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Disentangling the final–state interaction in the decay
of a tau lepton into a pion and a kaon
IPNO Participation: B. Moussallam
Les désintégrations semi hadroniques du lepton tau font intervenir à la fois les intéraction faibles et fortes.
Les états finaux d’étrangeté S=-1 permettent de mesurer l’élément de matrice CKM Vus et d’accéder à la
masse du quark étrange. Nous considérons ici le mode de désintégration en un kaon et un pion et montrons
comment déconvoluer l’effet de l’intéraction forte dans l’état final. Nous utilisons pour cela des propriétés
rigoureuses: analyticité des facteurs de forme, analyticité des amplitudes de diffusion et unitarité
ainsi que la symétrie de saveur et un traitement perturbatif de sa brisure. Nous montrons comment le
problème d’intéraction dans l’état final peut être résolu ici grâce aux proriétés “simples” de la diffusion pionkaon
dans les ondes S et P. Finalement, nous construisons une amplitude de désintégration ne contenant
qu’un seul parameter libre (lié à la brisure de symétrie de saveur) et comparons le résultat pour le spectre
en énergie de la paire pion-kaon à la mesure récente de haute statistique de la collaboration Belle.
Introduction
Decays of the tau lepton into hadrons probe the
weak as well as the strong interactions in the final
state. Cabbibo suppressed decays, in which the
final hadronic state has strangeness S=-1 are of
particular interest for the precision determination of
the CKM matrix element Vus (probing the unitarity
of the CKM matrix) and the determination of the
strange quark mass via sum rules. New data provided
by the Babar and Belle B factories have statistics
a thousand times larger than earlier generation
experiments and forthcoming tau/charm factories
will improve this further. In this work we focus on
the simple S=-1 decay mode involving one pion
and one kaon. We discuss how to disentangle the
effect of the strong final state interaction using model
independent rigorous methods.
Properties of the scalar and vector pion-kaon
form-factors
The tau decay amplitude can be expressed in
terms of two functions of t, the energy squared of
the pion-kaon system: f + (t) and f 0 (t) called the vector
and scalar form factor. The basic property of
these functions which we use is that they are analytic
in the complex plane of the variable t with a
right hand cut on the real axis. Furthermore, their
behaviour when t goes to infinity is known from
QCD to be of the form α s (t)/t, implying that they
can be written as unsubtracted dispersion relations
in terms of their imaginary parts on the cut.
Properties of pion-kaon scattering in the S and
the P wave
The final-state interaction appears upon expressing
these imaginary parts from unitarity: the scalar
and vector form-factors involve pion-kaon T-matrix
elements in the S-wave and inthe P-wave respectively.
Experimentally, pion-kaon scattering has
been studied in some detail in particular by the
LASS experiment [1]. The key feature which allows
to disentangle the effect of the final-state inte-
raction is that inelasticity remains « simple » up to
sufficiently large values of t, namelu it involves only
(to a good approximation) two-body channels. In
the S-wave the main inelastic channel is K-eta’
while in the P-wave the two main inelastic channels
are K*-eta’ and K-rho. Using this property the
dispersion relations are transformed into coupledchannel
integral equations of Muskhelishvili type.
Based on the LASS experimental results[1] we
have constructed a 2x2 T-matrix in the S-wave and
a 3x3 T-matrix in the P-wave. In order to solve the
equations there finally remain to specify the values
at t=0 of six form-factors ( in volving the elastic and
the inelastic channels). We determine these using
flavour symmetry and its breaking at first order in
terms of a single indetermined parameter. The result
of this construction is illustrated in the figure
below which shows the data measured by the Belle
collaboration.
References:
[1]Aston et al., Nucl. Phys. B296 (1988) 493
[2]Epifanov et al.,Phys. Lett. B654 (2007) 65
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