in principle , @xmath56 _ can _ distinguish systems reaching te .
it is easy to see from eq.([z3 ] ) that , cases a , b and g possess different values for @xmath56 and hence they can be identified by following the behavior of epr with the reaction rate .
the mm kinetics , shown in fig.[fig1 ] , with a single intermediate in the form of the es complex , is _ exactly _ solvable .
we now generalize this scheme to an enzyme catalysis reaction having n number of species .
these include the free enzyme e and ( n-1 ) intermediates , under similar chemiostatic condition as discussed in section ii .
the reaction scheme is depicted in fig.[fig6 ] .
essentially , the species @xmath82 refer to the various conformers of the enzyme - substrate complex .
the corresponding rate equations are given as @xmath83 with @xmath84 being the concentration of species @xmath85 at time @xmath86 .
the following periodic boundary conditions hold : @xmath87 @xmath88 we have set @xmath4 $ ] and @xmath89.$ ] the flux @xmath90 due to the i - th reaction is defined as @xmath91 the expression of epr then becomes @xmath92 it is generally not possible to solve the set of coupled equations analytically for a system of arbitrary size .
however , again focusing on a situation close to the ness , one can get some insights .
for that purpose , we define small deviations in species concentrations from their respective ness values as @xmath93 for a short time interval @xmath76 , using finite difference approximation , one gets @xmath94 putting eqs ( [ del])-([del1 ] ) in eq.([dai ] ) , we get @xmath95 as the reactions are coupled , so the @xmath96s are related to each other and can be expressed in terms of any one of them , say @xmath97 then , one can write @xmath98 next we will discuss the scheme to determine the @xmath99s .
the set of coupled equations ( [ delness ] ) , with the help of eq.([relatn ] ) , can be cast in the matrix form @xmath100 here @xmath101 is a @xmath102 matrix with @xmath103 and @xmath104 is a @xmath105 matrix with the property @xmath106 @xmath107 the non - zero matrix elements are @xmath108 @xmath109 @xmath110 from eq.([mat ] ) and eq.([matel ] ) , we obtain a recursion relation @xmath111 with the boundary conditions : @xmath112 the first of the relations becomes @xmath113 then , it is easy to follow from eq.([recur ] ) that , all the other @xmath114s can be expressed in terms of @xmath115 from the condition @xmath116 we get @xmath117 and using eq.([relatn ] ) , we have @xmath118 from eqs ( [ matel1])-([recur1 ] ) and eq.([sumf ] ) , one can determine the @xmath99s in eq.([relatn ] ) .
we are now ready to explore the epr near the ness . from eq.([dai ] ) , we have @xmath119 at ness . as we have chosen to express all the deviations in concentration from the ness in terms of @xmath120 ,
so we take the reaction rate as @xmath121 .
then , from eq.([dai ] ) with @xmath122 and using eq.([relatn ] ) along with the periodic boundary conditions , we get near ness @xmath123 where @xmath124 now putting eq.([del ] ) , eq.([relatn ] ) , eq.([jns ] ) and eq.([vns ] ) in eq.([eprc ] ) and also using the smallness of @xmath96s , the epr near ness becomes @xmath125 @xmath126 with @xmath127 @xmath128 @xmath129 eq.([epr1 ] ) is the generalized version of eq.([eprmm1 ] ) , confirming that expression of the epr as a functional of reaction rate possesses a universal character .
the next task is , whether states having the same sdr , _
i.e. _ , identical @xmath130 , can be generated for the n - cycle .
an obvious clue comes from the invariance of a cycle under rotation .
thus , if the steady concentrations @xmath131 are represented as n points uniformly placed on a circle , then rotations by an angle @xmath132 , defined as @xmath133 will just redistribute the @xmath131 values .
this keeps the steady flux @xmath134 in eq.([x1 ] ) unchanged .
therefore , for a n - cycle , there are _ at least _ ( n-1 ) ways to interchange the rate constants @xmath135 that will lead the reaction system to states with the same sdr .
we illustrate this result here by taking the simplest non - trivial case of a triangular network as an example .
one can see from eq.([ang ] ) that , for a triangular network with @xmath136 , _ at least _ two kinds of changes of the rate constants keep the sdr unchanged .
they are given below : case 1 .
any parent choice of rate constants .
case 2 . change @xmath137 with the boundary condition @xmath138 .
case 3 . change @xmath139 , @xmath140 and @xmath141 . one can generate additional ways to keep @xmath130 fixed with some added constraints on the rate constants .
two pairs of situations [ cases 4 and 5 , and 6 and 7 ] are the following : case 4 .
any parent choice of rate constants with @xmath142 .
change @xmath143 , @xmath144 , @xmath145 , @xmath146 , @xmath147 and @xmath148 .
any parent choice of rate constants with @xmath149 .
change @xmath150 , @xmath151 , @xmath152 , @xmath153 , @xmath154 and @xmath155 .
+ all the above variants have been numerically studied and shown in fig.[fig7 ] where the epr , determined exactly by eq.([eprc ] ) , is plotted as a function of reaction rate @xmath121 for each of the cases .
it is evident from the figure that the sdr are identical for the respective bunch of cases .
but they can be distinguished by following the @xmath156 vs. @xmath10 curve in the small-@xmath10 regime .
in summary , the present endeavor has been to characterize steady states with the same non - zero sdr .
we have found that the variation of epr with the reaction rate near completion of the reaction is a nice indicator to distinguish such states .
particularly important is the role of the slope of @xmath38 vs. @xmath10 curve near @xmath157 .
this has been substantiated by studying enzyme - catalysed reactions as an exactly - solvable test case .
we have also noticed , the leading term that accounts for the variation depends on the rate constants , more specifically on the enzyme efficiency .
it is gratifying to observe that the more efficient enzyme incurs higher total dissipation .
the physical appeal is immediate .
a more efficient enzyme approaches the steady state more quickly .
this implies the process becomes more irreversible .
hence , @xmath79 becomes higher .
one more notable point is the following .
the sdr is equal to the steady heat dissipation rate .
our study reveals that enzymes with very different efficiencies can show the same heat dissipation rate at steady state .
an extension to cases of higher complexities involving various conformers of the enzyme - substrate complex has also been envisaged .
further studies along this line on enzymes with multiple sites may be worthwhile .
k. banerjee acknowledges the university grants commission ( ugc ) , india for dr .
d. s. kothari fellowship .
k. bhattacharyya thanks crnn , cu , for partial financial support .
|
a non - equilibrium steady state is characterized by a non - zero steady dissipation rate .
chemical reaction systems under suitable conditions may generate such states .
we propose here a method that is able to distinguish states with identical values of the steady dissipation rate .
this necessitates a study of the variation of the entropy production rate with the experimentally observable reaction rate in regions close to the steady states . as an exactly - solvable test case
, we choose the problem of enzyme catalysis .
link of the total entropy production with the enzyme efficiency is also established , offering a desirable connection with the inherent irreversibility of the process .
the chief outcomes are finally noted in a more general reaction network with numerical demonstrations .
* states with identical steady dissipation rate : role of kinetic constants in enzyme catalysis * .1 in kinshuk banerjee and kamal bhattacharyya .05 in _ department of chemistry , university of calcutta , 92 a.p.c .
road , _
kolkata 700 009 , india .
pacs : 05.70.ln , 82.39.-k , 82.20.-w .05 in keywords : entropy production rate , dissipation , enzyme efficiency , + reaction network
|
for the group @xmath7 the irreducible representations are the @xmath8 . if @xmath9 , @xmath10 , @xmath11 are nonnegative integers such that @xmath12 , and if for @xmath13 and @xmath14 one puts @xmath15 then the map @xmath16 is a bilinear and @xmath17-equivariant map from @xmath18 onto @xmath19 .
the map @xmath20 is an isomorphism of @xmath17-modules ( `` clebsch - gordan decomposition '' ) , cf .
@xcite , p. 122 .
the maps @xmath21 are called transvectants ( _ berschiebungen _ in german ) .
their importance derives from the fact that they make the preceding isomorphism _
explicit_. now let @xmath22 , and let @xmath23 be the irreducible @xmath24-module whose highest weight has numerical labels @xmath25 where @xmath26 , @xmath27 are non - negative integers .
a representation @xmath28 decomposes similarly into irreducible summands , and the cartan - killing theory of highest weights allows us to compute the multiplicity with which @xmath29 occurs ( an entirely similar statement holds of course for @xmath30 or any semi - simple linear algebraic group ) ; in other words , the theory of highest weights asserts the existence of an isomorphism @xmath31 of irreducible representations of a semi - simple algebraic group , but does not give us the isomorphism , at least it is not easy to unravel from this theory . on the other hand ,
it is often important to know the isomorphism , e.g. * in _ the problem of rationality for fields of invariants _ , see @xcite for a survey .
here one almost always has to check certain nondegeneracy statements for maps of the form @xmath32 ( or similar maps constructed by representation theory ) , and for this one has to know the maps explicitly .
often one is dependent on computer aid when studying these maps , one needs fast methods for computing them . * in _ the geometry of syzygies _ ( see @xcite ) .
here one wants to understand differentials of certain chain complexes constructed by representation theoretic means , as for example by kempf s geometric technique based on taking direct images of koszul complexes ; here computational efficiency is again one of the desiderata . in the first sections of this article we give a very simple method , contained in theorem [ tclebschgordan ] , to obtain a basis for the space @xmath33 in particular
, it enables one to immediately write down matrix representatives for the occurring maps .
moreover , theorem [ tclebschgordan ] gives a factorization of all such maps into certain elementary building blocks and explicit formulas for them ; during the proof , which occupies sections 2 through 4 , we also set up a natural bijection between the basis maps and the expansions of young diagrams which occur in the combinatorics of the littlewood - richardson rule .
+ one is tempted to think that something of this sort should have been discovered before , but we could not find it in the classical or modern literature . in any event , for us
the main reason for introducing this computational scheme is that it is the one we use and found most convenient for applications to the problem of rationality for invariant function fields ; a sample of such applications is contained in section 5 .
first of all , theorem [ tclebschgordan ] allows one to prove rationality for many spaces @xmath34 via the double bundle method ( @xcite ) where @xmath23 is a space of mixed tensors .
we prove rationality of @xmath35 as an example
. for the double bundle method one uses linear fibrations over projective spaces ; one may also consider linear fibrations over more general grassmannians ; see proposition [ pfibrationovergrass ] .
this was already mentioned in @xcite , but has not yet found any application to our knowledge .
one problem is that one needs to know the stable rationality for quotients of grassmannians @xmath36 where @xmath37 is a linear representation of a reductive group @xmath38 . in proposition [ pstabratgrassmannians ]
we give a criterion for stable rationality that applies in some cases if @xmath38 is a group of type @xmath6 . using this and theorem [ tclebschgordan ]
, we prove the rationality of the moduli space of plane curves of degree @xmath5 , i.e. @xmath39 , in theorem [ trationalityv34 ] .
this case can not be handled by the double bundle method , cf .
remark [ rnoalternatives ] , nor has it been treated by any other method so far .
it is well known that isomorphism classes of irreducible @xmath41-modules correspond bijectively to @xmath11-tuples of integers @xmath42 with @xmath43 via associating to such a representation its highest weight @xmath44 where @xmath45 is the @xmath46-th coordinate function of the standard diagonal torus in @xmath41 .
the space of the corresponding irreducible representation will be denoted @xmath47 . here
@xmath48 is called the _ schur functor _ ( cf .
if all @xmath49 are non - negative , one associates to @xmath50 the corresponding _ young diagram _ whose number of boxes in its @xmath46-th row is @xmath51 ; @xmath50 will often be identified with this young diagram .
for example , ( 4,2 ) ( -3 , 1.5)@xmath52 ( -1 , 1.5)@xmath53 ( 1 , 1.5)@xmath54 ( 3 , 1.5)@xmath53 ( 4.5,1.7)(0.4,0.4 ) ( 4.5,1.3)(0.4,0.4 ) ( 4.5,0.9)(0.4,0.4 ) we list some properties of the schur functors for future use : * one has @xmath55 as @xmath30-representations if and only if @xmath56 is constant for all @xmath46 .
in fact , in this case @xmath57 * @xmath58 . *
the representation @xmath23 of @xmath59 is isomorphic to @xmath60 .
* for a young diagram @xmath50 with more than @xmath11 rows one has @xmath61 by definition .
the littlewood - richardson rule to decompose @xmath62 into irreducible factors where @xmath50 , @xmath63 are young diagrams ( cf .
a.1 ) says the following ( in this notation we suppress the space which the schur functors are applied to , since it plays no role ) : label each box of @xmath63 with the number of the row it belongs to .
then expand the young diagram @xmath50 by adding the boxes of @xmath63 to the rows of @xmath50 subject to the following rules : * the boxes with labels @xmath64 of @xmath63 together with the boxes of @xmath50 form again a young diagram ; * no column contains boxes of @xmath63 with equal labels .
* when the integers in the boxes added are listed from right to left and from top down , then , for any @xmath65 ( number of boxes of @xmath63 ) , the first @xmath66 entries of the list satisfy : each label @xmath67 ( @xmath68 ( number of rows of @xmath63)@xmath69 ) occurs at least as many times as the label @xmath70 .
we will call this configuration of boxes ( together with the labels ) a @xmath63-_expansion of _ @xmath50
. then the multiplicity of @xmath71 in @xmath72 is the number of times the young diagram @xmath73 can be obtained by expanding @xmath50 by @xmath63 according to the above rules , forgetting the labels .
for @xmath74 the following expansions are possible : ( 4,4.5 ) ( -4,4)(0.4,0.4 ) ( -3.6,4)(0.4,0.4 ) ( -3.2,4)(0.4,0.4)1 ( -2.8,4)(0.4,0.4)1 ( -4,3.6)(0.4,0.4 ) ( -3.6,3.6)(0.4,0.4)2 ( -1,4)(0.4,0.4 ) ( -0.6,4)(0.4,0.4 ) ( -0.2,4)(0.4,0.4)1 ( 0.2,4)(0.4,0.4)1 ( -1,3.6)(0.4,0.4 ) ( -1,3.2)(0.4,0.4)2 ( 2,4)(0.4,0.4 ) ( 2.4,4)(0.4,0.4 ) ( 2.8,4)(0.4,0.4)1 ( 2,3.6)(0.4,0.4 ) ( 2.4,3.6)(0.4,0.4)1 ( 2.8,3.6)(0.4,0.4)2 ( 4.5,4)(0.4,0.4 ) ( 4.9,4)(0.4,0.4 ) ( 5.3,4)(0.4,0.4)1 ( 4.5,3.6)(0.4,0.4 ) ( 4.9,3.6)(0.4,0.4)1 ( 4.5,3.2)(0.4,0.4)2 ( -3,2)(0.4,0.4 ) ( -2.6,2)(0.4,0.4 ) ( -2.2,2)(0.4,0.4)1 ( -3,1.2)(0.4,0.4)1 ( -3,1.6)(0.4,0.4 ) ( -2.6,1.6)(0.4,0.4)2 ( 0,2)(0.4,0.4 ) ( 0.4,2)(0.4,0.4 ) ( 0.8,2)(0.4,0.4)1 ( 0,0.8)(0.4,0.4)2 ( 0,1.6)(0.4,0.4 ) ( 0,1.2)(0.4,0.4)1 ( 3,2)(0.4,0.4 ) ( 3.4,2)(0.4,0.4 ) ( 3,1.2)(0.4,0.4)1 ( 3,1.6)(0.4,0.4 ) ( 3.4,1.6)(0.4,0.4)1 ( 3.4,1.2)(0.4,0.4)2 ( 5.5,2)(0.4,0.4 ) ( 5.9,2)(0.4,0.4 ) ( 5.5,0.8)(0.4,0.4)2 ( 5.5,1.6)(0.4,0.4 ) ( 5.9,1.6)(0.4,0.4)1 ( 5.5,1.2)(0.4,0.4)1 hence we have the following decomposition @xmath75 for @xmath76 the combinatorics of the littlewood - richardson rule can be handled explicitly . for this
let @xmath77 be a summand of @xmath78 .
in the following we set @xmath79 , @xmath80 and let @xmath81 be the unique young diagram corresponding to @xmath77 in the decomposition of @xmath82 .
[ lpq ] expand the young diagram of @xmath50 by adding @xmath83 boxes with label @xmath84 to row @xmath46 and afterwards @xmath85 boxes with label @xmath86 to row @xmath46 ( see figure [ fpq ] ) .
this is a @xmath63-expansion of @xmath50 if and only if the following inequalities hold : 1 .
1 . @xmath87 [ ipositivityp ] 2 .
@xmath88 [ ipositivityq ] 2 .
1 . @xmath89 [ ioverlap1row2 ] 2 .
@xmath90 [ ioverlap1row3 ] 3 .
@xmath91 [ ioverlap2row2 ] 4 .
@xmath92 [ ioverlap2row3 ] 3 .
1 . @xmath93 [ istring1 ] 2 .
@xmath94 [ istring2 ] 3 .
@xmath95 [ istring3 ] 4 .
1 . @xmath96 [ itotal1 ] 2 .
@xmath97 [ itotal2 ] the inequalities ( [ ipositivityp ] ) and ( [ ipositivityq ] ) are obvious positivity conditions .
( [ ioverlap1row2 ] ) and ( [ ioverlap1row3 ] ) ensure that the boxes of @xmath50 together with the boxes of @xmath63 with label @xmath84 form again a young diagram and there is at most one label @xmath84 in every column .
( [ ioverlap2row2 ] ) and ( [ ioverlap2row3 ] ) guarantee that the boxes of @xmath50 together with all boxes of @xmath63 form again a young diagram and there is at most one label @xmath86 in every column .
