The Chern-Simons Coefficient in the Higgs Phase

We study one-loop corrections to the Chern-Simons coefficient κ in abelian self-dual Chern-Simons Higgs systems and their N = 2 and N = 3 supersymmetric generalizations in both symmetric and asymmetric phases. One-loop corrections to the Chern-Simons coefficient of these systems turn out to be integer multiples of 1/4π in both phases. Especially in the maximally supersymmetric N = 3 case, the correction in symmetric phase vanishes and that in asymmetric phase is κ/(2π|κ|). Our results suggest that nonabelian self-dual systems might enjoy similar features. We also discuss various issues arising from our results. We consider one-loop correctons to the Chern-Simons coefficient in abelian Chern-Simons Higgs systems. In broken phase of these systems there are topological vortices of fractional spin and statistics. The quantum correction to the Chern-Simons coefficient could lead to a change in vortex dynamics given in the tree approximation. If there are charged particles in broken phase, there would be a nontrivial Aharonov-Bohm-type interaction between vortices and charged particles, which could also get a quantum correction. Thus, to find the quantum correction to the Chern-Simons coefficient would be interesting as a step to understand quantum physics in broken phase. There have been recently some controversies related to the nature of the correction to the Chern-Simons coefficient. In symmetric phase of abelian systems, there is a theorem that the correction originates from the fermion one-loop and no more [1]. When there are massless charged particles or when the gauge symmetry is spontaneously broken, the theorem breaks down and the loop corrections turn out to be in general complicated functions of couplings and particle masses [2,3]. When the gauge symmetry is nonabelian, the coefficient should be quantized to have a consistent quantum theory[4]. While this is indeed the case in the symmetric phase [5], the quantization is not respected by one-loop correction in asymmetric phase [6]. However, there has been no clear understanding of this puzzle. In this paper, we calculate one-loop correction to the Chern-Simons coefficient in abelian self-dual Chern-Simons Higgs systems and their N = 2, N = 3 supersymmetric generalizations [7,8,9] in both symmetric and asymmetric phases. The corrections in these systems turn out to be ‘quantized’ in the sense that they are integer multiples of the Dirac fermion correction, 1/4π. This suggests immediately an interesting possibility that the one-loop corrections in nonabelian self-dual systems [10] and their supersymmetric systems [8,9] would be also quantized. While the quantum consistency condition of nonabelian systems does not imply the quantization of the Chern-Simons coefficient in asymmtric phase, we will argue that our result is no coincidence and that there may be other reasons why the correction should be quantized. These points and other aspects arising from our calculation would be discussed at the end. We first start with a simple model of a gauge field Aμ, a complex Higgs field