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% QED and electroweak radiative corrections to polarized Bhabha scattering
% A.B. Arbuzov et al.
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\documentclass[a4paper]{jpconf}
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\begin{document}

\title{QED and electroweak radiative corrections to polarized Bhabha scattering}

\author{A B Arbuzov$^{1,2}$, S G Bondarenko$^1$, Ya V Dydyshka$^3$, \\
  L V Kalinovskaya$^3$, L A Rumyantsev$^{3,4}$, R R Sadykov$^3$}

\address{$^1$ Bogoliubov Laboratory of Theoretical Physics, JINR,
              Joliot-Curie str. 6, Dubna, 141980, Russia} 
\address{$^2$ Dubna State University, Universitetskaya str. 19,
              Dubna, 141982, Russia} 
\address{$^3$ Dzhelepov Laboratory of Nuclear Problems, JINR,
              Joliot-Curie str. 6, Dubna, 141980, Russia} 
\address{$^4$ Institute of Physics, Southern Federal University,
              Rostov-on-Don, 344090 Russia}

\ead{arbuzov@theor.jinr.ru}

\begin{abstract}
Complete one-loop electroweak radiative corrections to polarized Bhabha
scattering are presented. Numerical results are shown for the conditions of 
future circular and linear electron-positron colliders with polarized beams. 
A new Monte Carlo event generator for simulation of Bhabha scattering is created.
Higher order QED collinear radiation factors are evaluated in the next-to-leading 
logarithmic approximation. 
\end{abstract}


\section{Introduction}
\label{Sect:Intro}

Electron-positron (Bhabha) scattering is a basic process in
particle physics. It is widely used for the luminosity
monitoring at $e^+e^-$ colliders. Bhabha scattering with unpolarized initial particles
has been studied for many years since Ref.~\cite{Consoli:1979xw}.
Radiative corrections (RC) for the case of polarized $e^+e^-$ beams were
considered in Refs.~\cite{Hollik:1981bu,Hollik:1982wr}.
There are several Monte Carlo event generators for unpolarized Bhabha scattering,
see e.g.~\cite{Jadach:1996gu}.
In the {\tt SANC} computer system framework, the complete
one-loop electroweak (EW) RC to Bhabha scattering were recently
calculated~\cite{Andonov:2004hi} and expressed in terms of
helicity amplitudes (HA) and form factors (FF). 

In comparison with processes being studied at modern hing-energy hadron colliders, 
$e^{+}e^{-}$ interactions have a clean initial state, 
a lower multiplicity in the final state, and therefore provide the possibility
to perform much more precise measurements in most cases.
The substantially higher energy range of the future colliders also demands
re-estimation of various effects from both experimental and theoretical points of view.  
Precise measurements with polarized beams at future
$e^+e^-$ colliders like FCC-ee, CEPC, ILC and CLIC definitely require
an advanced theoretical 
support~\cite{Blondel:2018mad,Benedikt:2018qee,CEPCStudyGroup:2018ghi,CarloniCalame:2015zev,Riemann:2015zbi}.
In particular, physical programs of the future $e^{+}e^{-}$ linear 
colliders~\cite{Accomando:1997wt,Moortgat-Picka:2015yla} 
always demonstrated a great interest to the effects related
to the beam polarization. These facts motivate us to work on
high-precision theoretical predictions for processes which
will be studied at future $e^+e^-$ colliders. In order to match experimental
requirements, we have to implement our results into Monte Carlo event generators.
 
In Sect.~\ref{Sect:Bhabha} we present and discuss the results~\cite{Bardin:2017mdd} 
on the complete 
one-loop electroweak radiative corrections to polarized Bhabha scattering.
Tuned comparisons with results of alternative calculations partially available
in the literature were performed. In Sect.~\ref{Sect:NLO} we show preliminary
results on QED collinear radiative factors in the $\mathcal{O}(\alpha^3L^2)$ order.


