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\newcommand{\kppg}{K^{\pm} \rightarrow \pi^{\pm}\pi^{0}\gamma}
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\title{Highlights on rare charged Kaon decays}

\author{M. Raggi \address[MCSD]{Sezione Infn di Perugia, \\
        via Pascoli, 06100 Perugia, Italia}}

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\runtitle{Highlights on rare charged Kaon decays}

\runauthor{M. Raggi}

\begin{document}

\begin{abstract}
Two new results of NA48/2 in the charged K rare decays, based on a fraction of the 2003 data only, will be reported. The
first measurement of the DE and INT terms of the decay $\kppg$, based on 124K events, has been performed in the $\ecmk$
region $0<\ecmk<80$ MeV:
\begin{equation*}Frac(DE)_{0<\ecmk<80 MeV}=(3.35\pm0.35_{sta}\pm0.25_{sys})\%\end{equation*}
\begin{equation*}Frac(INT)_{0<\ecmk<80MeV}=(-2.67\pm0.81_{sta}\pm0.73_{sys})\%\end{equation*}
The best measurement of the $K^{00}_{e4}$ branching ratio has been performed leading to the result:
\begin{equation*} BR(K^{00}_{e4})=(2.587\pm0.026_{sta}\pm0.019_{sys}\pm0.029_{ext})\cdot10^{-5} \end{equation*}
%\vspace{1pc}
\end{abstract}

% typeset front matter (including abstract)
\maketitle

\section{NA48/2 experiment}

The NA48/2 detector is essentially based on the existing NA48\cite{Fanti:1999nm} one with a new beam line designed in order
to provide simultaneous $K^+$ and $K^-$ beams overlapping in the decay region. The beam, produced using primary 400 GeV
protons from the SPS accelerator at CERN, has narrow momentum band ($P_K$ = 60$\pm$3 GeV). The beam line has been equipped
with a new detector KABES (KAon BEam Spectrometer)\cite{Peyaud:2004tj} able to measure the Kaon momentum and direction with
high precision. The NA48/2 experiment data taking period extends in the summer of 2003 and 2004. What will be discussed in
this paper has been obtained using half of the 2003 data only.

\section{Introduction to $\kppg$ decay}

Nowadays the Chiral Perturbation Theory (ChPT) is one of the most reliable tool to describe low energy QCD dynamics. In this
theoretical framework the chiral anomaly plays a crucial role, and has been therefore object of intense studies.
Unfortunately precise experimental tests of the chiral anomaly are at the moment very few. Has been noticed that the genuine
manifestation of the chiral anomaly in non-leptonic decays, is restricted to the radiative decay of $\kppg$ in the charged
kaon sector.

Two different processes can be responsible for the origin of the gamma. It can be either produced by a final state radiation
of the $\pi^{\pm}$, Inner Bremsstrahlung (IB), or in the decay itself, Direct Emission (DE). Although, due to the dominant
IB, the DE component is very difficult to observe it can be isolated kinematically. The DE component consists of magnetic and
electric transitions.
%The magnetic part is dominated by reducible chiral anomalous
%amplitude, which is unambiguously determined by the Wess-Zumino-Witten functional, and has also a contribution of the direct
%anomalous amplitude, not calculable in a model-independent way. The electric DE amplitude also has two main contributions.
%The first one is a local scale independent contribution arising directly from $\cal L_{4}^{\Delta S=1}$ lagrangian, while the
%second one comes from chiral one-loops diagrams.
While the magnetic part can be evaluated using the Wess-Zumino-Witten functional, there is no definite prediction from ChPT
on the electric transition, whose amplitude depends on undetermined constants. The electric contribution is extremely
interesting since it interferes (INT) with the IB amplitude therefore it may be distinguished from the magnetic, which does
not.
%The measurement of both the DE and INT term will provide a stringent test of the chiral approach to the anomaly.

