On the Coleman-Hill Theorem

TL;DR

This paper revisits the Coleman-Hill theorem in 2+1 dimensional abelian gauge theories, reformulating it via the effective action, and explores conditions under which the topological mass remains uncorrected or receives radiative corrections.

Contribution

It re-expresses the Coleman-Hill theorem in terms of the effective action, strengthening the theorem and identifying new scenarios with radiative corrections to the topological mass.

Findings

The theorem is reformulated in terms of the effective action, leading to a stronger non-renormalization statement.

A known exception, spontaneously broken scalar Chern-Simons electrodynamics, obeys the new theorem.

A new scenario with scalar fields and parity-odd couplings allows radiative corrections to the topological mass.

Abstract

The Coleman-Hill theorem prohibits the appearance of radiative corrections to the topological mass (more precisely, to the parity-odd part of the vacuum polarization tensor at zero momentum) in a wide class of abelian gauge theories in 2+1 dimensions. We re-express the theorem in terms of the effective action rather than in terms of the vacuum polarization tensor. The theorem so restated becomes somewhat stronger: a known exception to the theorem, spontaneously broken scalar Chern-Simons electrodynamics, obeys the new non-renormalization theorem. Whereas the vacuum polarization {\sl does} receive a one-loop, parity-odd correction, this does not translate to a radiative contribution to the Chern-Simons term in the effective action. We also point out a new situation, involving scalar fields and parity-odd couplings, which was overlooked in the original analysis, where the conditions of the theorem are satisfied and where the topological mass does, in fact, get a radiative correction.