
\subsection{LO cross sections and NLO corrections \label{LONLO}}

As remarked in Section \ref{MOTIVATION}, the processes of interest for
the luminosity 
measurement at meson factories are Bhabha scattering and electron-positron annihilation into two photons and muon pairs. Here we present the LO formulae for the 
cross section of the processes  $e^+ e^- \to e^+ e^-$ and $e^+ e^- \to \gamma\gamma$, as
well as the QED corrections to their cross sections in the NLO 
approximation of perturbation theory. The reaction $e^+ e^- \to \mu^+ \mu^-$ is 
discussed in Section \ref{sec:2}.

\subsubsection{LO cross sections \label{LO}}

For the Bhabha scattering process
\begin{eqnarray}
e^-(p_-) + e^+(p_+)\to e^-(p_-') + e^+(p_+')
\end{eqnarray}
at Born level with %a pure photonic exchange we have
simple one-photon exchange (see Fig.~\ref{figlo:1}) the differential cross section reads
\begin{eqnarray}
\frac{\dd{\sigma}^{\mathrm{Bhabha}}_0}{\dd\Omega_-} 
= \frac{\alpha^2}{4s}\;\left(\frac{3+c^2}{1-c}\right)^2 
+  O \left(\frac{m_e^2}{s}\right),
\end{eqnarray}
where 
\begin{eqnarray}
s &=& (p_-+p_+)^2, \qquad c = \cos\theta_-.
\end{eqnarray}
The angle $\theta_-$ is defined between the initial and final electron three-momenta,
$\dd\Omega_-=\dd\phi_-\dd\cos\theta_-$, and $\phi_-$ is the azimuthal angle of the
outgoing electron. The small mass correction terms suppressed by the ratio $m_e^2/s$ are
negligible for the energy range and the angular acceptances 
which are of interest here.

\begin{figure}
\begin{center}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{Bhabha_s2.eps}~\hspace{2.5cm}\includegraphics{Bhabha_t2.eps}
}
\caption{LO Feynman diagrams for the Bhabha process in QED, corresponding 
to $s$-channel annihilation and $t$-channel scattering.}
\label{figlo:1}
\end{center}
\end{figure}

At meson factories the Bhabha scattering cross section is largely dominated by 
$t$-channel photon exchange, followed by $s$-$t$ interference and $s$-channel 
annihilation.
%with centre-of-mass energies of the order a few GeV, 
Furthermore, $Z$-boson exchange contributions and other electroweak
effects are suppressed at least by a factor $s/M_Z^2$. In particular,
for large-angle Bhabha scattering with a c.m. energy
$\sqrt{s} = 1$~GeV the $Z$ boson contribution amounts to about $-1\times 10^{-5}$.
For $ \sqrt{s} = 3$~GeV it amounts to $-1\times 10^{-4}$ 
and $-1 \times 10^{-3}$ for $\sqrt{s} = 10$~GeV. So only at $B$ factories the
electroweak effects should be taken into account at tree level, 
when aiming at a per mill precision level.

The LO differential cross section of 
the two-photon annihilation channel (see Fig.~\ref{figlo:2})
\begin{equation} \nonumber
e^+(p_+) + e^-(p_-) \to \gamma(q_1) + \gamma(q_2)
\end{equation}
can be obtained by a crossing relation from the Compton scattering
cross section computed by Brown and Feynman~\cite{Brown:1952eu}. It reads
\begin{eqnarray}
\frac{\dd\sigma^{\gamma\gamma}_0}{\dd\Omega_1}
= \frac{\alpha^2}{s}\left(\frac{1+c_1^2}{1-c_1^2}\right)
+ O \left(\frac{m_e^2}{s}\right),
\end{eqnarray}
where $\dd\Omega_1$ denotes the
differential solid angle of the first photon. It is assumed that both 
final photons are registered in a detector 
and that their polar angles with respect to the initial beam directions are
not small ($\theta_{1,2} \gg m_e/E$, where $E$ is the beam 
energy).

\subsubsection{NLO corrections \label{NLO}}

The complete set of NLO radiative 
corrections, emerging at  $O (\alpha)$ of perturbation theory, 
to Bhabha scattering and two-photon annihilation can be split into gauge-invariant subsets: 
QED corrections, due to emission of real photons off the charged leptons and 
exchange of virtual photons between them, and purely weak contributions arising from 
the electroweak sector of the SM. 

