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%--#[ Title page :
\begin{titlepage}
\vspace*{-3.5cm}
\vspace*{0.1cm} \rightline{KEK-CP-303,  NIKHEF 14-005}
\vspace*{0.1cm}\rightline{TTP14-004, SFB/CPP-14-06}
\vspace*{1.cm}
\begin{center}
{\Large{\bf Full $\mathcal{O}(\alpha)$ electroweak radiative corrections 
to $e^+e^- \rightarrow e^+e^-  \gamma$ at the ILC with GRACE-Loop}}\\

\vspace*{0.5cm}
  P.H. Khiem$^{A,B}$, Y. Kurihara$^{A}$, J. Fujimoto$^{A}$, T. Ishikawa$^{A}$,\\
 T. Kaneko$^{A}$, K. Kato$^C$, N. Nakazawa$^{C}$, Y. Shimizu$^A$, \\
 T. Ueda$^{D}$, J.A.M. Vermaseren$^E$, Y. Yasui$^{F}$\\
 
 \vspace*{1cm}

\textit{ $^{A)}$KEK, Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan.\\ 
  $^{B)}$SOKENDAI University, Shonan Village, Hayama, Kanagawa 240-0193 Japan. \\ 
  $^{C)}$Kogakuin University, Shinjuku, Tokyo 163-8677, Japan.\\ 
  $^{D)}$Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany. \\
  $^{E)}$Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands.\\ 
  $^{F)}$Tokyo Management College, Ichikawa, Chiba 272-0001, Japan.}
\end{center}
 \abstract{We present a calculation of the full 
 $\mathcal{O}(\alpha)$ electroweak radiative corrections to the process 
 $e^+e^- \rightarrow e^+e^-  \gamma$ at the International Linear Collider 
 (ILC). The computation is performed with the help of the GRACE-Loop system. 
 In this system, the calculations are checked numerically by three kinds of 
 consistency tests: ultraviolet finiteness, infrared finiteness, and gauge 
 parameter independence. The results show good numerical stability 
 when quadruple precision is used. We find that the electroweak corrections 
 range from $-2\%$ to $\sim -20\%$ when varying $\sqrt{s}$ from $250$ GeV to 
 $1$ TeV. Such corrections will play an important role for the high 
 precision program at the ILC.}
\end{titlepage}
%--#] Title page : 
%--#[ Introduction :

\section{Introduction}
The main goals of future colliders, such as the ILC, are not only to make 
precise measurements of the properties of the Higgs particle, top quarks and 
vector bosons interactions, but also to search for physics Beyond the 
Standard Model (BSM). The measurements will be performed to high 
precision with the statistical error typically below $0.1\%$ at the ILC. 
This will require a very precise measurement of the luminosity. At the ILC, 
the integrated luminosity is measured by counting Bhabha events and 
comparing it with the corresponding theoretical cross-section. Thus a 
precise calculation of Bhabha scattering is mandatory.

The full one-loop electroweak  corrections to the $e^+e^- \rightarrow 
e^+e^-$ reaction were calculated by K.~Tobimatsu et al in 
Ref~\cite{tobimatsu1,
tobimatsu2} and M.~Bohm et al in Ref~\cite{Bohm:1984yt,Bohm:1986fg} a long 
time ago. These calculations were independently calculated by F.A.~Berends 
et al in Ref~\cite{Berends:1987jm}, and  Fleischer et al in 
Ref~\cite{Fleischer:2006ht}.  The two-loop photonic corrections to this 
process were also completed by A.A.~Penin in Ref~\cite{Penin:2005kf,
Penin:2005eh}.

Beside a precise calculation of the Bhabha scattering cross-section, the 
evaluation of backgrounds to the Bhabha events is also important for the 
luminosity measurement at the ILC. With the misidentification of $e^+e^-  
\gamma$ as $e^+e^-$ events, the process $e^+e^- \rightarrow e^+e^-  \gamma$ 
is one of the channels that form a massive contribution to the background 
for Bhabha events. In order to correct the number of counted Bhabha events, 
theoretical calculations to the processes have to be considered.
The lowest-order calculation of this process was obtained in Ref~\cite{Tobimatsu:2001kb}.
A analytical calculation of one-loop QED corrections to the process $e^+e^- \rightarrow e^+e^-  \gamma$ 
is also performed in Ref~\cite{Igarashi}. 

