%Title: Radiative return at e+e- factories
%Author: H. Czyz, J. H. Kuehn, G. Rodrigo
%Published: Nucl.Phys.B(Proc.Suppl.) 116 (2003 Proceedings of RADCOR 2002 8-13 September 2002, Kloster Banz, Germany. ) 249-256.
%hep-ph/0211186
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\title{\phantom{}\vskip -1.2 cm
\hfill CERN-TH/2002-316
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Radiative return at \(e^+e^-\) factories.
\thanks{Work supported in part by BMBF under grant number 05HT9VKB0,
EC 5-th Framework EURIDICE network project HPRN-CT2002-00311 and
TARI project HPRI-CT-1999-00088;
presented by H. Czy{\.z} at RADCOR 2002, Kloster Banz,
September 8 - 13, 2002.}}
\author{
Henryk Czy\.z\address{{Institute of Physics, University of Silesia,
PL-40007 Katowice, Poland.}}$^{,}$
\thanks{Supported in part by
EC 5-th Framework, contract HPRN-CT-2000-00149; e-mail:czyz@us.edu.pl}%
Johann H. K\"uhn\address{Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany.}$^{,}$
\thanks{e-mail:jk@particle.uni-karlsruhe.de}%
Germ\'an Rodrigo\address{Theory Division,
CERN, CH-1211 Geneva 23, Switzerland.}$^{,}$
\thanks{Supported in part by
E.U. TMR grant HPMF-CT-2000-00989; e-mail: german.rodrigo@cern.ch}
}
\begin{document}
\begin{abstract}
The energy dependence of the electron - positron hadronic cross section
can be measured not only by a straightforward energy scan, but also
by means of the radiative return method. To provide extensive
comparisons between theory and experiment a Monte Carlo event generator
is an indispensable tool. We have developed such a generator called
PHOKHARA, which simulates
\(e^+e^-\to {\rm mesons} + {\rm photon(s)}\) processes.
In this paper we present its latest tests and upgrades.
\end{abstract}
% typeset front matter (including abstract)
\maketitle
\section{INTRODUCTION AND PHOKHARA UPGRADES}
In view of the precision of
the recent measurements of the muon anomalous magnetic moment
$a_{\mu} \equiv (g-2)_{\mu}/2$ at BNL~\cite{Brown:2001mg},
hadronic contributions are crucial for the interpretation of
this measurement, in particular for the isolation of
the electroweak or of non-Standard Model
physics contributions~\cite{Hughes:1999fp}.
Their understanding becomes even more important, as it seems
\cite{DEHZ} that the \(e^+e^-\) annihilation data are not consistent with
the \(\tau\) decay data.
A new \(a_{\mu}\) measurement, which is under way, will challenge the
theoretical predictions even more.
An important ingredient and the dominant source of uncertainties
in the theoretical prediction
for the muon anomalous magnetic moment
is the hadronic vacuum polarization ~\cite{hadronicmuon}.
It is in turn related via dispersion relations to the
cross section for electron--positron annihilation into hadrons
$\sigma_{had}=\sigma(e^+ e^- \rightarrow hadrons)$ and in some cases
via isospin symmetry to the decay width $\tau \to \nu_{\tau}+hadrons$.
This quantity plays an important role also in the
evolution of the electromagnetic coupling $\alpha_{QED}$ from the
Thompson limit to high energies~\cite{hadronicmuon,runningQED}.
The interpretation of improved measurements
at high energy colliders such as LEP, Tevatron, the LHC or TESLA
therefore depends significantly on the precise knowledge of $\sigma_{had}$.
The feasibility of using tagged photon events at high luminosity
electron--positron storage rings, such as the $\phi$-factory, DA$\Phi$NE,
CLEO-C or $B$-factories, to measure $\sigma_{had}$ over a wide range of
energies has been proposed and studied in detail
in~\cite{Binner:1999bt,Melnikov:2000gs,Czyz:2000wh}
(see also~\cite{Spagnolo:1999mt,Khoze:2001fs}).
The machine is operating at a fixed energy of the
\(e^+e^-\) centre-of-mass system (cms)
and the initial state radiation (ISR) is used to reduce
the invariant mass of the hadronic system.
The radiation of photons
from the hadronic system (the final state radiation, FSR)
should be considered as a background and can be suppressed by choosing
suitable kinematical cuts,
or controlled by the simulation, once a
suitable model for this amplitude has been adopted.
