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\title{Highlights on neutral K decays}

\author{C. Gatti\address{Laboratori Nazionali di Frascati - INFN, 
        Via E. Fermi 40  \\ 
        00044 Frascati (Rome) Italy}%
        \thanks{KLOE Collaboration}}
       
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\begin{document}
  
\begin{abstract}
  Recent results on kaon decays allow
  a new determination of the $CP$ and $CPT$ parameters
  $\Re{\epsilon}$ and $\Im{\delta}$ based on the unitarity relation. 
  In particular, the new experimental limit on 
  $BR(K_{S}\to3\pi^{0})$ and the determination of the charge asymmetry
  in $K_{S}\to\pi e \nu$ decays allow us to test the $CPT$ symmetry
  with a sensitivity of order $m_{K}/M_{Planck}$.
\vspace{1pc}
\end{abstract}

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\maketitle

\section{Introduction}
The neutral kaon system has long been used to test 
discrete symmetries. 
The most stringent limit on $CPT$-symmetry
violation is provided
by the mass difference 
$\left|m_{K^{0}}-m_{\bar{K}^{0}}\right|/m_{K^{0}}$.
This difference is related to the quantity
$\delta$ that parametrizes the $CPT$ violation in the mixing:
\begin{eqnarray}
  \nonumber
  \delta & = & 
  \frac{i\left(m_{K^{0}}-m_{\bar{K}^{0}}\right)
    +1/2\left(\Gamma_{K^{0}}-\Gamma_{\bar{K}^{0}}\right)}
	 {\Delta\Gamma} \times \\ 
	 & & \cos{\phi_{SW}} e^{i\phi_{SW}} 	 
	 \label{eq:defdelta}
\end{eqnarray}
where $\tan{\phi_{SW}}=2\Delta M/\Delta\Gamma$,
$\Delta\Gamma=\Gamma_{S}-\Gamma_{L}$, and 
$\Delta M=m_{L}-m_{S}$.
If we assume that the $CPT$ violation is negligible in the
decay width ($\Gamma_{K^{0}}-\Gamma_{\bar{K}^{0}}=0$), we have:
\begin{equation}
  \nonumber
  \frac{m_{K^{0}}-m_{\bar{K}^{0}}}{m_{K^{0}}}=3\times10^{-14}\Im{\delta}.
\end{equation}
Violation at first order
in $m_{K}/M_{Planck}\sim 10^{-19}$, 
coming from quantum gravity effects,
can thus be tested by
putting an upper limit on $\Im{\delta}$ of order $\sim 10^{-5}$.


$\Re{\delta}$ and $\Im{\delta}$ are measured in two different ways.
$\Re{\delta}$ is measured both 
from the difference $A_{S}-A_{L}$ of the charge asymmetries
in $K_{S}$ and $K_{L}$ semileptonic decays,  
and from the time-dependent asymmetry
\begin{equation}
  \nonumber
  A_{CPT}=\frac{P(\bar{K}^{0}\to\bar{K}^{0}(t))-P(K^{0}\to K^{0}(t))}
  {P(\bar{K}^{0}\to\bar{K}^{0}(t))+P(K^{0}\to K^{0}(t))},
\end{equation}
where $P(K^{0}\to K^{0}(t))$ ($P(\bar{K}^{0}\to\bar{K}^{0}(t))$) 
is the probability for a $K^{0}$ ($\bar{K}^{0}$) 
to be observed as a $K^{0}$ ($\bar{K}^{0}$) after a time t.

$Im{\delta}$ is obtained from the unitarity (Bell-Steinberger) relation:
\begin{eqnarray}
\nonumber
\left[\frac{\Gamma_{S}+\Gamma_{L}}{\Gamma_{S}-\Gamma_{L}}
  + i \tan{\phi_{SW}}\right]
\frac{\Re{\epsilon}-i\Im{\delta}}{1+\left|\epsilon\right|^{2}} &=& \\
 \frac{1}{\Gamma_{S}-\Gamma_{L}}\sum_{f}a^{*}_{S}(f)a_{L}(f) & = &
 \sum_{f}\alpha_{f}
 \label{eq:bellst}
\end{eqnarray}
where $a_{S,L}(f)$ are the $K_{S,L}$ decay amplitudes to the final
state $f$.

