Spontaneous Symmetry Breaking and the Renormalization of the Chern-Simons Term

We calculate the one-loop perturbative correction to the coefficient of the \cs 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 \cs term. In the case of an unbroken non-abelian subgroup, the coefficient of the relevant \cs 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 \cs term is a discontinuous function over the phase diagram of the theory.Comment: Version to be published in Phys Lett B., minor additional change