A Simple Proof of the Non-Renormalization of the Chern-Simons Coupling

TL;DR

This paper provides a straightforward proof that the Chern-Simons coupling in 2+1-dimensional gauge theories is only renormalized at one-loop order, regardless of the presence of complex interactions or couplings.

Contribution

It introduces a simple, general proof that small gauge invariance suffices to prevent higher-order renormalization of the Chern-Simons coupling.

Findings

Renormalization of the Chern-Simons coupling is limited to one-loop.

The proof applies to both abelian and nonabelian theories.

It holds even with non-renormalizable and non-minimal interactions.

Abstract

We give a very simple proof that the renormalization of the Chern-Simons coupling in the Wilsonian effective action is exhausted at one-loop. Our proof can apply to arbitrary 2+1-dimensional abelian as well as nonabelian gauge theories without a bare Chern-Simons coupling, including any non-renormalizable interactions and non-minimal couplings. Our proof reveals that small (but not large) gauge invariance is enough to ensure the absence of higher order corrections.