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{SAGA-HE-70-94  
   \hfill September, 1994}  \\
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\Large{{\bf $\bf \phi$ Radiative Decays}} \\

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\Large{{\bf and Scalar Meson Structure}} \\

 
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\Large
{S. Kumano $^\star$ }         \\
 
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\Large
{Department of Physics}         \\
 
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\Large
{Saga University}      \\
 
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\Large
{Saga 840, Japan} \\

 
\vspace{2.5cm}
 
\large
{Talk given at the Symposium on} \\

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{Quark Nuclear Physics with GeV Cooler-Ring Protons and Beyond} \\

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{RCNP, Osaka, Sept. 8--10, 1994}  \\
 
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{$\star$ Email: kumanos@himiko.cc.saga-u.ac.jp}  \\


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{to be published in proceedings}


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\begin{center}
 
\Large{$\phi$ Radiative Decays and Scalar Meson Structure} \\

\vspace{0.5cm}
 
{S. Kumano}             \\
 
{Department of Physics}    \\
 
{Saga University}      \\

{Saga 840, Japan} \\


\vspace{0.7cm}

\normalsize
Abstract
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\vspace{-0.46cm}

It is interesting to study radiative decays
of mesons in the mass region of 1--2 GeV at RCNP 
for investigating the details of hadron structure.
Among the decays, $\phi$ radiative decays into
f$_0$(975) and a$_0$(980) mesons
are not measured yet, and studies of these decays
could provide important clues on their substructures.
Furthermore, it is likely that the studies open up
the field of exotic mesons.

\vspace{0.8cm}

\noindent
{\bf 1. Introduction}

\vspace{0.2cm}


Quark models have been successful in explaining many properties
of hadrons. Most mesons in the 1--2 GeV region are considered
to be understood in nonrelativistic $q\bar q$ type models.
However, there are predictions of exotic mesons which are not
fitted into the $q\bar q$ category.
For example, they are bound-$K\bar K$, $q^2\bar q^2$,
glueball, and hybrid mesons. 
Especially, kaons are stable mesons so that it is no wonder
that they form a bound state.
Because the two-kaon mass is 2$m_{_{K^\pm}}$=987 MeV,
the bound state should appear in the mass 
region 980 MeV if it exists.
There are candidates of the kaon bound state: f$_0$(975) and a$_0$(980).
The $q\bar q$ type models have difficulty in explaining
narrow decay widths of these mesons. 
As discussed in section 2, 
calculated decay widths of $f_0$
in $q\bar q$ models are about twenty times
larger than the experimental one.
Because this difficulty is not enough to exclude 
the $q\bar q$ picture completely, 
it is important to test the substructure in other processes.

$\phi$ radiative decays into these mesons could be 
useful in discriminating among possible structure models.
It is because the decays are electric dipole (E1) transitions.
The E1 operator is sensitive to spatial distribution
of electric charges. Sizes of a $q\bar q$ bound state and
a $K\bar K$ bound state are expected to be much different.
This is obvious because the $K\bar K$ molecule is supposed to be
a loose bound state of $K$ and $\bar K$ 
by noting that the scalar-meson masses are slightly
below the $K\bar K$ threshold.
The analogy of the $K\bar K$ molecule
in nuclear physics is the deuteron, which exists just below the
n-p threshold and it is the loose bound state 
of the proton and the neutron.
From the above consideration, we expect that
the $\phi$-radiative-decay widths differ widely 
in various substructure models.

The best facility for measuring the $\phi$ radiative decays
is a $\phi$ factory.
There are $\phi$ factory proposals  at Frascati, KEK, 
Novosibirsk, and UCLA [1].
The Frascati $\phi$ factory is already 
approved and it will be built in a few years.
As it is shown in section 3, decay widths are
very small so that it would not be easy to measure the decays
at RCNP. However, it is still worth while measuring them
in an independent way,
e.g. p+d$\rightarrow ^3$He+M ($\rightarrow$M$'\gamma$)
[2].

The purpose of this talk is to point out why the $\phi$ radiative
decays are important and what the decay widths are in various
models [3]. 
However, our calculations are very naive estimates, 
so that much detailed theoretical analyses are needed
in comparison with future experimental data.

We first discuss the issue of strong decay widths 
of the scalar mesons in section 2.
Then, the radiative decay widths are estimated
by assuming various structures for the scalar mesons.


