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\begin{document}

\preprint{DESY 03-023}
%~~~~~\\
%hep-ph/0302xxx }

\title{A Model-Independent Treatment of FSR in Low Energy 
$\sigma_{\rm had}$ Measurements\footnote{Work supported in part by TMR, EC-Contract No.~HPRN-CT-2002-00311 (EURIDICE),EC-Contract No.~HPRN-CT-2000-00149 (Physics at Colliders), and the TARI project HPRI-CT-1999-0008.}}

\author{A.~Hoefer}
\affiliation{Institute of Nuclear Physics, Radzikowskiego 152, 
31-342 Krakow, Poland}
\author{J.~Gluza}
\affiliation{Institute of Physics, University of
Silesia, Uniwersytecka 4, 40-007 Katowice, Poland}
\affiliation{DESY Zeuthen, Platanenallee 6, D-15738 Zeuthen, Germany}
\author{S.~Jadach}
\affiliation{Institute of Nuclear Physics, Radzikowskiego 152, 
31-342 Krakow, Poland}
\author{F.~Jegerlehner}
\affiliation{DESY Zeuthen, Platanenallee 6, D-15738 Zeuthen, Germany}

%\date{\today}

%
%\preprint{DESY ...}
%

\begin{abstract}
A new strategy for extracting the total cross section
$\sigma(e^+e^-\to h^+h^-)$ for the production of a charged scalar
meson pair $h^+h^-$ from experimental $e^+e^-$ collision data is
proposed.  It is shown that by exploiting basic symmetries, like gauge
invariance and parity conservation, it is possible to circumvent the
need of ad hoc modeling to disentangle photonic radiation by the final
state hadrons from photons radiated off the initial $e^+e^-$ pair.
The presented essentially model--independent approach is expected to
be particularly interesting for the $\pi^+\pi^-$ production channel
since it provides a possibility to significantly reduce the model
uncertainty of the extraction of the electromagnetic pion form factor
$F_{\pi}$ which is mainly related to our deficient knowledge of
the hard photon radiation mechanism from the hadronic final
state. Especially, for $\pi^+\pi^-$ cross--section measurements via
radiative return experiments, like the ones performed currently at the
$\Phi$--factory DA$\Phi$NE, an appreciable reduction of the model
error could be achieved since here one is facing the problem of a
leading order final state radiation (FSR) background. Applying the
proposed method could be one step into the direction of a more
accurate theoretical prediction of the muon anomalous magnetic moment
$a_{\mu}$ for which the precise knowledge of $|F_{\pi}|^2$ is crucial.
Furthermore, the validity and limitations of FSR models including
scalar QED can hereby be tested.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pacs{13.40.Gp,13.66.Bc,Jn}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Precise measurements of the hadronic cross section $\sigma_{\rm had}$
at low energies currently play a key role in precision tests of the
Standard Model (SM). In fact, due to the breakdown of perturbative QCD
at low energies, the only way to obtain the hadronic contribution to
the running fine structure constant $\alpha(s)$ and to the muon
anomalous magnetic moment $a_\mu$ -- in an essentially model--independent
way -- is to utilize experimental data for $\sigma_{\rm had}$. The latter
is related to the hadronic contribution of the photon self--energy via
a suitable dispersion
integral~\cite{Cabibbo:1961sz,Eidelman:1995ny}. At present a reliable
comparison of the high precision $a_\mu$ data from the Brookhaven E821
experiment~\cite{Brown:2001mg,Bennett:2002jb} with the corresponding
theoretically predicted $a_\mu$ value is still limited by our
insufficient knowledge of $\sigma_{\rm had}$ at low
energies~\cite{Gourdin:1969dm,Kinoshita:1984xp,Kinoshita:1985it,Alemany:1998tn,Davier:1998si,Jegerlehner:2001wq,Narison:2001jt,DeTroconiz:2001wt,Cvetic:2001pg,Yndurain:2001qw,Melnikov:2001uw,Knecht:2001qf,Knecht:2001qg,Hayakawa:2001bb,Blokland:2001pb,Bijnens:2001cq,Kuhn:2003pu}.
Thereby, the $\pi^+\pi^-$ production cross section
$\sigma_{\pi\pi}=\sigma(e^+e^-\to \pi^+\pi^-)$ plays the most
important role. It accounts for more than 70 \% of $a_\mu^{\rm had}$
and therefore must be measured with the highest precision. Its
uncertainty should not exceed about $0.5$ \%. As a matter of fact,
presently, there is an unresolved discrepancy between
$\sigma_{\pi\pi}$ data from $e^+e^-$ experiments and corresponding
data obtained in an indirect way via an isospin rotation from
$\tau$--decay measurements (assuming approximative isospin
symmetry)~\cite{Eidelman:1995qm,Alemany:1998tn}.  While the
theoretical prediction of $a_\mu$ which is based on $e^+e^-$ data
deviates from the corresponding experimental value as much as 3
$\sigma$ the $\tau$--based value only yields an insignificant $0.9$
$\sigma$ discrepancy (for a detailed discussions see
\cite{Davier:2002dy,Leutwyler:2002hm}). 
With the objective to resolve this discrepancy and having in mind the
shortly expected experimental precision for $a_\mu$ of $0.35$ ppm, an
accurate analysis of all error sources related to $\sigma_{\rm had}$
measurements and additional cross checks among the different
experiments appear to be mandatory. In particular a comparison of more
precise $\pi^+\pi^-$ data from radiative return experiments at
DA$\Phi$NE~\cite{Maiani:1995ve,Cataldi:1999dc,Valeriani:2002yk,Denig:2002ps}
or BABAR~\cite{Solodov:2002xu} could provide for important tests of
the scan data from VEPP--2M and the forthcoming VEPP--2000
experiment~\cite{Akhmetshin:2001ig,Deile:2002vr,Eidelman:2002kg}.