In the $\kppg$ decay IB, INT, and DE can be distinguished kinematically using the variable W which is defined as
follows\cite{Good:1959}: \be W^2=\frac{(P^{*}_{K} \cdot P^{*}_{\gamma})(P^{*}_{\pi} \cdot P^{*}_{\gamma})}{(m_Km_{\pi})^2}
\ee with $P^{*}_{x}$ the 4-momentum of the x particle and $\gamma$ the radiative one. The decay rate depends only on $\ecmk$,
energy of the pion in the Kaon rest frame, and W. Integrating over $\ecmk$ an expression that separates the different
contributions into terms with different powers of W can be obtained: \be
\begin{align}
\frac{d\Gamma^{\pm}}{dW}&\simeq\left(\frac{d\Gamma^{\pm}}{dW}\right)_{IB}\left[1+\color{white}\right] \\
&+2\left(\frac{m_{\pi}}{m_K} \right)^2 W^2 |E| cos((\delta_1-\delta_0) \pm \phi)  \\
&\color{white}\left[ \color{black} +\left(\frac{m_{\pi}}{m_{K}}\right)^{4}W^{4}(|E|^{2}+|M|^{2})\right] \end{align} \ee
%\label{eqn:FUN_W_the}

The three terms represent IB, INT and DE contribution respectively.

\begin{table}[t]
%\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}{lccc}
\hline
exp.         & year & \#events & $BR(DE)\cdot10^{-6}$ \\
\hline
E787 \cite{Adler:2000it}  & 2000 & 20K  & $4.7\pm0.8\pm0.3$  \\
E470 \cite{Aliev:2002tq}  & 2003 & 4.5K & $3.2\pm1.3\pm1.0$  \\
E787 \cite{Tsunemi}       & 2005 & 20K  & $3.5\pm0.6\pm0.35$ \\
E470 \cite{Aliev:2005}    & 2005 & 10K  & $3.8\pm0.8\pm0.7$  \\
\hline
\end{tabular}
\caption{The $\kppg$ experimental results} \label{tab:exp_res}
%\\[2pt]
\end{table}

\subsection{Present experimental knowledge}

The IB component has been measured since the seventies by Abrams et al. \cite{Abrams:1972rj} achieving a good agreement with
solid QED theoretical predictions. The experimental measurement of the fractions of DE and INT is affected by very dangerous
BG sources, such as $\kpp$ and $\kpppn$ decays, suppressed in a kinematically BG free region, $\ecmk$ in the range 55-90 MeV.
Moreover a good measurement requires a very good reconstruction of both charged and neutral particles 4-momenta. The present
experimental knowledge on the $\kppg$ decay is summarized in tab.\ref{tab:exp_res}. All the results have been obtained in the
hypothesis of vanishing interference and in the $\ecmk$ region 55-90 MeV.

\begin{figure}[t]
\begin{center}
%\vspace{9pt} \framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\includegraphics[width=60mm,height=60mm]{mk_bg.eps}
\caption{Data-mc comparison of $M_K$ spectrum.} \label{fig:ppg BG}
\end{center}
\end{figure}

\subsection{Selection and backgrounds}

The main BG sources are $\kpp$ and $\kpppn$. The first decay needs an accidental photon or an hadronic extra cluster to mimic
the signal final state, while the second a lost or 2 fused gamma. The selection aims to suppress the contribution of them
both to less than 1\% of the DE component. The rejection of $\kpp$ relays on the $\ecmk$ cut. The request $\ecmk$ lower than
80 MeV allows to reject $\kpp$ and a part of the IB spectrum of $\kppg$ only, including region 0-55 MeV very rich of DE and
INT events. The upper cut at 80 MeV is due to trigger reasons. To suppress $\kpppn$ BG the very good kaon mass resolution,
(2.2 MeV), and the identification of fused gamma events constraints have been used. In fig. \ref{fig:ppg BG} the data kaon
mass spectrum is compared with the sum of $\kppg$ and $\kpppn$ MC.

A very important issue in the measurement of DE and INT is to identify the radiative $\gamma$ among the 3 available. A
dedicated set of cuts, based on the agreement of the vertex evaluated using the pion and kaon tracks, and the one evaluated
pairing the $\gamma$'s to form a $\pi^0$, has been implemented. Using this cuts a misidentification probabilities, computed
using MC simulation, of the order of the permille for all the components has been achieved. A total of $\sim220000$ candidate
events survived all selection cuts in the region of $\ecmk$ 0-80 MeV.