One finds that
selecting events with the tagged photons close to the beam axis and well
separated from the hadrons indeed reduces FSR drastically. As
demonstrated in Fig. \ref{fig:isrtofsr},
the FSR contribution to the total cross section
can be easily reduced to the 1\% level.
The model dependence of the FSR in the case of the \(\pi^+\pi^-\) final state
can be controlled by the same experiment through studies of the
forward--backward asymmetry of the angular charged pion distribution.
The asymmetry comes from FSR--ISR interference
and integrates to zero for 'charge blind' configurations. As a result
it does not contributes to rates in Fig. \ref{fig:isrtofsr}. It can be
used, however, to calibrate the FSR amplitude and more detailed
tests of its model dependence.
\vspace{-1. cm}
\begin{figure}[htb]
\vspace{9pt}
\epsfig{file=out1.ps,width=8 cm,}\vspace{-4. cm}
\caption{The role of the cuts in the suppression of the FSR contributions
to the cross section. Results from the PHOKHARA generator.
No cuts (upper curves) and suitable cuts applied (lower curves). }
\label{fig:isrtofsr}
\end{figure}
When running at higher energies,
the FSR is suppressed with respect to the ISR by the different
behaviour of various propagators and form factors relevant to the problem.
In practice it means that no special angular cuts are needed
to suppress the FSR contribution when running at high energies.
The suppression of the FSR overcomes the problem of its model
dependence, which must be taken into account in a completely inclusive
measurement~\cite{Hoefer:2001mx}.
Preliminary experimental results using this method have
been presented recently by the KLOE collaboration at
DA$\Phi$NE~\cite{Aloisio:2001xq,Denig:2001ra,Adinolfi:2000fv,Denig:2002}.
Large event rates were also observed by the BaBar
collaboration~\cite{babar}.
In the first version of the newly developed Monte Carlo program
PHOKHARA \cite{RCKS}
we have considered the full next-to-leading order (NLO) QED
corrections to the ISR in the annihilation
process $e^+ e^- \rightarrow \gamma + hadrons$, for the case
where the photon is
observed under a non-vanishing angle relative to the beam direction.
The virtual and soft photon corrections were presented in \cite{Rodrigo:2001jr}
and the contribution of the emission of a second hard photon
in \cite{RCKS}. The final hadronic state was limited to the \(\pi^+\pi^-\),
with the hadronic current modeled as in \cite{Kuhn:1990ad},
and the final state emission was not included. The program allowed also
for the generation of \(\mu^+\mu^-\gamma(\gamma)\) final states, again
limited to the emission of photon(s) from the initial leptons.
Radiative
corrections proportional to (\(\alpha m_e^2\) )
relevant to configurations with photons emitted at very
low (\(\simeq m_e/\sqrt{s}\))
angles relative to the beam direction were calculated in \cite{RK02}
and are included in the new version of PHOKHARA
\cite{CGKR}. The leading order corrections proportional to \(m_e^2\)
are typically of the order of a few per cent \cite{Rodrigo:2001cc}, while
the non-leading ones are of order of 0.1\%, as seen from
Fig.\ref{fig:nlo_mass}. They will be important when the precision of
the measurement will be below 1\%.
%
\begin{figure}[htb]
\epsfig{file=nlo_mass_1_4_10gev_n.ps,width=7.5 cm,}
\caption{The relative contributions of the non-leading mass corrections to the
differential cross section at \(\sqrt{s}\) = 1 GeV, 4 GeV and 10 GeV. }
\label{fig:nlo_mass}
\end{figure}
Their effect depends on \(Q^2\) and thus affects
the \(Q^2\) distribution from which the hadronic
cross section is extracted.
%
\begin{figure}[htb]
\epsfig{file=pipi_ll_nlo_diff_l.ps,width=7.5 cm,}
\caption{The relative
non-leading contributions to the differential cross section
at \(\sqrt{s}\) = 4 GeV. NLO - full next-to-leading result,
LL - leading logarithmic approximation. }
\label{fig:ll_nlo_1}
\end{figure}
Another new feature of the PHOKHARA event generator is the inclusion
of the four-pion hadronic final states (\(2\pi^+ 2\pi^-\) and
\( 2\pi^0\pi^+\pi^-\) ). The description of the hadronic current in that
case is based on the paper \cite{Fink}, with changes described in
\cite{Czyz:2000wh}.
