%Title: Electron-positron annihilation into three pions and the radiative return.
%Author: Henryk Czyz, Agnieszka Grzelinska, Johann H. K\"uhn, German Rodrigo
%Published: * Eur.Phys.J.C * ** 47 ** (2006) 617-624.
%hep-ph/0512180
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\headnote{
\hfill \parbox{4cm}{\tt \normalsize IFIC/05-38 \\ TTP05-28 \\}}
\title{
Electron-positron annihilation into three pions
and the radiative return.
\thanks{Work
supported in part by BMBF under grant number 05HT4VKA/3,
EC 5th Framework Programme under contract HPRN-CT-2002-00311
(EURIDICE network), TARI project RII3-CT-2004-506078,
Polish State Committee for Scientific Research (KBN)
under contract 1 P03B 003 28,
Ministerio de Educaci\'on y Ciencia under grant FPA2004-00996 (PARSIFAL),
and Generalitat Valenciana (GV05-015, GV04B-594 and GRUPOS03/013).}}
\author{Henryk Czy\.z\inst{1}
\thanks{\email{czyz@us.edu.pl}}
\and
Agnieszka Grzeli{\'n}ska\inst{2}
\thanks{\email{grzel@joy.phys.us.edu.pl}}
\and
Johann H. K\"uhn\inst{2}
\thanks{\email{johann.kuehn@uni-karlsruhe.de}}
\and
Germ\'an Rodrigo\inst{3}
\thanks{\email{german.rodrigo@ific.uv.es}}
}
\institute{
Institute of Physics, University of Silesia, PL-40007 Katowice, Poland.
\and
Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany.
\and
Instituto de F\'{\i}sica Corpuscular, CSIC-Universitat de Val\`encia,
Apartado de Correos 22085, E-46071 Valencia, Spain.
}
\date{\today}
\abstract{
The Monte Carlo event generator PHOKHARA, which simulates hadron
and muon production
at electron-positron colliders through radiative return, has been extended
to final states with three pions. A model for the form factor based
on generalized vector dominance has been employed, which is consistent
with presently available experimental observations.
}
\begin{document}
\authorrunning{H. Czy\.z et al. }
\titlerunning{Electron-positron annihilation into three pions ...
}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Measurements of form factors and cross sections for electron positron
annihilation in the low energy region provide important information
on hadron dynamics. At the same time, they are necessary ingredients
in dispersion relations, which are used to predict hadronic contributions
to the momentum dependent electromagnetic coupling and the anomalous
magnetic moment of the muon. The measurements are traditionally performed
by tuning the center of mass energy of an electron positron collider
to the point of interest. As an alternative, it has been advocated
to use the method of the radiative return \cite{Zerwas,Binner:1999bt} at
high luminosity $\phi$- and $B$-meson factories. In this second case
the collider energy remains fixed, while $Q^2$, the invariant mass
of the hadronic system, can be varied by considering events, where
one or several photons have been radiated. For a detailed and precise
analysis initial and final state radiation must be included
and a proper description of the various exclusive final states through
appropriate form factors is required. All these ingredients are contained
in the most recent version of the Monte Carlo event generator
PHOKHARA4.0 \cite{Nowak,Czyz:PH04},
which is based in particular on the virtual corrections described in
\cite{Rodrigo:2001jr,Kuhn:2002xg} and
which at present simulates production of
$\mu^+\mu^-$, $\pi^+\pi^-$, four pions
($2\pi^+2\pi^-$ and $\pi^+\pi^-2\pi^0$), $p\bar p$, and $n \bar n$
\cite{Nowak,Czyz:PH04,Rodrigo:2001jr,Kuhn:2002xg,Rodrigo:2001kf,Czyz:2002np,Czyz:PH03}.
In the present work the production of three pions is considered
on the basis of form factors, which include already available information
on the production cross section and on differential distributions
in two-pion subsystems. The model implements three-pion production
through $\omega$, $\phi$ and their radial excitations and the subsequent
decay of these resonances into $\rho\pi$, $\rho'\pi$ and $\rho''\pi$.
A small isospin-violating component $\gamma^*\to\omega(\to\pi^+\pi^-)\pi^0$
is added, which is needed to properly describe the data.
The total cross section and the distributions are well reproduced
within this model. It is furthermore demonstrated that the couplings
introduced and adopted for this purpose lead to a satisfactory
description of $\Gamma(\pi^0\to\gamma\gamma)$, of the slope parameter
of the $\pi^0\to\gamma\gamma^*$ amplitude and of the radiative
vector meson decays $\rho\to \pi^0\gamma$, $\phi\to \pi^0\gamma$,
but is in conflict with $\omega\to \pi^0\gamma$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A phenomenological description of three-pion production}
The amplitude for three-pion production through the electromagnetic current
is restricted by current conservation and negative parity
to the form
\bea
J_{\nu}^{\mathrm{em},3\pi} &=&
\langle\pi^+(q_+) \ \pi^-(q_-) \ \pi^0 (q_0)|J_{\nu}^{\mathrm{em}}
|0\rangle\nonumber \\
&=& \epsilon_{\nu \alpha \beta \gamma} q_{+}^{\alpha}
q_{-}^{\beta} q_{0}^{\gamma} \ F_{3\pi}(q_{+},q_{-},q_{0}) \ .
