On the two-loop contributions to the pion mass

TL;DR

This paper presents a simplified, explicit two-loop calculation of the pion mass within three-flavour chiral perturbation theory, including an approximation for the K-K-eta contribution and discussing numerical implications.

Contribution

It derives a minimal set of master integrals for two-loop pion mass calculations, providing an explicit analytic representation and an accurate approximation for the K-K-eta contribution.

Findings

Explicit two-loop pion mass expression in chiral perturbation theory.

Simplified representation of tensorial sunset integrals.

Approximate analytic form for K-K-eta contribution.

Abstract

We derive a simplified representation for the pion mass to two loops in three-flavour chiral perturbation theory. For this purpose, we first determine the reduced expressions for the tensorial two-loop 2-point sunset integrals arising in chiral perturbation theory calculations. Making use of those relations, we obtain the expression for the pion mass in terms of the minimal set of master integrals. On the basis of known results for these, we arrive at an explicit analytic representation, up to the contribution from K-K-eta intermediate states where a closed-form expression for the corresponding sunset integral is missing. However, the expansion of this function for a small pion mass leads to a simple representation which yields a very accurate approximation of this contribution. Finally, we also give a discussion of the numerical implications of our results.