The conserved axial current in the presence of multiple chiral symmetries

TL;DR

This paper investigates ambiguities in the conserved axial current in theories with multiple chiral symmetries, demonstrating that proper current construction removes dependence on the choice of symmetry, ensuring consistent physical observables.

Contribution

It introduces a renormalisation group approach to define a unique conserved current in continuum Ginsparg-Wilson theories, resolving ambiguities related to multiple chiral symmetries.

Findings

Proper current construction removes symmetry choice dependence

Standard expressions for Dirac operator and condensate are recovered

No observable depends on Mandula symmetry when currents are correctly defined

Abstract

In response to a recent work by Mandula, we investigate whether there are any ambiguities in the expression for the pion mass resulting from multiple chiral symmetries. If the conserved current for Ginsparg Wilson chiral symmetries is calculated in the usual way, different expressions of the chiral symmetry lead to different currents. This implies an ambiguity in the definition of the pion and pion decay constant for all Ginsparg-Wilson expressions of the Dirac operator, including the overlap operator on the lattice (although all these currents would have the same continuum limit). We use a renormalisation group mapping procedure to consider local chiral symmetry transformations for a continuum Ginsparg-Wilson "Dirac-operator." We find that this naturally leads to an expression for the conserved current which is independent of which of the Ginsparg-Wilson symmetries is chosen. We recover the standard expressions for the massive Dirac operator, propagator, and chiral condensate. Our main conclusion is that, when the currents are properly constructed and consistently applied, no observable depends on which Mandula symmetry is used; at least in these continuum Ginsparg-Wilson theories. We will consider whether the same argument applies to lattice theories in a subsequent paper.