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\def\NIM{Nucl.~Instrum.~ Methods}
\def\NIMA{Nucl.~Instrum.~ Methods A}
\def\NPA{Nucl.~Phys.~A~}
\def\NPB{Nucl.~Phys.~B~}
\def\PLB{Phys.~Lett.~B~}
\def\PRL{Phys.~Rev.~Lett.~}
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\newcommand{\ttbs}{\char'134}
\newcommand{\AmS}{{\protect\the\textfont2
  A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}

\newcommand{\ppz}{\pi^{0}\pi^{0}}
\newcommand{\ppc}{\pi^{+}\pi^{-}}
\newcommand{\pgg}{\pi^{0}\rightarrow \gamma \gamma}
\newcommand{\trepc}{\rm \pi^{\pm} \ppc}
\newcommand{\ktrepc}{\rm \kpm \rightarrow \trepc}
\newcommand{\ktrep}{\rm K \rightarrow 3\pi}
\newcommand{\duep}{\rm \pi^{\pm} \pi^{0}}
\newcommand{\trepn}{\rm \pi^{\pm} \ppz}
\newcommand{\ktrepn}{\rm \kpm \rightarrow \trepn}
\newcommand{\mz}{\rm M_{00}}
\newcommand{\mzq}{\rm M_{00}^{2}}
\newcommand{\mchp}{\rm m_{+}}
\newcommand{\mchpq}{\rm m_{+}^{2}}
\newcommand{\thr}{\rm \mz = \mchp}
\newcommand{\thrq}{\rm \mzq = \mchp^{2}}
\newcommand{\pp}{\pi\pi}
\newcommand{\chex}{\pi^{+}\pi^{-}\rightarrow \ppz}
\newcommand{\ama}{(a_{0}-a_{2})}
\newcommand{\kp}{K^{+}}
\newcommand{\km}{K^{-}}
\newcommand{\kpm}{K^{\pm}}
\newcommand{\keq}{\rm \kpm \rightarrow \ppc e^{\pm} \nu_{e}}

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\title{Low energy dynamics and the $\pi \pi$ scattering lengths from the NA48/2 experiment at CERN}

\author{Giuseppina Anzivino \address{Dipartimento di Fisica, Universit\`{a} di Perugia,
via A. Pascoli, 06123 Perugia, Italy}
        \thanks{on behalf of the NA48/2 Collaboration:
        Cambridge-CERN-Chicago-Dubna-Edinburgh-Ferrara-Firenze-Main-Northwestern-Perugia-Pisa-Saclay-Siegen-Torino-Wien}}

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\runtitle{$\pp$ scattering lengths}
\runauthor{G. Anzivino}

\begin{document}

\begin{abstract}
The observation of a cusp-like structure in the $\mz$ invariant mass
distribution in the decay $\ktrepn$ by the NA48/2 experiment at CERN
has given a renewed interest in the $\pp$ scattering lengths. We
report the result of a study of a partial sample of $2.3\times10^7$
decays, the observation of an anomaly in the $\ppz$ mass spectrum in
the region around $\thr$, where $\mchp$ is the mass of the charged
pion, and the subsequent interpretation in terms of the $\pp$
scattering lengths. This anomaly, never observed in previous
experiments, has been interpreted as an effect due to the final
state charge exchange scattering process $\chex$ in $\ktrepc$. From
the distribution we can determine precisely $\ama$, the difference
between the $\pp$ scattering lengths in the isospin $I = 0$ and $I =
2$ states. \vspace{1pc}
\end{abstract}

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\maketitle

\section{Introduction}

The NA48/2 experiment at CERN has been designed with the main
purpose of searching for direct CP violation in the decays of $\kpm$
in three pions. The experiment has collected data in 2003 and 2004,
using simultaneous and focused $\kp$ and $\km$ beams of 60 GeV/c
momentum, for a total of $4\times10^{9}$ fully reconstructed
$\ktrepc$ and $0.1\times10^{9}$ $\ktrepn$. The result reported is
based on a partial sample of the 2003 data and amounts to
$2.3\times10^{7} decays$.