$~~~~~$

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\noindent
{\bf 2. Strong decay widths}

\vspace{0.2cm}

There are ``evidences" against the $q\bar q$ picture
for the scalar mesons f$_0$ and a$_0$.
First, these mesons exist just below the $K\bar K$ threshold, so
that they could be $K\bar K$ bound states similar to
the deuteron in nuclear physics.
Second, experimental data indicate strong couplings of these mesons
to the $K\bar K$ channel, which violates the OZI rule.
Third, their $\gamma\gamma$ couplings calculated in $q\bar q$ models
are much larger than experimental data.
Forth, calculated strong-decay widths in $q\bar q$ models
are much larger than data.

In this section, we discuss the last ``evidence", 
the strong-decay-width problem. 
Decay widths of most mesons in the 1 GeV region are well described
by $q\bar q$ models. However, it is almost impossible to explain
the narrow decay widths of the scalar mesons.
In the following, the strong decay of f$_0$ is discussed
in the quark-model framework of Ref. [4].

It is not straightforward to describe strong-decay processes
because they are nonperturbative and because relativity could be
important in light-meson decays. 
Furthermore, final-state interactions would not be simple.
In Ref. [4], we simply extended existing decay models.
There are two major decay models:
a $^3S_1$ model for the $q\bar q$ creation
and a $^3P_0$ model.
If the $q\bar q$ pair is created from a gluon splitting,
a natural quantum state of the pair is the $^3S_1$ state,
as it was proposed in the original Cornell paper
for charmoniums. However, a phenomenological decay model $^3P_0$
has been widely used. As a physical interpretation, we may justify
the $^3P_0$-$q\bar q$ model by considering color-flux fluctuations. 
Because of the existence of strong final-state interactions,
it is rather difficult to discriminate between these models.
In Ref. [4], strong decays of mesons into two pions were investigated
in both models with consideration of the pion size and 
the final-state interactions. 


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%%%\begin{list}{}{\leftmargin 0.0in \rightmargin 3.7in}
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\item
$~~~~$
Our typical results are shown in Fig.1, where our theoretical
results for two-pion decays are compared with experimental data.
A coupling constant for the $q\bar q$ creation is determined by 
the experimental $\rho\rightarrow \pi\pi$ decay width.
As it was expected, the calculated decay widths are very
sensitive to the pion size. If we take the 0.29 fm pion size,
most decays are well explained. 
However, we still cannot explain the scalar meson decay 
$f_0 \rightarrow\pi\pi$ at all. Theoretical decay widths
are in the range of 500--1000 MeV, which is badly in contradiction
to the experimental one.
\end{list}
\vspace{-1.0cm}
\hspace{7.4cm}
Fig. 1 Strong decay widths of $M\rightarrow\pi\pi$. 

According to the version of 
the Review of Particle Properties in 1992 [5],
the total decay width is 47 MeV and the branching ratio of 
$f_0 \rightarrow\pi\pi$ is 78.1\%, so that the two pion decay width
is supposed to be 37 MeV.
As we mentioned, the decay models are not well established
due to problems of final-state interactions, hadron wave
functions, relativity, and decay models. However, it is peculiar
that only $f_0$ (and $a_0$) cannot be described by the models
even though almost all other mesons are explained.
The difference of an order of magnitude between the theoretical
result and the data is considered to be too large even if
these problems are taken into account.
Even if final state interactions are considered between final pions,
it is impossible to explain the narrow width, unless there exists
a strong repulsive interaction {\it only} 
in the S-wave $\pi\pi$ channel.
Our calculation is not the only one which suggests 
the issue of the $f_0$ decay. More than several people have investigated
the strong-decay problem, and they consistently obtained very large
decay widths. In these calculations, they used different
decay models and various wave functions; however, the results are
always in the range of 500--1000 MeV.

The situation is similar in the case of $a_0$ meson.
It decays into $\pi\eta$ dominantly and the total decay width
is 57 MeV experimentally [5].
In the $q\bar q$ model, the decay width ratio 
$\Gamma (f_0\rightarrow \pi\pi) / \Gamma (a_0\rightarrow \pi\eta)$
is approximately 4, so that the theoretical width 
is larger than the data.

From the difficulty of strong decay widths
together with other problems, it is natural
to consider the $f_0$ and $a_0$ mesons as non-$q\bar q$ type mesons.
For example, these narrow widths are understood in the $K\bar K$
picture as a natural consequence of weak binding.


