The KLOE experiment, in operation at the DA$\mathrm{\Phi}$NE $e^+e^-$ collider in
Frascati between 1999 and 2006, utilises radiative return to obtain precise
measurements of hadronic cross sections in the energy range below 1 GeV. As
the DA$\mathrm{\Phi}$NE machine was designed to operate as a meson factory with
collision energy equal to the mass of the $\phi$-meson ($m_\phi =$
1.01946 GeV), with limited possibility to change the energy of the
colliding beams while maintaining stable running conditions, the
use of events with initial state radiation of hard photons from the
$e^+$ or the $e^-$ is the only way to access energies below
DA$\mathrm{\Phi}$NE's nominal collision energy. These low-energy cross sections are
important for the theoretical evaluation of the muon magnetic moment
anomaly $a_\mu=(g_\mu-2)/2$~\cite{Eidelman:1995ny}, and high
precision is needed since the uncertainty on the cross section data
enters the uncertainty of the theoretical prediction. The channel
$e^+e^- \to \pi^+\pi^-$ gives the largest contribution to the hadronic
part $a_\mu^{\rm had}$ of the anomaly. Therefore, so far KLOE efforts have
concentrated on the derivation of the pion pair-production cross
section $\sigma_{\pi\pi}$ from measurements of the differential cross
section $\frac{{\rm d}\sigma_{\pi\pi\gamma(\gamma)}}{{\rm d}M^2_{\pi\pi}}$, in
which $M^2_{\pi\pi}$ is the invariant mass squared of the di-pion system in
the final state.
The KLOE detector (shown in Fig.~\ref{fig:radret_kloe1}), which
consists of a high
resolution drift chamber ($\sigma_{p} / p \leq 0.4\%$) and an
electromagnetic calorimeter with excellent time ($\sigma_t\sim
54 ~\mathrm{ps}/\sqrt{E~[\mathrm{GeV}]}$ $\oplus100~\mathrm{ps}$) and
good energy
($\sigma_E/E\sim 5.7\%/\sqrt
{E~[\mathrm{GeV}]}$) resolution, is optimally suited for this kind of
analyses.\\
%Exists a more clever way to do a \subsubsubsection???
{\noindent \it The KLOE $\pi\pi\gamma$ analyses\\}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=70mm] {radret_kloe1.eps}
%\vglue-0.5cm
\caption{KLOE detector with the selection regions for small
angle photons (narrow cones)
and for pion tracks and large angle photons (wide cones). Used with permission
of the KLOE collaboration.}
\label{fig:radret_kloe1}
\end{center}
\end{figure}
The KLOE analyses for $\sigma_{\pi\pi}$ use two
different sets of acceptance cuts:
\begin{itemize}
\item[$\bullet$] In the {\it small angle}
analysis, photons are emitted
within a cone of $\theta_\gamma<15^\circ$ around the
beamline (narrow cones in Fig.~\ref{fig:radret_kloe1}), and the two
charged pion tracks
have $50^\circ<\theta_\pi<130^\circ$. The photon is
not explicitly detected; its direction
is reconstructed from the track momenta
by closing the kinematics:
%~\footnote{For the clarity of presentation we neglect the boost of the $\phi$.}
$\vec{p}_\gamma\simeq\vec{p}_{miss}= -(\vec{p}_{\pi^+}
+\vec{p}_{\pi^-})$.
In this analysis, the separation of pion- and photon selection
regions greatly reduces the
contamination from the resonant process $e^+e^-\to
\phi\to\pi^+\pi^-\pi^0$ in which the $\pi^0$ mimics the missing
momentum of the photon(s) and from the final state
radiation process $e^+e^-\to \pi^+\pi^-\gamma_{\rm FSR}$.
Since ISR-photons are mostly collinear with the beam line, a high
statistics for the ISR signal events remains. On the other hand, a high
energy
photon emitted at angles close to the incoming beams forces the pions also to
have a small angle with respect to the beamline (and thus outside the selection cuts), resulting in a
kinematical suppression of events with $M^2_{\pi\pi}< 0.35$
GeV$^2$.