In this paper, we present the calculation of the full $\mathcal{O}(\alpha)$ 
electroweak radiative corrections to this process. The computation is 
performed with the help of the GRACE-Loop system. We then discuss the 
electroweak corrections to the total cross-section which are shown as a 
function of the center-of-mass energy, ($\sqrt{s}$). The distributions of 
the invariant masses, energy of final particles are also discussed in this 
paper.

The paper is organized as follows. In section 2, we present a short 
introduction to the GRACE-Loop system and the numerical tests of the 
results. In section 3 we present the physics results of the calculation. 
Conclusions and future plans will be addressed in section 4.

%--#] Introduction :
%--#[ GRACE-Loop and the calculation :

\section{GRACE-Loop and the process $e^+e^- \rightarrow e^+e^-  \gamma$ }

\subsection{GRACE-Loop}
GRACE-Loop is a generic program for the automatic calculation of high 
energy physics processes at the one-loop level. The program is described 
in detail in Ref~\cite{Belanger:2003sd} where a variety of $2\rightarrow 2 
$-body electroweak processes are presented and compared  
with other papers. The GRACE-Loop system has also been used to calculate 
$2\rightarrow 3$-body processes such as $e^+e^- \rightarrow 
ZHH$~\cite{Belanger:2003ya}, $e^+e^- \rightarrow t \bar{t} 
H$~\cite{Belanger:2003nm}, $e^+e^- \rightarrow \nu \bar{\nu} 
H$~\cite{Belanger:2002me}. The above calculations have been done 
independently by other groups, for example the processes $e^+e^- \rightarrow 
ZHH$~\cite{Zhang:2003jy}, $e^+e^- \rightarrow t \bar{t} 
H$~\cite{You:2003zq, Denner:2003ri, Denner:2003zp} and $e^+e^- \rightarrow 
\nu \bar{\nu} H$~\cite{Denner:2003yg, Denner:2003iy}. Also the 
$2\rightarrow 4$-body process $e^+e^- \rightarrow \nu_{\mu} 
\bar{\nu}_{\mu} HH$~\cite{Kato:2005iw} was calculated successfully with the 
use of the GRACE-loop system.

In the GRACE-Loop system the renormalization is carried out with the 
on-shell renormalization condition of the Kyoto scheme as described in 
Ref~\cite{kyotorc}. The ultraviolet (UV) divergences are regulated by 
dimensional regularization. While the infrared (IR) divergences are 
regularized by giving the photon an infinitesimal mass $\lambda$.

Ref~\cite{Belanger:2003sd} describes the method used in the GRACE-Loop 
system to reduce the tensor one-loop five- and six-point functions into 
one-loop four-point functions. The tensor one-, two-, three- and 
four-point functions are then reduced to scalar one-loop integrals which 
will be numerically evaluated by one of the FF~\cite{ff} or 
LoopTools~\cite{looptools} packages.

The program has been equipped with so-called non-linear gauge fixing 
terms~\cite{nlg-generalised} in the Lagrangian which are defined as
\begin{eqnarray}
{{\cal L}}_{GF}&=&-\frac{1}{\xi_W}
|(\partial_\mu\;-\;i e \tilde{\alpha} A_\mu\;-\;ig c_W
\tilde{\beta} Z_\mu) W^{\mu +} + \xi_W \frac{g}{2}(v
+\tilde{\delta} H +i \tilde{\kappa} \chi_3)\chi^{+}|^{2} \nonumber \\
& &\;-\frac{1}{2 \xi_Z} (\partial\cdot Z + \xi_Z \frac{g}{ 2 c_W}
(v+\tilde\varepsilon H) \chi_3)^2 \;-\frac{1}{2 \xi_A} (\partial \cdot A
)^2 \;.
\end{eqnarray}
We are working in the $R_{\xi}$-type gauges with the condition 
$\xi_{W}=\xi_{Z}=\xi_{A}=1$ (also called the 't Hooft-Feynman gauge), There 
is no longitudinal contribution in the gauge propagator. This choice has 
not only the advantage of making the expressions much simpler. It also 
avoids unnecessary large cancellations, high tensor ranks in the one-loop 
integrals and extra powers of momenta in the denominators which cannot be 
handled by the FF package. With the implementation of non-linear gauge fixing 
terms the system provides a powerful tool to check the results in a consistent 
way. After all, the results must be independent of the non-linear gauge 
parameters. This will be discussed in greater detail in the section on the 
numerical checks of the calculation.

