% gbr_final_prd.tex
% -- changes preceded by %! 
\documentstyle[preprint,aps]{revtex}
%\documentstyle{article}[12pt]
\tightenlines

%\pagestyle{plain}

%\renewcommand{\today}{26 February, 1997}

\newcommand{\nc}{\newcommand}
\nc{\be}{\begin{equation}}
\nc{\ee}{\end{equation}}
\nc{\bea}{\begin{eqnarray}}
\nc{\eea}{\end{eqnarray}}
\nc{\beas}{\begin{eqnarray*}}
\nc{\eeas}{\end{eqnarray*}}
\nc{\noi}{\noindent}
\nc{\sD}{\not \! \! D}
\nc{\s}[1]{\not \! #1}
\nc{\non}{\nonumber}
\nc{\bb}{\bibitem}
\nc{\lf}{\left}
\nc{\ri}{\right}
\nc{\mb}[1]{\makebox[#1]{}}
\nc{\pa}{\partial}
\nc{\sA}{\not \! \! A}
\nc{\newsec}[1]{\section{#1}\mb{0.5cm}}
\nc{\h}{\frac{1}{2}}
\nc{\ra}{\rightarrow}
\nc{\prw}{\widetilde{\Pi}_{\rho\omega}}
\nc{\la}{\leftarrow}
\nc{\etwopi}{$e^+e^-\ra\pi^+\pi^-\;$}
\nc{\ethrpi}{$e^+e^-\ra\pi^+\pi^0\pi^-\;$}
\nc{\lapp}{\hbox{$ {     \lower.40ex\hbox{$<$}
                   \atop \raise.20ex\hbox{$\sim$}
                   }     $}  }
\nc{\rapp}{\hbox{$ {     \lower.40ex\hbox{$>$}
                   \atop \raise.20ex\hbox{$\sim$}
                   }     $}  }
\nc{\M}{{\cal M}}
\nc{\rhoom}{$\rho^0$-$\omega\;$}
\nc{\rw}{$\rho$-$\omega\;$}
\def\mathunderaccent#1{\let\theaccent#1\mathpalette\putaccentunder}
\def\putaccentunder#1#2{\oalign{$#1#2$\crcr\hidewidth
\vbox to.2ex{\hbox{$#1\theaccent{}$}\vss}\hidewidth}}
\def\ttilde#1{\tilde{\tilde{#1}}}
\nc{\ti}{\mathunderaccent\tilde}
\def\hhha{\rule[-3.mm]{0.mm}{7.mm}}
\def\hhhb{\rule[-3.mm]{0.mm}{12.mm}}
\thispagestyle{empty}

\begin{document}
\preprint{\vbox{                        \hfill UK/TP 98-05 \\
                                        \null\hfill  \\
                                        \null\hfill September 1998\\
					Phys.Rev.D59:076002,1999}}

\title{Extracting Br$(\omega\ra\pi^+\pi^-)$ 
from the Time-like Pion Form-factor}

\author{S. Gardner\thanks{e-mail:
svg@ratina.pa.uky.edu} and H.B. 
O'Connell\thanks{e-mail: hoc@ruffian.pa.uky.edu}}
\address{Department of Physics and Astronomy,
University of Kentucky,\\
Lexington, KY 40506-0055 }
%\date{\today}
\maketitle

\begin{abstract}
We extract the G-parity-violating branching ratio 
Br$(\omega\ra\pi^+\pi^-)$ 
from the effective \rhoom mixing matrix element 
$\widetilde\Pi_{\rho\omega}(s)$, 
determined from \etwopi data. 
The $\omega\ra\pi^+\pi^-$ partial width can be determined either from the
time-like pion form factor or through
the constraint that the mixed physical
propagator $D_{\rho\omega}^{\mu\nu}(s)$ possesses no poles. 
The two procedures are inequivalent in
practice, and
we show why the first is preferred, to find finally 
Br$(\omega\ra \pi^+ \pi^-) = 1.9\pm 0.3\%$. 


\end{abstract}
\pacs{PACS numbers: 11.30.Hv, 12.40.Vv, 13.25.Jx, 13.65.+i}

\narrowtext

\newpage

\section{Introduction}

The presence of the $\omega$
resonance in \etwopi, in the region
dominated by the $\rho^0$, signals the presence of
the G-parity-violating decay 
$\omega\ra\pi^+\pi^-$. 
Our purpose is to extract the value of 
Br$(\omega\ra\pi^+\pi^-)$ from fits to \etwopi in the 
\rhoom interference region. To do so, we must consider the
relationship between the partial width 
$\Gamma(\omega \ra \pi^+\pi^-)$ and the 
effective \rhoom mixing matrix element 
$\widetilde\Pi_{\rho\omega}(m_\omega^2)$, 
determined in our earlier 
fits~\cite{GO} to \etwopi data~\cite{Bark,wd85}. 
The \etwopi cross section $\sigma(s)$
can be written as 
$\sigma(s)=\sigma_{\rm em}(s)|F_\pi(s)|^2$, where $\sigma_{\rm em}(s)$
is the cross section for the production of a structureless $\pi^+\pi^-$
pair and $s$ is the usual Mandlestam variable. 
The time-like pion
form factor $F_\pi(s)$ can in turn be written, to leading order
in isospin violation, as~\cite{GO}
%
\be
F_\pi(s)=F_\rho(s)\left[1+
\frac{1}{3}
\left(\frac{\widetilde{\Pi}_{\rho\omega}(s)}{s-m_\omega^2+
i m_\omega\Gamma_\omega}\right)\right]\;, 
\label{pionff}
\ee
%
where $F_\rho(s)$ parametrizes the $\rho^0$ resonance and 
$\widetilde\Pi_{\rho\omega}(s)$ is the effective
\rhoom mixing matrix element noted earlier. 
$\Gamma(\omega\ra\pi^+\pi^-)$ is determined by the effective
$\omega\ra\pi^+\pi^-$ coupling constant $g_{\omega\pi\pi}^{\rm eff}$, 
which can be 
extracted either from the
time-like pion form factor or from 
the relationship between the physical and isospin-perfect 
vector meson fields, determined through
the constraint that the mixed physical
propagator $D_{\rho\omega}^{\mu\nu}(s)$ possesses no poles. 
We evaluate not only the relationship
between these two different methods 
but also the impact of the 
uncertainty in the $\rho^0$ mass and width on Br$(\omega\ra\pi^+\pi^-)$
before reporting our final results. Despite the close connection 
between Br$(\omega\ra\pi^+\pi^-)$ and $\widetilde\Pi_{\rho\omega}(s)$,
we believe this work represents the first attempt to determine 
both simultaneously from \etwopi data. 