\section{One-loop corrections to polarized Bhabha scattering}
\label{Sect:Bhabha}

Consider  scattering of longitudinally polarized positrons and
electrons $e^++e^- \longrightarrow e^-+e^++(\gamma)$.
At large energies we will neglect contributions proportional to the ratio of 
the electron mass to the beam energy. 

The complete one-loop covariant amplitude comes out from
the straightforward calculation by means of the {\tt SANC} computer 
system~\cite{Andonov:2004hi}.
The amplitude is parameterized by a certain number of form factors (FFs)
${\cal F} = 1 + k {\tilde{\cal F}}$,
where ``1'' stands for the Born level and the term ${\tilde{\cal F}}$ with the factor
$k\equiv g^2/(16\pi^2)$ is the one-loop contribution.
After squaring the amplitude we neglect terms proportional to $k^2$ in order
to get the pure one-loop approximation without any admixture of higher-order
terms which can be added later if required.

The covariant amplitude for high-energy Bhabha scattering can be written 
in terms of the electromagnetic running coupling constant and four FFs 
with permuted arguments $s$ and $t$ as:
\begin{eqnarray}
{\cal A} &=&{\cal A}_{\gamma}(s)+{\cal A}_{\scriptscriptstyle{Z}}(s)
          -\left[{\cal A}_{\gamma}(t)+{\cal A}_{\scriptscriptstyle{Z}}(t)\right]
\\
&=& i\,e^2\Biggl\{ \left[ 
 \gamma_\mu \otimes \gamma_\mu \frac{{\cal F}_{\gamma}(s)}{s} 
 -\gamma_\mu \otimes \gamma_\mu \frac{{\cal F}_{\gamma}(t)}{t} \right]
+ \frac{\chi_{\scriptscriptstyle{Z}}(s)}{s}
 \biggl\{\hspace*{-.3mm} \left(I^{(3)}_{e}\right)^2\hspace*{-1mm}
 \gamma_\mu \gamma_6 \otimes\gamma_\mu \gamma_6 {\cal F}_{\scriptscriptstyle{LL}}(s,t,u)
\nonumber \\ %[1mm]
&+& 2\delta_{e} I^{(3)}_{e} \gamma_\mu \otimes \gamma_\mu \gamma_6 {\cal F}_{\scriptscriptstyle{QL}}(s,t,u)
+\delta^2_{e}\gamma_\mu\otimes\gamma_\mu \,
                   {\cal F}_{\scriptscriptstyle{QQ}}(s,t,u)
                   \biggr\}
- \frac{\chi_{\scriptscriptstyle{Z}}(t)}{t}\biggl\{ s\leftrightarrow t \biggr\},
\nonumber
\end{eqnarray}
see Ref.~\cite{Bardin:2017mdd} for the notation details.

The complete result for ${\cal O}(\alpha)$ corrections can be separated into 
the virtual (loop) contribution, the part due to the soft photon emission,
and the one due to the real hard photon Bremsstrahlung.
It is more convenient to calculate a cross-section by squaring non-interfering 
helicity amplitudes (HA). Complete analytic results were obtained
for HA of Bhabha scattering by means of the SANC system.
There are six non-zero HA, however, since for Bhabha scattering 
${\cal F}^{\scriptscriptstyle Z}_{\scriptscriptstyle LQ}={\cal F}^{\scriptscriptstyle Z}_{\scriptscriptstyle QL}$,
the number of independent HA is reduced to four.

