%%%%%%%%%% espcrc2.tex %%%%%%%%%%
%
% $Id: espcrc2.tex,v 1.2 2004/02/24 11:22:11 spepping Exp $
%
\documentclass[fleqn,twoside]{article}
% \usepackage{espcrc2}
% Use the option 'headings' if you want running headings
\usepackage[headings]{espcrc2}

% identification
\readRCS
$Id: espcrc2.tex,v 1.2 2004/02/24 11:22:11 spepping Exp $
\ProvidesFile{espcrc2.tex}[\filedate \space v\fileversion
     \space Elsevier 2-column CRC Author Instructions]

% change this to the following line for use with LaTeX2.09
% \documentstyle[twoside,fleqn,espcrc2]{article}

% if you want to include PostScript figures
\usepackage{graphicx}
% if you have landscape tables
\usepackage[figuresright]{rotating}

% put your own definitions here:
%   \newcommand{\cZ}{\cal{Z}}
%   \newtheorem{def}{Definition}[section]
%   ...
\newcommand{\ttbs}{\char'134}
\newcommand{\AmS}{{\protect\the\textfont2
  A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}
\newcommand{\mrm}{\mathrm}
\newcommand{\pppm}{{\rm\pi^{+}\pi^{-}}}
\newcommand{\pzpz}{{\rm\pi^{0}\pi^{0}}}
\newcommand{\eoe}{\mbox{${\rm Re}(\epsi'/\epsi)$}~}
\newcommand{\epsi}{\varepsilon}
\newcommand{\kl}{{\rm K_{L}}}
\newcommand{\ks}{{\rm K_{S}}}
\newcommand{\kp}{{\rm K^{+}}}
\newcommand{\km}{{\rm K^{-}}}
\newcommand{\kz}{\mbox{$\rm K^{0}$}~}
\newcommand{\kzb}{\mbox{$\rm \bar{K}^{0}$}~}
\newcommand{\kpm}{{\rm K^{\pm}}}
\newcommand{\ppm}{{\rm \pi^{\pm}}}
\newcommand{\pmp}{{\rm \pi^{\mp}}}
\newcommand{\epm}{{\rm e^{\pm}}}
\newcommand{\pim}{{\rm\pi^{-}}}
\newcommand{\pip}{{\rm\pi^{+}}}
\newcommand{\piz}{{\rm\pi^{0}}}

% add words to TeX's hyphenation exception list
\hyphenation{author another created financial paper re-commend-ed Post-Script}

% set the starting page if not 1
% \setcounter{page}{17}

% declarations for front matter
\title{Review of Chiral Perturbation Theory experimental
	results in kaon decays}

\author{R. Fantechi\address{INFN - Sezione di Pisa}
        \\NA48 Collaboration} 
       
% If you use the option headings,
% the title is also used as the running title,
% and the authors are also used as the running authors.
% You can change that by using \runtitle and \runauthor.


\begin{document}

\begin{abstract}
\vspace{1pc}
Chiral Perturbation Theory, as a low-energy effective field theory of
strong interactions, is a very useful tool to study the interactions of kaons.
Both in the semileptonic sector and in the non-leptonic one, several results
have been achieved by NA48, KTeV, KLOE and ISTRA+. Recent results from both 
neutral and charged kaons are reviewed.
\end{abstract}

% typeset front matter (including abstract)
\maketitle

\section{Chiral Perturbation Theory}
Chiral Perturbation Theory \cite{chpt} is the effective quantum 
field theory for hadronic 
interactions below the breaking scale of the chiral symmetry ($\approx$ 1 GeV).
At these energies, the spectrum of the theory is simple and it consists only
of the octet of light pseudoscalar mesons $\pi$, K and $\eta$.
$\chi$PT is now a mature technique able to provide high precision predictions for
the mesonic sector, for instance in K decays.

\section{Active kaon decay experiments}
Several experiments around the world are studying kaon decays, being able to
do accurate measurements to be compared with $\chi$PT predictions. 
Each of them has its own peculiar characteristics which provide complementary 
measurements. NA48 at CERN combines the use of high energy neutral and charged 
kaon beams with precision
measurement of photon energies. KLOE in Frascati is complementing high energy
measurement with low-energy, tagged kaon decays. KTeV at FNAL has collected
the greatest sample of $\kl$ decays in the world for rare decay studies. ISTRA+
in Protvino uses intermediate energy negative kaon beams to study rare decays.

