Finite volume effects for the pion mass at two loops

TL;DR

This paper calculates the finite volume effects on the pion mass at two loops using Chiral Perturbation Theory and compares them with an extended Luscher formula, finding negligible contributions at certain volume scales.

Contribution

It provides a detailed two-loop analysis of finite volume effects on the pion mass and validates the extended Luscher formula's accuracy at larger volumes.

Findings

Finite volume effects are small for Mpi*L>2 and L>2fm.

Two-loop calculations confirm the reliability of the extended Luscher formula.

Finite volume effects are well understood and controlled in the meson sector.

Abstract

We evaluate the pion mass in finite volume to two loops within Chiral Perturbation Theory. The results are compared with a recently proposed extension of the asymptotic formula of Luscher. We find that contributions, which were neglected in the latter, are numerically very small at the two-loop level and conclude that for Mpi*L>2, L>2fm the finite volume effects in the meson sector are analytically well under control.