We obtained the compact expression for the Born
(${\cal F}_{{\scriptscriptstyle QL},{\scriptscriptstyle LL},{\scriptscriptstyle QQ}} =1$) 
and the virtual (loop) part within the HA approach in the form
\begin{eqnarray}
\label{VIRT_HA}
{\cal H}_{++++} &=& {\cal H}_{----} =
-2 e^2\,\frac{s}{t} \Bigl[ {\cal F}^{(\gamma,Z)}_{\scriptscriptstyle QQ}(t,s,u) 
              - {\chi_{\scriptscriptstyle Z}}(t)\delta_{e} {\cal F}^{\scriptscriptstyle Z}_{\scriptscriptstyle QL}(t,s,u)\Bigr],
\nonumber \\
    {\cal H}_{+-+-} &=& {\cal H}_{-+-+}=
    - e^2\,c_{-}\Bigl[ {\cal F}^{(\gamma,Z)}_{\scriptscriptstyle QQ}(s,t,u)
      -{\chi_{\scriptscriptstyle Z}}(s)\delta_{e}{\cal F}^{\scriptscriptstyle Z}_{\scriptscriptstyle LQ}(s,t,u)\Bigr],
\nonumber \\
    {\cal H}_{+--+} &=& - e^2\,c_+
\Bigl( \Bigl[ {\cal F}^{(\gamma,Z)}_{\scriptscriptstyle QQ}(s,t,u)
+{\chi_{\scriptscriptstyle Z}}(s)\left( {\cal F}^{\scriptscriptstyle Z}_{\scriptscriptstyle LL}(s,t,u) \right.
%\nonumber \\
- 2\delta_{e} {\cal F}^{\scriptscriptstyle Z}_{\scriptscriptstyle LQ}(s,t,u)\left. \right) \Bigr]+\frac{\displaystyle s}{\displaystyle t} \Bigl[ s\leftrightarrow t  \Bigr]
\Bigr),
\nonumber \\
{\cal H}_{-++-} &=& - e^2\,c_+
\Bigl(\left[{\cal F}^{(\gamma,Z)}_{\scriptscriptstyle QQ}(s,t,u) \right]
 +\frac{\displaystyle s}{\displaystyle t} \left[  s\leftrightarrow t  \right] \Bigr),
\end{eqnarray}
where
$c_{\pm}= \cos\vartheta \pm 1$ and
$ {\cal F}^{(\gamma,Z)}_{\scriptscriptstyle QQ}(a,b,c)={\cal F}^{(\gamma)}_{\scriptscriptstyle QQ}(a,b,c)$
       $+{\chi_{\scriptscriptstyle Z}}(a)\delta_{e}^{2}{\cal F}^{(\scriptscriptstyle Z)}_{\scriptscriptstyle QQ}(a,b,c)$.

The Bremsstrahlung module of the {\tt SANC} system computes the contributions
due to the soft and inclusive hard photon emission. 
The soft photon contribution contains infrared divergences which 
are compensated by the corresponding divergences in the one-loop 
virtual QED (EW) corrections.
For a crosscheck, the contribution of the hard photon emission 
was obtained both by direct squaring of the matrix element and by the helicity 
amplitude approach.

To study the case of longitudinal polarization, we generated helicity amplitudes
and applied the formalism described in~\cite{MoortgatPick:2005cw}.
In our notation the Bhabha scattering cross-section with
longitudinally polarized initial particles can be expressed as
\begin{eqnarray}
\frac{d\sigma(P_{e^-},P_{e^+})}{d\cos\vartheta}
&=&\frac{1}{128\pi s}  \Bigl[
    (1-P_{e^-})(1-P_{e^+}) \sum_{ij}|{\cal H}_{++ij}|^2
  +(1-P_{e^-})(1+P_{e^+}) \sum_{ij}|{\cal H}_{+-ij}|^2
\nonumber \\
&+&(1+P_{e^-})(1-P_{e^+}) \sum_{ij}|{\cal H}_{-+ij}|^2
   +(1+P_{e^-})(1+P_{e^+}) \sum_{ij}|{\cal H}_{--ij}|^2\Bigr].
\label{PolXSec}
\end{eqnarray}
For a crosscheck we got an analytic zero for the difference 
between the square of the covariant amplitude (we introduced  
the spin density matrix into our procedures) and Eq.~(\ref{PolXSec}).