\section{Radiative decays}


\subsection{$\kl \rightarrow \gamma \gamma$}

In this decay O(p$^4$) contributions vanishes and O(p$^6$) ones mediated by 
pseudoscalar mesons depend on the value of singlet-octet mixing \cite{klggth}.
NA48 has performed a measurement of the branching ratio of 
$\kl \rightarrow \gamma \gamma$ normalized to $\kl \rightarrow 3\piz$ using
$\kl$ data collected in 2000. KLOE has made the same measurement with a 
sample of $\approx 1.6\cdot10^8 \kl$ tagged by $\ks \rightarrow \pip \pim$.

The results from NA48 \cite{klsgg} and KLOE \cite{kloeklgg} are respectively:
\begin{center}
$\Gamma( \kl \rightarrow \gamma \gamma)/\Gamma(\kl \rightarrow 3\piz)$ =
(2.81 $\pm$ 0.01$_{st} \pm$ 0.02$_{sy}$)$\cdot$10$^{-3}$
\end{center}

\begin{center}
$\Gamma( \kl \rightarrow \gamma \gamma)/\Gamma(\kl \rightarrow 3\piz)$ =
(2.79 $\pm$ 0.02$_{st} \pm$ 0.02$_{sy}$)$\cdot$10$^{-3}$
\end{center}

The corresponding decay width agrees with O(p$^4$) predictions, provided that 
the mixing angle is close to the value, measured by KLOE, of 
$-12.9^{+1.9}_{-1.6}$ degrees.

\subsection{$\ks \rightarrow \gamma \gamma$}
$\ks \rightarrow \gamma \gamma$ is interesting because it is calculable in
O(p$^4$) $\chi$PT with no counterterms and it is sensitive to loops 
\cite{ksgg1}\cite{ksgg2}\cite{ksgg3}. 
The theoretical
prediction for the BR is (2.1 $\pm$ 0.2)$\cdot$10$^{-6}$. 
Previous measurements by NA31 \cite{ksggna31} and NA48 \cite{ksggres}
gave respectively

\begin{center} 
BR($\ks \rightarrow \gamma \gamma$) = (2.4 $\pm$ 0.9)$\cdot$10$^{-6}$


BR($\ks \rightarrow \gamma \gamma$)=
(2.58 $\pm$ 0.36$_{st}$ $\pm$ 0.22$_{sy}$)$\cdot$10$^{-6}$.
\end{center} 

A better measurement has been performed by NA48 using data 
from the 2000 run. The normalisation channel is $\ks \rightarrow \piz \piz$.


The subtraction of the irreducible background from 
$\kl \rightarrow \gamma \gamma$, which amounts to 1.5 times the $\ks$ channel,
is done using the NA48 measurement of the $\kl \rightarrow \gamma \gamma$ 
branching ratio quoted above.

With a total number of 19916 K$_S$,$_L \rightarrow \gamma \gamma$ events, the
result is \cite{klsgg} 
\begin{center}
BR($\ks \rightarrow \gamma \gamma$) = 
(2.78 $\pm$ 0.06$_{st} \pm$ 0.03$_{sy} \pm$ 0.02$_{ext}$)$\cdot$10$^{-6}$
\end{center}
This result is compatible with previous measurements (Fig. \ref{fig1}), 
but it shows a 30\%
difference wrt to O(p$^4$) $\chi$PT predictions, indicating a possible
large O(p$^6$) contribution.
\begin{figure}
\begin{center}
%\begin{minipage}{0.45\textwidth}
\resizebox{0.45\textwidth}{!}{%
  \includegraphics{ksgg-comp.eps}
}
%\end{minipage}
\end{center}
\vspace*{-10mm}\caption{\it $\ks \rightarrow \gamma \gamma$: Theoretical predictions and measured values}\vspace*{-4mm}
\label{fig1}
\end{figure}


