\newcommand{\eeee}{\ensuremath{e^+e^-\to e^+e^-}\xspace}
\newcommand{\eemm}{\ensuremath{e^+e^-\to\mu^+\mu^-}\xspace}
\newcommand{\eepp}{\ensuremath{e^+e^-\to\pi^+\pi^-}\xspace}
%\newcommand{\ee}{\ensuremath{e^+e^-}\xspace}
\newcommand{\mm}{\ensuremath{\mu^+\mu^-}\xspace}
\newcommand{\pp}{\ensuremath{\pi^+\pi^-}\xspace}
The process $e^+e^- \to \pi^+\pi^-\gamma$ with final state radiation can be used to
answer the question whether one can treat pions as point-like particles and
apply scalar QED to calculate the radiative corrections to the cross
section. In particular, one can compare the photon spectra obtained using scalar
QED with those found in data.
The radiative corrections due to photon emission in the final state (FSR)
contribute about 1\% to the cross section. The hadronic contribution
of the process $e^+e^- \to \pi^+\pi^-$ to the value $a^{\rm had}_{\mu}$ amounts to
$\sim$50 ppm, while the anomalous magnetic moment of the muon was measured
in the E821 experiment at BNL with an accuracy of 0.5 ppm~\cite{Bennett:2006fi}.
Therefore the theoretical precision of the
cross section calculation for
this process should be several times smaller than 1\%. In this case we can
neglect the error of this contribution to the value $a^{\rm had}_{\mu}$
compared to 0.5 ppm. These facts are the main motivation to study this process.
\vspace{0.2cm}
{\noindent \it Event selection\\}
For the analysis, data were taken in a c.m. energy range from 720
to 780 MeV, with one photon detected in the CsI calorimeter.
Events from the processes $e^+e^- \to e^+ e^-\gamma$ and $e^+e^- \to \mu^+\mu^-\gamma$ have a
very similar topology in the detector, compared to $e^+e^- \to \pi^+\pi^-\gamma$ events.
In addition, the cross section of the process $e^+e^- \to \pi^+\pi^-\gamma$
with FSR is more than ten times smaller than the one for the similar process with ISR.
On the other hand, the cross section of the process
$e^+e^- \to \pi^+\pi^-\gamma$ has a strong energy dependence
due to the presence of the $\rho$-resonance. This fact allows to
significantly enrich the fraction of the events $e^+e^- \to \pi^+\pi^-\gamma$ with FSR
for energies below the $\rho$-peak. Indeed, ISR shifts the c.m. energy to smaller
values and, as a result, the cross section falls down dramatically,
whereas the process with FSR is almost energy-independent.
Several curves describing the ratio
$\sigma^{{\rm FSR} + {\rm ISR}}_{\pi^+\pi^-\gamma}/\sigma^{\rm ISR}_{\pi^+\pi^-\gamma}$
plotted against the c.m. energy, are presented in Fig.~\ref{isrfsr} (a) for different energy
thresholds for photons detected in the calorimeter. It is clearly visible that
the optimal energy range to be used in this study goes from 720 MeV up to 780 MeV.
It is also seen that this ratio increases with the threshold energy
for photons to be detected.
%%%TT
This means that the fraction of the
$\pi^+\pi^-\gamma$ events with FSR (with respect to events without FSR)
grows with increasing photon energy.
It allows to enrich the number of $\pi^+\pi^-\gamma$ events with FSR.
%%%TT
Let us recollect that the shape of the distribution of $\pi^+\pi^-\gamma$
events, at photon energies of the same order as the pion mass or larger,
is of special interest. First of all, namely in that part of
the photon spectrum we can meet a discrepancy with the sQED prediction.
\begin{figure}[!htb]
\begin{center}
\subfigure[]{\includegraphics[width=0.45\textwidth]{isrfsr_ratio.eps}}
\subfigure[]{\includegraphics[width=0.5 \textwidth]{Wsim.eps}}
\caption{\label{isrfsr} (a) Ratio
$\sigma_{{\rm ISR} + {\rm FSR}}/\sigma_{\rm ISR}$ vs the c.m. energy. The set of curves indicates
how this ratio depends on the threshold energy for the detected photons.
The threshold energy in MeV is stated over the curves.
%\caption{\label{Wsim}
(b) Distributions of the parameter $W$ for events of
the processes
$e^+e^- \to \pi^+\pi^-\gamma$,
$e^+e^- \to \mu^+\mu^-\gamma$ and
$e^+e^- \to e^+e^-\gamma$, for a c.m. energy of 780 MeV.}
\end{center}
\end{figure}
A typical $\pi^+\pi^-\gamma$ event in the CMD-2 detector has two tracks in the
drift chamber with two associated clusters in the CsI calorimeter
and a third cluster representing the radiated photon.
To suppress multi-photon events and significantly
cut off collinear $\pi^+\pi^-$ events the following
requirements were applied: the angle between the direction of photon momentum and
missing momentum must be larger than 1 rad and the angle between one of the two
tracks and the photon direction must be smaller than 0.2 rad.
To suppress $e^+e^-\gamma$ events, a parameter $W = p/E$
was used, in which the particle momentum $p$ (measured in the drift chamber)
is divided by the energy $E$
(measured in the CsI calorimeter). Simulation results are presented in
Fig.~\ref{isrfsr} (b). The condition $W < 0.4$ reduces the electron contribution to
the level of $\sim$ 1\%. The square of the invariant mass for electrons, muons and
pions is plotted in Fig.~\ref{M2sim} ~(a). The condition $M^2 > 10 000$~MeV$^2$
further rejects the number of electrons and muons by a factor of 1.5.
About 1\% of the pion events are lost with these cuts.
\begin{figure}[!htb]
\begin{center}
\subfigure[]{\includegraphics[width=0.45 \textwidth]{M2sim.eps}}
\subfigure[]{\includegraphics[width=0.45 \textwidth]{Ephcmp.eps}}
\caption{\label{M2sim} (a)
Distributions of the parameter $M^2$ for events of the processes
$e^+e^- \to \pi^+\pi^-\gamma$,
$e^+e^- \to \mu^+\mu^-\gamma$ and
$e^+e^- \to e^+e^-\gamma$ for a c.m. energy of 780 MeV.
%\caption{\label{ph-sp}
(b) Distribution of the $\pi^+\pi^-\gamma$ events against
the photon energy in relative units.
%%%TT
Also stated is the fraction of $\pi^+\pi^-\gamma$ events with FSR for each region as indicated by the vertical lines.}
\end{center}
\end{figure}
{\noindent \it Preliminary results of the analysis\\}
The histogram of the number $\pi^+\pi^-\gamma$ events against
the photon energy in relative units is presented in Fig.~\ref{M2sim} (b).
The histogram represents the simulation, while the points with error bars
show the experimental data. Vertical
dotted lines divide the plot area into three zones. The inscription inside each
zone indicates the fraction of $\pi^+\pi^-\gamma$ events with FSR with respect
to others. The number of the simulated events was normalised to the
experimental one. The average deviation between the two distributions was found
to be $(-2.1 \pm 2.3)\%$. Therefore, one can conclude that there is no
evidence that photon radiation by pions needs to be described beyond
the framework of scalar QED. In other words, pions can be treated as
point-like objects, and the application of scalar QED is found to be valid
within the stated accuracy. Unfortunately, the lack of statistics in the
energy range under study
does not allow us to check this assumption with better accuracy. Forthcoming
experiments at VEPP-2000 will significantly improve the
statistical error.