\section{$\Gamma(\omega\ra\pi^+\pi^-)$ and 
\rhoom Mixing}


If isospin symmetry were perfect, the $\rho$ and $\omega$ resonances would 
be exact eigenstates of G-parity, so that the $\rho$, of even 
G-parity, would 
decay to two, but not three, 
pions and the $\omega$, of odd G-parity, would decay 
to three, but not two, pions. 
Yet this is not strictly so, for \rhoom
 interference in \etwopi is
observed in nature~\cite{miller90}. Nevertheless, it 
is useful to introduce an isospin-perfect basis $\rho_I^0$ and 
$\omega_I$ in which to describe the physical $\rho^0$ and $\omega$. 
In this basis, 
G-parity can be violated either through ``mixing'', 
$\langle\omega_I| H^{\rm mix} |\rho_I\rangle$, where
$H^{\rm mix}$ represents isospin-violating terms in the effective
Hamiltonian in the vector meson sector,
or through the
direct decay $\langle\omega_I| H^{\rm mix} | \pi^+\pi^-\rangle$.
The vector mesons in \etwopi couple to 
a conserved current, so that 
we can write their propagators as 
$D_{VV}^{\mu\nu}(s)
\equiv g^{\mu\nu}D_{VV}(s)$, thereby defining
the scalar part of the propagator, $D_{VV}(s)$. The propagator
possesses a pole in the complex plane at $s=z_V$, so that
in the vicinity of this pole we have $D_{VV}(s)=1/(s-z_V)\equiv1/s_V$.
The difference between the diagonal scalar propagator 
in the physical and 
isospin-perfect bases, i.e., between  
$D_{VV}(s)$ and $D_{VV}^I(s)$, 
 is of non-leading-order in isospin violation, so that 
$D_{VV}^I(s)=1/s_V$ as well. 
Consequently, the pion form factor in the resonance region 
in the isospin-perfect basis can be 
written, to leading order in isospin violation, as 
%
\be
F_\pi(s)=\frac{g_{\rho_I\pi\pi}f_{\rho_I\gamma}}{s_\rho}+
\frac{g_{\omega_I\pi\pi}f_{\omega_I\gamma}}{s_\omega}+
\frac{g_{\rho_I\pi\pi}\Pi_{\rho\omega}^{I}(s)
f_{\omega_I\gamma}}{s_\rho s_\omega} \;,
\label{ff-iso}
\ee
%
where $g_{V_I\pi\pi}$ and $f_{V_I\gamma}$ are the 
vector-meson--pion-pion and vector-meson--photon 
coupling constants, respectively. The first term reflects the
dominant process $\gamma\ra\rho^0\ra\pi^+\pi^-$, whereas the
G-parity-violating terms reflect 
the direct decay $\omega\ra\pi^+\pi^-$
and \rhoom mixing, $\omega\ra\rho^0\ra\pi^+\pi^-$, respectively, 
noting
the mixing matrix element $\Pi_{\rho\omega}^{I}(s)$.
Defining $G\equiv g_{\omega_I\pi\pi}/g_{\rho_I\pi\pi}$ we can rewrite
Eq.~(\ref{ff-iso}) as 
%
\bea
\non
F_\pi(s)&=&
\frac{g_{\rho_I\pi\pi} f_{\rho_I\gamma}}{s_\rho} + 
\frac{g_{\rho_I\pi\pi} f_{\omega_I\gamma}}{s_\rho s_\omega}
(G(s-z_\rho) + \Pi_{\rho\omega}^{I}(s)) \\
\qquad &\equiv&
\frac{f_{\rho_I\gamma} g_{\rho_I\pi\pi}}{s_\rho}
\left[1+\frac{f_{\omega_I\gamma}}{f_{\rho_I\gamma}}
\left(
\frac{\widetilde{\Pi}_{\rho\omega}(s)}{s-z_\omega}
\right)
\right] \;.
\label{eq:formnew}
\eea
%
Note that we have defined the effective mixing matrix element 
%!
$\widetilde{\Pi}_{\rho\omega}(s)$, as $G$ and 
$\Pi_{\rho\omega}^{I}(s)$ cannot be 
meaningfully separated in a fit to data~\cite{MOW,OTW}, 
for both terms are $s$-dependent~\cite{GHT,OPTW}. 
As $\Gamma_\omega \ll m_\omega$
a Breit-Wigner lineshape may be used to model the $\omega$
resonance, but the large width of the $\rho$ relative to
its mass obliges a more sophisticated treatment. Rather than
adopting $s_\rho=s - z_\rho$, appropriate for $s\approx z_\rho$,
for the entire resonance region,
we replace 
$f_{\rho_I\gamma} g_{\rho_I\pi\pi}/{s_\rho}$ by 
$F_\rho(s)$, a function constructed 
to incorporate 
the constraints imposed on the form factor by time-reversal
invariance, unitarity, analyticity, and 
charge conservation. For further details, 
see Ref.~\cite{GO}
and references therein. Using
%
\be
F_\pi(s)=F_\rho(s)\left[1+ \frac{f_{\omega_I\gamma}}{f_{\rho_I\gamma}}
\left(
\frac{\widetilde{\Pi}_{\rho\omega}(s)}{s-m_\omega^2+
im_\omega\Gamma_\omega}
\right)\right] 
\label{ff-isofit}
\ee
%
with the SU(6) value of $f_{\omega_I\gamma}/f_{\rho_I\gamma}=1/3$,
we find
$\widetilde{\Pi}_{\rho\omega}(m_\omega^2)=-3500\pm 300$ MeV$^2$, where
the systematic error due to the $\rho^0$ parametrization adopted
is negligible~\cite{GO}. 
Note that both the imaginary part of 
$\widetilde{\Pi}_{\rho\omega}(s)$, 
${\rm Im}\widetilde \Pi_{\rho\omega}(m_\omega^2) = - 300 \pm 300$
MeV$^2$, and its 
$s$-dependence about $s=m_\omega^2$,
$\widetilde{\Pi}_{\rho\omega}(s) = \widetilde{\Pi}_{\rho\omega}(m_\omega^2) 
+ (s - m_\omega^2)\widetilde{\Pi}'_{\rho\omega}(m_\omega^2)$
%!
with $\widetilde{\Pi}'_{\rho\omega}(m_\omega^2)=0.03 \pm 0.04$,
are also negligible~\cite{GO}. 


Equation (\ref{eq:formnew}) can also
be used to define an effective, 
isospin-violating coupling constant, 
$g_{\omega\pi\pi}^{\rm eff}(s)$, such that 
%
\be
F_\pi(s)=
\frac{g_{\rho_I\pi\pi}f_{\rho_I\gamma} }{s_\rho} + 
\frac{g_{\omega\pi\pi}^{\rm eff}(s) f_{\omega_I\gamma}}{s_\omega} \;,
\label{ff-geff1}
\ee
%
so that 
$g_{\omega\pi\pi}^{\rm eff}(s)\equiv g_{\rho_I\pi\pi}
\widetilde{\Pi}_{\rho\omega}(s)/s_\rho$. 
To determine the partial width 
$\Gamma(\omega\ra\pi^+\pi^-)$, and hence 
Br$(\omega\ra\pi^+\pi^-)$, we must relate it to the 
effective coupling constant $g_{\omega\pi\pi}^{\rm{eff}}(s)$. 