\subsection{$\ks \rightarrow \piz \gamma \gamma$}
The study of $\ks \rightarrow \piz \gamma \gamma$, as in other radiative 
non-leptonic kaon decays, is useful to check $\chi$PT predictions for the 
low energy hadron dynamics. In the decay $\ks \rightarrow \piz \gamma \gamma$
the photon pair is produced by a pseudo-scalar meson. A measurement of the 
branching ratio can give information on possible extra non-pole contributions
\cite{ksp0gg1}. The prediction for the branching ratio is limited to the 
kinematical region $z=m^2_{\gamma\gamma}/m^2_K > 0.2$ and it is  
$3.8\cdot10^{-8}$ \cite{ksp0gg2}. Additional information from the 
z spectrum can determine, if sufficient statistics is available, the 
momentum dependence of the weak vertex.

Using data from the 1999 $\ks$ run, NA48 has published \cite{ksp0gg0} the 
upper limit  
\begin{center} 
BR($\ks \rightarrow \piz \gamma \gamma$)$_{z>0.2} < 3.3\cdot
10^{-7}$ 
\end{center} 
at 90\% confidence level.

Data has been collected by NA48 in the 2000 run using the same trigger for
$\ks \rightarrow \piz \piz$ and $\ks \rightarrow \piz \gamma \gamma$, so that
many systematic uncertanties cancel out.


After the selection the final sample consists of 31 events, 
with a background estimation is
13.7$\pm$3.2 events, half of which  due to pile-up accidentals. 
17.3$\pm$6.4 signal events are obtained
after background subtraction, which is the first observation of this decay. 
Using a normalization sample of about 285$\cdot$10$^3$ events, one gets
\begin{center}
$\Gamma( \ks \rightarrow \piz \gamma \gamma)_{z>0.2}/\Gamma(\ks \rightarrow \piz \piz)$ =
(1.57 $\pm$ 0.51$_{st} \pm$ 0.29$_{sy}$)$\cdot$10$^{-7}$
\end{center}
Using 
$BR(\ks \rightarrow \piz \piz)$ from PDG one obtains \cite{ksp0gg3}
\begin{center}
$BR(\ks \rightarrow \piz \gamma \gamma)_{z>0.2}$=
(4.9 $\pm$ 1.6$_{st}$ $\pm$ 0.9$_{sy}$)$\cdot$10$^{-8}$=(4.9 $\pm$ 1.8)$\cdot$10$^{-8}$
\end{center}
in agreement with \cite{ksp0gg2}.

The z distribution of the events after background subtraction has been 
compared with z distributions of simulated events using two different decay
generators, one with $\chi$PT matrix element and the other with pure 
phase space (Fig. \ref{ksp0gg2}). The experimental data agree with both 
calculations and none of the two predictions can be preferred.
\begin{figure}[ht]
\begin{center}
\begin{minipage}{0.46\textwidth}
\resizebox{0.75\textwidth}{!}{%
  \includegraphics{m34-pap.eps}
}
\end{minipage}
\end{center}
\vspace*{-10mm}\caption{\it $\ks \rightarrow \piz \gamma \gamma$: z$_q$ distribution for data and two theoretical models}\vspace*{-4mm}
\label{ksp0gg2}
\end{figure}

\subsection{$\kl \rightarrow \piz \gamma \gamma$}
$\kl \rightarrow \piz \gamma \gamma$ decay is important because the decay 
rate prediction from O(p$^4$) $\chi$PT is finite, but 2-3 times 
smaller than the 
measured one \cite{ksp0gg2}.
Going to  O(p$^6$) and including vector meson exchange, the 
measured rate is reproduced and a tail at low m$_{\gamma \gamma}$ is 
predicted \cite{klp0gg1}.
The vector meson dominance contribution is parametrized by 
a$_v$ \cite{ksgg1}\cite{klp0gg2}, 
to be measured, and the low m$_{\gamma \gamma}$ tail determines 
the CPC amplitude to the decay 
$\kl \rightarrow \piz e^+ e^-$ \cite{klp0gg3}. 
Predictions from $\chi$PT are 
BR($\kl \rightarrow \piz \gamma \gamma$) = $1.5\cdot10^{-6}$ and a$_v$ = -0.7.