\item[$\bullet$] The {\it large angle} analysis requires both photons and pions
to be emitted at $50^\circ<\theta_{\pi,\gamma}<130^\circ$ (wide
cones in Fig.~\ref{fig:radret_kloe1}), allowing for a detection of
the photons in the barrel of the calorimeter. This analysis allows to reach
the 2$\pi$ threshold region, at the price of higher background
contributions from the $\pi^+\pi^-\pi^0$ final state and events with
final state radiation. In addition, events from the
decays $\phi \to f_0\gamma \to \pi^+\pi^-\gamma$ and $\phi\to\pi^\pm\rho^\mp
\to \pi^\pm\pi^\mp\gamma$, which need to be described by model-dependent
parameterisations, contribute to the spectrum of the selected events
(running at the $\phi$ peak).
\end{itemize}
Two analyses based on the {\it small angle} acceptance cuts have been
carried out. The first one, using 140 pb$^{-1}$ of data taken in the
year 2001, was published in 2005~\cite{Aloisio:2004bu}. The second one,
based on 240 pb$^{-1}$ of data taken in 2002, was published in
2008~\cite{Ambrosino:2008en}.
The differential cross section is obtained from the spectrum of
selected events $N^{\mathrm{sel}}$ subtracting the residual background
(mostly $\mu\mu\gamma(\gamma)$, $\pi\pi\pi$ and radiative Bhabha
events) and dividing by the selection efficiencies and the integrated
luminosity:
\begin{equation}
\frac{{\rm d}\sigma_{\pi\pi\gamma(\gamma)}}{{\rm d}M^2_{\pi\pi}} =
\frac{N^{\mathrm{sel}}-N^{\mathrm{bkg}}}{\Delta
M^2_{\pi\pi}}\cdot\frac{1}{\varepsilon_{\mathrm{sel}}}\cdot\frac{1}{\int L{\rm d}t}~.
\label{eq:radret_kloe1}
\end{equation}
$\Delta M^2_{\pi\pi}$ is the bin width used in the analysis
(typically 0.01 GeV$^2$), and $\int L{\rm d}t$ is the integrated
luminosity obtained from Bhabha events detected at large angles
($55^\circ<\theta_e<125^\circ$) and
%compared to
the reference cross
section from the BabaYaga
generator~\cite{CarloniCalame:2000pz,Balossini:2006wc}
(discussed in Section~\ref{sec:1}).
The total cross section is then obtained from the formula
\begin{equation}
\sigma_{\pi\pi}(M^2_{\pi\pi})= s\cdot
\frac{{\rm d}\sigma_{\pi\pi\gamma(\gamma)}}{{\rm d}M^2_{\pi\pi}}
\frac{1}{H(s,M^2_{\pi\pi})}~.
\label{eq:radret_kloe2}
\end{equation}
In Eq.~(\ref{eq:radret_kloe2}), $s$ is the squared energy at which the DA$\mathrm{\Phi}$NE
collider is operated during data taking, and $H(s,M^2_{\pi\pi})$ is
the radiator function describing the emission of photons from the
$e^+$ or the $e^-$ in the initial state. Note that
Eq.~(\ref{eq:radret_kloe2}) does not contain the effects from final
state radiation from pions. These effects complicate the analysis, since the KLOE
detector can not distinguish whether photons in an event were emitted in the
initial or the final state. The PHOKHARA Monte Carlo
generator~\cite{Czyz:2003ue}, which includes final state radiation
at next-to-leading order and in the pointlike-pion approximation, is used
to properly take into account final state radiation in the
analyses. This is important because the {\it bare} cross section used
to evaluate $a_\mu^{\rm had}$ via an appropriate dispersion integral should be
inclusive with respect to final state radiation, and also needs to
be undressed from vacuum polarisation effects present in the virtual
photon produced in the $e^+e^-$ annihilation. For the latter, we
use a function provided by
F.~Jegerlehner~\cite{Jegerlehner:alphaWEB} (see Section~\ref{sec:4}), and correct the cross
section via
\begin{equation}
\sigma_{\pi\pi}^{\mathrm{bare}}(M^2_{\pi\pi})= \sigma_{\pi\pi}^{\mathrm{dressed}}(M^2_{\pi\pi}) \left( \frac{\alpha(0)}{\alpha(M^2_{\pi\pi})}
\right)^2\,.