The left-right asymmetry $A_{LR}$ and the relative correction $\delta$ 
are defined as
\begin{eqnarray}
A_{LR} =
 \frac{d\sigma(-1,1)-d\sigma(1,-1)}{d\sigma(-1,1)+d\sigma(1,-1)},\qquad
\delta =
 \frac{d\sigma^{\mathrm{1-loop}}(P_{e^-},P_{e^+})}{d\sigma^{\mathrm{Born}}(P_{e^-},P_{e^+})}-1,
\end{eqnarray}
where we dropped $d\cos\vartheta$ for the sake of brevity.

Here we present some numerical results for EW RC to
Bhabha scattering obtained by means of a new Monte Carlo 
event generator created by the {\tt SANC} group which was presented 
at this conference in the talk by R.~Sadykov. 
To crosscheck our results, we performed comparisons with 
the ones obtained by means of other modern packages. But actually,
it was possible to compare results only partially since there is
no any other code describing high-energy polarized Bhabha scattering 
with one-loop precision.

We found an agreement between our analytic result for 
the hard photon Bremsstrahlung contribution 
with the one obtained by means of the {\tt CalcHEP} computer 
system~\cite{Belyaev:2012qa}.
Numerical results for polarized Bhabha scattering with hard photon Bremsstrahlung
were compared with the ones of the {\tt WHIZARD} system~\cite{Kilian:2007gr}. 
A good agreement was observed as well. 
We obtained also a very good agreement (six significant digits) in the comparison 
of the {\tt SANC} and {\tt AItalc-1.4}~\cite{Fleischer:2006ht} results for 
the unpolarized differential Born cross-section and for the sum of 
the virtual and the soft photon contributions. 
Tables with the numerical results of the mentioned above comparisons can be 
found in ref.~\cite{Bardin:2017mdd}.

\begin{table}[!ht]
\caption{Born and 1-loop cross-sections of Bhabha scattering and the corresponding 
relative corrections $\delta$ for $\sqrt{s} = 250$, $500$, and $1000$~GeV.}
\label{Table:sanc_delta}
\begin{tabular}{|lcccc|}
\hline
$P_{e^-}$, $P_{e^+}$ & 0, 0 & -0.8, 0 & -0.8, -0.6 & -0.8, 0.6\\
\hline
\multicolumn{5}{|c|}{$\sqrt{s} = 250$ GeV}\\
\hline
$\sigma_{e^+e^-}^{\mathrm{Born}}$, pb & 56.6763(1) & 57.7738(1) & 56.2725(4) & 59.2753(5)\\
%\hline
$\sigma_{e^+e^-}^{\mathrm{1-loop}}$, pb & 61.731(6) & 62.587(6) & 61.878(6) & 63.287(7)\\
%\hline
$\delta$, \% & 8.92(1) & 8.33(1) & 9.96(1) & 6.77(1) \\
\hline
\multicolumn{5}{|c|}{$\sqrt{s} = 500$ GeV}\\
\hline
$\sigma_{e^+e^-}^{\mathrm{Born}}$, pb & 14.3789(1) & 15.0305(1) & 12.7061(1) & 17.3550(2)\\
%\hline
$\sigma_{e^+e^-}^{\mathrm{1-loop}}$, pb & 15.465(2) & 15.870(2) & 13.861(1) & 17.884(2)\\
%\hline
$\delta$, \% & 7.56(1) & 5.59(1) & 9.09(1) & 3.05(1)\\
\hline
\multicolumn{5}{|c|}{$\sqrt{s} = 1000$ GeV}\\
\hline
$\sigma_{e^+e^-}^{\mathrm{Born}}$, pb & 3.67921(1) & 3.90568(1) & 3.03577(3) & 4.77562(5)\\
%\hline
$\sigma_{e^+e^-}^{\mathrm{1-loop}}$, pb & 3.8637(4) & 3.9445(4) & 3.2332(3) & 4.6542(7)\\
%\hline
$\delta$, \% & 5.02(1) & 0.99(1) & 6.50(1) & -2.54(1)\\
\hline
\end{tabular}
\end{table}