In its latest version GRACE-Loop can use the axial gauge in the projection 
operator of external photons. It cures a problem with large numerical 
cancellations. This is very useful when calculating processes at small 
angles and energy cuts for the final state particles. It also provides a 
useful tool to check the consistency of the results which, due to the Ward 
identities, must be independent of the choice of the gauge. This method has 
been applied successfully to the process $e^+e^- \rightarrow t \bar{t} 
\gamma$ in Ref~\cite{Khiem:2012bp}. We also applied the method in the 
current calculation.

In the integration step we use a parallel version of BASES~\cite{bases} with
MPI~\cite{mpi} (Message Passing Interface) to reduce the calculation time.

\subsection{The process $e^+e^- \rightarrow e^+e^- \gamma$}

The full set of Feynman diagrams with the non-linear gauge fixing as 
described in the previous section consists of $32$ tree diagrams and  
$3456$ one-loop diagrams including the counterterm diagrams. In 
Fig~\ref{feynmandiagrams} we show some selected diagrams.
\begin{figure}[h]
 \begin{center}
 \includegraphics[width=6.3in, height=12cm]{grcfigeey.eps}
 \end{center}
\caption{\label{feynmandiagrams} Typical Feynman diagrams as 
generated by the GRACE-Loop system.}
\end{figure}

The calculation is checked numerically by three kinds of consistency 
tests. These are ultraviolet and infrared finiteness, and independence of 
the gauge parameters. In general the total cross-section with the full 
one-loop electroweak radiative corrections is written by
\begin{eqnarray}
\label{tot}
 \sigma^{e^-e^+ \gamma_ H}_{\textbf{\small tot}} &=& \int d\sigma^{e^-e^+ \gamma_H} _{\textbf{\small {T}}}
 +\int d\sigma^{e^-e^+ \gamma_H}_{\textbf{\small V}}(C_{UV},\{\tilde{\alpha}, \tilde{\beta}, \tilde{\delta},\tilde{\epsilon},  \tilde{\kappa} \}, \lambda)
 \nonumber \\ &&
 + \int d\sigma^{e^-e^+ \gamma_H}_{\textbf{\small T}} \delta_{\textbf{\small soft}}(\lambda \leq E_{\gamma_S} <k_c)
 + \int d\sigma^{e^-e^+ \gamma_H \gamma_S}_{\textbf{\small H}}(E_{\gamma_S} \geq k_c).
  \end{eqnarray}
In this formula $\sigma^{e^-e^+ \gamma_H}_{\textbf{\small T}} $ is the tree-level cross-section. $\sigma^{e^-e^+ \gamma_H}_{\textbf{\small V}}$ is 
the cross-section due to the interference of the one-loop (including 
counterterms) and the tree diagrams. The contribution
must be independent of the ultraviolet cutoff parameter ($C_{UV}$) and the 
non-linear gauge parameters ($\tilde{\alpha}, \tilde{\beta}, \tilde{\delta},
\tilde{\epsilon},  \tilde{\kappa}$). Because of the way we regularize the 
IR-divergences $\sigma^{e^-e^+ \gamma_H}_{\textbf{\small V}}$ depends on the photon mass $\lambda$. The 
$\lambda$ dependence of the result has to cancel against the soft 
photon contribution, the third term in Eq.\ref{tot}. The soft contribution 
can be factorised into a soft factor, which is written by the formula 
below,  and the crosssection from the tree diagrams.
\begin{eqnarray}
\delta_{\textbf{\small soft}}(\lambda \leq E_{\gamma_{S}} < k_c)&=&-e^2\;\int_{\lambda \leq q_0 \leq k_c} \frac{d^3q}{(2 \pi)^3 2q^0}
\left|\frac{p^-}{q\cdot p^-}- \frac{p^+}{q\cdot
p^+}\right|^2,
\end{eqnarray}
where $q$ and $p^{\pm}$ are the photon and the $e^{\pm}$ 4-momenta 
respectively.