\label{3pi_p}
\eea
%
G-parity dictates dominance of the isospin-zero component of
the electromagnetic current , which will be discussed in a first step.
The small isospin-one admixture will be discussed subsequently.
The form factor $F_{3\pi}^{I=0}$ is constructed under the
assumption that the virtual photon couples to the $\omega$-
and $\phi$-meson, whose subsequent transition to three pions
is dominated by the $\rho(\to 2\pi)\pi$ chain (Fig.\ref{3pi_diag})
\cite{Gell-Mann}.
Taking into account radial excitations of $\omega$, $\phi$, $\rho$
one arrives at the form factor
\bea
\kern-15pt F_{3\pi}^{I=0}&&(q_{+},q_{-},q_{0}) = \nonumber \\
&&\sum_{i,j}a_{ij} \cdot BW_{V_i}(Q^2)
\cdot H_{\rho_j}\left(Q_+^2,Q^2_-,Q_0^2\right)
\ ,
\label{3pi_pb}
\eea
%
%
where $V_i$ stands for either $\omega$- or $\phi$-resonances,
and $\rho_j$ represents contributions from $\rho$-mesons.
%
\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.5cm,height=4.5cm]{pi3_1.ps}
\caption{Diagrams contributing to the 3--pion current: $I=0$ component.}
\label{3pi_diag}
\end{center}
\end{figure}
%
From the PDG \cite{PDG04} it is clear that all
$\omega$- and $\phi$-resonances couple to the ground state
of the $\rho$-meson, whereas
there is no indication about couplings to the higher radial excitations
($\rho',\rho'',\cdots$). This missing piece of information
can, however, be obtained to large extent from the
known $e^+e^-\to \pi^+\pi^-\pi^0$ cross section, as shown below.
For the function $H_\rho$ we shall adopt the ansatz
\bea
H_{\rho}(Q_{+}^{2},Q_{-}^{2},Q_{0}^{2}) = BW_{\rho}(Q_{0}^{2})
+ BW_{\rho}(Q_{+}^{2}) + BW_{\rho}(Q_{-}^{2}) \ , \nonumber \\
\phantom{}\kern-15pt
\eea
\noindent
with
\bea
Q_{0}^{2} = (q_{+}+q_{-})^2, \
Q_{\pm}^{2} = (q_{\mp}+q_{0})^2, \
\eea
%
and the Breit-Wigner form factors are
\bea
BW_{V}(Q^2) &=& \biggl[ \frac{Q^2}{m_V^2} - 1 + i
\frac{\Gamma_V}{m_V} \biggr]^{-1} \ , \nonumber \\
BW_{\rho}(Q^{2}_{i}) &=& \biggl[ \frac{Q^{2}_{i}}{m_{\rho}^2} - 1
+ i \frac{\sqrt{Q^2_{i}}\Gamma_{\rho}(Q^{2}_{i},m_j,m_k) }{m_{\rho}^2} \biggr]^{-1}
,
\label{BW}
\eea
%
where $Q_i^2 = (q_j + q_k)^2$, and $m_{j} = m_{\pi^j}$, with
$i,j,k = 0,\pm$.
We use
propagators with constant widths for $\omega$'s and $\phi$, and
energy dependent widths for $\rho$- resonances as predicted by P-wave
$\rho\to\pi\pi $ decays:
%
\bea
\Gamma_{\rho}(Q^2_{i},m_j,m_k) = \Gamma_{\rho} \frac{m_{\rho}^2}
{Q^2_{i}}\biggl[
\frac{Q^2_{i} - (m_j+m_k)^2 }
{m_{\rho}^2 - (m_j+m_k)^2 } \biggr]^{3/2} \ .
\eea
%
The couplings $a_{ij}$ are taken as real constants and
we assume that the isospin symmetry is violated in this component only by the
$\pi^0-\pi^\pm$ mass difference.
%
\begin{figure}[ht]
\begin{center}
\epsfig{file=pi3_2.ps,width=8.cm}
\caption{Diagram contributing to the $I=1$ component
of the three-pion current.}
\label{3pi_diag1}
\end{center}
\end{figure}
%
The small isospin violating amplitude is mediated by the $I=1$
component of the electromagnetic current and is based on
the $4\pi$ current of Refs. \cite{Czyz:2000wh,Decker}
i.e. we take the $\rho-\gamma$ and $\rho\pi\omega$ couplings
from the $4\pi$ current and replace the $\omega\to 3\pi$ transition
by the isospin violating $\omega\to 2 \pi$ decay
as shown in Fig.\ref{3pi_diag1}.
