%Title: The radiative return at phi- and B-factories: a Status Report
%Author: Johann H.Kuhn
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\title{The radiative return at $\boldsymbol{\Phi}$- and $\boldsymbol{B}$-factories: a Status Report
\thanks{Work supported in part by BMBF under grant number 05HT9VKB0.}}

\author{
Johann H. K\"uhn
}

\institute{
Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany.}

\date{}

\abstract{ The measurement of the pion form factor and, more
generally, of the cross section for electron--positron annihilation
into hadrons through the radiative return has become an important
task for high luminosity colliders such as the $\Phi$- or
$B$-meson factories. For detailed understanding and analysis
of this reaction, the construction of a Monte Carlo program, 
PHOKHARA, has been undertaken. Version 2.0 was based on a 
next-to-leading order (NLO) treatment of the corrections from initial-state 
radiation (ISR) and leading order final-state radiation (FSR). 
The most recent extension of
PHOKHARA (version 3.0), which incorporates NLO
corrections to FSR and the impact of combined 
ISR and FSR on selected distributions, is presented.}


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Electron--positron annihilation into hadrons is one of the basic
reactions of particle physics, crucial for the understanding
of hadronic interactions. At high energies, around the $Z$ resonance,
the measurement of the inclusive cross section and its interpretation
within perturbative QCD~\cite{CKK} give rise to one of the most
precise and theoretically founded  determinations of the strong coupling
constant $\alpha_s$~\cite{EWG}. Also, measurements in the intermediate 
energy region, between 3 GeV and 11 GeV can be used to determine $\alpha_s$
and at the same time give rise to precise measurements of charm 
and bottom quark masses~\cite{KS}. The low energy region is crucial
for predictions of the hadronic contributions to $a_\mu$, the anomalous 
magnetic moment of the muon, and to the running of the electromagnetic
coupling from its value at low energy up to $M_Z$ 
(for reviews see 
e.g.~\cite{a_mu1,Jegetc.,Melnikov:2001uw,Davier:2002dy,HMNT02}).
Last, but not least, the investigation of the exclusive final states
at large momenta allows for tests of our theoretical 
understanding of form factors within the framework of perturbative QCD.

The recent advent of 
$\Phi$- and $B$-meson factories with their enormous luminosities
allows us to exploit the radiative return to explore the whole energy
region from threshold up to the nominal energy of the collider. 
Photon radiation from the initial state reduces the cross section by a factor
${\cal O}(\alpha/\pi)$. However, this is easily compensated by the 
enormous luminosity of `factories' and the advantage of performing
the measurement over a wide range of energies in one homogeneous data
sample~\cite{Binner:1999bt} (for an early proposal along these lines, 
see~\cite{Zerwas}). In principle the reaction 
$e^+e^- \to \gamma + {\mathrm {hadrons}}$ receives contributions from 
both initial- and final-state radiation. Only the former is of interest
for the radiative return; the latter has to be eliminated by
suitably chosen cuts. The proper analysis thus requires the construction
of Monte Carlo event generators.
The event generator EVA was based on a leading order treatment of ISR and FSR, 
supplemented by an approximate inclusion of additional collinear radiation 
based on structure functions and included two-pion~\cite{Binner:1999bt} and 
four-pion final states~\cite{Czyz:2000wh}.
Subsequently the event generator PHOKHARA was developed; it is based
on a complete next-to-leading order (NLO) treatment of 
ISR~\cite{Rodrigo:2001jr,Rodrigo:2001kf,Kuhn:2002xg,Czyz:2002np}.
In its version 2.0 it included ISR at NLO and FSR at leading order (LO)
for $\pi^+ \pi^-$ and $\mu^+ \mu^-$ final states and four-pion
final states without FSR.

Recent preliminary experimental results indeed demonstrate the power
of the method and seem to indicate that a precision of one per cent
or better is within 
reach~\cite{CDKMV2000}.
In view of this progress a further improvement of theoretical
understanding and MC-programs is necessary.  
The most recent version of PHOKHARA, version 3.0~\cite{Czyz:last}, 
allows for the `simultaneous' emission of one photon from 
the initial and one photon from the final state, requiring only one
of them to be hard. This includes in particular the radiative return
to $\pi^+ \pi^- (\gamma)$ and thus allows the measurement of the (one-photon) 
inclusive $\pi^+ \pi^-$ cross section. 

The issue of photon radiation from the final states is closely 
connected to the question of $\pi^+ \pi^- (\gamma)$ 
contributions to $a_{\mu}$ (for related discussions, 
see e.g.~\cite{Melnikov:2001uw,Gluza:rad}).
Soft photon emission is clearly described by
the point-like pion model. Hard photon emission, with 
$E_{\gamma} \geq {\cal O}$(100 MeV), however, might be sensitive
to unknown hadronic physics. Therefore the size of
virtual, soft and hard corrections has to be studied 
separately. It will be argued that contributions from  
the hard region, above 100 MeV, are small with respect 
to the present experimental and theory-induced uncertainty.