As has been discussed extensively in the past, the 
extraction of $\sigma_{\rm had}$ from experimental data 
with the desired precision requires 
an accurate treatment of QED corrections~\cite{Arbuzov:1997je,Arbuzov:1998te,Binner:1999bt,Czyz:2000wh,Rodrigo:2001kf,Hoefer:2001mx,Czyz:2002np,Khoze:2000fs,Khoze:2002ix,Gluza:2002ui}.  
Of the latter the dominant contribution is coming from 
initial state radiation (ISR) emitted 
by the colliding $e^+e^-$ pair.  
ISR can be calculated within perturbation theory without any 
auxiliary model assumptions. It is known 
up to $O(\alpha^2)$ including leading higher order contributions. 
Perturbative results for initial state fermion 
pair production exist as well. 
In contrast, the treatment of final state radiation (FSR)  
from the composite hadronic states   
so far required the use of theoretically not 
well motivated models like scalar QED (sQED), 
without having any stringent test verifying their validity.
Concerning the precise evaluation of $a_\mu^{\rm had}$ 
in fact the desired quantity to be determined from the 
true hadronic observables    
is the FSR--inclusive hadronic cross section 
$\sigma^{(\gamma)}_{\rm had}$ one obtains by subtracting 
the ISR corrections from the experimental data 
while keeping FSR~\cite{Hoefer:2001mx}. 
Including FSR to the hadronic cross section
yields via dispersion integration
the corresponding QED--corrected hadronic photon self--energy contributions to 
$\alpha(s)$ and $a_\mu$. 
Such QED corrections are numerically relevant as they are similar 
in size to the shortly expected experimental error 
related to the $a_\mu$ measurement.    

Since on an event by event basis it is not possible to distinguish
ISR-- from FSR--photons, in a recent paper an FSR--inclusive
measurement was proposed, with the aim to obtain
$\sigma^{(\gamma)}_{\rm had}$ directly from the experimental scan
data~\cite{Gluza:2002ui}. As was shown, for such a scenario the
handicap of having to rely on unsound FSR--models could in fact hereby
be avoided. Furthermore, it was pointed out that a similar strategy
could {\em not} be followed at radiative return experiments, like the
one performed at DA$\Phi$NE with the KLOE detector, which are running
at a fixed $e^+e^-$ center of mass energy $\sqrt{s}$ and where the
hadronic cross section $\sigma_{\rm had}(s')$ is obtained from the
spectral function $d\sigma_{\rm had}/ds'$ resulting from the radiation
of initial state photons. $\sqrt{s'}$ is the invariant mass of the
hadronic final state after emission of a hard photon. Given the fact
that for the radiative return scenario the observed spectral function
includes at least one real (ISR or FSR) photon it was shown that, in
contrast to scan measurements, FSR here plays the role of a disturbing
background, being moreover of leading order. This FSR background must
be subtracted.  Here FSR is under control only if either its
contribution can be neglected in respect to ISR --- being e.g.~true
for the $\rho$--peak region in case of $\pi^+\pi^-$--measurements at
DA$\Phi$NE ---, the emitted real photons are soft ($s' \simeq s$) ---
in which case FSR is known from theory --- or FSR can be suppressed
effectively by adequate angular cuts.  Concerning $\pi^+\pi^-$
measurements at DA$\Phi$NE energies ($s=M_\phi^2$) the latter is only
achievable in regions above $\sqrt{s'} \simeq 0.5$
GeV. FSR--suppressing cuts in addition lead to an unwanted reduction
of statistics, especially in the $s'$--region below the
$\rho$--resonance. Altogether, concerning the extraction of
$\sigma_{\pi\pi}$ at the $\Phi$--factory, we are presently facing an
insurmountable theoretical uncertainty being related to ad hoc FSR
modeling which in regions below $\sqrt{s'} \simeq 0.5$ GeV leads to a
large model error of several per cent.

Attempting to obtain reliable $\sigma_{\pi\pi}$ data from
DA$\Phi$NE over the whole available energy region ($2 m_\pi \leq
\sqrt{s'} \leq M_\phi^2$) --- last but not least with the aim to
cross-check current and future $e^+e^-$--scan-- and $\tau$--data ---
the obvious question arises whether there is no alternative,
theoretically well controlled, way of dealing with the $O(1)$ FSR
background. In particular it would be interesting to test to what
precision the true FSR--mechanism can in fact be approximated by
sQED. The latter question, although less urgent than for the radiative
return scenario, also appears to be vital concerning $\pi^+\pi^-$
measurements at scan experiments like CMD-2. Here, in the past, via a
dedicated event selection procedure, the total cross section that was
undressed from real FSR photons was extracted from the data. For
obtaining the desired FSR--inclusive $\pi^+\pi^-$--cross section
$\sigma^{(\gamma)}_{\pi\pi}$ subsequently real FSR as modeled by sQED
was added ``by hand'' resulting in a model error of sub-leading order.

