mu-Squared Dependent Deviation of the Non Perturbative ZA,MOM from the True Axial Renormalisation Constant, Implied by Ward Identity

TL;DR

This paper investigates the inconsistency in defining the non-perturbative axial current renormalisation constant using MOM schemes, revealing a mu^2 dependence linked to Ward identities and chiral symmetry breaking.

Contribution

It demonstrates that the axial Ward identity implies a mu^2 dependence of ZA,MOM, challenging the standard normalization condition used in lattice calculations.

Findings

ZA,MOM shows a mu^2 dependence near the chiral limit.

The mu^2 dependence is related to invariants in the pseudoscalar vertex.

This dependence persists due to spontaneous chiral symmetry breaking.

Abstract

It is recalled why, as already stated in a previous paper, there seems to be an inconsistency in identifying the non perturbative ZA,MOM as the renormalisation of the axial current, or equivalently, in setting as normalisation condition that the renormalised vertex=1 at p^2 = mu^2 at some renormalisation scale mu, where p is the momentum in the legs. Indeed, unlike the vector case, the Ward-Takahashi (WT) identity for the axial current is shown to imply both the renormalisation scale independence of ZA and a mu2 dependence of ZA,MOM. This mu^2 dependence is simply related to certain invariants in the pseudoscalar vertex and can persist in the chiral limit due to the spontaneous breaking of chiral symmetry (pion pole). It is seen clearly in the mu^2 dependence of some lattice calculations of ZA,MOM/ZV,MOM near the chiral limit.