The complete $O (\alpha)$ QED corrections to Bhabha
scattering are known since a long time~\cite{Redhead:1953xxx,Polovin:1956xxx}. 
The first complete NLO prediction in the electroweak SM was performed in \cite{Consoli:1979xw},
followed by  \cite{Bohm:1984yt} and several others. At NNLO, 
the leading virtual weak corrections from the top quark were derived first in \cite{Bardin:1990xe} and are available in the 
fitting programs ZFITTER \cite{Bardin:1999yd,Arbuzov:2005ma} and 
TOPAZ0 \cite{Montagna:1993py,Montagna:1993ai,Montagna:1998kp}, extensively used by the 
experimentalists for the 
extraction of the electroweak parameters at LEP/SLC. 
The weak NNLO corrections in the SM are also known for the $\rho$-parameter
\cite{%
Djouadi:1987gn,Djouadi:1988di,Kniehl:1988ie,vanderBij:1986hy,Barbieri:1993dq,Fleischer:1993ub,%
Fleischer:1994cb,Boughezal:2006xk,Chetyrkin:2006bj,Schroder:2005db,Chetyrkin:1995ix,%
Chetyrkin:1995js,Barbieri:1992nz,vanderBij:2000cg,Faisst:2003px,Boughezal:2004ef,%
Boughezal:2005eb}
and the weak mixing angle
 \cite{Awramik:2003rn,Awramik:2004ge,Hollik:2005va,Hollik:2006ma,Awramik:2006uz,Awramik:2006ar}, as well as corrections  from Sudakov logarithms
\cite{Kuhn:1999nn,Kuhn:2000hx,Kuhn:2001hz,Feucht:2004rp,Jantzen:2005xi,Jantzen:2005az,%
Denner:2006jr,Denner:2008hw}. 
Both NLO and NNLO weak effects are %of minor importance 
negligible at low energies and are not 
 implemented yet in numerical packages for Bhabha scattering at meson factories.
In pure QED, the situation is considerably different due to the remarkable progress 
made on NNLO 
corrections in recent years, as emphasised and discussed in detail in 
Section \ref{NNLO}.

As usual, the photonic corrections can be split into two parts according to
their kinematics. The first part preserves the Born-like kinematics and contains the effects
due to one-loop amplitudes (virtual corrections) and single soft-photon emission. 
Examples of Feynman diagrams giving rise to such corrections are represented in 
Fig.~\ref{fignlo:1}. The energy of a soft photon is assumed not to exceed an energy 
$\Delta E$, where
$E$ is the beam energy and the auxiliary parameter $\Delta \ll 1$ should be chosen
in such a way that the validity of the soft-photon approximation is guaranteed. 
The second contribution is due to hard photon emission, 
i.e. to single bremsstrahlung with photon energy above $\Delta E$ and 
corresponds to the radiative process $e^+ e^- \to e^+ e^- \gamma$.


\begin{figure}
\begin{center}
\resizebox{0.35\textwidth}{!}{%
\includegraphics{GammaGamma0.eps}~\hspace{3cm}\includegraphics{GammaGamma00.eps}
}
\caption{LO Feynman diagrams for the process $e^+e^- \to \gamma\gamma$.}
\label{figlo:2}
\end{center}
\end{figure}


