Search for Chiral Fermion Actions on Non-Orthogonal Lattices

TL;DR

This paper reviews the challenges in designing non-orthogonal lattice fermion actions that maintain chiral symmetry, minimal doubling, and avoid unwanted operators, highlighting the difficulty in achieving all three simultaneously.

Contribution

It provides an analysis of existing non-orthogonal fermion actions and discusses the complex balance needed to satisfy key lattice QCD properties.

Findings

Existing actions lack a perfect balance of symmetry, doubling, and operator suppression.

No current non-orthogonal lattice action achieves all desired properties simultaneously.

Radiative corrections generate unwanted operators due to symmetry limitations.

Abstract

The graphene-inspired fermion actions recently proposed by Creutz and Borici have sparked interest in the use of non-orthogonal lattices in lattice QCD. These fermion actions have the desired chiral symmetry and have the minimal doubling required by the Nielsen-Ninomiya no-go theorem. However, due to the lack of discrete symmetries, radiative corrections in the gauged lattice theory will lead to the generation of unwanted relevant and marginal operators. Other similarly motivated non-orthogonal fermion actions avoid these unwanted operators, but introduce incorrect continuum behavior or excessive fermion doubling. A delicate balance of symmetry is required for chiral symmetry, minimal doubling, and no relevant operators, and to date, no non-orthogonal lattice action has accomplished this balance.