%Title: Radiative return method as a tool in hadronic physics
%Author: Henryk Czyz, Agnieszka Grzelinska and Elzbieta Nowak-Kubat
%Published: * Acta Phys. Polon. B * ** 36 ** (2005) 3403-3412.
\documentclass{appolb}
\usepackage{epsfig}
%------------------------------------------------------
% Include epsfig package for placing EPS figures in the text
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% BEGINNING OF TEXT %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\ta}[1]{#1\hspace{-.42em}/\hspace{-.07em}} % SLASH
\newcommand{\taa}[1]{#1\hspace{-.55em}/\hspace{-.09em}} % SLASH
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\non}{\nonumber}
\newcommand{\eq}[1]{eq.~(\ref{#1})}
\newcommand{\Eq}[1]{Eq.~(\ref{#1})}
\newcommand{\eps}{\varepsilon}
\newcommand{\be}{\beta}
\begin{document}
\date{\today}
\pagestyle{plain}
%% uncomment the following line to get equations numbered by (sec.num)
%\eqsec
\newcount\eLiNe\eLiNe=\inputlineno\advance\eLiNe by -1
\title{ Radiative return method as a tool in hadronic physics
\thanks{Presented by H. Czy{\.z}
at XXIX International Conference of Theoretical Physics,
`Matter To The Deepest', Ustro{\'n}, 8-14 September 2005, Poland.
Work supported in part by
EC 5-th Framework Program under contract
HPRN-CT-2002-00311 (EURIDICE network), TARI project RII3-CT-2004-506078
and
Polish State Committee for Scientific Research
(KBN) under contracts 1 P03B 003 28 and 1 P03B 013 27.}
}
\author{Henryk Czy\.z$^a$
, Agnieszka Grzeli\'nska$^b$\\ and El\.zbieta~Nowak-Kubat$^a$
\address{ a: \ Institute of Physics, University of Silesia,
PL-40007 Katowice, Poland \\ b: \ Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe,\\ D-76128 Karlsruhe, Germany
}}
\maketitle
\vskip -5 cm
\hfill TTP05-20
\vskip 5 cm
\begin{abstract}
A short review of both theoretical and experimental aspects
of the radiative return method is presented. It is emphasised
that the method gives not only possibility of the independent
from the scan method measurement of the hadronic cross section,
but also can provide information concerning details of the hadronic
interactions. New developments in the PHOKHARA event generator
are also reviewed. The 3 pion and kaon pair production is implemented
within the version 5.0 of the program, together with contributions
of the radiative $\phi$ decays to the 2 pion final states.
Missing NLO radiative corrections to the $e^+e^- \to \mu^+\mu^- \gamma$
process will be implemented in the forthcoming version of the generator.
\end{abstract}
\PACS{13.40.Ks,13.66.Bc}
%***********************************************************************
\section{Introduction}
%***********************************************************************
Precise hadronic cross section measurements are 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 ($\alpha_{QED}$) from its value at low energy up
to $M_Z$ (for recent reviews look
\cite{Davier:2003pw,Jegerlehner:2003rx,Nyffeler:2004mw}).
When using the scan method one usually needs new experiments to
be performed with not negligible costs, while
the radiative return method proposed already years ago \cite{Zerwas}
allows for extracting the information on the hadronic cross section
basing on the measurements at the existing meson factories, profiting
from their huge luminosities and excellent detectors.
As it was shown in \cite{Nowak,Czyz:2004nq} the radiative
return method
is not only a powerful tool in the
measurement of $\sigma (e^+e^- \to \ {\mathrm{hadrons}})$,
but allows for detailed studies of hadronic interactions.
Due to a complicated experimental setup,
the use of Monte Carlo (MC) event generators
\cite{Szopa,rest,FSR,PHOKHARA_mu,actaHN,Czyz:2005},
which include various radiative corrections \cite{PHradcor}
is indispensable for both signal and background processes.
Some more detailed analysis of that subject can be found
also in \cite{proc,FSR1}.
This paper is aimed as a short review of the results obtained
by the radiative return method. It presents also a further
potential of the method and shows new developments in the
PHOKHARA event generator important for an extraction
of the $\sigma(e^+e^-\to {\rm hadrons})$ (and other physical
quantities) from the measured cross section
$\sigma(e^+e^-\to {\rm hadrons} + {\rm photons})$.
