Coupled channel effects in $\pi\pi$ S-wave interaction
F.Q.Wu and B.S.Zou
We study coupled channel effects upon isospin $I=2$ and $I=0$ $\pi\pi$ S-wave interaction. With introduction of the $\pi\pi\to\rho\rho\to\pi\pi$ coupled channel box diagram contribution into $\pi\pi$ amplitude in addition to $\rho$ and $f_2 (1270)$ exchange, we reproduce the $\pi \pi$ I=2 S-wave and D-wave scattering phase shifts and inelasticities up to 2 GeV quite well in a K-matrix formalism. For $I=0$ case, the same $\pi\pi\to\rho\rho\to\pi\pi$ box diagram is found to give the largest contribution for the inelasticity among all possible coupled channels including $\pi\pi\to\omega\omega\to\pi\pi$, $\pi\pi\to K \overline{K}\to\pi\pi$. We also show why the broad $\sigma$ appears narrower in production processes than in $\pi\pi$ scattering process.
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\markboth{F.Q.Wu}{Coupled channel effects
 in $\pi\pi$ S-wave interaction}

\catchline{}{}{}{}{}

\title{Coupled channel effects
 in $\pi\pi$ S-wave interaction}

\author{F.Q.Wu and B.S.Zou}
\address{CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, China ; \\
Institute of High Energy Physics, CAS, Beijing 100049, China}

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\maketitle

%\begin{history}
%\received{1 Sep 2004}
%\revised{1 Sep 2004}
%\end{history}

\begin{abstract}
We study coupled channel effects upon isospin $I=2$ and $I=0$
$\pi\pi$ S-wave interaction. With introduction of the
$\pi\pi\to\rho\rho\to\pi\pi$ coupled channel box diagram
contribution into $\pi\pi$ amplitude in addition to $\rho$ and
$f_2 (1270)$ exchange, we reproduce the $\pi \pi$ I=2 S-wave and
D-wave scattering phase shifts and inelasticities up to 2 GeV
quite well in a K-matrix formalism. For $I=0$ case, the same
$\pi\pi\to\rho\rho\to\pi\pi$ box diagram is found to give the
largest contribution for the inelasticity among all possible
coupled channels including $\pi\pi\to\omega\omega\to\pi\pi$,
$\pi\pi\to K \overline{K}\to\pi\pi$. We also show why the broad
$\sigma$ appears narrower in production processes than in $\pi\pi$
scattering process.
\end{abstract}

\keywords{$\pi\pi$ scattering, coupled channel, K-matrix }

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\section{INTRODUCTION}

As well known isospin $I=0 \ \ \pi \pi$ S-wave interaction gives a
good place to study the $I=0 \ \ J^{pc}=0^{++}$ particles such as
$\sigma$ and glueball.  However, to really understand the
isoscalar $\pi\pi$ S-wave interaction, one must first understand
the isospin I=2 $\pi\pi$ S-wave interaction due to the following
two reasons:  (1) There are no known s-channel resonances and less
coupled channels in I=2 $\pi\pi$ system, so it is much simpler
than the I=0 $\pi\pi$ S-wave interaction; (2) To extract I=0
$\pi\pi$ S-wave phase shifts from experimental data on
$\pi^+\pi^-\to\pi^+\pi^-$ and $\pi^+\pi^-\to\pi^0\pi^0$ obtained
by $\pi N\to\pi\pi N$ reactions, one needs an input of the I=2
$\pi\pi$ S-wave interaction.

Up to now, experimental information on the I=2 $\pi\pi$ scattering
mainly came from $\pi^+p\to\pi^+\pi^+n$ \cite{Hoogland77} and
$\pi^-d\to\pi^-\pi^-pp$ \cite{Durusoy73} reactions. In previous
analyses, the feature of inelasticity $\eta_0^2$ starting to
deviate from 1 for energies above 1.1 GeV was often overlooked.
Recently, with a K-matrix formalism \cite{lilong}, we show
\cite{Wu} that the feature can be well reproduced by
$\pi\pi$-$\rho\rho$ coupled-channel effect.  Here we extend the
study of the coupled channel effects to the I=0 case which allows
much more coupled channels. We also show why the broad $\sigma$
appears narrower in production processes than in $\pi\pi$
scattering process.




