Analytic representation of $F_K/F_\pi$ in two loop chiral perturbation theory

TL;DR

This paper provides an analytic formula for the ratio of kaon to pion decay constants in three-flavor two-loop chiral perturbation theory, using Kampé de Fériet series to express complex integrals, and demonstrates its effectiveness with lattice data fits.

Contribution

It introduces a novel analytic representation of $F_K/F_$ involving Kampe9 de Fe9riet series, simplifying complex two-loop calculations in chiral perturbation theory.

Findings

Good agreement with existing lattice data fits.

Compact analytic expressions derived for complex integrals.

Enhanced understanding of decay constant ratios in chiral perturbation theory.

Abstract

We present an analytic representation of $F_K/F_\pi$ as calculated in three-flavour two-loop chiral perturbation theory, which involves expressing three mass scale sunsets in terms of Kamp\'e de F\'eriet series. We demonstrate how approximations may be made to obtain relatively compact analytic representations. An illustrative set of fits using lattice data is also presented, which shows good agreement with existing fits.