In a Lagrangian model in which the pion is an elementary field and 
$g_{V\pi\pi}$ denotes the vector meson coupling constant to two
pions, the vector meson self-energy $\Pi_{VV}(s)$, noting 
$D_{VV}^{-1}(s)= s - m^2 - \Pi_{VV}(s)$, can be approximated
 as a sum of
iterated bubble diagrams, where each bubble contains a 
two-pion intermediate state~\cite{decay}. 
Here $g_{V\pi\pi}$ is a simple constant,
and direct calculation yields~\cite{decay}
%
\be
{\rm Im}\, \Pi_{VV}(s) = - g_{V\pi\pi}^2 
\frac{(s - 4m_\pi^2)^{3/2}}{48\pi \sqrt{s}} \Theta(s - 4m_\pi^2) \;.
\ee
%
Finally, noting $\lim_{s\ra m_V^2} {\rm Im}\, \Pi_{VV}(s)
= - m_V \Gamma (V\ra\pi^+\pi^-)$, then~\cite{decay,klingl}
%
\be
\Gamma(V\ra \pi^+\pi^-)=\frac{g^2_{V\pi\pi}}{48\pi}
\frac{(m_V^2-4m_\pi^2)^{3/2}}{m_V^2} \;.
\label{eq:gpp}
\ee
%
Replacing $g_{\omega\pi\pi}$ by 
$|g_{\omega\pi\pi}^{\rm{eff}}(s)|$, one finds that 
Br($\omega\ra\pi^+\pi^-$), to leading order in isospin
violation, is given by 
%
\be
{\rm Br}(\omega\ra\pi^+\pi^-)
=\frac{1}{48\pi}
\frac{(m_\omega^2-4m_\pi^2)^{3/2}}{m_\omega^2\Gamma_\omega}
\left| \frac{g_{\rho_I\pi\pi}\widetilde{\Pi}_{\rho\omega}(s)}
{s_\rho}
\right|^2 \Bigg|_{s=m_\omega^2} \;.
\label{Br1}
\ee
%

Another relation for $g_{\omega\pi\pi}^{\rm{eff}}(s)$ emerges
through consideration of the pion form factor in the 
physical basis \cite{basis1}. 
To leading order in isospin violation, 
we have~\cite{MOW,OTW}
%
\be
F_\pi(s) = g_{\rho\pi\pi} D_{\rho\rho} f_{\rho\gamma}
+ g_{\rho\pi\pi} D_{\rho\omega} f_{\omega\gamma}
 + g_{\omega\pi\pi} D_{\omega\omega} f_{\omega\gamma} \;,
\label{ff-phys}
\ee
%
where 
we introduce a \rhoom mixing  matrix element, $\Pi_{\rho\omega}(s)$,
such that~\cite{OPTW}
%
\be
D_{\rho\omega}^I(s)
=D^I_{\omega\omega}(s)\Pi_{\rho\omega}(s)D^I_{\rho\rho}(s).
\ee 
%
To relate the physical states $\rho$ and $\omega$ 
to the
isospin perfect ones $\rho_I$ and $\omega_I$, we introduce 
two mixing parameters,
$\epsilon_1$ and $\epsilon_2$, such that~\cite{MOW,OTW}
%
\be
\rho= \rho_I - \epsilon_1 \omega_I \quad ; \quad 
\omega= \epsilon_2 \rho_I + \omega_I \;.
\label{physdef}
\ee
%
Requiring the
mixed physical propagator 
$D_{\rho\omega}(s)$ to possess no poles, 
$\epsilon_1$ and $\epsilon_2$
are determined to be~\cite{MOW,OTW}:
%
\be
\epsilon_1=
\frac{\Pi_{\rho\omega}(z_\omega)}{z_\omega-z_\rho}
\,\,\,\,\,\,\,\,\,
\epsilon_2=
\frac{\Pi_{\rho\omega}(z_\rho)}{z_\omega-z_\rho} \;.
\ee
%
Using Eqs.~(\ref{ff-phys}) and (\ref{physdef}), 
and $D_{VV}=D_{VV}^I=1/s_V$ for $s$ in the vicinity of 
$z_\rho,z_\omega$ yields
%
\bea
\non
F_\pi(s)&=&\frac{g_{\rho_I\pi\pi}f_{\rho_I\gamma}}{s_\rho}
+
\frac{g_{\omega_I\pi\pi}}{s_\omega s_\rho} \\
&\times&
\left[
G(s - z_\rho) +  
\frac{(\Pi_{\rho\omega}(z_\rho) - \Pi_{\rho\omega}(z_\omega))}
{z_\omega - z_\rho} (s - z_\omega + s - z_\rho) + 
\Pi_{\rho\omega}(s)
\right]f_{\omega_I\gamma} \;. 
\label{ff-phys/iso}
\eea
%
Comparison with Eq.~(\ref{ff-iso}) shows that 
$\Pi_{\rho\omega}^I(s)$ and $\Pi_{\rho\omega}(s)$ are only
equivalent if $\Pi_{\rho\omega}(z_\rho) = \Pi_{\rho\omega}(z_\omega)$. 
We can then write $g_{\omega\pi\pi}^{\rm eff}$ found above 
as 
%
\bea
\non
g_{\omega\pi\pi}^{\rm eff}(s)
&=& \frac{g_{\rho_I\pi\pi}}{s_\rho}
\widetilde{\Pi}_{\rho\omega}(s) \\ 
&=& \frac{g_{\rho_I\pi\pi}}{s_\rho}
\left[
G(s - z_\rho) + 
\frac{(\Pi_{\rho\omega}(z_\rho) - \Pi_{\rho\omega}(z_\omega))}
{z_\omega - z_\rho} (s - z_\omega + s - z_\rho) + 
\Pi_{\rho\omega}(s)
\right] \;.
\label{geff1}
\eea
%
We could also have defined $g_{\omega\pi\pi}^{\rm eff}$ directly
from the relation between the physical and isospin perfect bases,
Eq.~(\ref{physdef}):
%
\bea
\non
g_{\omega\pi\pi}^{\rm eff}&=&
g_{\omega_I\pi\pi} + \epsilon_2 g_{\rho_I\pi\pi} \\
&=&
\frac{g_{\rho_I\pi\pi}}{z_\omega - z_\rho}
\left[
G(z_\omega - z_\rho) +
\Pi_{\rho\omega}(z_\rho)
\right] \;.
\label{geff2}
\eea
%
These two possible definitions of 
$g_{\omega\pi\pi}^{\rm eff}$ are {\it identical} at the 
$\omega$ pole, $s=z_\omega$. However, fits to the time-like
pion form factor data yield 
$\widetilde{\Pi}_{\rho\omega}(s)$ merely at real values of $s$, so
that Eq.~(\ref{geff1}) is the only practicable definition of
$g_{\omega\pi\pi}^{\rm eff}$. 
The two expressions differ in general as 
isospin-violating pieces are present in $f_{\rho\gamma}$ as well; 
they vanish, however, at $s=z_\omega$. 


Interestingly, if we were to demand as in Ref.~\cite{OTW} that 
$\Pi_{\rho\omega}^I(s)\equiv \Pi_{\rho\omega}(s)$, implying that 
Eq.~(\ref{physdef}) cannot be used to relate 
$f_{V\gamma}$ to $f_{V_I\gamma}$ and
$g_{V\pi\pi}$ to $g_{V_I\pi\pi}$ unless $\epsilon_1=\epsilon_2$~\cite{OTW}, 
then Eq.~(\ref{geff1}) would become 
$g_{\omega\pi\pi}^{\rm eff}(s) = g_{\rho_I\pi\pi}
(G(s - z_\rho) + \Pi_{\rho\omega}(s))/s_\rho$. This latter
definition of $g_{\omega\pi\pi}^{\rm eff}(s)$ would be inconsistent
with Eq.~(\ref{geff2}) at $s=z_\omega$. We prefer the analysis 
yielding Eq.~(\ref{geff1}). 

