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\begin{document}

\title{PION AND KAON VECTOR FORM FACTORS: A CHIRAL UNITARY APPROACH}

\author{J.E. PALOMAR, E. OSET}

\address{Departamento de F\'{\i}sica Te\'orica e IFIC, Centro Mixto
Universidad de Valencia-CSIC, 46100 Burjassot (Valencia), Spain} 

\author{J. A. OLLER}

\address{Forschungszentrum J\"ulich, Institut f\"ur Kernphysik (Theorie),  
  D-52425 J\"ulich, Germany}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle\abstracts{}

\vspace{-0.5cm}

\section{Unitarization}

Using an appropriate unitarization method we take into account the
final state interaction corrections to the tree level amplitudes
 calculated from lowest order $\chi PT$ \cite{GyL} and from the
inclusion of explicit resonance fields in a chiral symmetry fashion as
given in ref. \cite{ecker}. A similar procedure has already been used in the
scalar sector to describe the scalar form factor associated with the
strange-change scalar current $\overline{u}s$ in ref. \cite{jamin}.

We will work in the isospin limit, with $|\pi \pi\rangle$ and $|K \bar{K}\rangle$ 
states (and
the $\rho$ resonance) in the $I=1$ channel, and we will use matrix
notation, labeling pions with $1$ and kaons
 with $2$. In the $I=0$
case we only have kaons (and the
$\omega$ and $\phi$ resonances).
Starting from the unitarity of the $S$-matrix and the introduction
of the electromagnetic meson form factor $F_{MM'}(s)$: 
\be
\label{defiff}
\langle\gamma(q)|T|M(p)M'(p')\rangle=e\epsilon_{\mu}(p-p')^{\mu}F_{MM'}(s) 
\ee
with $q^2=s$, $e$ the modulus of the
electron charge and $\epsilon_{\mu}$ the photon
polarization vector, we can arrive at the expression:

\be 
\label{ff1} 
F^{I}(s) = \left[1+\widetilde{Q}(s)^{-1}\cdot
K^{I}(s) \cdot \widetilde{Q}(s)\cdot g^{I}(s) \right]^{-1}\cdot
R^{I}(s) 
\ee
where $\widetilde{Q}_{ij}(s) = p_{i}(s)\delta_{ij}$ and
$K^{I}(s)$ is the matrix collecting the tree level scattering amplitudes\footnote{Derived from lowest order
$\chi PT$ plus s-channel vector resonance exchange contributions \cite{ecker}
corresponding to the transition $i \rightarrow j$.}
between definite $\pi\pi$ and $K\bar{K}$ isospin states.
$F^{I}(s)$ is the column matrix $F^{I}(s)_{i}=F^{I}_{i}(s)$; 
$R^{I}(s)$ is a vector made up by functions without any cut and 
$g^{I}(s)$ is the diagonal matrix given by the loop with two
meson propagators (see ref. \cite{N/D}).

The expression for the P-wave amplitudes $T^{I}(s)$ that we have used to go
from eq.(\ref{defiff}) to eq.(\ref{ff1}) is 
 the one provided by the $N/D$ method adapted to the chiral framework (see ref. 
\cite{N/D})

In the large $N_{c}$ limit loop physics is supressed. Thus, in this limit 
 $F^{I}(s)=R_{N_{c}^{leading}}(s)=F_{t}^{I}(s)$, where $F_{t}^{I}(s)$ is 
 the tree level form
factor.\footnote{The tree level form factors are also evaluated using 
the lowest order $\chi PT$ Lagrangian \cite{GyL} plus the chiral resonance 
Lagrangian \cite{ecker}.}

This allows us to write:
\begin{equation}
\label{finalff2} 
F^{I}(s)=\left[1+\widetilde{Q}(s)^{-1}\cdot
K^{I}(s) \cdot \widetilde{Q}(s)\cdot g^{I}(s)\right]^{-1} \cdot
\left[ F_{t}^{I}(s)+R^{I}_{subleading}(s) \right]
\end{equation}
with $R^I_{subleading}(s)$ being  ${\mathcal{O}}(N_c^{-1})$. If we require 
that the vector form factor of eq. (\ref{finalff2}) vanishes when
$s\rightarrow \infty$ as is suggested by the experiments, we find
that the subleading part of $R^{I}(s)$, which at
first can be an arbitrary polynomial, must be a constant.
One can fix the values of $d^{I}_{i}$ (parameters appearing in the $g^{I}_{i}$
functions, see ref.\cite{N/D}) and $R^{I=0}_{subleading}(s)$ by
matching our results to those of one loop $\chi PT$, and the bare masses of the
resonances by the requirements that the moduli of the
scattering amplitudes have a maximum for $\sqrt{s}=M_{resonance}^{physical}$. We
take $R^{I=1}_{subleading}(s)=0$ in order to constrain further our approach, and
then all the parameters are fixed.

\section{Results and conclusions.}

If we compare our results with the
experiment we see that we can describe in a very
precise way the pion and kaon vector form factors and the P-wave $\pi\pi$ phase
shifts up to about s=1.44 GeV$^2$ (even for higher energies in the case of phase
shifts). If we compare our results with those of
$\chi PT$, we see that the resummation of our scheme leads to a much better
agreement with the two loop $\chi PT$ pion vector form factor than with the one
loop one. The resonance regions are also well reproduced.
%\begin{figure}[ht]
%\centerline{\includegraphics[width=0.7\textwidth]{pion.ps}}
%\caption{\footnotesize{$W$ is defined as $\sqrt{s}$ for $s>0$ and as
%$-\sqrt{-s}$ for $s<0$. a) $\pi^{+} \pi^{-}$
%vector form factor. b) $\pi\pi$ P-wave phase
%shifts. Both are compared with several experimental data.}}
%\end{figure}
%\begin{figure}[ht]
%\centerline{\includegraphics[width=0.7\textwidth]{paw4proc.ps}}
%\caption{\footnotesize{$W$ is defined as $\sqrt{s}$ for $s>0$ and as $-\sqrt{-s}$ for $s<0$. 
%From left to right and top to bottom: a) Vector pion form factor. The dashed-dotted 
%line represents one loop $\chi PT$ ref. \cite{GyL} and the dashed one the two loop $\chi PT$ 
%result ref. \cite{cola}. b) $K^+ K^-$ form factor. The meaning of the lines is the same as before. 
%c)Vector pion form factor in the $\rho$ region. Data from tau decay.
%d) $K^+K^-$ form factor. All the results are compared with data.}}
%\end{figure}
\vspace{-0.2cm}

\section*{References}
\begin{thebibliography}{200}
\bibitem{GyL} J. Gasser and H. Leutwyler, \NP{B250} (1985) 465, 517,
539.

\bibitem{ecker} G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de
Rafael,
\PL{B223} (1989) 425; G. Ecker, J. Gasser, A. Pich and E. de
Rafael, \NP{B321}
(1989) 311.

\bibitem{jamin} M. Jamin, J. A. Oller and A. Pich, in preparation.

\bibitem{N/D} J. A. Oller and E. Oset, \PR{D60} (1999)
074023.

\end{thebibliography}
\end{document}
