Gell-Mann-Oakes-Renner relation for multiple chiral symmetries

TL;DR

This paper investigates ambiguities in the pion mass expression arising from multiple chiral symmetries in Ginsparg-Wilson fermions, proposing a renormalisation group approach to define conserved currents and derive a consistent Gell-Mann-Oakes-Renner relation.

Contribution

It introduces a renormalisation group mapping method to resolve ambiguities in chiral symmetry currents for Ginsparg-Wilson fermions, leading to a unique pion mass expression.

Findings

Derived a conserved current independent of Ginsparg-Wilson symmetry choice.

Recovered standard expressions for Dirac operator, propagator, and chiral condensate.

Established a consistent Gell-Mann-Oakes-Renner relation with modified chiral condensate.

Abstract

As a first step towards considering a chiral perturbation theory for overlap fermions, we investigate whether there are any ambiguities in the expression for the pion mass resulting from multiple chiral symmetries. The concern is that, calculating the conserved current for Ginsparg Wilson chiral symmetries in the usual way, different expressions of the chiral symmetries 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. 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 that differs from the standard expression by cut-off artefacts, but 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. With this in place, we proceed to calculate the pion mass in the mapped theory as a function of the quark mass, and discover a unique expression for $F_\pi$ and $m_\pi$, recovering the usual Gell-Mann-Oakes-Renner relation, baring the substitution of the chiral condensate with its modified value. We hypothesise that the argument can be carried directly over to the lattice theory.