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\begin{center}
{\Large{\bf $\eta \to \pi^0 \gamma \gamma$ decay within a 
chiral unitary approach }}
\end{center}
\vspace{1cm}

\title{$\eta \to \pi^0 \gamma \gamma$ decay within unitarized chiral
perturbation theory }

\begin{center}
{\large{E. Oset$^1$, J. R. Pel\'aez$^2$ and L. Roca$^1$}}

 \vspace{.5cm}
{\it $^1$ Departamento de F\'{\i}sica Te\'orica and IFIC
 Centro Mixto Universidad de Valencia-CSIC\\
 Institutos de Investigaci\'on de Paterna, Apdo. correos 22085,
 46071, Valencia, Spain}    

\vspace{.5cm}Dip. di Fisica. Universita' degli Studi, Firenze, and INFN, Sezione di Firenze, Italy\\
   {\it $^2$ Departamento de
F\'{\i}sica Te\'orica II,  Universidad Complutense. 28040 Madrid,
Spain.
}  
\end{center}


\begin{abstract}
Together
with the VMD dominant terms for  $\eta \to \pi^0 \gamma \gamma$,
 we introduce the contribution of chiral loops 
from unitarized chiral perturbation theory,
thus generating the $a_0(980)$ resonance and fixing
the longstanding sign ambiguity on its contribution to this process.
 The same approach allows us to 
establish constraints in the decay from the
$\gamma \gamma \to \pi^0 \eta$ data. 
In particular, the $\eta VV$
couplings have to be reduced  to agree with
the experimental $V \to \eta \gamma$ partial decay widths.
We obtain 
for the $\eta \to \pi^0 \gamma \gamma$ decay width $0.47\pm 0.10$ eV, which
is in remarkable agreement with the most recent experimental measurement.

\end{abstract}


\section{Introduction}

The $\eta \to \pi^0 \gamma \gamma$ decay has attracted much theoretical 
attention, particularly in the last decade, since the attempts to obtain the
measured experimental width \cite{exp,Hagiwara:pw} within the formalism of chiral 
perturbation theory, 
$\chi PT$, have been rather problematic. 
The problem stems from the fact that the
tree level amplitudes, both at $O(p^2)$ and $O(p^4)$, vanish and the first
non-vanishing contribution comes at $O(p^4)$, but either from loops involving
kaons, which are
largely suppressed due to the kaon masses, or from pion loops, again
suppressed since they violate G parity and are thus
 proportional to $m_u -m_d$ \cite{Ametller:1991dp}. 
Thus, the first sizable contribution comes at 
$O(p^6)$ and the coefficients involved are not precisely determined. One must
recur to models: either Vector Meson Dominance (VMD) \cite{oneda,Ametller:1991dp,Picciotto:sn}, the 
Nambu-Jona-Lasinio model
(NJL) \cite{Bel'kov:1995fj}, or the extended Nambu-Jona-Lasinio model
(ENJL) \cite{Bellucci:1995ay,Bijnens:1995vg}, have
been used to determine these parameters. 
With the exception of models using quark box
diagrams to evaluate the eta decay rate \cite{Ng:sc,Nemoto:1996bh}, which 
obtain rates comparable with most
experiments \cite{exp}, those based in chiral perturbation theory produce systematically
smaller rates, about a factor two smaller in average. Attempts to consider more
mechanisms in the process have been made, like the inclusion of an axial vector
meson exchange \cite{Ko:1992zr,Ko:rg}, but there are 
large uncertainties in the couplings. Furthermore, 
values of these couplings that produce sizable effects in the eta
radiative decay lead to disagreement with experiment in the 
$\gamma \gamma\to \pi^0 \pi^0$ reaction \cite{Jetter:1995js}. A critical recent
discussion on the work done on the topic, both from the theoretical and
experimental point of view is done in \cite{Achasov:2001qm}, where at the same
time the authors
present some new experimental results consisting in an upper bound, which is consistent
with the earlier measurements.

  

The problem will have to be reconsidered 
if results from a recent experiment \cite{nefkens}, which give a decay width
about half the previous one, are confirmed. 
In this work the authors refine the background subtraction, which was known to be rather
problematic.
Yet, from the theoretical point of view, some extra work is mandatory
and we face this problem here. One of the sources of uncertainty quoted in
\cite{Ametller:1991dp} is due to the contribution of the $a_0(980)$ resonance, 
which is
taken into account approximately and has 
large uncertainties, including the sign
of its contribution. Another question is that no attempts have been done to
relate the process to the crossed channel, the $\gamma \gamma\to \pi^0 \eta$
reaction, although some consistency tests with $\gamma \gamma\to \pi^0 \pi^0$
have been carried out  as quoted above. 
The reason is not surprising since there are no
hopes within $\chi PT$ to reach the region of the $a_0(980)$ resonance where
there are measurements of the $\gamma \gamma\to \pi^0 \eta$ cross section 
\cite{Oest:1990ki,Antreasyan:1985wx}. The situation, however,
 has improved in recent years with the
advent of unitarized extensions of $\chi PT$, by means of which the results of 
$\chi PT$ can be extended to higher energies, 
and meson resonances up to 1.2 GeV
are nicely described 
\cite{Oller:1997ti,Kaiser:fi,OllOsePel,Oller:1999zr,Nieves:2000bx,GomezNicola:2001as}. 
In particular these
ideas were used to describe the $\gamma \gamma \to meson-meson$ reaction, with
good results in all the channels up to energies of around 1.2 GeV 
\cite{Oller:1997yg}. Work in a
similar direction for this latter reaction introducing nonperturbative 
techniques has also been done in \cite{Dobado:1992zs,Yamagishi:1995kr,Lee:1998mz}. 
For the problem of interest here
we just quote that the peak around the $a_0(980)$ resonance in the 
$\gamma \gamma\to \pi^0 \eta$ reaction was produced at the right place and with
about the experimental strength \cite{Oller:1997yg}, using the same input as in 
meson meson scattering without introducing any extra parameters .