( [ istring1 ] ) , ( [ istring2 ] ) and ( [ istring3 ] ) encode that the string of labels read from right to left and from top down always contains more @xmath84 s then @xmath86 s .
the last two equations reflect that the total number of @xmath84 s and @xmath86 s is given by the young diagram describing @xmath98 .
for given @xmath99 the equations above leave only one unknown : [ loneunknown ] let @xmath100 be the number of labelled boxes in the third row of the @xmath63-expansion . let furthermore @xmath101 be the number of @xmath84 s in the third row and @xmath102 the difference between the number of @xmath84 s in the second row and the number of @xmath86 s in the third row . with this
we obtain 1 .
@xmath103 [ ip3 ] 2 .
@xmath104 [ iq3 ] 3 .
@xmath105 [ ip2 ] 4 .
@xmath106 [ iq2 ] 5 .
@xmath107 [ ip1 ] 6 .
@xmath108 [ iq1 ] 7 .
@xmath109 [ is ] 8 . @xmath110
[ it ] ( [ ip3 ] ) , ( [ iq3 ] ) and ( [ ip2 ] ) follow from the definition of @xmath111 , @xmath66 and @xmath112 . since the total number of @xmath86 s is @xmath113 we obtain ( [ iq2 ] ) .
similarly @xmath96 implies ( [ ip1 ] ) . equation ( [ iq1 ] ) is true for all @xmath63-expansions . since we know that the number of labelled boxes is @xmath114 , the number of empty boxes is @xmath115 and the total number of boxes is @xmath116 , we obtain @xmath117 finally the total length of the first row is @xmath118 , on the one hand , and @xmath119 on the other .
this gives ( [ it ] ) . for given @xmath26 , @xmath27 , @xmath120 , @xmath113 , @xmath121 and @xmath122
there exists a @xmath63-expansion of @xmath50 of shape @xmath73 with @xmath123 if and only if @xmath111 satisfies the following inequalities : 1 .
1 . @xmath124 2 .
@xmath125 2 . 1 .
@xmath126 2 .
@xmath127 3 .
@xmath128 3 .
1 . @xmath129 2 .
@xmath130 substitute the expressions of lemma [ loneunknown ] into the inequalities of lemma [ lpq ] .
the inequality ( [ ioverlap2row2 ] ) gives @xmath131 which is always true since @xmath132 .
furthermore ( [ istring1 ] ) , ( [ itotal1 ] ) and ( [ itotal2 ] ) simplify to @xmath133 .
the numbering in the list above is taken from the corresponding inequalities in lemma [ lpq ] .
we put @xmath135 and denote by @xmath136 and @xmath137 dual bases in @xmath138 resp .
@xmath139 so that @xmath140 can be realized concretely as the kernel of the map @xmath141 we will always view @xmath140 in this way in the following . by @xmath142
we denote the equivariant projection from @xmath143 onto @xmath144 .
our purpose is to determine an explicit basis of the @xmath24-equivariant maps @xmath145 if @xmath144 is a subrepresentation of @xmath146 to this end we define the following elementary maps : @xmath147 @xmath148 @xmath149 @xmath150 note that an easier way of defining @xmath151 and @xmath152 is by saying that @xmath151 is multiplication by the determinant @xmath153 and @xmath152 multiplication by its inverse @xmath154 .
[ tclebschgordan ] suppose that @xmath144 occurs in the decomposition of @xmath155 and let @xmath66 and @xmath112 be defined as above .
let @xmath156 be the set of all integers @xmath111 satisfying the inequalities 1 .
1 . @xmath124 2 .
@xmath125 2 .
1 . @xmath126 2 .
@xmath127 3 .
@xmath128 3 .
1 . @xmath129 2 .
@xmath130 then a basis of @xmath157 is given by the restriction to @xmath158 of the maps @xmath159 if @xmath160 and @xmath161 if @xmath162
. a few explanatory remarks are in order .
[ rtacitmultiplication ] when writing a composition like @xmath163 , we suppress the obvious multiplication maps from the notation . for example if @xmath160 the map @xmath164 is composed with the multiplication map @xmath165 before applying @xmath166 to land in @xmath167 . before applying the equivariant projection @xmath142 we multiply again to map to @xmath168 which one , looking back at the definition of @xmath112 and @xmath66 , identifies as @xmath169
. this simplification of notation should cause no confusion .
note that the element @xmath170 is in the subspace @xmath171 by the definition of @xmath172 in formula ( [ formuladelta ] ) .
note also that the image of the map @xmath173 is a complement to the subspace @xmath144 in @xmath143 .
if @xmath160 we compute @xmath174 the inequalities above imply that this is a non - zero monomial in @xmath169 for all @xmath175 . if @xmath176 then @xmath177 is also a non - zero monomial in @xmath143 .
each nonzero bihomogeneous polynomial in the subspace @xmath178 contains monomials ( with nonzero coefficient ) divisible by @xmath179 . since the preceding monomials in cases @xmath180 resp
@xmath176 are not divisible by @xmath179 , a linear combination of them can be zero modulo @xmath181 only if this linear combination is already zero as a polynomial in @xmath169 .
but in both cases @xmath180 and @xmath176 , the degrees of the above monomials with respect to the variable @xmath182 are pairwise distinct , so they can not combine to zero nontrivially in @xmath169 .
to complete the picture , we will give in this section a method to compute the equivariant projection @xmath183 [ lpolynomialnature ] one has @xmath184 for some @xmath185 and certain @xmath186 ( the map @xmath187 is defined in formula [ formuladelta ] ) .
let us denote by @xmath188 the equivariant projection @xmath189 so that @xmath190 .
look at the diagram by schur s lemma , @xmath191 for some nonzero constants @xmath51 . on the other hand , @xmath192 therefore , since the assertion of the lemma holds trivially if one of @xmath26 or @xmath27 is zero , the general case follows by induction on @xmath193 .
note that to compute the @xmath194 in the expression of @xmath195 in lemma [ lpolynomialnature ] , it suffices to calculate the @xmath51 in formula [ formulalambdai ] which can be done by the rule @xmath196 which uses ( [ formulalambdai ] ) and the injectivitiy of @xmath197 .
in the following example we write down explicit matrix representatives for the maps given in [ tclebschgordan ] in one special case .
[ ematricesv(1,1 ) ] in the decomposition of @xmath198 , the representation @xmath199 occurs with multiplicity @xmath86 , corresponding to a two dimensional space @xmath200 of @xmath3-equivariant maps . here
@xmath201 and @xmath202 .
therefore a basis for this space of equivariant homomorphisms is given by @xmath203 and @xmath204 . to give matrix representatives of @xmath203 and @xmath204 we use the vectors @xmath205 ( in this order ) as a basis of the @xmath206-dimensional space @xmath199 .
using the definition of @xmath203 and @xmath204 we obtain : @xmath207 theorem [ tclebschgordan ] is of particular importance in applications to the question of rationality of quotient spaces @xmath209 . in the following , if a linear algebraic group @xmath38 acts on a variety @xmath210 , the quotient @xmath211 is always taken in the sense of rosenlicht : there is a non - empty @xmath38-invariant open subset @xmath212 for which a geometric quotient @xmath213 exists , and @xmath211 denotes any birational model for this quotient . + we need the following extension of the double bundle method of @xcite , see also @xcite . [ pfibrationovergrass ]
let @xmath37 and @xmath214 be representations of a connected reductive group @xmath38 with @xmath215 .
let @xmath216 be a subrepresentation of @xmath217 such that for generic @xmath218 the corresponding map in @xmath217 has full rank so that we get a rational map @xmath219&\mapsto \mathrm{ker}(u ) \ , .\end{aligned}\ ] ] let us assume furthermore that * @xmath220 is dominant ; this is equivalent to saying that a fibre @xmath221))$ ] has dimension @xmath222 .
* @xmath223 is stably rational in the sense that @xmath224 is rational for some @xmath225 .
* let @xmath226 be the kernel of the action of @xmath38 on @xmath227 : we require the existence of a @xmath228-linearized very ample line bundle @xmath229 on @xmath227 such that for the embedding @xmath227 in @xmath230 the locus of very stable points in @xmath227 ( i.e. stable with trivial stabilizer in @xmath231 ) is nonempty , @xmath226 acts trivially on @xmath232 , and there exists a @xmath228-linearized line bundle @xmath233 on the product @xmath234 cutting out @xmath235 on the fibres of the projection to @xmath227 .
let @xmath237the ( closure of ) the graph of @xmath220 , @xmath238 the restriction of the projection which ( maybe after shrinking @xmath227 ) we may assume to be a projective space bundle for which @xmath233 is a relatively ample bundle cutting out @xmath235 on the fibres .
the main technical point is the following result from descent theory ( @xcite , 7.1 , @xcite , thm .
1 ) : for all sufficiently large @xmath11 , putting @xmath239 the locus of very stable points in @xmath227 with respect to @xmath229 , @xmath240the locus of very stable points in @xmath210 w.r.t .
@xmath241 , one has @xmath242 , and a cartesian diagram @xmath243 such that @xmath233 descends to a line bundle @xmath244 on @xmath245 cutting out @xmath235 on the fibres of @xmath246 .
hence @xmath246 is also a zariski locally trivial projective bundle ( of the same rank as @xmath247 ) .
it then follows that @xmath236 is rational .
in fact @xmath249 and the multiplicity of @xmath250 in @xmath251 is @xmath86 . more precisely here @xmath252 and @xmath253 .
the most restrictive inequality of proposition [ tclebschgordan ] is @xmath254 in this situation .
therefore @xmath255 and @xmath256 are independent equivariant projections to @xmath257 .
we will use @xmath258 in this argument .
we now consider the induced map @xmath259 there are stable vectors in @xmath260 and on @xmath261 we can use @xmath262 as @xmath263-linearized line bundle .
moreover , @xmath260 is stably rational of level @xmath264 since the action of @xmath263 on pairs of @xmath265 matrices by simultaneous conjugation is almost free , and the quotient is known to be rational .
now consider a point @xmath266 . if the map @xmath267 has maximal rank @xmath268 , @xmath269 is well defined . in this situation
let @xmath270 be a generator of @xmath271 .
if the map @xmath272 has also rank @xmath268 we obtain that the fibre @xmath273))$ ] has the expected dimension . for a random @xmath274 it is straightforward to check all of this using a computer algebra program . notice that this can even be checked over a finite field , since the rank of a matrix is semicontinuous over @xmath275 .
see @xcite for a macaulay2-script
. we can therefore apply proposition [ pfibrationovergrass ] and obtain that @xmath248 is rational .
[ pstabratgrassmannians ] let @xmath37 be a ( finite dimensional as always ) representation of @xmath276 , @xmath247 prime .
let @xmath277 be the grassmannian of @xmath278-dimensional subspaces of @xmath37 .
assume : * the kernel @xmath226 of the action of @xmath38 on @xmath279 coincides with the center @xmath280 of @xmath281 and the action of @xmath231 on @xmath279 is almost free .
furthermore , the action of @xmath38 on @xmath37 is almost free and each element of @xmath226 not equal to the identity acts homothetically as multiplication by a primitive @xmath247th root of unity .
* @xmath282 .
* @xmath247 does not divide @xmath278 .
let @xmath284 be the affine cone over @xmath227 consisting of pure ( complety decomposable ) @xmath278-vectors .
we will show that under the assumptions of the proposition , the action of @xmath38 on @xmath285 is almost free .
this will accomplish the proof since @xmath286 is generically a torus bundle over @xmath223 hence zariski - locally trivial since tori are special groups , and the group @xmath287 is also special . recall that a linear algebraic group is called special if every tale locally trivial principal bundle for the group in question is zariski locally trivial .
see @xcite for the related theory .
so @xmath288 is birational to @xmath285 , hence rational , and @xmath38 is of course rational as a variety .
+ let @xmath289 be a general @xmath278-vector in @xmath290 . since @xmath282 and , in @xmath279 , @xmath291 ) = \dim \gamma$ ] since @xmath226 is finite and @xmath231 acts almost freely on @xmath279 , the @xmath292-dimensional projective linear subspace spanned by @xmath293 in @xmath279 will intersect the @xmath294 codimensional orbit @xmath295 $ ] only in @xmath296 $ ] .
hence , if an element @xmath297 stabilizes @xmath298 , it must lie in @xmath226 .
thus @xmath299 for a primitive @xmath247-th root of unity @xmath300 if @xmath301 .
but since @xmath247 does not divide @xmath278 , the case @xmath301 can not occur .
we have @xmath303 with multiplicity one and @xmath304 , @xmath305 . in this case @xmath306 and @xmath202 and the strongest restriction in theorem [ tclebschgordan ] is @xmath307 the projection @xmath308 is therefore given by @xmath309 . using this
we get an induced rational map @xmath310 with @xmath311 and @xmath312 .
moreover , proposition [ pstabratgrassmannians ] shows that @xmath313 is rational , and the action of @xmath314 , where @xmath226 is the center of @xmath3 , is almost free on @xmath315 .
moreover , for the @xmath3-linearized line bundle @xmath316 induced by the plcker embedding @xmath317 the locus of very stable points in the grassmannian is then nonempty ( one may choose @xmath86 linearly independent polynomial @xmath3-invariants @xmath318 , @xmath156 on @xmath319 of the same degree and gets a nonvanishing polynomial invariant via @xmath320 on the grassmannian ) . thus @xmath321 is @xmath263-linearized on @xmath322 , and if we choose on @xmath323 the bundle @xmath324 , all the assumptions of proposition [ pfibrationovergrass ] except the dominance of @xmath220 have been checked .
the latter dominance follows from an explicit computer calculation , as follows : choose a random point @xmath325 .
if the map @xmath326 has maximal rank @xmath327 , @xmath269 is well defined . in this case compute a basis @xmath328 , @xmath329 of @xmath330 .
compute then the two @xmath331-matrices @xmath332 resp . @xmath333 representing @xmath334 resp .
if @xmath336 which is a @xmath337 matrix , has maximal rank @xmath338 , the kernel of @xmath339 represents the fibre @xmath340))$ ] and is of expected dimension .
again one can easily do this calculation over a finite field using a computer algebra program .
see @xcite for a macaulay2 script .
[ rnoalternatives ] as far as we can see , the rationality of @xmath341 can not be obtained by direct application of proposition [ pfibrationovergrass ] with base of the projection a projective space .
in fact , a computer search yields that the inclusion @xmath342 is the only candidate to be taken into consideration for dimension reasons : @xmath343 and @xmath344 .
however , on @xmath345 there does not exist a @xmath263-linearized line bundle cutting out @xmath235 on the fibres of the projection to @xmath346 ; for such a line bundle would have to be of the form @xmath347 , @xmath348 , and none of these is @xmath263-linearized : since @xmath349 is @xmath263-linearized it would follow that the @xmath3 action on @xmath350 factors through @xmath263 which is not the case .
christian bhning and h .- chr.grafv.bothmer .
macaulay2 scripts for a clebsch - gordan formula for @xmath40 and applications to rationality .
available at http://www.uni-math.gwdg.de/bothmer/clebschgordan , 2008 .
shepherd - barron , n.i . ,
_ rationality of moduli spaces via invariant theory _ , topological methods in algebraic transformation groups ( new brunswick , nj , 1988 ) , progr . math .
80 * , birkhuser boston , boston , ma ( 1989 ) , 153 - 164
|
if @xmath0 , @xmath1 , @xmath2 are irreducible @xmath3-representations , we give an easy and explicit description of a basis of the space of equivariant maps @xmath4 ( theorem [ tclebschgordan ] ) . we apply this method to the rationality problem for invariant function fields . in particular , we prove the rationality of the moduli space of plane curves of degree @xmath5 .
this uses a criterion which ensures the stable rationality of some quotients of grassmannians by an @xmath6-action ( proposition [ pstabratgrassmannians ] ) .
|
the scalar mesons below @xmath2 gev are in center of debate since many years @xcite .
more states than expected from the quark - antiquark assignment are reported in particle data group ( pdg ) pdg , leading to the introduction of a scalar glueball @xcite , tetraquark states @xcite and mesonic molecules @xcite . in particular , the scalar resonances below 1 gev have appealing characteristics , such as the reversed level ordering of masses , expected from tetraquark states @xcite . in turn
, this scenario implies that quarkonia lie between @xmath3 and @xmath2 gev .
a complication in the analysis of scalar states is mixing : between 1 - 2 gev a quarkonia - glueball mixing in the isoscalar sector is considered , for instance , in refs . @xcite . mixing among tetraquark states below 1 gev and quarkonia above 2 gev
is studied in refs .
shechter , fariborz , tqchiral , where , however , the results do not coincide : while a large mixing is found in ref .
@xcite , a negligible mixing is the outcome of @xcite .
it should be stressed that different interpretations of scalar state are possible : a nonet of scalar quarkonia is settled below 1 gev in refs .
@xcite in agreement with the linear sigma model and the nambu jona - lasinio ( njl ) model , while in ref .
minkowski a broad glueball , to be identified with @xmath4 , is proposed .
we refer to refs .
@xcite for a discussion of arguments in favour and/or against the outlined assignments .
studies on scalar mesons have been extensively performed by using chiral perturbation theory @xcite , where a scalar resonance at about @xmath5 mev is inferred out of pion - pion scattering .
a full nonet of molecular - like scalar states is generated in the unitarized chiral perturbation theory of ref .
in particular , pelaez @xcite studied the large-@xmath6 dependence of the light scalar resonances finding that they do not scale as quarkonia but in agreement with a molecular or tetraquark composition ( see , however , also the discussions in refs .
