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{SAGA-HE-66-94  
   \hfill August, 1994}  \\
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\Large{{\bf Studies of f$_0$ and a$_0$ Meson Structures 
         at $\bf \phi$ Factory}} \\

\vspace{0.5cm}

\Large{{\bf and Their Effects on CP Violation Research}} \\

 
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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 workshop on} \\

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{B, D, K Meson Physics, Quark Mixing, and CP Violation} \\

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{YITP, Kyoto, July 18--19, 1994}  \\
 
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{$\star$ Email: kumanos@himiko.cc.saga-u.ac.jp}  \\


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\begin{center}
 
\Large
{Studies of f$_0$ and a$_0$ Meson Structures at $\phi$ Factory} \\

\Large
{and Their Effects on CP Violation Research}      \\
 
\vspace{0.5cm}
 
{S. Kumano $^*$}             \\
 
{Department of Physics}    \\
 
{Saga University}      \\

{Saga 840, Japan} \\


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\normalsize
Abstract
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Structures of f$_0$(975) and a$_0$(980) mesons
could be investigated at $\phi$ factories.
We discuss the possibility of discriminating
among various structure models by measuring
$\phi$ radiative decay widths.
We also discuss whether 
$\phi\rightarrow K^0 \bar K^0\gamma$
is a significant background to CP violation studies
at the $\phi$ factories.


\vspace{0.8cm}

\noindent
{\bf 1. Introduction}

\vspace{0.2cm}

There are $\phi$ factory proposals 
at Frascati, KEK [1], Novosibirsk, and UCLA.
The Frascati $\phi$ factory will be built
in a few years, so CP violation studies 
in the decay $\phi\rightarrow K^0\bar K^0$ become increasingly
interesting. There are two purposes for investigating
$\phi$ radiative decays into scalar mesons. One is to study
a possible CP background problem, and the other is to study
scalar meson structures [2]. 

One of the major purposes for building the $\phi$ factory
is to measure CP violation parameters 
by using the decay $\phi\rightarrow K_S K_L$.
CP violation effects are small, so 
we should be careful in excluding possible backgrounds.
The radiative decay 
$\phi\rightarrow S\gamma\rightarrow K^0 \bar K^0 \gamma$,
where $S=f_0(975)$ or $a_0(980)$,
could be an important background.
This is because the photon charge conjugation is negative, so that
we have $K_S K_S$ or $K_L K_L$ instead of $K_S K_L$.

On the other hand, it is interesting to investigate
scalar meson structures by using the radiative decays.
It is known that most mesons in the 1 GeV region are
well explained by $q\bar q$ type models.
However, structures of the scalar mesons $f_0$ 
and $a_0$ are not understood.
For example, we obtain 
500$-$1000 MeV strong-decay width of $f_0$ 
in the $q\bar q$ models [3].
This large decay width is in contradiction to 
the experimental value (47 MeV).
So, these mesons could be ``exotic'' mesons such as
$K\bar K$ molecule, $qq\bar q\bar q$, and glueball.

In this talk, the radiative decay widths are estimated
by assuming various structures for the scalar mesons.
Then, the CP background problem is discussed.


\vfill\eject

\noindent
{\bf 2. Radiative decay widths}

\vspace{0.2cm}

We first explain the reason why the radiative decay
is useful in discriminating among various models.
Because spin and parity of $\phi$ are $1^-$
and those of $S$ are $0^+$, the radiative decay
$\phi\rightarrow S\gamma$ is an electric-dipole (E1)
decay. The electric-dipole operator is given by
$\sum e_i \vec r_i$, where $e_i$ is the
charge of a constituent and $\vec r_i$ is 
the vector distance from the center of mass. 
Therefore, its matrix element is very sensitive 
to the scalar-meson size. If $S$ were the $q \bar q$ meson, 
charges should be confined in rather small space 
(about 0.5 fm size). 
On the other hand, if $S$ is
the $K\bar K$ molecule, the charges should be
distributed in larger space 
(about 1.5 fm size).
This is because scalar-meson masses are
just below the $K\bar K$ threshold 
(2$m_{_{K^\pm}}$=987 MeV)
and binding energies are small. 
From these discussions, we can reasonably
expect that charge distributions in the $q\bar q$ and
$K\bar K$ cases are very different. The E1-decay width
of $\phi\rightarrow S\gamma$ shall clearly reflect
the difference. 