  There is another point worth mentioning. On the one hand,
as it has become clear from former
theoretical studies, the chiral approach is useful 
and offers some guidelines in
the $\eta \to \pi^0 \gamma \gamma$ reaction,
but the strict chiral counting is not much helpful since the $O(p^6)$ terms are
larger than those of order  $O(p^4)$.  On the other hand it was also shown in
\cite{Ametller:1991dp} that the use of VMD to obtain the $O(p^6)$ chiral coefficients
by expanding the vector meson propagators lead to results about a factor 
of two smaller than the "all order" VMD term. Furthermore, 
recent studies on the
vector meson decay into two pseudoscalar mesons and one photon
\cite{Bramon:2001un,Palomar:2001vg,Marco:1999df} indicate that the combination of "all orders" 
vector meson  contribution plus the unitary summation of the
chiral loop functions leads to good agreement with data in a variety of
reactions. These include those where the chiral loops are dominant, 
$\phi \to \pi^0 \pi^0 \gamma$ \cite{Marco:1999df}, the VMD mechanism is dominant, 
$\omega \to \pi^0 \pi^0 \gamma$ \cite{Bramon:2001un,Palomar:2001vg}, or both 
mechanisms have about the same
strength and interfere constructively to give the right partial decay width,
$\rho \to \pi^0 \pi^0 \gamma$ \cite{Bramon:2001un,Palomar:2001vg}. Guided by 
the success of
this approach in the radiative decays of vector mesons we shall follow the same
approach for the double radiative decay of the eta.


\section{VMD contribution}
Following \cite{Ametller:1991dp} we consider the VMD
mechanism of Fig.~1
\begin{figure}[h]
\centerline{\hbox{\psfig{file=fig1.eps,width=12cm}}}
\caption{\rm \label{fig:diagrams1}
Diagrams for the VMD mechanism.}
\end{figure}
which can  be easily derived from the VMD Lagrangians involving
VVP and $V\gamma$ couplings \cite{Bramon:1992kr}
\begin{equation}
{\cal L}_{VVP} = \frac{G}{\sqrt{2}}\epsilon^{\mu \nu \alpha \beta}\langle
\partial_{\mu} V_{\nu} \partial_{\alpha} V_{\beta} P \rangle, \qquad
{\cal L}_{V \gamma} =-4f^{2}egA_{\mu}\langle QV^{\mu}\rangle,
\label{lagr}
\end{equation}
where $V_{\mu}$ and $P$ are standard $SU(3)$ matrices
constructed with the nonet of vector mesons containing the
$\rho$, and the nonet of pseudoscalar mesons containing the
$\pi$, respectively. For instance for pseudoscalar mesons
$P=\tilde{P}+\frac{1}{\sqrt{3}}\eta_1$, with $\tilde{P}$ given 
by \cite{Gasser:1983yg}
\begin{equation}
\tilde{P}=\left( \begin{array}{ccc}
\frac{1}{\sqrt{2}}\pi^0+\frac{1}{\sqrt{6}}\eta_8 & \pi^+  & K^+\\
\pi^- & -\frac{1}{\sqrt{2}}\pi^0+\frac{1}{\sqrt{6}}\eta_8 & K^0\\
K^- & \overline{K}^0 & -\frac{2}{\sqrt{6}}\eta_8 \\
\end{array}\right) ,
\end{equation}
and similarly for vector mesons \cite{Ecker:1988te}.
 We also assume the ordinary mixing for the $\phi$,
the $\omega$, the $\eta$ and $\eta'$:
\begin{eqnarray}
\omega &=& \sqrt{\frac{2}{3}}\omega_1 + \sqrt{\frac{1}{3}}\omega_8, \quad,
\qquad 
\phi = \sqrt{\frac{1}{3}}\omega_1   - \sqrt{\frac{2}{3}}\omega_8\\
\nonumber
\eta &=& \frac{1}{3}\eta_1 + \frac{2\sqrt{2}}{3}\eta_8, \quad
\qquad
\eta' = \frac{2\sqrt{2}}{3}\eta_1 - \frac{1}{3}\eta_8,
\label{eq:mezclas}
\end{eqnarray}
In Eq.~(\ref{lagr}) $G=\frac{3g^2}{4\pi^2f}$,
$g=-\frac{G_VM_{\rho}}{\sqrt{2}f^2}$ \cite{Bramon:1992kr}
and $f=93\,MeV$,
 with $G_V$ the
coupling of $\rho$ to $\pi\pi$ in the normalization of
\cite{Ecker:1988te}.
>From Eq.~(\ref{lagr}) one can obtain the radiative widths for
$V\to P\gamma$, which are given by
\begin{equation}
\Gamma_{V \to P\gamma}=\frac{3}{2} \alpha
 C_i^2 (G\frac{2}{3}\frac{G_V}{M_V})^2k^3
\end{equation}
with $k$ the photon momentum for the vector meson at rest and
$C_i$ an $SU(3)$ coefficient that we give in Table 1 
 for the different radiative decays, together
with the theoretical (using $G_V=69\,MeV$ and $f=93\,MeV$) and experimental \cite{Hagiwara:pw}
 branching ratios 

