Analytic representations of mK , FK , mη

Abstract

In this work, we consider expressions for the masses and decay constants of the pseudoscalar mesons in SUð3Þ chiral perturbation theory. These involve sunset diagrams and their derivatives evaluated at p 2 ¼ mRecalling that there are three mass scales in this theory, m π , m K and m η , there are instances when the finite part of the sunset diagrams do not admit an expression in terms of elementary functions, and have therefore been evaluated numerically in the past. In a recent publication, an expansion in the external momentum was performed to obtain approximate analytic expressions for m π and F π , the pion mass and decay constant. We provide fully analytic exact expressions for m K and m η , the kaon and eta masses, and F K and F η , the kaon and eta decay constants. These expressions, calculated using Mellin-Barnes methods, are in the form of double series in terms of two mass ratios. A numerical analysis of the results to evaluate the relative size of contributions coming from loops, chiral logarithms as well as phenomenological lowenergy constants is presented. We also present a set of approximate analytic expressions for m K , F K , m η and F η that facilitate comparisons with lattice results. Finally, we show how exact analytic expressions for m π and F π may be obtained, the latter having been used in conjunction with the results for F K to produce a recently published analytic representation of F K =F π .

FK/Fπ from Möbius domain-wall fermions solved on gradient-flowed HISQ ensembles

“…We find that the precision of our results requires including terms higher than NLO, and we have to work at a hybrid N 3 LO order to obtain a good description of our data. Therefore, we will begin with a discussion of the full N 2 LO χPT theory expression for F K =F π in the continuum limit [64][65][66][67].…”

New series representations for the two-loop massive sunset diagram

“…Driven by the needs of ChPT, in recent work [10,11] (see also [12,13]), analytic representations of the masses and decay constants of the light pseudoscalar mesons have been derived at the two-loop level. The relevant sunset diagrams that appear here have special values of the masses m 1 , m 2 and m 3 and squared external momentum p 2 (all in terms of m π , m K and m η ) that do not belong to the convergence regions of the analytic expressions given in [1].…”

“…The relevant sunset diagrams that appear here have special values of the masses m 1 , m 2 and m 3 and squared external momentum p 2 (all in terms of m π , m K and m η ) that do not belong to the convergence regions of the analytic expressions given in [1]. One of the aims of [10,11] was to give the ChPT results in terms of series representations which are easy to handle and computationally efficient at the same time. Therefore, to get the required results in [10,11] we have followed the spirit of [1].…”

“…One of the aims of [10,11] was to give the ChPT results in terms of series representations which are easy to handle and computationally efficient at the same time. Therefore, to get the required results in [10,11] we have followed the spirit of [1]. However, instead of considering the most general two-loop sunset integral as in [1], we have focused on the particular sunset configurations needed in ChPT: we have developed the Mellin-Barnes approach to evaluate the ChPT sunsets and analytical results were presented in terms of double series of Kampé de Fériet type (details of this derivation will be given in [14]).…”

Unveiling regions in multi-scale Feynman integrals using singularities and power geometry

“…Recent analytical progress can be found in refs. [4,5]. The MoR was applied by Kaiser and Schweizer [6] for studying π − k scattering processes in the context of ChPT.…”