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\noindent
{\bf 3. $\phi$ radiative decays}

\vspace{0.2cm}

There are evidences against the $q\bar q$ picture 
for the scalar mesons $f_0$ and $a_0$ as discussed
in the previous section.
However, these are not enough to exclude the $q\bar q$ picture
completely. As an independent test, we may use
$\phi$ radiative decays into the scalar mesons $f_0$ and $a_0$,
which we denote S in this section.
The radiative decays are, as we discussed in section 1,
electric dipole transitions; hence, their decay widths
are very sensitive to the internal structure.

Currently, the experimental upper bound of the branching ratio 
$BR(\phi\rightarrow f_0\gamma)$ is $2\times 10^{-3}$ and
it is $5\times 10^{-3}$ for $BR(\phi\rightarrow a_0\gamma)$ [5].
If the $\phi$ factory is completed, these decays will be
measured. 
However, it is also important to measure the decays in a completely
different method, for example, by using future RCNP facility.

In the following, we discuss how the decay widths differ
in various substructure models. This kind of research
is still in the early stage, 
so the following calculations are merely rough estimates.

{\bf [$\bf q\bar q$ structure]}
First, we assume that the $f_0$ meson is a $s\bar s$ bound state.
Then, the OZI-allowed E1 decay width 
$\Gamma[\phi(s\bar s)\rightarrow f_0(s\bar s)\gamma]$
is estimated from other E1 decay widths by taking into account
the expression $\Gamma(E1)\sim e^2 R^2 E_\gamma^3$, where
$e$ is the charge factor, $R$ is the hadron size, and 
$E_\gamma$ is the photon energy.
For example, we may use the observed charmonium decay
width $\Gamma(\chi_{c0}\rightarrow J/\psi\gamma)=0.092$ MeV.
Taking into the quark charge, the hadron size, the photon energy,
and a spin factor in the expression of the E1 width,
we obtain $BR(\phi\rightarrow f_0\gamma) \approx 1\times 10^{-5}$.
We also used other E1 decays for the estimate, and obtained values
are approximately equal to $10^{-5}$.

If $f_0$ and $a_0$ are both ordinary mesons, namely
$f_0=(u\bar u+d\bar d)/\sqrt{2}$ and $a_0=(u\bar u-d\bar d)/\sqrt{2}$,
the both radiative decays are OZI-violating processes.
Therefore, decay widths are supposed to be significantly smaller
than the above OZI-allowed estimate $1\times 10^{-5}$, and we
expect the branching ratio $BR\lapproxeq 10^{-6}$.

\vspace{-0.46cm}
%%%\begin{list}{}{\leftmargin 0.0in \rightmargin 3.7in}
\begin{list}{}{\leftmargin 0.0in \rightmargin 3.55in}
\item
$~~~~$
{\bf [$\bf K\bar K$ structure]}
Second, we discuss the case of $K\bar K$ molecules.
In this picture, we consider the following process.
$\phi$ decays into $K^+$ and $K^-$, 
either $K^+$ or $K^-$ radiates a photon, then
$K^+$ and $K^-$ form a bound state S as shown in
Fig. 2(b) or 2(c).
Because the $\phi K\bar K$ coupling is given by
$g_\phi \phi_\mu j^\mu$ with 
$j^\mu=iK^{\dagger} (\vec{\partial}^\mu -  
\stackrel{\leftarrow}{{\partial}^\mu})K$,
there is a process shown in Fig. 2(a) by the minimal 
substitution.
\end{list}
\vspace{-1.0cm}
\hspace{7.2cm}
Fig. 2 Radiative decays 
       $\phi \rightarrow K\bar K\gamma\rightarrow S\gamma$. 

\vspace{0.06cm}
\noindent
Physics meaning of Fig. 1(d) is not obvious at first hand.
Because S is a loose bound state of $K$ and $\bar K$,
the kaon loop momenta cannot be infinite quantities.
So we introduce a momentum cutoff $\phi(|\vec p|)$ given
by a $K\bar K$ wave function at the $SK\bar K$ vertex.
This extra momentum dependence produces a current by
the minimal substitution $\vec p \rightarrow \vec p -e\vec A$.
This current is called ``interaction current'', which
does not appear in point-like field theories.
Because the momentum cutoff $\phi(|\vec p|)$ was introduced
at the $SK\bar K$ vertex, $K$ and $\bar K$ annihilate into
$S$ with a finite distance controlled by the cutoff. 
The interaction current is associated with the current flow
over the finite distance at the vertex [6].
Fortunately, the photon energy in $\phi\rightarrow S\gamma$
is about 40 MeV, which is thought to be in the long wavelength region,
so that the details of the interaction current are not important.
We simply expand $\phi(|\vec p -e\vec A|)$ in the Taylor 
series of $e\vec A$, then take the first order term as
the interaction current.