\section{Coupled channel effects in $\pi\pi$ scattering }
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We follow the $K$-matrix formalism as in Ref.\cite{lilong}. For
$\pi\pi$ scattering, the scattering amplitude can be given in
K-matrix formalism as
\be
T_{el}=\frac{K}{1-i\rho K} = K +
K~i\rho~K + K~i\rho~K~i\rho~K + \cdots
\ee
which can be expressed
diagrammatically as in Fig.\ref{krk} for $\pi\pi$ scattering at
low energies with only one opening channel.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=10cm,height=2cm]{krk.eps}
\end{center}
\caption{Diagrammatic expression for $\pi\pi$ scattering in
K-matrix formalism}
 \label{krk}
\end{figure}

For the two-channel case, the two-dimensional $K$ matrix and phase
space $\rho(s)$ matrix are \be K=\left (
\begin{array}{ccc}
K_{11}  &  K_{12}  \\
K_{12}  &  K_{22}
\end{array} \right ),\hspace{1cm}
\rho(s)=\left (
\begin{array}{ccc}
\rho_1(s)  &  0  \\
0  &  \rho_2(s)
\end{array} \right ),
\ee with i=1,2 representing $\pi\pi$ and $\rho\rho$ channel,
respectively.  Ignoring the interaction between $\rho\rho$, we
have $K_{22}=0$; then
%
\be T_{11}=\frac{K_{11}+i K_{12} \rho_2 K_{21}}{1-i
\rho_1(K_{11}+iK_{12} \rho_2 K_{21} )}, \ee
%
where $iK_{12} \rho_2 K_{21}$ corresponds to the $\rho\rho$
on-shell contribution\cite{Lu} of the $\pi\pi\to\rho\rho\to\pi\pi$
box diagram as shown in Fig.\ref{block1}.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=10cm,height=2cm]{block_2_zou.eps}
\end{center}
\caption{The $\pi\pi\to\rho\rho\to\pi\pi$ box diagrams}
 \label{block1}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{rho1.eps}
\end{center}
\vspace{-4cm}
 \caption{The $I=2$ $\pi\pi$ $S$-wave ($\delta_0^2$,
$\eta_0^2$)
 phase shifts and inelasticities. Data are from
Ref.$^{1,2,3}$.  The solid curves represent the total contribution
of $\rho$, $f_2$ exchange and  the box diagram; dot-dashed curves
from $\rho $ and $f_2$ exchange; dashed curves from t-channel
$\rho$ exchange only.}
 \label{phase}
\end{figure}

With $K_{11}$ including contribution from $t$-channel $\rho$ and
$f_2(1270)$ exchange, we found that the basic features of I=2
$\pi\pi$ S-wave phase shifts and inelasticities are well
reproduced as shown in Fig.\ref{phase}. For details, see
Ref.\cite{Wu}.

With the success in reproducing the $I=2$ $\pi\pi$ S-wave
scattering, we extend our study of the coupled channel effects to
the isospin I=0 $\pi\pi$ scattering.  Here in additional to the
box diagrams shown in Fig. \ref{block1}, there are more other
coupled channels such as $\pi\pi\to\omega\omega\to\pi\pi$,
$\pi\pi\to\sigma\sigma\to\pi\pi$ and $\pi\pi\to K
\overline{K}\to\pi\pi$ as shown in Fig.\ref{block2}.


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=10cm,height=2cm]{boxdiagram.eps}
\end{center}
 \caption{The $\pi\pi\to\omega\omega\to\pi\pi$,
$\pi\pi\to\sigma\sigma\to\pi\pi$ and $\pi\pi\to K
\overline{K}\to\pi\pi$  box diagrams for I=0 case.}
 \label{block2}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.3]{new.eps}
\end{center}
\vspace{-3cm}
 \caption{The $I=0$ $\pi \pi$ S-wave inelasticity $\eta_{0}^{0}$
without including s-channel resonances in the $I=0$ amplitude.
``$\pi\pi-\rho\rho - \pi \pi$" means including
$\pi\pi-\rho\rho-\pi\pi$ box diagram; ``$\pi\pi-\rho\rho, \omega
\omega- \pi \pi$" means using three-dimensions K matrix to couple
$\pi\pi,\rho\rho$ and $\omega \omega$ channels together. }
 \label{sampl}
\end{figure}