In the tables (\ref{cuv}, \ref{gauge}, \ref{lambda}), we 
present the numerical results of the ultraviolet finiteness checks, the 
gauge invariance and the infrared finiteness at one random phase space 
point. This test is performed in quadruple precision. We find that the 
results are stable over a range of 19 digits. The different precisions are due to the 
ways these parameters occur in the formulas. $C_{UV}$ occurs only linearly as 
an extra term, The non-linear gauge parameters occur as products in terms 
that are by themselves typically much larger than the remaining terms. The 
infrared regulator $\lambda$ will mainly contribute due to its appearance 
in the denominators. As a result the $C_{UV}$ checks give a much better 
accuracy than the other checks.

\begin{table}[ht]
\centering
\begin{tabular}{ll } \hline 
$C_{UV}$       & $2\mathcal{\Re}(\mathcal{M}_{Loop}\mathcal{M}^+_{Tree})$ \\ \hline\hline
$0$            & $  -1.88001614070088633160096380252506 $ \\ \hline
$10^2$         & $  -1.88001614070088633160096380252504 $ \\ \hline
$10^4$         & $  -1.88001614070088633160096380252483 $ \\ \hline
\end{tabular}
\caption{\label{cuv} Test of the $C_{UV}$ independence of the amplitude. 
In this table, we take the non-linear gauge parameters to be $0$, 
$\lambda=10^{-17}$GeV and we use $1$ TeV for the center-of-mass energy.}
\end{table}

\begin{table}[ht]
\centering
\begin{tabular}{ll } \hline 
 $(\tilde{\alpha}, \tilde{\beta}, \tilde{\delta},\tilde{\kappa}, \tilde{\epsilon} )  
 $   &  $2\mathcal{\Re}(\mathcal{M}_{Loop}\mathcal{M}^+_{Tree})$         \\ \hline \hline
(0, 0, 0, 0, 0)             & $-1.88001614070088633160096380252506$ \\ \hline
(1.1,1.2,1.3,1.4,1.5)       & $-1.88001614070088633160096380252527$ \\ \hline
(11,12,13,14,15)            & $-1.88001614070088633160096380260499$ \\ \hline
\end{tabular}
\caption{\label{gauge} Gauge invariance of the amplitude. 
In this table, we set $C_{UV} = 0$, the photon mass is 
$10^{-17}$GeV and a $1$ TeV center-of-mass energy.}
\end{table}

\begin{table}[ht]
\centering
\begin{tabular}{lr} \hline 
 $\lambda$ [GeV]           &  $2\mathcal{\Re}(\mathcal{M}_{Loop}\mathcal{M}^+_{Tree})$+ soft contribution  \\ \hline \hline 
 $10^{-17}$ 		    & $ -0.392635564863145920331840202138979$  \\ \hline 
 $10^{-20}$ 		    & $ -0.392635564863145860698638985751228$  \\ \hline 
 $10^{-25}$ 		    & $ -0.392635564863145860639598148071754 $  \\ \hline 
\end{tabular}
\caption{\label{lambda} Test of the IR finiteness of the amplitude. 
In this table we take the non-linear gauge parameters to be $0$, 
$C_{UV}=0$ and the center-of-mass energy is $1$ TeV.}
\end{table}