@xcite ) . in the present paper we concentrate on an important aspect of light scalar resonances , namely the form of their spectral functions , in a simple theoretical context . in this study ,
relevant for both quarkonium or tetraquark assignment of light scalars @xcite , effects arising from loops of pseudoscalar mesons are considered : this leads to parametrizations of spectral functions beyond the ( usually employed ) breit - wigner and flatt distributions and allow to include finite - width effects in the evaluation of decay rates . in particular ,
we consider the following physical scenarios : ( i ) the case of a broad scalar resonance , strongly coupled to one decay channel , such as the @xmath7 in the pion - pion decay mode , for which the spectral function can deviate substantially form the breit - wigner form ; ( ii ) the case of two channels , one of which is sub - threshold for the mass and thus forbidden at tree - level , as the @xmath8 decay mode for the resonances @xmath9 and @xmath10 in the latter case a comparison with the usually employed flatt distribution is performed . the key - quantity of the discussion is the propagator of scalar resonances dressed by mesonic loops in one or more channels . when the kllen - lehmann representation is satisfied , as verified at one - loop level in the case of light scalar mesons for large ranges of parameters achasovprop , the spectral function ( proportional to the imaginary part of the propagator ) is correctly normalized and is interpreted as a ` mass distribution ' for the scalar state .
a general definition of the decay of a scalar state , involving the obtained mass distribution , is then possible . in this way one takes into account in a consistent fashion finite - width effects for the decay , hence allowing to study deviations form the usually employed tree - level formula for decay rates .
furthermore , the fulfillment of the kllen - lehmann representation offers a criterion to delimit the validity of our one - loop study : as soon as violations appear ( generally for large coupling constants ) the obtained spectral functions are no longer usable .
we regularize the mesonic loops by means of a cutoff function which in turn is equivalent to a nonlocal interaction lagrangian . in a phenomenological perspective
it is reasonable that the mesonic states in the loop can not have indefinitely high virtual momenta which are naturally limited due to the finite range of the meson - meson interaction .
we also show that the dependence on the chosen cutoff function and on the specific value of the cutoff is mild . in order to render the paper easily understandable and
self - contained we start in section ii with one - channel case by recalling the basic definitions and properties , then we apply the study to the scalar sigma @xmath11 and kaon @xmath12 resonances : the corresponding spectral function shows consistent deviations from the usual breit - wigner one . in section iii we turn to the two - channel case , with particular attention to the resonances @xmath13 and @xmath9 , their decay rates and spectral functions in comparison with the flatt distribution @xcite
implications of the results in view of a nonet of tetraquark states below 1 gev is discussed . in section
iv we drive our conclusions , emphasizing as in ref .
@xcite that the use of propagators fulfilling the kllen - lehmann representation , which implies normalized distributions and a correct definition of decay rates , should be preferable both in theoretical and experimental works .
we consider the scalar fields @xmath14 and @xmath15 described by the lagrangian@xmath16 in the limit @xmath17 the propagator of the field @xmath14 reads@xmath18we intend to study the modification to @xmath19 when @xmath20 which arises by considering the loop - diagram of fig . 1 and how this contribution affects the decay mechanism @xmath21 we recall that at tree - level the decay width reads @xmath22^{2}\theta ( m_{0}-2m)\hspace{0.2 cm } , \hspace{0.2 cm } p_{s\varphi \varphi } = \sqrt{\frac{m_{0}^{2}}{4}-m^{2}},\text { } g_{s\varphi \varphi } = \sqrt{2}g . \label{tl1}\]]where @xmath23 is the step function and ( the factor @xmath24 in the amplitude @xmath25 takes into account that the final state consists of two identical particles . ) in general , under the symbol @xmath26 the expression @xmath27i.e .
the momentum of the outgoing particle(s ) , is understood . at tree - level
the particle @xmath14 is treated as stable .
however , the very fact that the decay @xmath28 for @xmath29 means that @xmath14 is not stable and can not be considered as an asymptotic state of the lagrangian @xmath30 the tree - level expression @xmath31 is only valid in the limit @xmath32 the evaluation of the loop of fig .
1 offers a way to define and interpret the decay @xmath33 as we describe in the following .
the modified propagator of @xmath14 is obtained by ( re)summing the loop - diagrams of fig .
1:@xmath34where the self - energy @xmath35 reads : @xmath36 \left [ ( q - p/2)^{2}-m^{2}+i\varepsilon % \right ] } .\]]the integral defining @xmath35 is , as known , logarithmic divergent .
our intention is to consider the lagrangian @xmath37 as an effective low - energy description of the fields @xmath14 and @xmath38 and not as a fundamental theory valid up to indefinitely high mass scales .
we do not apply the renormalization scheme to @xmath37 but we introduce a regularization function @xmath39 which depends on a cut - off scale @xmath40 for the large momenta .
the self - energy @xmath35 is then modified to:@xmath41 \left [ ( q - p/2)^{2}-m^{2}+i\varepsilon \right ] } \label{loopf}\]]when choosing @xmath42 the covariance of the theory is preserved , otherwise is lost .
indeed , in many calculations related to mesonic loops a regularization of the kind @xmath43 is chosen , which leads to simple expressions for the self - energy contribution but breaks covariance explicitly , and thus is strictly valid only in the rest frame of the decaying particle .
in particular , the 3d - cutoff @xmath44 is often used .
we refer to appendix a for a closer analysis of the self - energy @xmath35 , where the case of unequal masses circulating in the loop is also presented .
the interaction strength among light mesons is suppressed for distances larger than @xmath45-@xmath3 fm : in this particular physical example it is then natural to implement a cutoff @xmath46 which varies between @xmath3 and @xmath2 gev .
the cutoff - function @xmath39 is not present in the lagrangian @xmath47 of eq .
( [ toy1ch ] ) . in this sense
the lagrangian is incomplete because it does not specify how to cut the high momenta .
one can take into account @xmath39 already at the lagrangian level by rendering the interaction term nonlocal:@xmath48the feynman rule for the 3-leg vertex is modified as:@xmath49where @xmath50 and @xmath51 are the momenta of the two particles @xmath52 the function @xmath39 enters directly into the expression of all amplitudes . in particular , the self - energy contribution of eq .
( [ loopf ] ) is now obtained by application of the ( modified ) feynman rules to the loop diagram of fig .
, the delocalization of the interaction term also induces a change of the tree - level result for the decay , which becomes ( for the case @xmath53):@xmath54^{2}\theta ( m_{0}-2m),\text { } g_{s\varphi \varphi } = \sqrt{2}g \label{tlnl}\ ] ] that is the function @xmath55 is explicitly present in the tree - level decay expression and can be interpreted as a phenomenological form factor the decay - amplitude takes the form @xmath56 $ ] ] if a step - function is used the local tree - level expression of eq .
( [ tl1 ] ) is recovered , provided that the cutoff @xmath40 is large enough . in this work
we use the following cutoff - function @xmath57 with this choice the fourier transform @xmath58 , see eq .
( [ ft ] ) , takes the form @xmath59
/\left\vert \overrightarrow{y}\right\vert , $ ] thus decreasing rapidly for increasing distance of the two interacting mesons @xmath60 the interaction range @xmath61 is of the order @xmath62 as discussed above based on general dimensional grounds . at each step of the forthcoming study we employed also different forms of @xmath39 , finding that the dependence on the precise form of @xmath63 affects only slightly the results .
notice that in refs .
@xcite a similar equation to ( [ tlnl ] ) ( where @xmath64 and @xmath65 in the notation of @xcite ) represents the starting point of the analysis .
the function @xmath66 in the above cited works is taken to be a gaussian , @xmath40 is of the order of 1 gev .
the present approach shows the link between such form factor @xmath65 and a nonlocal lagrangian .
however , we will not concentrate as in @xcite on scattering amplitudes but on spectral functions and decay widths . at the same time we do not relate the imaginary and real part of the propagator via kllen - lehmann dispersion relation , but we evaluate them independently and subsequently we check numerically if it satisfied , see details in the following discussion .
let us now turn to the self - energy @xmath67 a general property for @xmath68 follows from the optical theorem:@xmath69=x\gamma _
{ s\phi \phi } ^{\text{t - l}}(x ) .
\label{i(x)}\ ] ] the imaginary part of the self - energy diagram is zero for @xmath70 and nonzero starting at threshold . the real part @xmath71\]]is nonzero below and above threshold . in fig .
2 the functions @xmath72 and @xmath73 are plotted using eq .
( [ cutoff ] ) . a particular choice for the parameters @xmath74 gev and @xmath75 gev ( of the order of physical cases studied later ) is done .
anyway the plotted functions are qualitatively similar for large ranges of parameters . as noticeable , @xmath72 is continuous but not derivable in @xmath76 it has a cusp at @xmath77 : the left derivative is @xmath78 while the right derivative is finite and negative . in terms of the two functions @xmath72 and @xmath79 the propagator of eq .
( prop1 ) reads@xmath80we define the ( breit - wigner ) mass @xmath81 for the scalar field @xmath14 as the solution of the equation @xmath82when the function @xmath83 is positive , which is usually the physical case ( fig .
2 ) , the dressed mass @xmath81 is smaller than the bare mass @xmath84 showing that the quantum fluctuations tend to lower it .
gev and @xmath85 gev . ]
we now turn to the spectral function @xmath86 of the scalar field @xmath14 related to the imaginary part of the propagator as@xmath87\right\vert .
\label{dsdef}\]]in the limit @xmath88 we obtain the desired spectral function @xmath89 the normalization of @xmath86 holds for each @xmath90@xmath91the latter eq . is a consequence of the kllen - lehmann representation@xmath92}{y^{2}-x^{2}+i\varepsilon } \label{kl}\]]when taking the limit @xmath93 eqs .
( [ norm ] ) and ( [ kl ] ) hold in general for the full propagator . in our case
we check numerically the validity of the normalization condition ( [ norm ] ) at one - loop level of fig .
1 . we find that it is fulfilled to a high level of accuracy for large ranges of parameters , see also the discussion in @xcite and in the next subsection .
( dashed line ) and @xmath94 ( continuous line ) .
the coupling constant is @xmath95 gev and two values of the masses are chosen : @xmath96 gev and @xmath97 gev , @xmath74 gev . ]
let us consider @xmath86 in the two interesting cases @xmath98 and @xmath99 if @xmath98 eq .
( [ dsdef ] ) becomes : @xmath100where@xmath101when @xmath98 the constant @xmath102 is usually reabsorbed into the definition of the wave function renormalization , hence recovering the free propagator properly normalized as @xmath103 , corresponding to @xmath104 for @xmath105 as in the free case @xmath17 .
thus , we have still a stable particle with dressed mass @xmath81 instead of @xmath106 .
notice that @xmath107 because @xmath108 is a positive number : the quantity @xmath109 can be interpreted as the amount of virtual clouds of @xmath110 contributing to the wave - function .
if @xmath111 the spectral function reads@xmath112no delta - functions are present but typically a picked distribution @xmath86 is obtained , corresponding to a physical resonance .
the mass @xmath81 is not the maximum of the @xmath86 although in general very close to it .
consistent deviations can appear when @xmath81 is close to threshold and for large coupling constant , see next subsection for a more detailed discussion of this point .
notice moreover that @xmath86 is zero for @xmath113 we plot the typical behavior of the spectral function in both cases @xmath98 and @xmath111 in fig .
[ spectralexample ] .
we used the values @xmath96 gev and @xmath97 gev corresponding to the the two cases below and above the threshold , @xmath74 gev as before and @xmath114 gev .
the value of @xmath115 , for the sub - threshold case , is @xmath116 , we have numerically verified that the spectral functions are normalized in both cases .
when @xmath111 the function @xmath86 can be interpreted as the mass distribution of the resonance , see also appendix b for an intuitive discussion about this point .
we then define the decay rate for the process @xmath117 by implementing the distribution @xmath118 , and thus including finite width effects , as:@xmath119this formula reduces to the tree - level amplitude @xmath120 of eq .
( [ tlnl ] ) in the limit of small @xmath121 : @xmath122notice that in this limit @xmath123 .
however , even for finite @xmath124 when @xmath125 the formula @xmath126 offers a first approximation to the decay width of the state as long as the distribution is picked , i.e. the scalar state @xmath14 is not too broad . the definition eq .
( [ gendec ] ) for the decay @xmath127 is thus a generalization of the tree - level result of eq .
( [ tlnl ] ) and takes automatically into account that the state @xmath14 has a finite width parametrized by the mass distribution @xmath86 , which naturally arises by considering the self - energy of the scalar propagator .
notice that the real part of the propagator is necessary in order eq .
( [ norm ] ) to hold : its neglection would spoil the correct normalization .
evaluating the real and imaginary part at @xmath128 and neglecting their @xmath129-dependence , the distribution ( [ ds ] ) is approximated by @xmath130 which is the relativistic breit - wigner distribution for the resonance @xmath14 , usually employed in theoretical and experimental studies .
however , the distribution @xmath131 neglects the real part of the loop diagram and consequently the normalization of eq .
( [ norm ] ) does not hold , implying that @xmath131 has to be normalized by hand . at the same time
the mass @xmath81 does not coincide with the maximum of @xmath132 thus , we insist on that the usage of automatically normalized distribution emerging from propagators fulfilling kllen - lehmann should be preferable .
an interesting example for the one - channel case is the decay of the scalar meson @xmath7 .
as reported by the pdg @xcite , experimental data are affected by large uncertainties both for the value of the mass , @xmath134-@xmath135 gev , and the value of the breit - wigner width , @xmath136-@xmath3 gev .
the dominant channel , which we will consider here , is the decay into two pions , for which @xmath111 . by applying the formulas of section ii.a
we show in the left panel of fig .
[ spect_sigma ] the spectral functions @xmath137 of the @xmath11-resonance for the two boundary cases of pdg , namely @xmath134 gev and @xmath138 gev , respectively , for the coupling constant @xmath139 gev .
the spectral function assumes different shapes for different values of the mass . while for @xmath138
, far from the threshold , the spectral function has a regular ` breit - wigner - like ' form , in the case @xmath134 gev a distorted shape , with a narrow peak just above threshold , is visible the spectral function @xmath140 for @xmath141 due to the threshold enhancement . ] .
the employed value of the coupling constant , @xmath139 gev , serves as illustration and actually corresponds to a somewhat too narrow width .
the increase of @xmath142 leads , however , outside the range of validity of the normalization of eq .
( norm ) at one - loop level , see below .
the description of the scalar kaonic resonance @xmath12 follows the same line @xcite . as shown in the right panel of fig .
[ spect_sigma ] a strong deviation from the breit - wigner form is obtained when @xmath143 gev , corresponding to @xmath144 gev , while a less distorted shape is found for @xmath145 gev , for which @xmath146 gev . at this point a short discussion on the definition of the mass of an unstable particle
is needed : in the breit - wigner scheme , the mass of the particle is the value corresponding to the maximum of the spectral function and it is one of the parameter of the distribution ( the second one is of course the width ) . in our scheme , using the spectral functions coming from the loop evaluation this is no more the case : the mass @xmath81 defined in eq .
( [ polem ] ) is again a parameter of the distribution ( together with the coupling constant @xmath121 ) but it does not coincide with the maximum of the distribution . while for @xmath138 gev ( far from threshold ) the maximum of the spectral function occurs at @xmath147 gev , thus only slightly shifted from the mass , when @xmath134 gev the maximum of the spectral function ( apart form the threshold enhancement peak ) occurs at sizeable larger values respect to the mass , here @xmath148 gev mev , thus not far from threshold in the lower side of the pdg data : this is indeed the case of irregular form for the spectral function of this resonance , for which care is needed . ] .
( notice also that in the latter case the bare mass @xmath149 is @xmath150 gev , thus implying a strong influence of the pion loop to the sigma mass , see also appendix a for a comparison of different ` masses ' ) .
indeed , although the mass @xmath81 being the zero of the real part of the inverse propagator , see eq .
( [ polem ] ) , is referred to as breit - wigner mass @xcite , the best fit to the full spectral function @xmath137 by using a breit - wigner form is obtained for a breit - wigner mass @xmath151 coinciding with the maximum of the distribution .
it is also remarkable that in some cases the spectral function has not a maximum , a part from the threshold enhancement peak , as we can see for the @xmath12 meson in the right panel of fig .
[ spect_sigma ] for @xmath143 gev .
we now study closer the decay process @xmath152 using eq .
( [ gendec ] ) which implements the spectral function @xmath153 in fig .
[ sigmadecay ] we compare the full and the tree - level decay rates for different values of the cut - off and of the coupling constants ( a mass @xmath154 mev is used ) .
are shown as function of the coupling constant .
the cases @xmath155 and @xmath156 gev correspond to thin and thick lines respectively . ]
as expected , there is not a strong dependence on the choice of the cut - off .
moreover the results obtained with our formulae are well in agreement with the tree - level results for small values of @xmath121 ( since the spectral function tends to a delta function ) .
non - negligible differences instead occur for the larger values of @xmath121 .
this is due to a different analytic dependence of the two corresponding formulae from the coupling constant : while the tree - level expression depends quadratically on @xmath121 , in the loop formula @xmath121 appears also in the distribution @xmath86 .
the limit of the validity of the employed one - loop level analysis is an important aspect which deserves further discussion .
namely , when the coupling constant is too large and the mass is not far from threshold the normalization condition of eq .
( [ norm ] ) is lost , see also the corresponding discussion in @xcite .
this fact means that higher orders must be taken into account to satisfy the kllen - lehmann representation and thus to recover the correct normalization of eq .
( norm ) . at the same time
the violation of the normalization is a valid criterion to establish the limit of our study : for this reason in fig .
5 we stop the plot at @xmath157 gev ( corresponding to @xmath158 mev ) , in fact larger values imply @xmath159 at this point the full decay width ( using eq .
( [ gendec ] ) ) is already @xmath160 mev smaller than the tree - level counterpart
. a decay width of about @xmath161 mev is on the low side for the @xmath11 ( see ishida ) . in ref .