\label{eq:radret_kloe3}
\end{equation}
Here $\alpha(0)$ is the fine structure constant in the limit $q=0$,
%%%TT
and $\alpha(M^2_{\pi\pi})$ represents the value of the effective
coupling at the scale of the invariant mass of the di-pion system.
Since the hadronic contributions to $\alpha(M^2_{\pi\pi})$ are
calculated via a dispersion integral which includes the hadronic cross
section itself in the integrand (see Section~\ref{sec:4}), the correct
procedure has to be iterative and should include the same data that
must be corrected. However, since
the correction is at the few percent level, the inclusion of the new
KLOE data will not change $\alpha(M^2_{\pi\pi})$ at a level which would
significantly affect the analyses. We therefore have used the values
for $\alpha(M^2_{\pi\pi})$ derived from the existing hadronic cross
section database. As an example, Fig.~\ref{fig:radret_kloe2} shows the KLOE result
for ${{\rm d}\sigma_{\pi\pi\gamma(\gamma)}}/{{\rm d}M^2_{\pi\pi}}$ obtained from data taken in the year
2002~\cite{Ambrosino:2008en}. Inserting this differential cross section into Eq.~(\ref{eq:radret_kloe2}) and the result
into Eq.~(\ref{eq:radret_kloe3}),
one derives $\sigma_{\pi\pi}^{\mathrm{bare}}$. Using the {\it bare} cross section to get the $\pi\pi$-contribution to
$a_\mu^{\rm had}$ between 0.35 and 0.95 GeV$^2$ then gives the value (in units of $10^{-10}$)
\begin{displaymath}
a_\mu^{\pi\pi}(0.35 -0.95 {\rm GeV}^2) = (387.2\pm0.5_{\rm stat}\pm2.4_{\rm exp}\pm2.3_{\rm th})\,.
\end{displaymath}
Table~\ref{tab:radret_kloe1} shows the contributions to the systematic
errors on $a_\mu^{\pi\pi}(0.35 -0.95$ GeV$^2)$.\\
\begin{figure}[htb]
\begin{center}
\includegraphics[width=70mm] {radret_kloe2.eps}
%\vglue-0.5cm
\caption{Differential radiative cross section ${d\sigma_{\pi\pi\gamma(\gamma)}}/{dM^2_{\pi\pi}}$, inclusive in $\theta_\pi$ and with
$0^o<\theta_{\gamma}< 15^o$ or $165^o<\theta_{\gamma}< 180^o$ measured by the KLOE experiment~\cite{Ambrosino:2008en}. Used with permission
of the KLOE collaboration.}
\label{fig:radret_kloe2}
\end{center}
\end{figure}
%\begin{figure}[htb]
%\begin{center}
%\includegraphics[width=70mm] {radret_kloe2.eps}
%%\vglue-0.5cm
%\caption{{\it Bare} cross section for $e^+e^-\rightarrow \pi^+\pi^-$
% measured by the KLOE experiment using the radiative
%return~\cite{Ambrosino:2008en}.}
%\label{fig:radret_kloe2}
%\end{center}
%\end{figure}
\begin{table}
\begin{center}
\begin{tabular}{||l|c||}
\hline
Reconstruction Filter & negligible\\
Background subtraction & 0.3 \% \\
Trackmass & 0.2 \% \\
Particle ID & negligible\\
Tracking & 0.3 \% \\
Trigger & 0.1 \% \\
Unfolding & negligible \\
Acceptance ($\theta_{\pi\pi}$) & 0.2 \% \\
Acceptance ($\theta_\pi$) & negligible \\
Software Trigger (L3) & 0.1 \% \\
Luminosity ($0.1_{th}\oplus 0.3_{exp}$)\% & 0.3 \% \\
$\sqrt{s}$ dep. of $H$ & 0.2 \%\\
\hline
Total exp systematics & 0.6 \% \\
\hline
\hline
Vacuum Polarisation & 0.1 \% \\
FSR resummation & 0.3 \% \\
Rad. function $H$ & 0.5 \% \\