We discuss the widths for the $\phi$ radiative decays by
assuming various models for $S$
($n \bar n$, $s \bar s$, $qq\bar q\bar q$, 
 $K\bar K$, and a glueball), 
where $n$=$u$ or $d$-quark.
Our results are summarized in Table 1 [2].


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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. 2 \\ \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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent
The above results are explained in the following.

$~~~~~~~~~~$

\noindent
a) $K\bar K$ molecule

We discuss the radiative decay by assuming
a $K\bar K$ molecule picture for S.
We consider the decay through a $K\bar K$ loop, 
$\phi\rightarrow K\bar K \gamma \rightarrow S \gamma$.
For example, $\phi$ decays into $K^+$ and $K^-$,
then either $K^+$ or $K^-$ radiates a photon,
and finally $K^+$ and $K^-$ recombine to form a bound state S.
In this kind of calculation, we should be careful
in taking into account finite size effects.
Because S is a loose bound state of $K\bar K$,
the loop momentum cannot be infinite quantities.
The momentum should be restricted by the scalar meson size.
We simply use the dipole form factor $\phi(|\vec p|)$, 
which simulates the Toronto $K\bar K$ wave function, 
at the $K\bar K S$ vertex. 
Once the momentum dependence is introduced, 
current conservation or gauge invariance should be carefully
studied. This is because we have an extra current,
so called ``interaction current'', 
by the minimal substitution
$\phi(\vec k) \rightarrow \phi(\vec k -e\vec A)$.
The physics meaning of this current is 
explained in Ref. [4].

As a result of our numerical evaluation,
we obtain the width $6.3\times 10^{-4}$ MeV
in the pointlike limit ($R_{K\bar K}\rightarrow 0$).
However, if the $K\bar K$-molecule size
1.2 fm (the Toronto $K\bar K$ molecule) is used,
we obtain the width $1.8\times 10^{-4}$ MeV, 
which is significantly
smaller than the point-like result.
Using the total $\phi$ width 4.41 MeV, we  
get the branching ratio 
$B.R.(\phi\rightarrow f_0\gamma)=4 \times 10^{-5}$.
In the $K\bar K$-type picture for $a_0$, the decay width
for $\phi\rightarrow K\bar K\gamma \rightarrow a_0 \gamma$
is roughly the same as the one for 
$\phi\rightarrow f_0\gamma$. 


$~~~~~$

\noindent
b.1) $q^2\bar q^2$ ($K\bar K$-like ``bag'')

The constituent-quark contents of 
the $qq\bar q\bar q$ and $K\bar K$ systems
are identical, but their dynamical
structures are very different.
The essential difference is whether the multiquark
system is confined within a hadronic system
with a radius of the order of 1/$\Lambda_{QCD}$ or it is
two identifiable color singlets spread over
a region significantly greater than this.

If the scalar mesons are four-quark bound states 
($qq\bar q\bar q$), we find that the decay-width
ratio $R(a_0/f_0)$ provides us important information
on the internal structure.
The decay widths themselves are rather difficult to
be estimated due to poorly-understood 
decays of four-quark bound states.
In the following, we show that
the ratio $R(a_0/f_0)$ varies significantly with 
the $qq\bar q\bar q$ structure.
Because the electric-dipole operator
is given by $\sum e_i \vec r_i$,
we simply take into account constituent charges
for evaluating the widths.

If the $f_0$ and $a_0$ structures are  
$K \bar K$-like ``bag'', we denote
$S=(u\bar s)(\bar u s)\pm
           (d\bar s)(\bar d s)$,
where $+$ is for $f_0$ and $-$ for $a_0$.
In this case, E1 matrix elements are roughly given by
$M(E1)\sim (e_u+e_{\bar s})\pm
           (e_d+e_{\bar s})=1$.
Because the second term vanishes ($e_d+e_{\bar s}=0$),
we obtain the same matrix elements for both
decays. So, the decay-width ratio is
$R(a_0/f_0)\approx 1$.