@xcite a width @xmath162 mev larger is obtained . a study beyond one - loop level
would then be necessary to evaluate the spectral function for larger coupling ( i.e. larger width ) and represents a possible outlook of the present work .
surely the overestimation of the tree - level formula keeps growing for increasing interaction strengths .
similar considerations hold for the @xmath12 meson .
we now consider two channels for the scalar resonance @xmath14 described by the lagrangian density ( @xmath163 ) : @xmath164the processes @xmath165 and @xmath166 correspond to the tree - level decay rates@xmath167^{2}\theta ( m_{0}-2m_{1}),\text { } g_{s\varphi _ { 1}\varphi _ { 1}}=\sqrt{2}g_{1 } , \\
\gamma _ { s\varphi _ { 2}\varphi _ { 2}}^{\text{t - l}}(m_{0 } ) & = & \frac{p_{s\varphi _ { 2}\varphi _ { 2}}}{8\pi m_{0}^{2}}[g_{s\varphi _ { 2}\varphi _ { 2}}]^{2}\theta ( m_{0}-2m_{2}),\text { } g_{s\varphi _ { 2}\varphi _ { 2}}=\sqrt{2}g_{2}\text { .}\end{aligned}\]]the propagator is modified by loops of @xmath168 and @xmath169 , denoted as @xmath170 and @xmath171 and given by eq .
( [ loopf ] ) for @xmath172 and @xmath173 respectively .
a delocalization of the interaction , via a vertex - function @xmath58 and the corresponding fourier - transform @xmath174 is then introduced as in eq .
( [ deloc ] ) for both channels in order to regularize the self - energy contributions . as a consequence ,
the tree - level results are modified as@xmath175^{2}\theta ( m_{0}-2m_{1 } ) \\
\gamma _ { s\varphi _ { 2}\varphi _ { 2}}^{\text{t - l}}(m_{0 } ) & = & \frac{p_{s\varphi _ { 2}\varphi _ { 2}}}{8\pi m_{0}^{2}}[g_{s\varphi _ { 2}\varphi _ { 2}}f_{\lambda } ( % \overrightarrow{q}^{2}=p_{s\varphi _ { 2}\varphi _ { 2}}^{2})]^{2}\theta ( m_{0}-2m_{2}),\end{aligned}\]]and the propagator as @xmath80 where@xmath176+g_{s\varphi _ { 2}\varphi _ { 2}}^{2}re[\sigma _ { 2}(x=\sqrt{p^{2}})]\]]and@xmath177+g_{s\varphi _ { 2}\varphi _ { 2}}^{2}im[\sigma _ { 2}(x=\sqrt{p^{2 } } ) ] \\ & = & x\gamma _ { s\varphi _ { 1}\varphi _ { 1}}^{\text{t - l}}(x)+x\gamma _ { s\varphi _ { 2}\varphi _ { 2}}^{\text{t - l}}(x).\end{aligned}\]]in the last equation the optical theorem has been used .
the mass @xmath81 of the state @xmath14 is given by @xmath178 again , we have two cases : \(i ) @xmath179 : the distribution @xmath86 takes the form @xmath180 for @xmath181 the discussion is similar to the one - channel case .
( dsbt ) is still valid . at threshold
@xmath182 the continuum starts .
\(ii ) @xmath183 : as in eq .
( [ ds ] ) the distribution is @xmath184it vanishes for @xmath185 at @xmath186 the second channel opens . in the case
( ii ) we have a resonant state .
the decay rates into the two channels @xmath165 and @xmath187 are given by the integrals:@xmath188 a particularly interesting case takes place when @xmath189 : while the tree - level result for @xmath190 vanishes , we find that @xmath191 is not zero . in this case
the tree - level approximation is absolutely not applicable : the particle @xmath14 does decay in virtue of the high - mass tail of its distribution .
a physical example is well - known : the resonances @xmath9 and @xmath13 have a non - zero decay rate into @xmath192 although their masses are below the threshold @xmath193 clearly , a sizable decay rate @xmath194 is obtained only when @xmath81 is close to threshold .
a generalization to the present definitions to @xmath195 channels is straightforward @xcite .
when applying the decay formulas ( [ dec2c ] ) it is however important to verify numerically that the normalization of the distribution @xmath86 holds : in fact , as discussed in section ii.b , only in this case the formalism is self - consistent . in this subsection
we study the spectral functions of the scalar mesons @xmath196 and @xmath197 , whose masses are @xmath198 mev and @xmath199 mev @xcite . for both resonances
two decays have been observed : @xmath200 , @xmath201 and @xmath202 , @xmath203 notice that both masses are below threshold of kaon - antikaon production , @xmath204 mev , thus the decay of both resonances in @xmath205 vanishes at tree - level , while experimentally it was seen for both @xmath206 and @xmath207 states . for definiteness
we use the following ratios obtained in the experimental analysis of @xcite @xmath208 therefore leaving us with only one free parameter , chosen to be @xmath209 although experimental uncertainties are still large , the results of eq .
( [ bugg ] ) are qualitative similar to various studies , see @xcite and refs . therein , pointing to a large @xmath205 coupling for both resonances with a particular enhancement for @xmath207 ( see @xcite and refs .
therein for spectroscopic interpretations ) . for the typical value @xmath210 gev we report in fig .
distribuzioni - a0f0 the spectral functions of @xmath206 and @xmath211 there is a large probability , @xmath212 , in both cases , that these two mesons have a mass larger than the threshold of production @xmath213 and therefore the tree - level forbidden decay occurs . in the same figure
we compare our distribution with the flatt one @xcite , which is usually employed for the @xmath206 and @xmath207 mesons . at variance from our distribution ,
flatt distribution must be normalized by hand .
the two distributions are quite similar , only for the @xmath207 the values of the mass corresponding to the maximum of the distributions are slightly different .
this is due to the strong coupling of @xmath207 to kaons and , as already argued by achasov @xcite , the meson loop distributions coincide with the flatt ones only in the limit of weak coupling . in fig .
[ decaya0f0 ] we show the decay rates @xmath202 , @xmath214 and @xmath200 , @xmath201 as function of @xmath215 .
the dashed areas in both plots correspond to the total decay rate of @xmath207 and @xmath206 as indicated by the pdg ( notice , however , that in a note in pdg it is specified that the real width could be larger ) . to get agreement between our theoretical total decay rates and the measured ones , @xmath216 has to lye between @xmath217-@xmath218 gev .
the outcoming branching ratio @xmath219 is @xmath220-@xmath221 larger than the pdg average @xmath222 , while the obtained ratio @xmath223 is @xmath224-@xmath225 in qualitative agreement with the ( not bold ) results listed in pdg .
notice furthermore that the @xmath207 mesons turns out to have typically a larger width than @xmath206 meson in agreement with ref .
@xcite .
we finally comment on a possible tetraquark unified interpretation of the light scalar mesons as presented in ref .
@xcite . a too small decay constant @xmath215
would also imply a by far too narrow @xmath226 and @xmath12 mesons ( related by clebsh - gordon coefficients @xcite ) , thus against a tetraquark nonet . on the contrary ,
@xmath215 between @xmath217-@xmath218 gev is in agreement with a tetraquark nonet below @xmath3 gev , although problems , such as a too narrow @xmath12 , persist , see discussions in @xcite .
such a strong coupling in the @xmath205-channel implies that virtual cloud of kaon - antikaon pairs plays an important role , in particular for the @xmath207 resonance .
a heuristic indicator of the mesonic cloud can be given by the quantity @xmath227 where @xmath228 $ ] ( with @xmath229 ) refers to the kaonic loop only . as discussed in section ii in the subthreshold case ( which applies to the kaonic channel here ) the quantity @xmath109 varies between @xmath230 and @xmath3 and measures the mesonic cloud dressing of the original bare resonance @xmath231 in the @xmath207 case , by using eq .
( [ bugg ] ) together with @xmath232 gev , one finds @xmath233 hence implying a 38% of kaonic cloud .
this number increases for increasing coupling strength @xmath234 this discussion confirms the interpretation put forward in ref .
@xcite , where the light scalar mesons posses a tetraquark core but are dressed by kaonic clouds .
in this work we studied the spectral functions of scalar mesons in one- and two - channel cases suitable for the description of light scalar mesons below 1 gev .
we have computed , by using lagrangians with non - derivative couplings , the propagators of scalar mesons at one - loop level .
they satisfy for large ranges of parameters the kllen - lehmann representation , therefore implying normalized spectral functions . in this way a correct definition of decay amplitudes , weighted over the spectral function , is formulated : the finite - width effects are automatically taken into account .
the resulting decay rates are smaller than the tree - level ones with increasing mismatch for increasing interaction strength . on the other hand , a sub - threshold tree - level forbidden decay , such as
the @xmath205 mode for @xmath13 and @xmath235 becomes large .
the resulting spectral functions for the @xmath11 and @xmath12 mesons may deviate consistently from the breit - wigner form .
the flatt distribution , although it approximates to a good level of accuracy the @xmath236 and @xmath9 spectral functions , emerges as a small - coupling limit of our more general spectral function . as stressed by achasov @xcite it is important to use distributions satisfying kllen - lehmann representations in experimental and theoretical studies .
we thus believe that the use of distributions obtained from quantum field theoretical models fulfilling the correct normalization requirements can be helpful to correctly disentangle the nature of the scalar states .
future studies with derivative couplings , mixing effects and @xmath237-decays represent a possible interesting outlook . * acknowledgements * g.p
. acknowledges financial support from infn .
here we report basic formulas for the loop diagram of eq .
( [ loopf ] ) drawn in fig.1 for the vertex function @xmath43 . by evaluating the residua one obtains the one - dimensional integral@xmath238which can be easily evaluated numerically for each well - behaved @xmath239 .
we remind that within our conventions @xmath240 and that @xmath241 eq .
( [ lupo ] ) refers to a 3d - vertex function . in @xcite
the form @xmath242 is used and the limit @xmath243 is taken . as described in the text , we did not follow this procedure but we used definite form(s ) for the vertex function @xmath244 .
as remarked in the text , we performed the calculations also with different forms for @xmath245 ( different power form and exponential functions ) : the precise form of the cutoff function does not affect the physical picture .
when the scalar state @xmath14 couples to two particles of masses @xmath246 and @xmath247 the loop contribution is modified as following:@xmath248the choice @xmath249 with @xmath250-@xmath2 gev has been used in this work . in relation to the mass definition of section ii.b
we report and compare in table 1 the mass @xmath81 defined in eq .
( [ polem ] ) , the bare mass @xmath84 the maximum @xmath251 of the distribution @xmath252 and the average mass @xmath253 .
we use @xmath254 @xmath255 gev , @xmath256 gev . * tab.1 * : comparison of ` masses ' [ cols="<,<,<,<",options="header " , ] as expected , the larger the mass , the smaller the differences among the various mass - like quantities .
we present an intuitive argument for the correctness of interpretation of the spectral function @xmath86 as the ` mass distribution ' of the state @xmath14 . to this end
we introduce two scalar fields @xmath257 and @xmath258 the first massless and the second with @xmath259 and write down the interaction lagrangian@xmath260(for the following discussion the ` delocalization ' of eq .
( [ deloc ] ) is not important ) .
we suppose that the interaction strength @xmath261 is small enough to allow a tree - level analysis for the decay of the state @xmath262 the term @xmath263 generates the decay process @xmath264 which reads ( at tree - level)@xmath265^{2}. \label{bas}\]]however , when @xmath266 the state @xmath14 decays into @xmath267 that is the state @xmath14 is not an asymptotic state .
physically , we observe a tree - body decay @xmath268 whose decay - rate reads:@xmath269the tree - body decay is decomposed into two steps : @xmath270 and @xmath271 the quantity @xmath272 represents the probability for @xmath270 ( at a given mass @xmath273 for the state @xmath274 and @xmath275 is the corresponding weight , i.e. the probability that the resonance @xmath14 has a mass between @xmath129 and @xmath276 in this example @xmath86 emerges naturally as a mass distribution , correctly normalized , for the scalar state @xmath231 furthermore , notice that in virtue of the limit @xmath277 for @xmath88 one has @xmath278 in fact , if @xmath121 is very small the state @xmath14 is long - living and the equation ( [ bas ] ) is recovered .
the present analysis also shows that studies on the tree - body decay of the @xmath237 meson can be consistent only if propagators satisfying the kllen - lehmann representation are used .
99 c. amsler and n. a. tornqvist , phys .
* 389 * , 61 ( 2004 ) .
f. e. close and n. a. tornqvist , j. phys .
g * 28 * , r249 ( 2002 ) [ arxiv : hep - ph/0204205 ] ; r. l. jaffe , phys .
* 409 * ( 2005 )
1 [ nucl . phys .
* 142 * ( 2005 ) 343 ] [ arxiv : hep - ph/0409065 ] .
r. l. jaffe , arxiv : hep - ph/0701038 .
m. r. pennington , arxiv : hep - ph/0703256 .
n. n. achasov , arxiv : hep - ph/0609261 .
d. v. bugg , eur .
j. c * 47 * ( 2006 ) 57 [ arxiv : hep - ph/0603089 ] .
d. v. bugg , eur .
j. c * 47 * ( 2006 ) 45 [ arxiv : hep - ex/0603023 ] .
w. m. yao _ et al .
_ [ particle data group ] , j. phys .
g * 33 * ( 2006 ) 1 . c. amsler and f. e. close , phys .
b * 353 * , 385 ( 1995 ) [ arxiv : hep - ph/9505219 ] ; c. amsler and f. e. close , phys .
d * 53 * , 295 ( 1996 ) [ arxiv : hep - ph/9507326 ] .
r. l. jaffe , phys .
d * 15 * ( 1977 ) 267 .
r. l. jaffe , phys .
d * 15 * ( 1977 ) 281 .
r. l. jaffe and f. e. low , phys .
d * 19 * , 2105 ( 1979 ) .
m. b. voloshin and l. b. okun , jetp lett .
* 23 * , 333 ( 1976 ) [ pisma zh .
. teor .
* 23 * , 369 ( 1976 ) ] .
k. maltman and n. isgur , phys .
lett . * 50 * , 1827 ( 1983 ) .
k. maltman and n. isgur , phys .
d * 29 * , 952 ( 1984 ) .
l. maiani , f. piccinini , a. d. polosa and v. riquer , phys .
* 93 * ( 2004 ) 212002 [ arxiv : hep - ph/0407017 ] .
n. n. achasov and v. n. ivanchenko , nucl .
b * 315 * ( 1989 ) 465 .
f. giacosa , phys .
d * 74 * ( 2006 ) 014028 [ arxiv : hep - ph/0605191 ] .
f. e. close and a. kirk , eur .
j. c * 21 * , 531 ( 2001 ) [ arxiv : hep - ph/0103173 ] .
w. j. lee and d. weingarten , phys .
d * 61 * , 014015 ( 2000 ) [ arxiv : hep - lat/9910008 ] ; m. strohmeier - presicek , t. gutsche , r. vinh mau and a. faessler , phys .
d * 60 * , 054010 ( 1999 ) [ arxiv : hep - ph/9904461 ] .
f. giacosa , t. gutsche and a. faessler , phys .
c * 71 * , 025202 ( 2005 ) [ arxiv : hep - ph/0408085 ] .
f. giacosa , t. gutsche , v. e. lyubovitskij and a. faessler , phys .
d * 72 * , 094006 ( 2005 ) [ arxiv : hep - ph/0509247 ] .
f. giacosa , t. gutsche , v. e. lyubovitskij and a. faessler , phys .
b * 622 * , 277 ( 2005 ) [ arxiv : hep - ph/0504033 ] .
g. mosconi , private communications .
d. black , a. h. fariborz , s. moussa , s. nasri and j. schechter , phys .
d * 64 * ( 2001 ) 014031 [ arxiv : hep - ph/0012278 ] .
a. h. fariborz , r. jora and j. schechter , phys .
d * 72 * ( 2005 ) 034001 [ arxiv : hep - ph/0506170 ] .
a. h. fariborz , int .
j. mod .
a * 19 * ( 2004 ) 2095 [ arxiv : hep - ph/0302133 ] . a. h. fariborz , phys .
d * 74 * ( 2006 ) 054030 [ arxiv : hep - ph/0607105 ] .
m. napsuciale and s. rodriguez , phys . rev .
d * 70 * ( 2004 ) 094043 [ arxiv : hep - ph/0407037 ] .
f. giacosa , phys .
d * 75 * ( 2007 ) 054007 [ arxiv : hep - ph/0611388 ] .
e. van beveren , f. kleefeld , g. rupp and m. d. scadron , mod .
a * 17 * ( 2002 ) 1673 [ arxiv : hep - ph/0204139 ] .
m. d. scadron , g. rupp , f. kleefeld and e. van beveren , phys .
d * 69 * ( 2004 ) 014010 [ erratum - ibid .
d * 69 * ( 2004 ) 059901 ] [ arxiv : hep - ph/0309109 ] .
p. minkowski and w. ochs , nucl .
. suppl . * 121 * , 123 ( 2003 ) [ arxiv : hep - ph/0209225 ] ; i. caprini , g. colangelo and h. leutwyler , phys .
lett . * 96 * ( 2006 ) 132001 [ arxiv : hep - ph/0512364 ] .
g. colangelo , j. gasser and h. leutwyler , nucl .
b * 603 * ( 2001 ) 125 [ arxiv : hep - ph/0103088 ] .
j. a. oller , e. oset and j. r. pelaez , phys .
d * 59 * , 074001 ( 1999 ) [ erratum - ibid .
d * 60 * , 099906 ( 1999 ) ] [ arxiv : hep - ph/9804209 ] .
j. r. pelaez , phys .
* 92 * ( 2004 ) 102001 [ arxiv : hep - ph/0309292 ] .
a * 19 * ( 2004 ) 2879 [ arxiv : hep - ph/0411107 ] . j. r. pelaez and g. rios , phys .
* 97 * ( 2006 ) 242002 [ arxiv : hep - ph/0610397 ] . m. ishida , prog . theor .