\begin{figure*}
\[
\begin{array}{ccc}
  \includegraphics[width=75mm]{el_costh_NN_250.eps}
  &&
  \includegraphics[width=75mm]{delta_el_costh_250.eps}
  \end{array}
\]
\caption{The differential cross-section (left) [in pb] and 
the relative correction $\delta$ (right) [in \%] vs. the cosine 
of the electron scattering angle for $\sqrt{s}=250$~GeV.}
\label{el_delta_costh_250}
\end{figure*}

\begin{figure*}
\[
\begin{array}{ccc}
  \includegraphics[width=75mm]{a_el_costh_250.eps}
  &&
  \includegraphics[width=75mm]{a_el_costh_1000.eps}  
  \end{array}
\]
\caption{The left-right asymmetry $A_{LR}$ vs the cosine of the electron 
scattering angle at $\sqrt{s}=250$~GeV (left) and  $\sqrt{s}=1000$~GeV (right).}
\label{a_el_costh-250}
\end{figure*}

In Figs.~\ref{el_delta_costh_250}, we give an example of numerical
results produced by the new {\tt  SANC} Monte Carlo event generator for 
the unpolarized differential cross-section of Bhabha scattering
and the relative ${\mathcal{O}}(\alpha)$ 
correction $\delta$ (in percent) as a function
of the electron scattering angle
for $|\cos\theta|<0.9$ and $\sqrt{s}=250$~GeV.
The huge relative radiative corrections for the
backward scattering angles are due to the smallness of the Born cross-section 
in this domain. That does not mean any problem with the perturbation theory.
The integrated cross-section of the Bhabha scattering and the relative
correction $\delta$ are given in the Table~\ref{Table:sanc_delta} for
various energies and beam polarization degrees.
The $A_{LR}$ asymmetry at $\sqrt{s}=250$ and $1000$~GeV
is shown in Fig.~\ref{a_el_costh-250}.
One can see that the EW radiative corrections affect the asymmetry
considerably.


\section{Higher-order collinear NLO radiation functions}
\label{Sect:NLO}

At high energies, QED radiative corrections enhanced by large logarithms
$L=\ln(Q^2/m_e^2)$ are numerically very important. The so-called leading logarithmic (LO)
approximation takes into account terms proportional to $\alpha^2L^n$, $n=1,2,\ldots$
The next-to-leading logarithmic (NLO) approximation includes also terms of the order
$\mathcal{O}(\alpha^2L^{n-1}$. The factorization theorem well known in QCD allows us
to write down the Bhabha scattering cross-section in the next-to-leading logarithmic
approximation in the following form~\cite{Arbuzov:2006mu}:
\begin{eqnarray}
&& d\sigma = \sum\limits_{a,b,c,d=e,\bar{e},\gamma}
\int_{\bar{z}_1}^{1}dz_1 \int_{\bar{z}_2}^{1}dz_2
\mathcal{D}_{ae}^{\mathrm{str}}(z_1) \mathcal{D}_{b\bar{e}}^{\mathrm{str}}(z_2)
\biggl[ d\sigma^{(0)}_{ab\to cd}(z_1,z_2) 
      + d\bar{\sigma}^{(1)}_{ab\to cd}(z_1,z_2) \biggr]
\nonumber \\ && \qquad \times
\int_{\bar{y}_1}^{1}\frac{dy_1}{Y_1} \int_{\bar{y}_2}^{1}\frac{dy_2}{Y_2}
\mathcal{D}_{ec}^{\mathrm{frg}}\left(\frac{y_1}{Y_1}\right) 
\mathcal{D}_{\bar{e}d}^{\mathrm{frg}}\left(\frac{y_2}{Y_2}\right)
+\mathcal{O}(\alpha^nL^{n-2}), 
\nonumber
\end{eqnarray}
where $\mathcal{D}_{ba}^{\mathrm{str(frg)}}(z)$ are QED structure (fragmentation)
functions which provide the probability density to find parton (particle) $b$
in particle (parton) $a$ with the energy fraction $z$. 
Cross-sections $d\sigma^{(0)}_{ab\to cd}$ and correction to it
$d\bar{\sigma}^{(1)}_{ab\to cd}$ are computed for massless partons $a,b,c,d$ 
in the Born and one-loop approximations, respectively. 
Mass singularities are subtracted from the one-loop parton
cross-section $d\bar{\sigma}^{(1)}$ in the MSbar scheme.