\hline
Total theory systematics & 0.6 \% \\
\hline
\end{tabular}
\caption{List of systematic errors on the $\pi\pi$-contribution to
$a_\mu^{\rm had}$ between 0.35 and 0.95 GeV$^2$ when using the
$\sigma_{\pi\pi}$ cross section measured
by the KLOE experiment in the corresponding
dispersion integral~\cite{Ambrosino:2008en}.}
\label{tab:radret_kloe1}
\end{center}
\end{table}
%Radiator function
{\noindent \it Radiative corrections and Monte Carlo tools\\}
The radiator function is a crucial ingredient in this kind of
radiative return analyses, and is obtained using the
relation
\begin{equation}
H(s,M^2_{\pi\pi}) = s\cdot\frac{3
M^2_{\pi\pi}}{\pi\alpha^2\beta_\pi^3}\cdot\left.\frac{{\rm
d}\sigma^{\rm ISR}_{\pi\pi\gamma(\gamma)}}{{\rm d}M^2_{\pi\pi}}\right|_{|F_{2\pi}|^2=1},
\label{eq:radret_kloe4}
\end{equation}
in which $\left.\frac{{\rm d}\sigma^{\rm
ISR}_{\pi\pi\gamma(\gamma)}}{{\rm d}M^2_{\pi\pi}}\right|_{|F_{2\pi}|^2=1}$
is evaluated using the PHOKHARA Monte Carlo generator in
next-to-leading order ISR-only
configuration, with the squared pion form factor $|F_{2\pi}|^2$ set to $1$.
$\beta_\pi=\sqrt{1-\frac{4m_\pi^2}{M^2_{\pi\pi}}}$ is the pion
velocity. While Eq.~(\ref{eq:radret_kloe4}) provides a convenient mechanism to
extract the dimensionless quantity $H(s,M^2_{\pi\pi})$ also for
specific angular
regions of pions and photons by applying the relevant cuts to
$\left.\frac{{\rm d}\sigma^{\rm ISR}_{\pi\pi\gamma(\gamma)}}{{\rm d}M^2_{\pi\pi}}\right|_{|F_{2\pi}|^2=1}$,
in the published KLOE analyses. $H(s,M^2_{\pi\pi})$ is evaluated
fully inclusive for pion and photon angles in the range
$0^\circ < \theta_{\pi,\gamma} < 180^\circ$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=70mm] {radret_kloe3.eps}
%\vglue-0.5cm
\caption{The dimensionless radiator function $H(s,M^2_{\pi\pi})$,
inclusive in $\theta_{\pi,\gamma}$. The value used for $s$ in the
Monte Carlo production was
$s = M_\phi^2=(1.019456$ GeV$)^2$.}
\label{fig:radret_kloe3}
\end{center}
\end{figure}
Figure~\ref{fig:radret_kloe3} shows the radiator function in the range
of $0.35 < M^2_{\pi\pi} < 0.95$ GeV$^2$. As can be seen from
Table~\ref{tab:radret_kloe1}, the 0.5\% uncertainty of the radiator function
quoted by the authors of PHOKHARA translates into an
uncertainty of 0.5\% in the $\pi\pi$-contribution to
$a_\mu^{\rm had}$ between 0.35 and 0.95 GeV$^2$, giving the
largest individual contribution and dominating the theoretical
systematic error. \\
%Final State Radiation
The presence of events with final state radiation in the data sample
affects the analyses in several ways:
\begin{itemize}
\item[$\bullet$] {Passing from $M^2_{\pi\pi}$ to $(M^0_{\pi\pi})^2$.}
The presence of final state radiation
shifts the observed value of $M^2_{\pi\pi}$ (evaluated from the momenta of the
two charged pion tracks in the events) away from the value of the
invariant mass squared of the virtual photon
produced in the collision of the electron and the
positron, $(M^0_{\pi\pi})^2$.
