An asymptotic formula for the pion decay constant in a large volume

TL;DR

This paper derives an asymptotic formula for finite volume corrections to the pion decay constant, expressed via an integral over the three-pion to vacuum amplitude, and analyzes it numerically within chiral perturbation theory.

Contribution

It introduces a new asymptotic formula for finite volume effects on the pion decay constant using an integral over the three-pion amplitude, extending Luscher's approach.

Findings

The formula accurately predicts finite volume corrections at leading order.

Numerical analysis confirms the formula's validity at next-to-leading order.

The approach provides a practical method for lattice QCD calculations of decay constants.

Abstract

We derive an asymptotic formula a la Luscher for the finite volume correction to the pion decay constant: this is expressed as an integral over the < 3 \pi | A_\mu|0 > amplitude after proper subtraction of the pion pole contribution. We analyze the formula numerically at leading and next-to-leading order in the chiral expansion.