The conserved axial current in the presence of multiple chiral symmetries

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.