Employing the dipole cutoff, which is similar to 
the $K\bar K$ molecule wavefunction of a Toronto group,
we obtain $BR(\phi\rightarrow f_0\gamma) \approx 4 \times 10^{-5}$.
The branching ratio $BR(\phi\rightarrow a_0\gamma)$ is roughly equal
to the above value.
Momentum cutoff effects are significant, and the ratio becomes
much smaller than the calculated one in a point-like theory.
However, it is still much larger than the $q \bar q$ model 
predictions, so that we could at least discriminate between
the $q\bar q$ model and the $K\bar K$ one.

{\bf [$\bf qq\bar q\bar q$ structure]}
Even though quark contents of the $K\bar K$ molecule 
and $ns\bar n\bar s$ (where $n$ denotes $u$ or $d$ quark)
are the same, their dynamical structures
are very different. If two color singlets are well separated and
if they could be identified, the meson is a $K\bar K$ bound state
rather than a four-quark bound state.
If the color singlets are not identified, the meson should be
considered as a four-quark state.
It is unfortunate that strong decays of
the four-quark bound state are little understood,
so we cannot address ourselves to the radiative decay
widths. However, we could study the relation between the decay-width ratio
$\Gamma(\phi\rightarrow f_0\gamma) / \Gamma(\phi\rightarrow a_0\gamma)$
and the internal structure.
The E1 operator is given by the constituent charge $e_i$ and
the vector distance $\vec r_i$ as $\vec D=\sum e_i \vec r_i$.
Neglecting the spatial structure and 
simply counting constituent charges, we obtain rough estimates
of the ratio.

If the scalar mesons are $K \bar K$-like ``bags'',
they are expressed 
$f_0=(u\bar s)(\bar u s)+(d\bar s)(\bar d s)$ and
$a_0=(u\bar s)(\bar u s)-(d\bar s)(\bar d s)$.
Then, E1 matrix elements are given by
$M(f_0)\sim (e_u+e_{\bar s})+(e_d+e_{\bar s})=1$ and
$M(a_0)\sim (e_u+e_{\bar s})-(e_d+e_{\bar s})=1$. 
Therefore, the decay width ratio is $R(a_0/f_0)\approx 1$.

If the mesons are diquark-antidiquark-like ``bags'',
they are expressed 
$f_0=(us)(\bar u \bar s)+(ds)(\bar d \bar s)$ and
$a_0=(us)(\bar u \bar s)-(ds)(\bar d \bar s)$.
Then, E1 matrix elements are given by
$M(f_0)\sim (e_u+e_s)+(e_d+e_s)=-1/3$ and
$M(a_0)\sim (e_u+e_s)-(e_d+e_s)=1$.
Therefore, the decay width ratio is $R(a_0/f_0)\approx 9$.
This ratio is much different from the $K\bar K$-like-bag result,
so that not only the absolute decay width but also the
decay-width ratio provides important information on
the internal structure.

If the scalar mesons are $\pi\eta$ or $\eta\eta$-like ``bags'', 
they are expressed 
$f_0=(u\bar u)(s\bar s)+(d\bar d)(s\bar s)$ and
$a_0=(u\bar u)(s\bar s)-(d\bar d)(s\bar s)$.
Then, E1 matrix elements are given by
$M(f_0)\sim   [(e_u+e_{\bar u})-(e_s+e_{\bar s})]
            + [(e_d+e_{\bar d})-(e_s+e_{\bar s})]=0$ and
$M(a_0)\sim   [(e_u+e_{\bar u})-(e_s+e_{\bar s})]
            - [(e_d+e_{\bar d})-(e_s+e_{\bar s})]=0$.
In this case, more sophisticated analysis are needed.

{\bf [glueball  structure]}
If $f_0$ is a glueball, we estimate the quarkonium-glueball
mixing by the observed $f_0\rightarrow\pi\pi$ decay width and
the calculated $^3P_0(q\bar q)\rightarrow\pi\pi$ width 
in section 2:
the mixing$\lapproxeq \Gamma (f_0\rightarrow\pi\pi)/
                      \Gamma (^3P_0(q\bar q)\rightarrow\pi\pi)
           =26/(500-1000)
           \lapproxeq 1/20$.
Therefore, the decay width is 
      $\Gamma(\phi\rightarrow f_0(glueball)\gamma)
           \lapproxeq 
       \Gamma(\phi\rightarrow f_0(q\bar q)\gamma)/20$.
Using the results
for $\Gamma(\phi\rightarrow f_0(q\bar q)\gamma)$, 
we obtain
$
B.R.(\phi \rightarrow f_0\gamma) 
           \lapproxeq 10^{-6} .
$



Our estimates are summarized in the following table.

