Pseudoscalar Decays into Two Photons in Chiral Perturbation Theory

Abstract

We study the one-loop corrections in chiral perturbation theory to the yy and yy* decays of the pseudoscalar mesons. The latter are divergent, thus requiring the existence of dimension-six terms contributing to the anomalous Lagrangian. Some examples of such terms are given. We also discuss the experimental consequences of the next-to-leading terms. Chiral perturbation theory' starts with the Lagrangian X2 and proceeds to a higher-order expansion in terms of momenta and quark masses. The resulting efkctive low-energy theory is completely known to nextto-leading order, once the g~h as been integrated out. It contains two distinct types of terms. The first ones are originated by pseudoscalar loops with vertices deducible from L2 giving rise to analytic and nonanalytic contributions. The latter are particularly interesting, since they are considered to be the dominant ones and, on the other hand, involve a dependence on the renormalization scale p. As a result of this, the second type of (counter)terms, eliminating that p dependence and the associated divergences, is required. The set of all the counterterms appearing in the next-order Lagrangian, L4, has been identified and extensively analyzed by Gasser and Leutwyler.

Decayπ0→γγto next to leading order in chiral perturbation theory

“…3 reside in the terms displayed explicitly through their dependence on the masses and decay constants, as well as in pieces that stem from continuum and excited states. In works that preceded that of Moussallam [12], such as references [9] and [10], such mixing corrections as well as the NLO effects of the latter kind were disregarded. Ignoring such effects implies that only NLO corrections which are absorbed into the π 0 decay constant remain, as it was shown in [9]; in that case and taking F π 0 = F π + , the predicted π 0 width is the previously mentioned 7.725 eV.…”

“…Within the two-flavor framework, wherein the strange quark is integrated out, any corrections to the amplitude are O(p 6 ) and reside entirely in F π 0 [9] or in the O(p 6 ) odd-intrinsic parity Chiral Lagrangian [11], also known as the O(p 6 ) Wess-Zumino (WZ) Lagrangian. At leading order, the ensuing theoretical prediction (taking F π 0 = F π + ) is Γ π 0 →γγ = 7.725 eV, a result that agrees well with the experimental world average within its generous error.…”

Working group on goldstone boson production and decay

“…This disagrees at the two sigma level with the experimental measurement of Antipov et al (1987); F 3π = 12.9 ± 0.9 ± 0.5. The chiral loop corrections have been estimated by Bijnens, Bramon and Cornet (1990) and are 6 to 12 % over the phase space relevant for this experiment. The loop parts are fairly small and the major part of the upwards corrections comes from the tree level contributions at order p 6 estimated via Vector Meson Dominance.…”

“…The loop parts are fairly small and the major part of the upwards corrections comes from the tree level contributions at order p 6 estimated via Vector Meson Dominance. Higher order corrections have been estimated by Holstein (1996) by unitarizing the CHPT calculation of Bijnens, Bramon and Cornet (1990). An alternative way to estimate this is to use models of the meson structure.…”

“…Measurements of the double Dalitz decays and the formfactors in there as well as double tagged production experiments of π 0 , η and η ′ are therefore very welcome. The decay η → π + π − γ has a width of 58 ± 6 eV while the one-loop CHPT result is about 47 eV (Bijnens, Bramon and Cornet (1990)). Improved measurements of the form-factors and the decay with the photon off-shell would provide information about precisely which type of contributions are the reason for this discrepancy.…”