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\title{Properties of scalar--isoscalar mesons from
multichannel interaction analysis below 1800 MeV}

\author{R. Kami\'nski
%\thanks{presently at LPNHE et LPTPE Universit\'es
%D.~Diderot P.~et~M.~Curie, 4, Place Jussieu,
%75252 Paris CEDEX 05, France}
$^{\mbox{\scriptsize a, b}}$,
L. Le\'sniak 
\address{Department of Theoretical Physics,
The Henryk Niewodnicza\'nski Institute of Nuclear Physics,
 PL 31-342 Krak\'ow, Poland}
and
B.\ Loiseau\address{LPTPE Universit\'e P.~et M.~Curie,
4, Place Jussieu, 75252 Paris CEDEX 05, France}}
      
\begin{document}

\maketitle

\begin{abstract}
Scalar-isoscalar mesons
 are studied using an unitary model
%with separable potentials
in three channels: $\pi\pi$, \kk and an
effective $2\pi2\pi$.
%, in a mass range from the \pp threshold up to 1800 MeV.
All the solutions, fitted to the \pp and \kk data, exhibit 
a wide $f_0(500)$, a narrow \fo and two relatively 
narrow resonances, lying on different sheets
%, in the energy region
between  1300 MeV and 1500 MeV. These latter states are similar to 
the $f_0(1370)$ and $f_0(1500)$ seen in experiments at CERN.
Branching ratios are compared with
available data. 
%A new solution, constrained with \pp on-shell
%and \kk off-shell scattering lengths as predicted by chiral symmetry,
%satisfies in a better way Roy
%equations, implementing  better
%crossing symmetry properties.
We have started investigations of some crossing symmetry and chiral constraints
imposed near the \pp threshold on the scalar-isoscalar, scalar-isotensor and
P-wave \pp amplitudes.
\end{abstract}

\section{INTRODUCTION}
Study of scalar-isoscalar mesons is an important issue of QCD
%and many theo\-re\-ti\-cal and experimental efforts have been 
%recently made for 
%a better understanding of their classifications and properties
%\cite{hadron97}.% From QCD One
: one expects presence of some scalar
($J^{PC} = 0^{++}, I=0$) glueballs~\cite{pdg98}
%which can be mixed with ordinary $q\overline{q}$ scalar states \cite{close97}
. Here we shall  try to see what
 can be learned from the
present experimental knowledge of the scalar-isoscalar \pp and \kk
phase shifts.  We consider an unitary model  with
separable interactions  in three channels: $\pi\pi$, \kk and an effective
$2\pi2\pi$, denoted \roro, in a mass range from the \pp threshold up to 1800
MeV \cite{kll297,kll299}.
Several solutions are obtained by fitting \pp phase shifts 
from the CERN-Cracow-Munich analysis
of the \reactpol reaction on a polarized target
\cite{klr} together with lower energy \pp  %\cite{rosbelsri}
and \kk data from reactions on unpolarized target %\cite{cohen}.
(see references given in~\cite{kll299}).
%We shall describe in the next section some properties of our meson-meson
%multichannel amplitudes such as the possible resonances and their branching
%ratios. Section III is devoted to the study of chiral and crossing symmetry
%constraints one can try to impose to these solutions and section IV gives some
%concluding remarks.
 
%Ana\-ly\-ti\-cal structure of the
%meson-meson multichannel amplitudes is studied with a special emphasis
%on the important role played by the $S$-matrix zeroes.
%Poles, located in the complex energy plane, not
%too far from the physical region, are interpreted as
%scalar resonances.
%We find a wide $f_0(500)$, a narrow \fo and a relatively 
%narrow $f_0(1400)$.
%The dependence of the positions of the $S$-matrix singularities  on the
%interchannel coupling strengths is investigated to find origin
%of resonances.
%In all our solutions two resonances, lying on different sheets, in the energy 
%region between  1300 MeV and 1500 MeV are found and
%can be compared with the resonances $f_0(1370)$ and $f_0(1500)$ 
% in the experiments at CERN.
%Total, elastic and inelastic channel cross sections, 
%branching ratios and coupling constants are evaluated and
%compared with available data. 
%We show that new measurements at the \kk threshold are needed to precise
%the experimental values of the \fo branching ratios.

%Comparison with chiral symmetry predictions for low energy \pp and
%\kk amplitudes is studied.
% shall present new solutions with \pp on-shell and
%\kk off-shell scattering lengths as predicted by chiral symmetry.
%We shall furthermore consider the crossing conditions that the 
%Roy equations can impose to our model.