* 101 * ( 1999 ) 661 [ arxiv : hep - ph/9902260 ] .
z. h. guo , l. y. xiao and h. q. zheng , arxiv : hep - ph/0610434 . n. n. achasov and a. v. kiselev , phys .
d * 70 * ( 2004 ) 111901 [ arxiv : hep - ph/0405128 ] .
s. m. flatte , phys .
b * 63 * ( 1976 ) 228 .
v. baru , j. haidenbauer , c. hanhart , a. kudryavtsev and u. g. meissner , eur .
phys . j. a * 23 * ( 2005 ) 523 [ arxiv : nucl - th/0410099 ] .
n. a. tornqvist , z. phys .
c * 68 * ( 1995 ) 647 [ arxiv : hep - ph/9504372 ] .
m. boglione and m. r. pennington , phys .
d * 65 * ( 2002 ) 114010 [ arxiv : hep - ph/0203149 ] .
the resonance @xmath279 is now listed in the compilation of the particle data group @xcite but it still needs confirmation and is omitted from the summary table . the resonance is also found in many recent theoretical and experimental works ( @xcite , e. van beveren , t. a. rijken , k. metzger , c. dullemond , g. rupp and j. e. ribeiro , z. phys .
c * 30 * ( 1986 ) 615 .
s. ishida , m. ishida , t. ishida , k. takamatsu and t. tsuru , prog .
* 98 * ( 1997 ) 621 [ arxiv : hep - ph/9705437 ] .
d. black , a. h. fariborz , f. sannino and j. schechter , phys . rev .
d * 58 * ( 1998 ) 054012 [ arxiv : hep - ph/9804273 ] and refs .
therein . )
|
in this work we study the spectral functions of scalar mesons in one- and two - channel cases . when the propagators satisfy the kllen - lehman representation a normalized spectral function
is obtained , allowing to take into account finite - width effects in the evaluation of decay rates . in the one - channel case ,
suitable to the light sigma and k mesons , the spectral function can deviate consistently from a breit - wigner shape . in the two - channel case with one subthreshold channel
the evaluated spectral function is well approximated by a flatte distribution ; when applying the study to the @xmath0 and @xmath1 mesons the tree - level forbidden kk decay is analysed .
|
the @xmath3ray sky has been successfully observed by satellite based missions in the energy range between 50 @xmath4 and 30 @xmath5 @xcite . at higher energies
the steeply falling @xmath3ray spectrum makes it impossible to observe by satellite based detectors , primarily because of the limited detector size and exposure time available for a given source .
alternatively , the ground based atmospheric erenkov technique has been successfully exploited at these energies . over the years
, this technique has proved to be the most sensitive in studying the celestial @xmath3rays in the very high energy ( vhe ) range , @xmath6 - 50 @xmath7 .
celestial vhe @xmath3rays initiate an electromagnetic cascade in the atmosphere as they enter the earth s atmosphere .
the electrons and positrons in the cascade , being relativistic , emit erenkov light as they propagate down the atmosphere resulting in a faint flash of bluish light lasting a few nanoseconds .
this directional erenkov flash is detected on the ground by conventional optical light detectors during moon - less clear nights .
taking advantage of the fact that the erenkov flash is highly collimated , within a cone of half angle @xmath8 , along the direction of the incident particle most experiments simply limit the optical field of view of the erenkov telescopes to a small region of the sky .
however the main drawback of this technique is the presence of abundant cosmic ray background which severely limit the sensitivity of this technique .
numerous efforts to develop methods to distinguish the erenkov light pool produced by cosmic @xmath9rays from that by the cosmic rays led to two important techniques based on complementary ways of viewing the cascade , _ viz .
_ the angular sampling or the _ imaging technique _ and spatial sampling or the _ wavefront sampling technique_. both these techniques are currently being employed by different groups @xcite . the possibility of timing the erenkov light front for the estimation of arrival direction of the primary has been realized as early as 1968 by tornabene and cusimano @xcite .
it was later demonstrated by the durham group @xcite that while viewing crab pulsar the on - axis events were richer in @xmath10-rays than off - axis events .
it was also used by gupta _
et al._@xcite to improve the signal to noise ratio of atmospheric erenkov telescopes consisting of large mirrors with poor optical quality .
the basic principle of all these experiments is the fast timing technique using spatially separated erenkov telescopes .
the signal to noise ratio in such an experiment is given by @xcite @xmath11 where _
@xmath12 _ and _ @xmath13 _ are the fluxes of omni - directional cosmic rays and @xmath14 rays from a point source respectively .
_ a _ is the collection area of the array , _ t _ is the time of observations , _
@xmath15 _ is the solid angle of acceptance , _
@xmath16 _ is the fraction of showers due to cosmic rays that are identified and rejected as background and _ @xmath17 _ is the fraction of showers due to @xmath3rays that are identified as signal and hence retained . in order to achieve high @xmath18 apart from increasing @xmath17 and _ @xmath16
_ one could either increase the collection area and the observation time or decrease the solid angle of acceptance . for a given exposure time and available hardware resources one can possibly increase _
s / n _ by only reducing @xmath15 as celestial @xmath14 rays from point sources are directional while cosmic rays are isotropic . due to the finite opening angle of the erenkov cone and the spread in the arrival angle of erenkov photons the aperture of the telescopes
have to be restricted to few degrees , which sets a lower limit to _
@xmath15_. however , it is possible to improve the @xmath18 for point sources without losing erenkov light if the direction of arrival of primary particles is estimated accurately @xcite .
the shower axis retains the original direction of the primary . to estimate the arrival direction accurately
, the error in the reconstructed direction , _ i.e. _ the angular resolution has to be very small .
the two dominant factors which contribute to the angular resolution are the average distance , _ d _ , between the telescopes and _ @xmath19 t _ , the uncertainty in the measurement of arrival time of photons at the telescopes .
for example , the angular resolution in the zenith angle , @xmath20 for @xmath21 detectors is given by @xcite : @xmath22 therefore , a large number of telescopes with instrumentation to measure the relative arrival time of photons and separated by large distances are needed to reconstruct the shower front and estimate the direction of arrival of the shower fairly accurately .
on the other hand , in case of the imaging technique the angular resolution is limited by pmt(pixel ) size which is typically of the order of a quarter degree .
the improvement in the signal to noise ratio by restricting the angle of acceptance to @xmath23 would be : @xmath24 this is a very significant advantage of non - imaging arrays with high angular resolution .
moreover , this method of reducing background is independent of the primary species .
hence it is extremely useful at very low primary energies where the cosmic ray electrons form a significant source of background .
electrons can not be discriminated easily , unlike hadrons , since they too undergo electromagnetic interactions in the atmosphere akin to @xmath14 rays .
however one has to bear in mind that the angular resolution could be poorer at lower primary energies due to paucity of erenkov photons at the ground level . in the rest of the paper
we discuss the method of obtaining the arrival direction of the primary radiation using our array of erenkov telescopes and estimate its error .
the experiment at pachmarhi ( longitude : 78@xmath25 26@xmath26 e , latitude : 22@xmath25 28@xmath27 and altitude : 1075 @xmath28 ) , is based on the wavefront sampling technique and employs an array of erenkov telescopes , called the pachmarhi array of erenkov telescopes(pact ) , to sample the erenkov light pool .
pact is now fully functional .
the experimental set - up of pact has been explained in detail elsewhere @xcite .
briefly , it consists of a @xmath29 array of atmospheric erenkov telescopes deployed in the form of a rectangular matrix with a separation of 25 _ m _ in the n - s direction and 20 _ m _ in the e - w direction .
each telescope consists of 7 parabolic mirrors of 0.9 _ m _ diameter mounted paraxially and having a focal length of 90 _ cm_. each mirror is viewed by a fast photo - multiplier tube ( pmt , emi 9807b ) at the focus behind a circular mask of @xmath30 3@xmath31 diameter .
however the field of view is limited by the diameter of the pmt photo - cathode to @xmath30 2.9@xmath31 _
fwhm_. the array has been divided into 4 sectors with six telescopes in each .
the pulses from 7 pmts in a telescope are added linearly to form a telescope sum pulse called _
royal sum_. each _ royal sum _ from the 6 telescopes in a sector are suitably discriminated ( typical _ royal sum _
rates @xmath30 30 - 50 khz . ) and a trigger is generated by a coincidence of any 4 of these 6 _ royal sums_. the typical event rate is @xmath30 2 - 5 hz . for every event ,
information on the relative arrival times and density of erenkov photons are recorded for the 6 peripheral mirrors / pmt in each telescope in each sector .
the relative arrival times of _ royal sum _
pulses are recorded both in the respective sector and in the central data processing station .
thus , pact measures the arrival time of shower front at various locations within the erenkov light pool at two distance scales , _ short range _
( intra - telescope ) and _ long range _
( inter - telescope ) .
the arrival direction of the shower is estimated from the measured arrival times of erenkov photon front at each of the spatially separated telescopes while the six adjacent measurements in a given telescope could be used to study the fluctuations in the measured shower parameters .
the dispersion of photon arrival times at each telescope contain the signature of the primary species @xcite and hence could be used to distinguish between @xmath14 rays from the background @xcite .
the density measurements on the other hand enable us to estimate the energy of the primary species as well as to reject cosmic ray background @xcite .
the telescopes are equatorially mounted and each telescope is independently steerable in right ascension and declination within @xmath32 .
the movement of the telescopes is remotely controlled by automatic computerized telescope orientation system ( actos ) @xcite .
the hardware for actos consists of a semi - intelligent closed loop stepper motor drive system which senses the angular position using a gravity based angle transducer called _ clinometer _ with an accuracy of 1@xmath33 .
the two clinometers , one each in n - s and e - w direction , are accurately calibrated using stars at various hour - angles and declinations . in order to estimate the source pointing error of the system
, the telescopes were oriented to different bright stars at random positions in the sky and the offsets in orthogonal directions ( ascension and declination ) with respect to the star were noted . using this data
it is concluded that the system can orient the telescopes to the putative source with a mean offset of @xmath34 .
the source pointing is monitored with an accuracy of @xmath35 and corrected in real time during tracking .
the uncertainty in pointing represents the subjective error in manual reading of the star position at the cross wire of the guiding telescope field of view ( @xmath36 ) .
as the seven mirrors of a telescope are mounted on a single mount it is necessary to ensure that all their optic axes are parallel to each other so that they view the same part of the sky .
this alignment of mirrors is done manually by positioning the star image at the centre of the focal plane . to check the accuracy of alignment of the mirrors and telescopes a bright star drift scan
is carried out .
the telescope is aligned to an isolated bright star ( typically of visual magnitude 2 to 3 ) .
then the telescope is moved to the west by @xmath37 and at a suitable time the telescope tracking is switched off .
the counting rates from each of the pmts are monitored every second and recorded .
the count rates stay reasonably constant until the star walks into the field of view when they increase , go through a maximum and return to the background value as shown in figure [ fig : bscan ] .
the background count rates before and after the star transit are fitted to a linear function .
the background is subtracted from the count rates during the star transit by interpolation .
the background corrected count rates are then fitted to a quadratic function .
figure [ fig : bscan ] shows one such fit to a typical count rate profile of a mirror . table [ tab : bscan ] shows the summary of results of a typical bright star scan .
it shows in column 3 the fwhm of the drift scan profile of the count rates due to the star .
the offset of the centroid of this profile with respect to the centre of the field of view ( _ i.e. _ expected transit time ) is shown in the last column .
this offset is the error in the alignment in right ascension .
the relative fwhm s of the profiles could indicate the misalignment in declination , if any . on the other hand the absolute values of fwhm of the count rate profiles are a function of the pmt gains , image quality ( point source image size @xmath38 ) , star brightness @xmath39 . using this method
, it is ensured that the optical axes of all the 7 mirrors in a telescope are parallel to each other within an error of @xmath40 or less .
if the error exceeds this value the particular mirror is re - aligned and re - checked .
this method of alignment is similar to that adapted for aligning the imaging telescope arrays @xcite .
0.25 cm .bright star scan results of @xmath41 ursa major for a telescope [ tab : bscan ] [ cols="<,<,<,<",options="header " , ] from the results of the present study it is seen that two independent measurements of the angular resolutions are @xmath42 and @xmath43 resulting from the timing measurements of telescopes and mirrors of the entire array respectively for similar energy events .
the essential difference between the two measurements is an increase in the number of degrees of freedom by a factor of @xmath44 in the latter case . from this consideration
the expected angular resolution for individual mirrors is @xmath45 which is almost twice the value of @xmath46 mentioned above .
these are consistent since the timing uncertainty for _ royal sums _ is higher by the same factor .
figure [ fig : dndom ] shows differential plots of the number of events per solid angle @xmath47 as a function of the zenith angle .
the panel @xmath48 shows plots for events whose arrival directions have been estimated using timing signals from _ royal sums _ while the panel @xmath49 shows the same for individual mirror timing information .
the two curves shown in panel @xmath48 correspond to events detected simultaneously in all the 4 sectors ( solid line ) and in one sector ( # 3 ) only ( dashed line ) .
this provides a method of measuring the effective opening angle of the array @xcite . from these plots
it can be seen that the array field of view is around @xmath50 .
this is significantly lower than the geometric opening angle of @xmath51 .
it may be noted that the effective field of view of the array is expected to be lower than the geometric value primarily because the event trigger efficiency falls with increasing core distance and zenith angle . the other possible factors which could potentially reduce the effective field of view like the residual alignment errors and non - ideal optical quality of the mirrors are relatively less significant .
the effective fwhm of the array ( @xmath52 ) for which the mean telescope separation is @xmath53 is lower than that for a sector ( @xmath54 ) for which the mean telescope separation is @xmath55 supporting the above argument . the data used for estimating the angular resolution of pact reported here consists of cosmic rays .
it is well known that the protons exhibit a larger ( by almost a factor of 4 ) intrinsic timing jitter relative to @xmath10-ray primaries @xcite .
while the contribution of systematic effects to the timing resolution ( which is estimated to be small ) is independent of the primary species , the decreased intrinsic timing jitter is expected to improve the angular resolution for @xmath10-ray primaries significantly .
table [ tab : comp ] summarizes the angular resolution of imaging as well as non - imaging atmospheric erenkov telescopes currently in operation and under construction .
it can be seen that the angular resolutions of most of the single imaging telescopes are modest , in the range @xmath56 .
wavefront sampling experiments using the modified solar concentrator arrays like the stacee , celeste , graal @xmath57 , will not be able to exploit their angle reconstruction technique to reject a significant cosmic ray background primarily because of their rather small field of view , in the range @xmath58 ( fwhm ) which are comparable to their angular resolutions .
pact , on the other hand , has the best angular resolution among the non - imaging experiments which can be used to enhance the signal to noise ratio significantly .
pact has been able to achieve this because of multiple fast timing measurements at each telescope .
this has two - fold advantages : firstly , it provides an increased number of degrees of freedom which in turn improves the accuracy of angle reconstruction and secondly , it provides a means of computing the erenkov photon arrival time dispersion at each telescope .
the timing jitter is a species sensitive parameter and hence will enable us reject a significant fraction of on - axis hadronic showers .
the signal to noise ratio could also be improved by using the normalized @xmath59 values resulting from a spherical fit to the timing information of a shower .
this is expected to be larger for hadron initiated showers @xcite .
the future imaging telescope arrays are expected to achieve unprecedented angular resolution by employing the stereoscopic technique .
they are able to reconstruct the shower in 3-dimensions using the multiple images in 3 or more imaging telescopes . from table [
tab : comp ] it can be seen that both the hegra and the shalon - alatoo arrays are able to improve the angular resolution of their single imaging telescopes by a factor of @xmath60 by using 2 or more of them in stereoscopic mode .
it may be mentioned here that the quoted angular resolutions for the future projects listed in the table are from simulation results for @xmath10-ray primaries while the rest are derived from measurements from cosmic ray events .
it is well known that the erenkov light front has a curvature .
it was seen that when two well separated erenkov telescopes were tilted towards each other by about a degree the coincidence rate increased and also reduced the spread in the time separation between them @xcite .
this indicated , as claimed by the authors , the presence of curvature in the photon front .
more recently , it was shown that the radius of curvature of the front is equal to the height of the shower maximum from the observation altitude @xcite .
hence it is clear that a plane front approximation of the erenkov light front will introduce a systematic error in the arrival angle reconstruction .
the large separation of telescopes , to some extent , offsets the worsening of resolution due to the curvature of the light front .
it may also be pointed out that the effect of curvature on the angular resolution is more significant for near vertical showers . in the case of inclined showers ,
the arrival time differences at spatially separated telescopes due to shower axis inclination far exceed the differences arising out of the shower front curvature .
however it has been argued that the angular resolution of given array will improve if one corrects for the wavefront curvature @xcite .
the details of the systematic effects due to the plane front approximation and the improvement in the accuracy of estimated arrival directions when the curvature of the shower front is taken into account are currently under investigation .
a paper based on the results of these simulation studies is under preparation .
a detailed analysis of the angular resolution of pact using data collected with telescopes pointing to zenith is presented .
the improvement in the angular resolution with larger separation between detectors ( _ d _ ) and with increase in the number of degrees of freedom ( _ n _ ) has been verified .
the angular resolution @xmath61 , is found to improve according to the relation : @xmath62 there are two types of angular resolutions that could be defined for pact .
firstly , the fast timing information from the 25 telescopes spread in an area of about @xmath63 yield an angular resolution of @xmath64 .
secondly the timing information from the mirrors constituting the 25 telescopes yield the best angular resolution of @xmath0 at a @xmath10-ray energy of around 3 @xmath7 .
this is the best angular resolution achieved for any ground based atmospheric telescope system which will probably be superseded by the proposed veritas or hess imaging telescope array .
the angular resolution for @xmath10-ray primaries could be significantly better than what is presented here , which is based on cosmic ray primaries , because of two main reasons .
firstly , photon arrival time jitter for @xmath10-ray primaries is far less than that for charged cosmic rays and secondly the radius of curvature for @xmath10-ray primaries is more than that for cosmic rays which consequently reduces the effects of the shower front curvature for the former .
tornabene , h. s. and cusimano , f. j. , can .
j of phys .