This leads to the following ansatz
%
\bea
F_{3\pi}^{I=1}(q_{+},q_{-},&&q_{0}) = G_{\omega}
\cdot BW_{\omega}(Q^2)/\tilde m^2_{\omega}\nonumber \\
&&\kern-15pt\left[BW_{\rho}(Q_0^2)/\tilde m^2_\rho+\sigma BW_{\rho''}(Q_0^2)/
\tilde m^2_{\rho''}\right] \ \ ,
\label{i0}
\eea
%
\noindent
where
\bea
G_{\omega}= \frac{1.55}{\sqrt{2}}\ 12.924\ {\rm GeV}^{-1} \ 0.266 \ m_\rho^2
\ g_{\omega\pi\pi}
\label{i0p}
\eea
%
\noindent
and $\tilde m_\rho = 0.77609$ GeV, $\tilde\Gamma_\rho = 0.14446$ GeV,
$\tilde m_{\rho''} = 1.7$ GeV, $\tilde\Gamma_{\rho''} = 0.26$ GeV,
$\sigma=-0.1$, where the parameters are taken directly
from \cite{Decker}.
At the present level of experimental accuracy it is not clear
whether the $\rho''$ term is necessary for the description of
the 3$\pi$ current (see
below). However, as it is a prediction coming from the 4$\pi$ current
we consider its contribution also here.
%
\begin{figure*}[ht]
\begin{center}
\epsfig{file=v_660_760_n.ps,width=8.25cm}
\epsfig{file=v_760_800_n.ps,width=8.5cm}
\caption{$e^+e^-\to \pi^+\pi^-\pi^0$ cross section obtained with
fitted parameters (solid line, see text for details) vs. experimental data.}
\label{date:1}
\end{center}
\end{figure*}
%
%
\begin{figure*}[ht]
\begin{center}
\epsfig{file=v_800_1000_n.ps,width=8.5cm}
\epsfig{file=v_1000_1050_n.ps,width=8.5cm}
\caption{$e^+e^-\to \pi^+\pi^-\pi^0$ cross section obtained with
fitted parameters (solid line, see text for details) vs. experimental data.}
\label{date:2}
\end{center}
\end{figure*}
%
\begin{figure}[ht]
\begin{center}
\epsfig{file=v_1050_3000_n.ps,width=8.5cm}
\caption{$e^+e^-\to \pi^+\pi^-\pi^0$ cross section obtained with
fitted parameters (solid line, see text for details) vs. experimental data. }
\label{date:3}
\end{center}
\end{figure}
%
The coupling $g_{\omega\pi\pi}$ can be extracted from the
decay rate $\Gamma(\omega\to\pi\pi)$
(see Eq.(\ref{gamrhopipi})).
Using the world average value from the PDG \cite{PDG04}
one gets $g_{\omega\pi\pi}= 0.185(15)$.
The total form factor is of course given by the coherent sum
\bea
\kern-15pt F_{3\pi}(q_{+},q_{-},q_{0}) = F_{3\pi}^{I=0}(q_{+},q_{-},q_{0})
+ F_{3\pi}^{I=1}(q_{+},q_{-},q_{0}) \ .
\label{3pi_pa}
\eea
Data on $\sigma(e^+e^-\to\pi^+\pi^-\pi^0)$, from
energy scan experiments
\cite{CMD2,SND,Dol:91,DM1,Antonelli:92}
consist of 217 data points, covering the energy range from $~660$ MeV
to $~2400$ MeV. Moreover, by using the radiative return method,
BaBar \cite{babar3pi} has obtained additional 78 data points,
covering
the energy from 1.06 GeV up to almost 3 GeV.
Data from the DM2 collaboration
\cite{Antonelli:92} being inconsistent
with the more accurate BaBar data, were not used in the fit.