To demonstrate the significance of coupled channel effects in the
I=0 $\pi\pi$ scattering, we  do not include any s-channel
resonances in the $I=0$ amplitude, just introduce $\pi\pi\to
\rho\rho \to\pi\pi$ ,$\pi\pi \to \omega\omega \to \pi \pi$,
$\pi\pi\to K \overline{K}\to\pi\pi$ and $\pi\pi\to \sigma \sigma
\to \pi\pi$ box diagrams respectively into the $I=0$ amplitude
which includes t-channel $\rho$ and $f_{2}(1270)$ exchange
\cite{lilong}. The result shows $\pi\pi\to \rho\rho \to\pi\pi$
(solid line in Fig. \ref{sampl}) gives the most important
contribution. We also use three-dimensions K matrix to couple
$\pi\pi,\rho\rho$, and $\omega \omega$ channels together. The
result is shown by the dotted line in Fig.\ref{sampl}.


\section{$\pi\pi$ S-wave interaction in production processes}

$\pi\pi$ production processes can be regarded as a special case
for coupled channels. For the $\pi\pi$ S-wave interaction in
production processes, it can be expressed diagrammatically as in
Fig.\ref{prk}.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=10cm,height=2cm]{prk.eps}
\end{center}
\caption{Diagrammatic expression for $\pi\pi$ scattering in
K-matrix formalism}
 \label{prk}
\end{figure}

Compared with elastic scattering shown in Fig.\ref{krk}, the only
difference is the first interaction vertex. So the production
amplitude can be expressed similar to Eq.(1) as

\be T_{prod}=\frac{P}{1-i\rho K} = P + P~i\rho~K +
P~i\rho~K~i\rho~K + \cdots \ee

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8cm,height=6cm]{sigma.eps}
\end{center}
\caption{Amplitude squared for $\pi\pi$ S-wave in production
process (dashed curve) compared with in elastic scattering (solid
curve), assuming P=1 and K from Ref.$^{9}$. } \label{sigma}
\end{figure}

For the I=0 $\pi\pi$ S-wave scattering at low energies ($\leq
2m_K$, the K-matrix and its corresponding elastic scattering
amplitude $T_{el}$ can be well determined by the $\pi\pi$
scattering phase shift data \cite{CM,Becker,Kaminski}. The solid
line in Fig.\ref{sigma} shows a solution \cite{BSZ} for
$|\rho_1T_{el}|^2$ from fitting the well-known CERN-Munich
$\pi\pi$ scattering data \cite{CM,Becker}. The solution has a
T-matrix pole at $571-i420$ MeV for the broad $\sigma$ which
provides the broad background for two narrow dips caused by its
interference with the narrow $f_0(980)$ and $f_0(1500)$. With the
same K-matrix, if we assume P-matrix to be 1 for $\pi\pi$
production, we can get $|\rho_1T_{prod}|^2$ as shown by the dashed
line in Fig.\ref{sigma}. A much narrower peak at lower energy is
appearing although in fact it has the same broad pole as in the
$\pi\pi$ elastic scattering process. This gives a clear
demonstration why the broad $\sigma$ appears narrower in
production processes than in $\pi\pi$ scattering process. The
reason is $T_{prod}\sim T_{el}/K$ here. This has also been noted
by Ref.\cite{Oset} in a slightly different language.

Some recent analyses \cite{E791,BES} of various $\pi\pi$
production processes gave a much narrower $\sigma$ pole than
Ref.\cite{BSZ,zou1} from $\pi\pi$ scattering. A reason is that it
is assumed a direct production of $\sigma$ with production vertex
$P=1/(m^2_\sigma-m^2_{\pi\pi})$ instead of considering the
$\sigma$ due to final state $\pi\pi$ scattering with $P=1$ or some
smooth function of $m_{\pi\pi}$. With the same production data but
with different production vertex $P$, one will get different
$\sigma$ pole. See Refs.\cite{E791,FOCUS} for an example.