\section{Experimental inputs to $\Re{\delta}$}
\label{sec:redelta}

Using the time-dependent asymmetry in semileptonic decays~\cite{cplearpr} 
the CPLEAR Collaboration has obtained $\Re{\delta}$.
They measure the distribution
\begin{equation}
  A_{\delta}(t) = 4 \Re{\delta} + {\cal F}\left( \Im{\delta},y,\Re{x_{-}},\Im{x_{+}}\right) 
\end{equation}
that depends, beside the $CPT$-violating parameter $\delta$, on the 
parameters $y$ and $x_{\pm}$ that describe the $CPT$ violation in the decays
and the violation of the $\Delta S = \Delta Q$ rule, respectively.
They obtain:
\begin{eqnarray}
  \nonumber
  \Re{\delta} & = & (3.0\pm3.3\pm0.6)\times 10^{-4}
  \\
  \nonumber
  \Im{\delta} & = & (-1.5\pm2.3\pm0.3)\times 10^{-2}
  \\
  \nonumber
  \Re{x_{-}} & = & (0.2\pm1.3\pm0.3)\times 10^{-2}
  \\
  \Im{x_{+}} & = & (1.2\pm2.2\pm0.3)\times 10^{-2}  
\end{eqnarray}
This result is improved by adding as a constraint
the measurement~\cite{kspen,PDG}
$A_{S}-A_{L}=4\left[\Re{\delta}+\Re{x_{-}}\right]=(-1.8\pm10.0)\times 10^{-3}$
to the original CPLEAR fit, obtaining:
\begin{eqnarray}
  \nonumber
  \Re{\delta} & = & (3.3\pm2.8)\times 10^{-4}
  \\
  \nonumber
  \Im{\delta} & = & (-1.1\pm0.7)\times 10^{-2}
  \\
  \nonumber
  \Re{x_{-}} & = & (-0.03\pm0.25)\times 10^{-2}
  \\
  \Im{x_{+}} & = & (0.8\pm0.7)\times 10^{-2}  
  \label{eq:imxpiu}
\end{eqnarray}
The uncertainty on $\Im{x_{+}}$ is reduced by a factor three.
As we will see in the next section, this parameter enters in the
unitarity relation and is one of the limiting factors in the determination
of $\Im{\delta}$ and $\Re{\epsilon}$.

\section{Experimental inputs to $\Im{\delta}$ and $\Re{\epsilon}$} 
\label{sec:imdelta}

The only decay modes relevant for the evaluation of
Eq.~(\ref{eq:bellst}) are the $\pi\pi(\gamma)$, $\pi\pi\pi$,
and semileptonic modes. The product of amplitudes $\alpha_{f}$
are obtained from the lifetimes of the kaons, the ratios
of $K_{S}$ and $K_{L}$ amplitudes $\eta$, and the branching ratios,
in the following way:
\begin{eqnarray}
\nonumber
\alpha_{\pi\pi}&=&\eta_{\pi\pi}BR(K_{S}\to\pi\pi)
\\
\nonumber
\alpha_{\pi\pi\gamma(DE)}&=&\sqrt{\frac{\tau_{S}}{\tau_{L}}BR(K_{L})BR(K_{S})}
e^{i\phi+-\gamma}
\\
\nonumber
\alpha_{\pi\pi\pi}&=&\frac{\tau_{S}}{\tau_{L}}\eta_{\pi\pi\pi}BR(K_{L}\to\pi\pi\pi)
\end{eqnarray} 
where $\pi\pi=\pi^{+}\pi^{-},\pi^{0}\pi^{0}$, 
and $\pi\pi\pi=\pi^{+}\pi^{-}\pi^{0},\pi^{0}\pi^{0}\pi^{0}$. The $\pi^{+}\pi^{-}$ mode
includes the inner bremsstrahlung (IB) emission. On the contrary,
the direct emission (DE) component is added independently since there is no 
interference with the IB term.

The contribution of semileptonic decays to Eq.~(\ref{eq:bellst})
is given by:
\begin{eqnarray}
  \nonumber
  \alpha_{\pi l\nu}&=&2\frac{\tau_{S}}{\tau_{L}}BR(K_{L}\to\pi l \nu)\times
  \\
  \nonumber
    & & \left[\Re{\epsilon}-\Re{y}-i\Im{\delta}-i\Im{x_{+}}\right]
  \\
  &=&\left[(A_{S}+A_{L})/4-i\Im{\delta}-i\Im{x_{+}}\right]
\end{eqnarray}
where $\Im{x_{+}}$ is given by Eq.~(\ref{eq:imxpiu}), and 
$(A_{S}+A_{L})/4=\Re{\epsilon}-\Re{y}=(-4.8\pm19.0)\times10^{-3}$,
while $\Im{\delta}$ is the unknown  of Eq.~(\ref{eq:bellst}).
The determination of $\Re{y}$ is of primary importance for the measurement of 
$A_{S}$~\cite{kspen}.