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c||c||c|c|}\hline  
$i$ &$C_i$  &$B_i^{th}$ & $B_i^{exp}$ \\ \hline \hline  
 $\rho\pi^0\gamma$  & $\sqrt{\frac{2}{3}}$  & $7.1\times 10^{-4}$ &$(7.9\pm 2.0)\times 10^{-4}$ \\ \hline  
 $\rho\eta\gamma$ & $\frac{2}{\sqrt{3}}$  &  $5.7\times 10^{-4}$   &  $(3.8\pm 0.7)\times 10^{-4}$ \\ \hline  
 $\omega\pi^0\gamma$ &$\sqrt{2}$  & $12.0$\%  &  $8.7\pm 0.4$\% \\ \hline  
$\omega\eta\gamma$ & $\frac{2}{3\sqrt{3}}$ & $12.9\times 10^{-4}$  & $(6.5\pm 1.1)\times 10^{-4}$ \\ \hline  
 $\phi\eta\gamma$& $\frac{2}{3}\sqrt{\frac{2}{3}}$ & $0.94$\% &
  $1.24\pm 0.10$\% \\ \hline  
 $\phi\pi^0\gamma$ &0 &-- &-- \\ \hline  
${ {K^{*+} \to K^+\gamma} \atop {K^{*-} \to K^-\gamma}}$ 
&$\frac{\sqrt{2}}{3}(2-\frac{M_{\omega}}{M_{\phi}})$  & $13.3\times 10^{-4}$  &  $(9.9\pm 0.9)\times 10^{-4}$ \\ \hline  
 ${ {K^{*0}\to K^{0}\gamma} \atop 
 {\overline K^{*0}\to \overline K^{0}\gamma} }$
 &$-\frac{\sqrt{2}}{3}(1+\frac{M_{\omega}}{M_{\phi}})$ & $27.3\times 10^{-4}$ & $(23\pm 2)\times 10^{-4}$ \\   
\hline
\end{tabular}
\caption{   SU(3) $C_i$ coefficients together with
 theoretical and experimental branching ratios for different 
vector meson decay processes. }
\end{center}
 \end{table}

The agreement with the data is fair but the
results can be improved if $SU(3)$ breaking mechanisms are
incorporated as done in \cite{Bramon:1994pq}.
For the purpose of
improving on the VMD amplitude for the diagram of Fig.~1
using the universal $SU(3)$ coupling, we can fine tune the
couplings such that the $\omega\to\pi^0\gamma$,
$\omega\to\eta\gamma$, $\rho\to\pi^0\gamma$ and
$\rho\to\eta\gamma$ branching ratios agree with experiment.
This is accomplished by multiplying the amplitude 
by normalization factors. We find that the 
diagram of Fig.~1 with an intermediate $\rho$ has to be multiplied by
$0.861$ whereas the one with the $\omega$ by  $0.604$ to reproduce the center values of the
experimental data of Table 1.

Once the $VP\gamma$ couplings have been fixed,  we can use
them in the VMD amplitude corresponding to the diagram
of Fig.~1, which is given by
\begin{eqnarray}
-it^{VMD}&=& \{ i\sqrt{6}\frac{1}{q^2-M_{\rho}^2+iM_{\rho}\Gamma(q^2)}
\left( G\frac{2}{3} e \frac{G_V}{M_V} \right) ^2
\cdot
\left\vert \begin{array}{ccc}
q\cdot q & q\cdot k_2  & q\cdot \epsilon_2\\
k_1\cdot q & k_1\cdot k_2 & k_1\cdot \epsilon_2\\
\epsilon_1\cdot q & \epsilon_1\cdot k_2 & \epsilon_1\cdot
\epsilon_2
\nonumber
\end{array} \right\vert \\
&& + (k_1\leftrightarrow k_2,\,q\to q') \} 
+ \{ \rho \to \omega
\}, 
\label{tvmd}
\end{eqnarray} 
where $q=P-k_1$, $q'=P-k_2$, with $P,k_1,k_2$ the momentum of the
$\eta$ and the two photons. We have parametrized
the $\rho$ width phenomenologically as: 
$\Gamma_\rho(q,s)=\frac{(6.14)^2}{48\pi s}(s^2-4 s\,m_\pi^2)^{3/2}$
whereas for the $\omega$ we have considered a constant
$\Gamma_\omega=8.44\,$ MeV. Nevertheless, our results are rather 
insensitive to these details. From the above amplitude,
the $\eta$ decay width is easily calculated, as well as the
$\gamma\gamma$ invariant mass distribution, $M_I$. In particular:
\begin{equation}
\frac{d\Gamma}{dM_I}=\frac{1}{16(2\pi)^4m_{\eta}^2}M_I
\int_0^{m_\eta-\omega}dk_1\int_0^{2\pi}d\phi\ \, \Theta(1-A^2)
\, \Sigma\mid t\mid^{2},
\end{equation} 
where we take for reference the momentum of the pion,
$\vec{p}$, in the $z$ direction, and then 
\begin{eqnarray}
\vec{p} = p \, \left(
\begin{array}{c}
0\\
0\\
1\\
\end{array} \right),
\quad
\vec{k_1} = k_1 \left(
\begin{array}{c}
sin\,\theta \ \cos\,\phi \\
sin\,\theta\ sin\,\phi\\
cos\,\theta\\
\end{array} \right),
\quad
\vec{k_2}=-(\vec{k_1}+\vec{p}),\hspace{2cm}\\
p=\frac{\lambda^{1/2}(m_{\eta}^2,M_I^2,m_{\pi}^2)}{2m_{\eta}},\quad
A\equiv
cos(\gamma_1\pi^0)=\frac{1}{2k_1p}[(m_{\eta}-
\omega-k_1)^2-k_1^2-\vert\vec{p}\vert^2]
\end{eqnarray}
with $\omega$ the energy of the $\pi^0$.