The transition from $M^2_{\pi\pi}$ to $(M^0_{\pi\pi})^2$ is performed
using a modified version of the PHOKHARA Monte Carlo
generator, which allows to (approximately) determine whether a generated
photon comes from
the initial or the final
state~\cite{Czyz:PHOKHARA_omega}. Figure~\ref{fig:radret_kloe4} shows
the probability matrix relating $M^2_{\pi\pi}$ to
$(M^0_{\pi\pi})^2$ by giving the probability for an event in a bin of $M^2_{\pi\pi}$ to end up in a bin of $(M^0_{\pi\pi})^2$.
It can be seen that the shift is only in one
direction, $(M^0_{\pi\pi})^2 \ge M^2_{\pi\pi}$, so events with one
photon from initial state radiation and one photon from final state
radiation move to a higher value of $(M^0_{\pi\pi})^2$. The entries lining up
above $(M^0_{\pi\pi})^2 \simeq 1.03$ GeV$^2$ represent events with two
pions and only one photon, emitted in the final
state. Events of this type have $(M^0_{\pi\pi})^2= s$, there is no
hard photon from initial state radiation present. Since in the KLOE analyses,
the maximum value of $(M^0_{\pi\pi})^2$ for which the cross sections
are measured is $0.95$ GeV$^2$ and sufficiently smaller than $s\simeq M^2_\phi$ of the
DA$\mathrm{\Phi}$NE collider, these {\it leading-order} final state radiation
events need to be taken out in the analysis. By moving these events to
$(M^0_{\pi\pi})^2 = s$, the passage from $M^2_{\pi\pi}$ to
$(M^0_{\pi\pi})^2$ automatically performs this
task. Figure~\ref{fig:radret_kloe5} shows the fraction of events from
{\it leading-order} final state radiation contributing to the total
number of events, evaluated with the PHOKHARA event generator. Since
in the {\it small angle} analysis the angular regions for pions and
photons are separated, final state radiation, for which the photons
are emitted preferably along the direction of the pions, is suppressed to
less than 0.5\%. Using {\it large angle} acceptance cuts, the effect
is much bigger, especially above and below the $\varrho$-resonance,
where it can reach 20-30\%.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=55mm] {unshifting3d_new.eps}
%\vglue-0.5cm
\caption{Probability matrix relating the measured quantity
$M^2_{\pi\pi}$ to $(M^0_{\pi\pi})^2$. To produce this plot, a
private version of the
PHOKHARA Monte Carlo
generator was used~\cite{Czyz:PHOKHARA_omega}. The photon angle is
restricted to $\theta_\gamma <15^\circ$ ($\theta_\gamma >165^\circ$).}
\label{fig:radret_kloe4}
\end{center}
\end{figure}
The correction of the shift in $ M^2_{\pi\pi}$ depends on the implementation of final state
radiation in the Monte Carlo generator in terms of model dependence
and missing contributions. It also relies on the correct assignment of
photons coming from the initial or the final state; however, in case
of symmetrical cuts in $\theta_\gamma$, interference effects between
the two states vanish and the separation of initial and final state
amplitudes is feasible.
\item[$\bullet$] The acceptance in $\theta_\gamma$. Since the direction of the photons
emitted in the final state is peaked along the direction
of the pions, and the photons are emitted in the initial state along
the $e^+$/$e^-$ direction, the choice of the acceptance cuts affects
the amount of final state radiation in the
analyses. Using the {\it small angle} analysis cuts, a large part of
final state radiation is suppressed by the separation of the pion
and photon acceptance regions, and consequently needs to be
reintroduced using corrections obtained from Monte Carlo
simulations to arrive at a result which is inclusive with respect to
final state radiation (as needed in the dispersion integral for
$a_\mu^{\pi\pi}$). Even if in the {\it large angle} analysis the fraction
of events with final state radiation surviving the selection is larger, again the missing
part has to be added using Monte Carlo simulations. The acceptance
correction for the cut in $\theta_\gamma$ is evaluated for initial and
final state radiation using the PHOKHARA generator, and the
small differences found in the comparison of data and Monte Carlo
distributions contribute to the systematic uncertainty of the
measurement (see Table~\ref{tab:radret_kloe1} and~\cite{KLOE_Note221}).