$~~~~~$

\noindent
b.2) $q^2\bar q^2$ ($D\bar D$-like ``bag'')

If the structures are diquark-antidiquark-like 
``bag'', we denote
        $S=(us)(\bar u \bar s)\pm
           (ds)(\bar d \bar s)$.
The E1 matrix elements are given by
$M(E1)\sim (e_u+e_s)\pm
           (e_d+e_s)=-1/3 ~(f_0),~1~(a_0)$ and
the ratio is
$R(a_0/f_0)\approx 9$,
which is very different from the one 
in the $K\bar K$-like bag case.
Therefore, we find that the decay-width ratio 
is also an important quantity for investigating 
the structures.


$~~~~~$

\vfill\eject
\noindent
b.3) $q^2\bar q^2$ ($\pi\eta$, $\eta\eta$-like ``bag'')
 
If the structures are 
$\pi\eta$ or $\eta\eta$-like ``bag'',
we denote $S=(u\bar u)(s\bar s)\pm
           (d\bar d)(s\bar s)$.
The E1 matrix elements are given by
$M(E1)\sim [(e_u+e_{\bar u})-(e_s+e_{\bar s})]
       \pm [(e_d+e_{\bar d})-(e_s+e_{\bar s})]=0$, 
so that we cannot obtain the ratio by the above simple
method. Much more detailed analyses are needed for
the decay-width ratio and the widths themselves.

$~~~~~$

\noindent
c.1) $f_0=s\bar s$ and $a_0=(u\bar u-d\bar d)/\sqrt 2$

If $f_0$ is a $s\bar s$ bound state and 
   $a_0$ is an ordinary $n\bar n$ state, we first estimate
the decay width of $\phi \rightarrow f_0(s\bar s)\gamma$
by using observed E1-decay widths.
For example, the charmonium E1 decay 
$\chi_{c0}\rightarrow J/\psi \gamma$
is observed and the experimental width is 0.092 MeV.
E1 decay widths are in general expressed as
$\Gamma (E1) \sim e^2 R^2 E_\gamma^3$, where
$e$ is the charge factor, $R$ is the hadron size, and
$E_\gamma$ is the emitted-photon energy.
Taking into account these factors, we obtain
$
B.R.(\phi\rightarrow f_0\gamma)\approx
{1 \over {4.41}} \cdot {1 \over 3} \cdot
\bigl ( {{e_s} \over {e_c}} \bigr )^2 \cdot
\bigl ( {{R_s} \over {R_c}} \bigr )^2 \cdot
\bigl ( {{44}  \over {320}} \bigr )^3 \cdot
0.092 \simeq 1 \times 10^{-5},
$
where 1/3 is a $\phi$-spin factor and 4.41 MeV is
the total $\phi$ decay width. 
The radiative decay into $a_0$ is an OZI violating
process, so that the branching ratio should be significantly
smaller:
$
R(a_0/f_0)\equiv \Gamma (\phi \rightarrow a_0\gamma)/
                  \Gamma (\phi \rightarrow f_0\gamma)
 << 1 .
$

$~~~~~$

\noindent
c.2) $f_0=(u\bar u+d\bar d)/\sqrt 2$ and
          $a_0=(u\bar u-d\bar d)/\sqrt 2$

If both $f_0$ and $a_0$ are ordinary $n\bar n$ type mesons,
both radiative decays are OZI-violating processes.
Therefore, the decay widths are much smaller than 
the above value ($10^{-5}$):
$
\Gamma (\phi\rightarrow f_0\gamma), 
\Gamma (\phi\rightarrow a_0\gamma) \lapproxeq 10^{-6} .
$

$~~~~~$

\noindent
d) $f_0$=glueball

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 [3]:
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 previous results
for $\Gamma(\phi\rightarrow f_0(q\bar q)\gamma)$, 
we obtain
$
B.R.(\phi \rightarrow f_0\gamma) 
           \lapproxeq 10^{-6} .
$