\section{Beam and detector}

Two simultaneous focused kaon beams of opposite charge, with a
central momentum of 60 GeV/c and a momentum band of $\pm 3.8 \%$ are
produced by a 400 GeV proton beam impinging on a 40 cm Be target.
The decay volume is a 114 m long vacuum tank; the $\trepn$ final
state is reconstructed by a magnetic spectrometer and a liquid
krypton calorimeter (LKr). Charged particles are measured by the
magnetic spectrometer, consisting of four drift chambers and a
dipole magnet located between the second and the third chamber; the
momentum resolution is $\sigma (p)/p=1.02\% \oplus 0.044\%p$ (p in
GeV/c). The magnetic spectrometer is followed by a scintillator
hodoscope consisting of two planes segmented into horizontal and
vertical strips and arranged in four quadrants (charged hodoscope).
The $\pgg$ decays are reconstructed using the LKr, an almost
homogeneous ionization chamber with an active volume of $\sim 10~
m^{3}$ and a $27~X_0$ thickness; the energy resolution is
$\sigma(E)/E=0.032/\sqrt{E}\oplus0.09/E\oplus0.0042$ (E in GeV). The
space resolution for single electromagnetic shower can be
parametrized as $\sigma_x=\sigma_y=0.42/\sqrt{E}\oplus0.06$ cm (E in
GeV). At a depth of $\sim 9.5 X_0$ inside the active volume of the
calorimeter, a hodoscope consisting of a plane of scintillating
fibres is installed (neutral hodoscope); the signals from the four
quadrants are used to give a fast trigger.

A more detailed description of the beams and detector can be found
elsewhere \cite{techart}.

\section{Trigger and event selection}

The $\ktrepn$ decays are selected by a two level trigger. A signal
in at least one quadrant of the charged hodoscope in coincidence
with an energy deposition in the calorimeter consistent with at
least two photons is required by the first level trigger. The second
level is a fast on-line processor that reconstructs the momentum of
charged particles and calculates the missing mass under the
assumption that the particle is a $\pi^\pm$ originating from the
decay of a 60 GeV $\kpm$ travelling along the beam axis. The main
$\duep$ background is rejected by the requirement that the missing
mass is not consistent with the $\pi^0$ mass. For further analysis
selection criteria on tracks and clusters are applied: at least one
charged track with momentum above 5 GeV/c and at least four energy
clusters in the calorimeter, each consistent with a photon and above
an energy threshold of 3 GeV are required. Additional cuts on time
consistency between tracks and photons and on the distance between
any two photons and each photon and the impact point of the track on
the LKr are applied in order to ensure full containment of the
electromagnetic showers. For each possible pair of photons we assume
that it originates from $\pgg$ decay and we calculate the distance
$D_{ik}$ between the $\pi^0$ decay vertex and the LKr:

\begin{eqnarray}
\nonumber
D_{ik}=\frac{\sqrt{E_{i}E_{k}[(x_{i}-x_{k})^{2}+(y_{i}-y_{k})^{2}]}}{m_{\pi^0}}
\end{eqnarray}

where $E_{i},E_{k}$ are the energies of the two photons and $x, y$
their impact point coordinates on LKr. The two photon pairs with the
smallest $D_{ik}$ difference are selected as the best combination
consistent with two $\pi^0$ mesons from $\ktrepn$ decay and the
arithmetic average of the two $D_{ik}$ values is used as the decay
vertex position. The final event selection requires that the
reconstructed $\trepn$ invariant mass differs from the nominal
$\kpm$ mass quoted in the PDG by at most $\pm$ 6 MeV. This
requirement is satisfied by $2.287 \times10^{7}$ events. The
fraction of events with wrong photon pairing in this sample is $\sim
0.25 \%$ as estimated by a high statistics Monte Carlo simulation.

\section{The cusp effect}

The invariant mass of the $\ppz$ system has been investigated in
order to study the formation of pionium atoms in $\ktrepn$ decays.
Thanks to the high statistics, the very good $\mzq$ resolution, due
the excellent intrinsic energy and spatial resolution of the LKr
calorimeter, and the proper $\mz$ reconstruction strategy, the data
revealed a structure in the region $\thr$, where $m_+$ is the mass
of the charged pion. Fig.\ref{fig:cusp} (upper part) shows the
$\mzq$ mass distribution; a sudden change of slope near $\thrq =
0.07792 (GeV/c^{2})^{2}$ is clearly visible. Fig.\ref{fig:zoom}
(lower part) is an enlargement of the region around $\thrq$. Such an
anomaly has not been observed in previous experiments.