%*******************************************************************
\section{Hadronic cross section \label{sec2}}
The extraction of the cross section $\sigma(e^+e^-\to {\rm hadrons})$
from the measured cross section
$\sigma(e^+e^-\to {\rm hadrons} + {\rm photons})$ relies on the
factorisation
%
\bea
{ d\sigma(e^+e^- \to \mathrm{hadrons} + n \gamma) = }
{\ H }{ \ d\sigma(e^+e^-\to \mathrm{hadrons})}
\label{fac}
\eea
%
valid at any order for photons emitted from initial leptons, where
the function $H$ contains QED radiative corrections. It is known
analytically, if no cuts are imposed on photons, at next to leading order
(NLO) and has to be provided
in form of an event generator of the reaction
$e^+e^-\to {\rm hadrons} + {\rm photons}$
for a realistic experimental setup. The emission of photons from
the final state hadrons has to be controlled as well \cite{FSR,FSR1},
with an accuracy
which allows for an error small enough not to spoil the accuracy
of the $\sigma(e^+e^-\to {\rm hadrons})$ extraction.
Comparing the
scan method and the radiative return method one has to say that
they are in many aspects complementary. It is due to the fact that
many experimental systematic errors are completely different in both cases.
The radiative return method has though the advantage that most of them are
the same for all the values of the invariant mass of the hadronic system
for which the measurement is performed. That is not true for the scan method,
where for each energy one has to perform a separate analysis of many
systematical errors (an energy calibration etc.). It is also important
that using the radiative return method one can use machines, which were
built for other purposes ($\phi$ and B factories) and
one has to `invest' only in the experimental analysis. In many aspects
both methods encounter however the same problems, which have to be solved.
The already mentioned
final state emission has to be studied carefully in both cases,
the photon vacuum polarisation with its fast varying behaviour
nearby resonances has to be taken into account,
higher orders radiative corrections have to be properly implemented
in event generators etc. Even if
the details are different for scan and the radiative return method
the main features remain the same.
The most
important hadronic channel,
from the point of the hadronic contributions to $a_\mu$,
mainly $\pi^+\pi^-$, is an example of that complementarity.
Very accurate KLOE measurement \cite{KLOE1} provided an important
cross check of the CMD-2 data \cite{Akhmetshin:2003zn}
and even if the agreement
is not excellent it has allowed to conclude that the disagreement
between $e^+e^-$ data and the $\tau$ data concerning the pion form factor
is not of the experimental origin. Further, mostly theoretical work,
will be required to solve that puzzle and one will have to
find new sources of the
isospin violation effects, which finally will explain that disagreement.
Already now many new valuable physical information was obtained by means
of the radiative return method. The BaBar measurement of the
$\sigma(e^+e^- \to 2\pi^+2\pi^-,2K^+2K^-,K^+K^-\pi^+\pi^-)$
\cite{Aubert:2005eg}
are the most accurate to date
results, with the $2\pi^+2\pi^-$ mode also extremely important
for hadronic contributions to $a_\mu$ and $\alpha_{QED}$.
The BaBar measurement of the $\sigma(e^+e^- \to \pi^+\pi^-\pi^0)$
\cite{Aubert:2004kj}
has shown that the cross section
around the $\omega$'' resonance is actually much bigger as compared
to an
old measurement by DM2 collaboration \cite{Antonelli:92}.
Results coming from BaBar on narrow resonances
\cite{Aubert:2003sv,Aubert:2004kj,Aubert:2005eg}
are also very competitive to the ones coming from the scan method
(for a review see \cite{Brambilla:2004wf}).
Not to mention the forthcoming results \cite{anulli,Marianna}, which
show still growing potential for the radiative return method.
%*******************************************************************
\section{Not only the hadronic cross section - looking inside the hadronic
interactions}
The radiative return method originally proposed for the hadronic cross
section measurements \cite{Zerwas,Binner} can be used to much more
detailed studies of the hadronic interaction. The first investigations
along these lines were done in \cite{Nowak}, where it was shown that
it is feasible to measure separately the nucleon form factors in the
time-like region at B-factories. That measurements are important for
the understanding of the experimental situation
in the space-like region (for a review look \cite{arrington}), where
two different type of measurements lead to different results for
the ratio of the magnetic and electric proton form factors.