Having sketched the major difficulties and aims related  
to FSR in low energy $\sigma_{\rm had}$ measurements,
in the remaining part of this paper we will develop a  
new, model--independent strategy for treating the leading order 
FSR contribution to the production of a charged 
meson pair $h^+h^-$.  
For this purpose some results which are derived in the Appendix, 
concerning the general structure of the hadronic tensor 
related to the $\gamma^\ast\gamma^\ast h^+h^-$ Feynman diagram,  
are utilized. 
It turns out that in fact a  model--independent way of 
treating FSR in $h^+h^-$ production is   
feasible by taking basic symmetries like Lorentz covariance, 
gauge invariance and parity conservation into account. 
The obtained results are expected to be especially 
important for the $\pi^+\pi^-$ production channel. 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Model--Independent FSR Measurement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The leading order bremsstrahlung amplitude ${\cal M}^{(1)}$ corresponding
to the process $e^-(p_1)\;e^+(p_2)\to\gamma^\ast \to 
h^-(k_1)\;h^+(k_2)\;\gamma(k)$, 
$h^+h^-$ being a meson pair with charge $\pm e$ 
and $\gamma$ a single real (ISR or FSR) 
photon, can be written as the sum of an $O(\alpha)$ 
ISR-- and FSR--amplitude:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba
{\cal M}^{(1)} &=& 
{\cal M}^{(1)}_{\rm ISR} + {\cal M}^{(1)}_{\rm FSR} \;.
\ea 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
${\cal M}^{(1)}_{\rm ISR}$ and 
${\cal M}^{(1)}_{\rm FSR}$ can here be expressed as a contraction of 
a leptonic tensor ($E_{\mu\nu}^{(1)}$ or $E_{\mu}^{(0)}$) 
containing the initial $e^+e^-$ part with a 
hadronic tensor ($F^{\mu}_{(0)}$ or $F^{\mu\nu}_{(1)}$) 
containing the $h^+h^-$ part of the amplitude and a covariant polarization vector 
$\epsilon^{\ast\mu}(k,\lambda)$ 
corresponding to the emitted real photon 
($\lambda$ being the photon polarization), respectively:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba
{\cal M}^{(1)}_{\rm ISR} &=& E_{\mu\nu}^{(1)}(p_1,p_2,k)
\;F^{\mu}_{(0)}(k_1,k_2)\;\epsilon^{\ast\nu}(k,\lambda) 
\label{MISR} \\
{\cal M}^{(1)}_{\rm FSR} &=& E_{\mu}^{(0)}(p_1,p_2)\;
\epsilon^\ast_{\nu}(k,\lambda)\;
F^{\mu\nu}_{(1)}(k_1,k_2,k),
\label{MFSR}
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where [$s=(p_1+p_2)^2$, $s'=(k_1+k_2)^2$, $l^\mu = k_1^\mu-k_2^\mu$] 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba
E_{\mu}^{(0)} &=&  
-\frac{e}{s}\;\bar{v}_{s_2}(p_2)\gamma_{\mu}u_{s_1}(p_1),\nn\\
E^{(1)}_{\mu\nu} &=& \frac{e^2}{s'}\;\left[
\bar{v}_{s_2}(p_2)\gamma_\mu u_{s_1}(p_1)
\left(\frac{p_{1\nu}}{p_1k}-\frac{p_{2\nu}}{p_2k}\right) \right.\nn\\
&& \quad\qquad\quad
-\frac{\bar{v}_{s_2}(p_2)\gamma_\mu \sla{k} \gamma_\nu u_{s_1}(p_1)}{2p_1k}
\nn\\
&& \quad\qquad\quad \left.
+\frac{\bar{v}_{s_2}(p_2)\gamma_\nu \sla{k} \gamma_\mu u_{s_1}(p_1)}{2p_2k}
\right],\nn\\
F_{(0)}^\mu &=& - ie\;F_h(s')\;l^\mu , \nn\\
F^{\mu\nu}_{(1)} &=& F_1(s',k_1k)\;T_1^{\mu\nu} + 
F_2(s',k_1k)\;T_2^{\mu\nu},
\label{ISRandFSRtensors}
\ea
%%%%%%%
and 
%%%%%%%
\ba
T_1^{\mu\nu} &=& -ie^2\;\left[l^\mu-\frac{ql}{kq}\;k^{\mu}\right] \nn\\
&& \times \;
\left[\frac{k_1^\nu}{k_1k}-\frac{k_2^\nu}{k_2k}+\frac{1}{2}
\left(\frac{1}{k_1k}-\frac{1}{k_2k}\right)k^\nu\right] ,
\nn\\
T_2^{\mu\nu} &=& 2ie^2 \;\left[g^{\mu\nu}-\frac{k^\mu q^\nu}{kq}\right] .
\label{tensorT1T2}
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The spinor indices $s_{1,2}$ refer to the $e^{\pm}$ spins. 
In (\ref{ISRandFSRtensors}) the dynamics of 
the scattering process under consideration has been parameterized 
by the lowest order meson form factor $F_h(s')$ (which is the 
pion form factor $F_\pi(s')$ in case of $\pi^+\pi^-$ production) 
and the two FSR--form factors $F_1(s',k_1k)$ and $F_2(s',k_1k)$ 
(see Fig.~\ref{leading}). 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\mbox{\epsfxsize 8cm  \epsffile{leadingISRFSR.eps}}
%\vspace{-1.7cm}
\caption{Feynman diagrams corresponding to the process 
$e^-(p_1)\;e^+(p_2) \to \gamma^\ast \to h^-(k_1) \;h^+(k_2) \;\gamma(k)$.
Here $s'$ and $s$ are the squared invariant mass of the virtual photon 
for the ISR-- and the FSR--diagrams respectively. 
The ISR amplitude ${\cal M}^{(1)}_{\rm ISR}$ also contains the 
contribution of a second diagram with a photon emitted from the initial 
$e^+$.  The $\gamma^\ast h^+h^-$ vertex is parameterized by the meson 
form factor $F_h(s')$ and the $\gamma^\ast\gamma\, h^+h^-$ vertex by the 
FSR form factors $F_1(s',k_1k)$ and $F_2(s',k_1k)$.} 
\label{leading}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As we prove in the Appendix such a parameterization is in fact 
model--independent in the sense that it requires no other 
theoretical input than the utilization of Lorentz covariance, 
gauge invariance and parity conservation. 
Note that the Born amplitude (no real photon emission) corresponding to 
the leading order contribution  to $e^+e^- \to \gamma^\ast \to h^+h^-$ 
is just given by 
\ba
{\cal M}^{(0)}=E_{\mu}^{(0)}(p_1,p_2)\;F^{\mu}_{(0)}(k_1,k_2) .
\ea 
The goal will now be to derive from the above expressions  
sets of observable quantities which allow for a measurement of 
$F_h$, $F_1$ and $F_2$.
As was motivated in the introduction a true measurement of FSR 
is expected to be of particular importance for the radiative 
return scenario since here the leading order amplitude is 
given by ${\cal M}^{(1)}$ and therefore one has to deal with 
a leading order FSR contribution which is supposed to be  
known as precisely as possible. 
Hence as a first step we are going to analyze possible observables 
related to a $h^+h^-\gamma$ final state.

