As it comes out, no experiment so far has seen direct traces of the Higgs boson. This should be not too big a deception, since the Higgs mass is big - and, as it seems, too high for the Higgs to be produced in today's particle colliders. However, there are indirect means to estimate a probable range for the mass of the Higgs boson. The current expectations about the Higgs mass are encoded in the following figure:
A least-square fit to a number of precisely known data in electroweak physics using the Standard Model as theoretical framework and the Higgs mass as a free parameter yields an expectation value for the Higgs mass around the minimum of the parabola. [Source: Precision Electroweak Measurements and Constraints on the Standard Model by the LEP Collaborations and the LEP Electroweak Working Group, arXiv: 0712.0929v2, Figure 5.]
For while the Higgs mass could not be measured directly so far, it's a parameter which, within the framework of the Standard Model, influences all kinds of measurable quantities. An example of this reasoning is the analysis of the Z pole, the resonance peak of the Z boson in the cross section for the production of particles in electron-positron annihilation. There are many details of the Z pole that have been determined very precisely in experiment, for example the width, shape, asymmetry, or background slope - about a dozen or so parameters altogether. On the other hand, all these measured parameters can also be calculated within the Standard Model. In these calculations, there are five quantities that have to be assumed as input parameters: the coupling constants of QED and QCD at the Z pole, α(m²Z) and αs(m²Z), the masses of the Z boson and the top quark, and the Higgs mass.
Now, one can search for those values of these five parameters that reproduce best the data measured in experiment. To do so, one minimises the squares of the differences between measured data and calculated values, weighted by the experimental error. This quantity to be minimised as a function of the free parameters is called χ² ("chi-square"). It is given in general by the formula
where Xi and &sigmai are the experimental data and their respective errors, while the fi are the theoretical predictions, depending on the free parameters p1 ... pN. In this case, there are five free parameters. Fixing the values for the coupling constants and the Z and top quark masses (parameters p1 ... p4) at their best fit values, one can then check how χ² changes if the Higgs mass (parameter p5) is varied.
This yields the plot above: The parabolic curves show, as a function of the Higgs mass mH, the corresponding χ² above its minimum at a mass of about 80 GeV/c². Note that the Higgs mass is shown on a logarithmic scale. The blue band represents theoretical uncertainties within the Standard Model calculations, and the dashed and dotted parabolas correspond to a different parametrisation for the running of the coupling constant α and the inclusion of more data into the fit, respectively. The area marked in yellow corresponds to Higgs masses which can be excluded by available experimental data: A Higgs boson with a mass below 114.4 GeV/c² would have been discovered at the Large Electron-Positron Collider LEP at CERN before it was shut down to give place for the construction of the Large Hadron Collider LHC.
Now, supposing that the Standard Model is the correct theory to describe electroweak data such as the Z pole, the fitting procedure predicts the Higgs mass to be around the minimum of the χ² curve. Together with the experimentally excluded mass range, this yields an upper limit on the Higgs mass of 182 GeV/c² at a 95% confidence level, meaning that if the Standard Model is right, there is a 95% probability that the mass of the Higgs boson is between 114 and 182 GeV/c².
This is a predictions that will be
The Higgs mass plot is taken from the preprint Precision Electroweak Measurements and Constraints on the Standard Model by the The LEP Collaborations: ALEPH Collaboration, DELPHI Collaboration, L3 Collaboration, OPAL Collaboration, the LEP Electroweak Working Group, arXiv: 0712.0929v2. More plots and details can be found on the web site of The LEP Electroweak Working Group (LEP EWWG), which "combines the measurements of the four LEP experiments ALEPH, DELPHI, L3 and OPAL on electroweak observables, such as cross sections, masses and various couplings of the heavy electroweak gauge bosons, properly taking into account the common systematic uncertainties" to confront "theories such as the Standard Model of particle physics."
Details of the fitting procedure (Which data have been fitted, actually? What are the fitting parameters?) are described in Section 8 (see specifically 8.5 and 8.6) of the paper Precision Electroweak Measurements on the Z Resonance by the ALEPH Collaboration, the DELPHI Collaboration, the L3 Collaboration, the OPAL Collaboration, the SLD Collaboration, the LEP Electroweak Working Group, the SLD electroweak, heavy flavour groups, arXiv:hep-ex/0509008v3, Physics Reports 427 (2006) 257.
χ² minimisation is done using the software package MINUIT. Different minimisation techniques and the χ² minimisation are described in a tutorial (PDF file).
Results of the direct Higgs search at CERN before the decommission of the LEP which have established the lower bound for the Higgs mass at 114.4 GeV/c² are described in Search for the Standard Model Higgs Boson at LEP by G. Abbiendi, et al., arXiv: hep-ex/0306033v1, Phys. Lett. B565 (2003) 61-75.
Physics World has a portrait of Peter Higgs, after whom the Higgs field has its name.
This post is a latecomer to our A Plottl A Day series.