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\begin{document}

\input{FEYNMAN}
\title{The $\phi\rightarrow\pi^+\pi^-$ 
decay}

\author{J. R. Pel\'aez$^1$, J.A. Oller$^2$ and E. Oset$^3$}

\address{$^1$ Dept. F\'{\i}sica Te\'orica. Universidad Complutense. 
Madrid 28040, SPAIN.\\
$^2$ Forschungszentrum J\"ulich, Institut f\"ur
Kernphysik (Theorie).
 D-52425 J\"ulich, GERMANY.\\
$^3$ Dept. F\'{\i}sica Te\'orica and I.F.I.C.
Centro Mixto Universidad de Valencia- C.S.I.C.
46100 Burjassot (Valencia), SPAIN.}

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% You may repeat \author \address as often as necessary      %
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\maketitle

\section{Introduction}
The interest of the $\phi\rightarrow\pi^+\pi^-$ decay is due
to the fact that it violates isospin, it is OZI suppressed, and 
thus it could shed some light on both $\phi-\rho$ and $\phi-\omega$
mixing. 
In addition, at present there is a strong experimental conflict
between the two most recent and precise measurements at VEPP-2M.
On the one hand, CMD-2 \cite{CMD2} reports 
a value of $BR=(2.20\pm0.25\pm0.20)\times 10^{-4}$
and on the other hand SND \cite{SND} provides 
$BR=(0.71\pm0.11\pm0.09)\times 10^{-4}$.

There are two main contributions to this process:
\begin{itemize}
\item From $\phi-\rho$ mixing, where, once more, we distinguish:
  \begin{itemize}
  \item The electromagnetic $\phi-\gamma-\rho-\pi\pi$ contribution.
    The $g^{\gamma}_{\phi\pi\pi}$ effective coupling 
is obtained from the vector
    resonance chiral lagrangian \cite{Ecker}
   \item The strong contribution due to kaon loops, where we
    have used the chiral unitary approach to obtain an effective
    coupling\cite{nosotros} $g^{K\bar{K}}_{\phi\pi\pi}$.
  \end{itemize}
\item From the $\phi-\omega-\rho$ process. We have calculated
a $g^{\phi\omega}_{\phi\pi\pi}$ coupling for
three different scenarios in the literature: the 
``Strong $\phi-\omega$ mixing'' \cite{achasov}, the
``Weak $\phi-\omega$ mixing'' \cite{achasov} and the
``Hidden Local Symmetry'' scenario \cite{oconel}.
\end{itemize}
\vskip-0.4cm
\vskip-0.1cm
%\vskip-0.5cm
\section{The kaon-loop contribution to the $\phi\pi\pi$ effective
  coupling.}
\vspace{-.2cm}

The inverse amplitude method (IAM), based on the standard Chiral
Perturbation Theory (ChPT) $O(p^4)$ SU(3) Lagrangian \cite{ChPT}, provides
a coupled channel unitary $T$ matrix respecting the constraints of
chiral symmetry at low energies. It has been very succesful \cite{IAM} in describing
the meson-meson interactions up to 1.2 GeV, generating six resonances.
In addition, in the $I=0$, $J=0$ partial wave it develops 
a pole that corresponds to the octet $\omega_8$ resonant state.

By including isospin breaking effects in the ChPT lagrangian, 
the chiral unitary approach provides a nonvanishing
$K^+K^-\rightarrow \pi^+\pi^-$  $P-wave$ amplitude.
Reinterpreting these amplitudes at the $\omega_8$ pole,
in terms of $\omega_8$ exchange, we can 
obtain the effective $g_{\phi\pi\pi}^{K\bar{K}}$, schematically, as follows:

\vspace{.5cm} 

\hspace*{-.7cm}
\begin{picture}(35000,8000)

\THICKLINES

\drawline\fermion[\SE\REG](2000,8000)[2500]
\drawline\fermion[\NE\REG](2000,4500)[2500]
\put(5200,6250){\circle{3000}}
\global\advance \pbackx by 3000
\drawline\fermion[\SE\REG](\pbackx,\pbacky)[2500]
\drawline\fermion[\NE\REG](\pfrontx,\pfronty)[2500]

\put (8700,7300) {$\pi^+$}
\put (8700,4500) {$\pi^-$}
\put (200,7300) {$K^+$}
\put (200,4500) {$K^-$}
\put(4300,5850){IAM}

\drawline\fermion[\E\REG](0,4000)[11000]

\drawline\fermion[\SE\REG](2000,3500)[2500]
\drawline\fermion[\NE\REG](2000,0)[2500]
\put(5200,1750){\circle{3000}}
\global\advance \pbackx by 3000
\drawline\fermion[\SE\REG](\pbackx,\pbacky)[2500]
\drawline\fermion[\NE\REG](\pfrontx,\pfronty)[2500]

\put (8700,2700) {$K^+$}
\put (8700,0) {$K^-$}
\put (200,2700) {$K^+$}
\put (200,0) {$K^-$}
\put(4300,1350){IAM}

