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

TL;DR

This paper computes one-loop quantum corrections to the Chern-Simons term in non-abelian gauge theories with Higgs fields, revealing how symmetry-breaking patterns affect the term's coefficient and its discontinuities across phases.

Contribution

It provides a detailed calculation of the one-loop correction to the Chern-Simons coefficient considering various symmetry-breaking scenarios in non-abelian gauge theories.

Findings

Residual U(1) symmetry does not alter the Chern-Simons coefficient.

Unbroken non-abelian subgroups induce integral corrections to the coefficient.

The Chern-Simons coefficient depends discontinuously on the phase of the theory.

Abstract

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.