To determine Br$(\omega\ra\pi^+\pi^-)$ using Eq.~(\ref{Br1}) 
we must evaluate $g_{\rho_I\pi\pi}/s_\rho$ at $s=m_\omega^2$. 
As $s_\rho=s-z_\rho$ only for $s\approx z_\rho$, it is 
appropriate to replace 
$g_{\rho_I\pi\pi}/s_\rho$
by $F_\rho(s)/f_{\rho_I\gamma}$, noting
Eqs.~(\ref{eq:formnew}) and (\ref{ff-isofit}), 
to yield finally 
%
\be
{\rm Br}(\omega\ra\pi^+\pi^-)
=\frac{(m_\omega^2-4m_\pi^2)^{3/2}}{48\pi
m_\omega^2\Gamma_\omega f_{\omega_I\gamma}^2}
\left|F_\rho(m_\omega^2)\;
\frac{1}{3}\;\widetilde{\Pi}_{\rho\omega}(m_\omega^2)\right|^2 \;.
\label{Br_final}
\ee
%
In the fit to data using Eq.~(\ref{ff-isofit}), 
$(f_{\omega_I\gamma}/f_{\rho_I\gamma})\widetilde{\Pi}_{\rho\omega}(s)$
appears as a single fitting parameter. Choosing 
$f_{\omega_I\gamma}/f_{\rho_I\gamma}=1/3$, then, allows us to 
use our earlier value of 
$\widetilde{\Pi}_{\rho\omega}=-3500$ MeV$^2$~\cite{GO}. 
Equation (\ref{Br_final}) defines the branching ratio in terms of the 
phenomenologically well-constrained
fitting functions $F_\rho(s)$ 
%!
and $\prw/3$ and thus avoids the explicit introduction of
$\rho$ resonance parameters. The model dependence 
of Eq.~(\ref{Br_final}) is therefore minimal, and 
for this reason it is our  preferred definition. 

%!
To assess its utility we shall compare it with other definitions
in the literature. We may also 
use Eq.~(\ref{eq:gpp}) to replace $g_{\rho_I\pi\pi}$ and 
write $s_\rho= s - m_\rho^2 + i m_\rho \Gamma_\rho$ to find
%
\be
{\rm Br}^{(2)}(\omega\ra\pi^+\pi^-)=
\frac{m_\rho^2(m_\omega^2-4m_\pi^2)^{3/2}}{m_\omega^2(m_\rho^2-4m_\pi^2)^{3/2}}
\frac{\Gamma_\rho}{\Gamma_\omega}
\left|\frac{\prw(m_\omega^2) }
{m_\omega^2 - m_\rho^2+im_\rho\Gamma_\rho}
\right|^2 \;,
\label{Br2}
\ee
%
where we have used $\Gamma_\rho=\Gamma(\rho\ra\pi^+\pi^-)$.
If we set $m_\omega=m_\rho$, 
Eq.~(\ref{Br2}) becomes that used in Ref.~\cite{CB} to extract 
$\prw(m_\omega^2)=-4520$ MeV$^2$~\cite{newCB}
a value commonly used in the literature~\cite{quoted}.
We prefer determining both $\prw(m_\omega^2)$ and 
Br($\omega\ra\pi^+\pi^-$) 
directly from our fits to 
the \etwopi data. 
Yet another expression for Br($\omega\ra\pi^+\pi^-)$ results
if we consider Eq.~(\ref{geff2}) in place of Eq.~(\ref{geff1}) for 
$g_{\omega\pi\pi}^{\rm eff}$; that is, 
%
\be
{\rm Br}^{(3)}(\omega\ra\pi^+\pi^-)=
\frac{m_\rho^2(m_\omega^2-4m_\pi^2)^{3/2}}{m_\omega^2(m_\rho^2-4m_\pi^2)^{3/2}}
\frac{\Gamma_\rho}{\Gamma_\omega}
\left|\frac{\overline{\Pi}_{\rho\omega}(m_\omega^2)}
{z_\omega - z_\rho}
\right|^2 \;,
 \label{Br3}
\ee
%
where $\overline{\Pi}_{\rho\omega}(m_\omega^2)\equiv 
G(z_\omega - z_\rho) + \Pi_{\rho\omega}(z_\rho)$. 
$\overline{\Pi}_{\rho\omega}(m_\omega^2)$ is not determined directly
in fits to \etwopi data and thus we favor Eqs.~(\ref{Br_final}) or
(\ref{Br2}). Nevertheless, 
as we found no significant $s$-dependence to
$\prw$ 
in our fits to
\etwopi data~\cite{GO}, we will replace 
$\overline{\Pi}_{\rho\omega}(m_\omega^2)$
by $\widetilde{\Pi}_{\rho\omega}(m_\omega^2)$ in our
subsequent numerical estimates. 
Neglecting terms of ${\cal O}((m_\omega-m_\rho)/m^{\rm av})$, with 
$m^{\rm av}=(m_\rho + m_\omega)/2$, and setting 
$z_\rho=m_\rho^2 + i m_\rho\Gamma_\rho$,
Eq.~(\ref{Br3}) yields
%
\be
\Gamma(\omega\ra\pi^+ \pi^-) = 
\frac{\left|\overline{\Pi}_{\rho\omega}(m_\omega^2)\right|^2}
{4 m_\rho^2((m_\omega - m_\rho)^2 + 
{1\over 4} (\Gamma_\omega - \Gamma_\rho)^2)}\Gamma_\rho \;,
\ee
%
and is thus equivalent to Eq.(B.12) in Ref.~\cite{GL}.
So far we have freely changed from one realization of $s_\rho$
to another; i.e., we have written both $s_\rho=s - z_\rho$ and
$s_\rho=s- m_\rho^2 + i m_\rho \Gamma_\rho$. Yet it is important
to recognize that for a broad resonance, such as the $\rho$ (but 
unlike the $\omega$), these
realizations are not necessarily equivalent. A parametrization of 
$F_\rho(s)$ which explicitly suits the constraint of 
unitarity and time-reversal
invariance, obliging 
its phase to be that of $l=1$, $I=1$ $\pi$-$\pi$ scattering for $s$ where
the scattering is elastic~\cite{gas66,HL,GO}, 
results in an $s$-dependent width~\cite{GS}. Effectively, then,
$(F_\rho(s))^{-1}\propto s - m_\rho^2 + i m_\rho \Gamma_\rho(s)$, where
the $m_\rho$ and $\Gamma_\rho$
we have used thus far satisfy $\Gamma_\rho\equiv \Gamma_\rho(m_\rho^2)$.
However,
the $\rho$ pole, $z_\rho$, in the complex $s$ plane is 
determined by requiring $(F_\rho(z_\rho))^{-1}=0$. 
Thus, in the presence of a $s$-dependent width, $\Gamma_\rho(s)$, 
$z_\rho \ne m_\rho^2 - im_\rho \Gamma_\rho$. 
If we parametrize $z_\rho$ as 
%
\be
z_\rho\equiv\overline{m}^2_\rho-i\overline{m}_\rho\overline{\Gamma}_\rho\;,
\label{pole}
\ee
%
then 
$\overline{m}_\rho$ and $\overline{\Gamma}_\rho$ differ substantially
from $m_\rho$ and $\Gamma_\rho$~\cite{lang79}, as illustrated in 
Table \ref{one}. Moreover, 
$\overline{m}_\rho$ and $\overline{\Gamma}_\rho$ 
are independent
of the parametrization of $F_\rho(s)$~\cite{Levy,Smatrix,pdg72,lang79}, 
whereas
$m_\rho$ and $\Gamma_\rho$ are {\it not}~\cite{pisut68,ben93,PDG,GO}.
In marked contrast to $m_\rho$ and
$\Gamma_\rho$ given in Table~\ref{one},
the average values of $\overline{m}_\rho$
and  $\overline{\Gamma}_\rho$, 
%
\be
\overline{m}_\rho=757.0 \pm 1.1\,{\rm MeV}\;,\,\,\,\,\,\,\,\,\,\,\,\,
\overline{\Gamma}_\rho=141.3 \pm 3.1\,{\rm MeV} \;,
\ee
%
are within one standard deviation of the 
$\overline{m}_\rho$ and $\overline{\Gamma}_\rho$ found in 
each and every model. 
This is in excellent agreement with Ref.~\cite{BCP}, where the
$\rho$ parameters are found to be 
$\overline{m}_\rho=757.5\pm1.5$ MeV and 
$\overline{\Gamma}_\rho=142.5\pm3.5$ MeV. 
The stability shown here is that of the
S-matrix
pole position, $z_\rho$, which is 
model independent~\cite{Levy,Smatrix,pdg72,lang79}.
The separation of $z_\rho$ into a ``mass'' and
``width'', as in Eq.~(\ref{pole}),
though useful \cite{Levy}, 
is somewhat artificial, as Re($\sqrt{z_\rho}$) and
$\sqrt{{\rm Re}\,z_\rho}$ could equally well serve as 
the mass~\cite{Stuart}. 
We shall consider the consequence of 
$z_\rho \ne m_\rho^2 - im_\rho \Gamma_\rho$ on the numerical values of
${\rm Br}^{(3)}(\omega\ra\pi^+\pi^-)$. 