%
\begin{table}[hb]
\begin{center}
\caption{Values of the couplings masses and widths obtained in the fit;
couplings $A-F$ in ${\rm GeV}^{-3}$ masses and widths
in ${\rm MeV}$ (see text for details).}
\label{tab:cons_F_c}
\begin{tabular}{l|l|l|l}\hline
$m_{\omega(782)}$ & 782.4(4) & $A $ & 18.20(8) \\ \hline
$\Gamma_{\omega(782)}$ & 8.69(7)& $B $ & -0.87(5) \\ \hline
$m_{\phi(1020)}$ & 1019.24(3) & $C $ & -0.77(5) \\ \hline
$\Gamma_{\phi(1020)}$ & 4.14(5) & $D $ & -1.12(4) \\ \hline
$m_{\omega(1420)}$ & 1375(1) & $E $ & -0.72(10) \\ \hline
$\Gamma_{\omega(1420)}$ & 250(5)& $F $ & -0.59(4) \\ \hline
$m_{\omega(1650)}$ & 1631(6) & & \\ \hline
$\Gamma_{\omega(1650)}$ & 245(13)& $\chi^2/\mathrm{d.o.f}$& 1.14 \\ \hline
\end{tabular}
\end{center}
\end{table}
%
\begin{figure}[ht]
\begin{center}
\epsfig{file=test_v_dane_0.75_plpm.ps,width=4.1cm}
\epsfig{file=test_v_dane_0.75_pl0.ps,width=4.cm} \\
\epsfig{file=test_v_dane_1._plpm.ps,width=4.05cm}
\epsfig{file=test_v_dane_1._pl0.ps,width=4.cm}\\
\epsfig{file=test_v_dane_1.1_plpm.ps,width=4.0cm}
\epsfig{file=test_v_dane_1.1_pl0.ps,width=4.2cm}\\
\includegraphics[width=3.8cm,height=4.55cm]{test_v_dane_1.4_plpm.ps}
\includegraphics[width=3.8cm,height=4.7cm]{test_v_dane_1.4_pl0.ps}
\caption{Two pion invariant mass distributions for four different
ranges of $\pi^+\pi^-\pi^0$ invariant mass.
The BaBar data points, given as events/bin,
are superimposed
on plots obtained by PHOKHARA
(see text for details). }
\label{subdistr}
\end{center}
\end{figure}
To fit the experimental data we include the following
resonances:
$\omega(782)$, $\omega'\equiv \omega(1420)$, $\omega''\equiv\omega(1650)$,
$\phi(1020)$, $\rho(770)$,
$\rho'\equiv \rho(1450$) and $\rho''\equiv \rho(1700)$.
The strategy was to minimize the number of
contributions in Eq.(\ref{3pi_pb}) and arrive
at a good fit in the same time.
Our best fit for
the isospin zero component reads
%
\begin{align}
& F_{3\pi}^{I=0}(q_{+},q_{-},q_{0}) =
H_{\rho(770)} (Q_{+}^{2},Q_{-}^{2},Q_{0}^{2})\nonumber \\ &
\kern+50pt \cdot \biggl[
{ A \cdot BW_{\omega(782)}}(Q^2)
+ B \cdot { BW_{\phi(1020)}} (Q^2) \nonumber \\ &
\kern+50pt + C \cdot BW_{\omega(1420)} (Q^2)
+ D \cdot BW_{\omega(1650)} (Q^2)
\biggr] \nonumber \\ &
\kern+20pt
+ E \cdot { BW_{\phi(1020)}} (Q^2)\cdot
H_{\rho(1450)} (Q_{+}^{2},Q_{-}^{2},Q_{0}^{2})\nonumber \\ &
\kern+20pt + F \cdot BW_{\omega(1650)} (Q^2)
\cdot H_{\rho(1700)} (Q_{+}^{2},Q_{-}^{2},Q_{0}^{2})
\ ,
\label{3pi_fac}
\end{align}
%
with the couplings and masses
given in Table \ref{tab:cons_F_c}.
The errors are the
parabolic errors calculated by the MINUIT processor MINOS and correspond
to the change of $\Delta\chi^2=1$.
The value
$\chi^2/\mathrm{d.o.f} =1.14$ is at the edge of the 95\% confidence
interval. It is to large extent a result of the fact that the data
of different experiments are only marginally mutually consistent
as evident from Figures \ref{date:1}--\ref{date:3}.
Lack of published numerical information on differential distributions
does not allow for further refinements
of the model. In particular, indications for additional contributions
from higher radial excitations in the region of large $Q^2$ cannot be
substantiated by more detailed fits.
Access to distributions in invariant
masses of pion pairs and to pion angular distributions would allow
to study these effects.
\begin{table}[ht]
\begin{center}
\caption{The masses and widths of $\rho$ resonances used in the fits}
\label{rho}
\begin{tabular}{l|l|l|l}
& $\rho(770)$& $\rho(1450)$&$\rho(1700)$ \\ \hline
$m$ (GeV) & 0.77609& 1.465& 1.7\\ \hline
$\Gamma$ (GeV) & 0.14446& 0.31&0.235 \\ \hline
\end{tabular}
\end{center}
\end{table}
%
For the moment we
set the masses and widths of the $\rho$ mesons to the values
collected in Table \ref{rho} and assume equal masses and widths
for the neutral
and charged $\rho$ mesons.
Nevertheless qualitative
comparisons of two--pion invariant mass distributions
can be performed by superimposing the experimental and the model
(data from Fig. 15 of \cite{babar3pi} and differential cross sections
generated with
PHOKHARA 5.0)
distributions (see Fig.\ref{subdistr}).