Following~\cite{Berends:1973jb,Beenakker:1990mb},
%we write the first part in the form
the soft plus virtual (SV) correction can be cast into the form 
\begin{eqnarray}
%\label{bsv}
\frac{\dd\sigma^{\mathrm{Bhabha}}_{B+S+V}}{\dd\Omega_-}
&=& \frac{\dd{\sigma}^{\mathrm{Bhabha}}_{0}}{\dd\Omega_-}
\biggl\{1+ \frac{2\alpha}{\pi}(L-1)\left[2\ln\Delta
+ \frac{3}{2}\right] \nonumber \\ \label{bsv}
&-& \frac{8\alpha}{\pi}\ln(\mbox{ctg}\frac{\theta}{2})
\ln\Delta + \frac{\alpha}{\pi}K^{\mathrm{Bhabha}}_{SV} \biggr\},
\end{eqnarray}
where the factor $K^{\mathrm{Bhabha}}_{SV}$ is given by
\begin{eqnarray}
&& K^{\mathrm{Bhabha}}_{SV} = -1-2\mathrm{Li}_2(\sin^2\frac{\theta}{2}) 
+ 2\mathrm{Li}_2(\cos^2\frac{\theta}{2})
\nonumber \\ %\nonumber
&& \quad + \frac{1}{(3+c^2)^2}\biggl[\frac{\pi^2}{3}(2c^4 - 3c^3 - 15c)
+ 2(2c^4 - 3c^3 + 9c^2 
\nonumber \\ %\nonumber
&& \quad + 3c + 21)\ln^2(\sin\frac{\theta}{2})
- 4(c^4+c^2-2c)\ln^2(\cos\frac{\theta}{2}) 
\nonumber \\ \nonumber
&& \quad - 4(c^3+4c^2+5c+6)\ln^2(\mbox{tg}\frac{\theta}{2})
+ 2(c^3-3c^2+7c
\nonumber \\% \nonumber
&& \quad -5)\ln(\cos\frac{\theta}{2}) 
+2(3c^3+9c^2+5c+31)\ln(\sin\frac{\theta}{2}) \biggr],
\label{eq:svbh}
\end{eqnarray}
and depends on the scattering angle, due to the contribution from 
initial-final-state interference and box diagrams (see Fig.~\ref{fignlo:2}). 
It is worth noticing that the SV correction contains a leading logarithmic 
(LL) part enhanced by
the collinear logarithm $L = \ln(s/m_e^2)$. Among the virtual corrections 
there is also a numerically important effect due to
vacuum polarisation in the photon propagator. Its contribution 
is omitted in Eq.~(\ref{eq:svbh}) but can be taken into account
in the standard way by insertion of the resummed vacuum polarisation
operators in the photon propagators of the Born-level Bhabha amplitudes.

The differential cross section of the single hard %photon 
brems\-strah\-lung process
\begin{eqnarray*}
e^+(p_+) + e^-(p_-) \to e^+(p_+') + e^-(p_-') + \gamma(k)
\end{eqnarray*}
for scattering angles up to corrections of order $m_e/E$ reads
\begin{eqnarray} \label{bere}
\dd\sigma^{\mathrm{Bhabha}}_{\mathrm{hard}} &=& \frac{\alpha^3}{2\pi^2s}\;
R_{e\bar{e}\gamma}\dd \Gamma_{e\bar{e}\gamma} , 
\\ \nonumber 
\dd\Gamma_{e\bar{e}\gamma} &=& \frac{\dd^3p_+'\dd^3p_-'\dd^3k}{\eps_+'\eps_-'k^0}
\delta^{(4)}(p_++p_--p_+'-p_-'-k), 
\\ \nonumber
R_{e\bar{e}\gamma} &=& \frac{WT}{4}
- \frac{m_e^2}{(\chi_+')^2}\left(\frac{s}{t}+\frac{t}{s}+1\right)^2
\\ \nonumber 
&-& \frac{m_e^2}{(\chi_-')^2}\left(\frac{s}{t_1}+\frac{t_1}{s}+1\right)^2
- \frac{m_e^2}{\chi_+^2}\left(\frac{s_1}{t}+\frac{t}{s_1}+1\right)^2
\\ \nonumber
&-& \frac{m_e^2}{\chi_-^2}\left(\frac{s_1}{t_1}+\frac{t_1}{s_1}+1\right)^2,
\end{eqnarray}
where
\begin{eqnarray}
W &=& \frac{s}{\chi_+\chi_-} + \frac{s_1}{\chi_+'\chi_-'}
- \frac{t_1}{\chi_+'\chi_+} \nonumber
%\\ \nonumber
- \frac{t}{\chi_-'\chi_-}
+ \frac{u}{\chi_+'\chi_-} + \frac{u_1}{\chi_-'\chi_+}\, , \\ \nonumber
T &=& \frac{ss_1(s^2+s_1^2) + tt_1(t^2+t_1^2)+uu_1(u^2+u_1^2)}{ss_1tt_1}\, ,
\end{eqnarray}
and the invariants are defined as
\begin{eqnarray}
\nonumber
&& s_1 = 2p_-'p_+',\qquad t=-2p_-p_-',\qquad
t_1=-2p_+p_+', \\ \nonumber
&& u = -2p_-p_+',\quad u_1=-2p_+p_-',\quad \chi_{\pm}=kp_{\pm},\quad
\chi_{\pm}'=kp_{\pm}'.
\end{eqnarray}
%Explicit expressions for the differential distribution of single hard photon emission with vacuum polarisation taken into account can be found in  Ref.~\cite{Arbuzov:1997pj}.

