
beam energy spectrum for the ILC at this energy, and the effect of polarized 
beams. The energy range and top quark mass follow the choices in 
Reference~\cite{Seidel:2013sqa}.


\begin{table}[htb] 
\caption{Single top content of the $W^+bW^-\bar{b}$ sample, for unpolarized
   and polarized beams, as a function of the center-of-mass energy.}
\begin{center} 
\begin{tabular}{|c|c|c|c|c|c|c|} 
\hline 
   $\sqrt{s}$  &   $e^-e+$     &    $e_L^-e_R^+$  &   $e^-e+$   &   $e^-e+$   &   $e^-e+$  & $e^-_Le+_R$  \\ 
    (GeV)      &    (LO)       &      (LO)       &    (NLO)   &    (NLO)  & (NLO, ISR, LS)  & (NLO, ISR, LS)   \\ 
\hline
   344      & 23\%   & 36\%    & 32\%    & 31.1\% & 38.2\% &  37.6\% \\ 
   345      & 19\%   & 30\%    & 22\%    & 22.3\% & 27.5\% &  27.8\%\\ 
   346      & 13\%   & 26\%    & 15\%    & 16.3\% & 20.1\% &  19.9\%\\ 
   347      &  9\%   & 19\%    &  9\%    & 9.9\% & 12.4\% &  11.5\%\\ 
   348      &  7\%   & 14\%    &  6\%    & 5.9\% & 7.5\% &  7.7\%\\ 
   349      &  5\%   & 10\%    &  5\%    & 4.4\% & 5.5\% &  6.1\%\\ 
\hline
\end{tabular}
\end{center} 
\label{table02} 
\end{table} 
 
\begin{figure}[htb]
\begin{center}
  \includegraphics[width=0.9\linewidth]{threshold_scan_NLO_beam.pdf}
  \caption{Cross-sections at NLO, according to WHIZARD, for $W^+bW^-\bar{b}$ and $t\bar{t}$ 
           production in $e^+e^-$ annihilations with unpolarized beams, as
           a function of center-of-mass energy, around the double-top
           threshold region. The 1S-top mass is set to 174 GeV.
    \label{fig:threshold}}
\end{center} 
\end{figure}


\begin{figure}[htb]
\begin{center}
  \includegraphics[width=0.9\linewidth]{threshold_scan_NNLO.pdf}
  \caption{Cross-sections at NNLL, according to 
A.H.Hoang et al.~\cite{Hoang:2010gu}, for $t\bar{t}$ 
           production in $e^+e^-$ annihilations with unpolarized beams
           a function of center-of-mass energy. The 1S-top mass is set to 174 GeV.     \label{fig:threshold2}}
\end{center} 
\end{figure}

\begin{figure}[htb]
\begin{center}
  \includegraphics[width=0.9\linewidth]{single_top_pedro.pdf}
  \caption{The effect on the cross-section of non-resonant corrections, 
according to M.~Beneke et al.~\cite{Beneke:2010mp}, for $e^+e^-$ annihilation to $W^+bW^-\bar{b}$ in the $t\bar{t}$ threshold region.
          The cross-section for unpolarized beams is given as a function of 
center-of-mass energy. With respect to the original publication, the curve is shifted such that the 1S peak coincides approximately with those of Figures~\ref{fig:threshold2} and~\ref{fig:threshold}. 
    \label{fig:threshold3}}
\end{center} 
\end{figure}

We note that the presence of single top events modifies 
significantly the cross-section. Moreover, the fraction
of single top events varies rapidly across the threshold region.
This does not mean, however,  that the top mass measurement
is affected. According to the method proposed in~\cite{Seidel:2013sqa},
a fit is performed to the cross-section measurement, using a predefined
function obtained from the double-top calculation at NNLO.
As long as the shape of the curve is not modified, the result is
not affected. We observe that according to the NLO calculation of 
the $W^+bW^-\bar{b}$ cross-section, the 
$t\bar{t}$ cross-section may be obtained quite exactly by shifting
down the $W^+bW^-\bar{b}$ cross-section by some 0.020 pb. 
There is only a residual shift at 348 GeV of 0.004 pb that would
shift the top mass by at most 20 MeV. The $W^+bW^-\bar{b}$ and
$t\bar{t}$ curves are displayed Figure~\ref{fig:threshold}. 


This result is also supported by the 
NNLL result in Reference~\cite{Hoang:2010gu}, that we reproduce in
Figure~\ref{fig:threshold2}. The calculations are not directly
comparable, as initial state radiation and beam energy spread are not
included in Reference~\cite{Hoang:2010gu}. 
The authors find that the impact impact of a cut on the $Wb$ mass 
is an approximately constant shift. The cross-section for $t\bar{t}$ events 
with $Wb$ massin a range -15 $< m_{Wb} - m_t < $ 15 is, as before, shifted down 
from the inclusive  cross-section by 0.050 pb.
Finally, Figure~\ref{fig:threshold3} shows the effect of non-resonant
effects on the $WbWb$ cross-section. Again, the shift in the cross section
is essentially constant across the threshold region.

The single top content of the samples depends strongly
on the energy, as seen in table~\ref{table02}. In analyses where
the top quark is reconstructed and cuts are applied on the mass
and/or energy of the candidates, the selection
efficiency can be quite different for top quark pair and single
top events (up to 50\% smaller for single top in the example above). 
This means that, after correcting for efficiency,
the shape of the curve will be deformed if single top events are ignored.  
Note that this does not apply to Reference~\cite{Seidel:2013sqa}, since
in this analysis no selection cuts based on reconstructed
top quark mass and energy are applied.
 
As the most precise calculations are only available for the 
$e^+e^- \rightarrow t\bar{t}$
process, none of the studies of future 
collider prospects so far has taken into account the full
$W^+bW^-\bar{b}$  process. As soon as the necessary tools have been developed
we suggest to repeat the
study of the top quark mass extraction of future lepton
colliders using $W^+bW^-\bar{b}$ events and polarized beams.



\section{Energy dependence}
\label{sec:energy}


\begin{figure}[htb]
\begin{center}
  \includegraphics[width=1.0\linewidth]{single_top_lc.pdf}
  \caption{Cross-section for $W^+bW^-\bar{b}$ and $t\overline{t}$
           events produced in unpolarized electron-positron
           annihilations as a function of the 
           center-of-mass energy. The curves are obtained using
           WHIZARD at LO.
    \label{energydep}}
\end{center} 
\end{figure}

The composition of the $W^+bW^-\bar{b}$ sample is energy dependent
as shown in Fig.~\ref{energydep}, that presents the 
unpolarized LO cross-sections for $e^+e^- \rightarrow W^+bW^-\bar{b}$.
The solid line corresponds to the full 2 $\rightarrow$ 4 process, 
including single top production and non-top production. The dashed line
presents the results for the $e^+e^- \rightarrow t\bar{t} \rightarrow 
W^+bW^-\bar{b}$ process. All results have been obtained 
using WHIZARD. The difference between the $t\bar{t}$ 
cross-section and the full 2 $\rightarrow$ 4 result is 10\% at 500 \gev{} 
and rises