Following the work of Khare et al., we show that the generalization to systems with spontaneous symmetry breaking of the Coleman–Hill theorem to one-loop order can be extended to the case including fermions with the most general interactions. Although the correction to the parity-odd part of the vacuum polarization looks complicated in the Higgs phase, it turns out that the correction to the Chern–Simons term is identical to that in the symmetric phase, with the difference coming only from the contribution of the would be Chern–Simons term. We also discuss the implication of our result to non-Abelian systems.