Taking the absolute square of the amplitude ${\cal M}^{(1)}$,
subsequently averaging over the $e^+e^-$ spins $s_{1,2}$, summing 
over the photon helicities $\lambda$ (which results in 
$\overline{|{\cal M}^{(1)}|}^2$) 
and integrating over the $h^+h^-\gamma$ phase space yields the
corresponding total cross section 
\ba
\sigma^{(1)}(s) &=& \frac{1}{F}\int dLips^{(1)}
\;\overline{|{\cal M}^{(1)}|}^2 .
\ea 
[$dLips^{(1)}$ denotes the Lorentz--invariant phase space 
integration and $F=2s\beta_e$ (with $\beta_e=(1-4m_e^2/s)^{1/2}$) 
is the flux factor].
Here we are obviously not interested in the total cross section but 
in differential cross sections since our aim is to measure 
the three complex form factors which are functions of $s'$ and $k_1k$. 
We therefore first consider 
the totally differential cross section which we can write as
a sum of six products:
\ba
\frac{d\sigma^{(1)}}{dLips^{(1)}} &=& 
\frac{1}{F}\;\overline{|{\cal M}^{(1)}|}^2 \nn\\
&&  \hspace{-1cm}=\;\frac{1}{F}\sum_{i=1}^{6} c_i(s',k_1k)\;
f_i^{(1)}(s',k_1k,p_1k,p_1k_1) ,
\label{dsigmadlips}
\ea
with
\ba
c_1 &=& |F_h|^2 ,\quad c_2 \;=\;\re(F_h F_1^\ast), \quad 
c_3 \;=\; \re(F_h F_2^\ast) , \nn\\ 
\quad c_4 &=& |F_1|^2, \quad 
c_5 \;=\; \re(F_1 F_2^\ast), \quad
c_6 \;=\; |F_2|^2  .
\label{ci}
\ea 
and 
\ba
f_1^{(1)} &=& -e^2\;E_{\mu\nu}^{(1)}\;[E^{(1) \ast}]^{\nu}_{\mu'}
\;l^\mu\;l^{\mu'} , \nn\\
f_2^{(1)} &=& 2ie\;E_{\mu\nu}^{(1)}\;E_{\mu'}^{0\ast}\;l^\mu\;T_1^{\ast\mu'\nu} ,
\nn\\
f_3^{(1)} &=& 2ie\;E_{\mu\nu}^{(1)}\;E_{\mu'}^{0\ast}\;l^\mu\;T_2^{\ast\mu'\nu} ,
\nn\\
f_4^{(1)} &=& -E_{\mu}^{0}\;E_{\mu'}^{0\ast}\;T_1^{\mu\nu}T_{1\nu}^{\ast\mu'},
\nn\\
f_5^{(1)} &=&  - 2 E_{\mu}^{0}\;E_{\mu'}^{0\ast}\;T_1^{\mu\nu}T_{2\nu}^{\ast\mu'},
\nn\\
f_6^{(1)} &=& -E_{\mu}^{0}\;E_{\mu'}^{0\ast}\;T_2^{\mu\nu}T_{2\nu}^{\ast\mu'}.
\ea 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Note that the six real functions $f_i^{(1)}$ ($i=1\dots 6$) are 
unambiguously given by 
the kinematics of the process 
which can be expressed in terms of the 
four independent measurable invariants 
$s'$, $k_1k$, $p_1k$ and $p_1k_1$, corresponding to the 
four phase space degrees of freedom. Therefore the $f_i^{(1)}$ 
are uniquely determined by the kinematics of a given $h^+h^-\gamma$ event.
The $c_i$ ($i=1\dots 6$) on the other hand are functions of the 
three a priori unknown complex form factors 
$F_h$, $F_1$ and $F_2$ which are supposed to be 
determined from the data by choosing appropriate observables.  
Here the fact that, in contrast to the $f_i^{(1)}$, the $c_i$ are  
depending on only two of the four phase space variables  
($s'$ and $k_1k$, see the Appendix) indeed offers the possibility for 
achieving this aim as will be shown in the following. 

Let us consider for the moment the idealized scenario that 
we could actually measure accurately all kinds of kinematically allowed 
$h^+h^-\gamma$ events being determined by the variables   
$s'$, $k_1k$, $p_1k$ and $p_1k_1$. Then we   
obviously could choose to apply a sorting procedure such that we 
put events of given $s'$ and $k_1k$ into the same kinematical  
set ${\cal A}=\{s',k_1k\}$ without discarding events. 
For each two--dimensional data set ${\cal A}$ we then have two phase 
space degrees of freedom left which can be related to the subset of 
the remaining two invariants  ${\cal B}=\{p_1k,p_1k_1\}$. 
Each single event is then given by the respective values 
of ${\cal A}$ and ${\cal B}$. 
The two--dimensional differential cross section corresponding 
to a given data set ${\cal A}$ can then be written as 
%%%%%%%%%%%%%%%%%
\ba
\left(\frac{d\sigma}{dLips}\right)^{{\cal A}}({\cal B})
&=& 
\frac{1}{F} \sum_{i=1}^6 c_i^{{\cal A}} 
\;f_i^{{\cal A}}({\cal B}) ,
\label{subset}
\ea
%%%%%%%%%%%%%%%%%  
where the suffix ``${\cal A}$'' here denotes that the value of $s'$ 
and $k_1k$ is fixed for a given set ${\cal A}$. 
Within the considered idealized scenario it is now possible to determine 
via (\ref{subset}) for a given set ${\cal A}$
the six unknown constants $c_i^{{\cal A}}$ ($i=1\dots 6$)
solely from experimental data.  
For this we have to take for each set ${\cal A}$ at least six 
independent events $\{{\cal A},{\cal B}_j\}$ 
($j=1\dots n$, $n\geq 6$) for which the functions 
$f^{{\cal A},{\cal B}_j}_i 
=f_i^{{\cal A}}({\cal B}_j)$ are obviously 
uniquely defined. 
Taking the corresponding measured value of 
$d\sigma^{{\cal A},{\cal B}_j}/dLips= 
(d\sigma/dLips)^{{\cal A}}({\cal B}_j)$ 
--- being obviously also fixed if 
${\cal A}$ and ${\cal B}_j$ are given ---
then yields a system of $n\geq 6$ independent equations
for the $6$ unknown variables  $c_1^{{\cal A}}\dots c_6^{{\cal A}}$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba
c_1^{{\cal A}}\;f_1^{{\cal A},{\cal B}_1}+\dots 
+ c_6^{{\cal A}}\;f_6^{{\cal A},{\cal B}_1} &=&  
d\sigma^{{\cal A},{\cal B}_1}/dLips \nn\\
c_1^{{\cal A}}\;f_1^{{\cal A},{\cal B}_2}+\dots 
+ c_6^{{\cal A}}\;f_6^{{\cal A},{\cal B}_2} &=&  
d\sigma^{{\cal A},{\cal B}_2}/dLips \nn\\
&\vdots& \nn\\
%&&\hspace{-4cm} 
%\dots \dots \dots \dots\dots \dots \dots \dots
%\dots \dots\dots \dots  
%\nn\\
c_1^{{\cal A}}\;f_1^{{\cal A},{\cal B}_n}+\dots 
+ c_6^{{\cal A}}
\;f_6^{{\cal A},{\cal B}_n} &=&  d\sigma^{{\cal A},{\cal B}_n}/dLips .
\label{eqsystem}
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We thus can determine now 
for each data set ${\cal A}$ the six unknown constants 
$c_i^{{\cal A}}$ ($i=1\dots 6$) simply by solving the equation system  
(\ref{eqsystem}).
This immediately yields the three complex form factors
$F_h(s')$, $F_1(s',k_1k)$ and $F_2(s',k_1k)$ (with $\{s',k_1k\} ={\cal A}$)
since they are related to the $c_i^{{\cal A}}$ via (\ref{ci}). 
%????
%As a consequence the theoretical error for the extraction of 
%$|F_h(s')|^2$ at radiative return experiments, being 
%related to ad hoc FSR modeling, can be 
%avoided by the proposed extraction method.  

