Analytic representations of $m_K$, $F_K$, $m_\eta$ and $F_\eta$ in two loop $SU(3)$ chiral perturbation theory

TL;DR

This paper derives exact analytic expressions for the masses and decay constants of kaons and eta mesons in two-loop SU(3) chiral perturbation theory, using Mellin-Barnes methods, and analyzes their numerical significance.

Contribution

It provides fully analytic, exact expressions for kaon and eta masses and decay constants, improving upon previous approximate or numerical methods in chiral perturbation theory.

Findings

Derived exact analytic formulas for $m_K$, $F_K$, $m_ ext{eta}$, and $F_ ext{eta}$.

Numerical analysis of loop contributions and low-energy constants.

Presented approximate formulas for comparison with lattice results.

Abstract

In this work, we consider expressions for the masses and decay constants of the pseudoscalar mesons in $SU(3)$ chiral perturbation theory. These involve sunset diagrams and their derivatives evaluated at $p^2=m_P^2$ ($P=\pi, K, \eta$). Recalling that there are three mass scales in this theory, $m_\pi$, $m_K$ and $m_\eta$, 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 $m_\pi$ and $F_\pi$, the pion mass and decay constant. We provide fully analytic exact expressions for $m_K$ and $m_\eta$, the kaon and eta masses, and $F_K$ and $F_\eta$, 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 low-energy constants is presented. We also present a set of approximate analytic expressions for $m_K$, $F_K$, $m_\eta$ and $F_\eta$ that facilitate comparisons with lattice results. Finally, we show how exact analytic expressions for $m_\pi$ and $F_\pi$ may be obtained, the latter having been used in conjunction with the results for $F_K$ to produce a recently published analytic representation of $F_K/F_\pi$.