The $I=0$ component alone predicts identical distributions in
$M_{\pi^+\pi^-}$ and $M_{\pi^\pm\pi^0}$. The small $I=1$ component
is concentrated in the spike at $M_{\pi^+\pi^-}=m_\omega$.
This channel starts to contribute for $Q^2$ values above 1~GeV only.
The model seems to describe the distributions reasonably well.
However the following deviations are observed:
In the lowest region (0.75~GeV $< M_{3\pi} <$ 0.82~GeV) the experimental
results for the distributions in $M_{\pi^+\pi^-}$ and $M_{\pi^\pm\pi^0}$,
respectively,
seem to differ in the upper range, an indication of isospin-violation,
that cannot be reproduced by our ansatz. In the large $Q^2$ range
(1.4~GeV $< M_{3\pi} <$ 1.8~GeV) an excess is observed in both charge
modes for masses of the two-pion system between 1~GeV and 1.2~GeV.
A similar excess is not observed in the pion form factor.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Meson couplings and partial decay widths.}
From the results of the fit we can evaluate the
meson couplings separately, combining the following relations
%
\bea
A&=&2 g_{\omega\gamma}\cdot g_{\omega\pi\rho}\cdot g_{\rho\pi\pi}
\nonumber \\
B&=&2 g_{\phi\gamma}\cdot g_{\phi\pi\rho}\cdot g_{\rho\pi\pi}
\nonumber \\
C&=&2 g_{\omega'\gamma}\cdot g_{\omega'\pi\rho}\cdot g_{\rho\pi\pi}
\nonumber \\
D&=&2 g_{\omega''\gamma}\cdot g_{\omega''\pi\rho}\cdot g_{\rho\pi\pi}
\nonumber \\
E&=&2 g_{\phi\gamma}\cdot g_{\phi\pi\rho'}\cdot g_{\rho'\pi\pi}
\nonumber \\
F&=&2 g_{\omega''\gamma}\cdot g_{\omega''\pi\rho''}\cdot g_{\rho''\pi\pi} \ ,
\label{couplings}
\eea
%
with information about partial decay widths.
From the known decay widths of $\rho^0,\omega$ and $\phi$ to two pions
one determines the couplings $g_{V\pi\pi}$ ($V = \rho^0,\omega,\phi$):
%
\bea
\Gamma_{V\to \pi^+\pi^-} = g_{V\pi\pi}^2 \frac{m_V}{48\pi}
\left[1-\frac{4m_{\pi^+}^2}{m_V^2}\right]^{3/2} \ .
\label{gamrhopipi}
\eea
%
Similarly, the $g_{V\gamma}$ couplings can be found from the measured
values of $\Gamma(V\to e^+e^-)$:
%
\bea
\Gamma_{V\to e^+e^-} = g_{V\gamma}^2 \frac{4\pi\alpha^2}{3m_V^3} \ .
\label{gamrhoee}
\eea
%
For the $\omega'$ and $\omega''$ we do not know
the partial decay widths and the couplings cannot be extracted
separately. The numerical values of the couplings
obtained from Eq.(\ref{couplings})
and partial decay widths \cite{PDG04} are collected in Table
\ref{tab:couplings}.
%
\begin{table}[hb]
\begin{center}
\caption{Values of the three-- and two--particle couplings;
\ \ $g_{V\pi\pi}$ is dimensionless, $g_{V\gamma}$ in ${\rm GeV}^{2}$,
$g_{V\pi\rho}$ in ${\rm GeV}^{-5}$.}
\label{tab:couplings}
\begin{tabular}{l|l|l|l}\hline
$g_{\rho\pi\pi}$ & 5.997(32) & $g_{\rho\gamma} $ & 0.1212(13)
\\ \hline
$g_{\omega\pi\pi}$ & 0.185(15) & $g_{\omega\gamma} $ & 0.03591(37)
\\ \hline
$g_{\phi\pi\pi}$ & 0.0072(6) & $g_{\phi\gamma} $ & 0.0777(7)
\\ \hline
$g_{\omega\pi\rho}$ & 42.3(5) &$g_{\omega'\gamma}\cdot g_{\omega'\pi\rho}$ & -0.064(8) \\ \hline
$g_{\phi\pi\rho}$ & -0.93(5)
& $g_{\omega''\gamma}\cdot g_{\omega''\pi\rho}$ & -0.093(3) \\ \hline
\end{tabular}
\end{center}
\end{table}
%
Having extracted the couplings, we are able to predict many
physical quantities that have been measured
already and check the model.
In particular we will investigate if various meson--photon interactions
can be modeled, as in the $3\pi$ case, by three--meson couplings and
vector meson-photon mixings with the couplings and propagators as
introduced above
(for a review of alternative models see \cite{BGP_DAPHNE,Harada}).