It should also be noted that 
the value of 
$\widetilde{\Pi}_{\rho\omega}(m_\omega^2)$
to be used in Eqs.~(\ref{Br2}) and (\ref{Br3}) 
can be determined 
from our previous, averaged 
result~\cite{GO}, noting Eq.~(\ref{ff-isofit}),
through
%
\be
\widetilde{\Pi}_{\rho\omega}(m_\omega^2)=
\frac{1}{3}\frac{f_{\rho_I\gamma}}{f_{\omega_I\gamma}}
\left( -3500\, {\rm MeV}^2
\right) \;, 
\label{replace}
\ee
%
We must therefore now determine the leptonic couplings 
$f_{\rho_I\gamma}$ and $f_{\omega_I\gamma}$. 


%
\section{Vector meson electromagnetic couplings}
%
We have related the branching ratio Br$(\omega\ra\pi^+\pi^-)$
to the effective mixing term $\prw(s)$ and 
various vector-meson parameters, yet in order 
to fix $\prw$ in a 
fit to \etwopi data, 
we need to determine the ratio $r_\gamma\equiv f_{\rho_I\gamma}/
f_{\omega_I\gamma}$. 
In the SU(6) limit
$r_\gamma=3$, but
this relation is broken
at the $\sim 10\%$ level \cite{klingl} by the large $\rho$ width 
\cite{GS,renard}. 
In this section we discuss the
extraction of $f_{\rho_I\gamma}$ and $f_{\omega_I\gamma}$.

The vector-meson--photon coupling constant
$f_{V\gamma}$ is
related to the leptonic decay width $\Gamma(V\ra \ell^+\ell^-)$ through 
%
\be
\Gamma(V\ra \ell^+\ell^-)
=\frac{4\pi\alpha^2}{3m_V^3}f^2_{V\gamma } \;,
\label{eq:lep}
\ee
%
noting
that lepton masses enter at ${\cal O}((m_\ell/m_V)^4)$~\cite{klingl}.

The cross-section for \etwopi, proceeding solely through 
$e^+e^-\ra \rho^0 \ra \pi^+\pi^-$, that is, assuming no background,
for $s=m_\rho^2$ is 
%
\bea\non
\sigma(e^+e^-\ra\rho_I\ra\pi^+\pi^-) 
&=&\left.\frac{\pi\alpha^2}{3}\frac{(s-4m_\pi^2)^{3/2}}{s^{5/2}}
\frac{\left(f_{\rho_I\gamma}g_{\rho_I\pi\pi}\right)^2}{(s-m_\rho^2)^2
+m_\rho^2\Gamma^2_\rho}\right|_{s=m_\rho^2} \\
&=&12\pi\frac{\Gamma(\rho_I\ra e^+e^-)\Gamma(\rho_I\ra \pi^+\pi^-)}
{m_\rho^2\Gamma_\rho^2} \;,
\label{picase}
\eea
%
where we have used Eqs.~(\ref{eq:gpp}) and (\ref{eq:lep}).
This is a particular case of 
the Cabibbo--Gatto relation 
for a resonant, spin-one interaction \cite{CG}, valid for any
hadronic final state. Thus, an analogous ``Cabibbo-Gatto'' 
formula exists for $e^+e^-\ra \omega \ra \pi^+\pi^0\pi^-$. In this
manner, $\Gamma(\omega\ra e^+e^-)$ and 
$f_{\omega_I \gamma}$, via Eq.~(\ref{eq:lep}), can both be 
inferred from the \ethrpi data~\cite{3pi}. We use 
$\Gamma({\omega \ra e^+e^-})=0.60\pm.02$ keV~\cite{PDG} in what follows.


We can now calculate $\Gamma({\rho^0\ra e^+e^-})$ and hence
$f_{\rho_I\gamma}$.
Recalling 
Eq.~(\ref{ff-isofit}) we find
%
\be
\sigma(e^+e^-\ra\rho_I\ra\pi^+\pi^-)
=\frac{\pi\alpha^2}{3}\frac{(s-4m_\pi^2)^{3/2}}{s^{5/2}}
|F_\rho(s)|^2 \Bigg|_{s=m_\rho^2} \;,
\label{rhoIcross}
\ee
%
which when combined with
Eq.~(\ref{picase}) 
yields
%
\be
\Gamma(\rho_I^0\ra e^+e^-)=\frac{\alpha^2}{36}
\frac{(m_\rho^2-4m_\pi^2)^{3/2}}{m_\rho^3}|F_\rho(m_\rho^2)|^2\Gamma_\rho \;,
\label{Gree}
\ee  
%
where $\Gamma_\rho=\Gamma(\rho\ra\pi^+\pi^-)$,
allowing us to determine $f_{\rho_I\gamma}$
from Eq.~(\ref{eq:lep}).

We note in passing that it
is quite common in the literature to see the $\omega$ contribution
to the pion form-factor
expressed in terms of $\omega$ partial widths \cite{Bark,Ben}. Such
an expression follows 
from our Eq.~(\ref{ff-geff1}), in concert with Eqs.~(\ref{eq:gpp}) 
and (\ref{eq:lep}), to yield
%
\bea
F_\pi(s)
=F_\rho(s)
+ \sqrt{\frac{36\Gamma({\omega\ra e^+e^-})\Gamma({\omega\ra\pi^+\pi^-})}{
m_\omega^2\alpha^2\beta_\omega^3}}\frac{m_\omega^2}{s-
m_\omega+im_\omega\Gamma_\omega},
\label{eq:rit}
\eea
%
where $\beta_\omega=(1-4m_\pi^2/m_\omega^2)^{1/2}$ and
we replace 
$f_{\rho_I\gamma}g_{\rho_I\pi\pi}/(s-m_\rho+im_\rho\Gamma_\rho)$ with
$F_\rho(s)$ as earlier. 
Thus, our Eq.~(\ref{Br_final}) is explicitly
equivalent to the determinations of Br($\omega\ra\pi^+\pi^-$) found
in Refs.~\cite{Bark,Ben}. 