\item[$\bullet$] The distributions of kinematical variables. Cuts on
the kinematical {\it trackmass} variable
%\footnote{Assuming the
% presence of one photon in the event
%and that the tracks belong to particles of the same mass, $M_{\rm trk}$ is computed from energy and momentum conservation:
%$\left(\sqrt{s}-\sqrt{|\mathbf{p_+}|^2 + M^2_{\rm trk}}-
%\sqrt{|\mathbf{p_-}|^2 + M^2_{\rm trk} }\right)^2-\left(\mathbf{p_+}
%+\mathbf{p_-}\right)^2 = 0$
%where $\mathbf{p_\pm}$ is the measured momentum of the positive (negative) particle,
%and only one of the four solutions is physical.}
$M_\mathrm{trk}$ (see Eq.~(\ref{eq:mtrkdef})),
introduced in the analyses to remove background from the process
$e^+e^- \to \phi \to \pi^+\pi^-\pi^0$, take out also a fraction of
the events with final state radiation, necessitating a correction to
obtain an inclusive result. Figure~\ref{fig:radret_kloe45} shows the
effect final state radiation has on the distribution of the
trackmass variable. The radiative tail of multi-photon events to the
right of the peak at the $\pi^\pm$ mass increases because the additional
radiation moves events from the peak to higher values in
$M_\mathrm{trk}$. The width of the peak at $M_{\pi^\pm}$ is due to the
detector resolution; the plot was produced using the PHOKHARA event
generator interfaced with the KLOE detector
simulation~\cite{Ambrosino:2004qx}. Between 150 and 200 MeV, an
$M_{\pi\pi}^2$-dependent cut is used in the event selection to
reject the $\pi^+\pi^-\pi^0$ events which have a value of
$M_\mathrm{trk} > M_{\pi^\pm}$. In this region, the cut also acts on
the signal events. Missing terms concerning final state radiation in
the Monte Carlo simulation or the non-validity of the pointlike-pion
approximation used in PHOKHARA may affect the shape of the radiative
tail in the trackmass variable. To overcome this, in the KLOE
analyses, small corrections are applied to the momenta and the
%%%TT
angles of the charged particles in the event in the simulation, and
good agreement in the shape of $M_\mathrm{trk}$ is obtained between
Monte Carlo simulation and data~\cite{KLOE_Note221}.
\begin{figure}
\begin{center}
\subfigure[]{\includegraphics[width=80mm]{sma_fsr_contrib2.eps}}
\hspace*{0.cm}
\subfigure[]{\includegraphics[width=80mm]{la_fsr_contrib2.eps}}
\caption{(a) Fraction of events with leading order final state radiation in the
{\it small angle} selection: $50^\circ<\theta_\pi<130^\circ$ and
$\theta_\gamma<15^\circ$ ($\theta_\gamma>165^\circ$). (b) Fraction of events with leading
order final state radiation in the
{\it large angle} selection: $50^\circ<\theta_\pi<130^\circ$ and
$50^\circ<\theta_\gamma<130^\circ$. The PHOKHARA generator was used to produce the plots.}
\label{fig:radret_kloe5}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=90mm]{isr_fsr_mtrk2.eps}
\caption{Modification of the distribution of the trackmass variable due
to the presence of final state radiation (dark grey triangles) compared to
the one
with initial state radiation only (light grey triangles). The arrows indicate
the region in which the $M_{\pi\pi}^2$-dependent cut is applied in the
analysis. The plot was created with the PHOKHARA generator interfaced
to the KLOE detector simulation~\cite{Ambrosino:2004qx}.}
\label{fig:radret_kloe45}
\end{center}
\end{figure}
\item[$\bullet$] The division by the radiator function $H(s,M^2_{\pi\pi})$. In
this case,
one assumes perfect
factorisation between the ISR and the FSR process. This has been
tested by performing the analysis in an inclusive and exclusive
approach with respect to final state
radiation.