As one can recognize our model is an extension to higher radial
excitations of the model outlined in \cite{Gell-Mann}.
Let us start with the decay width of $\pi^0\to\gamma\gamma$.
The model is based on the diagrams
shown in Fig.\ref{pi0gamgma}. As indicated by the fit,
contributions from $\rho'-\gamma$ and $\rho''-\gamma$ mixings are
not required at the present level of precision.
%
\begin{figure}
\begin{center}
\includegraphics[width=4.cm,height=4cm]{pi3_3.ps}
\caption{Diagrams contributing to $\pi^0\to\gamma\gamma$ decay.}
\label{pi0gamgma}
\end{center}
\end{figure}
\begin{table}[hb]
\begin{center}
\caption{ Mean life time for $\pi^0\to 2\gamma$ in seconds and decay rates
for $\rho,\omega,\phi \to \pi^0\gamma$ in MeV as
obtained within our model compared to experimental
results \cite{PDG04}.}
\label{direct}
\begin{tabular}{l|l|l}
& model & experiment \\ \hline
$\tau(\pi^0\to\gamma\gamma)$ & 6.6(3)$\cdot 10^{-17}$
& 8.3(6)$\cdot 10^{-17}$\\ \hline
$\Gamma(\rho^0\to\pi^0\gamma)$ & 0.078(3)
& 0.090(20) \\ \hline
$\Gamma(\omega\to\pi^0\gamma)$
& 1.31(4)
& $0.757^{+25}_{-22}$\\ \hline
$\Gamma(\phi\to\pi^0\gamma)$ & 4.2(5)$\cdot 10^{-3}$
& 5.2(4)$\cdot 10^{-3}$\\ \hline
\end{tabular}
\end{center}
\end{table}
%
%
The partial decay width of the decay $\pi^0\to\gamma\gamma$ is thus given
in our model by
%
\bea
\Gamma_{\pi^0\to \gamma\gamma} &=& \pi\alpha^2 m_{\pi^0}^3\
g^2_{\rho\gamma} \
T^2\ ,
\label{gampi0gg}
\eea
%
with
\bea
T=
g_{\omega\pi\rho}\cdot g_{\omega\gamma}
+g_{\phi\pi\rho}\cdot g_{\phi\gamma}
+g_{\omega'\pi\rho}\cdot g_{\omega'\gamma}
+g_{\omega''\pi\rho}\cdot g_{\omega''\gamma} \nonumber \\
\label{TT}
\eea
%
This ansatz leads to
$\tau_{\pi^0} = 6.6(3) \cdot 10^{-17}$~s
for the $\pi^0\to\gamma\gamma$ lifetime
to be compared with the PDG \cite{PDG04} value
$\tau_{\pi^0} = 8.3(6) \cdot 10^{-17}$~s.
The $\omega'$ and $\omega''$ contributions are small but not negligible and
the value of the $\pi^0$ lifetime obtained without them reads
$\tau_{\pi^0}(\omega \ {\rm and} \ \phi \ {\rm only})=
5.21(13)\cdot 10^{-17}$~s.
Within two and a half
standard deviations experiment and the model are in agreement
($experiment-model = (1.7\pm 0.7)\cdot 10^{-17}$~s)
and the remaining discrepancy can be attributed to the missing
$\rho$-resonances or a more complicated $Q^2$-dependence
of the propagators. One has to remember that we make here
extrapolation from the region of $\omega,\phi,\cdots$ resonances
where the fit was performed
to the region of the applicability of the chiral limit.
Thus we should recover here the chiral theory result for the
$\pi^0\to\gamma\gamma$ lifetime (see \cite{BGP_DAPHNE})
\bea
\Gamma(\pi^0\to\gamma\gamma)= \frac{\alpha^2m_{\pi^0}^3}{32\pi^3f^2}\ ,
\eea
\noindent
which gives $\tau_{\pi^0} =8.69\cdot 10^{-17}$~s for f=132~MeV.
Rephrasing the same statement using the coupling constants one approximately
expects
\bea
\frac{1}{32\pi^4f^2} \simeq g^2_{\rho\gamma} T^2 + \cdots \ ,
\eea
where the dots correspond to terms neglected in modelling of the $3\pi$
current. The discrepancy indicates that extending the validity of our
model beyond the description of the $3\pi$ current one should probably take
into account further contributions. However the 2.5 $\sigma$ discrepancy
does not allow to draw any final conclusion.
The amplitude of the process $\pi^0 \to \gamma \gamma^*$ for small
values of the four momentum of the off shell photon can be
parameterized by a slope parameter $\alpha$
%
\bea
{\cal M}_{\pi^0 \to \gamma \gamma^*} = {\cal M}_{\pi^0 \to \gamma \gamma}
\left(1 + \alpha k^{*2}\right) \ .