The nucleon electromagnetic current is defined by the
form factors as follows
\bea
J_\mu = - i e \cdot \bar u(q_2)
\left({ F_1^N(Q^2)}\gamma_\mu
- \frac{{ F_2^N(Q^2)}}{4 m_N} \left[\gamma_\mu,\taa Q \ \right]
\right) v(q_1) \ , \
\label{ff}
\eea
with electric and magnetic form factors
$G_M^N = F_1^N + F_2^N~, \ G_E^N = F_1^N + \tau F_2^N~$.
The statistics is not a problem
for a measurement at B-factories with hundreds of fb$^{-1}$ accumulated
luminosity as the integrated cross section
for the event selection corresponding
to lower curve in
Fig.~\ref{nucleon}a (angular cuts close to BaBar angular acceptance)
is about 59.3~fb for protons and 125~fb for neutrons
in the final state. The separation of the electric and magnetic form factors
is also possible for quite a big range of the nucleon pair
invariant mass. It is particularly easy if one performs analysis
in the nucleon
pair rest frame as shown in Fig.~\ref{nucleon}b, where the proton
polar angle distribution is plotted both for a model which predict
the ratio of form factors in agreement with measurements using the
Rosenbluth method ($G_M^p = \mu_p G_E^p$, triangles) and a model which
predict the ratio of the form factors with agreement with the measurements
using polarisation method (squares).
For details concerning both methods see a review
article \cite{arrington} and references therein.
For the
$4 \ {\rm GeV^2} < Q^2 < 4.5 \ {\rm GeV^2}$ one expects about 2000 events
per 100~fb$^{-1}$ accumulated luminosity and clearly two parameter
($G_M^N,G_E^N$) fit is possible and its accuracy will be limited mostly
by systematic errors.
%
\begin{figure}[ht]
\begin{center}
\epsfig{file=praca_protNLO_zcieciami_i_bez.ps,width=6.09cm}
\epsfig{file=mQframe_GR_GMGE_qq4-4.5_fot25-155_hist-p.ps,width=6.29cm} \\
\hskip 1.5 cm (a) \hskip 5.6 cm (b)
\caption{ (a) The differential, in $Q^2$, cross section for the reaction
$e^+e^- \to p \bar {p} \gamma$ for two different sets of event selections.
(b) The differential, in $\cos\hat\theta$
($\hat\theta$ - proton polar angle in the proton pair rest frame),
cross section for the reaction
$e^+e^-\to \bar p p \gamma$ for
two theoretical models (see text for details).}
\label{nucleon}
\end{center}
\end{figure}
%
Already now BaBar collaboration has preliminary results
for the proton electromagnetic form factor measurements \cite{anulli},
which when completed will allow for extensive tests of the theoretical models.
Another example \cite{Czyz:2004nq}, very specific for the
radiative return method
at DAPH\-NE energy,
is the study of radiative $\phi$ decays at KLOE (for present status of
the experimental situation see \cite{Marianna}). The $\phi$ decay
($\phi\to f_0(\to\pi^+\pi^-)\gamma$) contributing to the reaction
$e^+e^-\to\pi^+\pi^-\gamma$ might in principle cause some problems
in the pion form factor extraction from the
$\sigma(e^+e^-\to\pi^+\pi^-\gamma)$ measurement. However, as it was shown
in \cite{Czyz:2004nq}, the detailed studies of the charge asymmetries
allow not only to control that contribution, but also to distinguish
between different models of the radiative $\phi$ decays. That is clearly seen
in Fig.~\ref{asym}, where the charge asymmetry for an event selection
enhancing the FSR contributions is presented. The differences between
predictions coming from different models of the radiative $\phi$ decays
and also from scalar QED (see \cite{Czyz:2004nq} for details) are
sizable in the region of large and small values of the pion pair
invariant mass ($Q^2$) and a measurement can easily distinguish
between them leading to tests with unprecedented accuracy.