\section{Results and Discussion}

We now use our fits of Ref.~\cite{GO} to compute 
Br($\omega\ra\pi^+\pi^-$), $\Gamma(\rho\ra e^+e^-)$, and other
associated parameters, in addition to their errors. 
Our fits to the pion form-factor 
data~\cite{wd85}, noting Eq.~(\ref{ff-isofit}), adopt parametrizations of
$F_\rho(s)$ consistent with the following theoretical constraints.
That is, analyticity requires that 
$F_\rho(s)$ be real below threshold,
$s=4 m_\pi^2$, charge conservation requires $F_\rho(0)=1$, and
unitarity and time-reversal invariance 
requires its phase be that of $l=1$, $I=1$ $\pi$-$\pi$
scattering for $s$ where the latter is elastic~\cite{HL}. 
For the present work we shall concentrate on four of
these choices for $F_\rho(s)$, labeled, as per Ref.~\cite{GO}, 
A, B, C, and D, in which
$\prw$
is an explicit fitting parameter. These four 
fits assume $\prw$ to be 
a real constant in the resonance region, for the current \etwopi data 
supports neither a phase nor $s$-dependent pieces~\cite{GO}.

Table \ref{one} shows our results for $\Gamma({\rho^0\ra e^+e^-})$ 
and Br$(\rho^0\ra e^+e^-)\equiv\Gamma({\rho^0\ra e^+e^-})/\Gamma_\rho$. 
as determined
from Eq.~(\ref{Gree}).
We find the following average values:
%
\be
\Gamma({\rho^0\ra e^+e^-})
\!=\!7.11\pm 0.08\pm 0.25\;{\rm keV},\, {\rm Br}(\rho^0\ra e^+e^-)\!=\!
(4.63\pm 0.05 \pm 0.07)\times10^{-5},
\ee
%
where the second error on $\Gamma({\rho^0\ra e^+e^-})$ is 
the theoretical systematic error associated with
model choice~\cite{syserr}, and all other errors are statistical.
$\Gamma_\rho$ from Fit D is significantly lower
than those from the other fits and leads to a significantly lower value for
$\Gamma({\rho^0\ra e^+e^-})$, indeed one commensurate with the value  of 
$6.77\pm 0.10 \pm 0.30$  keV
reported in Ref.~\cite{Bark}. This is likely consequent to
the choice of the Gounaris--Sakurai form factor \cite{GS} in both fits; our
other fits use a Heyn--Lang form factor \cite{HL}.
Such model
dependence also plagues the extraction of the $\rho$ parameters 
$m_\rho$ and $\Gamma_\rho$, as discussed following Eq.~(\ref{pole}). 

Using $\Gamma(\rho_I\ra e^+ e^-)$ of Table \ref{one} and Eq.~(\ref{eq:lep})
yields $f_{\rho_I\gamma}$ and $r_\gamma$, using
 $f_{\omega_I\gamma}$ computed from $\Gamma(\omega\ra e^+e^-)$ of 
Ref.~\cite{PDG}.
In the SU(6) limit $r_\gamma$ is 3; 
the ``finite width'' correction~\cite{GS,renard}, as 
seen in Table \ref{two}, is $\sim10$\%, as also found in Ref.~\cite{klingl}, 
and hence significant. 
Including this correction as per Eq.~(\ref{replace}) 
gives us perhaps a 
more realistic value of 
$\prw(m_\omega^2)$~\cite{footnote}, 
and its model dependence appears to be modest, allowing
us to determine an average value of 
%
\be
\prw(m_\omega^2)=-3900\pm 300\,{\rm MeV}^2\;,
\label{newpirw}
\ee
%
again some 10\% larger than our value of 
$\prw(m_\omega^2)=-3500\pm 300\,{\rm MeV}^2$ in Ref.~\cite{GO}
using $r_\gamma=3$.



Our preferred determination of 
Br$(\omega\ra\pi^+\pi^-)$, Eq.~(\ref{Br_final}), does not require
$r_\gamma$, and we find 
%
\be
{\rm Br}(\omega\ra\pi^+\pi^-)=1.9\pm0.3\%\;.
\ee
%
Barkov {\it et al.}, noting Eq.~(\ref{eq:rit}) and the discussion
thereafter, obtain Br$(\omega\ra\pi\pi)=2.3\pm0.4\%$~\cite{Bark} 
with the same data set~\cite{wd85} used here. 
%!
We agree closely, however, with 
the result of Bernicha {\it et al.}, $1.85\pm0.30\%$~\cite{BCP}, obtained 
from the same data~\cite{wd85}. 
Their relation for the branching ratio, 
Eq.(42)~\cite{BCP}, is our Eq.~(\ref{Br2}), though they use 
the parameters ${\overline m}_\rho$ and ${\overline \Gamma}_\rho$,
noting Eq.~(\ref{pole}), in place of $m_\rho$ and $\Gamma_\rho$ and
use $\Gamma_{\rho\ra e^+e^-} = 6.77$ keV to compute the leptonic
coupling $f_{\rho_I\gamma}$~\cite{BCP}. The latter effects compensate, so that
we would expect to find a branching ratio comparable to theirs. 
The data set we have adopted~\cite{wd85}
contains 30 data points for center of mass energies between
750 and 810 MeV, the region likely most relevant for the determination of
$\Gamma(\omega\ra\pi^+\pi^-)$. The older work of 
Benaksas {\it et al.}~\cite{Ben} and
Quenzer {\it et al.} \cite{Q} find 
Br$(\omega\ra\pi\pi)=3.6\pm0.4\%$ and 
Br$(\omega\ra\pi\pi)=1.6\pm0.9\%$, respectively, though both experiments
possess less than 10 data points in the energy region of interest.

We can also compute Br($\omega\ra\pi^+\pi^-$) using
Eqs.~(\ref{Br2}) or (\ref{Br3}) and (\ref{replace}), 
as shown in Table \ref{three}. 
Apparently it makes little difference whether we use
Eq.~(\ref{Br_final}) or Eq.~(\ref{Br2}), though the former, 
our preferred analysis, possesses essentially no parametrization
dependence. Br$^{(3)}(\omega\ra\pi\pi)$, from Eq.~(\ref{Br3}), is
substantially larger, though this may be an artifact of using
the true S-matrix pole position $z_\rho$ in Eq.~(\ref{Br3}). 
If we were to replace $z_\rho$ with
$m_\rho^2 - i m_\rho \Gamma_\rho$, noting the discussion surrounding
Eq.~(\ref{pole}), the values, as shown in parentheses,
%!
would differ less, even though we were obliged to 
assume that
$\overline{\Pi}_{\rho\omega}(m_\omega^2)$ and
$\widetilde{\Pi}_{\rho\omega}(m_\omega^2)$ are the same. 

In summary, we have elucidated the connection between 
$\prw(m_\omega^2)$ and Br($\omega\ra\pi^+\pi^-$) and shown 
how different methods of determining 
Br($\omega\ra\pi^+\pi^-$) would be equivalent
were it possible to evaluate $\prw(z_\omega)$. In practice, the
methods are different, yet, nevertheless,
it seems that a plurality of methods of computing
Br($\omega\ra\pi^+\pi^-$) yield roughly comparable results. 