\section{RESULTS}

The different solutions A, B, E and F of our model
are characterized by  presence or absence of \kk and \roro bound states
 when all the interchannel couplings are switched off (see
Table 2 of ~\cite{kll299}). For the fully coupled case, poles of the
$S$-matrix, located in the complex energy plane not
too far from the physical region, are interpreted as
scalar resonances. In all our solutions we find a wide $f_0(500)$,
a narrow \fo and a relatively
narrow $f_0(1400)$ which splits into
 two resonances, lying on different sheets classified according to the sign
 of $Imk_{\pi \pi}, Imk_{K \overline{K}}, Imk_{\sigma \sigma}$.
 Their average masses and widths are
 summarized in Table \ref{resonances}. The finding of  the
 two states near 1400 MeV 
  seems to indicate that
the \pp data with polarized target are quite compatible with  the 
Crystal-Barrel and other LEAR data which need, in order to be explained,
a broad $f_0(1370)$ and a narrower $f_0(1500)$ 
\cite{pdg98}. In \cite{kll299} we have furthermore studied the
dependence of the positions of the $S$-matrix singularities on the
interchannel coupling strengths to find origin of resonances. We have also
looked at the interplay between $S$-matrix zeroes and poles.

 
\begin{table}[htb]
\caption{Average masses and widths of resonances}
%\epsig, \fo and \epw found in solutions A, B, E and F. Here errors represent 
%the
\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{2pc} % enlarge column spacing
\renewcommand{\arraystretch}{1.2} % enlarge line spacing
\begin{tabular}{@{}cccc}
\hline
resonance & mass (MeV) & width (MeV) & sheet \\
\hline 
\epsig or $\sigma$ & $523 \pm 12$ & $518 \pm 14$ & $-++$ \\
\hline
\fo & $991 \pm 3$ & $71 \pm 14$ &  $-++$ \\
\hline
& $1406 \pm 19$ & $160 \pm 12$ & $---$ \\
\epw & $1447  \pm 27$ & $108 \pm 46$ & $--+$ \\
\hline 
%\fj & 1703 & 542 & $---$ \\
% & 1624 & 350 & $+--$ \\
%\hline
\end{tabular}
\label{resonances}
\end{table}

%Let us here recall that one
In the \pp channel ($j= 1$) one can define three
%the following
 branching ratios
$b_{1j} = \sigma_{1j}/\sigma^{tot}_{11}~, j=1,2,3$.
Our model has in total nine such ratios.
%\be 
%b_{1j}=\frac{\sigma_{1j}}{\sigma^{tot}_{11}}~, ~~j = 1,2,3,\label{branch} 
%\eb
Here  $\sigma_{11}$ is the elastic \pp cross section, $\sigma_{1j}$
are the transition cross sections to %channels
\kk ($j=2$) and \roro ($j=3$)
and $\sigma^{tot}_{11}$ is the total \pp cross section.
%the \pp channel.
%These branching ratios obviously satisfy
%\be b_{11}+b_{12}+b_{13}=1.                                \label{bsum}   \eb
One has $b_{11}+b_{12}+b_{13}=1.$
The energy dependence of these ratios is plotted in Fig.~1 for
solution B.

%\begin{figure}[h]
%\vspace{15.0cm}
%\special{psfile=fig6a.ps   voffset=-250
%hoffset=-20  angle=0   vscale=80  hscale=80 }
%\caption{ Transition form factor.}
%\label{fig1}
%\end{figure}