%
\begin{figure}[ht]
\begin{center}
\epsfig{file=v_as_pipl_all_c1.ps,width=8cm}
\caption{ The pion charge asymmetry for different
radiative $\phi$ decay models \cite{Czyz:2004nq}.}
\label{asym}
\end{center}
\end{figure}
%
%*******************************************************************
%*******************************************************************
\section{New developments in the PHOKHARA event generator}
%*******************************************************************
\subsection{PHOKHARA 5.0: $\pi^+\pi^-\pi^0$ and $KK$ final states}
%
\begin{figure}[ht]
\begin{center}
\epsfig{file=test_v_dane_1.1_plpm.ps,width=6.05cm}
\epsfig{file=test_v_dane_1.1_pl0.ps,width=5.95cm}
\caption{ The two-pion invariant mass distributions:
BaBar data \cite{Aubert:2004kj} (points with error bars) and PHOKHARA5.0
predictions \cite{new_3pi} (filled circles) .}
\label{BaBar}
\end{center}
\end{figure}
%
In the newly released version of
the PHOKHARA Monte Carlo generator (5.0)
three new hadronic channels were added: $\pi^+\pi^-\pi^0$,
$K^+K^-$ and $\bar K K$. For the $K^+K^-$ both initial and final
state photon(s) radiation was taken into account,
while for the $\pi^+\pi^-\pi^0$
and the $\bar K K$
only initial state photon(s)
emission was considered. The kaon hadronic current
was adopted from \cite{Bruch:2004py}, while a detailed analysis
of all existing data on the $e^+e^-\to \pi^+\pi^-\pi^0$ cross section
was performed in \cite{new_3pi}.
The constructed model allows not only for an excellent
fit to the cross section and a good description of two pion invariant
mass distributions (see Fig.~\ref{BaBar}), but many three-meson couplings
were extracted separately from that fit making possible predictions
of various decay rates and cross sections \cite{new_3pi}.
\subsection{Next to leading order radiative corrections to
the reaction $e^+e^-\to \mu^+\mu^-\gamma$}
The reaction
%
\bea
e^+(p_1)e^-(p_2) \to \mu^+(q_1)\mu^-(q_2)\gamma(k)
\label{mumu}
\eea
%
may serve as a luminosity
monitoring process for the radiative return method. If this method is used
one measures
the ratio
%
\bea
{{\cal R}(s)
=\frac{\sigma(e^+e^-\rightarrow hadrons +\gamma)}
{ \sigma(e^+e^-\rightarrow \mu^+ \mu^- +\gamma)}} \ ,
\label{Rmu}
\eea
%
and for the extraction of the $\sigma(e^+e^-\rightarrow {\rm hadrons})$
from the data the theoretical knowledge of both
processes is needed. Due to a complicated experimental setup that
piece of information
has to be provided in a form of event generators and the NLO
radiative corrections to both processes are indispensable to provide
accurate theoretical predictions. Already in \cite{PHOKHARA_mu} a part
of the NLO(FSR) radiative corrections was implemented and the missing parts
of the generator consist of diagrams shown schematically in
Fig.~\ref{diag_mu}.
%
\begin{figure}[ht]
\begin{center}
\epsfig{file=d8.ps,width=4cm}
\epsfig{file=d9.ps,width=4cm}
\epsfig{file=d10.ps,width=4cm}
\caption{ NLO contributions to the $\sigma(e^+e^- \to \mu^+\mu^-\gamma)$
missing in the PHOKHARA 5.0 code.}
\label{diag_mu}
\end{center}
\end{figure}
%
All details of the calculations and implementation in the Monte Carlo event
generator PHOKHARA will be presented in separate publications \cite{new_mu}
while in this paper
few details and the present status of this project is sketched.
The contributions from diagrams in Fig.~\ref{diag_mu}a are implemented
using helicity amplitude method in an analogous way as it was done
for two photon ISR contributions \cite{Szopa}. The contributions
from Fig.~\ref{diag_mu}b, even if in principle are analogous to the
ISR virtual corrections calculated in \cite{PHradcor}, have to be treated
in a different way as the generator has to work also for DAPHNE energy
where the muon mass is not a small parameter. The expansions used
for the ISR, which shorten the final result enormously, cannot be applied
and even if the whole result is known further work is
required to construct formulae for calculation of the radiative corrections
fast enough for a Monte Carlo event generator. In both cases the
helicity amplitudes can be written as a combination of 14 terms if
the gauge invariance and the current conservation is used. The way
one writes the result is not unique and we have chosen to write explicitly
the part proportional to the Born amplitude plus suitably chosen symmetric
and antisymmetric coefficients. The ISR corrections
read then
\bea
{\cal M}_{ISR} \sim \bar u(q_2)\gamma_\mu v(q_1)\cdot
\sum_{i=1}^{14} F_i \cdot \bar v(p_1) S_i^\mu u(p_2) \ ,
\label{ampISR}
\eea
%
where
the coefficients $F_i$ contain loop corrections. The FSR corrections
have analogous structure.