\subsection{Measurement technique}

The extraction of the fractions of DE and INT relies on the fact that different components show quite different W
distributions. An extended maximum likelihood technique, assigning weights to MC W distributions of the 3 components to
reproduce data spectrum, has been used to get the fractions.
%we can write a binned maximum likelihood in term of the fraction of IB
%DE INT, $\alpha$ $\beta$ and $\gamma$ respectively, imposing $\alpha+\beta+\gamma=1$ as constraint.

\begin{table}[t]
\begin{center}
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%\renewcommand{\tabcolsep}{2pc} % enlarge column spacing
%\renewcommand{\arraystretch}{1.2} % enlarge line spacing
\begin{tabular}{lcc}
\hline
Effect         & syst. DE & syst. INT \\
\hline
Energy scale                    & 0.09      & -0.21  \\
LKr non linearity               & $<0.05$   & $<0.05$ \\
$\gamma$ misidentification      & -         & $\pm0.2$  \\
Fitting procedure               & 0.02      & 0.019   \\
Resolution difference           & $<0.05$   & $<0.1$ \\
LVL1 trigger                    & $\pm0.17$ & $\pm0.43$ \\
LVL2 Trigger                    & $\pm0.17$ & $\pm0.52$ \\
BG contributions                & $<0.05$   & $<0.05$ \\
\hline
TOTAL                  & $\pm0.25$ & $\pm0.73$ \\
\hline
\end{tabular}
\caption{Systematic checks results} \label{tab:sys_res}
%\\[2pt]
\end{center}
\end{table}

Many checks to verify the result were made which are summarized in table \ref{tab:sys_res}. They include checks on the
$\gamma$ energies reconstructed by the calorimeter, on trigger efficiencies and on BG contribution. The systematic
uncertainties are dominated by the trigger.
\begin{figure}[t]
\begin{center}
%\vspace{9pt} \framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\includegraphics[width=60mm,height=60mm]{result_3p.eps}
\caption{Contour plot for DE and INT components} \label{fig:cont}
\end{center}
\end{figure}

The fit has been performed in the interval 0.2-0.9 in the W variable using only 124K events from the total sample. After
correcting for different acceptances we get, in the region $0<\ecmk<80 MeV$, the following values for the fractions of DE and
INT with respect to IB: \bea
Frac(DE)=(3.35\pm0.35_{sta}\pm0.25_{sys})\% \\
Frac(INT)=(-2.67\pm0.81_{sta}\pm0.73_{sys})\% \eea This is the first measurement of a non vanishing interference term in the
$\kppg$ channel. The contour plot in figure \ref{fig:cont} shows the very high correlation of the two contributions.


\section{The Ke4 neutral decay}

The $K\to\pi\pi e \nu$ decays are a very interesting field to study many properties of QCD dynamics. In fact the decay form
factors provide useful constraints on the parameters of the ChPT Lagrangian. They are also clean sources of low energy of
$\pi-\pi$ pairs, from which one can extract the scattering lengths of significance for models of hadron dynamics. In the
charged kaon sector there are two kind of Ke4 decays, $K^{\pm}_{e4}$, $K^{00}_{e4}$. NA48/2 plans to study in detail both of
them with very high statistic data samples. Since Ke4 decays are theoretically related to each other by isospin arguments,
experimental studies for all channels will provide a complete understanding of Ke4 physics. Among the Ke4 decays, the
$K^{00}_{e4}$ is the simplest because the decay kinematics can be described by only one form factor, due to the presence of
two identical $\pi^0$'s in the final state.

Up to now the best measurement of the properties of the $K^{00}_{e4}$, based on 216 events, has been performed by the E470 at
KEK\cite{Shimizu:2004it}.

