On the Coleman-Hill theorem

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 mementum) 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 electro-dynamics, obeys the new non-renormalization theorem. Whereas the vacuum polarization 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 orginal analysis, where the conditions of the theorem are satisfied and where the topological mass does, in fact, get a radiative correction.