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\begin{document}

\title{$K_{\ell4}$ at two-loops and CHPT predictions for $\pi\pi$-scattering}
%Talk presented at Chiral Dynamics 2000, Jefferson Lab, Newport News,
%Virginia, USA, July 17-22, 2000.

\author{Gabriel Amor\'os, Johan Bijnens}

\address{Department of Theoretical Physics 2, Lund University\\
S\"olvegatan 14A, S 22362 Lund, Sweden}

\author{Pere Talavera}

\address{Groupe de Physique Th\'eorique,
Institut de Physique Nucl\'eaire,\\
Universit\'e Paris Sud,
F-91406 Orsay Cedex, France}

\maketitle

\abstracts{We present the results from our two-loop calculations
of masses, decay-constants, vacuum-expectation-values
and the $K_{\ell4}$ form-factors in three-flavour Chiral Perturbation
Theory (CHPT).
We use this to fit the $L_i^r$ to two-loops and discuss the ensuing
predictions for $\pi\pi$-threshold parameters.} 

%\section{Introduction}

We present the main results of a rather long program of
two-loop calculations in three flavour CHPT in the meson sector.
%For an introduction to CHPT we refer to the plenary talks by
%J.~Gasser and H.~Leutwyler.
The Lagrangian at lowest order $p^2$ has two-parameters and two quark masses.
In practice, in the isospin limit, this leads to $F$, $B_0 \hat{m}$
and $B_0 {m_s}$ as the three free parameters. At order $p^4$
there are an additional\cite{GL1} 10 and at order $p^6$ there are\cite{BCElag}
90. An overview of the calculations at two-loops in
the two-flavour sector can be found in \cite{Mainz}, presented in the
previous Chiral Dynamics meeting. Since then the 
$\pi\pi$ scattering\cite{pipi},
the vector and scalar form-factor\cite{BCT}
and the expressions in terms of the $p^6$ low-energy
constants have been published\cite{BCEinf}. In the three-flavour
sector the generalized double-log contributions were found to
be large\cite{BCEdbl}. This prompted us to start our
program so a major refit of CHPT parameters to two-loop order would be
possible. Other three flavour $p^6$ work is in Ref.\cite{Other}.

%\section{Calculations and fits}

In Ref.\cite{ABT1} we calculated the $VV$ and $AA$ two-point functions
to two-loop order and in Refs.\cite{ABT2,ABT3} the $K_{\ell4}$
form-factors and the vacuum-expectation-values. Future work is concentrating
on isospin violating effects.

There is a large number of parameters
that needs determination. We have been forced to use assumptions to
estimate the $p^6$ constants but checked that varying them within a factor
of two didn't change results significantly. In addition we
used $L_4^r=L_6^r=0$ and $m_s/\hat{m}=24$ as theoretical input.
Experimental input is $F_\pi$, $F_K/F_\pi$, $M_\pi$, $M_K$,
$M_\eta$ and from $K_{\ell4}$, $F(0)$, $G(0)$ and the slope $\lambda$.
The resulting fit for the $L_i^r$ is quite stable. 
Fit results without the $\eta^\prime$ contribution in the $p^6$ estimate
are in \cite{ABT2} and with it in \cite{ABT3}.
The main results and the variation with some of the input
are shown in Fig. \ref{tabLi}. Notice the variation with $K_{\ell4}$ input
in fit 9. 
We look forward to more $K_{\ell4}$ data that would add more
precision, especially in the value of the slopes for $F$ and $G$, to have more
redundancy in the input.
The excellent
prediction for $\pi\pi$ is shown with data,
including the new BLN data\cite{Zeller}, in Fig.~\ref{figpipi}.
\begin{figure}
\footnotesize
\begin{minipage}{0.61\textwidth}
\caption{The fit of $L_i^r$ described in the text.
We have shown some of the input variations used.
\label{tabLi}}
\begin{tabular}{|ccccc|}
\hline
		& Main Fit &fit 2&
  fit 8 & fit 9\\
\hline
$10^3\,L^r_1$&   0.52$\pm$0.23&   0.53&
   0.63&   0.65\\	       
$10^3\,L^r_2$&   0.72$\pm$0.24&   0.73&
   0.73&   0.85\\      
$10^3\,L^r_3$&$-$2.70$\pm$0.99&$-$2.71&
$-$2.67&$-$3.27\\	       
$10^3\,L^r_5$&   0.65$\pm$0.12&   0.62&
   0.51&   0.60\\ 	       
$10^3\,L^r_7$&$-$0.26$\pm$0.15&$-$0.20&
$-$0.25&$-$0.26\\	       
$10^3\,L^r_8$&   0.47$\pm$0.18&   0.35&
   0.44&   0.48\\
\hline	     
changed      & &$m_s/\hat m$&
$\mu$&$g(0)$\\ 
quantity     &          & 26  
&1.0&4.93\\
\hline
\end{tabular}
\end{minipage}
\begin{minipage}{0.39\textwidth}
\caption{The results for $\pi\pi$ scattering and
 the new data  (Pislak).\label{figpipi}}
\epsfxsize=\textwidth
\epsfbox{bijnens.pipi.eps} 
\end{minipage}
\end{figure}