, 46 , ( 1968 ) , s81 .
dowthwaite , j. c. _ et.al._ , astrophys .
, 286 , ( 1984 ) , l35 .
gupta , s. k. _ et.al._ , astrophys . and sp .
, 115 , ( 1985 ) , 163 .
acharya , b.s .
_ et.al._ , j.phys.g , 19 , ( 1993 ) , 1053 .
tornabene , h. s. , @xmath65 int .
cosmic ray conf . ,
kyoto , 1 , ( 1979 ) , 139 .
vishwanath , p. r. , proc .
workshop in vhe gamma - ray astronomy , ed .
p. v. ramana murthy and t. c. weekes ( tata institute of fundamental research , bombay ) 1982 , 21 .
bhat , p.n .
_ et.al._ , bull .
india , 28 , ( 2000 ) , 455 .
chitnis , v.r . and bhat , p.n . , astroparticle physics , 12 , ( 1999 ) , 45 .
chitnis , v.r . and bhat , p.n . , astroparticle physics , 15 , ( 2001 ) , 29 .
chitnis , v.r . and bhat , p.n . ,
proc . of int .
symp . on gamma - ray astrophysics through multi - wavelength experiments , game-2001 , mt.abu , india , ( ed : r.k.kaul and c.l.kaul ( 2001 ) ; to appear in bull .
soc . of india 2002 .
gothe , k.s .
_ et.al._ , indian jour .
pure & applied phys . , 38 , ( 2000 ) , 269 .
kohnle , a. _ et.al._ , astroparticle physics , 5 , ( 1996 ) , 119 .
oser , s. _ et.al._ , apj . , 547 , ( 2001 ) , 949 .
s. , ph.d .
thesis , university of bombay , unpublished , ( 1987 ) miller and r. s. and westerhoff , astroparticle physics , 11 , ( 1999 ) , 379 .
aharonian , f. _ et.al._ , astroparticle physics , 6 , ( 1997 ) , 343 .
biller , s. _ et.al._ , veritas proposal document , ( 1999 ) .
aharonian , f. _ et.al._ , hess proposal document , ( 1997 ) , http://www.mpi-hd.mpg.de/hfm/hess/hess.html .
chitnis , v.r . and bhat , p.n . ,
proc . of int
. symp . on gamma - ray astrophysics through multi - wavelength experiments , game-2001 , mt.abu , india , ( ed : r.k.kaul and c.l.kaul ( 2001 ) ; to appear in bull .
astr . soc . of india 2002 .
antonov , r. a. _ et.al._ , proc . of @xmath66 icrc , dublin , 2 , ( 1991 ) , 664 .
goret , p. _ et.al._ , nucl .
. & meth . ,
a270 , ( 1988 ) , 550 .
krenrich , f. _ et.al._ , astroparticle physics , 8 , ( 1998 ) , 213 .
piron , f. _ et.al._ , 374 , ( 2001 ) , 895 .
kranich , d. , _
et.al._ , 1997 , @xmath67 compton symp . ,
williamsburg , va , ed .
: dermer , c. d. , strickman , m. s. and kurfess , j. d. , aip conf .
vol . 410 .
tanimori , t. _ et.al._ , proc . of @xmath68 icrc , utah , og4.3.04 , ( 1999 ) .
sapru , m. l. _ et.al._ , towards a major atmospheric erenkov detector - v , berg - en - dal , ed .
o. c. de jager , ( 1997 ) , 329 .
sinitsyana , v. g. _ et.al._ , proc . of 27th icrc , hamburg , og .
222 , ( 2001 ) goetting , n. _ et.al._ , proc . of 27th icrc , hamburg ,
199 , ( 2001 ) daum , a. _ et.al._ , astroparticle physics , 8 , ( 1997 ) , 1 .
baillon , p. _ et.al._ , astroparticle physics , 1 , ( 1993 ) , 341 .
de naurois , m. _ et.al._ , astrophysical j. , february , ( 2002 ) , astro - ph/0107301 .
arqueros , f. _ et.al._ , proc . of 27th icrc , hamburg , og .
158 , ( 2001 ) karle , a. _ et.al._ , astroparticle physics , 3 , ( 1995 ) , 321 .
atkins , r. _
et.al._ , astrophysical j. lett . , 533 , ( 2000 ) ,
amenomori , m. _ et.al._ , proc . of 27th icrc , hamburg ,
1.6 , ( 2001 ) barrio , j. a. _ et.al._ , magic proposal document , ( 1998 ) .
mori , m. _ et.al._ , _ the cangaroo - iii project _ ,
_ tev gamma ray workshop _ , snowbird , utah , usa , ( 1999 ) .
williams , d. a. _ et.al._ , astrophysics preprint library , astro - ph/0010341 .
|
the pachmarhi array of erenkov telescopes consists of a distributed array of 25 telescopes that are used to sample the atmospheric erenkov photon showers .
each telescope consists of 7 parabolic mirrors each viewed by a single photo - multiplier tube .
reconstruction of photon showers are carried out using fast timing information on the arrival of pulses at each pmt .
the shower front is fitted to a plane and the direction of arrival of primary particle initiating the shower is obtained .
the error in the determination of the arrival direction of the primary has been estimated using the _ split _ array method .
it is found to be @xmath0 for primaries of energy @xmath1 .
the dependence of the angular resolution on the separation between the telescopes and the number of detectors are also obtained from the data .
vhe @xmath2 - rays , atmospheric erenkov technique , angular resolution , non - imaging telescope array , pact .
|
most of low - mass x - ray binaries ( lmxbs ) , namely x - ray emitting binaries with roche - lobe filling low mass stars , are considered to involve neutron stars ( nss ) with low magnetic field strengths , @xmath12 @xmath1310@xmath14 g. they make a contrast to high - mass x - ray binaries ( hmxbs ) , mostly containing nss with high magnetic field strengths ( @xmath12 @xmath1510@xmath16 g ) which capture stellar winds from their companions .
in fact , among @xmath1550 known lmxbs , only three are known to have nss with strong magnetic fields ; her x-1 , 4u 1626@xmath067 and gx 1 + 4 @xcite .
these distinct combinations of the mass - donating and mass accreting components may be generally interpreted as population effects , that older nss have weaker fields .
however , it is not necessarily clear whether nss gradually lose their magnetic fields ( e.g. , )
. then , it is worth while searching for other lmxbs that involve nss with strong magnetic fields .
evidently , the magnetic field affects accretion mechanisms in binaries .
when the ns has strong magnetic fields , the accreting matter is funneled onto the two magnetic poles , leading to strong pulsations , and the production of very hard x - ray spectra which is often accompanied by cyclotron resonance scattering features ( crsfs ; @xcite ) .
if , in contrast , the ns is weakly magnetized , an accretion disk is considered to extend down to vicinity of the ns , and the emergent x - ray spectra will exhibit characteristic bimodel behavior between so - called soft state and hard state ( e.g. @xcite ) . these spectral properties , together with the presence / absence of x - ray pulsations , will conversely allow us to tell whether the ns in a mass exchanging binary is strongly or weakly magnetized . in an attempt of looking for lmxbs that involve strongly magnetized nss ,
the present paper focuses on the dipping x - ray binary 4u 1822 - 37 , located at an estimated distance of 2.5 kpc @xcite .
the companion star in 4u 1822@xmath037 is estimated to have a mass of 0.44 - 0.56 _ m@xmath17 _ @xcite , with @xmath18 being the solar mass , and hence this system is clarified as an lmxb .
x - ray and optical light curves of this source both show intensity modulations and dips synchronize with its orbital period , @xmath19 @xmath15 5.7 hr .
these effects are attributed to either gradual occultation ( by the companion star ) of a large x - ray scattering corona above the accretion disk ( e.g. , @xcite ) , or variable attenuation of direct x - rays from the ns by ionized humps on the disk ( e.g. , ) . from dip properties ,
the orbital inclination is constrained between 76@xmath20 and 84@xmath20 @xcite .
although 4u 1822@xmath037 is classified as an lmxb , its x - ray properties appear rather different from those of other dipping lmxbs .
first , it shows clear x - ray pulsations with a period of @xmath21 = 0.5924 s which is much longer than those of other typical lmxbs , a few milliseconds @xcite if ever detected .
figure [ corbet_diagram ] is so - called corbet diagram ( ) , which displays neutron - star binaries on a plane of orbital period and rotation period . like 4u 1626@xmath067 and her x-1 , 4u 1822@xmath22
is thus located on this plot between typical lmxbs and hmxbs . combining the estimated x - ray luminosity ( 10@xmath23 erg s@xmath3 ) with @xmath21 and its change rate , @xmath24 = @xmath25 s s@xmath3 , the magnetic field strength of the ns in this source
was actually estimated as @xmath26 g ( @xcite ) .
although the error is very large , it is noteworthy that rather high field strength are allowed .
second , this object exhibits much harder spectra together with a lower cutoff energy than typical lmxbs in the hard state . from the measured cutoff energy ( @xmath15 6 kev ) , obtained an estimate as @xmath12 @xmath15 10@xmath16 g. third , @xcite , using the _ chandra hetgs _ , detected from this source narrow iron k@xmath27 and k@xmath28 lines , which are generally absent or much weaker in other lmxbs @xcite . in the present study , we examine whether the ns in 4u 1822@xmath037 has _
@xmath15 10@xmath16 g or not . for this purpose , it is necessary to obtain spectra with a good energy resolution and a wide energy coverage .
the fifth japanese x - ray satellite suzaku @xcite , carrying onboard the xis @xcite and the hxd @xcite , is best suited for the requirements .
we hence utilized an archival suzaku data set of 4u 1822@xmath037 , and study its spectral and pulsation properties .
the mechanism of the periodic x - ray and optical dips , though interesting , is beyond the scope of the present paper .
with suzaku , 4u 1822@xmath037 was observed for an exposure 37 ks on 2006 october 20 ( i d 401051010 ) . in this observation ,
the target was placed at the xis nominal position . of the four xis cameras ,
xis0 was operated in the 1/4 window mode , and the others in the full window mode .
the hxd was operated in the normal mode .
using heasoft ver 6.12 , we analyzed the data from xis0 , xis2 , xis3 and hxd - pin . in this study , we did not use xis1 which is the back - illuminated camera , because it has a higher background and is subject to larger calibration uncertainties than the other xis cameras .
as the average counts rate of the xis , @xmath15 10 cts s@xmath3 , was high enough to cause pile up under the full window mode , the xis events were accumulated over an annular region with the inner and outer radii of [email protected] and [email protected] , respectively .
this allowed us to reduce pile up effects to within 1% @xcite .
the response and arf files of the xis were generated using xisrmfgen and xissimarfgen , respectively .
the hxd - pin data were analyzed using a `` tuned '' non x - ray background file and the epoch2 response file , both officially released by the hxd team .
we did not use the hxd - gso data because the source was undetectable therein .
the present suzaku data were already analyzed by . combined with previous data
, they determined the orbital period to be @xmath19 = 20054.2049 s = 5.57 hr , but did not report on pulse detection , or spectral analysis .
figure [ lightcurve ] shows background - subtracted light curves of 4u 1822@xmath037 obtained with the xis ( 1 to 10 kev ) and hxd - pin ( 15 to 60 kev ) , together with the hxd - pin vs. xis hardness ratio .
the gross exposure of this observation ( @xmath1590 ks ) covered about four orbital cycles , in which we detected about 4 dips in both the xis and hxd - pin bands .
however , unlike the cases of many other dipping sources ( e.g. xb 1916@xmath0053 , xb 1323@xmath0619 and exo 0748@xmath0176 ) , no x - ray bursts were detected .
the same light curves , folded at @xmath19=20054.2049 s , are shown in figure [ foldedlightcurve ] .
they are consistent with those of previous studies ( e.g. , ) . from the bottom panel of figure [ foldedlightcurve ] ,
the hardness ratio is observed to decrease during the dips .
we also tried to detect the pulsation at a period @xmath150.59 s @xcite , using only the hxd - pin data because of the low time resolution of the xis . after applying barycentric corrections to the individual hxd - pin events
, we further corrected the event arrive times for the expected orbital delay , @xmath30 , in 4u 1822@xmath037 .
this @xmath30 is calculated with the pulsar s semi - major axis @xmath31 and the inclination @xmath32 as @xmath33,\ ] ] where @xmath34 ( 0 @xmath35 1 ) is the initial orbital phase .
we fixed @xmath36 at 1.006 lt - s , after an accurate measurement by @xcite , and chose @xmath37 so that @xmath30 becomes maximum at the observed x - ray dips .
after these corrections , we calculated a 15 - 40 kev periodogram .
as shown in figure [ efsearch](a ) , the pulsation was detected with a high ( @xmath7 99% confidence ) significance at a period of @xmath21 = 0.5924337 @xmath8 0.000001 s. the pulse profile folded at @xmath21 is presented in figure [ efsearch](b ) .
as already reported @xcite , the pulse fraction is rather small , @xmath38% in the relative peak amplitude . in figure
[ spinhistory ] , the measured value of @xmath21 is compared with previous pulse - period measurements . over the past @xmath15 6 years
, the object has thus been spinning up monotonically with an approximately constant rate of @[email protected]@xmath80.05@xmath4110@xmath5 s s@xmath3 , or @xmath42=6.7 kyr .
this value of @xmath39 reconfirms the previous measurements by @xcite . 37 with 128 s bins , obtained with xis0+xis3 ( top panel ; 1 - 10 kev ) and hxd - pin ( middle panel ; 15 - 60 kev ) .
the bottom panel shows the hxd - pin vs xis hardness ratio.,width=302,height=302 ] 37 as in figure[lightcurve ] , folded at the orbital period of 20054 s. the three panels correspond to those of figure [ lightcurve ] .,width=302,height=302 ] of 4u 1822@xmath037 based on @xcite and the present work .
the triangle indicates the present result.,width=302,height=302 ] figure [ compsepctra](a ) shows 1 - 50 kev spectra of 4u 1822@xmath037 obtained with the xis and hxd - pin .
the same spectra , normalized to a power - law model of photon index 2.0 , are compared in figure [ compsepctra](b ) with those of another dipping source exo 0748@xmath0676 and the x - ray pulsar her x-1 .
the spectral shape of 4u 1822@xmath037 is similar to that of her x@xmath01 while different from that of exo 0748@xmath0676 , in the following two points .
first , the spectrum of 4u 1822@xmath037 shows a very hard slope ( with a power - law photon index of @xmath431 ) below 10 kev and a steep cutoff at @xmath15 15 kev , in resemblance to the her x-1 spectrum , while that of exo 0748@xmath0676 exhibits a softer ( @xmath432 ) slope without clear high - energy cutoff .
second , complex fe @xmath44-@xmath45 lines are seen at @xmath15 6.4 kev in the spectra of 4u 1822@xmath037 and her x-1 , while they are absent in the case of exo 0748@xmath0676 .
these suggest that the ns in 4u 1822@xmath037 has strong magnetic fields , like that in her x@xmath01 @xcite . to quantify the xis and hxd - pin spectra of 4u 1822@xmath037 , we fitted them jointly with an absorbed cutoff - power - law ( cutoffpl ) model , including three narrow gaussians to represent fe - k@xmath6 , fe xxv k@xmath6 , and fe xxvi k@xmath6 lines , at @xmath156.4 , @xmath156.7 and @xmath156.9 kev , respectively . the photon index @xmath46 , the cutoff energy @xmath47 , and normalization of cutoffpl were left free , as well as the column density of absorption
similarly , the three gaussians were allowed to have free center energies , free normalizations , and free widths .
the hxd vs. xis cross normalization was fixed at 1.16 @xcite . however , a large data excess below 3 kev made the fit unacceptable , with a reduced chi - square of @xmath483.0 for @xmath49=287 degrees of freedom . to account for this `` soft excess '' feature , we chose a blackbody ( bb ) , and fitted the data with wabs@xmath41(bb+cutoffpl+3gaussians ) .
a bb with a temperature of @xmath15 0.1 kev improved the fit to @xmath50=1.21 ( @xmath49=284 ) .
however , as shown in figure [ spectrafit](b ) , there still remained negative residuals at @xmath15 35 kev . since they are suggestive of a crsf , we employed a model of the form of wabs@xmath41(bb+cutoffpl+3gaussians)@xmath41cyclabs , where cyclabs represents the cyclotron resonance absorption factor given , e.g. , in @xcite . by allowing the resonance energy , width , and the depth of cyclabs to vary freely
, the fit has been improved to @xmath51 for @xmath49=280 .
as summarized in table 1 , the center energy of crsf was obtained as @xmath52=33@xmath82 kev . to make the model more physical , we once excluded the cyclabs factor , and tried a thermal comptonizetion ( comptt ; e.g. , @xcite ) and a negative - positive power - law with exponential cutoff " ( npex ; e.g. , @xcite ) continua instead of cutoffpl , while retaining the bb component and the three gaussians .
as summarized in table 1 , the comptt continuum was less successful than that with cutoffpl , while the npex continuum was better .
as shown in figure [ spectrafit](e ) , however , the bb+npex continuum again left the negative residuals at @xmath1535 kev like the case of cutoffpl .
we hence revived the cyclabs factor , and fitted the data with wabs@xmath41(bb+npex+3gaussian)@xmath41cyclabs .
the fit has become fully acceptable , @xmath50=1.13 with @xmath49=279 .
the obtained cyclabs parameters , shown in table 1 , are not much different from those with the cutoffpl continuum case , except that the width has now been constrained as 5.0@xmath53 .
while the error ranges in table 1 refer to 90% confidence limit , the cyclabs depth still remains positive , @xmath540.24 , if we employ more conservatively 99% limit ( i.e. , @xmath55=6.63 for a single parameter ) .
since the inclusion of the cyclabs factor improved the fit by @[email protected] for @xmath57=@xmath03 , or @xmath56/@xmath57 = 4.56 , an f - test indicates that the fit improvement is significant at a confidence level of 99% . using the gabs model for the crsf instead of the cyclabs model gave nearly the same @xmath50 and a consistent value of @xmath52 from these results
, we can claim the detection of a crsf .
the 1 - 50 kev flux of this source becomes 1.5@xmath58 erg s@xmath3 @xmath2 , and the luminosity is @xmath59 = 1.0@xmath60 erg s@xmath3 at the distance of 2.5 kpc @xcite .
these are consistent with previous reports @xcite .
spectra of 4u 1822@xmath037 , fitted with wabs@xmath41(bb+npex+3@xmath41gaussian)@xmath41cyclabs .
the softer and harder continuum components represent the negative and positive power - laws of the npex model , with a common exponential cutoff factor .