Explicit expression for the QED structure functions are well known in the leading logarithmic
approximation~\cite{Kuraev:1985hb,Skrzypek:1992vk,Arbuzov:1999cq}. The NLO contributions to 
for QED structure and fragmentation functions in the $\mathcal{O}(\alpha^2L^1)$ order
were specified in Refs.~\cite{Berends:1987ab,Arbuzov:2002cn}. Continuously increasing
experimental precision makes it worth to include also the  $\mathcal{O}(\alpha^3L^2)$ order
contributions. They can be found by means of iterative solutions of the QED DGLAP evolution
equations in the NLO approximation. Here we present new results for 
the $\mathcal{O}(\alpha^3L^2)$ contributions to non-singlet electron structure function. 
Taking into account the pure photonic contributions we get
\begin{eqnarray} \nonumber
&& \mathcal{D}_{ee}^{(\gamma)}(x,\mu_f,m_e) = \delta(1-x)
+ \frac{\alpha}{2\pi}d_1(x,\mu_0,m_e)
+ \frac{\alpha}{2\pi}L_f P_{ee}^{(0)}(x)
\nonumber \\ \nonumber && \quad
+ \left( \frac{\alpha}{2\pi} \right)^2\biggl(
\frac{1}{2}L_f^2 P_{ee}^{(0)}\otimes P_{ee}^{(0)}(x)
+ L_f P_{ee}^{(0)}\otimes d_1(x,\mu_0,m_e) 
+ L_f P_{ee}^{(1,\gamma)}(x)\! \biggr)
\nonumber \\ \nonumber && \quad
+ \left( \frac{\alpha}{2\pi} \right)^3\biggl(
\frac{1}{6}L_f^3 P_{ee}^{(0)}\otimes P_{ee}^{(0)}\otimes P_{ee}^{(0)}(x)
+ L_f^2 P_{ee}^{(0)}\otimes P_{ee}^{(1,\gamma)}(x) 
\nonumber \\ && \quad
+ L_f^2 P_{ee}^{(0)}\otimes P_{ee}^{(0)}\otimes d_1(x,\mu_0,m_e) 
 \biggr)
+ \mathcal{O}(\alpha^3L^1).
\end{eqnarray}
Here $\mu_f$ and $\mu_0$ are the factorization and renormalization scales;
$P_{ee}$ are electron splitting functions; $d_1(x)$ is the initial condition
for evolution equations, see details in Ref.~\cite{Arbuzov:2002cn}.
Pair contributions to the structure function will be considered elsewhere. 
The $\mathcal{O}(\alpha^3L^2)$ contributions are
\begin{eqnarray} \label{PP1}
&&\!\!\! P_{ee}^{(0)}\otimes P^{(1,\gamma){\mathrm{str}}}_{ee}(x)
= \Bigg[ \frac{1+x^2}{1-x} \biggl(
  - 4\mathrm{S}_{12}(1-x)
  + 4\mathrm{Li}_2(1-x)(\ln(1-x)-\ln(x))
\nonumber \\ \nonumber && \quad
  - 4\ln(x)\ln^2(1-x)
  + 4\ln^2(x)\ln(1-x)
  - \frac{2}{3}\ln^3(x)
  + 3\mathrm{Li}_2(1-x)
  - 3\ln(x)\ln(1-x)
\nonumber \\ \nonumber && \quad
  + \frac{3}{2}\ln^2(x)
  + 4\zeta(2)\ln(x)
  + 6\zeta(3)
  - 3\zeta(2)
  + \frac{3}{8}
  \biggr)
  + 4(1+x)\mathrm{S}_{12}(x)
  + \frac{1+x}{2}\ln^3(x)