\acknowledgments
This work was supported by the U.S. Department of Energy 
under grant \# DE--FG02--96ER40989.



\begin{thebibliography}{99}
\bb{GO} S.~Gardner and H.B.~O'Connell,  Phys. Rev. D {\bf 57}, 2716 (1998). 
\bb{Bark} L.M.~Barkov {\it et al.}, Nucl. Phys. {\bf B256}, 365 (1985).
        The expression after Eq.~(5) in this reference, 
        noting our Eq.~(\ref{eq:rit}), 
        is missing an overall square root on the right-hand side.
\bb{wd85} OLYA: A.~D. Bukin {\it et al.}, Phys. Lett. {\bf 73B}, 226 (1975);
        L.M.~Kurdadze {\it et al.}, JETP Lett. {\bf 37}, 733 (1983); 
        L.M.~Kurdadze {\it et al.}, Sov. J. Nucl. Phys. {\bf 40}, 286 (1984); 
        I.B.~Vasserman {\it et al.}, Sov. J. Nucl. Phys. {\bf 30}, 519 (1979);
        CMD: G.V.~Anikin {\it et al.}, Novosibirsk preprint INP 83-85 (1983). 
        Note also 
        S.R.~Amendolia {\it et al.}, Phys. Lett. B {\bf 138}, 454 (1984); 
        I.B.~Vasserman {\it et al.}, Sov. J. Nucl. Phys. {\bf 33}, 368 (1981); 
        L.M.~Barkov {\it et al.}, Novosibirsk preprint INP 79-69 (1979);
        A.~Quenzer  {\it et al.}, Phys. Lett. {\bf 76B}, 512 (1978); 
        G.~Cosme {\it et al.}, ORSAY preprint LAL-1287 (1976). 
\bb{miller90} G.A.~Miller, B.M.K.~Nefkens, and I.~Slaus, 
        Phys. Rep. {\bf 194}, 1 (1990).
%!
\bb{MOW} K.~Maltman,
        H.B.~O'Connell, and A.G.~Williams,  Phys. Lett. {\bf B376}, 19 (1996).
\bb{OTW}H.B.~O'Connell, A.W.~Thomas, and A.G.~Williams, 
        Nucl. Phys. {\bf A623}, 559 (1997).
\bb{GHT} S.~Coleman and H.J.~Schnitzer, Phys. Rev. {\bf 134}, B863 (1964);
        T.~Goldman, J.A.~Henderson, and A.W.~Thomas,
        Few Body Syst. {\bf 12}, 123 (1992) and
        Mod. Phys. Lett. {\bf A7}, 3037 (1992).
\bb{OPTW}H.B.~O'Connell, B.C.~Pearce, A.W.~Thomas, and A.G.~Williams, 
        Phys. Lett. {\bf B336}, 1 (1994) and
        Prog. Part. Nucl. Phys. {\bf 39}, 201 (1997).
\bb{decay} B.W.~Lee and M.T.~Vaughn, Phys. Rev. Lett. {\bf 4}, 578 (1960).
\bb{klingl}F.~Klingl, N.~Kaiser, and W.~Weise, 
        Z. Phys. {\bf A356}, 193 (1996).
\bb{basis1} F.~M. Renard, Springer Tracts in 
        Modern Physics {\bf 63}, 98 (1972) and references therein.
\bb{CB} S.A.~Coon and R.C.~Barrett, Phys. Rev. {\bf C36}, 2189 (1986) and
        references therein. 
\bb{newCB} This value arises from 
Br($\omega\ra\pi^+\pi^-$), $m_\rho$, and 
$\Gamma_\rho$ as determined in Ref.~\cite{Bark}.
Note that these parameters with 
        $m_\omega=781.94$ MeV~\cite{PDG} yield
        $|\prw(m_\omega^2)|=4180$ MeV$^2$. 
\bb{quoted} Y.~Wu, S.~Ishikawa, and T.~Sasakawa,
        Phys. Rev. Lett. {\bf 64}, 1875 (1990); 
        T.~Hatsuda, E.M.~Henley, Th.~Meissner and
        G.~Krein, Phys. Rev. C {\bf 49}, 452 (1994); 
        H. Genz and S. Tatur,
        Phys. Rev. {\bf D50}, 3263 (1994); M.J.~Iqbal and J.A.~Niskanen,
        Phys. Lett. {\bf B322}, 7 (1994); K.L.~Mitchell, P.C.~Tandy, 
        C.D.~Roberts, and R.T.~Cahill, Phys. Lett. {\bf B335} (1994) 282; 
        M.J.~Iqbal, X.~Jin, and D.B.~Leinweber, Phys. Lett. {\bf B367}, 45
        (1996); 
         S.~Gao, C.M.~Shakin, and W.D.~Sun,
        Phys. Rev. C {\bf 53}, 1374 (1996).
\bb{GL} J.~Gasser and H.~Leutwyler, Phys. Rep. {\bf 87}, 77 (1982). 
        Note $\prw\equiv2m_\rho M_{\rho\omega}$ as per Eq.~(F1) of this
        reference.
        See
        also Ref.~\cite{urech}. 
\bb{urech} R.~Urech, Phys. Lett. {\bf B355}, 308 (1995);
        D.-N.~Gao and M.-Y.~Yan, 
        ``\rhoom Mixing in U$(3)_L\otimes$U$(3)_R$ Chiral Theory of Mesons,"
         to appear in Euro. Phys. J. {\bf A}. 
        These papers are inconsistent with 
        Eq.~(\ref{Br3}) if terms of ${\cal O}((m_\omega-m_\rho)/m^{\rm av})$ 
        are included. 
\bb{gas66} S.~Gasiorowicz, Elementary Particle Physics (John Wiley \& Sons,
        New York, 1966).
\bb{HL} M.F.~Heyn and C.B.~Lang, Z.~Phys. {\bf C7}, 169 (1981).
\bb{GS} G.J.~Gounaris and J.J.~Sakurai, Phys. Rev. Lett. {\bf 21},
        244 (1968). 
\bb{lang79} C.B. Lang and A. Mas-Parareda, Phys. Rev. D {\bf 19}, 956 (1979).
\bb{Levy} M. L\'evy, Nuovo Cimento {\bf 13}, 115 (1959). See also 
        R. E. Peierls, Proc. 1954 Glasgow Conf. on Nuclear and Meson Physics
        (Pergamon, New York, 1955) p. 296. 
\bb{Smatrix} R.J.~Eden, P.V.~Landshoff, P.J.~Olive, and J.C.~Polkinghorne,
        {\it The Analytic S-Matrix} (Cambridge University Press, 1966).
\bb{pdg72}  P.~S\"{o}ding {\it et al.} (Particle Data Group), 
        Phys. Lett. {\bf 39B}, 1 (1972).