Data from 1998 and 1999 NA48 runs were selected using the same trigger as 
the normalization channel $\kl \rightarrow \piz \piz$ 
to cancel out many systematic uncertainties.


\begin{figure}[ht]
\begin{center}
\begin{minipage}[c]{0.24\textwidth}
\resizebox{0.95\textwidth}{!}{%
  \includegraphics{back12.eps}
}
\end{minipage}%
\begin{minipage}[c]{0.24\textwidth}
\resizebox{0.95\textwidth}{!}{%
  \includegraphics{back4.eps}
}
\end{minipage}
\end{center}
\vspace*{-10mm}\caption{\it $\kl \rightarrow \piz \gamma \gamma$: Events in the signal region and m$_{\gamma \gamma}$ distribution}\vspace*{-4mm}
\label{klpi0gg2}
\end{figure}

$\approx$ 2500 events were found in the signal region 
132 MeV $<$ M$_{\piz}$ $<$ 138 MeV \cite{klpres}. Fig. \ref{klpi0gg2}
shows the distribution of events in the signal region and of the invariant mass
of the two photons.
Fitting the M$_{\gamma \gamma}$ distribution,
one obtains 
\begin{center}
a$_v$ = -0.46 $\pm$ 0.03$_{st}$ $\pm$ 0.03$_{sy}$$\pm$ 0.02$_{th}$
\end{center}
Using the fitted a$_v$ value, the branching ratio  is 
\begin{center}
BR($\kl \rightarrow \piz \gamma \gamma$)
= (1.36 $\pm$ 0.03$_{st}$ $\pm$ 0.03$_{sy}$$\pm$ 0.03$_{norm}$)$\cdot 10^{-6}$.
\end{center}
and the CP conserving part of the $\kl \rightarrow \piz e^+ e^-$  decay is
\begin{center}
BR($\kl \rightarrow \piz e^+ e^-$)$_{CPC}$ = (4.7 $\pm$ 2.2)$\cdot$ 10$^{-13}$
\end{center}

KTev has obtained with a partial data sample \cite{ktevp0gg}
\begin{center}
BR($\kl \rightarrow \piz \gamma \gamma$)
= (1.68 $\pm$ 0.10)$\cdot 10^{-6}$ and a$_v$ = -0.72 $\pm$ 0.08.
\end{center} 
They plan to complete the analysis with the full data sample.
 
\subsection{$\kl \rightarrow \pip \pim \piz \gamma$ and 
$\kl \rightarrow \pip \pim \piz e^+ e^-$}

Up to now there were no published results on these decays.
$\kl \rightarrow \pip \pim \piz \gamma$ is expected to be dominated by 
inner bremsstrahlung \cite{ktev1} with a BR(E$_{\gamma}>$10 MeV) = 
(1.65 $\pm$ 0.03)$\cdot$ 10$^{-4}$. Direct emission is estimated to be 
very small \cite{ktev2}. Instead, no theoretical predictions for 
$\kl \rightarrow \pip \pim \piz e^+ e^-$ are available.
KTeV have observed for the first time these decays from the analysis of 
40\% of their large $\kl$ sample.

For $\kl \rightarrow \pip \pim \piz \gamma$ they have obtained
2853 candidates, normalized to  $\kl \rightarrow \pip \pim \piz $. The final
branching ratio is 
BR(E$_{\gamma} >$ 10 MeV) = 
(1.70 $\pm$ 0.03$_{st}$ $\pm$ 0.04$_{st}$ $\pm$ 0.03$_{norm}$)$\cdot$10$^{-4}$.

For $\kl \rightarrow \pip \pim \piz e^+ e^-$,
132 candidates, normalized to  $\kl \rightarrow \pip \pim \piz_D$,
with an estimated background of 1.2 $\pm$ 0.9 events, give a branching 
ratio of BR(E$_{ee}>$20 MeV) = (1.60 $\pm$ 0.18$_{st}$)$\cdot$ 10$^{-7}$.

The KTeV Collaboration plans to measure Direct Emission and Charge radius 
in the near future with their full dataset.