\vspace{-0.3cm}
\includegraphics[width=18pc]{cusp}
  \begin{figure}[htb]
%  \vspace{9pt}\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\vspace{-2.2cm}
%  \caption{Distribution of the $\ppz$ invariant mass squared ($\mzq$)}
  \label{fig:cusp}
%\end{figure}
\hspace{0.2cm}
\includegraphics[width=18pc]{zoom}
%  \begin{figure}[htb]
%  \vspace{pt}\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\vspace{-1.9cm}
  \caption{Distribution of the $\ppz$ invariant mass squared,
  $\mzq$, (upper part). Enlargement of the region around $\thrq$, indicated by the arrow (lower part).}
  \label{fig:zoom}
  \end{figure}

\section{Theoretical interpretation}

The observed sudden change of slope suggests the presence of a
threshold "cusp" effect from the decay $\ktrepc$ contributing to the
$\ktrepn$ amplitude through the charge exchange process $\chex$. The
presence of a cusp at $\thrq$ in $\ppz$ elastic scattering due to
the effect of virtual $\ppc$ loops has been discussed first by
Meissner et al. \cite{meissner}. This contribution is directly
proportional to $\ama$ and displays a characteristic behavior when
the $\ppz$ mass is in the vicinity of the $\ppc$ threshold, where it
goes from dispersive to (dominantly) absorptive. Cabibbo recently
computed \cite{cabibbo} the $\ktrepn$ amplitude taking into account
the 1-loop diagram shown in Fig. \ref{fig:one_loop}.

\vspace{0.3cm}

\includegraphics[width=15pc]{one_loop}
  \begin{figure}[htb]
%  \vspace{9pt}\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\vspace{-2cm}
  \caption{The $\pi\pi$ rescattering diagram.}
  \label{fig:one_loop}
  \end{figure}

\vspace{-1cm}
In the Cabibbo theory, the $\ktrepn$ decay amplitude
is given by the sum of two terms: the "unperturbed" amplitude
$\mathcal{M}_{0}$ $\grave{a}$ la PDG plus a new term,
$\mathcal{M}_{1}$, proportional to the I=0 and I=2 S-wave $\pp$
scattering lengths (in the limit of exact isospin symmetry). The new
term changes from real to imaginary at $\thr$. The destructive
interference of $\mathcal{M}_{0}$ and $\mathcal{M}_{1}$ in the total
amplitude causes the cusp and the apparent lack of events below the
threshold. More recently Cabibbo and Isidori \cite{cabisi} have
computed $\mathcal{O}(a_{i}^{2})$ corrections to $\ktrep$ amplitudes
including one-loop (other rescattering processes) and two-loop
diagrams. In the limit of exact isospin symmetry the amplitude
depends on five S-wave scattering lengths, that can be expressed as
a linear combination of $a_0$ and $a_2$. This model has been used to
extract $\ama$ from the $\ppz$ invariant mass distribution with high
precision.

\section{Determination of $\ama$}

The experimental $\mzq$ spectrum has been fitted using different
theoretical models. In the first attempt to fit the data the
unperturbed amplitude $\mathcal{M}_{0}$ has been used:

\begin{eqnarray}
\nonumber \mathcal{M}_{0}=1+\frac{1}{2}g_{0}u
\end{eqnarray}

where $u=(s_{3}-s_{0})/\mchpq$ is the Lorentz-invariant variable,
$s_{i}=(P_{K}-P_{i})^2$ $(i=1,2,3)$, $P_{K}$ and $P_{i}$ are the
4-mumentum vectors of the initial kaon and of the three outgoing
pions (i=3 corresponds to the $\pi^\pm$), respectively;
$s_{0}=(s_{1}+s_{2}+s_{3})/3$ and $\mzq=s_3$. The free parameters of
the fit are $g_{0}$ and an overall normalization constant. Because
of the anomaly at $\thr$ it is impossible to find a reasonable fit
to the data. However fits with acceptable $\chi^2$ values are
obtained if the lower edge of the fit interval is raised few bins
above $\thr$. The quality of this fit is displayed in Fig.
\ref{fig:fit1}, where the quantity $\Delta$=(data-fit)/data is
plotted as a function of $\mzq$; it can be seen that in the region
$\mzq <(2m_+)^2$ the data fall below the prediction based on the
same parameters obtained from the fit region.

\includegraphics[width=18pc]{fit1}
  \begin{figure}[htb]
%  \vspace{9pt}\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\vspace{-2.2cm}
  \caption{$\Delta$=(data-fit)/data versus $\mzq$.}
  \label{fig:fit1}
  \end{figure}

In order to investigate this deficit, several checks against
instrumental effects have been performed. We find no evidence for
either resolution effects or acceptance non-linearities in the cusp
region. In addition, variation in shape of photon energy
distribution across the cusp agrees with MC prediction without cusp
and no difference is observed between $\kp$ and $\km$ nor between
data taken with opposite directions of the magnetic field. We
conclude that the deficit of events in the data in the region
$\mzq<\mchp^2$ is due to a real physical effect.

