Spontaneous symmetry breaking and the renormalization of the Chern-Simons term

We calculate the one-loop perturbative correction to the coefficient of the Chern-Simons term in non-abelian gauge theory in the presence of Higgs fields, with a variety of symmetry-breaking structures. In the case of a residual U(1) symmetry, radiative corrections do not change the coefficient of the Chern-Simons term. In the case of an unbroken non-abelian subgroup, the coefficient of the relevant Chern-Simons term (suitably normalized) attains an integral correction, as required for consistency of the quantum theory. Interestingly, this coefficient arises purely from the unbroken non-abelian sector in question; the orthogonal sector makes no contribution. This implies that the coefficient of the Chern-Simons term is a discontinuous function over the phase diagram of the theory.