The functions $S_i^\mu$ read
\bea
S_1^\mu = \frac{1}{2}
\left(\frac{2p_1\epsilon^*-\ta\epsilon^*\ta k}{p_1k}\gamma^\mu
-\gamma^\mu\frac{2p_2\epsilon^*-\ta\epsilon^*\ta k}{p_2k}\right)\ , \
S_2^\mu = \ta k\ta\epsilon^*\gamma^\mu\ ,
\nonumber \\
\ S_3^\mu = \ta k\ta\epsilon^*p_+^\mu\ , \
\ S_4^\mu = \ta k\ta\epsilon^*p_-^\mu\ , \
\ S_5^\mu = \ta\epsilon^*p_+^\mu-\ta k\epsilon^{*\mu}\ , \
\nonumber \\
\ S_6^\mu = \left[ \ta k \ p_+\epsilon^* -
\ta\epsilon^* p_+k\right]\gamma^\mu\ ,
\ S_7^\mu = \left[ \ta k \ p_-\epsilon^* -
\ta\epsilon^* p_-k\right]\gamma^\mu\ ,
\nonumber \\
\ S_8^\mu = \ta k
\left[ p_1\epsilon^*\ p_2k - p_2\epsilon^*\ p_1k\right]p_-^\mu\ , \
\ S_9^\mu = \ta k
\left[ p_1\epsilon^*\ p_2k - p_2\epsilon^*\ p_1k\right]p_+^\mu\ , \
\nonumber \\
\ S_{10}^\mu = \ta\epsilon^* p_+^\mu - \ta k
\left[ \frac{p_1\epsilon^*}{p_1k}p_2^\mu
+ \frac{p_2\epsilon^*}{p_2k}p_1^\mu\right]\ , \
\ S_{11}^\mu = \ta\epsilon^* p_-^\mu - \ta k
\left[ \frac{p_1\epsilon^*}{p_1k}p_2^\mu
- \frac{p_2\epsilon^*}{p_2k}p_1^\mu\right]\ , \
\nonumber \\
\ S_{12}^\mu = \epsilon^{*\mu}- \frac{p_1\epsilon^*}{p_1k}p_2^\mu
- \frac{p_2\epsilon^*}{p_2k}p_1^\mu\ , \
\nonumber \\
\ S_{13}^\mu =
\left[ p_1\epsilon^*\ p_2k - p_2\epsilon^*\ p_1k\right]p_-^\mu\ , \
\ S_{14}^\mu =
\left[ p_1\epsilon^*\ p_2k - p_2\epsilon^*\ p_1k\right]p_+^\mu\ , \
\nonumber
\eea
%
where $p_\pm = p_1\pm p_2$ and $\epsilon$ is the photon polarisation vector.
Within that
parameterisation all the coefficients, but the $F_1$,
are free from ultraviolet and infrared singularities, as the $S_1$
has exactly the spinor structure of the Born (ISR) amplitude. Moreover
most of the coefficients vanish in the massless limit and they are numerically
important only for the configurations with the photon collinear to one of the
initial leptons.
The diagrams in Fig.~\ref{diag_mu}c consist of box and pentabox
diagrams. The pentabox tensor integrals (up to the third rank), which
has appeared in the calculations, were reduced in D-dimension
to standard box diagrams (tensor integrals up to rank two)
with method equivalent to
the one presented in \cite{DenDit}, even if in this particular case
the reduction is simple due to the symmetry of the integrals.
Further reduction
is done using standard Passarino-Veltman reduction to scalar integrals.
The reduction of two remaining pentabox scalar integrals to the
box scalar integrals has introduced simple denominators ($kp_1-kp_2$)
and ($kq_1-kq_2$), which have zeros in the physical phase space.