In Fig.~2 we show the results of the mass distribution with
and without the radiative widths normalization factors.
The integrated width is given by
$\Gamma=0.57\,$eV (universal couplings); 
$\Gamma=0.30 \pm 0.06\,$eV (normalized couplings),
where the error has been calculated from a Montecarlo Gaussian sampling
of the normalization parameters within the errors of the experimental branching ratios of Table 1.
We should mention that, had we used the data of the PDG 2000, the value obtained would have been
$0.21 \pm 0.05\,$eV. It is interesting to compare these results with those in
\cite{Ametller:1991dp}. There the only coupling adjusted to the experimental data was the
one  for the $\omega\to \pi^0\gamma$ decay and the underlying $SU(3)$ symmetry generated
the other ones. Hence, it is accidental that the choice of these couplings gave rise to a
contribution in the "all orders" case of $0.31~eV$ \cite{Ametller:1991dp}, very similar
to the one we obtain here adjusting to the most recent data \cite{Hagiwara:pw} and to
more branching ratios.


\begin{figure}[h]
\centerline{\hbox{\psfig{file=fig2.eps,width=9cm}}}
\label{fig:fig2}
\caption{\rm 
Invariant mass distribution of the two photons 
with VMD terms only. The solid curve
has been calculated with an universal coupling, whereas the dashed
one  has the couplings  normalized differently
to fit the radiative decays.}
\end{figure}


Our  VMD normalized
 result is within three standard deviations from the value presently given
in \cite{exp},\cite{Hagiwara:pw}:  $\Gamma=0.84 \pm 0.18\,$eV,
but within one sigma of the more recent one 
presented in \cite{nefkens}, $\Gamma=0.42 \pm 0.14\,$eV.
There are, however, other contributions that we consider next.


\section{Chiral loops}
The contribution of the chiral loops to the present problem was calculated
in
\cite{Ametller:1991dp,Bel'kov:1995fj,Bellucci:1995ay,Bijnens:1995vg,Ko:1992zr,Jetter:1995js}
 and proceeds via the charged kaon loops,
through $\eta \to \pi^0 K^+ K^-$.
A smaller contribution from G-parity violating pion loops through $\eta\to
\pi^0\pi^+\pi^-$, proportional to $m_u-m_d$, is also evaluated in \cite{Ametller:1991dp}.
Its contribution to the total decay rate is very small and we think that if such terms
are included, other isospin violating terms proportional to $m_u-m_d$, and isospin
violating corrections to our amplitudes should also be included. Rather than undertaking
this delicate test, we use the results of \cite{Ametller:1991dp} to estimate uncertainties
from all these sources. 
 The evaluation of the kaon loops requires the
$K^+K^-\to\gamma\gamma$ amplitude as well as the
$\eta\pi^0\to K^+K^-$ transition which is taken from the lowest
order chiral Lagrangian. At this point we shall already
introduce elements of unitarized ChPT to make a resummation of loop
diagrams, which allows us to study  the $\gamma\gamma\to
\pi^0\eta$ reaction around the $a_0(980)$ region \cite{Oller:1997yg}. 
The $\gamma\gamma\to\pi^0\eta$ amplitude is given by
\begin{equation}
-it=(\tilde{t}_{AK^+K^-} + \tilde{t}_{\chi
K})t_{K^+K^-,\pi^0\eta}
\label{eq:tNPA629}
\end{equation}
diagrammatically represented in Fig.~3.
\begin{figure}[h]
\centerline{\hbox{\psfig{file=fig3.eps,width=12cm}}}
\caption{\rm \label{fig:diagrams1}
Diagrams for the chiral loop contribution  }
\end{figure}
The first three diagrams correspond to $\tilde{t}_{\chi K}\,
t_{K^+K^-,\pi^0\eta}$ of Eq.~(\ref{eq:tNPA629}), already evaluated in 
\cite{Bijnens:1987dc,Donoghue:ee}. In
our case $\tilde{t}_{\chi K}$, written in a general
 gauge to be also used for the $\eta\to
\pi^0 \gamma\gamma$ reaction,  is given by
\begin{equation}
\tilde{t}_{\chi K} = - \frac{2 e^2 }{16 \pi^2}
\left( g^{\mu\nu}-\frac{k_{2\mu}k_{1\mu}}
{ k_1 \cdot k_2 } \right) \epsilon_{1\mu}\epsilon_{2\mu}
\,
\left\{ 1 + \frac{m_K^2}{s} \, \left[ \log \left( \frac{1 + (1 - 4 m_K^2 
/ s)^{\frac{1}{2}}}
{1 - (1 - 4 m_K^2/s)^{\frac{1}{2}}} \right
) - i \pi\right]^2 \right\},
\end{equation}
above the $K^+K^-$ threshold, with the $-i\pi$ term removed below
threshold. Note that the full $t_{K^+K^-,\pi^0\eta}$
transition matrix, not just the lowest order chiral amplitude,
is factorized outside the loop integral. This on shell
factorization was shown in \cite{Oller:1997yg}
 by proving  that the off shell
part of the meson-meson amplitude did not contribute to the
integral.

