
\begin{figure*}[hbtp]
\begin{center}
\mbox{
\resizebox{0.81\columnwidth}{0.86\columnwidth}{
\includegraphics{polaraccept.eps}}
\hspace*{0.5cm}
\resizebox{0.85\columnwidth}{0.83\columnwidth}{
\includegraphics{acollincya.eps}}
}
\caption{Comparison between large-angle Bhabha KLOE data
(points) and MC (histogram) distributions for the $e^\pm$ polar angle
$\theta$ (left) and for the acollinearity,
$\zeta=|\theta_{e^+}+\theta_{e^-}-180^\circ|$ (right), where
the flight direction of the $e^\pm$ is given by the 
position of clusters in the calorimeter. In each case, MC and data histograms are normalised
to unity. From~\cite{Ambrosino:2006te}.}
\label{figmotiv:1}
\end{center}
\end{figure*}

\subsection{Motivation \label{MOTIVATION}}

The luminosity of a collider is the normalisation constant between 
the event rate and the cross section of a given process. 
For an accurate measurement of the cross section
of an electron-positron ($e^+e^-$) annihilation 
process, the precise knowledge of the collider
luminosity is mandatory.

The luminosity depends on three factors:
beam-beam crossing
frequency, beam currents and the beam overlap area in the
crossing region. However, the last quantity is difficult to
determine accurately from the collider optics.
Thus, experiments prefer to determine the luminosity
by the counting rate of well
selected events whose cross section is known with good accuracy,
using the formula~\cite{Ambrosino:2006te}
\begin{equation}
\int\!\mathcal{L}\,{\rm d} t = \frac{N}{\epsilon \, \sigma} ,
\label{eq:1}
\end{equation}
where $N$ is the number of events of the chosen reference process, $\epsilon$ the 
experimental selection efficiency and $\sigma$ the theoretical cross section of the reference 
process. 
Therefore, the total luminosity
error will be given by the sum in quadrature of the fractional experimental and theoretical
uncertainties. 

%\begin{figure*}[htbp]

Since the advent of low luminosity $e^+e^-$ colliders, a great
effort was devoted to obtain good precision in the cross section
of electromagnetic processes, extending the pioneering work of the
earlier days~\cite{Cabibbo:1961sz}. At the $e^+e^-$ colliders 
operating in the c.m. energy range $1\mbox{ GeV}<\sqrt{s}<3\mbox{ GeV}$,
such as ACO at Orsay, VEPP-II at Novosibirsk and Adone at Frascati, the
luminosity measurement was based on
Bhabha scattering~\cite{Bhabha:1936xxx,Barbiellini:1975}
with final-state electrons and positrons detected at small angles,
or single and double 
bremsstrahlung processes~\cite{Dehne:1974bc},
thanks to their high statistics. The electromagnetic cross
sections scale as $1/s$, while elastic
$e^+e^-$ scattering has a steep dependence on the polar angle,
$\sim 1/\theta^3$, thus providing a high rate for small values of
$\theta$.

Also at high-energy, accelerators running in the '90s around the $Z$ pole 
to perform precision tests of the Standard Model (SM), such 
as LEP at CERN and SLC at Stanford, the experiments used small-angle Bhabha
scattering events as a luminosity monitoring process. Indeed, for the very forward angular 
acceptances considered by the LEP/SLC collaborations, the Bhabha 
process is dominated by the electromagnetic interaction and, therefore, calculable, 
at least in principle, with very high accuracy.
At the end of the LEP and SLC operation,
a total (experimental plus theoretical) precision of 
one per mill (or better) was
achieved~\cite{Jadach:1996gu,Arbuzov:1995qd,Montagna:1996gw,Arbuzov:1996eq,Montagna:1998sp,Ward:1998ht,Jadach:2003zr}, 
thanks to the work of
different theoretical groups and 
the excellent performance of precision luminometers.

At current low- and intermediate-energy high-lu\-mi\-no\-si\-ty meson
factories, the small polar angle region is difficult
to access due to the presence of the low-beta insertions close to the
beam crossing region, while wide-angle Bhabha scattering produces a
large counting rate and can be exploited for a precise
measurement of the luminosity.