The comparison with the Monte Carlo, which simulates
the same process at leading order \cite{Czyz:2000wh}
and includes additional collinear
radiation through structure function (SF) techniques, shows typical difference
of order of 1\% as seen in Fig. \ref{fig:ll_nlo_1} (a similar behaviour can be
observed for \( 2\pi^0\pi^+\pi^-\) final state).
The non-leading contributions to the cross section of the four-pion
final states are of the expected
size and of the same order as for the two-pion final state \cite{RCKS}.
The program now includes also the contributions from the final state
emitted photons together with the ISR--FSR interference calculated at
the lowest order for \(\pi^+\pi^-\) and \(\mu^+\mu^-\) final states,
while for the four-pion final states the FSR contribution is not taken into
account.
\section{TESTS OF PHOKHARA}
An obvious and one of the most important tasks
in the construction of a Monte Carlo event generator is to demonstrate
that its technical accuracy is much better than the desired physical
accuracy. The tests that were performed for the previous version of
PHOKHARA \cite{RCKS} are still valid, within the limitations
of this version, which was applicable for non-vanishing photon angles.
%
\begin{figure}[htb]
\epsfig{file=e43_pi0_diff.ps,width=7.5 cm,}
\caption{The relative difference of the differential cross sections for two
different values of the separation parameter \(\epsilon\).}
\label{fig:e43}
\end{figure}
%
\begin{figure}[htb]
\epsfig{file=e54_pi0_diff.ps,width=7.5 cm,}
\caption{The relative difference of the differential cross sections for two
different values of the separation parameter \(\epsilon\).}
\label{fig:e54}
\end{figure}
In the first step we demonstrate
the independence of total and differential cross sections
of the separation parameter \(\epsilon\) (called \(w\) in \cite{RCKS})
between soft and hard photon
regions. The soft photon contribution is calculated analytically,
while the additional hard photon is treated
via Monte Carlo simulation. The parameter that specifies the
separation between the
two regions of the phase space
has to be kept small enough to validate the soft photon
approximation and large enough to avoid negative weights.
We performed the tests for
a \(\pi^+\pi^-\) hadronic final state in \cite{RCKS}, while for one of the
four-pion modes the results are collected in Figs. \ref{fig:e43} and
\ref{fig:e54}. From Fig. \ref{fig:e43} it is clear that the choice
\(\epsilon=10^{-3}\) is still too big, whereas Fig. \ref{fig:e54} demonstrates
the stability of the results between \(\epsilon=10^{-4}\)
and \(\epsilon=10^{-5}\). This also proves that the Monte Carlo integration
works well in the soft photon region.
%
\begin{figure}[htb]
\epsfig{file=Fig3.ps,width=7.5 cm,}
\caption{The relative difference between differential cross sections
obtained by the PHOKHARA Monte Carlo generator (MC) and a Gauss numerical
integration (Gauss).}
\label{fig:Fig3}
\end{figure}
The Monte Carlo integration of the part of the program that simulates
one hard large-angle photon
emission was tested in \cite{RCKS}
against a Gauss numerical integration. As shown in Fig. \ref{fig:Fig3}
a technical precision of the program at the level of \(10^{-4}\) was
demonstrated.
%
\begin{figure}[htb]
\epsfig{file=small_1gev.ps,width= 8 cm,}
\caption{A comparison between PHOKHARA and analytical \cite{BNB} results. }
\label{fig:small_1gev}
\end{figure}
Analytical results exist for the differential (in \(Q^2\)) cross section
integrated over the whole angular range of the photon(s) for both one and
two emitted photons \cite{BNB}. The comparison with PHOKHARA
can be found in Fig. \ref{fig:small_1gev}, demonstrating again an excellent
technical precision also for the two-photon final state.
The results presented in Fig. \ref{fig:small_1gev}
refer to the sum of virtual and hard corrections to the
\(e^+e^-\to\pi^+\pi^-\gamma\) cross section, while
more detailed tests can be found in \cite{CGKR}.
\section{CONCLUSIONS}
The PHOKHARA Monte Carlo event generator was upgraded, allowing
for simulation in the small photon angles region.
Besides the \(\pi^+\pi^-\) hadronic final state,
its present version includes also
\(2\pi^+2\pi^-\) and \(2\pi^0\pi^+\pi^-\) final states.
For the \(\pi^+\pi^-\) and \(\mu^+\mu^-\) final states, the FSR photonic
contributions were implemented at the lowest order,
including ISR--FSR interference.
Further upgrades
are in progress.
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\end{document}