\subsection{BR and form factor measurement}

The $K^{00}_{e4}$ is affected by two main BG sources $\kpppn$ and $K^{\pm}\to \pi^0 e^{\pm} \nu (\gamma)$. In the first case
the pion has to be misidentified as an electron, while for the radiative Ke3 another accidental gamma is required, to
reproduce the signal final state. The selection uses the compatibility of 2 the neutral vertices, evaluated using $\gamma$
pairs, to assign two $\gamma$s to each of the two $\pi^0$, and assume the Kaon momentum to be 60 GeV/c to compute the
transverse momentum of the neutrino, and then the kaon mass. At the end of the selection, on 2003 data only, a sample of 9642
candidate events has been identified with an estimated background of $260\pm96$ events, coming form $\kpppn$, and $16\pm2$
from Ke3$\gamma$. Using the $\kpppn$ decay as normalization channel to compute the total kaon flux, the $BR(K^{00}_{e4})$ has
been evaluated to be:

\be BR(K^{00}_{e4})=(2.587\pm0.026_{sta}\pm0.019_{sys}\pm0.029_{ext})\cdot10^{-5} \ee

Due to the presence of two identical $\pi^0$'s in the final state in this Ke4 channel there is only one form factor F: \be
F=f_s+f'_sq^2 + f''_sq^4+f_{e}(S_e/4m^2_\pi)+ ...\ee To enhance the sensitivity to the form factors the fit has been
performed using both 2003 and 2004 data ($\sim$38K events). In the first fit attempt unfortunately no sensitivity to $f_e$
has been reached and then a second fit has been performed assuming $f_e=0$. Under this assumption we get: \be
f'_s/f_s=0.129\pm0.036_{sta}\pm0.020_{sys}\ee \be f''_s/f_s=-0.040\pm0.034_{sta}\pm0.020_{sys}\ee Those values are consistent
with the ones measured by NA48/2, with higher statistical sensitivity, in the $K_{e4}^{\pm}$ channel.

\begin{thebibliography}{9}

%\cite{Fanti:1999nm}
\bibitem{Fanti:1999nm}
  V.~Fanti {\it et al.}  [NA48 Collaboration],
%   ``A new measurement of direct CP violation in two pion decays of the  neutral
  %kaon,''
  Phys.\ Lett.\ B {\bf 465}, 335 (1999)
% [arXiv:hep-ex/9909022].
%%CITATION = HEP-EX 9909022;%%

%\cite{Peyaud:2004tj}
\bibitem{Peyaud:2004tj}
  B.~Peyaud,
%  ``KABES: A novel beam spectrometer for NA48,''
  Nucl.\ Instrum.\ Meth.\ A {\bf 535}, 247 (2004).

%\cite{Good:1959}
\bibitem{Good:1959}
  J. D.~Good,
%  ``Pion spectrum in Radiative $K^{\pi}+$ Decay,''
  PHYS. REV. \textbf{113},  352 (1959).

\bibitem{Abrams:1972rj}
  R.~J.~Abrams {\it et al.},
%  ``Evidence For Direct Emission In The Decay $K^{\pm}\to\pi^{\pm}\pi^{0}\gamma$,''
  Phys.\ Rev.\ Lett.\  {\bf 29}, 1118 (1972).

%\cite{Adler:2000it}
\bibitem{Adler:2000it}
  S.~C.~Adler {\it et al.}  [E787 Collaboration],
  Phys.\ Rev.\ Lett.\  {\bf 85}, 4856 (2000)

%\cite{Aliev:2002tq}
\bibitem{Aliev:2002tq}
  M.~A.~Aliev {\it et al.}  [E470 Collaboration],
  Phys.\ Lett.\ B {\bf 554}, 7 (2003)

%\cite{Tsunemi}
\bibitem{Tsunemi}
T. Tsunemi, ``New Results on $K^+ \to \pi^+ \pi^0 \gamma$ from E787'', talk given at Kaon 2005 Chicago, June 2005.

%\cite{Aliev:2005}
\bibitem{Aliev:2005}
  M.~A.~Aliev {\it et al.}  [E470 Collaboration],
  [arXiv:hep-ex/0511060 v1], 2005


%\cite{Shimizu:2004it}
\bibitem{Shimizu:2004it}
  S.~Shimizu {\it et al.}  [E470 Collaboration],
  Phys.\ Rev.\ D {\bf 70}, 037101 (2004)
\end{thebibliography}
\end{document}