The meson scattering amplitude was evaluated in
\cite{Oller:1997ti} by summing
the Bethe Salpeter (BS) equation with a kernel formed from the
lowest order meson chiral Lagrangian amplitude and
regularizing the loop function with a three momentum cut off.
Subsequently, other approaches like the inverse amplitude
method \cite{OllOsePel,GomezNicola:2001as} or the $N/D$ method
\cite{Oller:1999zr} were used and all of them
gave the same results in the meson scalar sector,
$L=0,\,I=0,1$. For the present case only the 
$L=0,\,I=1$ amplitude is needed. The BS equation sums the
diagrammatic series of Fig.~4,
\begin{figure}[h]
\centerline{\hbox{\psfig{file=fig4.eps,width=12cm}}}
\caption{\rm \label{fig:diagrams1}
Diagrams summed in the BS equation, using the $O(p^2)$ ChPT vertices.  }
\end{figure}
which implies that in the $\gamma\gamma\to\pi^0\eta$ transition
of Fig.~3 one is resumming the diagrams of Fig.~5.
\begin{figure}[h]
\centerline{\hbox{\psfig{file=fig5.eps,width=12cm}}}
\caption{\rm \label{fig:diagrams1}
Resummation for $\gamma\gamma\rightarrow\pi^0\eta$.}
\end{figure}
Furthermore, the same on-shell factorization of the $t$ matrix
in the loops found for $\gamma\gamma\to\pi^0\eta$
was also justified for meson-meson scattering in
\cite{Oller:1997ti}.  Thus, the BS equation with coupled channels can be 
 solved algebraically, leading to the following solution in matrix form 
\begin{equation}
t(s)=[1-t_2(s)G(s)]^{-1}t_2(s),
\end{equation}
with $s$ the invariant mass of the two mesons, $t_2$ the lowest
order chiral amplitude and $G(s)$ a diagonal matrix,
$\mbox{diag}(G_{\overline{K}K},G_{\eta\pi})$, accounting for the loop
functions of two mesons, and which was regularized in
\cite{Oller:1997ti} by 
means of a cut off, or dimensional regularization in 
\cite{Oller:1999zr}. Its analytic expression can be found in
\cite{OllOsePel}.

In the case of $\eta\to\pi^0\gamma\gamma$ the series involved
corresponds to the diagrams in Fig.~6.
\begin{figure}[h]
\centerline{\hbox{\psfig{file=fig6.eps,width=12cm}}}
\caption{\rm \label{fig:diagrams1}
Resummation for $\eta\rightarrow\pi^0\gamma\gamma$.}
\end{figure}
Hence the $\pi\eta\to K^+K^-$ $t$ matrix factorizes
with an argument $s\to(p_{\eta}-p_{\pi})^2\equiv M_I^2$, where $M_I$
is nothing
but the invariant mass of the two photons. The amplitude
of the $\eta\to\pi^0\gamma\gamma$ reaction is then given by
Eq.~(\ref{eq:tNPA629}) simply substituting $s$ by $M_I^2$.
In Eq.~(\ref{eq:tNPA629}) there is another term, $\tilde{t}_{AK^+K^-}
t_{K^+K^-,\pi^0\eta}$, which corresponds to the last two
diagrams of Fig.~3 where an axial vector is exchanged. For
this term we follow 
\cite{Donoghue:1993kw} and, given the large mass of the axial
vector, both the factorization of the on shell meson scattering
amplitude outside the loop, as well as that of the $\gamma\gamma\to
K^+K^-$ amplitude are also justified
\cite{Oller:1997yg}. Hence one has now
\begin{equation}
t_{AK^+K^-} = - 2 e^2 \left( g^{\mu\nu}-\frac{k_{2\mu}k_{1\mu}}
{ k_1 \cdot k_2 } \right) \epsilon_{1\mu}\epsilon_{2\mu} 
\frac{\displaystyle{ (L^r_9 + L^r_{10})}}{\displaystyle{f^2}}
\left[ \frac{\displaystyle{s_A}}{\displaystyle{2 \beta (s)}} l n 
\left( \frac{\displaystyle{1 + \beta (s) + \frac{s_A}{s}}}
{\displaystyle{1 - \beta (s) + \frac{s_A}{s}}}
\right) + s\right],
\end{equation}
with $s_A=2(m_A^2-m_K^2)$, and 
$\beta(s)=(1-\frac{4m_K^2}{s})^{1/2}$.

First of all we show in Fig.~7 the result for the 
$\gamma\gamma\to\pi^0\eta$ cross section obtained up to here, which
coincides with that obtained in \cite{Oller:1997yg}.
\begin{figure}[h]
\centerline{\hbox{\psfig{file=fig7.eps,width=10cm}}}
\caption{\rm \label{fig:fig7}
$\gamma\gamma\rightarrow\pi^0\eta$ cross section, using Eq.~(\ref{eq:tNPA629}).
Z is the maximum value of $cos(\theta)$ integrated.
The experimental data come from \cite{oest,antre}, the latter one normalized in the
$a_2(1320)$ peak region. 
The dashed histogram corresponds to the convolution
over an experimental resolution of 40 MeV.}
\end{figure}
 We see in Fig. 7 the peak of the $a_0(980)$. We show the results obtained 
by means of Eq.~(\ref{eq:tNPA629}) for the cross section and also 
to ease the comparison with experimental data we show
 the events concentrated in bins of 40 MeV, roughly like the
experimental ones.
The contribution of the $a_2(1320)$ resonance (second peak) is included
 here as in ref.~\cite{Oller:1997yg}.
 The agreement with experiment is fair 
but some discrepancies can be noticed in the low energy region. The generation
of the peak in the $\gamma\gamma\rightarrow\pi^0\eta$ cross section in 
our approach
is guaranteed by the resummation of diagrams in Fig.5, since the resummed
$t_{K^+K^-\rightarrow\pi^0\eta}$ amplitude contains the pole 
of the $a_0(980)$ resonance with the properties
well described by \cite{Oller:1997ti}. Now let us see 
how this cross section is 
changed if one includes the VMD tree level amplitude calculated in section I, 
substituting the outgoing
$\pi^0$ by an incoming pion in Fig~1. 

