The $\zeta$-function answer to parity violation in three dimensional gauge theories

TL;DR

This paper investigates parity violation in 2+1-dimensional gauge theories with fermions, using $$-function regularization to analyze how parity-breaking effects induce a Chern-Simons term in the effective action.

Contribution

It introduces a $$-function regularization method to evaluate fermion currents, revealing two distinct sources of parity violation in gauge theories.

Findings

Identifies classical parity breaking due to fermion mass.

Discovers a parity violation contribution present even for massless fermions.

Shows how gauge-invariant regularization affects parity properties.

Abstract

We study parity violation in $2+1$-dimensional gauge theories coupled to massive fermions. Using the $\zeta$-function regularization approach we evaluate the ground state fermion current in an arbitrary gauge field background, showing that it gets two different contributions which violate parity invariance and induce a Chern-Simons term in the gauge-field effective action. One is related to the well-known {\em classical} parity breaking produced by a fermion mass term in 3 dimensions; the other one, already present for massless fermions, is related to peculiarities of gauge invariant regularization in odd-dimensional spaces.