Finally, we consider the contribution of the hard photon bremsstrahlung, 
$\sigma^{e^-e^+ \gamma \gamma}_{\textbf{\small H}}(k_c)$. 
This part is the process 
$e^+e^- \rightarrow e^- e^+ \gamma \gamma$ and is generated by the tree 
level version of
GRACE~\cite{grace} with the use of the phase space integration by BASES. By 
adding this contribution to the total cross-section, the final results have 
to be independent of $k_c$. In table (\ref{kc}) the numerical result of the 
$k_c$ stability check is shown. By changing the value of $k_c$ from 
$10^{-3}$ GeV to $0.1$ GeV, we find that the results are in agreement to an 
accuracy which is better than $0.05\%$. For the $k_c$ stability checks, it
is important to note that we have two photons at the final state. One of
them is the hard photon which will be applied an energy cut of
 $E_{\gamma}^{cut} \geq 10$ GeV and an angle cut of
$10^{\circ}\leq \theta_{\gamma}^{cut} \leq 170^{\circ}$.
The other one is a soft photon of which the energy is greater than $k_c$
and smaller than the first photon's energy.
\begin{table}[ht]
\centering
\begin{tabular}{llll} \hline 
 $k_c$ [GeV]     &$\sigma_{S}$ [pb] & $\sigma_{H}$ [pb] &$\sigma_{S+H}$ [pb]  \\ \hline \hline 
$10^{-1}$ 		    & 6.829 &1.454  & 8.284  \\ \hline 
$10^{-2}$ 		    & 6.302 &1.983  & 8.286  \\ \hline 
$10^{-3}$ 		    & 5.776 &2.512  & 8.289  \\ \hline 
\end{tabular}
\caption{\label{kc} Test of the $k_c$-stability of the result. We 
choose the photon mass to be $10^{-17}$ GeV and the center-of-mass energy 
is $1$ TeV. The second column presents the hard photon cross-section and 
the third column presents the soft photon cross-section. The final column 
is the sum of both.}
\end{table}

The reduction method for the one-loop five point function in GRACE-Loop is 
also cross-checked with the one in Ref~\cite{Denner:2005nn} by calculating 
the product of diagram 813 and diagrams 5 and 13 in 
figure~(\ref{feynmandiagrams}). Table~\ref{reduction} shows the numerical 
check at an arbitrary phase space point. The two methods are in agreement 
over a range of 19 digits.
\begin{table}[ht]
\centering
\begin{tabular}{ll } \hline 
 &  $2\mathcal{\Re}(\mathcal{M}_{Loop}\mathcal{M}^+_{Tree})$  \\ \hline \hline 
GRACE-Loop & 4.227391116214468972776738502979044$\cdot 10^{-6}$   \\ \hline
Ref~\cite{Denner:2005nn}    & 4.227391116214468972272266356271230$\cdot 10^{-6}$
   \\ \hline
\end{tabular}
\caption{\label{reduction} Test of the reduction of the one loop five point 
functions in GRACE-Loop in comparison with the method in 
Ref~\cite{Denner:2005nn}.}
\end{table}

After checking the results successfully, we can proceed with the 
computations of the physics results of the process. Hereafter we set 
$\lambda=10^{-17}$ GeV, $C_{UV}=0$, $k_c=10^{-3}$ GeV and $\tilde{\alpha}= 
\tilde{\beta}= \tilde{\delta}= \tilde{\kappa}=\tilde\varepsilon=0$. In order to reduce the calculation time,
we neglected the diagrams which contained the coupling of Higgs boson to electron and positron ($\lambda_{He^-e^+}$)
in the integration step. Because it's contribution is smaller than the statistical error of Monte Carlo integration.

%--#] GRACE-Loop and the calculation : 
%--#[ The physics results :

\section{The physics results}
Our input parameters for the calculation are as follows. The fine structure 
constant in the Thomson limit is $ \alpha^{-1}=137.0359895$. For the boson masses,
we input $M_H=126$ GeV, $M_Z=91.1876$ GeV and $M_W=80.385$ GeV. 
For the lepton masses we take $m_e=0.510998928$ MeV, $m_{\tau}=1776.82$ 
MeV and $m_{\mu}=105.6583715$ MeV. For the quark masses we take $m_u=2.3$ 
MeV, $m_d=4.8$ MeV, $m_c=1.275$ GeV, $m_s=95$ MeV, $m_t=173.5$ GeV and 
$m_b=4.18$ GeV. The decay width of the Z boson is $2.3549$ GeV which 
is calculated by using the tree version of GRACE~\cite{grace} with the same 
input parameters. We consider all the decay channels of the Z boson 
within the Standard Model to generate this decay width. Then, we applied it to 
the propagators with Z boson exchange (it is important note that 
we did not apply it to the propagators which appear in one-loop integrals). 
It helps to avoid the singularity due to Z boson resonance.