That problem will be solved by means of the expansion of the resulting
around the mentioned zeros. Again here the remaining problem is
the size of the result after the tensor reduction and further
work is required to produce formulae, which can be used within a Monte
Carlo program.
\section{Summary}
A short review of experimental results obtained by means of the radiative
return method is presented, with an extensive discussion of
the theoretical basis of the method. The new version of the PHOKHARA
Monte Carlo event generator (PHOKHARA5.0) is presented. The status
of the work on the NLO radiative corrections to the reaction
$e^+e^- \to \mu^+\mu^- \gamma$, a luminosity monitoring
process for the radiative return method, is also outlined.
\vskip 0.1 cm
{\bf Acknowledgements}
The publication is based in a big part on results obtained in collaboration
with
J.~H.~K\"uhn and G.~Rodrigo. The authors are grateful for many
useful discussions concerning experimental aspects of the radiative
return method to members of the KLOE collaboration, mainly
Cesare Bini, Achim Denig, Wolfgang Kluge, Debora Leone, Stefan Miller,
Federico Nguyen and Graziano Venanzoni.
%*******************************************************************
\begin{thebibliography}{99}
%*******************************************************************
\bibitem{Davier:2003pw}
M.~Davier, S.~Eidelman, A.~Hocker and Z.~Zhang,
%``Updated estimate of the muon magnetic moment using revised results
% from e+e- annihilation,''
Eur.\ Phys.\ J.\ C {\bf 31}, 503 (2003)
[hep-ph/0308213].
%%CITATION = HEP-PH 0308213;%%
\bibitem{Jegerlehner:2003rx}
F.~Jegerlehner,
%``The role of sigma(e+ e- $\to$ hadrons) in precision tests of the standard
%model,''
Nucl.\ Phys.\ Proc.\ Suppl.\ {\bf 131}, 213 (2004)
[hep-ph/0312372].
%%CITATION = HEP-PH 0312372;%%
\bibitem{Nyffeler:2004mw}
A.~Nyffeler,
%``The muon g-2 in the standard model and beyond,''
Nucl.\ Phys.\ Proc.\ Suppl.\ {\bf 131}, 162 (2004).
%%CITATION = NUPHZ,131,162;%%
\bibitem{Zerwas}
Min-Shih Chen and P.~M.~Zerwas,
%"SECONDARY REACTIONS IN ELECTRON - POSITRON (ELECTRON) COLLISIONS"
Phys. Rev. D {\bf 11} (1975) 58.
%%CITATION = PHRVA,D11,58;%%
\bibitem{Nowak}
H.~Czy{\.z}, J.~H.~K{\"u}hn, E.~Nowak and G.~Rodrigo,
% ``NUCLEON FORM-FACTORS, B MESON FACTORIES AND THE RADIATIVE RETURN.''
Eur.~Phys.~J.~C {\bf 35} (2004) 527 [hep-ph/0403062].
%%CITATION = HEP-PH 0403062;%%
\bibitem{Czyz:2004nq}
%{\bf ``Charge asymmetry and radiative Phi decays''}
H.~Czy{\.z}, A.~Grzeli{\'n}ska and J.~H.~K{\"u}hn
Phys.\ Lett.\ B {\bf 611} (2005) 116
[hep-ph/0412239].
%%CITATION = HEP-PH 0412239;%%
\bibitem{Binner}
S.~Binner, J.~H.~K\"uhn and K.~Melnikov,
%``Measuring sigma(e+ e- --> hadrons) using tagged photon,''
Phys.\ Lett.\ B {\bf 459} (1999) 279
[hep-ph/9902399].
%%CITATION = HEP-PH 9902399;%%
\bibitem{CK}
H.~Czy\.z and J.~H.~K\"uhn,
%``Four pion final states with tagged photons at electron positron colliders,''
Eur.\ Phys.\ J.\ C {\bf 18} (2001) 497
[hep-ph/0008262].
%%CITATION = HEP-PH 0008262;%%
\bibitem{Szopa}
G.~Rodrigo, H.~Czy\.z, J.H.~K\"uhn and M.~Szopa,
%``Radiative return at NLO and the measurement of the hadronic cross-section in electron positron annihilation,''
Eur.\ Phys.\ J.\ C {\bf 24} (2002) 71
[hep-ph/0112184].
