exotic nuclei structure and reaction noyaux exotiques ... - IPN - IN2P3

Chiral expansions of the π 0 lifetime
IPNO Participation: B. Moussallam
Collaboration : K. Kampf (Charles University, Prague)
L’amplitude de désintégration du pion neutre en deux photons est une observable clé pour la brisure spontannée
de la symétrie chirale en QCD , la nature de quasi boson de Nambu-Goldstone du pion et l’anomalie.
L’amplitude est connue exactement dans la limite chirale et l’amplitude physique s’exprime comme un
développement en fonction des masses des quarks dans le cadre d’une théorie effective: ChPT. Dans ce
travail nous avons a) calculé pour la première fois la correction à deux boucles en ChPT montrant, en particulier
que des termes logarithmiques apparaissent à cet ordre et b) nous avons étudié le matching, à l’ordre
d’une boucle entre les développements à deux et trois saveurs ce qui permet d’estimer les constantes
de couplage chirales et le role des mélanges π-η et π-η’ . Ce travail est motivé par l’expérience PRIMEX
qui cherche à mesurer la durée de vie avec une précision de 1% (contre 10% actuellement) et par les
simulations numériques de QCD sur réseau qui permettent de faire varier les masses des quarks.
Introduction
The decay amplitude of the neutral pion into two
photons is a key observable with respect to the
spontaneous breaking of chiral symmetry in QCD,
the Nambu-Goldstone nature of the neutral pion
and the anomaly. In this work we compute, firstly,the
corrections of order NNLO in the quark
mass expansion of the amplitude. This is useful in
view of forthcoming results from lattice QCD simulations
in which the quark masses can be varied.
We have furthermore displayed the presence of
chiral logarithms in the expansion, which are absent
at NLO. Secondly, we study the matching, at
order NLO, between the two-flavour and the three
flavour expansions. This allows us to refine the
phenomenological estimates of the chiral coupling
constants at NLO and the numerical prediction of
the decay amplitude in view of the forthcoming
measurement, at the 1% level, of the PRIMEX collaboration.
Two-flavour NNLO expansion of π 0 amplitude
We consider first the expansion of the decay amplitude
as a function of the two quark masses
mu,md which can be performed using ChPT. The
leading order result is given exactly by the ABJ
anomaly
At NLO, one must compute one-loop diagrams and
use the NLO chiral Lagrangian which was classified
by Bijnens et al.[1]. When expressed in terms
of the physical pion mass and decay constant the
NLO contribution is a polynomial (no logarithms)
controled by two combinations of chiral coupling
constants. In this work, we have considered the
NNLO contributions, which could be enhanced by
chiral logarithms and are also necessary for matching
with lattice QCD simulations. The evaluation
involves computing a set of two-loop graphs, as
well as a set of one-loop graphs containing one
NLO vertex. The explicit result is the following:
It indeed contains two kinds of chiral logarithms,
with the coefficient of the quadratic logarithm depending
only on F as expected from Weinberg’s
arguments.
Matching two and three-flavour expansions
For physical quark masses we find that the NNLO
chiral logarithms are numerically small and it is
necessary to estimate the coupling constants involved
in the NLO contribution. We reconsider this
issue by matching the two-flavour and the threeflavour
expansions. A new method to perform this
directly at the level of the generating functionals
was recently proposed[2] . It amounts to express
the two-flavour coupling constants in terms of the
three-flavour ones as an expansion in inverse powers
of the strange quark mass. It turns out that
only two of these constants are involved. Using
experimental inputs from η decay, from η’ decay as
well as from π(1300) decay it is possible to determine
all the chiral couplings at NLO. We make the
following prediction for the width
Γ = (8.09 ± 0.08 ± 0.10) eV
Where the first uncertainty comes from md-mu and
the second from contributions quadratic in ms.
References
[1]J. Bijnens et al., Eur. Phys. J. C23 (2002) 539
[2]J. Gasser et al., Phys. Lett. B652 (2007) 21
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