For the final state particles we apply an energy cut of $E^{cut} \geq 10$ 
GeV and an angle cut of $10^{\circ}\leq \theta^{cut} \leq 170^{\circ}$ with 
respect to the beamline. In order to isolate the photon from the electron 
(or positron) we apply an opening angle cut between the photon and the 
$e^-$($e^+$) of $10^{\circ}$. Moreover, to distinguish $e^- e^+ \gamma$ 
events from $\gamma \gamma$ events, we apply an angle cut between the final 
state electron and positron of $10^{\circ}$. 

The full $\mathcal{O}(\alpha)$ electroweak cross-section $\sigma(\alpha)$ 
takes into account the tree graphs and the full one-loop virtual corrections 
as well as the soft and hard bremsstrahlung contributions. The relative 
correction is defined as
\begin{eqnarray}
 \delta_{EW} & = & \frac{\sigma(\alpha)}{\sigma_{Tree} }-1.
\end{eqnarray}

In order to obtain the genuine weak corrections, we first calculate the 
pure QED corrections which are defined by
\begin{eqnarray}
	\delta_{QED} & = & 
          \frac{\sigma^{QED}(\alpha)-\sigma_0^{QED}}{\sigma_{Tree}}.
\end{eqnarray}
Here $\sigma_0^{QED}$ is the cross-section of the tree-level QED diagrams 
and $\sigma^{QED}(\alpha)$ is the cross-section due to the interference of 
the QED one-loop (including counterterm) diagrams and the QED tree 
diagrams. This part can be calculated separately in the GRACE-Loop system.

After obtaining the QED corrections, 
we evaluate the genuine weak corrections in the 
$\alpha$ scheme. These corrections are defined by
\begin{eqnarray}
 \delta_{W} & = & \delta_{EW}-\delta_{QED}. 
\end{eqnarray}

In fig~\ref{ew}, the cross-section and the electroweak corrections are shown
as a function of $\sqrt{s}$. The center-of-mass energy ranges from 250 GeV 
(which is near the threshold of $M_H+M_Z$) to $1$ TeV. In the left part of 
fig~\ref{ew} the corrections make the cross-section decrease more and more 
with increasing center-of-mass energy. We find that the electroweak 
corrections are from $-2\%$ to $\sim -20\%$ when varying $\sqrt{s}$ from 
$250$ GeV to $1$ TeV. The right part of fig~\ref{ew} clearly shows that 
QED provides the dominant contribution as compared to the weak corrections.
It goes to $-14\%$ at $\sqrt{s}=1$ TeV, while the weak corrections change 
from $\sim2\%$ to $\sim 6\%$ when $\sqrt{s}$ changes from 250 GeV to 1 
TeV. It is clear that the corrections are a sizable contribution to the 
total cross-section and cannot be ignored for the high precision program at 
the ILC.
\begin{figure}[ht]
\begin{center}$
\begin{array}{cc}
\hspace{-0.5cm}\includegraphics[width=3.2in,height=9.0cm,angle=-90]{sigmaeeg.eps} &
\hspace{-1.cm}\includegraphics[width =3.2in,height=9.0cm,angle=-90]{delta-eeg.eps}
\end{array}$
\end{center}
\caption{\label{ew}In this figure, the cross-section (left) and full 
electroweak corrections (right)
are presented as a function of the center-of-mass energy.}
\end{figure}

In fig~\ref{photonenergy}, the distribution of the cross-section as a 
function of the photon energy are presented for $\sqrt{s}=250$ GeV and 
$\sqrt{s}=1$ TeV. The dashed line is the result of the tree level 
calculation and the solid line is the result of including the full 
radiative corrections. On the whole the cross-section decreases with 
increasing photon energy. At $\sqrt{s} = 250$ GeV we observe two peaks at 
$E_{\gamma}=\frac{s-M_Z^2}{2\sqrt{s}}$ and $\frac{\sqrt{s}}{2}$. The first 
peak corresponds to the photon recoiling against an on-shell $Z$ boson, and 
the second has the photon recoil against a virtual photon that creates a 
small mass electron-positron pair. Due to the high energy the peaks overlap 
within our resolution at $\sqrt{s}=1$ TeV. The distributions also clearly 
show that the radiative corrections are visible and are important for the 
luminosity monitor at the ILC.
\begin{figure}[ht]
\begin{center}$
\begin{array}{cc}
\hspace{-1.5cm}\includegraphics[width= 3.5in,height=9.4cm,angle=-90]{E3-250.eps} &
\hspace{-2cm}\includegraphics[width= 3.5in,height=9.4cm,angle=-90]{E3-1000.eps}
\end{array}$
\end{center}
\caption{\label{photonenergy} The differential cross-section as a function 
of the photon energy at $\sqrt{s}=250$ GeV and $\sqrt{s}=1$ TeV.}
\end{figure}

