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\begin{document}

\title{A Chiral Lagrangian Framework for Scalar Mesons}

\author{Deirdre Black, Amir H. Fariborz$^{*,}$ \footnote{Speaker} and Joseph
Schechter}

\address{Department of Physics, Syracuse University, Syracuse, NY 13244-1130.} 

\address{$^*$Department of Applied Mathematics, State University of New York
Institute of Technology, Utica, NY 13504-3050.}

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\maketitle\abstracts{The experimental and theoretical status of the light scalar
mesons is summarized, and the main aspects of their inclusion into a non-linear
chiral Lagrangian framework of refs. \cite{San,Blk,Far,Mec} is briefly discussed.}


Scalar mesons play important roles in low-energy QCD, and are at the focus of many
theoretical and experimental investigations.  Scalars are important from the
theoretical point of view because they are Higgs bosons of QCD and induce chiral
symmetry breaking, and therefore, are probes of the QCD vacuum.  Scalars are also
important from a phenomenological point of view, as they are very important
intermediate states in Goldstone boson interactions away from threshold, where
chiral perturbation theory is not applicable. There are 9 candidates for the
lowest-lying scalar mesons ($ m < 1$ GeV): $f_0(980)$ [I=0] and $a_0(980)$ [I=1]
which are well established experimentally \cite{PDG};  $\sigma (560)$ or
$f_0(400-1200)$ [I=0] with uncertain mass and decay width \cite{PDG}; and
$\kappa(900)$ [I=1/2] which is not listed but mentioned, in PDG 2000 \cite{PDG}.  
The $\kappa(900)$ is only observed in some theoretical models.  It is known that a
simple $q {\bar q}$ picture does not explain the properties of these mesons.  
Different theoretical models that go beyond a simple $q {\bar q}$ picture have been
developed, including: MIT bag model \cite{Jaf}, $K {\bar K}$ molecule \cite{Isg},
unitarized quark model \cite{Tor}, QCD sum-rules \cite{Eli}, and chiral Lagrangians
\cite{San,Blk,Far,Mec}.



Chiral Lagrangians, the main focus of this article, provide a powerful framework for
studying scalars probed in different Goldstone boson interactions ($\pi\pi$, $\pi
K$, $\pi\eta$,...) away from threshold \cite{San,Blk,Far,Mec}. In this approach, a
description of
scattering amplitudes which are, to a good approximation, both crossing symmetric
and unitary is possible.  To construct scattering amplitudes, all contributing
intermediate resonances up to the energy of interest are considered, and only tree
diagrams (motivated by large $N_c$ approximation) are taken into account.  In this
way, crossing symmetry is satisfied, but the constructed amplitudes should be
regularized.  The regularization procedure in turn unitarizes the scattering
amplitude.  
Finally, by fitting the resulting scattering amplitude to experimental data, the
unknown physical properties (mass, decay width, ...) of the light scalar mesons can
be extracted. A $\sigma(560)$ and a $\kappa(900)$ are required in order to describe
experimental data on $\pi\pi$ and $\pi K$ scattering amplitudes, respectively.
Therefore, this approach suggests the existence of a scalar meson nonet, $N$, below
1 GeV (containing $\sigma(560)$, $f_0(980)$, $\kappa(900)$, and $a_0(980)$).  
Rewriting the Lagrangian in terms of $N$ involves terms that mix the two isosinglet
combinations $(N_1^1+N_2^2)/\sqrt{2}$ and $N_3^3$.  This mixing can be parametrized
in terms of an a priori unknown scalar mixing angle, the value of which can be
determined by requiring consistency with different available experimental data.  It
is shown in \cite{Blk} that 
\[ \left( \begin{array}{c} \sigma \\ f_0 \end{array}
\right) = \left( \begin{array}{c c} \sim 0.94 & \sim -0.34 \\ \sim 0.34 & \sim 0.94
\end{array} \right) \left( \begin{array}{c} N_3^3 \\ \frac { N_1^1 + N_2^2}{\sqrt 2}
\end{array} \right) \] 
This is  closer to a 4-quark assignment for the lowest-lying scalars, and
therefore, agrees with a non $q \bar q$ nature of these light scalars. 


The next-to-lowest scalars (1 GeV $< m <$ 2 GeV) are: $K_0^*(1430)$ [I=1/2],
$a_0(1450)$ [I=1], $f_0(1370)$, $f_0(1500)$, $f_J(1710)$ [I=0], and are all listed
in \cite{PDG}. These states are generally believed to be closer to $q {\bar q}$
objects, except $f_0(1500)$ which is believed to contain a large glue component and
therefore a good candidate for the lowest scalar glueball state. Within the chiral
Lagrangian framework described above, the properties of the next-to-lowest-lying
scalar nonet which includes $K_0^*(1430)$ [I=1/2] and $a_0(1450)$ [I=1] can also be
studied. Some puzzling features on their mass and decay widths can be explained
based on a mixing mechanism between underlying $q{\bar q}$ and $qq{\bar q}{\bar q}$
nonets \cite{Mec}.


%\section{}
\begin{thebibliography}{99}

\bibitem{San}
F. Sannino and J. Schechter, 
\Journal{\PRD}{52}{96}{1995}, and
M. Harada, F. Sannino and J. Schechter, 
\Journal{\PRD}{54}{1991}{1996}; \Journal{\PRL}{78}{1603}{1997}.

\bibitem{Blk}D. Black, A.H. Fariborz, F. Sannino and J. Schechter,
\Journal{\PRD}{58}{054012}{1998}; \Journal{\PRD}{59}{074026}{1999}.

\bibitem{Far}A.H. Fariborz and J. Schechter, \Journal{\PRD}{60}{034002}
{1999}.

\bibitem{Mec}D. Black, A.H. Fariborz and J. Schechter,
\Journal{\PRD}{61}{074001}{2000}; \Journal{\PRD}{61}{074030}{2000}.

\bibitem{PDG}
Review of Particle Physics, 
\Journal{\EJPC}{15}{1}{2000}.

\bibitem{Jaf}
R.L. Jaffe, 
\Journal{\PRD}{15}{267}{1977}.

\bibitem{Isg}
J. Weinstein and N. Isgur, 
\Journal{\PRD}{41}{2236}{1990}.

\bibitem{Tor}
N.A. T\"ornqvist, 
\Journal{\ZPC}{68}{647}{1995}; E. van Beveren et al, 
\Journal{\ZPC}{30}{615}{1986}.

\bibitem{Eli}
V.Elias et al,
\Journal{\NPA}{633}{279}{1998}; 
\Journal{\NPA}{671}{416}{2000}.



\end{thebibliography}
\end{document}