Using the Cabibbo one-loop model, a fit to the $\mzq$ distribution
in the interval $0.074<\mzq<0.097$ $(GeV/c^{2})^{2}$ has been
performed; the result, shown in Fig. \ref{fig:fit4}a, gives
$\chi^2=420.1$ for 148 degrees of freedom. One can see that this
model provides a much better but still unsatisfactory description of
the data. In particular the data points are systematically above the
fit in the region near $\thrq$. A significant improvement is
obtained when the Cabibbo-Isidori model is used. This fit has five
free parameters: $\ama\mchp$, $a_{2}\mchp$, the linear slope $g_0$,
the quadratic slope $h$ and an overall normalization constant. The
quality of the fit ($\chi^2=154.8$ for 146 degrees of freedom) is
shown in Fig. \ref{fig:fit4}b. A better fit ($\chi^2=149.1$ for 145
degrees of freedom, see Fig. \ref{fig:fit4}c) is obtained by adding
to the model a term describing the expected formation of the $\ppc$
atom ("pionium"), decaying to $\ppz$ at $\thr$. The best fit value
for the rate $\kpm \rightarrow \pi^\pm$+ pionium decay, normalized
to the $\ktrepc$ dacay rate, is ($1.6 \pm 0.66)\times 10^{-5}$, in
reasonable agreement with the predicted value $\sim 0.8 \times
10^{-5}$ \cite{silagadze}. Finally, since the rescattering model of
ref. \cite{cabisi} does not include radiative corrections, we prefer
to exclude from the final fit a region of seven consecutive bins
centered at $\thr$. The quality of this fit ($\chi^2=145.5$ for 139
degrees of freedom) is shown in Fig. \ref{fig:fit4}d. Taking into
account all systematic and external uncertainties, the final result,
taken as the arithmetic average of two independent analysis
\cite{paper}, is:
%\vspace{-0.5cm}
\begin{eqnarray}
\nonumber \ama\mchp=0.268\pm0.010(stat)\pm0.004(syst)\\
\nonumber
          \pm0.013(ext)\\
%\end{eqnarray}
%\begin{eqnarray}
\nonumber
a_{2}\mchp=-0.041\pm0.022(stat)\pm0.014(syst)
\end{eqnarray}

The external error is an additional theoretical error of $\pm 5\%$
on $\ama$ estimated in ref. \cite{cabisi} as the result of
neglecting higher order terms in the rescattering model. This
uncertainties have no significant effect on $a_{2}\mchp$. The two
statistical errors from the fit are strongly correlated, with a
correlation coefficient of -0.858. Performing the fit with
constraints imposed on $a_{0}$ and $a_{2}$ by analyticity and chiral
symmetry \cite{cgl} we obtain
\begin{eqnarray}
\nonumber a_{0}\mchp=0.220\pm0.006(stat)\pm0.004(syst)\\
\nonumber \pm0.011(ext)
\end{eqnarray}
which corresponds to
\begin{eqnarray}
\nonumber
\ama\mchp=0.264\pm0.006(stat)\pm0.004(syst) \\
\nonumber
          \pm0.013(ext)
\end{eqnarray}

\hspace{-0.8cm}
\includegraphics[width=18pc]{fit4.eps}
  \begin{figure}[htb]
%  \vspace{9pt}\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\vspace{-2cm}
  \caption{$\Delta$=(data-fit)/data versus $\mzq$ for various theoretical models.
            See text for explanation.}
  \label{fig:fit4}
  \end{figure}

It is worth to note that this analysis gives the first direct
measurement of $a_{2}$, though not as precise as that of $\ama$.
This result is compatible, within the errors, with the results
obtained by the BNL E865 \cite{E865} and the CERN DIRAC experiments
\cite{dirac}. It is also in very good agreement with theoretical
calculations performed in the framework of Chiral Perturbation
Theory (ChPT) \cite{colangelo}, which predict $\ama\mchp=0.265\pm
0.004$. Another theoretical calculation based on direct analysis of
$\pp$ scattering data without using chiral symmetry \cite{pelaez}
leads to the result ($\ama\mchp=0.278\pm0.016$), slightly different
and with a larger uncertainty, which also agrees with our result.

