An analytic analysis of the pion decay constant in three-flavoured chiral perturbation theory

Abstract

A representation of the two-loop contribution to the pion decay constant in SU(3) chiral perturbation theory is presented. The result is analytic up to the contribution of the three (different) mass sunset integrals, for which an expansion in their external momentum has been taken. We also give an analytic expression for the two-loop contribution to the pion mass based on a renormalized representation and in terms of the physical eta mass. We find an expansion of and in the strange-quark mass in the isospin limit, and we perform the matching of the chiral SU(2) and SU(3) low-energy constants. A numerical analysis demonstrates the high accuracy of our representation, and the strong dependence of the pion decay constant upon the values of the low-energy constants, especially in the chiral limit. Finally, we present a simplified representation that is particularly suitable for fitting with available lattice data.

Analytic representations of mK , FK , mηAnanthanarayan1, Bijnens2, Friot3 et al. 2018Phys. Rev. DSelf Cite160201View full textAdd to dashboardCite

“…In [12], m π was treated by taking an approximation at the level of the loop integral [12]. In [3], some of the present authors extended the former work to the case of F π , and were able to integrate out the s-quark from the expressions of the pion mass and decay constant, in the same way as the SUð3Þ ChPT reduces to the SUð2Þ version, reproducing known results in the chiral limit, as well as evaluating the departure from the chiral limit to leading order in the light quark mass.…”

“…(i) To extend the work of [3,12] to the case of the kaon and eta, and provide approximate analytic expressions for m K , m η , F K and F η that are easily amenable to fitting with lattice data. (ii) To provide exact (nonapproximate) two loop analytic expressions for all the pseusoscalar meson masses and decay constants.…”

“…(iii) To perform a first order study of the numerics of m K , m η , F K and F η to determine the relative contributions to them of their different components, as well as their dependence on the values of parameters such as the low energy constants of ChPT. Although this work is a sequel to [3], the approach taken here is completely different and novel. In the aforementioned work, as well as in [12], the three mass scale sunset integrals were approximated by taking an expansion in the external momentum p 2 ¼ m 2 π around zero.…”

“…In addition to the full results, we therefore also present a set of analytic approximations that may easily be fitted with data from lattice simulations. These approximations are obtained by suitably truncating the infinite series so that their omitted tails are numerically smaller than a chosen percentage of the central value obtained from the exact expression 3 for the inputs being considered. And as these inputs will depend on the precise lattice set being used, the approximations will need to be changed accordingly.…”

“…In a prior work, a different scheme was employed to obtain analytic approximations of m π and F π in SUð3Þ ChPT at two loops [3]. Recall that ChPT is an effective field theory for the pseudoscalar octet degrees of freedom, namely the pions, kaons and eta.…”