$~~~~~$


%%%\vfill\eject
\noindent
{\bf 3. CP background problem}

\vspace{0.2cm}

One of the major projects at the $\phi$ factory
is to measure CP violation parameters accurately
by using the decay $\phi\rightarrow K^0 \bar K^0$.
If the charge conjugation is conserved in strong
and electromagnetic interactions, 
the final kaon state is $K_S K_L$ (C=odd).
In order to measure the CP-violation-parameter ratio 
$\varepsilon ' /\varepsilon$ accurately
in the decay $\phi \rightarrow K_0 \bar K_0$,
it is essential to understand C=even backgrounds.
For example, $K_S K_S$ or $K_L K_L$ 
could be produced instead of $K_S K_L$ 
in a $\phi$ radiative decay
because the photon charge conjugation is negative.
If the branching ratio for the radiative decay is large
[B.R.$(\phi\rightarrow K^0 \bar K^0 \gamma) >10^{-6}$],
it is a significant background 
to the studies of CP violation.

The resonant contribution to 
$\phi\rightarrow K^0\bar K^0\gamma$ 
($\phi\rightarrow S\gamma\rightarrow K^0 \bar K^0 \gamma$)
is especially problematic.
As discussed in the previous section, the structures
of the scalar mesons ($f_0$ and $a_0$) are little understood.
Consequently, we cannot calculate the decay width
$\Gamma (\phi\rightarrow S\gamma
                \rightarrow K^0 \bar K^0 \gamma)$
accurately. In fact, prior to our publication,
very different widths were obtained by several groups.
The large differences are due in part to
errors and in part to modeling, and we refer to the
Brown-Close report [2] on this point for the interested
reader. We estimate the decay width by
using the results in the previous section 
[$\Gamma (\phi\rightarrow S\gamma)$]
and also the Breit-Wigner form for the
$S$ propagation.
We should note that there is a destructive 
interference between the $f_0$ and $a_0$ amplitudes, and
the result indicates
$B.R.(\phi\rightarrow K^0\bar K^0\gamma)\lapproxeq 10^{-7}$.
This is too small to be a serious background to
the CP-violation experiment.

$~~~~~$

\noindent
{\bf 4. Conclusions}

\vspace{0.2cm}

We showed that structures of controversial
$f_0$(975) and $a_0$(980) mesons could be
investigated by the $\phi$ radiative decays.
However, our estimates for the decay widths are so naive that
much detailed theoretical studies are needed.
According to our result,
the decay $\phi\rightarrow S\gamma \rightarrow K^0\bar K^0\gamma$
is not a serious background to 
the CP-violation studies at $\phi$ factories.


\vspace{0.2cm}

\begin{center}
{\bf Acknowledgment} \\
\end{center}
\vspace{-0.2cm}

This talk is based on the research done with F. E. Close and
N. Isgur.
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.

 
$~~~$

\noindent
* E-mail: kumanos@himiko.cc.saga--u.ac.jp.

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\begin{center}
{\bf References} \\
\end{center}

\vspace{0.1cm}
 
\vspace{-0.30cm}
\vspace{-0.38cm}
\begin{description}{\leftmargin 0.0cm}
\item{[1]}
M. Fukawa et al., KEK report 90-12 (1990).

\vspace{-0.38cm}
\item{[2]}
F. E. Close, N. Isgur, and S. Kumano, 
Nucl. Phys. {\bf B389}, 513 (1993);
N.  Brown and F. E. Close, report RAL-91-085;
S. Kumano, pp. 283--292 in 
{\it Flavour and Spin in Hadronic and Electromagnetic Interactions},
edited by F. Balestra, R. Bertini, and R. Garfagnini,
SIF, Bologna (1993).

\vspace{-0.38cm}
\item{[3]}
S. Kumano and V. R. Pandharipande, Phys. Rev. {\bf D38}, 146 (1988)
and references therein.


\vspace{-0.38cm}
\item{[4]}
L. Heller, S. Kumano, J. C. Martinez, and E. J. Moniz,
           Phys. Rev. {\bf C35}, 718 (1987).


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