Some of the most recent measurements entering in the determination
of the $\alpha$ parameters are described in the following sections.
The numerical results are shown in Tab.~\ref{table:alphas}.
\begin{table*}[htb]
\caption{Values of the $\alpha$ parameters.}
\label{table:alphas}
\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}{@{}lll}
\hline
$\alpha$ parameters        &Real Part $\times 10^{3}$&Imaginary Part$\times 10^{3}$\\
\hline
$\alpha(\pi^{+}\pi^{-})$           &\m$1.126\pm0.014$&\m$1.064\pm0.014$\\
$\alpha(\pi^{0}\pi^{0})$           &\m$0.494\pm0.007$&\m$0.472\pm0.008$\\
$\alpha(\pi^{+}\pi^{-}\gamma(DE))$ &\m$0.000\pm0.002$&\m$0.000\pm0.002$\\
$\alpha(\pi l\nu)_{\Im{\delta}=0}$ &\m$0.003\pm0.002$&\m$-0.019\pm0.017$\\
$\alpha(\pi^{+}\pi^{-}\pi^{0})$    &\m$0.001\pm0.002$&\m$-0.001\pm0.002$\\
$\alpha(\pi^{0}\pi^{0}\pi^{0})$    &\m$<0.007$ at 95\% C.L.&\m$<0.007$ at 95\% C.L.\\
\hline
\end{tabular}\\[2pt]
\end{table*}

\subsection{$\pi\pi$}
The two body decays give the largest contribution to Eq.~(\ref{eq:bellst}).
The $K_{S}$ branching ratios are obtained from the recent
KLOE measurement of the ratio 
$\Gamma(K_{S}\to\pi^{+}\pi^{-})/\Gamma(K_{S}\to\pi^{0}\pi^{0})=(2.2549\pm0.0054)$~\cite{kspipi} 
with a sample of $400\times10^{6}$ $\phi\to K_{S}K_{L}$ decays.
The $K_{L}\to\pi^{+}\pi^{-}$ branching ratio has been recently measured by 
KLOE~\cite{klpipi} with $\sim45000$ events, they  find 
$BR(K_{L}\to\pi^{+}\pi^{-}(\gamma_{IB+DE}))=(1.963\pm0.021)\times10^{-3}$,
where both the IB and DE components are included.
The DE component is then subtracted using the KTeV result described
in the following.
KTeV~\cite{ktevkl} measured both the charged and neutral decays
with $\sim10^{5}$ events each. They find
$BR(K_{L}\to\pi^{+}\pi^{-}(\gamma_{IB}))=(1.975\pm0.012)\times10^{-3}$,
and
$BR(K_{L}\to\pi^{0}\pi^{0})=(0.865\pm0.010)\times10^{-3}$.
The measurement of charged mode includes only the IB component.
The two new measurements of $BR(K_{L}\to\pi^{+}\pi^{-})$ are in agreement
but disagree with the PDG value of 2004. 

\begin{figure}[htb]
\centering
\vspace{-15pt}
\includegraphics[height=5.5cm]{egprlppg.eps}
\vspace{-20pt}
\caption{Spectrum of photon energy in $K_{L}\to\pi^{+}\pi^{-}\gamma$ events,
measured by the KTeV Collaboration.}
\label{fig:ppgamma}
\end{figure}

The DE component is measured with two different methods. 
With the first method, the photon spectrum is measured directly
in $K_{L}\to\pi^{+}\pi^{-}\gamma$ decays. KTeV~\cite{ktevppgde} has 
measured it from a sample of $\sim10^{5}$ $\pi\pi\gamma$ events
with $E_{\gamma}^{*}>20~\mbox{MeV}$ (Fig.~\ref{fig:ppgamma}).
They find that the fraction of DE component for $E_{\gamma}^{*}>20~\mbox{MeV}$
is $DE/(DE+IB)=0.689\pm0.021$.
With the second method, the internal conversion of the photon into
a pair $e^{+}e^{-}$ is used. The resulting decay 
$K_{L}\to\pi^{+}\pi^{-}e^{+}e^{-}$ shows an asymmetry in the  
distribution of the angle between the $\pi^{+}\pi^{-}$ and 
$e^{+}e^{-}$ planes. This asymmetry is due to the interference between
the IB and DE components. KTeV~\cite{ktevppee} measures this
asymmetry with $\sim5000$ selected events and extracts
the information on the DE. The results are in agreement
with the previous method, but less precise.