%%%%%%%--------------------------------------------------------%%%%%%%%
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\begin{center}
\normalsize
\begin{tabular}{|cc|c|l|}   \hline
~~~            &  ~~~  & Ratio & ~~~ \\
\multicolumn{2}{|c|}{Constitution}
                & $\Gamma (\phi\rightarrow\gamma a_0)/~$
                & \multicolumn{1}{|c|}{Absolute B.R.}            \\ 
~~~ & ~~~ & $~\Gamma (\phi\rightarrow\gamma f_0)$ & ~~~   \\ \hline
$K\bar K$      & molecule=``diffuse $K\bar K$''
               & 1
               & $a_0\simeq f_0\simeq 4\times 10^{-5}$       \\  \hline
$q^2\bar q^2$  & $K\bar K$-like ``bag''$=(n\bar s)(\bar n s)$
               & 1
               & $a_0,f_0\lapproxeq 10^{-6}$ ?  \\ \cline{2-3}
~~~            & $D\bar D$-like ``bag''$=(ns)(\bar n \bar s)$
               & 9 
               & see Ref. 3 \\ \cline{2-3}
~~~            & $\pi\pi,\pi\eta$-like ``bag''$=(n\bar n)(s\bar s)$
               & $-$
               & ~~~~~ \\ \hline
$^3P_0(q\bar q)$  & $f_0(n\bar n)$, $a_0(n\bar n)$
                  & $-$
                  & $a_0,f_0\lapproxeq 10^{-6}$   \\ \cline{2-4}
~~~               & $f_0(s\bar s)$, $a_0(n\bar n)$
                  & $\approx$ 0
                  & $f_0 \simeq 1 \times 10^{-5}$        \\ \hline
glueball          & $f_0$
                  & $-$
                  & $f_0\lapproxeq 10^{-6}$     \\ \hline
\end{tabular}

\vspace{1.0cm}

Table 1. Estimated decay widths in various models.
\end{center}
\vspace{0.5cm}

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\noindent
{\bf 4. Conclusions}

\vspace{0.2cm}

Internal structures of the scalar mesons $f_0$ and $a_0$ 
have been controversial. 
Strong decay widths are too large to explain experimental ones
in $q\bar q$ type quark models.
$\phi$ radiative decays could be used for discriminating
among various models because 
the decay widths differ widely: 
B.R.=$10^{-4} - 10^{-6}$ depending on the structure.
It is useful to study the radiative decays
at the future RCNP facility.



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\begin{center}
{\bf Acknowledgment} \\
\end{center}

\vspace{-0.30cm}

This research was partly supported by the Grant-in-Aid for
Scientific Research from the Japanese Ministry of Education,
Science, and Culture under the contract number 06640406.
S.K. would like to thank RCNP for the financial support
for his participating in this workshop.
 
\vspace{0.4cm}

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\begin{center}
{\bf References} \\
\end{center}
 
\vspace{-0.15cm}
\vspace{-0.38cm}
\begin{description}{\leftmargin 0.0cm}
\item{[1]}
A. Zallo, report LNF-91-078-R (1991);
M. Fukawa {\it et al.}, KEK report 90-12 (1990).

\vspace{-0.38cm}
\item{[2]}
See other contributions to this symposium
by M. Ishida and Y. Mizuno.

\vspace{-0.38cm}
\item{[3]}
F. E. Close, N. Isgur, and S. Kumano, 
Nucl. Phys. {\bf B389}, 513 (1993);
S. Kumano, pp. 283--292 in 
{\it Flavor and Spin in Hadronic and Electromagnetic Interactions},
edited by F. Balestra, R. Bertini, and R. Garfagnini,
SIF, Bologna (1993).

\vspace{-0.38cm}
\item{[4]}
S. Kumano and V. R. Pandharipande, Phys. Rev. {\bf D38}, 146 (1988).

\vspace{-0.38cm}
\item{[5]}
Particle Data Group, Phys. Rev. {\bf D45}, I.1 (1992).

\vspace{-0.38cm}
\item{[6]}
L. Heller, S. Kumano, J. C. Martinez, and E. J. Moniz,
           Phys. Rev. {\bf C35}, 718 (1987).


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