Realization of Chiral Symmetry in the ERG

TL;DR

This paper explores how chiral symmetry is realized in a linear sigma model within the ERG framework, deriving a generalized Ginsparg-Wilson relation and constructing non-perturbative solutions that clarify symmetry realization.

Contribution

It introduces a family of solutions to the Ward-Takahashi identities in the ERG framework, demonstrating different realizations of chiral symmetry, including a continuum analog of lattice operators.

Findings

Derived a generalized Ginsparg-Wilson relation from Ward-Takahashi identities.

Constructed non-perturbative solutions illustrating various chiral symmetry realizations.

Showed existence of Dirac fields with standard chiral transformations in the continuum.

Abstract

We discuss within the framework of the ERG how chiral symmetry is realized in a linear $\sigma$ model. A generalized Ginsparg-Wilson relation is obtained from the Ward-Takahashi identities for the Wilson action assumed to be bilinear in the Dirac fields. We construct a family of its non-perturbative solutions. The family generates the most general solutions to the Ward-Takahashi identities. Some special solutions are discussed. For each solution in this family, chiral symmetry is realized in such a way that a change in the Wilson action under non-linear symmetry transformation is canceled with a change in the functional measure. We discuss that the family of solutions reduces via a field redefinition to a family of the Wilson actions with some composite object of the scalar fields which has a simple transformation property. For this family, chiral symmetry is linearly realized with a continuum analog of the operator extension of $\gamma_5$ used on the lattice. We also show that there exist some appropriate Dirac fields which obey the standard chiral transformations with $\gamma_5$ in contrast to the lattice case. Their Yukawa interactions with scalars, however, becomes non-linear.