\section*{\vskip-0.7cm Acknowledgments\vskip-0.2cm}

Work partially supported by the EU TMR Network
EURODAPHNE (Contract No. ERBFMX-CT98-0169) and by the Swedish Science
Foundation.


\begin{thebibliography}{99}

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J.~Gasser and H.~Leutwyler,
%``Chiral Perturbation Theory: Expansions In The Mass Of The Strange Quark,''
Nucl.\ Phys.\  {\bf B250} (1985) 465.
%%CITATION = NUPHA,B250,465;%%

\bibitem{BCElag}
J.~Bijnens et al., % G.~Colangelo and G.~Ecker,
%``The mesonic chiral Lagrangian of order p**6,''
JHEP {\bf 9902} (1999) 020
[hep-ph/9902437].
%%CITATION = HEP-PH 9902437;%%

\bibitem{Mainz}
J.~Bijnens,
%``Goldstone boson production and decay,''
in ``Mainz 1997, Chiral dynamics: Theory and experiment,
eds. A.~Bernstein, D.~Drechsel and T.~Walcher ``,
hep-ph/9710341.
%%CITATION = HEP-PH 9710341;%%

\bibitem{pipi}
J.~Bijnens et al., % G.~Colangelo, G.~Ecker, J.~Gasser and M.~E.~Sainio,
%``Pion pion scattering at low energy,''
Nucl.\ Phys.\  {\bf B508} (1997) 263
[hep-ph/9707291].
%%CITATION = HEP-PH 9707291;%%
%J.~Bijnens, G.~Colangelo, G.~Ecker, J.~Gasser and M.~E.~Sainio,
%``Elastic $\pi\pi$ scattering to two loops,''
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[hep-ph/9511397].
%%CITATION = HEP-PH 9511397;%%

\bibitem{BCT}
J.~Bijnens et al., % G.~Colangelo and P.~Talavera,
%``The vector and scalar form factors of the pion to two loops,''
JHEP {\bf 9805} (1998) 014
[hep-ph/9805389].
%%CITATION = HEP-PH 9805389;%%

\bibitem{BCEinf}
J.~Bijnens et al., % G.~Colangelo and G.~Ecker,
%``Renormalization of chiral perturbation theory to order p**6,''
Annals Phys.\  {\bf 280} (2000) 100
[hep-ph/9907333].
%%CITATION = HEP-PH 9907333;%%

\bibitem{BCEdbl}
J.~Bijnens et al., % G.~Colangelo and G.~Ecker,
%``Double chiral logs,''
Phys.\ Lett.\  {\bf B441} (1998) 437
[hep-ph/9808421].
%%CITATION = HEP-PH 9808421;%%

\bibitem{ABT1}
G.~Amoros et al., % J.~Bijnens and P.~Talavera,
%``Two-point functions at two loops in three flavour
%chiral perturbation  theory,''
Nucl.\ Phys.\  {\bf B568} (2000) 319
[hep-ph/9907264].
%%CITATION = HEP-PH 9907264;%%

\bibitem{ABT2}
G.~Amoros et al., % J.~Bijnens and P.~Talavera,
%``Low energy constants from K(l4) form-factors,''
Phys.\ Lett.\  {\bf B480} (2000) 71
[hep-ph/9912398].
%%CITATION = HEP-PH 9912398;%%


\bibitem{ABT3}
G.~Amoros et al., % J.~Bijnens and P.~Talavera,
%``K(l4) form-factors and pi pi scattering,''
Nucl.\ Phys.\  {\bf B585} (2000) 293
[hep-ph/0003258].
%%CITATION = HEP-PH 0003258;%%

\bibitem{Zeller} M.~Zeller, these proceedings.

\bibitem{Other}
B.~Moussallam,
%``Flavor stability of the chiral vacuum and scalar meson dynamics,''
JHEP {\bf 0008} (2000) 005
[hep-ph/0005245];
%%CITATION = HEP-PH 0005245;%%
E.~Golowich and J.~Kambor,
%``Two-loop analysis of axialvector current propagators in chiral  perturbation theory,''
Phys.\ Rev.\  {\bf D58} (1998) 036004
[hep-ph/9710214],
%%CITATION = HEP-PH 9710214;%%
%\bibitem{Golowich:1995kd}
%E.~Golowich and J.~Kambor,
%``Two loop analysis of vector current propagators in chiral perturbation theory,''
Nucl.\ Phys.\  {\bf B447} (1995) 373
[hep-ph/9501318].
%%CITATION = HEP-PH 9501318;%%
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%``Higher-order corrections to Sirlin's theorem in O(p**6) chiral  perturbation theory,''
Phys.\ Rev.\ Lett.\  {\bf 79} (1997) 4088
[hep-ph/9701422];
%%CITATION = HEP-PH 9701422;%%
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Phys.\ Rev.\  {\bf D61} (2000) 114025
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%%CITATION = HEP-PH 9907539;%%



\end{thebibliography}
\end{document}