Let us start with the process in leading order:
\begin{equation}
e^+ \ e^- \to \pi^+(p^+) \  \pi^-(p^-) \ \gamma~. 
\end{equation}
 The amplitude  of interest  describes  radiation  from  
the  initial state  (ISR). It is  proportional  to 
the  pion  form factor, evaluated  at
$Q^2=(p^+  + p^-)^2$. However,  radiation from  the charged  pions (FSR)
evidently leads to the same final state and must be suppressed
and controlled with sufficient precision.  A variety of methods 
are  described in more detail in 
~\cite{Binner:1999bt,Rodrigo:2001kf,Czyz:2002np,Czyz:last}.

The fully differential cross section describing photon emission can 
be split into three pieces
\begin{equation}
d\sigma = d\sigma_\mathrm{ISR} + d\sigma_\mathrm{FSR} + d\sigma_\mathrm{INT}~,
\end{equation}
which originate  from the  squared ISR and  FSR amplitudes respectively,
plus the  interference term. They  depend on two Dalitz  variables, which
characterize the energies  of $\pi^+$ and $\pi^-$ and  the photon, and on
the   three   Euler   angles   describing   the   orientation   of   the
$\pi^+\pi^-\gamma$ production plane in  the centre-of-mass system (cms).

The  interference  term, $d\sigma_\mathrm{INT}$,  is  antisymmetric under
the exchange of $\pi^+$ and $\pi^-$ or  $e^+$ and $e^-$. This allows us to test
the model for FSR. 

The charge-symmetric pieces can be separated on the basis of their 
markedly different angular distribution, either by fitting the two
components~\cite{Czyz:last} or by employing suitable cuts.

Let us summarise the main points of the `leading order' 
discussion~\cite{Czyz:last}:
\begin{itemize}
\item[1.]{The cross sections for photon emission from 
   ISR and FSR can be disentangled as a consequence of the marked 
   difference in angular distributions between the two processes. 
   This observation is completely general
   and does not rely on any model like sQED for FSR. This
   allows us to measure the cross section for
   $\sigma(e^+e^-\to\gamma^*\to\pi^+\pi^-\gamma)$ directly for fixed $s$ as
   a function of $E_\gamma$, an important ingredient in the analysis of
   hadronic contributions to $a_\mu$. 
   Alternatively we can employ suitable angular 
   cuts to separate the two components.}

\item[2.]{Various charge-asymmetric distributions can be used for independent
   tests of the FSR model amplitude, with the forward--backward
   asymmetry as simplest example. Typically the charge asymmetry is
   large in the region where ISR  and FSR are comparable in size and
   small when the separation is clean and simple from general
   considerations. A typical case for this second possibility is the
   radiative return at $\sqrt{s} = 10.52$~GeV, at the $B$-meson
   factories, where the $\pi^+\pi^-\gamma$ final state is completely
   dominated by ISR.}
\end{itemize}




At present hadronic contributions to
\(a_\mu\) are known to roughly half to one percent. Photonic corrections to 
final states with hadrons thus start to become relevant. 
Qualitatively the order of magnitude of this effect can be
estimated either by using the quark model with $m_u\approx m_d\approx
m_s \approx 180$~MeV (adopted to describe the lowest order 
contribution~\cite{piv}), or with  $m_u\approx m_d\approx m_s \approx 66$~MeV
(adopted to describe the lowest order contribution to $\alpha(M_Z)$) 
or by using $\pi^+\pi^-$ as dominant intermediate hadronic state plus 
photons coupled according to sQED. The three estimates 
\bea
\delta a_\mu({\rm quark},\gamma,m_q=180~{\rm MeV})
 = \ 1.880 \times 10^{-10}~, \nonumber \\
\delta a_\mu({\rm quark},\gamma,m_q=66~{\rm MeV})
 = \ 8.577 \times 10^{-10}~, \nonumber \\
 \delta a_\mu(\pi^+\pi^-,\gamma) = \ 4.309 \times 10^{-10}~,
\eea
are comparable in magnitude and may be relevant at the present
level of precision. 

In \cite{Czyz:last} it has been shown that the  
measurement of $R(\pi^+\pi^-(\gamma))$ is
indeed possible with the technique of the radiative return.
The following strategy for the exclusive measurement 
was proposed:

The contribution from $R^{\mathrm{V+R}}(\pi^+\pi^-(\gamma), E^{\mathrm{cut}})$
as a function of $E^\mathrm{cut}$ can be measured by analysing final states 
where hard-photon radiation is suppressed by suitable chosen cuts on energy 
and collinearity of the pions. The $E^\mathrm{cut}$ dependence of 
$R^{\mathrm{H}}(\pi^+\pi^-\gamma,E^{\mathrm{cut}})$ can be measured
by collecting $\pi^+\pi^-\gamma$ final states (and correcting of
course for ISR). 
Alternatively we may correct for (unmeasured) hard photon events by 
employing a model like sQED, which has to be checked
experimentally --- at least for a few selected kinematic
configurations.