We can see in Fig.8 that the results obtained adding the VMD 
amplitude normalized to the $\omega, \rho$ radiative decay rates are acceptable
around the peak of the  $a_0(980)$ resonance.
Let us also note that the inclusion of these terms improves the 
description of the low energy region
The binning of the theoretical results would make again
the apparent agreement with data to look much better, but for the sake of
clarity we have not added more lines to the figure, as long as
 the binning effect has already been illustrated 
in Fig.~\ref{fig:fig7}.
Since this region is dominated by the $a_0(980)$ resonance the effect of renormalizing the
couplings of the vector meson radiative decays is not as drastic as seen in 
Fig.~2 for the $\eta$ decay, where only the VMD mechanisms was considered.

\begin{figure}[h]
\centerline{\hbox{\psfig{file=fig8.eps,width=10cm}}}
\caption{\rm \label{fig:diagrams1}
$\gamma\gamma\rightarrow\pi^0\eta$ cross section, using eq.(8) (continuous line), adding the universal VMD contribution (dotted line) or
the normalized VMD contribution (dashed line).}
\end{figure}


One may wonder whether one should also add loops to the VMD tree level 
amplitude. In fact some of the uncertainties estimated in 
\cite{Ametller:1991dp} were linked to those loops. 
We are in a position to do so now.

One can estimate that the loops from the VMD terms must be quite smaller 
than those of the kaon loops studied so far by comparing the strength of the 
$K^+K^-\gamma\gamma$ contact term with the VMD amplitude. The size of the latter 
is about 2\% that of the former. Yet, since now we would 
have $\eta\pi^0$ loops instead of $K^+K^-$ which have larger mass, the strength of the loops from the VMD amplitude will be bigger than 2\%. 
Once again we can sum
the series obtained by iterating the loops in the four meson
vertex as shown in Fig.~9.


The new amplitude which we shall call $t^{VMDL}$ will be given by:
\begin{equation}
  t^{VMDL}=t_{\eta\pi^0,\eta\pi^0}(M_I)G_{\eta\pi}
\tilde t^{VMD}_{\eta\pi}(M_I)
\left[ \epsilon_1\epsilon_2-\frac{(k_2\epsilon_1)(k_1\epsilon_2)}{k_1 k_2}\right]
\end{equation}
where now $ \tilde t^{VMD}_{\eta\pi}$ is the factor that multiplies
the $\epsilon_1\epsilon_2$ product in the $s$-wave projection of the
$ t^{VMD}_{\eta\pi}$ amplitude in the $\gamma\gamma\rightarrow\pi^0\eta$ CM. 
Although the Lorentz structure
of polarization vector products may seem rather complicated
from Eq.~(\ref{tvmd}), it is easy to show that after the s-wave projection
the polarization vectors factorize indeed as $\epsilon_1\epsilon_2$.
In a general frame
the $\epsilon_1\epsilon_2$ factor has to be replaced by 
$\epsilon_1\epsilon_2-(k_2\epsilon_1)(k_1\epsilon_2)/(k_1 k_2)$.
Once again we have factorized the amplitudes for the same reasons
 as done with the other terms.

\begin{figure}[h]
\centerline{\hbox{\psfig{file=fig9.eps,width=12cm}}}
\caption{\rm \label{fig:diagrams10}
Loop diagrams for VMD terms.}
\end{figure}
Of course, when introducing the $\eta\pi^0$ loop corrections, we also 
have to include those
involving $K^+K^-$ or $K^0\bar{K}^0$ in the loop function with a $K^{*+}$
or a $K^{*0}$ exchanged between the photons (see Fig.9.b). 
These would be given by
\begin{eqnarray}
  \label{eq:kstar}
  t^{VMDL}_{K\bar{K}}&=&\left(t_{\eta\pi^0,K^+K^-}(M_I)G_{K\bar{K}}(M_I)
\tilde T^{VMD}_{K^+K^-}(M_I)\right.\\ &&+\left.
t_{\eta\pi^0,K^0\bar{K}^0}(M_I)G_{K\bar{K}}(M_I)
\tilde T^{VMD}_{K^0\bar{K}^0}(M_I)\right)\left[ \epsilon_1\epsilon_2-\frac{(k_2\epsilon_1)(k_1\epsilon_2)}{k_1 k_2}\right] \nonumber
\end{eqnarray}

\begin{figure}[h]
\centerline{\hbox{\psfig{file=fig10.eps,width=12cm}}}
\caption{\rm \label{fig:diagrams11}
Diagrams with two anomalous $\gamma\rightarrow 3 M$ vertices.}
\end{figure}
Finally, there is another kind of loop contribution considered in 
\cite{Ametller:1991dp} which involves two anomalous 
$\gamma\rightarrow 3 M$ vertices. This leads to the two loop 
functions arising from the diagrams in Fig.10.
They are of $O(p^8)$ and are evaluated in \cite{Ametller:1991dp}
were it was found that they can have a non negligible effect on the $\eta$ decay,
of the order of the other kaon loops actually discussed. Yet, the crossed
character of the loop in the
$\gamma\gamma\rightarrow\eta\pi^0$ reaction makes its contribution
 completely negligible  
in the region of the $a_0(980)$ resonance, where loops in the s-channel have a
large strength, enough to generate  that resonance dynamically. Here we 
take the results
from \cite{Ametller:1991dp} where it is found that their largest contribution
comes from the kaon loops, whereas pionic loops are negligible. 
We use eqs.(12),(13) and (27) of that reference and note that there is a 
global change of sign
with respect to our notation.
The further rescattering of the mesons in the diagrams of Fig.~10, given the structure of
the $\gamma MMM$ vertex \cite{Ametller:1991dp} in the momenta of the
particles, would be
suppressed by factors of $\vec{p}\,^2_{\gamma}/\vec{q}\,^2$ (with $q$ the loop variable) with
respect to those considered for the VMD mechanism. 
This and the fact that these anomalous mechanisms provide a small
contribution with respect to the one of the VMD mechanism,
 makes the consideration of this extra
rescattering loops superfluous.