\subsection{$\eta \rightarrow \piz \gamma \gamma$}

The decay $\eta \rightarrow \piz \gamma \gamma$ has been extensively studied by
theorists and many predictions within $\chi$PT has been 
published \cite{etath}. 
All span in the interval 0.11 eV $< \Gamma <$ 0.92 eV, which translates in 
a value for the branching ratio between 8.5 and 71  in units of 10$^{-5}$.

KLOE has measured this branching ratio using 
68 $\pm$ 23 signal events, normalized to $\eta \rightarrow \piz \piz \piz$.

The measured branching ratio \cite{etap0gg} is 
BR($\eta \rightarrow \piz \gamma \gamma$) = 
(8.4 $\pm$ 2.7$_{st}$ $\pm$ 1.4$_{sy})\cdot10^{-5}$, which is in 
agreement with O(p$^6$) calculations  with VMD resonance saturation and 
also with Belkov NJL.

\subsection{$\kpm \rightarrow \ppm \gamma \gamma$}
The decay $\kpm \rightarrow \ppm \gamma \gamma$ has no tree O(p$^2$) 
contribution; the leading contribution is at order  O(p$^4$) with an
undetermined constant \^c of order 1 \cite{kpiggth}(Fig. \ref{kpiggth}). 
\begin{figure}
\begin{center}
\begin{minipage}{0.46\textwidth}
\resizebox{0.75\textwidth}{!}{%
  \includegraphics{mgg_theory.eps}
}
\end{minipage}
\end{center}
\vspace*{-10mm}\caption{\it $\kpm \rightarrow \ppm \gamma \gamma$: theoretical predictions of the m$_{\gamma\gamma}$ distribution 
for two different values of \^c}\vspace*{-4mm}
\label{kpiggth}
\end{figure}
O(p$^6$) corrections to the branching 
ratio are expected to be around 30-40 \%. Different models give a branching 
ratio of about $7\cdot10^{-7}$.
The E787 experiment at BNL observed this decay for the first time \cite{kpigg},
with $5.1 \pm 3.3$ events in the range 100 MeV/c $< P_{\pi} <$ 180 MeV/c and a 
branching ratio of $(6.0 \pm 1.5_{st} \pm 0.7_{sy})\cdot10^{-7}$. NA48 
is performing an analysis of this decay with data collected in 2003 and in
2004. A first sample from 2003 consists of about 2000 events with very small
backgrounds. 
There will be at least as many events in 2004 data.


\subsection{$\kp \rightarrow \pip e^+ e^-$}
The form factor in the amplitude for this decay contains two parameters 
$a_i, b_i$ to parametrize respectively contributions at O(p$^4$) 
and O(p$^6$) \cite{kpieeth}.
Data from E865 at BNL \cite{kpiee} ($\approx$ 10000 events with 1.2\% 
background) have been used to fit $a_i, b_i$ finding $a_+ = -0.587 \pm 0.010, 
b_+ = -0.655 \pm 0.044$ and a branching ratio of 
$(2.94 \pm 0.05_{st} \pm 0.13_{sy} \pm 0.05_{model})\cdot10^{-7}$. All these
results are consistent with $\chi$PT predictions.
NA48 is doing an analysis of this decay: the sample from 2003 data contains
about 4000 events, 
which will at least double with the addition of 2004 data.
The background is less than 1\%.

\section{Radiative semileptonic decays}
$\kl$ and $\kpm$ radiative semileptonic decays give information on the kaon 
structure. The ratio $R = \frac{\Gamma(K_{e3\gamma},E^*_{\gamma}>30 MeV,
\theta^*_{e\gamma}>20^0)}{\Gamma(K_{e3})}$ is predicted to be between 0.95\% 
and 0.99\%. Two different theoretical approaches are used: 
current algebra \cite{krads1} and $\chi$PT \cite{krads2}. 
Both basically agree and the latter is continuosly improving
accuracy: the latest estimate is $(0.96 \pm 0.01)$. A recent measurement from
KTeV \cite{krads3} gives R=$(0.908 \pm 0.008^{+0.013}_{-0.012})\%$, 
in disagreement with the predictions.
NA48 has analyzed the decay $\kl \rightarrow \ppm e^{\mp} \nu \gamma$ finding
18977 candidates and 6 million $\kl \rightarrow \ppm e^{\mp} \nu$ 
normalization events. The background is less than 1\%. 
The result \cite{krads4}, in full 
agreement with the predictions, is R=$(0.964 \pm 0.008^{+0.011}_{-0.009})\%$.
An analysis is in progress by NA48 on the $\kpm$ radiative semileptonic decays.
On this subject, it should be noted that radiative  $\km$ semileptonic 
decays are studied also at ISTRA+, as it has been presented in this 
Conference \cite{duk}.