The DE component in $K_{S}$ decay is suppressed by $CP$ violation. 
The $K_{S}$ photon spectrum has been measured in~\cite{ramberg}.

\subsection{$\pi l\nu$}

The semileptonic decays of $K_{L}$ have been recently measured 
by KLOE~\cite{kloekl}, KTeV~\cite{ktevkl}, and NA48~\cite{na48kl}
in order to determine the Cabibbo angle. At this conference 
this item has been discussed in a different talk~\cite{antonella}.

\begin{figure}[htb]
\centering
%\vspace{-10pt}
\includegraphics[height=5cm]{emisspmiss_eca2_ch2_periotot_come2_smo0_peso0.eps}
\vspace{-20pt}
\caption{$E_{miss}-P_{miss}$ distribution for $K_{S}\to\pi^{-}e^{+}\nu$ decays.}
\label{fig:kssemi}
\end{figure}

The $K_{S}\to\pi e\nu$ branching ratio and its charge asymmetry
have been recently measured by KLOE~\cite{kspen}
with a sample of 6500 selected events for each charge mode. 
The distribution of the missing
energy minus the missing momentum $\Delta E_{\pi e}$ is shown
in Fig.~\ref{fig:kssemi} for the $\pi^{-}e^{+}\nu$ sample. The signal is centered at zero
as expected for a missing neutrino. KLOE finds
$BR(K_{S}\to\pi e\nu)=(7.028\pm0.092)\times10^{-4}$ 
and $A_{S}=(1.5\pm10.0)\times10^{-3}$. 

\subsection{$\pi\pi\pi$}


The amplitudes for $K\to3\pi$ decays can be decomposed
according to the $CP$ state of the three pion system:
\begin{eqnarray}
  \nonumber
  a_{L}&=&a_{L}^{CP-}(X,Y)
  \\
  a_{S}&=&a_{S}^{CP+}(X,Y)+a_{S}^{CP-}(X,Y)
  \label{eq:threepi}
\end{eqnarray}
where $a^{CP+(-)}$ is the amplitude for the
decay to the final state with $CP=+1(-1)$,
and $X\propto E_{+}-E_{-}$ and $Y\propto E_{0}$.
The decay to the $CP=+1$ state is suppressed
by the centrifugal barrier. Therefore, $a_{L}^{CP+}$ is
suppressed both by the centrifugal barrier and by $CP$ violation
and can be neglected. On the contrary, both $K_{S}$ amplitudes
are retained being one suppressed by $CP$ violation and the
other one by the centrifugal barrier.
The amplitudes have the following symmetry property:
\begin{equation}
  a^{CP\pm}(X,Y)=\mp a^{CP\pm}(-X,Y)
  \label{eq:symm3pi}
\end{equation}
Hence, the only contribution to Eq.~(\ref{eq:bellst}) 
comes from:
\begin{equation}
  \eta_{+-0}=\frac{\int dX dY a_{L}^{*}a_{S}^{CP-}}
      {\int dX dY |a_{L}|^{2}}
      \label{eq:eta}
\end{equation}
while 
\begin{equation}
  \nonumber
  \frac{\int dX dY a_{L}^{*}a_{S}^{CP+}}
      {\int dX dY |a_{L}|^{2}}=0
\end{equation}

CPLEAR measures $\eta_{+-0}$ from the time-dependent
asymmetry~\cite{cplearpr}:
\begin{equation}
  \nonumber
  \frac{\bar{N}_{3\pi}(t)-N_{3\pi}(t)}{\bar{N}_{3\pi}(t)+N_{3\pi}(t)}
\end{equation}
where $N_{3\pi}(t)$ ($\bar{N}_{3\pi}(t)$) is the number
of decays to three pions observed at time $t$ for 
kaons tagged as $K^{0}$ ($\bar{K}^{0}$) at time $t=0$.
They find:
\begin{equation}
  \nonumber
\eta_{+-0}=(-2\pm7)\times10^{-3}+i(-2\pm9)\times10^{-3}
\end{equation}