With all these new ingredients we look again at the 
$\gamma\gamma\rightarrow\pi^0\eta$ cross section and find that the
extra mechanisms beyond those considered in Fig.~8  barely change
the results shown in that figure.

\vspace{0.9cm}
\begin{figure}[h]
\centerline{\hbox{\psfig{file=fig11.eps,width=12cm}}}
\label{fig:fig11}
\caption{\rm 
Different contributions to the invariant mass distribution of
the two photons. From bottom to top,
short dashed line: only chiral loops; 
long dashed line: only VMD tree level terms;
dashed-dotted line:
coherent sum of the two latter mechanisms; double
dashed-dotted line: idem  but adding loop diagrams for VMD terms of
Fig.~9;
continuous line: idem but adding also the anomalous terms of Fig.~10,
which is the full model presented in this work. 
(we are also showing as a dotted line, the
full model but substituting the full 
$t_{K^+K^-,\eta\pi^0}$
scattering matrix by its lowest order $O(p^2)$).
}
\end{figure}


Back to the $\eta$ decay, in Fig.~11
we plot the different contributions to $d\Gamma/dM_I$.
We can see that the largest contribution is that of the VMD
mechanism (long dashed line). The chiral loops by themselves
(short dashed line) give a small
contribution ($0.011\,$eV), but when added coherently to the VMD
mechanism, they lead to an increase of $30\,$\% in the $\eta$ decay
rate (dashed-dotted line). More interesting, the shape of the $\gamma\gamma$ 
invariant mass distribution is appreciably changed with respect to
the one of the VMD mechanism alone, with the strength moving to
higher invariant masses. The consideration of meson rescattering
of the VMD leads, upon coherent sum with the other mechanisms,
to a moderate increase of the $\eta$ decay rate (double dashed
dotted line), smaller in
size of that of the chiral loops considered before. The last
ingredient considered is the contribution of the anomalous
mechanisms of Fig.~10 which lead again to a moderate increase of
the $\eta$ decay rate (continuous line), also smaller than 
that of the chiral loops.
The different shape of the anomalous mechanism compared to the
chiral loops has as a consequence the interference with the VMD
mechanism in the whole range of invariant masses.
  
Altogether, when integrating over the invariant mass, we get:
\begin{equation}
  \Gamma(\eta\rightarrow\pi^0\gamma\gamma)=0.47\pm 0.08 \,\mbox{eV}
\end{equation}
and we see that the inclusion of the loops has increased the
contributions of the renormalized VMD term alone by 50\%. 
For the moment, the theoretical error is obtained only from the
experimental errors in the vector meson radiative decay branching
ratios in Table~1.
For comparison, we quote  here what
we would obtain using the universal coupling in the VMD terms: $0.80\,$eV.

We come now to estimate the uncertainties considered in  
\cite{Ametller:1991dp}. One of the sources was the contribution of 
the $a_0(980)$ and  $a_2(1320)$  resonances. 
%We can now say 
%what is the contribution of the tail of the $a_0(980)$ to the $\eta$ decay.
In our approach the $a_0(980)$ is generated dynamically from the multiple 
scattering of the mesons implied in the Bethe-Salpeter equation,
showing up indeed in the $\gamma\gamma\rightarrow\pi^0\eta$
amplitude around 1 GeV. In the $\eta$ decay the $a_0$
contribution appears through the loop terms
in Fig.6 (also in the smaller contributions in Fig.9). This can be seen 
explicitly in Eq.~(\ref{eq:tNPA629}), where the transition
matrix $t_{K^+ K^-,\eta\pi^0}$, which contains the $a_0(980)$
 pole, appears explicitly. Since the $a_0(980)$ is generated
by multiple scattering (iterated loops in the BS equation), 
we can remove it simply by substituting the full  $t_{K^+K^-,\eta\pi^0}$
scattering matrix by its lowest order $O(p^2)$. 
In such case we simply reproduce
the standard ChPT results. The difference between using the full
$t$ matrix and its lowest order can be seen in Fig.11
(difference between continuous and dotted lines).
The contribution of the $a_0(980)$ resonance tail is rather small and increases
the $\eta$ decay rate from $0.47$eV to $0.48\,$eV. The sign of its
contribution is unambiguously determined. Here we can see an improvement
with respect to ref.~\cite{Ametller:1991dp} where this contribution was included
as a source of theoretical error contributing mostly to the $\pm0.2\,$eV
accepted uncertainties in that work. Thus, the present calculation removes
completely this source of error. The explicit calculation of the $a_0(980)$ 
contribution giving such a small effect justifies the neglect of the
$a_2(1320)$ resonance contribution which lies much further away in energy than
the $a_0(980)$. 

The other source of uncertainty in  \cite{Ametller:1991dp}
was the contribution of the loops from the VMD term. 
We have been able to calculate them in this work and, as seen in Fig.11,
these effects are also rather small. They increase the $\eta$ decay rate in
$0.02\,$eV. Altogether the $a_0(980)$ plus rescattering terms in the VMD
mechanisms increase the $\eta$ decay
rate in $0.03\,$eV. We thus eliminate these two sources of previous
uncertainties in the calculations  while, at the same time, we realize that
the uncertainties of $0.2\,$eV attributed to these sources in 
\cite{Ametller:1991dp} were indeed a generous upper bound.