The assumption was found to be valid within $0.2\%$~\cite{Aloisio:2004bu,KLOE_Note192}.
\end{itemize}
It has been argued that contributions from events with two hard photons in the final
state, which are not included in the PHOKHARA generator, may have an effect on the
analyses~\cite{Jegerlehner:2008zza}.\\
%f0+rhopi (off peak large angle)
The effect of the direct decay $\phi\to \pi^+\pi^-\gamma$ on the
radiative return analysis has been addressed already
in~\cite{Melnikov:2000gs}. Running at $\sqrt{s}\simeq1.02$ GeV, the
amplitude for the processes $\phi\to (f_0(980)+f_0(600))\gamma \to
\pi^+\pi^-\gamma$ interferes with the amplitude for the final state
radiation process. Due to the yet unclear nature of the scalar states
$f_0(980)$ and $f_0(600)$, the effect on the $\pi^+\pi^-\gamma(\gamma)$
cross section depends on the model used to describe the scalar
mesons. The possibility to simulate $\phi$ decays together with the
processes for initial and final state radiation has been implemented
in the PHOKHARA event generator in~\cite{Czyz:2004nq}, using two
characteristic models for the $\phi$ decays: the ``no structure''
model of~\cite{Bramon:1992ew} and the $K^+K^-$ loop model
of~\cite{LucioMartinez:1994yu}. A refined version of the $K^+K^-$ loop
model~\cite{Achasov:2005hm} and the double vector resonance
$\phi\to\pi^\pm\varrho^\mp(\to \pi^\mp\gamma)$ have been included as
described in~\cite{Pancheri:2007xt}. Using parameter values for the
different $\phi$ decays found in the analysis of the neutral channel
$\phi\to (f_0(980)+f_0(600))\gamma \to
\pi^0\pi^0\gamma$~\cite{Achasov:2005hm,Ambrosino:2006hb}, one can
estimate the effect on the different analyses. While in the {\it small
angle} analysis there is no significant effect due to the choice of
the acceptance cuts, in the {\it large angle} selection the effect is
of the order of several percent and can reach up to 20\% in the
vicinity of the $f_0(980)$, see
Fig.~\ref{fig:radret_kloe6} (a). While this allows to study the
different models for the direct decays of $\phi$-mesons (see also
Section~\ref{radret:KLOEasymFSR}), it prevents a precise measurement of
$\sigma_{\pi\pi}$ until the model and the parameters are understood
with better accuracy. An obvious way out is to use
data taken at a value of $\sqrt{s}$ outside the narrow peak of the
$\phi$ resonance ($\Gamma_\phi = 4.26\pm0.04$
MeV~\cite{Amsler:2008zzb}).