\section{K$_{e4}$}
The K$_{e4}$ decays ($\kl \rightarrow \pi^{\pm} \piz e^{\mp} \nu_e$,
$\kpm \rightarrow \pi^{\pm} \pim e^{\mp} \nu_e$, 
$\kpm \rightarrow \piz \piz e^{\pm} \nu_e$) are known to be a good test
of ChPT predictions for long-distance meson interactions. They are used to
determine the  $\pi \pi$ partial wave expansion parameters, like threshold
parameters, slopes and scattering lengths, which can be further related to
the quark condensate \cite{ke4th1}. 
The complete set of the parameters has been computed
at order O(p$^4$) and the form factors at O(p$^6$) \cite{ke4th2}.
NA48 has studied all these three decays. The report of the results for the 
$\kpm$ decays have been presented in this Conference \cite{anzivino}. 
For the $\kl$ decay,
NA48 has got 5500 events with $\approx 1\%$ of background, obtaining the best
measurement of the branching ratio up to now \cite{kle4} (Fig. \ref{ke4}):

\begin{center}
BR($\kl\rightarrow\pi^{\pm} \piz e^{\mp} \nu_e$)=
(5.21$\pm$0.07$_{st}\pm$0.09$_{sy}$)$\cdot$10$^{-5}$
\end{center}

together with the values of the form factors
\begin{center}
\begin{description}
\item[ ]\=f$_s = 0.052\pm0.006_{st}\pm0.002_{sy}$
\item[ ]\=f$_p = -0.051\pm0.011_{st}\pm0.005_{sy}$
\item[ ]$\lambda_g = 0.087\pm0.019_{st}\pm0.006_{sy}$
\item[ ]\=h$ = -0.32\pm0.12_{st}\pm0.07_{sy}$
\end{description}
\end{center}

\begin{figure}
\begin{center}
%\begin{minipage}{0.46\textwidth}
\resizebox{0.45\textwidth}{!}{%
  \includegraphics{kle4.eps}
}
%\end{minipage}
\end{center}
\vspace*{-10mm}\caption{\it $\kl \rightarrow \pi^{\pm} \piz e^{\mp} \nu_e$: Theoretical predictions and measured values}\vspace*{-4mm}
\label{ke4}
\end{figure}