\nonumber \\ \nonumber && \quad
  - 2(1+x)\ln^2(x)\ln(1-x)
  + (6x-2)\mathrm{Li}_2(1-x)
  + (6-2x)\ln(x)\ln(1-x)
\nonumber \\  && \quad
  + \left(\frac{11}{4}x - \frac{9}{4}\right)\ln^2(x)
  + (6-4x)\ln(1-x)
  + (5x-3)\ln(x)
  + 2x
  - \frac{1}{2}
  \Bigg]_+,
\end{eqnarray}
where the standard plus prescription is applied.
The second relevant new contribution is
\begin{eqnarray} \label{PPd1}
&& P_{ee}^{(0)}\otimes P_{ee}^{(0)}\otimes d_1(x,m_e,m_e) 
= \biggl[ \frac{1+x^2}{1-x} \biggl(
   16\mathrm{S}_{12}(1-x)
 + 8\mathrm{Li}_2(1-x)\ln(x)
 - 16\ln^3(1-x)
\nonumber \\ \nonumber && \quad
 + 20\ln(x)\ln^2(1-x) 
 - 4\ln^2(x)\ln(1-x) 
 + \zeta(2)(40\ln(1-x)-16\ln(x))
 - 32\zeta(3)
\nonumber \\ \nonumber && \quad
 - 30\ln^2(1-x)
 - 2\ln^2(x)
 + 24\ln(x)\ln(1-x)
 + 24\zeta(2)
 - \frac{17}{2}\ln(1-x)
 + 2\ln(x)
 + \frac{15}{4}
\biggr)
\nonumber \\ \nonumber && \quad
 + (1+x)\biggl(
   20\mathrm{Li}_3(1-x)
 - 10\mathrm{S}_{12}(1-x)
 - 20\mathrm{Li}_2(1-x)\ln(1-x)
 - 2\mathrm{Li}_2(1-x)\ln(x)
\nonumber \\ \nonumber && \quad
 - 10\ln(x)\ln^2(1-x)
 + 3\ln^2(x)\ln(1-x)
 + 8\zeta(2)\ln(x)
 \biggr)
 - 2(1+x)\mathrm{Li}_2(1-x)
\nonumber \\ \nonumber && \quad
 + 20(1-x)\ln^2(1-x)
 + \frac{3}{2}(1-x)\ln^2(x)
 - 18\ln(x)\ln(1-x)
 + 14x\ln(x)\ln(1-x)
\nonumber \\ && \quad
 + 4(1-x)\ln(1-x)
 - 2\ln(x)
 - 16(1-x)\zeta(2)
 - 2(1-x)
\biggr]_+.
\end{eqnarray}
Note that expressions (\ref{PP1}) and (\ref{PPd1}) represent contributions
to universal collinear QED radiation factors which can be useful for description
of a wide class of processes at high energies.


\section{Conclusions}

The SANC computer system is upgraded to facilitate description of processes at 
electron-colliders with polarized beams. The results on complete electroweak
one-loop radiative corrections to polarized Bhabha scattering are presented.
A new Monte Carlo event generator to simulate this process is created.

The $\mathcal{O}(\alpha^3L^2)$ contributions to the electron structure function
will be implemented into the ZFITTER code~\cite{Arbuzov:2005ma} in order to perform 
realistic numerical estimates of the corresponding effect in measurable cross-sections.
Further, these contributions will be used in Monte Carlo event generators being under
development by the SANC group for future $e^+e^-$ colliders like FCC-ee and CEPC.


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\end{document}

