A non-renormalization theorem in gapped quantum field theory

We discuss the two-point functions of the U(1) current, and energy-momentum tensor in certain gapped three-dimensional field theories, and show that at zero momentum, the parity-odd part in both of these correlation functions is one-loop exact. In particular, we find a new and simplified derivation of the Coleman-Hill theorem that also clarifies several subtleties in the original argument. For the energy momentum tensor, our result means that the gravitational Chern-Simons term for the background metric does not receive quantum corrections.