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\section*{Acknowledgements}

The work is partly supported by CAS Knowledge Innovation Project
(KJCX2-SW-N02) and the National Natural Science Foundation of
China.
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\bibitem{Hoogland77} W. Hoogland et al., Nucl. Phys. {\bf B 126} (1977)
109.
%
\bibitem{Durusoy73}
N.B.Durusoy et al., Phys. lett. {\bf 45B} (1973) 517; D.Cohen et
al., Phys. Rev.{\bf D7}(1973) 661.
%
\bibitem{lilong} L. Li, B.S. Zou, G.L. Li, Phy. Rev. {\bf D
63} (2001) 074003; Phy. Rev. {\bf D 67} (2003) 034025; B.S.Zou,
D.V.Bugg, Phys. Rev. {\bf D 50} (1994) 591.
\bibitem{Wu} F.Q.Wu, B.S.Zou, L.Li, D.V.Bugg, Nucl. Phys. {\bf
A735} (2004) 111.
\bibitem{Lu} Y.Lu, B.S.Zou and M.P.Locher, Z. Phys. {\bf A345}
(1993) 207;  Z. Phys. {\bf A347} (1994) 281.
\bibitem{CM} B. Hyams et al., Nucl. Phys. {\bf B 64} (1973) 134.
%
\bibitem{Becker} H. Becker et al., Nucl. Phys. {\bf B151} (1979) 46.
%
\bibitem{Kaminski} R.Kaminski et al., Z.Phys. {\bf C74} (1997) 79.
\bibitem{BSZ} D.V. Bugg, A.V. Sarantsev, B.S. Zou, Nucl. Phys.
                 {\bf B 471} (1996) 59.
%
\bibitem{Oset} L. Roca, J.E.Palomar, E.Oset, H.C. Chiang,
hep-ph/0405228.
%
\bibitem{E791} E791 Collaboration, E.M. Aitala et al., Phys. Rev. Lett.
{\bf 86} (2001) 770.
\bibitem{BES} BES Collaboration, M. Ablikim et al., Phys. Lett. {\bf B598} (2004)
149.
\bibitem{zou1} B.S.Zou, D.V.Bugg, Phys. Rev. {\bf D48} (1993)
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\bibitem{FOCUS} FOCUS Collaboration, J.M. Link et al., Phys. Lett.
{\bf B585} (2004) 200.

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\bibitem{Hoogland77} W. Hoogland et al., Nucl. Phys. {\bf B 126} (1977)
109.
%
\bibitem{Durusoy73}
N.B.Durusoy et al., Phys. lett. {\bf 45B} (1973) 517; D.Cohen et
al., Phys. Rev.{\bf D7}(1973) 661.
%
\bibitem{lilong} L. Li, B.S. Zou, G.L. Li, Phy. Rev. {\bf D
63} (2001) 074003; Phy. Rev. {\bf D 67} (2003) 034025; B.S.Zou,
D.V.Bugg, Phys. Rev. {\bf D 50} (1994) 591.
\bibitem{Wu} F.Q.Wu, B.S.Zou, L.Li, D.V.Bugg, Nucl. Phys. {\bf
A735} (2004) 111.
\bibitem{Lu} Y.Lu, B.S.Zou and M.P.Locher, Z. Phys. {\bf A345}
(1993) 207;  Z. Phys. {\bf A347} (1994) 281.
\bibitem{CM} B. Hyams et al., Nucl. Phys. {\bf B 64} (1973) 134.
%
\bibitem{Becker} H. Becker et al., Nucl. Phys. {\bf B151} (1979) 46.
%
\bibitem{Kaminski} R.Kaminski et al., Z.Phys. {\bf C74} (1997) 79.
\bibitem{BSZ} D.V. Bugg, A.V. Sarantsev, B.S. Zou, Nucl. Phys.
                 {\bf B 471} (1996) 59.
%
\bibitem{Oset} L. Roca, J.E.Palomar, E.Oset, H.C. Chiang,
hep-ph/0405228.
%
\bibitem{E791} E791 Collaboration, E.M. Aitala et al., Phys. Rev. Lett.
{\bf 86} (2001) 770.
\bibitem{BES} BES Collaboration, M. Ablikim et al., Phys. Lett. {\bf B598} (2004)
149.
\bibitem{zou1} B.S.Zou, D.V.Bugg, Phys. Rev. {\bf D48} (1993)
3948.
\bibitem{FOCUS} FOCUS Collaboration, J.M. Link et al., Phys. Lett.
{\bf B585} (2004) 200.

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