\label{slope}
\eea
%
Expanding the Breit-Wigner propagators one obtains
%
\bea
\alpha = \frac{1}{2}\biggl( &&\frac{1}{m_\rho^2}
+ \frac{g_{\omega\pi\rho}\cdot g_{\omega\gamma}}{T m_\omega^2}
+ \frac{g_{\phi\pi\rho}\cdot g_{\phi\gamma}}{T m_\phi^2}\nonumber\\
&& + \frac{g_{\omega'\pi\rho}\cdot g_{\omega'\gamma}}{T m_{\omega'}^2}
+ \frac{g_{\omega''\pi\rho}\cdot g_{\omega''\gamma}}{T m_{\omega''}^2}
\biggr) \ ,
\label{slope1}
\eea
%
where $T$ is defined in Eq.(\ref{TT}). Numerically this
gives $\alpha = 1.74(2)$ GeV$^{-2}$,
or $\alpha\cdot m_{\pi^0}^2 = 0.0317(5)$,
to be compared with the experimental value \cite{PDG04}
$\alpha\cdot m_{\pi^0}^2 = 0.032(4)$.
The model also predicts the decay rates for the $\rho^0\to\pi^0\gamma$,
$\omega\to\pi^0\gamma$ and $\phi \to\pi^0\gamma$:
%
\bea
\Gamma_{\rho^0\to \pi^0\gamma} = \frac{\alpha}{24} m_{\rho}^5
\biggl[1-\frac{m^2_{\pi^0}}{m_\rho^2}\biggr]^3 \cdot T^2
\label{rhopig}
\eea
%
\bea
\Gamma_{V\to \pi^0\gamma} = \frac{\alpha}{24} m_{V}^5
\biggl[1-\frac{m^2_{\pi^0}}{m_V^2}\biggr]^3
g^2_{V\pi\rho}\cdot g^2_{\rho\gamma} \ ,
\label{omegapig}
\eea
%
\noindent
where $V$ stands for $\omega$ or $\phi$. The results are collected in
Table \ref{direct} and compared with the respective experimental values.
In the $\rho^0\to \pi^0\gamma$ decay
our
prediction for this branching ratio 5.2(2)$\cdot 10^{-4}$
is in agreement with the
6.0(1.3)$\cdot 10^{-4}$ of \cite{PDG04} within 1$\sigma$.
As the only isospin violating effect in our model for this
process is the charged--neutral pion mass difference,
our prediction for the
${\rm{ Br}}(\rho^{\pm}\to\pi^\pm\gamma) = 5.2(2)\cdot 10^{-4}$ is identical
(within the errors) with the one for the
neutral mode and is also in agreement with
the data \cite{PDG04} ${\rm{ Br}}(\rho^{\pm}\to\pi^\pm\gamma) = 4.5(5)\cdot 10^{-4}$.
For the
$\phi\to \pi^0\gamma$ decay the
branching ratio ${\rm{ Br}}(\phi\to\pi^0\gamma) = 0.99(12)\cdot 10^{-3}$
is in agreement within 2$\sigma$ with the value
$1.23(12)\cdot 10^{-3}$ from Ref. \cite{PDG04}. Our result for
${\rm{ Br}}(\omega\to\pi^0\gamma)= 15.4(5)\%$
overestimates however the measured value (($8.92^{+0.28}_{-0.24}$)\%)
by factor 1.7.
The extracted couplings
determine also the cross section of the reaction $e^+e^-\to \pi^0\gamma$,
%
\bea
&&\kern-10pt\sigma(e^+e^- \to \pi^0 \gamma) = \frac{2\pi^2\alpha^3}{3}
\left(1-\frac{m_{\pi^0}^2}{s}\right)^3g^2_{\rho\gamma}\nonumber \\
\cdot && |\
\bigl[
g_{\omega\pi\rho}\cdot g_{\omega\gamma}
+g_{\phi\pi\rho}\cdot g_{\phi\gamma} \nonumber \\
&&+g_{\omega'\pi\rho}\cdot g_{\omega'\gamma}
+g_{\omega''\pi\rho}\cdot g_{\omega''\gamma}
\bigr] \cdot BW_{\rho}(s) \nonumber\\
&&+g_{\omega\gamma}\cdot
g_{\omega\pi\rho}
\cdot BW_{\omega}(s)
+g_{\phi\gamma}\cdot
g_{\phi\pi\rho}
\cdot BW_{\phi}(s) \nonumber \\
&&+g_{\omega'\pi\rho}\cdot g_{\omega'\gamma}
\cdot BW_{\omega'}(s)
+g_{\omega''\pi\rho}\cdot g_{\omega''\gamma}
\cdot BW_{\omega''}(s)
|^2 \ ,\nonumber \\
\label{sigeepi0g}
\eea
%
with $BW_i \ , i =\rho,\omega,\phi,\omega',\omega'' $
defined in Eq.(\ref{BW}).