\bb{pisut68} J.~Pi$\check{{\rm s}}$\'ut and M.~Roos, 
        Nucl. Phys. {\bf B6}, 325 (1968).
\bb{ben93} M.~Benayoun {\it et al.}, Z.~Phys. {\bf C58}, 31 (1993) and
        Eur. Phys. J. {\bf C2}, 269 (1998).
\bb{PDG} R.M.~Barnett {\it et al.} (Particle Data Group), 
        Phys. Rev. D {\bf 54}, 1 (1996).
\bb{BCP}  A. Bernicha, G. Lopez Castro, and J. Pestieau,
        Phys. Rev. D {\bf 50}, 4454 (1994).
\bb{Stuart} R.G.~Stuart, Phys. Lett.  {\bf B262}, 113 (1991); 
        Phys. Lett. {\bf B272}, 353 (1991); 
        Phys. Rev. Lett. {\bf 70}, 3193 (1993);
        Phys. Rev. D {\bf 52}, 1655 (1995). 
\bb{renard} J.P.~Perez-y-Jorba and F.M.~Renard, Phys. Rep.
        {\bf 31}, 1 (1977); F.M.~Renard, Nucl. Phys. {\bf B82}, 1 (1974).
\bb{CG} N.~Cabibbo and R.~Gatto, Phys. Rev. {\bf 124}, 1577 (1961).
\bb{3pi} J.E.~Augustin {\it et al.}, Phys. Lett. {\bf 28B}, 513 (1969);
        A.M.~Kurdadze {\it et al.}, Pisma Zh. Eksp. Teor. Fiz. {\bf 36}, 221
        (1982) [JETP Lett. {\bf 36}, 274 ( 1982)];
        L.M.~Barkov {\it et al.}, Pisma Zh. Eksp. Teor. Fiz. {\bf 46}, 132
        (1987) [JETP Lett. {\bf 46}, 164 (1987)]; 
        S.I.~Dolinsky {\it et al.}, Z.~Phys. {\bf C42}, 511 (1989).
\bb{Ben} D.~Benaksas {\it et al.}, Phys. Lett. {\bf 39B}, 289 (1972).
\bb{syserr} As in Ref.~\cite{GO}, we 
assume that the central values computed 
in the various models, fitted to the same data set, 
are normally distributed, so that we
compute their variance to infer the 
theoretical systematic error in a particular parameter.
\bb{footnote} Note that $f_{\rho_I\gamma}$ is determined at 
$s=m_\rho^2$, whereas we determine 
$\prw(s)$ at $s=m_\omega^2$. 
%!
%\bb{details} To see this, note 
%Eq.~(\ref{eq:gpp}), with Eqs.~(\ref{eq:formnew}) and
%(\ref{ff-isofit}), 
%and the relationship between the
%leptonic coupling, $f_{V\gamma}$, used by us and that 
%used by Bernicha {\it et al.}, which we denote as $F_{V\gamma}$, 
%$f_{V\gamma}=m_V^2/F_{V\gamma}$.

\bb{Q} A.~Quenzer {\it et al.}, Phys. Lett. {\bf 76B}, 512 (1978).



\end{thebibliography}



\nc{\m}{$m_\rho$ (MeV)}
\nc{\G}{$\Gamma_\rho$ (MeV)}
\nc{\gree}{$\Gamma({\rho\ra e^+ e^-})$ (keV)}
\nc{\Bree}{Br$({\rho\ra e^+e^-})\times (10^{5})$}
\nc{\frg}{$f_{\rho_I\gamma}$ (GeV$^2$)}
\nc{\zm}{$\overline{m}_\rho$ (MeV)}
\nc{\zg}{$\overline{\Gamma}_\rho$ (MeV)}
\begin{table}[htb]
\caption{
The results of our fits~\protect{\cite{GO}} to the pion form-factor 
and the corresponding values of
$\Gamma(\rho^0\ra e^+e^-)$, noting Eq.~(\protect{\ref{Gree}}), and 
Br$(\rho^0\ra e^+e^-)$.
Also shown are the $\rho$ parameters, $\overline{m}_\rho$
and $\overline{\Gamma}_\rho$,
defined from the pole position $z_\rho$, as in Eq.~(\protect{\ref{pole}}).
}
\begin{tabular}{ccccccc}
Fit & \m\cite{GO} & \G\cite{GO} &\gree & \Bree & \zm & \zg          \\
\hline
A&$763.1\pm3.9$&$153.8\pm1.2$&$7.27\pm0.08$&$4.73\pm0.05$&
        $756.3\pm1.2$ & $141.9\pm3.1$\\
B&$771.3\pm1.3$&$156.2\pm0.4$&$7.24\pm0.08$&$4.63\pm0.06$&
        $757.0\pm1.0$ & $141.7\pm3.0$\\
C&$773.9\pm1.2$&$157.0\pm0.4$&$7.19\pm0.08$&$4.58\pm0.05$&
        $757.0\pm1.0$ & $141.7\pm3.0$\\
D&$773.9\pm1.2$&$146.9\pm3.4$&$6.73\pm0.10$&$4.58\pm0.05$&
        $757.0\pm1.0$ & $141.7\pm3.0$\\
\end{tabular}
\label{one}
\end{table}

\vspace{2cm}

\nc{\prwm}{$\widetilde{\Pi}_{\rho\omega}(m_\omega^2)$}
\nc{\ff}{$f_{\rho_I\gamma}/f_{\omega_I\gamma}$}
\nc{\B}{Br$(\omega\ra\pi^+\pi^-)$ }
\nc{\BB}{Br$^{(2)}(\omega\ra\pi^+\pi^-)$ }
\nc{\BBB}{Br$^{(3)}(\omega\ra\pi^+\pi^-)$ }
\begin{table}[htb]
\caption{
Results for the effective \rhoom mixing element, $\prw$,
and the branching ratio \B, from Eq.~(\protect{\ref{Br_final}}), using the
fits of Ref.~\protect{\cite{GO}}. 
$f_{\omega_I\gamma}$ follows from
Eq.~(\protect{\ref{eq:lep}}) 
and the parameters of 
Ref.~\protect{\cite{PDG}}. We also show the value of $\prw$ 
which results from using Eq.~(\protect{\ref{replace}}) with 
$f_{\rho_I\gamma}/f_{\omega_I\gamma}$, as per 
Eqs.~(\protect{\ref{eq:lep}}) and (\protect{\ref{picase}}), again
using the fits of Ref.~\protect{\cite{GO}}. 
}
\begin{tabular}{cccccc}
Fit&\prwm (MeV$^2$)\cite{GO}&\frg  &  \ff  &\prwm (MeV$^2$) &\B\\    
\hline
A&$-3460\pm290$&$0.120\pm0.001$&$3.36\pm0.07$&$-3870\pm320$&$1.87\pm0.30\%$\\
B&$-3460\pm290$&$0.122\pm0.001$&$3.40\pm0.06$&$-3920\pm330$&$1.87\pm0.30\%$\\
C&$-3460\pm290$&$0.122\pm0.001$&$3.41\pm0.06$&$-3930\pm330$&$1.87\pm0.30\%$\\
D&$-3460\pm290$&$0.118\pm0.001$&$3.30\pm0.06$&$-3800\pm330$&$1.87\pm0.30\%$\\
\end{tabular}
\label{two}
\end{table}

\vspace{2cm}

\begin{table}[htb]
\caption{
The branching ratio \B from 
our preferred method, Eq.~(\protect{\ref{Br_final}}), 
compared with the alternatives \BB,
Eq.~(\protect{\ref{Br2}}), and \BBB, Eq.~(\protect{\ref{Br3}}). 
In parentheses
we give the values for the branching ratio as determined by 
Eq.~(\protect{\ref{Br3}})
but replace $z_\rho$ with $m_\rho^2-im_\rho\Gamma_\rho$, noting
the discussion preceding Eq.~(\protect{\ref{pole}})
and the results of Table~\protect{\ref{one}}.
}
\begin{tabular}{ccccc}
Fit & \B          & \BB          & \BBB\\
\hline
A&$1.87\pm0.30\%$&$1.93\pm0.32\%$&$2.41\pm0.39\%\,\,(2.15\pm0.35\%$)\\
B&$1.87\pm0.30\%$&$1.97\pm0.32\%$&$2.50\pm0.40\%\,\,(2.19\pm0.35\%$) \\
C&$1.87\pm0.30\%$&$1.96\pm0.32\%$&$2.51\pm0.40\%\,\,(2.19\pm0.35\%$) \\
D&$1.87\pm0.30\%$&$1.95\pm0.32\%$&$2.20\pm0.37\%\,\,(2.20\pm0.35\%$) \\
\end{tabular}
\label{three}
\end{table}

\end{document}





















