Constraints on the Existence of Chiral Fermions in Interacting Lattice Theories

TL;DR

This paper demonstrates that under certain conditions, interacting lattice theories cannot host chiral fermions, implying constraints on lattice formulations of chiral gauge theories.

Contribution

It provides a proof that lattice theories satisfying locality, a relativistic continuum limit without massless bosons, and pole-free current vertices must have a vector-like spectrum, ruling out chiral fermions.

Findings

Interacting lattice theories with specified conditions are vector-like.

Chiral fermions cannot exist under these lattice constraints.

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

Abstract

It is shown that an interacting theory, defined on a regular lattice, must have a vector-like spectrum if the following conditions are satisfied: (a)~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.