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%  Contribution of H. Sazdjian to the Conference `Quark Confinement and
%              the Hadron Spectrum', Wien 2000
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%       `Relationship of pionium lifetime with pion scattering lengths
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\begin{document}

\title{RELATIONSHIP OF PIONIUM LIFETIME WITH PION SCATTERING LENGTHS
IN GENERALIZED CHIRAL PERTURBATION THEORY}  

\author{H. SAZDJIAN}

\address{Groupe de Physique Th\'eorique, Institut de Physique Nucl\'eaire,\\
Universit\'e Paris XI, F-91406 Orsay Cedex, France\\
E-mail: sazdjian@ipno.in2p3.fr}

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\maketitle\abstracts{The pionium lifetime is calculated in the framework
of the quasipotential-constraint theory approach, including the sizable
electromagnetic corrections. The framework of generalized chiral
perturbation theory allows then an analysis of the lifetime value as a
function of the $\pi\pi$ $S$-wave scattering lengths with isospin $I=0,2$,
the latter being dependent on the quark condensate value.}

\noindent 

The DIRAC experiment at CERN is expected to measure the pionium
lifetime with a 10\% accuracy. The pionium is an atom made of $\pi^+\pi^-$,
which decays under the effect of strong interactions into $\pi^0\pi^0$.
The physical interest of the lifetime is that it gives us information
about the $\pi\pi$ scattering lengths. The nonrelativistic formula of the 
lifetime was first obtained by Deser {\it et al.} \cite{dgbt}:
\begin{equation} \label{e1}
\frac{1}{\tau_0} = \Gamma_0 = \frac{16\pi}{9}\sqrt
{\frac{2\Delta m_{\pi}}
{m_{\pi^+}}} \frac{(a_0^0-a_0^2)^2}{m_{\pi^+}^2} |\psi_{+-}(0)|^2,\ \ \
\ \ \Delta m_{\pi}=m_{\pi^+}-m_{\pi^0},
\end{equation}
where $\psi_{+-}(0)$ is the wave function of the pionium at the origin
(in $x$-space) and $a_0^0$, $a_0^2$, the $S$-wave scattering lengths with
isospin 0 and 2, respectively.
\par
The evaluation of the relativistic corrections to this formula can be
done in a systematic way in the framework of chiral perurbation theory
($\chi PT$) \cite{gl}, in the presence of electromagnetism \cite{uk}. 
There arise essentially two types of correction. (i) The pion-photon 
radiative corrections, which are similar to those met in conventional QED.
(ii) The quark-photon radiative corrections, which appear through terms
where the photon field is not explicitly present and which are mainly 
responsible for the pion mass difference at lowest-order. 
\par
Three different methods of evaluation have been used for the study of the
pionium bound state in the framework of $\chi PT$. The first method uses a 
three-dimensionally reduced form of the Bethe--Salpeter equation within
the quasi\-poten\-tial--constraint theory approach  \cite{jss}. The second
method uses the Bethe--Salpeter equation with the Coulomb gauge \cite{illr}.
The third one uses the approach of nonrelativistic effective theory 
\cite{gglr}. All the above approaches lead to
similar estimates, on the order of $6\%$, for the relativistic corrections
to the nonrelativistic formula of the pionium decay width. 
\par
The theoretical interest of the $\pi\pi$ scattering lengths is that they
allow us to estimate the value of the quark condensate in QCD. The 
fundamental order parameter of spontaneous chiral symmetry breaking being
$F_{\pi}$, the pion weak decay constant, other order parameters may 
eventually vanish in the chiral limit without contradicting chiral symmetry
breaking, as long as $F_{\pi}$ remains different from zero in that limit.
Such an issue is intimately dependent on the mechanism of chiral symmetry 
breaking. In standard $\chi PT$ \cite{gl}, it is assumed that the quark 
condensate parameter, defined as $<0|\overline qq|0>/F_{\pi}^2$, is on the
order of the hadronic mass scale ($\sim 1$ GeV). This hypothesis is verified 
in the sigma-model and the Nambu--Jona-Lasiono model. The vacuum state here
is similar to a ferromagnetic type medium. On the other hand, in an 
antiferromagnetic type medium, one would have a vanishing quark condensate 
and yet chiral symmetry would still be broken \cite{l,s}. An intermediate 
possibility, due to an eventual phase transition in QCD for large values of
the light quark flavor number, was also advocated recently \cite{dgs}.
\par
Generalized $\chi PT$ is a framework in which the quark condensate value is
left as a free parameter subjected to an experimental evaluation \cite{fss}.
The Goldstone boson scattering amplitudes are sensitive to the quark
condensate value and hence their experimental measurment gives us an
estimate of the latter quantity. Thus, in the $\pi\pi$ scattering amplitude
relatively small values of the $S$-wave isospin-0 scattering length $a_0^0$,
on the order of, say, 0.21-0.22, correspond to the predictions of standard 
$\chi PT$, while relatively large values of $a_0^0$, on the
order of, say, 0.28-0.36, correspond to small values of the quark condensate
parameter. 
\par
We have redone the analysis of the pionium lifetime in the framework of
generalized $\chi PT$ \cite{sz}. Eliminating the quark condensate parameter
in favor of the combination $(a_0^0-a_0^2)$ we have calculated the pionium
lifetime as a function of $(a_0^0-a_0^2)$. The corresponding curve is
presented in Fig. \ref{f1}.
\par
\bfg
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\end{picture}
\caption{The pionium lifetime as a function of the combination
$(a_0^0-a_0^2)$ of the $S$-wave scattering lengths (full line). 
The band delineated by the dotted lines takes into account the estimated
uncertainties (2-2.5\%).}
\lb{f1}
\end{center}
\efg
Values of the lifetime close to 3 fs, lying above 2.9 fs, say, would 
confirm the scheme of standard $\chi PT$. Values of the lifetime lying 
below 2.4 fs remain outside the domain of predictions of standard $\chi PT$
and would necessitate an alternative scheme of chiral symmetry breaking.
Values of the lifetime lying in the interval 2.4-2.9 fs, because of the
possibly existing uncertainties, would be more difficult to interpret and
would require a more refined analysis.
\par

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\par

\end{thebibliography}

\end{document}

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