In figure~\ref{invariance} the distributions of the invariant mass of the 
$e^-$, $e^+$ pair are shown at $\sqrt{s}=250$ GeV and $\sqrt{s}=1$ TeV.
The dashed lines present the result of the tree-level calculations and the 
solid lines are the result of including the full radiative corrections. We 
find the peaks at the $M_Z$ pole and at the high
mass region (corresponding to the radiative tail). Again the radiative 
corrections are clearly visible.

\begin{figure}[ht]
\begin{center}$
\begin{array}{cc}
\hspace{-1.5cm}\includegraphics[width= 3.5in,height=9.5cm,angle=-90]{m45-250.eps} &
\hspace{-2.5cm}\includegraphics[width= 3.5in,height=9.5cm,angle=-90]{m45-1000.eps}
\end{array}$
\end{center}
 \caption{\label{invariance} The differential cross-section as a function of 
the invariant mass of the $e^-$, $e^+$ pair.
At the left $\sqrt{s} = 250$ GeV and at the right $\sqrt{s} = 1$ TeV.}
\end{figure}

%--#] The physics results : 
%--#[ Conclusions :

\section{Conclusions}

The QED and full $\mathcal{O}(\alpha)$ electroweak radiative corrections to 
$e^+e^- \rightarrow e^+e^-  \gamma$ at the International Linear Collider 
have been calculated successfully by using the GRACE-Loop system.

This system incorporates a generalised non-linear gauge fixing 
condition which includes five gauge parameters. Together with the UV, IR 
finiteness this provides a powerful tool to test the consistency of the 
results. In the numerical checks of this calculation, we find that the 
results are numerically stable when quadruple precision is used. 

%Using the axial gauge for the external photon, we can calculate the process 
%at small energy and angle cuts of the final state particles. The results 
%for the QED corrections are investigated with various energy and angle cuts 
%for the final state particles. They clearly show that the process can be 
%numerically evaluated over the full kinematic region. 

In the physics results of the full electroweak corrections we find that the 
numerical value of the full electroweak radiative corrections vary from 
$-2\%$ to $\sim-20\%$ in the range of $250$ GeV to $1$TeV for the 
center-of-mass energy. Therefore this calculation is important for the 
determination of the luminosity at the ILC.

%--#] Conclusions : 
%--#[ Bibliography :

\section*{ Acknowledgments} We wish to thank Prof. F.~Yuasa and Dr. 
N.~Watanabe for valuable discussions and comments. The authors are grateful
to Prof.~K. Tobimatsu and Prof.~M. Igarashi for useful discussions and their contributions.
The work of T.U. was supported by the DFG through SFB/TR 9 ``Computational Particle Physics''.

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K.Tobimatsu and Y.Shimizu, 
% Radiative corrections to $e^+e^-\to e^+e^-$ in electroweak theory I,
Prog. Theor. Phys. {\bf 74} (1985), 567-575.
\bibitem{tobimatsu2}
K.Tobimatsu and Y.Shimizu, 
% Radiative corrections to $e^+e^-\to  e^+e^-$ in electroweak theory II,
Prog. Theor. Phys. {\bf 75} (1986), 905-913.

%\cite{Bohm:1984yt}
\bibitem{Bohm:1984yt}
  M.~Bohm, A.~Denner, W.~Hollik and R.~Sommer,
  %``Radiative Corrections to Bhabha Scattering in SU(2) X U(1),''
  Phys.\ Lett.\ B {\bf 144} (1984) 414.
  %%CITATION = PHLTA,B144,414;%%
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%\cite{Bohm:1986fg}
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% %\cite{Penin:2011aa}
% \bibitem{Penin:2011aa}
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%\cite{Belanger:2002me}
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% \bibitem{supplement100}
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\end{thebibliography}

%--#] Bibliography : 

\end{document}