The comparison with existing data \cite{Achasov:03pig,Achasov:00}
is shown in Fig. \ref{cspigam}, where
one standard
deviation bands are given.
Reasonable agreement is observed around the $\phi$ resonance
in contrast to the $\omega$ region. This is of course a reflection
of the agreement and disagreement of the corresponding decay rates.
A similar behaviour was also observed
in Ref. \cite{RFPP}.
Their
model predict a value for $\Gamma(\omega\to\pi^0\gamma)$ well in accordance
with the experimental value, but could not easily reproduce
$\Gamma(\omega\to \pi^+\pi^-\pi^0)$.
Further, both theoretical
and experimental, studies of reactions $e^+e^- \to \pi^+\pi^-\pi^0$
and $e^+e^- \to \pi^0\gamma$ are thus required.
\begin{figure}[ht]
\begin{center}
\epsfig{file=v_SND1_pik_om_n.ps,width=7.5cm}
\epsfig{file=v_SND1_dol_om_n.ps,width=7.5cm}
\epsfig{file=v_SND2_pik_fi_n.ps,width=7.5cm}
\caption{Differential cross section of the process
$e^+ e^- \to \pi^0\gamma $. Data from Refs.\cite{Achasov:03pig} (SND 2003) and
\cite{Achasov:00} (SND 2000) are shown together with 1 $\sigma$ allowed
bands for our model predictions.
}
\label{cspigam}
\end{center}
\end{figure}
The branching ratios of the $\omega$, $\phi$ and $\rho$ decays to
$\pi^+\pi^-\pi^0$ obtained within our model
can be found in Table \ref{br3pi}. They are in agreement with
the PDG values \cite{PDG04} for $\omega$, $\phi$
within 1-2 standard deviations. However the predicted
value for ${\rm{ Br}}(\rho^0\to\pi^+\pi^-\pi^0)$ is more then two orders
of magnitude smaller than the experimental value
${\rm{ Br}}(\rho^0\to\pi^+\pi^-\pi^0) \simeq 1\cdot 10^{-4}$, which is consistent with
zero at two sigma level.
\begin{table}[hb]
\begin{center}
\caption{Branching ratios of the $\omega$, $\phi$ and $\rho$ to
$\pi^+\pi^-\pi^0$ decays: model vs. experiment \cite{PDG04}.}
\label{br3pi}
\begin{tabular}{l|l|l}
& model& experiment \\ \hline
${\rm{ Br}}(\omega\to\pi^+\pi^-\pi^0)$ & 95.1(22)\%& 89.1(7)\%
\\ \hline
${\rm{ Br}}(\phi\to\pi^+\pi^-\pi^0)$ & 14.5(22)\%& 15.4(5)\%\\ \hline
${\rm{ Br}}(\rho\to\pi^+\pi^-\pi^0)$ & 1.9(3)$\cdot$ 10$^{-6}$&
($1.01^{+0.54}_{-0.36}$$\pm$0.34)$\cdot$ 10$^{-4}$\\ \hline
\end{tabular}
\end{center}
\end{table}
\section{Tests of the MC code }
In comparison to the previous versions of PHOKHARA
the default random number generator was changed to the double precision
version of RANLUX \cite{RANLUX} written in C by Martin L\"uscher. A
C--FORTRAN interface is provided with the PHOKHARA 5.0 distribution.
To assure a technical precision of the
code better than a fraction of one per mile
a number of tests were performed for the new hadronic state
in the generator. Among other tests, the initial state emission
was tested against known analytical results, where all photon angles
are integrated, similarly to previously performed tests for other
hadronic channels (see \cite{Rodrigo:2001kf,Czyz:2002np,Czyz:PH03,Czyz:PH04}).
The independence of the result from the soft photon separation parameter
was also checked.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and Conclusions}
The Monte Carlo event generator PHOKHARA
has been extended to the three-pion mode. The model
adopted for the hadronic form factor properly describes the currently
available data for the cross section and the distributions. The ansatz
is based on generalized vector dominance and is consistent with various other
measurements like $\Gamma(\pi^0\to\gamma\gamma)$, the slope parameter
of the $\pi^0\to\gamma\gamma^*$ amplitude and radiative vector meson
decays $\rho\to \pi^0\gamma$, $\phi\to \pi^0\gamma$,
but is in conflict with $\omega\to \pi^0\gamma$.
The current version of the computer program (PHO\-KHA\-RA 5.0) is
available at
{\tt http://cern.ch/german.rodrigo/phokhara}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}
The authors thank J. Portol\'es for very helpful discussions
and comments on the manuscript.
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\end{document}