\put (11300,3700) { \Large =}

\drawline\fermion[\SE\REG](15000,8000)[2500]
\drawline\fermion[\NE\REG](15000,4500)[2500]
\drawline\fermion[\E\REG](16800,6450)[3000]
\drawline\fermion[\E\REG](16800,6050)[3000]
\global\advance \pbacky by 200
\drawline\fermion[\SE\REG](\pbackx,\pbacky)[2500]
\drawline\fermion[\NE\REG](\pfrontx,\pfronty)[2500]

\put (21700,7300) {$\pi^+$}
\put (21700,4500) {$\pi^-$}
\put (13200,7300) {$K^+$}
\put (13200,4500) {$K^-$}
\put (18000,6850) {$\omega_8$}

\drawline\fermion[\E\REG](13000,4000)[11000]

\drawline\fermion[\SE\REG](15000,3500)[2500]
\drawline\fermion[\NE\REG](15000,0)[2500]
\drawline\fermion[\E\REG](16800,2050)[3000]
\drawline\fermion[\E\REG](16800,1650)[3000]
\global\advance \pbacky by 200
\drawline\fermion[\SE\REG](\pbackx,\pbacky)[2500]
\drawline\fermion[\NE\REG](\pfrontx,\pfronty)[2500]

\put (21700,2700) {$K^+$}
\put (21700,0) {$K^-$}
\put (13200,2700) {$K^+$}
\put (13200,0) {$K^-$}
\put (18000,2350) {$\omega_8$}


\put (24000,3700) {
\Large $=\frac{g_{\omega_8\pi\pi}^{K\bar{K}}}{g_{\omega_8 K\bar{K}}}\simeq
\frac{g_{\phi\pi\pi}^{K\bar{K}}}{g_{\phi K\bar{K}}}$,}

%\put (28800,2000) {\LARGE $\uparrow$}
%\put (27200,700) {ideal mixing}
\end{picture}    
where the first amplitude fraction comes from the chiral unitary
approach and $g_{\phi K\bar{K}}$ is known from the vector
resonance chiral Lagrangians.

There are other isospin breaking terms including photons, 
which would involve chiral parameters which are presently unknown. 
From absorptive calculations \cite{Bramon} 
their contributions have been estimated
to be at most 25\% of those due to kaon loops, but with opposite sign.
\vspace{-.4cm}
\section{Results}
\vspace{-.2cm}
They depend on the $\phi-\omega$ mixing scenario.

$\bullet$ ``Strong mixing'': 
$BR= (1.7\pm0.3)\times 10^{-4}$ to $(2.5\pm0.3)\times 10^{-4}$.

$\bullet$ ``Weak mixing'': $BR\simeq(0.38\pm0.12)\times 10^{-4}$.

$\bullet$ ``Hidden Local Symmetry'': $BR\simeq(0.3\pm0.1\pm0.3)\times
10^{-4}$.

The first one is perfectly compatible with the CMD-2 experiment,
whereas the other two seem to prefer a value somewhat lower (but still 
compatible) than that of SND. Therefore, it seems that once the 
experimental issue is settled it could be used to differentiate
between $\phi-\omega$ mixing scenarios.

\vspace{-.3cm}
\section*{Acknowledgments}
\vspace{-.2cm}
Work partially supported by DGICYT contracts PB96-0753,
AEN97-1693 and PB98-0782 and EU TMR Eurodaphne contract 
ERBFMRX-CT98-0169.

\vspace{-.3cm}
\section*{References}
\vspace{-.2cm}
\begin{thebibliography}{99}
\bibitem{CMD2} G. V. Fedotovich for the CMD-2 Collaboration at
the 8th Intl. Conf. on Hadron Spectroscopy. HADRON 99,
24-28 August 1999, Beijing China.
\bibitem{SND} S. Burdin for the SND Collaboration.  hep-ex/9910070. 
N. N. Achasov et al., Nucl. Phys. A675 (2000) 320.
\bibitem{Ecker} G. Ecker {\it et al.}, Nucl. Phys.
 B321,  (1989) 311.
\bibitem{nosotros} J.A. Oller, E. Oset and J.R. Pel\'aez, hep-ph/9911297 
\bibitem{achasov} N.N. Achasov {\it et al.}, Int. J. Mod. Phys. A7
  (1992) 3187; Phys. Lett.
  B233, (1989) 474; Int. Jour. Phys. A7, (1992) 4825.
\bibitem{oconel} M. Benayoun {\it et al.}, nucl-th/0004005.
\bibitem{ChPT} J. Gasser and H. Leutwyler, Nucl. Phys. B250, (1985) 465.
\bibitem{IAM}  J. A. Oller, E. Oset and J. R. Pel\'aez, 
Phys. Rev. Lett. 80, (1998)
3452; Phys. Rev. D59, (1999) 074001; Erratum-ibid. D60, (1999) 099906.
\bibitem{Bramon} A. Bramon and A. Varias, Phys. Rev. D20, (1979) 2262.
\end{thebibliography}

\end{document}
