# An Analytic Analysis of the Pion Decay Constant in Three-Flavoured Chiral Perturbation Theory

## TL;DR

This paper provides an analytic two-loop calculation of the pion decay constant and mass in three-flavor chiral perturbation theory, including expansions in the strange quark mass and a simplified form for lattice data fitting.

## Contribution

It introduces an analytic two-loop representation of the pion decay constant in SU(3) chiral perturbation theory, with explicit expressions and a form suitable for lattice data analysis.

## Key findings

- High accuracy of the analytic representation demonstrated.
- Strong dependence of F_pi on low energy constants.
- Effective simplified form for lattice data fitting provided.

## 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 upto 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 $F_{\pi}$ and $M_{\pi}^2$ in the strange quark mass in the isospin limit, and 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.