Therefore, also in this latter case of $e^\pm$ scattered at large angles,
e.g.  larger than 55$^\circ$ for the KLOE experiment~\cite{Ambrosino:2006te}
running at DA$\mathrm{\Phi}$NE in Frascati, and 
larger than 40$^\circ$ for the CLEO-c
experiment~\cite{cleo:2007zt} running
at CESR in Cornell, the main advantages of Bhabha scattering are preserved:
\begin{enumerate}
%\renewcommand{\labelenumi}{\roman{enumi})}
\item large statistics. For example at DA$\mathrm{\Phi}$NE, a statistical error
$\delta\mathcal{L}/\mathcal{L}\sim0.3\%$ is reached in
about two hours of data taking, even at the lowest luminosities;
\item high accuracy for the calculated cross section;
\item clean event topology of the signal and
small amount of background. 
\end{enumerate}

In Eq.~(\ref{eq:1}) the cross section is usually evaluated by inserting
event generators, which include radiative corrections at a high level of
precision, into
the MC code simulating the detector response. 
The code has to be developed to reproduce the detector performance
(geometrical acceptance, reconstruction efficiency and
resolution of the measured quantities) to a high level of confidence.

In most cases the major sources of the systematic errors of the
luminosity measurement are differences of efficiencies and resolutions
between data and MC.

In the case of KLOE, the largest experimental error of the
luminosity measurement is due to a different polar angle resolution
between data and MC which is observed at the edges of the accepted
interval for Bhabha scattering events. Fig.~1 shows a comparison
between large angle Bhabha KLOE data and MC, at left for the polar
angle and at right for the acollinearity $\zeta=|\theta_{e^+}+\theta_{e^-}-180^\circ|$. 
One observes a very good agreement between data and MC, but also 
differences (of about 0.3 \%) at the sharp interval edges. The analysis cut, 
$\zeta<9^\circ$, applied to the acollinearity distribution is very far
from the bulk of the distribution and does not introduce noteworthy
systematic errors.
%\begin{figure}[htbp]
\begin{figure}[h]
\begin{center}
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figcleo.eps}%%~\hspace{3cm}\includegraphics{Bhabha_t2}
}
\caption{Distributions of CLEO-c $\sqrt{s} = 3.774$ GeV
data (circles) and MC simulations (histograms) for the polar angle
of the positive lepton (upper two plots) in $e^+e^-$ and $\mu^+\mu^-$
events, and for the mean value of $|\cos\theta_\gamma|$ of the two photons
in $\gamma\gamma$ events (lower panel). MC histograms are normalised to the number of
data events. From~\cite{cleo:2007zt}.}
\label{figmotiv:2}
\end{center}
\end{figure}
Also in the CLEO-c luminosity measurement with Bhabha scattering
events, the detector modelling is the main
source of experimental error. In particular, uncertainties
include those due to finding and reconstruction of the electron
shower, in part due to the nature of the electron shower, as well as
the steep $e^\pm$ polar angle distribution.

The luminosity measured with Bhabha scattering events is often checked
by using other QED processes, such as $e^+e^-\to\mu^+\mu^-$
or $e^+e^-\to\gamma\gamma$. In KLOE, the luminosity measured
with $e^+e^-\to\gamma\gamma$ events differs by $0.3\%$
from the one determined from Bhabha events. In CLEO-c,
$e^+e^-\to\mu^+\mu^-$ events 
are also used, and the luminosity determined from $\gamma\gamma$ ($\mu^+\mu^-$)
is found to be $2.1\%$ ($0.6\%$) larger than that from Bhabha events.
Fig.~\ref{figmotiv:2} shows the CLEO-c data for the polar angle distributions 
of all three processes, compared with the corresponding MC predictions. 
The three QED processes are also used by the BaBar experiment at the 
PEP-II collider, Stanford, yielding a luminosity determination with an 
error of about 1\% \cite{andreas:2009xxxx}. 
Large-angle Bhabha scattering is the normalisation process adopted
by the CMD-2 and SND collaborations at VEPP-2M, Novosibirsk, while both
BES at BEPC in Beijing and Belle at KEKB in Tsukuba measure luminosity using 
the processes $e^+ e^- \to e^+ e^-$ and $e^+ e^- \to \gamma\gamma$ with 
the final-state particles detected at wide polar angles and an experimental 
accuracy of a few per cent. However, BES-III aims at reaching an error
of a few per mill in their luminosity measurement in the near future
\cite{Asner:2008nq}. 

The need of precision, namely better than $1\%$, 
and possibly redundant
measurements of the collider luminosity is of utmost importance to perform accurate measurements
of the $e^+ e^- \to {\rm hadrons}$ cross sections, which are the
key ingredient for evaluating the hadronic contribution
to the running of the electromagnetic coupling constant $\alpha$ and the muon anomaly $g-2$.
