An Introduction to Chiral Symmetry on the Lattice

TL;DR

This paper reviews how chiral symmetry is implemented on the lattice in quantum chromodynamics, highlighting the Ginsparg-Wilson relation and its solutions like overlap fermions, which preserve chiral symmetry without fermion doubling.

Contribution

It provides a comprehensive introduction to lattice chiral symmetry, emphasizing the Ginsparg-Wilson relation and recent fermion formulations that maintain chiral symmetry exactly.

Findings

Ginsparg-Wilson relation ensures exact lattice chiral symmetry.

Overlap fermions satisfy the Ginsparg-Wilson relation.

Chiral symmetry can be realized without fermion doubling on the lattice.

Abstract

The $SU(N_f)_L \otimes SU(N_f)_R$ chiral symmetry of QCD is of central importance for the nonperturbative low-energy dynamics of light quarks and gluons. Lattice field theory provides a theoretical framework in which these dynamics can be studied from first principles. The implementation of chiral symmetry on the lattice is a nontrivial issue. In particular, local lattice fermion actions with the chiral symmetry of the continuum theory suffer from the fermion doubling problem. The Ginsparg-Wilson relation implies L\"uscher's lattice variant of chiral symmetry which agrees with the usual one in the continuum limit. Local lattice fermion actions that obey the Ginsparg-Wilson relation have an exact chiral symmetry, the correct axial anomaly, they obey a lattice version of the Atiyah-Singer index theorem, and still they do not suffer from the notorious doubling problem. The Ginsparg-Wilson relation is satisfied exactly by Neuberger's overlap fermions which are a limit of Kaplan's domain wall fermions, as well as by Hasenfratz and Niedermayer's classically perfect lattice fermion actions. When chiral symmetry is nonlinearly realized in effective field theories on the lattice, the doubling problem again does not arise. This review provides an introduction to chiral symmetry on the lattice with an emphasis on the basic theoretical framework.