\begin{figure}[h]
 \vspace{6.cm}
% \begin{center}
\special{psfile=fig6a.ps   voffset=-72
hoffset=0  angle=0   vscale=39  hscale=80 }
%\xslide{fig6a.ps}{7cm}{16}{170}{540}{650}{11.5cm}
\caption{
%Energy dependence of 
Branching ratios for \pp transitions to
 \pp $(b_{11})$, \kk $(b_{12})$ and \roro $(b_{13})$}
%for the solution B}
%    \end{center}
                \label{br}
  \end{figure}
Above the \kk threshold one can define an average branching ratio:
%$\overline b_{12}= (1/(M_{max}-M_{min})
%$\int_{M_{min}}^{M_{max}}b_{12}(E) dE $
%\be
$$\overline b_{12}= \frac{1}{M_{max}-M_{min}}
\int_{M_{min}}^{M_{max}}b_{12}(E) dE.$$
%\label{nbav12}
%\eb 
In   \cite{amsler97} the branching ratios for the $f_0(1500)$ decay into
five channels, \pp, $\eta\eta$, $\eta\eta^{,}$, \kk and $4\pi$, are given as
29, 5, 1, 3 and 62 \%, respectively. The two main
disintegration channels are \pp and $4\pi$. In our model the $4\pi$ channel
is represented by the effective
\roro channel and we also obtain large fractions for the  averaged branching
 ratios $\overline{b}_{11}$ and $\overline{b}_{13}$. If we calculate the ratios
 $b_{13}/b_{11}$ exactly at 1500 MeV then we obtain numbers
 2.4, 1.2 and 2.3 for the solutions A, B and E, respectively. These numbers
 show the
 importance of the $4\pi$ channel in agreement with the experimental result of
 \cite{amsler97}. If we choose the energy interval from 
 1350 MeV to 1500 MeV our
 average branching ratios near $f_0(1400)$ for our solution B
 are $\overline{b}_{11}=0.61$ ,
 $\overline{b}_{12}=0.16$  and
 $\overline{b}_{13}=0.23$ .
%compared with experimental values for $f_0(1500)$~\cite{amsler97} in
% Table~\ref{table_br}
%In \cite{abele98} the ratio $r = (B[f_0 \rightarrow
% K\overline{K}]/B[f_0 \rightarrow \pi\pi])\,k_1/k_2 = 0.24 \pm 0.09$
% is calculated (here by $f_0$ we mean $f_0(1500)$). If we define the ratio
% $\overline{b}_{12}/\overline{b}_{11}$ then we obtain values
% 0.20, 0.27 and 0.22 for the solutions
% A, B and E, respectively. These values are close to $r$. From the partial
% decay widths of the $f_0(1500)$ given in \cite{bugg96} one can calculate
% $\Gamma_{K\overline{K}}/\Gamma_{\pi\pi} \approx 0.10 \pm 0.05$ which is
% smaller than $r$ but still consistent within the experimental errors.
We
 know that the extraction of the branching ratios from experiment is a
 difficult task \cite{abele96bis}.
 The average branching ratios depend quite
 sensitively on the energy bin chosen in the actual calculation as
 seen in Fig.~1. 
  In particular  the branching ratio
 $b_{12}$ (\pp$\rightarrow$\kk transition) is very
 small around 1420 MeV, close to the position of our \epw resonance poles.
 This is in qualitative agreement with the small number for the \kk
 branching ratio (3 \%) given in \cite{amsler97}.



%\begin{table}[htb]
%\caption{Averaged branching ratios near \epw for solution B
%compared with experimental values for \fgg [Amsler]}
%\label{table_br}
%\newcommand{\m}{\hphantom{$-$}}
%\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
%\renewcommand{\tabcolsep}{2pc} % enlarge column spacing
%\renewcommand{\arraystretch}{1.2} % enlarge line spacing
%\begin{tabular}{@{}cccc}
%\hline
%decay channel             & Experiment & branching ratio  & solution B \\
%\hline
%$f_0 \rightarrow \pi\pi$  & 0.29 & $\overline b_{11}$ & 0.61 \\
%$f_0 \rightarrow \eta\eta$  & 0.05 & ----- & ------ \\
%$f_0 \rightarrow \eta\eta'$  & 0.01 & ----- & ------ \\
%$f_0 \rightarrow K\overline K$ & 0.03 &  $\overline b_{12}$ & 0.16 \\
%$f_0 \rightarrow 4\pi$ $(\sigma\sigma)$ & 0.62 &  $\overline b_{132}$ & 0.23 \\
%\hline
%\end{tabular}\\[2pt]
%\end{table}


\section{CROSSING SYMMETRY AND CHIRAL CONSTRAINTS}

%Let us recall that at leading order in the chiral expansion the $T$-matrix
%are given by~\cite{donoghue90}
%\be
%T_{11} = \frac{2s-m_{\pi}^2}{32\pi F_{\pi}^2} +O(s^2),
%\label{t11}
%\be
%T_{12} = T_{21} = \frac{\sqrt{3}s}{64\pi F_{\pi}^2} + O(s^2) and
%\eb
%\be
%T_{22} = \frac{3s}{64\pi F_{\pi}^2} + O(s^2).
%\label{tij}
%\eb
%where $m_{\pi}$ is the pion mass, $s$ the total energy square
%and $F_{\pi}^2$ is the pion decay constant.
We have looked for a new
solution fitting the previous \pp and \kk data and satisfying some chiral
constraints
%~\ref{tij}
at the \pp threshold  $s=4m_{\pi}^2$ ~\cite{donoghue90}. We have fixed 2
parameters of our model in such a way that the \pp scattering amplitude 
$T_{11}(4m_{\pi}^2) = 0.21m_{\pi}^{-1}$ and the \kk amplitude
$T_{22}(4m_{\pi}^2) = 0.13m_{\pi}^{-1}$.
%and $T_{12}(4m_{\pi}^2) = 0.072m_{\pi}^-1$.
The first value corresponds to a scalar-isoscalar scattering length $a_0^0$
close to those obtained in two loop calculations in chiral perturbation theory
\cite{bijnens97} and the second, for the \kk, to the leading order
value~\cite{donoghue90}.
%Our model is unitary, however it does not satisfy crossing symmetry.
Our unitary model for the $J=I=0$ amplitudes should be supplemented by a
suitable
parameterization of $J=0, I=2$ and $J=I=1$ waves in order to satisfy some
minimum crossing symmetry properties.
Parameters of our separable potentials can  be constrained in
such a way that the above set of amplitudes satisfies in an approximate way
Roy's equations~\cite{roy71}.
In order to do so we have used the equations
given in ~\cite{basdevant72} with the higher energy and $J \ge$ 2
contributions as
estimated in ~\cite{pennington73}. We have integrated the partial wave
spectral functions up to $s=46m_{\pi}^2$.
 The parameterization given in ~\cite{klr} has been used for the $I=J=1$ wave.
 For the scalar isotensor  wave we
have built a fit to the available phases~\cite{hoogland74}
 as in ~\cite{klr}
but with a rank 2 separable potential imposing a scattering length,
$a_0^2= -0.045m_{\pi}^{-1}$, close to the two loop results of
\cite{bijnens97}. With such a value the new set of the three
amplitudes satisfies better
 Roy's equations as can be seen in Fig.~2.
\begin{figure}[h]