Note that the normalization was chosen such that
for the case of sQED the related 
form factors $F_h^S$, $F_1^S$ and $F_2^S$ and via (\ref{ci}) 
the corresponding $c_i^S$ ($i=1\dots 6$) are $1$, being true
for each point of phase space (see the Appendix). 

In realistic experiments, however, the measured events obviously    
have an experimental error. Furthermore, due to the limitation of the  
angular and energy--momentum resolution of the particle detectors,    
a binning of events is required. 
The idealized ``local'' data sets ${\cal A}$ and ${\cal B}$ 
therefore have to be replaced by finite phase space regions (bins) 
for which ${\cal A}$ and ${\cal B}$ are the corresponding central values. 
Such a sorting procedure can indeed be achieved by utilizing a dedicated 
program package like PAW~\cite{Brun:1988pg}.   
In a realistic experiment evaluating the $c_i^{{\cal A}}$ 
by solving the equation systems (\ref{eqsystem}) has to be 
replaced by a ``best--fit'' procedure. The $c_i^{{\cal A}}$  
corresponding to the different data bins can so be 
determined by ``tuning'' them such that the agreement with the 
experimental data is best while also taking into account 
the experimental error. We would like to stress again that
in principle all measured $h^+h^-\gamma$ events can be
taken into account for the fitting procedure which means
that the available data can be exploited in an optimal way.

The four invariants $s'$, $k_1k$, $p_1k$ and $p_1k_1$ can of course 
be expressed in terms of the usual phase space variables in the   
$e^+e^-$ center of mass system. 
E.g.~we can express $k_1k$ as a function of the $h^+h^-$--invariant
mass $\sqrt{s'}$ and the $h^-$--energy $E^-$ or $h^+$--energy $E^+$:
\ba
k_1k &=& -\frac{s'}{2}+\sqrt{s}\;E^-=
\frac{s}{2}-\sqrt{s}\;E^+.
\ea
Hence measuring the invariant mass $\sqrt{s'}$ of the meson pair and 
the energy of alternatively the $h^-$ or the $h^+$ 
uniquely defines the corresponding data set ${\cal A}$. 
Tagging in addition the real photon $\gamma$ 
allows for a closure of phase space such that processes including 
more than one hard photon can be excluded from the data. 
Since the contributions of additional real photons with sufficiently 
low energy factorize and are theoretically known 
we can treat these corrections in a model--independent way.
Including soft real photons of course also requires to take into account 
virtual corrections of the same order
to obtain physical (infrared--finite) quantities.
For details concerning the treatment of soft photons we refer to 
the literature~\cite{Bloch:1937pw,Yennie:1961ad}. 
It should be noted that virtual FSR corrections 
cannot be applied in a model--independent way since the corresponding 
loop integrals contain virtual photons 
of high energy and momentum. Virtual FSR corrections therefore 
depend on the short distance structure of the hadronic final state. 
Only the soft virtual photon contribution is, as the real
soft photon part, uniquely determined by theory.
The treatment of the remaining (infrared--finite)
short distance virtual FSR correction part, in contrast, is model--dependent.
It can be absorbed into the definition of $F_h$, $F_1$ and $F_2$ 
or, since it is of higher order, modeled 
by sQED (see e.g.~\cite{Hoefer:2001mx}).

Up to now we only considered a $h^+h^-\gamma$ final state. 
The described strategy for the extraction of $F_h$, $F_1$ and $F_2$ 
can however be extended to the case of more than one ISR photon. 
Such an extension would especially be desirable for the scenario that the 
photon is not tagged (no closure of phase space by event selection) 
and contributions of 
more than one real hard photon are not negligible. 
We therefore will briefly sketch how higher 
order ISR contributions can be included.  
Obviously the dominant higher order contribution is related to 
pure ISR emission. Amplitudes corresponding to higher order 
(real + virtual) ISR can be written in a 
similar way as the leading ISR amplitude in (\ref{MISR}). 
E.g.~we can write the two--photon ISR--amplitude as 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba
{\cal M}_{\rm ISR}^{(2)} &=& E_{\mu\nu\rho}^{(2)}(p_1,p_2,k,k')
\;F^{\mu}_{(0)}(k_1,k_2)\;\epsilon^{\ast\nu}(k,\lambda)\;\nn\\
&& \times  \epsilon^{\ast\rho}(k',\lambda') , 
\label{MISR2} 
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $E_{\mu\nu\rho}^{(2)}$ -- in analogy to 
the one--photon initial state tensor
$E_{\mu\nu}^{(1)}$ -- is the initial state 
tensor corresponding to the emission of two real photons with the
momenta $k$ and $k'$ and polarizations $\lambda$ and $\lambda'$.
Approximating the  amplitude corresponding to a $h^+h^-\gamma\gamma$ 
final state by its ISR contribution we may write the corresponding
totally differential cross section as
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba
\frac{d\sigma^{(2)}}{dLips^{(2)}} &\simeq& \frac{1}{F}\;
 |F_h(s')|^2\;f_1^{(2)} ,
\label{dsigma2}
\ea 
with 
\ba 
f_1^{(2)} = e^2\; 
E^{(2)}_{\mu\nu\rho}\;[E^{(2)\ast}]_{\mu'}^{\nu\rho}\;l^{\mu} \;l^{\mu'}. 
\ea
In a photon--inclusive measurement $d\sigma^{(2)}/dLips^{(2)}$
has to be added to $d\sigma^{(1)}/dLips^{(1)}$ to obtain the 
next-to-leading-order-ISR--improved differential cross section. 
Into the 1--photon distribution function $d\sigma^{(1)}/dLips^{(1)}$ 
then also the leading order virtual ISR corrections have to be 
included such that $d\sigma^{(1)}/dLips^{(1)}+d\sigma^{(2)}/dLips^{(2)}$
yields a meaningful (infrared--finite) result.
Only taking into account ISR in the second order contribution,
as done in (\ref{dsigma2}), we obviously only have to replace  
$f_1^{(1)}$ by $\tilde{f}_1^{(2)}=f_1^{(1)}+f_1^{(2)}$ in 
(\ref{dsigmadlips}), where -- as mentioned before -- to $f_1^{(1)}$  
the first order virtual ISR corrections have to be added.   
The same is true if we add ISR contributions of higher
order ($d\sigma^{(n)}/dLips^{(n)}$, with $n\geq 3$).   
Then the following replacement in (\ref{dsigmadlips}) has to be applied:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba
f_1^{(1)} &\rightarrow& \tilde{f}_1^{(n)} = 
f_1^{(1)}+f_1^{(2)}+\dots + f_1^{(n)} \nn\\
f_i^{(1)} &\rightarrow& \tilde{f}_i^{(n)} = f_i^{(1)}
,\quad i = 2\dots 6.
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Again we have to add the corresponding virtual ISR corrections such that 
$f_1^{(1)} \dots  f_1^{(n)}$ all contain contributions up to 
$n^{\rm th}$ order and $\tilde{f}_1^{(n)}$ then is infrared--finite. 
Hence also if we include higher order ISR corrections 
we again end up with an equation system of the form
(\ref{eqsystem}) but with $f_i^{(1)}\to\tilde{f}_i^{(n)}$ 
($i= 1\dots 6$) and the form factor extraction procedure
obviously stays the same as described before.  