The consistency of the input for this process with that of 
$\gamma\gamma\rightarrow\pi^0\pi^0$ was studied in 
\cite{Ko:1992zr,Bellucci:1995ay,Jetter:1995js,Bijnens:1995vg,Bel'kov:1995fj,
Ko:rg}. We performed such a test in \cite{Oller:1997yg} and found good 
results for
the $\gamma\gamma\rightarrow\pi^0\pi^0$ cross section up to 1.4 GeV, 
much beyond where ChPT can be applied. We should also mention that
the VMD terms with $\rho$ and $\omega$  in the intermediate
 states were explicitly
considered in \cite{Oller:1997yg}, hence our 
present approach is definitely consistent with
 $\gamma\gamma\rightarrow\pi^0\pi^0$.

We have not considered in our approach the contribution of the axial
resonances discussed in \cite{Ko:1992zr} in the VMD terms. According to 
 \cite{Ko:1992zr} that would increase the decay width by about $0.07\,$eV.
However, as shown in  \cite{Ko:1992zr,Jetter:1995js}, their inclusion in 
$\gamma\gamma\rightarrow\pi^0\pi^0$ with the couplings used in 
 \cite{Ko:1992zr} would overestimate the $\gamma\gamma\rightarrow\pi^0\pi^0$
cross section. We will nevertheless accept a theoretical uncertainty from
this source of the order of $0.05\,$eV which, in view of the discrepancies
mentioned in the $\gamma\gamma\to\pi^0\pi^0$ reaction, 
should still be a
generous upper bound. On the other hand, as commented above, we also accept
as uncertainties due to isospin violation the contribution found in 
\cite{Ametller:1991dp} for pionic loops of the type of Fig.~3 which, from
the results obtained there, can be seen to give a contribution of
$0.05\,$eV to the total $\eta$ decay rate. Taking into account the
uncertainties in Eq.~15 from the experimental errors in the vector meson
radiative decay branching ratios plus these two other sources of
uncertainty and summing them in quadrature we obtain a final result of

\begin{equation}
\Gamma(\eta\to\pi^0\gamma\gamma)=0.47\pm 0.10\,eV
\label{resultfinal}
\end{equation}

Altogether we still have reduced the uncertainty from previous calculations,
in spite that we have considered the uncertainties coming from the
experimental errors of the vector meson radiative decays, which were
neglected in the former works, and we proved here to be the largest source
of uncertainty in our case.

The result of Eq.~(\ref{resultfinal}) is in remarkable agreement 
with the latest experimental numbers \cite{nefkens},
and lie within two sigmas from the 
 earlier ones in \cite{exp,Hagiwara:pw}. Confirmation
of those preliminary results  would therefore be important to test
the consistency of this new approach. Furthermore 
the $\gamma\gamma$ invariant mass distributions would 
be of much help given the differences found
 with and without loop contributions.

\section{Conclusions}

We have reanalyzed the $\eta\rightarrow\pi^0\gamma\gamma$
decay to the light of the work done in the framework
of Chiral Perturbation Theory (ChPT) and included the elements
found relevant there: the VMD term plus the loop contributions. 
We have 
introduced two new elements in the study:
First we have connected the problem of $\eta\to\pi^0\gamma\gamma$
decay with the related problem of $\gamma\gamma\to\eta\pi^0$ which could be
addressed thank to the use of techniques of unitarized ChPT. These
techniques were essential to reproduce the $a_0(980)$ excitation clearly
visible in the $\gamma\gamma\to\eta\pi^0$ cross section.
The analytical continuation of this amplitude to the low energies involved
in the $\eta\to\pi^0\gamma\gamma$ decay allowed us to quantify unambiguously the
contribution of the tail of the $a_0(980)$ resonance to this process,
eliminating a former source of uncertainty.

Another novelty in the present work is
the  realization that the VMD  amplitude with a universal SU(3) coupling 
for the VVP vertices was not consistent with the experimental radiative 
decays of the $\rho$ and $\omega$, and
in particular in the $\rho\rightarrow\eta\gamma$ and 
$\omega\rightarrow\eta\gamma$ decays. We renormalized  these couplings
to get the experimental branching ratios and this resulted in a reduction
of the standard VMD contribution to the $\eta\to\pi^0\gamma\gamma$ decay.

We have also introduced new chiral loop terms accounting for meson
rescattering through the Bethe-Salpeter equation stemming from VMD
mechanisms, which was also a source of uncertainty in former works.

Furthermore, we have done an error analysis of our results considering the
experimental errors in the vector meson radiative decay widths not
considered in the past and which turns out to be the largest source of
uncertainty in our study. Simultaneously we also considered uncertainties
from other mechanisms like isospin breaking and exchange of axial
resonances in the VMD coupling directly to $\pi\eta$ which we have neglected
here but which, with some uncertainties, had been shown to be quite small
in former works.

Altogether we have found a result of
$\Gamma(\eta\to\pi^0\gamma\gamma)=0.47\pm0.10\,eV$.  

The agreement of this result with
the new preliminary data from \cite{nefkens} is remarkable, but
the test of the invariant mass distribution would be
more stringent. Confirmation of the preliminary results of  \cite{nefkens}
and the measurement of the $\gamma\gamma$ invariant mass distribution should 
then be the experimental priorities 
to clarify the situation.


\section*{Acknowledgments}
We are specially grateful to J.A. Oller for fruitful discussions, 
technical help and  his
careful reading of the manuscript.
We would also like to acknowledge useful discussions with J. Bijnens.
One of us, L.R., acknowledges support from the Consejo Superior de
Investigaciones Cient\'{\i}ficas. J.R.P. acknowledges financial
support from a CICYT-INFN collaboration grant as well as a
Marie Curie fellowship MCFI-2001-01155. He also thanks the
Dipartimento de Fisica, Universita' de Firenze-INFN Sezione di Firenze
for its hospitality. 
This work is also
partly supported by DGICYT contract numbers BFM2000-1326, PB98-0782, and the 
E.U. EURIDICE network contract no. HPRN-CT-2002-00311.
  


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