On the absence of chiral fermions in interacting lattice theories

TL;DR

This paper proves that under certain conditions, interacting lattice theories with internal symmetries cannot host chiral fermions, implying they are necessarily vector-like in the continuum limit.

Contribution

It demonstrates that lattice theories satisfying locality, a relativistic continuum limit without massless bosons, and pole-free current vertices cannot support chiral fermions, extending the Nielsen-Ninomiya theorem.

Findings

Spectrum is necessarily vector-like under specified conditions.

Chiral fermions cannot exist in these lattice theories.

The proof uses the inverse retarded propagator and Nielsen-Ninomiya theorem.

Abstract

We consider interacting theories with a compact internal symmetry group on a regular lattice. We show that the spectrum is necessarily vector-like provided the following conditions are satisfied: (a)~weak form of locality, (b)~relativistic continuum limit without massless bosons, and (c)~pole-free effective vertex functions for conserved currents. The proof exploits the zero frequency inverse retarded propagator of an appropriate set of interpolating fields as an effective quadratic hamiltonian, to which the Nielsen-Ninomiya theorem is applied. The main results of this paper have been reported in WIS-93/56-JUNE-PH, hep-lat/9306023.