%%CITATION = HEP-PH 0112184;%%
\bibitem{rest}
H.~Czy{\.z}, A.~Grzeli{\'n}ska, J.~H.~K{\"u}hn and G.~Rodrigo,
%``The radiative return at Phi- and B-factories: Small-angle photon emission at next to leading order,''
Eur.\ Phys.\ J.\ C {\bf 27} (2003) 563
[hep-ph/0212225];
%%CITATION = HEP-PH 0212225;%%
\bibitem{FSR}
H.~Czy{\.z}, A.~Grzeli{\'n}ska, J.~H.~K{\"u}hn and G.~Rodrigo,
Eur.\ Phys.\ J.\ C {\bf 33} (2004) 333
[hep-ph/0308312];
%``The radiative return at Phi and B-factories: FSR at next-to-leading order,''
%%CITATION = HEP-PH 0308312;%%
\bibitem{PHOKHARA_mu}
H.~Czy{\.z}, A.~Grzeli{\'n}ska, J.~H.~K{\"u}hn and G.~Rodrigo,
%"The radiative return at Phi- and B-factories: FSR for muon
% pair production at next-to-leading order"
Eur. \ Phys. \ J. \ C {\bf 39} (2005) 411 [hep-ph/0404078].
%%CITATION = HEP-PH 0404078;%%
\bibitem{actaHN}
H.~Czy{\.z} and E.~Nowak,
%``e+ e- $\to$ pi+ pi- e+ e-: A potential background for sigma(e+ e- $\to$ pi+
%pi-) measurement via radiative return method,''
Acta Phys.\ Polon.\ B {\bf 34} (2003) 5231
[hep-ph/0310335].
%%CITATION = HEP-PH 0310335;%%
\bibitem{Czyz:2005}
H.~Czy{\.z}, E.~Nowak-Kubat, these proceedings.
\bibitem{PHradcor}
G.~Rodrigo, A.~Gehrmann-De Ridder, M.~Guilleaume and J.~H.~K\"uhn,
%``NLO QED corrections to ISR in e+ e- annihilation and the measurement of sigma(e+ e- $\to$ hadrons) using tagged photons,''
Eur.\ Phys.\ J.\ C {\bf 22} (2001) 81
[hep-ph/0106132].
J.~H.~K\"uhn and G.~Rodrigo,
%``The radiative return at small angles: Virtual corrections,''
Eur.\ Phys.\ J.\ C {\bf 25} (2002) 215
[hep-ph/0204283].
%%CITATION = HEP-PH 0106132;%%
%%CITATION = HEP-PH 0204283;%%
\bibitem{proc}
J.~H.~K{\"u}hn
%"Measuring sigma(e+ e- --> hadrons) with tagged photons at
% electron positron colliders"
Nucl. Phys. Proc. Suppl. {\bf 98} (2001) 289 [hep-ph/0101100].
G.~Rodrigo,
%``Radiative return at NLO and the measurement of the hadronic cross section,''
Acta Phys.\ Polon.\ B {\bf 32} (2001) 3833 [hep-ph/0111151].
G.~Rodrigo, H.~Czy\.z and J.~H.~K\"uhn,
%``Radiative return at NLO: The PHO\-KHA\-RA Monte Carlo generator,''
hep-ph/0205097; Nucl.Phys.Proc.Suppl.123(2003)167 [hep-ph/0210287];
Nucl.Phys.Proc.Suppl. {\bf 116} (2003) 249 [hep-ph/0211186].
H.~Czy{\.z}, A.~Grzeli{\'n}ska, J.~H.~K{\"u}hn and G.~Rodrigo,
%``Perspectives for the radiative return at meson factories,''
Nucl.~Phys.~Proc.~Suppl. {\bf 131} (2004) 39 [hep-ph/0312217].
%%CITATION = HEP-PH 0101100;%%
%%CITATION = HEP-PH 0111151;%%
%%CITATION = HEP-PH 0205097;%%
%%CITATION = HEP-PH 0210287;%%
%%CITATION = HEP-PH 0211186;%%
%%CITATION = HEP-PH 0312217;%%
\bibitem{FSR1}
H.~Czy{\.z} and A.~Grzeli{\'n}ska,
%``FSR at leading and next-to-leading order in the radiative return at meson
%factories,''
Acta Phys.\ Polon.\ B {\bf 34} (2003) 5219
[hep-ph/0310341].