\begin{thebibliography}{9}

\bibitem{chpt} G. D'Ambrosio and G. Isidori, Int. J. Mod. Phys. \textbf{A13}
(1998) 1; S. Weinberg, Physica \textbf{A96} (1979) 327; J. Gasser and 
H. Leutwyler, Ann. Phys. (N.Y.) \textbf{158} (1984) 142; A.V. Manohar and
H. Georgi, Nucl. Phys. \textbf{B234} (1984) 189.
\bibitem{klggth} E. Ma and A. Pramudita, Phys. Rev. \textbf{D24} (1984) 2476;
J.F. Donoghue, B.R. Holstein, Y.-C.R. Lin, Nucl.Phys. \textbf{B277} (1986) 651
\bibitem{klsgg} A. Lai et al., Phys. Lett. \textbf{B551} (2003) 7.
\bibitem{kloeklgg} M. Adinolfi et al., Phys. Lett. \textbf{B566} (2003) 61.
\bibitem{ksgg1} J.Kambor and B.R. Holstein, Phys. Rev. \textbf{D49} (1994) 2346.
\bibitem{ksgg2} J.L.Goity, Z. Phys. \textbf{C34} (1987) 341.
\bibitem{ksgg3} G. D'Ambrosio, D.Espriu, Phys. Lett. \textbf{B175} (1986) 237.
\bibitem{ksggna31} G.D. Barr et al., Phys. Lett. \textbf{B351} (1995) 579.
\bibitem{ksggres} A. Lai et al., Phys. Lett. \textbf{B493} (2000) 29.
\bibitem{ksp0gg1} J. Bijnens, E. Pallante and J. Prades, Nucl. Phys. 
\textbf{B521} (1998) 305.
\bibitem{ksp0gg2} G. Ecker, A. Pich and E. de Rafael, Phys. Lett \textbf{B189}
(1987) 363.
\bibitem{ksp0gg0} A. Lai et al., Phys. Lett. \textbf{B556} (2003) 105.
\bibitem{ksp0gg3} A. Lai et al., Phys. Lett. \textbf{B578} (2004) 276.
\bibitem{klp0gg1} G. D'Ambrosio and J. Portoles, Nucl. Phys. \textbf{B492} 
(1997) 417; G. Ecker, A. Pich and E. de Rafael, Phys. Lett \textbf{B237}
(1990) 481; P. Heiliger and L.M. Sehgal, Phys. Rev. \textbf{D47} (1993) 4920.
\bibitem{klp0gg2} A.G. Cohen, G. Ecker and A. Pich, Phys. Lett. \textbf{B304}
(1993) 347; L. Cappiello, G. D'Ambrosio and M. Miragliuolo, Phys. Lett.
\textbf{B298} (1993) 423.
\bibitem{klp0gg3} J.F. Donoghue and F. Gabbiani, Phys. Rev. \textbf{D51} (1995)
2187.
\bibitem{klpres} A. Lai et al., Phys. Lett. \textbf{B536} (2002) 229.
\bibitem{ktevp0gg} A. Alavi-Harati et al., Phys. Rev. Lett. \textbf{83} 
(1999) 917.
\bibitem{ktev1} G. D'Ambrosio et al., Z. Phys. \textbf{C76} (1997) 301.
\bibitem{ktev2} G. Ecker et al., Nucl. Phys. \textbf{B413} (1994) 321.
\bibitem{etath} Ll. Ametller et al., Phys. Lett. \textbf{B276} (1992) 185;
E. Oset et al., Phys. Rev. \textbf{D67} (2003) 073013 and references therein.
\bibitem{etap0gg} B. Di Micco, Acta Physica Slovaca, \textbf{56} (2006) 403.
\bibitem{kpiggth} G. D'Ambrosio and J. Portoles, Phys. Lett. \textbf{B389}
(1996) 770 .
\bibitem{kpigg} P. Kitching et al., Phys. Rev. Lett. \textbf{79} (1997) 4079.
\bibitem{kpieeth} G. Ecker, A. Pich and E. de Rafael, Nucl. Phys. \textbf{B291}
(1987) 692; G. D'Ambrosio et al., J. High En. Phys. \textbf{JHEP08} (1998)004.
\bibitem{kpiee} R. Appel et al., Phys. Rev. Lett. \textbf{83} (1999) 4482.
\bibitem{krads1} H. Fearing et al., Phys. Rev. Lett. \textbf{24} (1970) 189;
H. Fearing et al., Phys. Rev. \textbf{D2} (1970) 542; M.G. Doncel, Phys. Lett.
\textbf{B32} (1970) 623.
\bibitem{krads2} B.R. Holstein, Phys. Rev. \textbf{D41} (1990) 2829; J. Bijnens
et al., Nucl. Phys. \textbf{B369} (1993) 81; V. Cirigliano et al., Eur. Phys.
J. \textbf{C23} (2002) 121.
\bibitem{krads3} The KTeV Collaboration, Phys. Rev. \textbf{D64} (2001) 112004.
\bibitem{krads4} A. Lai et al., Phys. Lett. \textbf{B605} (2005) 247.
\bibitem{duk} V. Duk, Proceedings of this Conference.
\bibitem{ke4th1} G. Colangelo, J. Gasser and H. Leutwyler, Phys. Rev. Lett.
\textbf{86} (2001) 5008.
\bibitem{ke4th2} G. Amor\'os, J. Bijnens, P. Talavera, Nucl. Phys. 
\textbf{B585} (2000) 293.
\bibitem{anzivino} G. Anzivino, Proceedings of this Conference.
\bibitem{kle4} A. Lai et al., Phys. Lett. \textbf{B595} (2004) 75.

\end{thebibliography}


\end{document}