In 2006, the KLOE experiment has taken $\sim$ 250
pb$^{-1}$ of data at $\sqrt{s}=1$ GeV, 20 MeV below $M_\phi$. As can
be seen in Fig.~\ref{fig:radret_kloe6} (b), this reduces the effect
due to contributions from $f_0\gamma$ and $\varrho\pi$ decays of the
$\phi$-meson to be within $\pm 1\%$.\\
\begin{figure}[tb]
\centering
\subfigure[]{\includegraphics[width=80mm]{f0_kloe_onpeak.eps}}
%\hspace*{1.cm}
\subfigure[]{\includegraphics[width=80mm]{f0_kloe_offpeak.eps}}
%\vglue-0.5cm
\caption{(a): ${{\rm d}\sigma_{\pi\pi\gamma}^{({\rm ISR}+{\rm FSR}+f_0+\varrho\pi)}}/{{\rm d}\sigma_{\pi\pi\gamma}^{({\rm ISR}+{\rm FSR})}}$ for $\sqrt{s}=1.019$ GeV. (b):
${{\rm d}\sigma_{\pi\pi\gamma}^{({\rm ISR}+{\rm
FSR}+f_0+\varrho\pi})}/{{\rm d}\sigma_{\pi\pi\gamma}^{({\rm ISR}+{\rm FSR})}}$
for $\sqrt{s}=1$ GeV. Both plots were produced with the PHOKHARA v6.1 event
generator using {\it large
angle} acceptance regions for pions and photons, with model
parameters for the $f_0$ and $\varrho\pi$ contributions from~\cite{Achasov:2005hm,Ambrosino:2006hb}.}
\label{fig:radret_kloe6}
\end{figure}
%Outlook (method for ratio)
{\noindent \it Normalisation with muon events\\}
An alternative method to extract the pion form factor is to normalise
the differential cross section
${{\rm d}\sigma_{\pi\pi\gamma(\gamma)}}/{{\rm d}M^2_{\pi\pi}}$ directly to
the process $e^+e^- \to \mu^+\mu^-\gamma(\gamma)$, ${{\rm
d}\sigma_{\mu\mu\gamma(\gamma)}}/{{\rm d}M^2_{\mu\mu}}$, in
each bin of $\Delta M^2_{\pi\pi}= \Delta M^2_{\mu\mu}$. Radiative
corrections like the effect of vacuum polarisation, the radiator
function and also the integrated luminosity $\int L{\rm d}t$ cancel out in
the ratio of pions over muons, and only the effects from final state
radiation (which is different for pions and muons) need to be taken
into account consistently. An approach currently under way at KLOE
uses the following equation to obtain $|F_{2\pi}|^2$:
\begin{equation}
|F_{2\pi}(s')|^2\cdot(1+\eta(s'))=\frac{4(1+2m_\mu^2/s')\beta_\mu}{\beta^3_\pi}
\cdot\frac{(\frac{{\rm d}\sigma_{\pi\pi\gamma(\gamma)}}
{{\rm d}M^2_{\pi\pi}})^{{\rm ISR}+{\rm FSR}}}{(\frac{{\rm d}\sigma_{\mu\mu\gamma(\gamma)}}{{\rm d}M^2_{\mu\mu}})^{{\rm ISR}}}
\label{}
\end{equation}
In this formula, the measured differential cross section
${{\rm d}\sigma_{\pi\pi\gamma(\gamma)}}/{{\rm d}M^2_{\pi\pi}}$ should be
inclusive with respect to pionic final state radiation, while the
measured cross section
${{\rm d}\sigma_{\mu\mu\gamma(\gamma)}}/{{\rm d}M^2_{\mu\mu}}$ should be
exclusive for muonic final state radiation.
$s'=M^2_{\pi\pi}=M^2_{\mu\mu}$ is the squared invariant mass of the
di-pion or the di-muon system after the respective corrections
for final state radiation. Using this approach, one gets on the
left-hand side the pion form factor times the factor $(1+\eta(s'))$,
which describes the effect of the pionic final state radiation. This
{\it bare} form factor is the quantity needed in the dispersion
integral for the $\pi\pi$-contribution to $a_\mu^{\rm had}$. While
the measurement of ${{\rm d}\sigma_{\pi\pi\gamma(\gamma)}}/{{\rm d}M^2_{\pi\pi}}$
and its corrections for pionic final state radiation
are very similar to the one using the normalisation with
Bhabha events already performed at KLOE, the corrections needed to
subtract the muonic final state radiation from the
${{\rm d}\sigma_{\mu\mu\gamma(\gamma)}}/{{\rm d}M^2_{\mu\mu}}$ cross section are
pure QED and can be obtained from the PHOKHARA generator, which
includes final state radiation for muon pair production at
next-to-leading order~\cite{Czyz:2004rj}. Due to the fact that
the KLOE detector does
not provide particle IDs, pions and muons have to be separated and
identified using kinematical variables (e.g. the aforementioned
trackmass variable)~\cite{Muller:2006bk}. The analysis is in progress and a systematic precision similar to the one obtained in the absolute measurement
is expected.