Finally some comments about the limitations of the described 
procedure being related to ISR$\otimes$FSR diagrams in which 
photons are emitted from the initial as well as from
the final state. The leading one of these diagrams, with one ISR and
one FSR photon,  
is shown in Fig.~\ref{ISRotimesFSR}. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\mbox{\epsfxsize 6cm  \epsffile{ISRotimesFSR.eps}}
%\vspace{-1.7cm}
\caption{ISR$\otimes$FSR Feynman diagram corresponding to the 
process
$e^-(p_1) e^+(p_2) \to \gamma^\ast(s_V) \to h^-(k_1) h^+(k_2) \;
\gamma(k_i) \gamma(k_f)$.} 
\label{ISRotimesFSR}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Since also this diagram contains a $h^+h^-\gamma^\ast\gamma$ 
vertex we can use as before the parameterization of the
corresponding coupling by the form factors $F_1$ and $F_2$. 
These, however, are now functions of the invariant mass of
the virtual photon $\sqrt{s_V}$ and the scalar product 
$k_1k_f$ which both are not observable quantities, being due to
the fact that the initial state photons $\gamma(k_i)$ cannot
be distinguished from the final state photons $\gamma(k_f)$
on an event basis. ISR$\otimes$FSR contributions therefore cannot 
be fitted to experimental data but they can (since these
contributions are small in size) e.g.~be included as ``known''
contributions into the fitting procedure by using models
like sQED. Alternatively we can apply an iterative procedure
where in a first step ISR$\otimes$FSR is neglected and the
such fitted form factors $F_1$ and  $F_2$ are then used in a second step 
for the parameterization of the ISR$\otimes$FSR contribution.  

Diagrams with more than one real photon emitted from 
the final state would require a more involved FSR parameterization 
than before and the fitting procedure of the corresponding 
form factors would become very difficult.  
The corresponding contributions, however, can be estimated to be below 
one per mill (estimation within sQED) and they are therefore expected 
to be negligible. 

Obviously the described form factor extraction strategy is not
restricted to radiative return experiments where the $e^+e^-$ center
of mass energy $\sqrt{s}$ is fixed but it can also be applied to 
scan experiments with varying $\sqrt{s}$. Here the proposed method not 
only allows for a scan of $|F_h|^2$ but of the complex form factor 
$F_h$ itself as well as of the complex FSR form factors $F_1$ and $F_2$, 
leading to a very attractive test of how valid the previously 
used sQED--approximation of FSR is.     
Scanning the FSR form factors finally also allows to obtain the 
FSR--inclusive $h^+h^-$ cross section $\sigma_{hh}^{(\gamma)}$ in a 
model independent way.

As a cross check of how precisely $F_h$, $F_1$ and $F_2$ 
can be obtained from real experimental data 
(e.g.~from $\pi^+\pi^-$ production measurements at DA$\Phi$NE)
by using the proposed method, one can apply a similar 
form factor parameterization to the muon pair production channel 
($e^+e^-\to \gamma^* \to \mu^+\mu^-\gamma$) which is
again based only on Lorentz covariance, gauge invariance and
parity conservation. 
Since on the other hand the muon form factors are obviously given 
by QED one can test the precision of the form factor extraction
procedure by comparing the extracted form factors to
the true QED results.  

\section{Summary}

A model--independent treatment of the leading order FSR contribution
to $e^+e^-$ collision processes with a charged meson pair $h^+h^-$ 
in the final state was introduced. 
For this purpose a parameterization of FSR by the two form factors 
$F_1$ and $F_2$, being based only on Lorentz covariance,
gauge invariance and parity conservation, was utilized.  
It was shown that, as a consequence, the leading order FSR background 
at radiative return experiments does not have to be modeled anymore 
but can in fact be determined directly from experimental data.  
Avoiding the problem of FSR modeling 
could e.g.~lead to significantly more precise measurements of the 
electromagnetic pion form factor $F_\pi$ at $\Phi$ factories
like DA$\Phi$NE. Furthermore the validity of previously used 
crude FSR approximations like scalar QED can be tested.
      
%-------------------------------------------------------------------

\begin{acknowledgments}
%\section{Acknowledgments}
It is our pleasure to thank Achim Denig, Stefano DiFalco, Simon
Eidelman, Paolo Franzini, Wolfgang Kluge, Andreas Nyffeler and
Graziano Venanzoni for stimulating discussions as well as Mark
Tyburski for carefully reading the manuscript. 
\end{acknowledgments}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\appendix

\section{Parameterization of the $\gamma^\ast\gamma^\ast h^-h^+$ Sub--Process}

\begin{figure}[h]
\begin{center}
\mbox{\epsfysize 3cm  \epsffile{blob.eps}}
%\vspace{-1.7cm}
\caption{$\gamma^\ast \gamma^\ast h^+h^-$ diagram corresponding
to the hadronic tensor $F^{\mu\nu}$. 
The momenta $Q_1$, $Q_2$, $K_1$ and $K_2$ are chosen
to point inwards the hadronic blob. \label{blob}}
\end{center}
\end{figure}

Let us consider the Feynman sub--diagram with two (in general) virtual
photon lines  
[$\gamma_1^\ast(Q_1^{\mu})$ and $\gamma_2^\ast(Q_2^{\nu})$]
and two on-shell scalar meson lines [$h^-(K_1)$ and $h^+(K_2)$]  
entering a hadronic blob as shown in 
Fig.~\ref{blob}. Since here the relevant couplings are determined by 
non-perturbative QCD effects such a contribution cannot be 
calculated in a straightforward way within perturbation theory.
Thus in the following we will treat the hadronic blob as 
a black box while avoiding the use of any ad hoc models. 
The goal will be to derive the general structure
of the corresponding rank-2 tensor $F^{\mu\nu}(Q_1,Q_2,K_1,K_2)$ 
by exploiting basic 
symmetries like 
Lorentz covariance, gauge invariance and parity conservation. 
The obtained result will then be used for the  
parameterization of the leading FSR contribution to the 
process $e^+e^-\to\gamma^\ast\to h^+ h^-$.