( b ) fit residuals with the wabs@xmath41(bb+cutoffpl+3@xmath41gaussian ) model .
( c ) those with wabs@xmath41(bb+cutoffpl+3@xmath41gaussian)@xmath41cyclabs .
( d ) those with wabs@xmath41(bb+comptt+3@xmath41gaussian ) .
( e ) those with wabs@xmath41(bb+npex+3@xmath41gaussian ) .
( f ) those with wabs@xmath41(bb+npex+3@xmath41gaussian)@xmath41cyclabs.,width=302,height=302 ] 37.,width=302,height=302 ]
in figure [ compsepctra](b ) , the spectra of 4u 1822@xmath037 were visually compared with those of her x-1 and exo 0748@xmath0676 . for doing this more quantitatively , we fitted the cutoffpl model to suzaku spectra of other hard - state lmxbs ( aql x@xmath01 , exo 0748@xmath0676 , xb 1323@xmath0619 , 4u 1636 + 536 and 4u 1608@xmath052 ) , and accreting pulsars which have strong magnetic fields ( 4u 1626@xmath067 , cen x-3 , smc x-1 , her x-1 , gro j1008@xmath057 , 1a 1118@xmath061 , 1a 0535 + 26 and lmc x-4 ) . figure [ gammacutoff ] summarized those fit results , on a plane of @xmath46 vs @xmath47 .
the hard state lmxbs typically have softer ( @xmath431.7 ) slopes and higher cutoff energies ( @xmath61 50 kev ) , while the accreting pulsars show harder ( @xmath430.6 ) slopes together with lower cutoff energies ( @xmath6210 kev ) . clearly , 4u 1822@xmath037 is more similar to the accreting pulsars than to the hard - state lmxbs . in terms of the continuum slope , 4u 1822@xmath037
is thus inferred to have common characteristics of accreting nss with @xmath63 g. as argued in @xcite , the high - energy cutoff of the spectra of accreting pulsars , which are often steeper than a thermal rollover as pointed out by @xcite , is presumably caused by the presence of a crsf , which is empirically thought to appear at energies of @xmath64 .
then the npex fit results , @xmath47=4.8 - 6.2 kev , predict a crsf to appear at @xmath52=16 - 23 kev .
this argument was already employed by @xcite to suggest @xmath65 g. indeed , as expected , we have detected a crsf at @xmath66 kev with a high significance . using the basic relation of @xmath12=@xmath67 g ( e.g. , @xcite ) , where @xmath68 is the gravitational redshift of the ns
, we obtain @xmath12=([email protected])@xmath10 g assuming @xmath68=0 for simplicity .
this gives the most convincing evidence that the ns in 4u 1822@xmath037 is strongly magnetized .
just to make the crsf detection more convincing , let us examine the derived crsf parameters . as already confirmed ,
the resonance energy @xmath52 is consistent with the continuum shape .
the depth , @xmath69 , is reasonable , in comparison with typical values of @xmath70=0.1 - 1.7 found in other objects .
in addition , @xmath71=2 - 10 kev=(0.06 - 0.30)@xmath52 agrees , within errors , with the general scaling of @xmath71=(0.27 - 0.50)@xmath52 found by @xcite .
thus , the present crsf interpretation of the @xmath1533 kev spectral feature is considered reasonable .
the present hxd data yielded @xmath21 = 0.5924337 s and @xmath39 = -2.43@xmath72 s s@xmath3 .
when these values , together with @xmath12=2.8@xmath10 g , are substituted into the accretion torque formula by @xcite , namely @xmath73 where @xmath74 is the luminosity in units of 10@xmath11 erg s@xmath3 and @xmath75 is the magnetic field in @xmath76 g , we obtain @xmath77 erg s@xmath3 .
this luminosity is much higher than the value of @xmath151.0@xmath78 erg s@xmath3 derived from the observed flux and an assumed distance of 2.5 kpc @xcite .
one possible cause of this discrepancy is that the object is in really located at @xmath79 kpc distance , instead of the 2.5 kpc which is based on some assumptions .
an alternative possibility , already pointed out previously ( ) , is that the object appears unusually x - ray faint due , e.g. , to x - ray obscuration by some ionized materials on the accretion disk .
indeed , its x - ray to optical luminosity ratio of @xmath1520 for @xmath801@xmath4110@xmath81 erg s@xmath3 ( @xcite ) is much lower than a typical value of @xmath15500 for lmxbs ( @xcite ) .
the ratio of 4u 1822@xmath037 will increase to @xmath15600 if we employ @xmath803@xmath4110@xmath11 erg s@xmath3 .
another interesting discussion may be performed on the iron lines in the spectra . generally ,
lmxbs have weak , sometimes broad @xcite iron lines with small equivalent width ( ews ) ; e.g. , @xmath82 ev @xcite .
strongly magnetized nss , including her x@xmath01 and gx 1 + 4 , in contrast show narrow iron lines with significantly larger ews ( @xmath7 50 ev ; @xcite ) .
these differences can be employed as an additional empirical argument to strengthen the high @xmath12 scenario of 4u 1822@xmath037 , although theoretical account of this issue is beyond the scope of the present paper . in this respect ,
the narrow ( @xmath83=0.04 kev ) and strong ( ew@xmath15 50 ev ) fe k@xmath6 line , detected at 6.4 kev with the xis , clearly classifies this ns into the high - field category . to be somewhat more quantitative , the width of the fe k@xmath6 line , [email protected] kev
, implies that the velocity @xmath84 of the gases which emit the iron line is @xmath15 0.6% of the light speed . assuming that the gas obeys keplar rotation , its distance becomes @xmath85 cm ( @xmath86 being the gravitational constant and @xmath87 the ns mass ) , which is comparable to the alfven radius ( @xmath88 cm ) for @xmath63 g while much larger than the ns radius ( @xmath89 cm ) .
therefore , the matter creating the fluorescent fe - k line is likely to be stored on the alfven surface . in the present data , we detected not only the nearly neutral fe k@xmath6 line , but also the other two lines which are identified as helium like ( fe xxv ) line and hydrogen like iron ( fe xxvi ) line as judged from their line energies .
empirically , such ionized iron lines are observed from x - ray pulsars mainly when the source luminosity is high ; e.g. , @xmath901@xmath91 erg s@xmath3 .
such examples include cen x-3 @xcite , her x-1 @xcite , lmc x-4 @xcite , and be pulsars in luminous outbursts @xcite .
quantitatively this is reasonable , because ionization of circum - source materials is determined by so - called ionization parameter @xmath92=@xmath93 , where @xmath94 is the electron density of the line emitter while @xmath95 is its distance from the ns ( ionizing photon source ) . assuming that the two ionized lines in 4u 1822@xmath037 are emitted by the same material that is different from those emitting the 6.4 kev line , we estimate as @xmath96 , from @xcite and the comparable ews of the two ionized lines ( table1 ) . employing the same argument as for @xmath97
, we may obtain @xmath98 cm from the line width of @xmath99 kev . adopting @xmath94=@xmath100 @xmath101 from ,
the luminosity is 1@xmath91 erg s@xmath3 , in agreement with estimate using equation ( 2 ) .
thus , the source is considered to be relatively luminous in x - rays . during the net exposure of 37 ks ( @xmath102h )
, no x - ray bursts were detected .
similarly , there have been not previous reports of burst detection from this objects .
while this could be due to rather infrequent burst occurrence at @xmath103 erg s@xmath3 ( ) , a more appropriate explanation would be to regard 4u 1822@xmath037 as an x - ray pulsar , which do not produce x - ray bursts .
we hence conclude that 4u 1822@xmath037 is another example of lmxb that contains a strongly magnetized ( @xmath1043@xmath4110@xmath16 g ) ns . furthermore , its luminosity is likely to be @xmath153@xmath4110@xmath11 erg s@xmath3 , instead of the previously reported value of @xmath151@xmath4110@xmath81 erg s@xmath3 , although it is at present unclear whether this discrepancy is due to inaccurate distance estimate , on due to the edge - on source geometry which could reduce the x - ray flux reaching us .
we thank all members of the suzaku hardware and software teams and the science working group .
and t.e . are supported by the japan society for the promotion of science ( jsps ) research fellowship for young scientists , the grant - in - aid for scientific research ( a ) ( 23244024 ) from jsps , and grant - in - aid for jsps fellows , 24 - 3320 , respectively .
|
the dipping x - ray source 4u 1822@xmath037 was observed by _ suzaku _ on 2006 octrober 20 for a net exposure of 37 ks .
the source was detected with the xis at a 1 - 10 kev flux of 5.5@xmath1 erg @xmath2 s@xmath3 , and with the hxd ( hxd - pin ) at a 10 - 50 kev flux of 8.9@xmath1 erg @xmath2 s@xmath3 .
with hxd - pin , the pulsation was detected at a barycentric period of 0.592437 s , and its change rate was reconfirmed as @xmath410@xmath5 s s@xmath3 .
the 1 - 50 kev spectra of 4u 1822 - 37 were found to be very similar to those of her x-1 in the slopes , cutoff and iron lines .
three iron lines ( fe k@xmath6 , fe xxv , and fe xxvi ) were detected , on top of a 1 - 50 kev continuum that is described by an npex model plus a soft blackbody .
in addition , a cyclotron resonance scattering feature was detected significantly ( @xmath799% confidence ) , at an energy of 33@xmath82 kev with a depth of 0.4@xmath9 .
therefore , the neutron star in this source is concluded to have a strong magnetic field of 2.8@xmath10 g. further assuming that the source has a relatively high intrinsic luminosity of several times 10@xmath11 erg s@xmath3 , its spectral and timing properties are consistently explained .
|
a quantum point contact ( qpc ) is the simplest mesoscopic device that directly shows quantum mechanical properties .
it is a short ballistic transport channel between two electron reservoirs , which shows quantized conductance as a function of the width of the channel @xcite .
a widely applied approach for implementing qpcs is using a split - gate structure on the surface of a heterostructure with a two - dimensional electron gas ( 2deg ) at about 100 nm beneath its surface .
the conventional design of such a split - gate qpc has two metallic gate fingers ( fig .
[ fig1sempic]a ) . operating this device with a negative gate voltage @xmath0 results in the formation of a barrier with a small tunable opening between two 2deg reservoirs , because the 2deg below the gate fingers gets depleted over a range that depends on @xmath0 . for electrons in the 2deg
, this appears as an electrostatic potential @xmath1 that is a large barrier with a small opening in the form of a saddle - point potential ( fig .
[ fig3saddlepotential ] ) .
the saddle - point potential gives transverse confinement in the channel that is roughly parabolic , which results for this transverse direction in a discrete set of electronic energy levels .
for electron transport along the channel this gives a discrete set of subbands with one - dimensional character .
quantized conductance appears because each subband contributes @xmath2 to the channel s conductance @xcite , where @xmath3 is the electron charge and @xmath4 is planck s constant .
the length of the qpc channel is fixed and can be parameterized by the lithographic length @xmath5 of the gate structure .
the diagram also illustrates the measurement scheme ( see main text for details ) .
( b ) sem image of a length tunable qpc with 6 gate fingers ( qpc@xmath6 device ) . here
the effective length @xmath7 of the qpc can be tuned by changing the ratio of the gate voltages on the central gates ( @xmath8 ) and side gates ( @xmath9 ) . ]
we present here the design and experimental characterization of qpcs which offer additional control over the shape of the saddle - point potential .
we focused on developing devices for which the effective length of the saddle - point potential ( along the transport direction ) can be tuned _ in situ_. the additional control is implemented with a symmetric split - gate design based on 6 gate fingers ( fig .
[ fig1sempic]b ) .
such devices will be denoted as qpc@xmath6 , and conventional devices with 2 gate fingers ( fig .
[ fig1sempic]a ) as qpc@xmath10 .
these qpc@xmath11 are operated with the gate voltage on the outer fingers ( @xmath9 ) less negative than the gate voltage on the central fingers ( @xmath8 ) to avoid quantum dot formation .
sweeping @xmath8 from more to less negative values opens the qpc@xmath11 .
by co - sweeping @xmath9 at fixed ratio @xmath12 it behaves as a qpc with a certain length for the saddle - point potential , and this length can be chosen by setting @xmath12 : it is shortest for @xmath13 and longest for @xmath14 . for our design
the effective length could be tuned from about 200 nm to 600 nm .
the motivation for developing these length - tunable qpcs comes from studies of electron many - body effects in qpcs .
a well - known manifestation of these many - body effects is the so - called 0.7 anomaly @xcite , which is an additional shoulder at @xmath15 in quantized conductance traces .
these many - body effects are , despite many experimental and theoretical studies since 1996 @xcite , not yet fully understood .
recent theoretical work @xcite suggested that many - body effects cause the formation of one or more self - consistent localized states in the qpc channel , and that these effects result in the 0.7 anomaly and the other signatures of many - body physics .
this theoretical work predicted a clear dependence on the length of the qpc channel , and testing this directly requires experiments where this length is varied . the work by koop _ et al . _
@xcite already explored the relation between the device geometry and parameters that describe the many - body effects in a large set of qpc@xmath10 devices .
this work compared nominally identical devices , and devices for which the lithographic length @xmath5 ( see fig .
[ fig1sempic]a ) and width of the channel in the split - gate structure was varied .
these results were , however , not conclusive .
the parameters that describe the many - body effects showed large , seemingly random variation , not correlated with the device geometry . at the same time , the devices showed ( besides the 0.7 anomaly ) clean quantized conductance traces , and the parameters that reflect the non - interacting electron physics did show the variation that one expects when changing the geometry ( for example , the channel pinch - off gate voltage @xmath16 and subband spacing @xmath17 ) .
this confirms that these qpcs had saddle - point potentials that were smooth enough for showing quantized conductance , while it also shows that the many - body effects are very sensitive to small static fluctuations on these saddle - point potentials or to nanoscale device - to - device variations in the dimensions of the potentials .
this picture was confirmed by shifting the channel position inside a particular qpc@xmath10 device .
this can be implemented by operating a qpc@xmath10 with a difference @xmath18 between the values of @xmath0 on the two gate fingers in fig .
[ fig1sempic]a .
such a channel shift did not change the quantized conductance significantly , but did cause strong variation in the signatures of many - body physics .
earlier work had established that such device - to - device fluctuations can be due to remote defects or impurities , a slight variation in electron density , or due to the nanoscale variation in devices that is inherent to the nanofabrication process @xcite . consequently , studying how the many - body effects depend on the length of the qpc channel requires qpcs for which the channel length can be tuned continuously _ in situ _ , and where this can be operated without a transverse displacement of the qpc channel in the semiconductor material .
the work that we report here aimed at realizing such devices .
this article is organized as follows : section [ 5sec : designconsids ] starts with a short overview of the options and the choices we made for realizing the qpc@xmath11 devices .
next , in section [ 5sec : simulations ] , we present the results of electrostatic simulations . in section [ 5sec :
samplefabandtech ] , we describe the sample fabrication and measurement techniques .
this is followed by comparing results from simulations and experiments for qpc@xmath6 devices in section [ 5sec : experimentalrelization ] , and section [ 5sec : conclusion ] summarizes our conclusions .
device with 6 rectangular gate fingers .
( b ) design of a qpc@xmath11 device with the 4 outer gates in a shape that explicitly induces a funnel shape for the entry and exit of the qpc transport channel . ]
we designed our qpc@xmath11 devices with 6 rectangular gate fingers , in a symmetric layout with two sets of 3 parallel gate fingers ( fig .
[ fig1sempic]b and fig . [ fig2designqpc6f]a ) .
sem inspection of fabricated devices yields that the central gate finger is 200 nm wide ( as measured along the direction of channel length @xmath7 ) .
the outer gate fingers are 160 nm wide , and the narrow gaps between gate fingers are 44 nm wide .
this yields @xmath19 for the total distance between the outer sides of the 3 parallel gate fingers .
the lithographic width of the qpc channel ( distance between the two sets of 3 gate fingers ) is 350 nm .
an example of alternative designs for the gate geometries that we considered is in fig .
[ fig2designqpc6f]b .
this design has a two - sided funnel shape for the channel and this could result in length - tunable qpc operation that better maintains a regular shape for the saddle - point potential .
however , the electrostatic simulations in section [ 5sec : simulations ] show that the rectangular gate fingers as in fig .
[ fig2designqpc6f]a also give a length - tunable saddle - point potential that maintains a regular shape while tuning the length .
this observation holds for a range of device dimensions similar to our design . for our particular design , the lithographic length and width ( 350 nm ) of the channel are comparable , and the 2deg is as far as 110 nm distance below the surface ( and the part in the center of the channel that actually contains electrons is very narrow , about 20 nm ) . in this regime , the saddle - point potential is strongly rounded with respect to the lithographic shapes of the gates ( see for example fig .
[ fig3saddlepotential]c , d ) .
an important advantage of the rectangular design is that it provides two clear points for calibrating the effective channel length @xmath7 : operating at @xmath20 gives @xmath21 for the central gate finger alone ( 200 nm , see fig .
