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\begin{document}

\title{Radiative $\phi$ decays}

\author{E. Oset$^1$, S. Hirenzaki$^2$, E. Marco$^3$ and H. Toki$^4$}

\address{$^1$Departamento de F\'{\i}sica Te\'orica and I.F.I.C.,
  Centro Mixto,Universidad de Valencia-C.S.I.C.,
  46100 Burjassot (Valencia), Spain\\
  $^2$Department of Physics, Nara Women's University,
  Nara 630-8506, Japan\\
  $^3$Physik Department, Technische Universit\"at M\"unchen,
  D-85747, Germany\\
  $^4$RCNP, Osaka University, Ibaraki, Osaka 567-0047, Japan} 
  
\maketitle\section{Radiative $\phi$ decay} 

  The $\phi$  decay into $\pi^0 \pi^0 \gamma$ and $\pi^0 \eta \gamma$ is
forbidden at tree level but can proceed via kaonic loops, which involve the 
$K^+K^-$ transition to $\pi^0 \pi^0$ and  $\pi^0 \eta$ \cite{Bra}, where the 
$f_0(980)$ and $a_0(980)$ resonances appear. This is the reason why these
reactions have been for long envisaged as a source of information on the nature
of these resonances, for which there have been many suggestions, like 
$q \bar{q}$, $qq \bar{q}\bar{q}$ or $K \bar{K}$ molecules. The advent of recent
developments in chiral unitary theory has brought new light into this question
as was evident in the animated discussion between N.Isgur and U.G. Meissner in
the group report of U.G. Meissner \cite{ulf}. According to the works
\cite{oop,ollerpaco,oo}, these resonances would be ordinary meson meson
scattering resonances coming from multiple scattering of the mesons,  the
genuine $q \bar{q}$ states appearing at higher energies, around 1.4 GeV.

  We follow the steps of \cite{Bra}, however, using the amplitudes generated in
the approaches of \cite{oop,ollerpaco,oo}, or the more simple one, which proves
to be good for the s-waves needed in this problem, of \cite{ollos}. In addition
we also use the tensor formulation for the coupling of vector mesons to pairs
of pseudoscalar mesons of \cite{ecker}.
     

 
The diagrammatic representation of the mechanisms involved in the radiative
 decay process are shown in fig. 1 of \cite{frascati}. The diagrams involve
 loops with charged kaons with the photon coupled in all possible ways to the
 loops and $\pi^0 \pi^0$ or $\pi^0 \eta$ in the final state.


 The evaluation of these diagrams  is done in \cite{marco}. An
 interesting feature of these reactions is that the $K^+K^- \to \pi^0 \pi^0$
 and $K^+K^- \to \pi^0 \eta$
 transition amplitudes account rather accurately for the $f_0(980)$ and 
 $a_0(980)$ excitation in the chiral unitary approaches mentioned above.





 We have evaluated the invariant
mass distribution for these decay channels and in Fig.\ 2 of \cite{frascati}
one can see 
the distribution $dB/dM_I$ for $\phi \rightarrow \pi^0 \pi^0 \gamma$ 
which allows us to see the
$\phi \rightarrow f_0 \gamma$ contribution since the $f_0$ is
the important scalar resonance appearing in the
$K^+ K^- \rightarrow \pi^0 \pi^0$ amplitude \cite{ollos}. The results are
obtained using $G_V$=55 MeV and $F_V$=165 MeV, which are suited to describe
the $K \bar{K}$ and $e^+ e^-$ decay of the $\phi$, and  
 $G_V$=69 MeV,  $F_V$=154 MeV, which are suited for $\rho$ decay. Both the size
 and the shape of the theoretical results compare well with the results 
 of the Novosibirsk experiment \cite{Novo}, or
the similar ones of \cite{CMDpi0pi0}.
The branching ratio obtained for the case
$\phi \rightarrow \pi^0 \eta \gamma$ is $0.87 \times 10^{-4}$. The
results obtained at Novosibirsk are \cite{pi0eta} $(0.83 \pm 0.23)\times 10^{-4}$
and \cite{Akhmetshin}
$(0.90\pm 0.24\pm 0.10)\times 10^{-4}$. Should we use the values for $F_V$ and  
$G_V$ of the $\rho$ decay we would obtain
$1.6 \times 10^{-4}$.
The spectrum, not shown, is dominated by the $a_0$ contribution. The results 
obtained using the different $G_V$ and $F_V$ couplings give us an idea of our
theoretical uncertainties. More precise data in the future can help us
determine more universal values for these couplings free of the
renormalization corrections which make them different in the $\phi$ or $\rho$
cases.



 The results reported here are an example of the successful application of the 
 chiral unitary techniques. A recent review of multiple applications of these
 methods can be seen in \cite{oor}. 




\vspace{0.1cm}

\section{References}

 
\begin{thebibliography}{99}

%\footnotesize

\bibitem{Bra} A. Bramon, A. Grau and G. Pancheri, Phys.\ Lett.\ B {\bf 289},
  97 (1992).
\bibitem{ulf} U.G. Meissner, group report discussion session.
\bibitem{oop} J. A. Oller, E. Oset and J. R. Pel\'aez, Phys.\ Rev.\ Lett.\
  {\bf 80}, 3452 (1998); Phys.\ Rev.\ D {\bf 59}, 074001 (1999); 
 Erratum-ibid. D {\bf 60}, 099906 (1999).
\bibitem{ollerpaco} F. Guerrero and J. A. Oller, Nucl.\ Phys.\ B {\bf 537}, 
 459 (1999).
\bibitem{oo} J. A. Oller and E. Oset, Phys.\ Rev.\ D {\bf 60}, 074023 (1999).
\bibitem{ollos} J. A. Oller and E. Oset, Nucl.\ Phys.\ A {\bf 620},
  438 (1997); erratum Nucl.\ Phys.\ A {\bf 624}, 407 (1999).
\bibitem{ecker} G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl.\ Phys.\
 B {\bf 321}, 311 (1989).
\bibitem{frascati}  E. Oset, S. Hirenzaki, E. Marco, J.A. Oller,
 J.R. Pelaez and H. Toki. 
 Frascati Physics Series Vol. XVI, p. 581, hep-ph/0002189. 
\bibitem{marco} E. Marco, S. Hirenzaki, E. Oset and H. Toki, Phys.
Lett.\ B {\bf 470} 20 (1999). 
\bibitem{Novo} M. N. Achasov et al., Phys.\ Lett.\ B {\bf 440}, 442 (1998).
\bibitem{CMDpi0pi0} R. R. Akhmetshin et al.,Phys. \ Lett. \ B {\bf 462}, 380
(1999).
\bibitem{pi0eta} M. N. Achasov et al., Phys.\ Lett.\ B {\bf 438}, 441 (1998).
\bibitem{Akhmetshin} R. R. Akhmetshin et al., Phys.\ Lett.\ B {\bf 415},
  452 (1997).
\bibitem{oor} J. A. Oller, E. Oset and A. Ramos, Prog.\
 Part.\ Nucl.\ Phys.\  {\bf 45}, 157 (2000).
\end{thebibliography}  



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