Most generally the amplitude corresponding to a
``complete'' Feynman diagram (only external on-shell particles) 
containing the considered $\gamma^\ast\gamma^\ast h^-h^+$ sub diagram 
can be written as a contraction of some Lorentz 
tensor $E_{\mu\nu}$ with the hadronic tensor $F^{\mu\nu}$: 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba
{\cal M} &=& E_{\mu\nu}(Q_1,Q_2,\{p_i\})\;F^{\mu\nu}(Q_1,Q_2,K_1,K_2),
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\{p_i\}=\{p_1 \dots p_n\}$ are some not further specified
external on-shell momenta. 
Here the Lorentz indices $\mu$ and $\nu$ 
correspond to the momenta $Q_1^\mu$ and $Q_2^\nu$ of the photons 
$\gamma_1^\ast$ and $\gamma_2^\ast$ connecting the $\gamma^\ast\gamma^\ast h^-h^+$ 
sub diagram with the remaining part of the complete Feynman diagram. 
Obviously, if one of the photons is real $E_{\mu\nu}$ contains the 
corresponding on-shell photon polarization vector with the corresponding
Lorentz index ($\mu$ or $\nu$).
Taking into account that an amplitude in general can 
always be decomposed into two separately gauge invariant Lorentz tensors
by cutting the connecting photon lines we can 
make use of the following Ward identities:
%
\ba
Q_{1\mu}\;F^{\mu\nu} &=& 0 \quad,\quad Q_{2\nu}\;F^{\mu\nu} \;=\; 0 \;,
\label{wardF} \\
Q_{1\mu}\;E^{\mu\nu} &=& 0 \quad,\quad Q_{2\nu}\;E^{\mu\nu} \;=\; 0 \;.
\label{wardE}
\ea
%
As a consequence of (\ref{wardE}) and the mentioned
separability of the amplitude ${\cal M}$ there 
are no tensor contributions to $F^{\mu\nu}$ containing the momenta $Q_1^\mu$  
and $Q_2^\nu$. In other words such terms would correspond to longitudinal 
polarization contributions of the virtual photons which 
drop out because of gauge invariance.  
Further we use that 
because of 4-momentum conservation 
($Q_1^\mu+Q_2^\mu+K_1^\mu+K_2^\mu=0$)  
$F^{\mu\nu}$ can be written as a function of three linearly 
independent momenta 
which we choose to be $Q_1$, $Q_2$ and 
$L \equiv K_1-K_2$. 
We can then make the most general tensor ansatz by writing  
$F^{\mu\nu}$ as a linear combination 
of all linearly independent rank-2 tensors  
{\em not} containing $Q_1^\mu$ and $Q_2^\nu$:
\ba
F^{\mu\nu} &=& f_{LL}\;L^\mu L^\nu + f_{21}\;Q_2^{\mu}Q_1^{\nu}
+f_{L1}\;L^{\mu}Q_1^{\nu}+f_{2L}\;Q_2^{\mu}L^\nu \nn\\
&+& f_g\;g^{\mu\nu} + 
f_{12}^{\epsilon}\;\epsilon^{\alpha\beta\mu\nu} Q_{1\alpha} Q_{2\beta} 
+
f_{1L}^{\epsilon}\;\epsilon^{\alpha\beta\mu\nu} Q_{1\alpha} L_\beta
\nn\\
&+&
f_{2L}^{\epsilon}\;\epsilon^{\alpha\beta\mu\nu} Q_{2\alpha} L_\beta \;.
\label{tensor1}
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here we have taken into account that the mesons $h^+$ and $h^-$ 
are asymptotically  
free hadronic states with spin 0. For the case of spin-1/2 
particles like e.g. protons 
we would have to make a similar tensor ansatz 
but here taking a linear combination of Dirac-current tensors. 
Applying the Ward identities (\ref{wardF}) to (\ref{tensor1}) 
leads to the following six equations:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba
f_{2L}^{\epsilon} = 0,\;\;\;f_{1L}^{\epsilon} &=& 0 ,\nn\\
f_{LL} (Q_1L)+f_{2L} (Q_1Q_2) &=& 0 , \nn\\
f_{LL} (Q_2L)+f_{L1} (Q_1Q_2) &=& 0 ,\nn\\
f_{21} (Q_1Q_2)+f_{L1} (Q_1L)+f_g &=& 0 ,\;\;\; \nn\\
f_{21} (Q_1Q_2)+f_{2L} (Q_2L)+f_g &=& 0 ,
\label{eqsys1}
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
which after simple manipulations yield the following 
dependences between five of the unknown parameters:  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba
f_{2L} &=& -\frac{(Q_1L)}{(Q_1Q_2)}\;f_{LL} 
\quad,\quad
f_{L1} \;=\; -\frac{(Q_2L)}{(Q_1Q_2)}\;f_{LL}, 
\nn\\
f_{21} &=& -\frac{1}{(Q_1Q_2)} \left[f_g-\frac{(Q_1L)(Q_2L)}{Q_1Q_2}\;
f_{LL}\right]. 
\label{eqsys2}
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Inserting (\ref{eqsys2}) into (\ref{tensor1}) and 
taking into account parity conservation (thus dropping 
the tensor $\propto f_{12}^{\epsilon}$) 
yields the general expression for $F^{\mu\nu}$ obeying gauge 
invariance and parity conservation: 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba 
F^{\mu\nu} &=& f_{LL} \left[L^\mu-\frac{(Q_1L)}{(Q_1Q_2)}Q_2^{\mu}\right]
\left[L^\nu-\frac{(Q_2L)}{(Q_1Q_2)}Q_1^{\nu}\right] \nn\\
&+&f_g \left[g^{\mu\nu}-\frac{Q_2^\mu Q_1^\nu}{(Q_1Q_2)}\right] \;.
\label{tensor2}
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Thus we finally expressed the hadronic tensor in terms of 
the two form factors $f_{LL}$ and $f_g$. They are  
scalar functions of in general four independent kinematical 
variables which can be chosen to be the invariants $Q_1^2$, 
$Q_2^2$, $L^2$ and $Q_1L$.
(Note that in contrast the rank-1 tensor corresponding to  
a 3-particle $\gamma^\ast h^- h^+$ sub diagram  