The $CP$-conserving component of the decay,
parametrized by the quantity: 
\begin{equation}
  \lambda=\frac{\int_{X>0} dX dY a_{L}^{*}a_{S}^{CP+}}
	       {\int_{X>0} dX dY |a_{L}|^{2}}
	       \label{eq:lambda}
\end{equation}
has been measured by NA48~\cite{na48ks3pi} by fitting the time-dependent
asymmetry:
\begin{equation}
  \nonumber
  \frac{N_{3\pi}^{X>0}(t)-N_{3\pi}^{X<0}(t)}
       {N_{3\pi}^{X>0}(t)+N_{3\pi}^{X<0}(t)}
\end{equation}
They find:
\begin{equation}
  \nonumber
\lambda=(0.038\pm0.010)+i(-0.013\pm0.007)
\end{equation}
A similar result has been obtained also by CPLEAR~\cite{cplearpr}.


\begin{figure}[htb]
  \centering
%  \vspace{-15pt}
  \includegraphics[height=5cm]{eta000.eps}
  \vspace{-20pt}
  \caption{90\% C.L. limits on the plane 
    $\Im{\eta_{000}}$-$\Re{\eta_{000}}$ obtained
    from the measurements of NA48 and KLOE.}
  \label{fig:eta000}
\end{figure}
Because of its symmetry property $a_{S}^{CP+}(X,Y)=0$
for three equal pions, as in $K_{S}\to3\pi^{0}$ decays.
Therefore, this decay is $CP$ violating, and its
branching ratio is expected to be $\sim10^{-9}$.
Before the recent measurements of KLOE and NA48,
$\Im{\delta}$ was limited by the poor knowledge
of $\eta_{000}=a_{S}(000)/a_{L}(000)$.

There are two different ways of measuring this quantity.
NA48 measures the interference in $K\to3\pi^{0}$ decays as a function of the
proper time, with $5\times10^{6}$ decays from the 
{\it near target}, normalized to the rate of $10^{8}$ $K_{L}\to3\pi^{0}$
from the {\it far target}~\cite{na48ks3pi0}:
\begin{eqnarray}
  \nonumber
  f_{3\pi^{0}}(t)&\propto& 1+\left|\eta_{000}\right|^{2} 
  e^{-(\Gamma_{S}-\Gamma_{L})t}
  \\
  \nonumber
  & & + 2D(p)\left[\Re{\eta_{000}}\cos{\Delta m t} - \Im{\eta_{000}}\right.
\\
& & \left. \sin{\Delta m t} \right] e^{-1/2(\Gamma_{S}-\Gamma_{L})t}
\end{eqnarray}
where $D(p)$ is the {\it dilution} factor that describes the momentum dependent
production asymmetry between $K^{0}$ and $\bar{K}^{0}$ at the target.
NA48 measures:
\begin{equation}
  \eta_{000}=(-0.002\pm0.019)+i(-0.003\pm0.021)
\end{equation}

KLOE uses a pure $K_{S}$ beam tagged with a $K_{L}$.
This allows KLOE to directly search
$K_{S}\to3\pi^{0}$ decays~\cite{kloeks3pi0}.
The selection is done on a sample of $\sim5\times10^{8}$
$K_{L}-K_{S}$, looking for six photon clusters in the 
calorimeter and applying a veto for charged tracks in the
drift chamber. For each decay both $2\pi^{0}$ and
$3\pi^{0}$ hypotheses are tested defining two $\chi^{2}$-like variables.  
The events are counted in the signal box in the plane defined by
these two variables. KLOE finds 2 events with 3 background events expected,
leading to the following 90\% limits on the branching ratio and
on the module of $\eta_{000}$:
\begin{eqnarray}
  \nonumber
  BR(K_{S}\to3\pi^{0}) &\leq& 1.2\times10^{-7}
  \\
  \left|\eta_{000}\right|&\leq& 0.018
\end{eqnarray}
The comparison of NA48 and KLOE results is shown in Fig.~\ref{fig:eta000}.


The $K_{L}$ decays to three pions have been measured
by KLOE and KTeV, and a preliminary result for the $3\pi^{0}$ decay
has been also presented by NA48. 
These measurements are described in~\cite{antonella}.