Using sQED as a model, one finds~\cite{Czyz:last} that contributions from 
hard photon radiation, with a cut at $100$~MeV, are still small relative 
to the present uncertainty $\delta a_\mu^\mathrm{had,LO} = 7 \times 10^{-10}$
\cite{Davier:2002dy,HMNT02}. However, cuts on the photon
energy around 50~MeV or below might well lead to important shifts 
in \(a_\mu\). Of course we have assumed that
hard radiation is not grossly underestimated by sQED, an assumption to
be tested by experiment.

Intuitively it should also be possible to exploit the radiative return
for the extraction of FSR for arbitrary invariant mass $\sqrt{s'}$
of the $\pi^+\pi^-\gamma$ system through the reaction
\begin{equation}
e^+e^-\to \gamma\gamma^*(\to \pi^+\pi^-\gamma) \ .
\label{eq:twostep}
\end{equation}
The implementation of this two-step process has been achieved in
of the newest version of PHO\-KHA\-RA (PHO\-KHA\-RA 3.0).


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The impact of the new corrections on a 
few selected distributions has been discussed in~\cite{Czyz:last}.
Before entering the discussion, let us recall the meaning of various 
abbreviations, which will be used in the following.
ISRNLO corresponds to the ISR cross section calculated at NLO without
any FSR. IFSLO includes in addition FSR at LO. Finally, IFSNLO stands for ISR 
and FSR at NLO as implemented in the new version of PHOKHARA (version 3.0).

For a cms energy of 1.02 GeV, relevant for the 
KLOE experiment, the IFSNLO correction to the cross section
is relatively big
at low \(Q^2\), if no cuts are applied (Fig. \ref{fig:ifs+lo_cut}a).  
Below the \(\rho\) resonance, they grow from zero at the resonance to 
10\% of the IFSLO cross section near the production threshold, 
while they remain small above the \(\rho\) resonance. This is due to the fact
that the ISR leading to the \(\rho\) meson is strongly enhanced.
Subsequently the \(\rho\) decays into \(\pi^+\pi^-\gamma\), with a large 
contribution from the region where \(Q^2\), the
invariant mass of the \(\pi^+\pi^-\), is low. Of course
this is smeared by the width of the \(\rho\), but
the above discussion remains valid and the contribution from the 
newly implemented diagrams, through the reaction 
\(e^+e^-\to\gamma\rho(\to \gamma\pi^+\pi^-)\), is sizeable.

The effect of several `standard cuts' at \(\sqrt{s}=1.02\)~GeV
is shown in Figs.~\ref{fig:ifs+lo_cut}b and~\ref{fig:ifs+lo_cut}c. The 
sensitivity of the newly implemented contributions to these cuts can be 
exploited to test, control or experimentally eliminate FSR at NLO.
It is relatively easy to find cuts that lower the correction to 
a level of 2\%--3\%. In fact all the cuts which were used to eliminate
FSR at LO are also effective here. The standard KLOE cut on the
track mass~\cite{Aloisio:03} reduces FSR further, to less than 1\% for most
of the \(Q^2\) range, apart of the high values of  \(Q^2\), where the
corrections remain at the level of 2\%-3\% (Fig.~\ref{fig:ifs+lo_cut}c,
lower curve). The behaviour of the additional contribution at 10.52 GeV is
qualitatively similar. 


%14
\begin{figure}[ht]
\begin{center}
\epsfig{file=me_lo_nlo-1.ps,height=5.9cm,width=8.5cm}
\epsfig{file=me_c_ifsnlo_isrnlo.ps,height=5.9cm,width=8.5cm}
\epsfig{file=v_trkmass_50.ps,height=5.9cm,width=8.5cm}
\caption{Comparison of the \(Q^2\) differential cross sections
for \(\sqrt{s} = 1.02\)~ GeV: IFSNLO contains the complete NLO contribution, 
while IFSLO has FSR only at LO. The pion and photon(s) angles are not
restricted in (a), and restricted in (b). In (c) the cuts are imposed on
the missing momentum direction and the track mass (see text for description).}
\label{fig:ifs+lo_cut}
\end{center}
\end{figure}


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\section*{Acknowledgements}

This contribution is based on work performed in collaboration with
H.~Czy{\.z}, A.~Grzeli{\'n}ska and G.~Rodrigo.
I would like to thank A.~Denig, W.~Kluge, and S.~M\"uller  for discussion of 
experimental aspects.

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