can be parameterized by a single meson form factor $F_h(Q^2)$
which is just a function of the invariant mass $Q^2$ of the photon.) 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the special case of a minimal coupling of the two photons to 
point-like scalar particles with charge $\pm e$
the form factors $f_{LL}$ and $f_g$ could simply be calculated from scalar QED
which yields
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\be
f_{LL}^S = ie^2\;\frac{2(Q_1Q_2)}{[Q_1(Q_2-L)][Q_2(Q_1-L)]} \quad,\quad
f_g^S= 2ie^2.
\ee
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Coming back to the general structure for $F^{\mu\nu}$ in 
(\ref{tensor2}) we can ask the question how this expression would look like 
if one of the photons was on-shell (see also~\cite{Diehl:2000uv}). This in fact is just what we
need to obtain the leading FSR contribution to the process 
$e^-(p_1)\;e^+(p_2) \to \gamma^\ast(q) \to h^-(k_1)\; h^+(k_2)\; \gamma(k)$
(see FIG.~\ref{eeTogHH}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\mbox{\epsfysize 5.5cm  \epsffile{eeTogHH.eps}}
%\vspace{-1.7cm}
\caption{Feynman diagram corresponding to the leading FSR 
contribution to $e^-\;e^+ \to \gamma^\ast \to h^-\; h^+\; 
\gamma$. The sub diagram below the dashed line corresponds
to the one shown in Fig.~\ref{blob}.
 \label{eeTogHH}}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Making the identifications 
\ba
K_{1,2} &=& -k_{1,2},\quad Q_1\;=\;q,\quad Q_2\;=\;-k , 
\nn\\ 
F_1 &=& \frac{2i}{e^2}\;\frac{(k_1k)(k_2k)}{qk}\;f_{LL}, 
\;\;\; F_2 \;=\; -\frac{i}{2e^2}\;f_g ,
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
we can write the general FSR tensor as 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba 
F^{\mu\nu}_{\rm FSR} &=& F_1\;T_1^{\mu\nu} + F_2\;T_2^{\mu\nu},
\label{tensorFSR}
\ea
%%%%%%%
with ($l=k_1-k_2$)
%%%%%%%
\ba
T_1^{\mu\nu} &=& -ie^2\;\left[l^\mu-\frac{ql}{kq}\;k^{\mu}\right] \nn\\
&& \times \;
\left[\frac{k_1^\nu}{k_1k}-\frac{k_2^\nu}{k_2k}+\frac{1}{2}
\left(\frac{1}{k_1k}-\frac{1}{k_2k}\right)k^\nu\right] , 
\nn\\
T_2^{\mu\nu} &=& 2ie^2\;\left[g^{\mu\nu}-\frac{k^\mu q^\nu}{kq}\right] .
\label{tensorT1T2app}
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It is important to note that the two FSR form factors $F_1$ and $F_2$ 
now effectively depend on only two independent kinematical variables  
which is due to the fact that one of the photons is on--shell 
($k^2=0$) and the invariant mass $\sqrt{s}=\sqrt{q^2}$ of the virtual photon 
is given by the $e^+e^-$ collision energy in the corresponding
center of mass frame.   
For given $s$ we can e.g.~choose to write $F_1$ and $F_2$ as 
functions of the two kinematic variables $s'=(k_1+k_2)^2$ and $k_1k$ 
since $l^2=s'-4m_h^2$ ($m_h$ being the meson on--shell mass), 
$k l=-ql$ and $k_1l = -k_2l = -s'/2+2m_h^2$:
%%%%%%%%%%%%%%%%%%%%%%
\ba
F_{1,2} &=& F_{1,2}(s',k_1k) .
\ea
%%%%%%%%%%%%%%%%%%%%%%
We can now easily express the amplitude 
${\cal M}_{\rm FSR}$ corresponding to the complete 
leading order FSR diagram as a contraction 
of the FSR tensor with the tensor
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ba
E_{\mu}^0\;\epsilon^\ast_{\nu}(k,\lambda) &=&  
-\frac{e}{s}\;\bar{v}_{s_2}(p_2)\gamma_{\mu}u_{s_1}(p_1)\;
\epsilon^\ast_{\nu}(k,\lambda) ,
\label{tensorE0}
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$s_1$, $s_2$ and $\lambda$ being the electron-, positron- 
and real-photon-spin, respectively and $\epsilon^\ast_{\nu}$ being
the covariant real photon polarization vector:  
\ba
{\cal M}_{\rm FSR} &=& E_{\mu}^0\;\epsilon^\ast_{\nu}(k,\lambda)
\;F^{\mu\nu}_{\rm FSR} \nn\\
&=& F_1(s',k_1k)\;E_{\mu}^0\;\epsilon^\ast_{\nu}(k,\lambda)\;T_1^{\mu\nu} \nn\\
&+& F_2(s',k_1k)\;E_{\mu}^0\;\epsilon^\ast_{\nu}(k,\lambda)\;T_2^{\mu\nu} .
\ea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As $E_{\mu}^0$ and $T_{1,2}^{\mu\nu}$ 
correspond to known tensor expressions 
[see (\ref{tensorT1T2app}) and (\ref{tensorE0})] 
we finally have managed to shift our lack of detailed 
knowledge about the FSR mechanism to the two unknown form factors 
$F_1$ and $F_2$. At the end the goal will be to measure, if possible,
these form factors which would immediately yield the true leading order 
FSR contribution.   
For the sake of completeness we finally give the simple result 
for $F_1$ and $F_2$ we would obtain for the special case of a minimal
photon coupling to point-like scalar mesons as can be calculated
from sQED: 
%%%%%%%%%%%%%%%%%%%%%%
\ba
F_1^S &=& F_2^S = 1 .
\ea
In fact the normalization of the tensors $T_1^{\mu\nu}$ has  
been chosen such that in sQED $F_1$ and $F_2$ just yield 
unity. 

%\newpage

\bibliographystyle{apsrev}

\bibliography{references}

\end{document}