%%CITATION = HEP-PH 0310341;%%
\bibitem{KLOE1}
A.~Aloisio {\it et al.}, [KLOE Collaboration],
Phys.~Lett. B {\bf 606} (2005) 12, [hep-ex/0407048].
%"Measurement of sigma(e+ e- --> pi+ pi- gamma) and
% extraction of sigma(e+ e- --> pi+ pi-) below 1-GeV with
% the KLOE detector"
%%CITATION = HEP-EX 0407048;%%
\bibitem{Akhmetshin:2003zn}
R.~R.~Akhmetshin {\it et al.} [CMD-2 Collaboration],
%``Reanalysis of hadronic cross section measurements at CMD-2,''
Phys.\ Lett.\ B {\bf 578} (2004) 285
[hep-ex/0308008].
%%CITATION = HEP-EX 0308008;%%
\bibitem{Aubert:2005eg}
B.~Aubert {\it et al.} [BABAR Collaboration],
%``The e+ e- $\to$ pi+ pi- pi+ pi- , K+ K- pi+ pi- , and K+ K- K+ K- cross
%sections at center-of-mass energies 0.5-GeV - 4.5-GeV measured with
%initial-state radiation,''
Phys.\ Rev.\ D {\bf 71} (2005) 052001
[hep-ex/0502025].
%%CITATION = HEP-EX 0502025;%%
\bibitem{Aubert:2004kj}
B.~Aubert {\it et al.} [BABAR Collaboration],
%``Study of e+ e- $\to$ pi+ pi- pi0 process using initial state radiation
%with BaBar,''
Phys.\ Rev.\ D {\bf 70} (2004) 072004
[hep-ex/0408078].
%%CITATION = HEP-EX 0408078;%%
\bibitem{Antonelli:92}
A.~Antonelli {\it et al.} [DM2 Collaboration]
%Measurement of the e+ e- $\to$ pi+ pi- pi0 and e+ e- $\to$
%omega pi+ pi- reactions in the energy interval 1350-MeV - 2400-MeV"
Z.\ Phys.\ C {\bf 56} (1992) 15.
%%CITATION = ZEPYA,C56,15;%%
\bibitem{Aubert:2003sv}
B.~Aubert {\it et al.} [BABAR Collaboration],
%``J/psi production via initial state radiation in e+ e- $\to$ mu+ mu- gamma
%at an e+ e- center-of-mass energy near 10.6-GeV,''
Phys.\ Rev.\ D {\bf 69} (2004) 011103
[hep-ex/0310027].
%%CITATION = HEP-EX 0310027;%%
\bibitem{Brambilla:2004wf}
N.~Brambilla {\it et al.},
%``Heavy quarkonium physics,''
CERN Yellow Report, CERN-2005-005, hep-ph/0412158.
%%CITATION = HEP-PH 0412158;%%
\bibitem{anulli}
F.~Anulli, these proceedings
\bibitem{Marianna}
M.~Testa, these proceedings
\bibitem{arrington}
J.~Arrington, Phys. Rev. {\bf C 68} (2003) 034325 [nucl-ex/0305009].
%"How well do we know the electromagnetic form factors of the proton?"
%%CITATION = NUCL-EX 0305009;%%
\bibitem{Bruch:2004py}
C.~Bruch, A.~Khodjamirian and J.~H.~K{\"u}hn,
%``Modeling the pion and kaon form factors in the timelike region,''
Eur.\ Phys.\ J.\ C {\bf 39}, (2005) 41
[hep-ph/0409080].
%%CITATION = HEP-PH 0409080;%%
\bibitem{new_3pi}
H.~Czy{\.z}, A.~Grzeli{\'n}ska, J.~H.~K{\"u}hn and G.~Rodrigo,
in preparation.
\bibitem{new_mu}
H.~Czy{\.z}, A.~Grzeli{\'n}ska, J.~H.~K{\"u}hn and G.~Rodrigo,
in preparation.
\bibitem{DenDit}
A.~Denner, and S.~Dittmaier, hep-ph/0509141.
%%CITATION = HEP-PH 0509141;%%
\end{thebibliography}
\end{document}