\section{Results}

Following the method outlined in Secs.~\ref{sec:redelta} and~\ref{sec:imdelta}
and the values in Tab.~\ref{table:alphas} 
the KLOE Collaboration together with G.~Isidori and G.~D'Ambrosio obtained
the following results:
\begin{eqnarray}
  \nonumber
  \Re{\epsilon} &=& (161.0\pm1.0)\times10^{-5}
  \\
  \Im{\delta} &=& (1.3\pm2.0)\times10^{-5}
  \label{eq:reepsimdelta}
\end{eqnarray} 
and from Eq.~(\ref{eq:defdelta}):
\begin{eqnarray}
  \nonumber
  \Gamma_{K^{0}}-\Gamma_{\bar{K}^{0}} &=& (5\pm4)\times10^{-18}~\mbox{GeV}
  \\
  m_{K^{0}}-m_{\bar{K}^{0}} &=& (-2.2\pm2.0)\times10^{-18}~\mbox{GeV}
  \label{eq:deltamdeltagamma}
\end{eqnarray}
The 90\% and 68\% C.L. regions are shown in Fig.~\ref{fig:deltavsespilon}.
\begin{figure}[htb]
%  \centering
  \vspace{-15pt}
  \includegraphics[height=3.8cm]{reeimd.eps}%
  \includegraphics[height=3.8cm]{dmdg.eps}
  \vspace{-20pt}
  \caption{Allowed region at 68\% and 95\% C.L. in the $\Re{\epsilon}$,
    $Im{\delta}$ plane, and in the 
    $\Gamma_{K^{0}}-\Gamma_{\bar{K}^{0}}$,
    $m_{K^{0}}-m_{\bar{K}^{0}}$ plane.}
  \label{fig:deltavsespilon}
\end{figure}

Assuming no $CPT$ violation in the decay width ($\Gamma_{K^{0}}-\Gamma_{\bar{K}^{0}}=0$), they find the 95\% C.L.:
\begin{equation}
  -4\times10^{-19} < m_{K^{0}}-m_{\bar{K}^{0}} < 7\times10^{-19}~\mbox{GeV}
  \label{eq:kloeresult}
\end{equation}
to be compared with the previous determination due to the
CPLEAR Collaboration:
\begin{equation}
  -10\times10^{-19} < m_{K^{0}}-m_{\bar{K}^{0}} < 17\times10^{-19}~\mbox{GeV}
  \label{eq:cplearresult}
\end{equation}

Eq.~(\ref{eq:kloeresult}) is at present the most stringent test of 
$CPT$ symmetry. 
The uncertainty is dominated by the knowledge of
the phases $\phi_{+-}$ and $\phi_{00}$, and of the
parameter $\Im{x_{+}}$. Further improvements are expected from
the analysis of the complete data sample of KLOE ($\sim2.5~\mbox{fb}^{-1}$),
in particular for what concerns the $K_{S}$ rare decays.    
However, there are no plans
to further improve the determination of $\phi_{+-,00}$.

\begin{thebibliography}{9}
\bibitem{cplearpr} CPLEAR Collaboration, Physics Report No. 374 (2003) 165.
\bibitem{kspen}    KLOE Collaboration, Physics Letters B636 (2006) 166.
\bibitem{PDG} Particle Data Group, S. Eidelman et al., Physics Letters B592 (2004) 1.
\bibitem{kspipi} KLOE Collaboration, hep-ex/0601025. Submitted to EPJC.
\bibitem{klpipi} KLOE Collaboration, Physics Letters B638 (2006) 140.
\bibitem{ktevkl} KTeV Collaboration, Physics Review D70 (2004) 092006.
\bibitem{ktevppgde} KTeV Collaboration, hep-ph/0604035. Submitted to PRL.
\bibitem{ktevppee} KTeV Collaboration, Physics review Letters 96 (2006) 101801.
\bibitem{ramberg} E731 Collaboration, Phyics Review Letters 70 (1993) 2525.
\bibitem{kloekl} KLOE Collaboration, Physics Letters B632 (2006) 43.
\bibitem{na48kl} NA48 Collaboration, Physics Letters B602 (2004) 41.
\bibitem{antonella} A.~Antonelli, these proceedings.
\bibitem{na48ks3pi} NA48 Collaboration, Physics Letters B630 (2005) 31.
\bibitem{na48ks3pi0} NA48 Collaboration, Physics Letters B610 (2005) 165.
\bibitem{kloeks3pi0} KLOE Collaboration, Physics Letters B619 (2005) 61.
\end{thebibliography}


\end{document}