%    \begin{center}
\vspace{12.0 cm}
    
%\xslide{f00.ps}{6.5cm}{40}{183}{511}{632}{10.5cm}
%\xslide{f20.ps}{6.5cm}{40}{183}{511}{632}{10.5cm}
%\xslide{f11.ps}{7.2cm}{40}{127}{511}{632}{10.5cm}
\special{psfile=f00.ps   voffset=130
hoffset=-20  angle=0   vscale=35  hscale=80 }
\special{psfile=f20.ps   voffset=-50
hoffset=-20  angle=0   vscale=35  hscale=80 }

\vspace{-0.4cm}

\caption{ Tests of Roy's equations for $Re f^0_0 (s)$ and $Re f^2_0(s)$
(see text)}
%Comparison of results obtained using Roy equations for
%solutions A and for new solution
%with those coming from relation \ref{real} (see text)}
%    \end{center}
                \label{roy}
  \end{figure}
There we have
compared (for $J=0$ and $I=0,2$)  the real parts of the partial wave
amplitudes $Re f^I_J(s)$ as calculated from
$Re f^I_J = (1/2) \sqrt{s/(s-4)} sin 2 \delta^I_J(s)$
%Eq.~\ref{real}
%\be 
%Re f^I_J = \frac{1}{2}\sqrt{\frac{s}{s-4}} sin 2 \delta^I_J(s)
%\label{real} 
%\eb
for solution A~\cite{kll297} (dash-dot line)  and for the
new solution (solid line) to those given by Roy's equations for the
solution A (short-dashed line) and for the new solution (long-dashed line).
%In the case of $Re f^1_1(s)$  solid and dash-dot lines are identical as we
%use the same $I=J=1$ wave~~\cite{klr} in Eq.~\ref{real}.
%As
%in~\cite{patarakin96} 
We find that if  $a_0^0$ is close to 0.2 $m_{\pi}^{-1}$
then $a_0^2$ should be in the vicinity of -0.04 $m_{\pi}^{-1}$ in order 
to satisfy Roy's equations for
the isoscalar and isotensor waves.
The  $I=J=1$ wave, not shown here, is less sensitive to these values
and does satisfy Roy's equation relatively well.
%Such a
%result has also been obtained by J. Gasser and coworkers~\cite{gasser99}.
This preliminary study 
can be further extended by inclusion of other possible
chiral constraints  such
as those on the transition \pp to \kk.
One can also try to improve treatment of high partial
waves and high energy contributions to Roy's equations.

%which we can
%calculate in part from our model.

%\be 
%Re f^I_J = \frac{1}{2}\sqrt{\frac{s}{s-4}} sin 2 \delta^I_J(s)
%\label{real} 
%\eb
%\be
%Re(f_J^I(s)) = \lambda_J^I(s) + \sum_{I'=0}^2 \sum_{l'=0}^2
%\int_{4m_{\pi^2}}^{50m_{\pi^2}} K_{I,l}^{I',l'}(s,s')Imf_{l'}^{I'}(s')
%+ \phi_J^I(s) 
%\eb
%
%\be
%\lambda_0^0(s) = a_0^0 +\frac{s-4}{12}(2a_0^0 -5a_0^2)
%\eb
%%
%\be
%\lambda_0^2(s) = a_0^2 +\frac{s-4}{24}(2a_0^0 -5a_0^2)
%\eb
%
%\be
%\lambda_1^1(s) = a_0^2 +\frac{s-4}{72}(2a_0^0 -5a_0^2)
%\eb
\vspace{1mm}
{\it Acknowledgments:} We thank B. Moussallam, J. Stern and R. Vinh Mau 
for helpful
discussions. This work has been supported by IN2P3-Polish laboratories
Convention (project No. 99-97). R. Kami\'nski thanks NATO for a grant.

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\end{document}









