Revisiting the Vector and Axial-vector Vacuum Susceptibilities

TL;DR

This paper analytically revisits the vector and axial-vector vacuum susceptibilities using Ward-Takahashi identities, confirming known results and deriving the axial susceptibility as three-fourths of the pion decay constant squared.

Contribution

It provides an analytical derivation of vacuum susceptibilities in the chiral limit, confirming the vector susceptibility as zero and the axial susceptibility as related to the pion decay constant, also reproducing the Weinberg sum rule.

Findings

Vector vacuum susceptibility is zero in the chiral limit.

Axial-vector vacuum susceptibility equals three-fourths of the pion decay constant squared.

Reproduction of the Weinberg sum rule.

Abstract

We re-investigate the vector and axial-vector vacuum susceptibilities by taking advantage of the vector and axial-vector Ward-Takahashi identities. We show analytically that, in the chiral limit, the vector vacuum susceptibility is zero and the axial-vector vacuum susceptibility equals three fourths of the square of the pion decay constant. Besides, our analysis reproduces the Weinberg sum rule.