\begin{figure}
\begin{center}
%%%\resizebox{0.475\textwidth}{!}{%
\resizebox{0.5\textwidth}{!}{
\includegraphics{bhabha_isr.eps}~\hspace{0.75cm}\includegraphics{bhabha_vertex.eps}
}
\caption{Examples of Feynman diagrams for real and virtual NLO QED 
initial-state corrections to the $s$-channel contribution of the Bhabha process.}
\label{fignlo:1}
\end{center}
\end{figure}

NLO QED radiative corrections to the two-photon annihilation channel were 
obtained in~\cite{terentyev:1969xxx,Berends:1973tm,Eidelman:1978rw,Berends:1981uq}, 
while weak corrections were computed in~\cite{bohm:1986bs}. 

In the one-loop approximation the part of the differential cross section with
the Born-like kinematics reads
\begin{eqnarray}
\label{bsvgg}
&& \frac{\dd\sigma^{\gamma\gamma}_{B+S+V}}{\dd\Omega_1} 
= \frac{\dd{\sigma}^{\gamma\gamma}_0}{\dd\Omega_1}
\Biggl\{1 + \frac{\alpha}{\pi}
\biggl[(L-1)\biggl(2\ln\Delta+\frac{3}{2}\biggr)
\nonumber \\ %\label{gg_bsv}
&&\qquad\qquad\quad + \, K^{\gamma\gamma}_{SV}\biggr]\Biggr\}, 
\nonumber \\ %\nonumber
&& K^{\gamma\gamma}_{SV} = \frac{\pi^2}{3}+\frac{1-c_1^2}{2(1+c_1^2)}\biggl[\biggl(1
+ \frac{3}{2}\,\frac{1+c_1}{1-c_1}\biggr)\ln\frac{1-c_1}{2} 
\nonumber\\
&& \quad + \biggl(1+\frac{1-c_1}{1+c_1}+\frac{1}{2}\frac{1+c_1}{1-c_1}\biggr)
\ln^2\frac{1-c_1}{2}
+ (c_1\to -c_1)\biggr], \nonumber \\
&& c_1 = \cos\theta_1,\qquad
\theta_1=\widehat{\vec{q}_1\vec{p}}_-\, .
\end{eqnarray}

In addition, the three-photon production process
\begin{eqnarray*}
e^+(p_+)\ +\ e^-(p_-)\ \to\ \gamma(q_1)\ +\ \gamma(q_2)\ +\ \gamma(q_3)
\end{eqnarray*}
must be included. Its cross section is given by
\begin{eqnarray} \label{3gamma}
&& \dd\sigma^{e^+e^-\to 3\gamma} = \frac{\alpha^3}{8\pi^2 s}
R_{3\gamma}\,\dd\Gamma_{3\gamma}\, , \\ \nonumber
&& R_{3\gamma} = s\;\frac{\chi_3^2+(\chi_3')^2}{\chi_1\chi_2\chi_1'\chi_2'}
- 2m_e^2\biggl[\frac{\chi_1^2+\chi_2^2}{\chi_1\chi_2(\chi_3')^2}
+ \frac{(\chi_1')^2+(\chi_2')^2}{\chi_1'\chi_2'\chi_3^2}\biggr] 
\\ \nonumber 
&&\quad \quad + \, \mbox{(cyclic permutations)}, 
\\ \nonumber
&& \dd \Gamma_{3\gamma} = \frac{\dd^3q_1\dd^3q_2\dd^3q_3}{q_1^0q_2^0q_3^0}
\delta^{(4)}(p_++p_--q_1-q_2-q_3),
\end{eqnarray}
where
\begin{eqnarray*}
\chi_i=q_ip_-,\qquad \chi_i'=q_ip_+,\qquad i=1,2,3\, .
\end{eqnarray*}
The process has to be treated as a radiative correction to
the two-photon production. The energy of the third photon should 
exceed the soft-photon energy threshold $\Delta E$. In practice, the tree photon
contribution, as well as the radiative Bhabha process 
$e^+ e^- \to e^+ e^- \gamma$, should be simulated with the help of a MC event generator
%according to Eq.~(\ref{3gamma})
 in order to take into account the proper experimental criteria of a given event selection.
 