[ fig1sempic]b ) , while operating at @xmath22 gives @xmath7 equal to the lithographic distance between the outer sides of the 3 parallel gate fingers ( 608 nm ) .
a point of concern for this design that deserves attention is whether the narrow gaps between the 3 parallel gate fingers induce significant structure on the saddle - point potential .
the electrostatic simulations show that this is not the case ( see again the examples in fig .
[ fig3saddlepotential]c , d ) .
the part of the channel that contains electrons is relatively far away from the gate electrodes , and the potential @xmath1 at this location is strongly rounded .
notably , the full height of the potentials in fig .
[ fig3saddlepotential ] is about 1 ev , while the occupied subbands are at a height of only about 10 mev above the stationary point of the saddle - point potential ( in the center of the channel ) .
such gaps between parallel gate electrodes can be much narrower when depositing a wider gate on top of the central gate , with an insulating layer between them .
we chose against applying this idea since we also aimed to have devices with a very low level of noise and instabilities from charge fluctuations at defect and impurity sites in the device materials . in this respect ,
we expect better behavior when all gate fingers are deposited in a single fabrication cycle , and when deposition of an insulating oxide or polymer layer can be omitted .
this section presents results of electrostatic simulations of the saddle - point potentials that define the qpc channel .
the focus is on the design with 6 rectangular gate fingers ( fig .
[ fig2designqpc6f]a ) , with gate dimensions as mentioned in the beginning of section [ 5sec : designconsids ] .
the simulations are based on the modeling approach that was introduced by davies _
_ @xcite .
davies _ et al . _
@xcite introduced a method for modeling the electrostatics of gated 2deg .
it calculates the electrostatic potential @xmath1 for electrons in the 2deg regions around the gates ( the approach only applies to the situation where the 2deg underneath the gates is depleted due to a negative voltage on gate electrodes ) .
there are other models and approaches @xcite for calculating such potential landscapes , but these are all more complicated and computationally more demanding .
the approach by davies _
et al . _ is relatively simple .
it does not account for electrostatic screening effects , and , notably , it does not account for the electron many - body interactions that were mentioned earlier .
still , it was shown that it is well suited for calculating a valid picture of a qpc saddle - point potential near the channel pinch - off situation @xcite .
felt by electrons in the 2deg plane .
the plots represent an area of 1000@xmath231000 @xmath24 , centered at the middle of a qpc channel with a length @xmath5 of 200 nm ( a ) and 600 nm ( b ) of a qpc@xmath10 device with a lithographic channel width of 350 nm .
it is calculated for the material parameters that are valid for the measured devices .
see fig .
[ fig1sempic]b for relating the @xmath25- and @xmath26-direction to the gate geometry .
( c),(d ) similar saddle - point potentials @xmath1 calculated for qpc@xmath11 devices ( with material parameters and geometry as the measured devices ) .
the effective channel length is shorter for the case that is calculated for @xmath27 ( c ) than for the case @xmath28 ( d ) ( also note that qpc@xmath6 results for @xmath29 are the same as plot ( a ) ) .
panel ( c ) and ( d ) also show that the narrow gaps between 3 parallel gate fingers do not induce significant structure at low energies in the saddle - point passage ( it only induces a weak fingerprint off to the side in the channel , at energies that are much higher than the occupied electron levels , see panel ( d ) ) . ] the negative voltage on a gate that is needed to exactly deplete 2deg underneath a large gate is called the threshold voltage @xmath30 , and it is to a good approximation given by @xmath31 here @xmath32 is the electron density in the 2deg ( at zero gate voltage ) , @xmath33 is the depth of the 2deg , @xmath34 is the relative dielectric constant of the material below the gate , and @xmath35 is the dielectric constant of vacuum ( for details see ref . ) .
the value of @xmath36 for a certain 2deg material defines the value @xmath37 where the electrostatic potential @xmath1 for electrons in the 2deg becomes higher than the chemical potential of the 2deg . in turn , this can be used to define in an arbitrary potential landscape @xmath1 ( for arbitrary gates shapes and for arbitrary gate voltages ) the positions where @xmath38 .
that is , one can calculate the positions in a gated device structure where there is a boundary between depleted and non - depleted 2deg , and also calculate the electrostatic potential @xmath1 around such points .
when the center of the qpc has @xmath38 , the channel is at pinch - off and no electrons can pass through the qpc . the gate voltage at which this happens is called the pinch - off voltage @xmath16 .
notably , the calculated value of @xmath1 at a certain position is simply the superposition of all the contributions to @xmath1 from different gate electrodes , and it is linear in the gate voltage on each of these electrodes @xcite . figure [ fig3saddlepotential ] presents examples of saddle - point potentials @xmath1 that are calculated with davies method , both for qpc@xmath10 and qpc@xmath11 devices .
the calculations are for material parameters and geometries of measured devices ( as described in detail in the next sections ) .
figures [ fig3saddlepotential]c , d show that the length of the transport channel depends on the applied ratio @xmath12 , and that the narrow gaps between 3 parallel electrodes in qpc@xmath11 devices do not give significant structure on the saddle - point potential in the operation regime that we consider .
the focus of this work is on realizing qpc channels with a tunable length .
the channels are in fact saddle - point potentials ( see fig .
[ fig3saddlepotential ] ) , and it is for such a smooth shape not obvious what the value is of the channel length .
we therefore characterize this channel length with the parameter @xmath7 , which corresponds to the value of the lithographic length @xmath5 of a qpc@xmath10 type device ( with rectangular gate electrodes , see fig .
[ fig1sempic]a ) that gives effectively the same saddle - point potential .
device , illustrating length variables that are introduced in the main text .
( b ) the calculated length @xmath39 for a range of values of the lithographic length of qpc@xmath10 devices , for three values of @xmath40 .
( c ) the calculated effective length @xmath39 for @xmath41 for a qpc@xmath11 device , as a function of the ratio @xmath12 . ]
we implemented this as follows .
we calculated the saddle - point potential @xmath42 for the pinch - off situation ( see fig .
[ fig1sempic]b and fig .
[ fig3saddlepotential]a for how the @xmath25- and @xmath26-directions are defined ) .
the transverse confinement in the middle of the qpc ( defined as @xmath43 , @xmath44 ) is parabolic to a very good approximation .
when moving out of the channel along the @xmath25-direction , the transverse confinement becomes weaker , but remains at first parabolic .
notably , the energy eigenstates for confinement in such a parabolic potential , described as @xmath45 have a width that is ( for all levels ) proportional to @xmath46 . in this expression @xmath47 is the effective mass of the electron and @xmath48 is the angular frequency of natural oscillations in this potential .
the parameter @xmath48 defines here the steepness of @xmath49 , and we obtain @xmath50 values from fitting eq .
[ 5equ : parabollauy ] to potentials @xmath42 obtained with davies s method .
we use this and investigate the width @xmath51 in @xmath26-direction for the lowest energy eigenstate , at all positions @xmath25 along the channel ( see fig .
[ 5fig : lalpha]a ) . for parabolic confinement
this wavefunction in @xmath26-direction has a gaussian shape and has a width @xmath52 with this approach we analyzed that the distance from @xmath43 to the @xmath25-position @xmath53 where the value @xmath51 increased by a factor @xmath54 defines a suitable point for defining the value of @xmath7 .
that is , we define @xmath55 and find @xmath53 by solving @xmath56 for a certain @xmath40 .
subsequently , @xmath7 is defined by using the suitable @xmath40 value , @xmath57 we came to this parameterization as follows .
we used this _ ansatz _
first in simulations of qpc@xmath10 devices . here , we explored for different values of @xmath40 the relation between @xmath5 and @xmath39 .
results of this for @xmath58 , 1.1 and 1.2 are presented in fig .
[ 5fig : lalpha]b .
for the range of @xmath5 values that is of interest to our study ( @xmath59100 nm to @xmath59500 nm ) , we find the most reasonable overall agreement between the actual value for @xmath5 ( input to the simulation ) and the value @xmath39 ( derived from the simulation ) for @xmath41 .
the agreement is not perfect , but we analyzed that the deviation is within an uncertainty that we need to assume because the exact shapes of saddle - point potentials in different device geometries do show some variation , and because the limited validity of davies method .
nevertheless , it provides a reasonable recipe for assigning a value @xmath7 to any saddle - point potential , with at most 20% error .
[ 5fig : lalpha]c presents results of calculating @xmath60 for @xmath41 from simulations of a qpc@xmath11 device , operated at different values for @xmath12 .
the results show a clear monotonic trend , with @xmath61 for @xmath29 to @xmath62 for @xmath63 .
this is for a qpc@xmath11 device for which we expect @xmath64 for @xmath29 and @xmath65 for @xmath63 ( see section [ 5sec : designconsids ] ) . in section [ 5sec : experimentalrelization ]
we discuss how this latter point is used for applying a small correction to the simulated values for @xmath7 .
these simulations show that the qpc@xmath11 that we consider allows for tuning @xmath7 by about a factor 3 .
it is worthwhile to note that our current design showed optimal behavior in the sense that it can tune @xmath7 from about 200 nm to 600 nm , while the dependence of @xmath7 on @xmath12 is close to linear
. we also simulated qpc@xmath11 devices with wider gate electrodes for the outer gates , and ( as mentioned in section [ 5sec : designconsids ] ) devices with gate geometries as in fig .
[ fig2designqpc6f]b .
these devices showed a steeper slope for part of the relation between @xmath12 and @xmath7 , which is not desirable .
we fabricated qpc devices with a gaas/@xmath66 mbe - grown heterostructure , which has a 2deg at 110 nm depth below its surface from modulation doping .
the layer sequence and thickness of the materials from top to bottom ( _ i.e. _ going into the material ) starts with a @xmath67 nm gaas capping layer , then a 60 nm @xmath68 layer with si doping at about @xmath69 , which is followed by an undoped spacer layer of 45 nm .
the 2deg is located in a heterojunction quantum well at the interface with the next layer , which is a 650 nm undoped gaas layer .
this heterostructure was grown on a commercial semi - insulating gaas wafer , after first growing a sequence of 10 gaas / alas layers for smoothing the surface and trapping impurities .
the 2deg had an electron density @xmath70 and a mobility @xmath71 .
we fabricated both conventional qpc@xmath10 devices and qpc@xmath11 devices by standard electron - beam lithography and clean - room techniques .
the gate fingers were deposited using 15 nm au on top of a 5 nm ti sticking layer . for measuring transport through the qpcs we realized ohmic contacts to the 2deg reservoirs by annealing of a auge / ni / au stack that was deposited on the wafer surface @xcite .
the geometries of the fabricated devices were already described in the beginning of section [ 5sec : designconsids ] .
the measurements were performed in a he - bath cryostat and in a dilution refrigerator , thus getting access to effective electron temperatures from 80 mk to 4.2 k. we used standard lock - in techniques with an a.c .
excitation voltage @xmath72 rms at @xmath73 hz .
[ fig1sempic]a shows the 4-probe voltage - biased measurement scheme , where both the current and the actual voltage drop @xmath74 across the qpc channel are measured such that any influence of series resistances could be removed unambiguously .
the gate voltages are applied with respect to a single grounded point in the loop that carries the qpc current .
and @xmath9 for a qpc@xmath11 device , presented in the form of iso - conductance lines at integer @xmath75 levels . the conductance was measured at 4.2 k where the quantized conductance is nearly fully washed out by temperature . the two operational regimes above and below the line @xmath76 yield qpc and quantum dot behavior , respectively . ]
this section presents an experimental characterization of the qpc@xmath11 devices that we designed ( fig [ fig1sempic]b ) and we compare the results to our simulations .
figure [ 5fig : isoconductance2d ] presents measurements of the conductance @xmath77 as a function of @xmath8 and @xmath9 .
several labels in the plot illustrate relevant concepts , which were partly discussed before .
for the area in this plot with @xmath9 more negative than @xmath8 we expect some quantum - dot like localization in the middle of the channel and this regime should therefore be avoided in studies of qpc behavior .
further , the plot illustrates that operation for a particular value of @xmath7 requires co - sweeping of @xmath8 and @xmath9 from a particular point below pinch - off in a straight line to the pivot point .
this corresponds to opening the qpc at a fixed ratio for @xmath12 .
the pivot point is the point where the gate voltages do not alter the original electron density of the 2deg . for this measurement
this is for @xmath78 , but this is different for the case of biased cool downs .
we carried out biased cool downs for suppressing noise from charge instabilities in the donor layer @xcite .
for such experiments the qpcs were cooled down with a positive voltage on the gates .
we typically used + 0.3 v , and observed indeed better stability with respect to charge noise .
the effect of such a cool down can be described as a contribution to the gate voltage of -0.3 v that is frozen into the material @xcite .
consequently , co - sweeping of @xmath8 and @xmath9 for maintaining a fixed channel length must now be carried out with respect to the pivot point @xmath79 instead of @xmath78 . and the lithographic length of qpc@xmath10 devices .
points are experimental results .
the solid line is a phenomenological expression that was used for parameterizing the relation between @xmath16 and the lithographic length .
( b ) comparison between measured and simulated values of the effective channel length for a qpc@xmath11 device . ]
the theory behind the davies method illustrates why operation at fixed effective lenght requires a fixed ratio @xmath12 .
all points in the potentials landscapes @xmath1 for qpc@xmath10 devices as in fig .
[ fig3saddlepotential]a , b have a height that scales linear with the gate voltage @xmath0 .
thus , when opening the qpc the full saddle - point potential changes height at a fixed shape .
mimicking this situation with qpc@xmath11 devices requires a fixed ratio @xmath12 , again because @xmath8 and @xmath9 influence @xmath1 in a linear manner .
the plot also illustrates the two special operation lines where the effective length of the channel is unambiguous , and we used these points to better calibrate the relation between @xmath12 and @xmath7 .
the first case is the line at @xmath80 , which yields @xmath64 , as defined by the central gates alone .
the second case is the line @xmath76 . here
@xmath7 is 608 nm , as defined by the full lithographic length of the 3 gate fingers . as a function of @xmath8 for a qpc@xmath11 device , measured at 80 mk .
different traces are for fixed ratio @xmath29 to @xmath63 ( left to right , traces _ not _ offset ) , which corresponds to increasing the effective channel length @xmath7 from 200 nm to 608 nm . ] we improved and further checked our calibration of the relation between @xmath12 and @xmath7 as follows .
we used the trend that came out of the simulations ( fig .
[ 5fig : lalpha]c ) , but pinned the curve at 200 nm for @xmath29 and at 608 nm for @xmath63 ( black line in fig .
[ 5fig : lalphavsmeasured]b ) .
this trace shows good agreement with results from an independent check ( dashed line ) that used the pinch - off gate voltage @xmath16 as an identifier for the effective length .
this independent check used data from a set of qpc@xmath10 devices for calibrating the relation between @xmath5 and @xmath16 ( fig .
[ 5fig : lalphavsmeasured]a ) .
this shows the trend that shorter qpc@xmath10 devices require a more negative gate voltage to reach pinch - off @xcite .
we related this to the pinch - off values in qpc@xmath11 devices .
in particular , we analyzed the pinch - off points on the @xmath8 axis , and its dependence on @xmath12 ( see also fig .
[ 5fig : linearg ] ) .
the results of using this for assigning a certain @xmath7 to each @xmath12 is the dashed line in fig .
[ 5fig : lalphavsmeasured]b , and shows good agreement with the values that were obtained from simulations . we can thus assign a value to @xmath7 for each @xmath12 with an absolute error that is at most 50 nm
notably , the relative error when describing the increase in @xmath7 upon increasing @xmath12 is much smaller . the results in fig .
[ 5fig : linearg ] provide an example of linear conductance measurements on a qpc@xmath11 device at 80 mk .
the traces show clear quantized conductance plateaus for all settings of @xmath7 .
several of these linear conductance traces also show the 0.7 anomaly , and the strength of its ( here rather weak ) expression shows a modulation as a function of @xmath7 over about 3 periods .
a detailed study of this type of periodicity can be found in ref . .
this example illustrates the validity and importance of our type of qpcs in studies of length - dependent transport properties .
we have developed and characterized length - tunable qpcs that are based on a symmetric split - gate geometry with 6 gate fingers .
gate structures with different shapes and dimensions can be designed depending upon the required range for length tuning and for optimizing the tuning curve . for our purpose ( qpcs with an effective channel length between about 200 nm and 600 nm , and 350 nm channel width ) we found that simple rectangular gate fingers are an attractive choice .
our simulations and experimental results are in close agreement .
we were able to tune the effective length by about a factor 3 , from 200 nm to 608 nm .
qpcs are the simplest devices that show clear signatures of many - body physics , as for example the @xmath81 anomaly and the zero - bias anomaly ( zba ) @xcite .
our length - tunable qpcs provide an interesting platform for systematically investigating these many - body effects .
in particular , these qpcs provide a method for studying the influence of the qpc geometry without suffering from device - to - device fluctuations that hamper such studies in conventional qpcs with 2 gate fingers .
studies in this direction are presented in ref . .
we thank y. meir for discussions , b. h. j. wolfs for technical assistance , and the german programs dfg - spp 1285 , research school ruhr - universitt bochum and bmbf quahl - rep 01bq1035 for financial support .
mji acknowledges a scholarship from the higher education commission of pakistan .
|
we report on developing split - gate quantum point contacts ( qpcs ) that have a tunable length for the transport channel .
the qpcs were realized in a gaas / algaas heterostructure with a two - dimensional electron gas ( 2deg ) below its surface .
the conventional design uses 2 gate fingers on the wafer surface which deplete the 2deg underneath when a negative gate voltage is applied , and this allows for tuning the width of the qpc channel .
our design has 6 gate fingers and this provides additional control over the form of the electrostatic potential that defines the channel .
our study is based on electrostatic simulations and experiments and the results show that we developed qpcs where the effective channel length can be tuned from about 200 nm to 600 nm .
length - tunable qpcs are important for studies of electron many - body effects because these phenomena show a nanoscale dependence on the dimensions of the qpc channel .
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.