Recently Colangelo et al. (CGKR) \cite{cgkr} calculated the $\ktrep$
amplitudes within a non-relativistic effective Lagrangian framework,
by a double expansion in $a$ (scattering lengths) and $\epsilon$
(kinetic energies) at order $\epsilon^2$, $a\epsilon^3$,
$a^2\epsilon^2$; CGKR representation is valid in the whole decay
region. The amplitudes agree with Cabibbo-Isidori calculation up to
$a\epsilon^3$ and differ away from threshold at order $a^2$.


\section{Conclusions}

The cusp anomaly observed in the $\ppz$ invariant mass distribution
at $\thr$ can be interpreted in terms of an additional contribution
to the decay amplitude from the charge exchange process $\chex$.
From the spectrum we extracted $\ama$, the difference between the
I=0 and I=2 S-wave $\pp$ scattering lengths.

The determination of $\pp$ scattering lengths relies on a variety of
methods, such as the measurement of the $\keq$ decay, the
measurement of the pionium lifetime as well as the pion-nucleon
scattering near threshold. Our collaboration has studied the $\keq$
decay and recently has reported a preliminary result based on a
sample of $3.7 \times 10^{5}$ events; using the universal band
constraint a value of
$a_{0}\mchp=0.256\pm0.008_{stat}\pm0.007_{syst}\pm0.018_{theory}$
has been measured \cite{brigitte}. It is important to note that care
must be taken in comparing results obtained with different
techniques. They may be compared only in the same theoretical
framework.

The cusp method is very accurate, sensitive to the sign of $\ama$
and model independent, since it is based only on general assumption
of unitarity and analyticity. We note that a similar effect arises
in the interference between $K_{L}\rightarrow \pi^{0}\pi^{0}\pi^{0}$
and $K_{L}\rightarrow \pi^{+}\pi^{-}\pi^{0}$, followed by a $\chex$
process. The effect is smaller than in the charged kaon decay, but
could also lead to a determination of $\ama$. We are studying this
effect using a set of $0.1 \times 10^{9}$ events collected in 2000
by NA48, when the experiment was taking data to measure the direct
CP violation parameter $\varepsilon'/\varepsilon$.

To conclude, the study of a large sample of $\ktrepn$ decays with
excellent resolution on the $\ppz$ invariant mass has provided a
novel, precise determination of $\ama$, independent of other
methods. In the near future, the full data sample collected in 2003
and 2004 by NA48/2, which represents an increase of a factor 5 in
statistics, will be analyzed resulting in a further reduction of the
statistical error of the measurement. An improvement of the
rescattering model to include higher order terms and radiative
correction is expected on the theoretical part.

\begin{thebibliography}{9}

\bibitem{techart} R. J. Batley et al.,\PLB 634 (2006) 474.
\bibitem{meissner} \"{U}. G. Meissner, G. M\"{u}ller, S. Steininger, \PLB 406 (1997) 154;\\
                    \"{U}. G. Meissner, \NPA 629 (1998) 72.
\bibitem{cabibbo} N. Cabibbo \PRL 93 (2004) 121801.
\bibitem{cabisi} N. Cabibbo and G. Isidori JHEP 503 (2005) 21.
\bibitem{silagadze} Z.K. Silagadze, JETP Lett. 60 (1994) 72.
\bibitem{cgl} G. Colangelo, J. Gasser, H. Leutwyler \PRL 86 (2001) 5008.
\bibitem{E865} S. Pislak et al. (BNL E865 Collaboration), \PRD 67 (2003) 072004.
\bibitem{dirac} B. Adeva et al. (DIRAC Collaboration), \PLB 619 (2005) 50.
\bibitem{paper} R.J. Batley et al., (NA48/2 Collaboration), \PLB 633 (2006) 173.
\bibitem{colangelo} G. Colangelo, J. Gasser and H. Leutwyler \PLB 488 (2000) 261; \\
                    G. Colangelo, J. Gasser and H. Leutwyler \NPB 603 (2001) 125.
\bibitem{pelaez} J. R. Pelaez and F. J. Yndurain \PRD 71 (2005) 074016.
\bibitem{cgkr} G. Colangelo, J. Gasser, B. Kubis and A. Rusetsky, hep-ph/0604084.
\bibitem{brigitte} B. Bloch-Devaux "Recent results from NA48/2 on $Ke4$ and $\ktrepn$.
                    Interpretation in terms of $\pp$ scattering lengths", talk given at
                    QCD06, 3-7 July 2006, Montpellier, France.

\end{thebibliography}

\end{document}