\begin{figure}
\begin{center}
\resizebox{0.5\textwidth}{!}{%
\includegraphics{bhabha_box}~\hspace{0.75cm}\includegraphics{bhabha_crossbox}
}
\caption{Feynman diagrams for the NLO QED box corrections to the $s$-channel contribution of the Bhabha process.}
\label{fignlo:2}
\end{center}
\end{figure}

In addition to the corrections discussed above, also the effect of vacuum polarisation, due
to the insertion of fermion loops inside the photon propagators, must be included in the precise 
calculation of the Bhabha scattering cross section. Its
theoretical treatment, which faces the non-trivial 
problem of the non-perturbative contribution due to hadrons, 
is addressed in detail in Section \ref{sec:4}. However, 
numerical results for such a correction are presented in Section \ref{NUMERICS} 
and Section \ref{TH}.

\begin{figure}[h]
\begin{center}
%%%%%%%%\resizebox{0.475\textwidth}{!}{%
\resizebox{0.475\textwidth}{!}{%
%%%\includegraphics{plot_christopher}%%~\hspace{3cm}\includegraphics{Bhabha_t2}
\includegraphics{fig-wg.eps}%%~\hspace{3cm}\includegraphics{Bhabha_t2}
}
\caption{Cross sections of the processes $e^+e^- \to e^+e^-$ and $e^+e^- \to \gamma\gamma$ 
in LO and NLO approximation as a function of the c.m. energy at meson factories (upper panel).
In the lower panel, the relative contribution due to the NLO QED corrections 
(in per cent) to the two processes is shown.}
\label{fignlo:3}
\end{center}
\end{figure}

In Fig.~\ref{fignlo:3} the cross sections of the Bhabha and two-photon production processes in LO and
NLO approximation are shown as a function of the c.m. energy between $\sqrt{s} \simeq 2\, m_{\pi}$ and
$\sqrt{s} \simeq 10$~GeV (upper panel). The results were obtained imposing the following cuts 
for the Bhabha process:
\begin{eqnarray}%\label{eq:KLOE}
%    \left \{
%\begin{aligned}
%    \sqrt{s} &= 1.02\ \textrm{GeV}\\
&& \theta_{\pm}^{\textrm{min}}  = 45^{\circ}\,, 
    \qquad \quad \, \, \, \theta_{\pm}^{\textrm{max}} = 135^{\circ}\,,  \nonumber \\
&& E_{\pm}^{\textrm{min}} = 0.3\sqrt{s}\,,  \qquad \xi_{\textrm{max}} =10^{\circ}\,,
%\end{aligned}
%  \right.
\label{eq:cutsmf}
\end{eqnarray}
where $\theta_{\pm}^{\textrm{min,max}}$ are the angular acceptance cuts, 
$E_{\pm}^{\textrm{min}}$ are the minimum energy thresholds for the detection of the 
final-state electron/positron and $\xi_{\textrm{max}}$ is the maximum $e^+ e^-$ acollinearity.
For the photon pair production processes we used correspondingly:
\begin{eqnarray}%\label{eq:KLOE}
%%    \left \{
%%\begin{aligned}
%%    \sqrt{s} &= 1.02\ \textrm{GeV}\\
&& \theta_{\gamma}^{\textrm{min}}  = 45^{\circ}\,, 
\qquad \quad \, \, \, \theta_{\gamma}^{\textrm{max}} = 135^{\circ}\,,  \nonumber \\
&& E_{\gamma}^{\textrm{min}} = 0.3\sqrt{s}\,,  \qquad \xi_{\textrm{max}} =10^{\circ}\,,
%%\end{aligned}
%%  \right.
\label{eq:cutsmfg}
\end{eqnarray}
where, as in Eq.~(\ref{eq:cutsmf}), $\theta_{\gamma}^{\textrm{min,max}}$ are the angular acceptance cuts, $E_{\gamma}^{\textrm{min}}$ is the minimum energy threshold for the detection of at 
least two photons and $\xi_{\textrm{max}}$ is the maximum acollinearity between the most energetic and next-to-most energetic photon.

The cross sections display the typical $1/s$ QED behaviour. The relative 
effect of NLO corrections is shown in the lower panel. It can be seen that the NLO corrections are 
 largely negative and increase with increasing
c.m. energy, because of the growing importance of the 
collinear logarithm $L= \ln(s/m_e^2)$. The corrections to $e^+e^- \to \gamma\gamma$ are 
about one half of those to Bhabha scattering, because of the absence of final-state radiation 
effects in photon pair production.
