The Pion Sum Rule beyond the Chiral Limit

TL;DR

This paper extends the pion sum rule beyond the chiral limit by incorporating finite quark mass effects using nonperturbative techniques and tests it with tau-decay data.

Contribution

It introduces a generalized form of the pion sum rule that accounts for linear quark-mass effects, beyond the traditional chiral limit approximation.

Findings

The generalized sum rule accurately predicts pion mass differences.

The dispersion relation is validated with experimental tau-decay data.

Nonperturbative methods effectively incorporate finite-mass effects.

Abstract

The difference between the charged and neutral pion masses can be predicted from a well-known dispersion relation involving an infinite-energy integral over experimental data, the pion sum rule. This relation, however, holds only in the chiral limit. Here we combine several nonperturbative techniques to determine the form of the physical finite-energy integral, thereby generalizing this sum rule to include effects linear in the light-quark masses. We test the dispersion relation using hadronic tau-decay data.