The Non-abelian Chern-Simons Coefficient in the Higgs Phase

We calculate the one loop corrections to the Chern-Simons coefficient $\kappa$ in the Higgs phase of Yang-Mills Chern-Simons Higgs theories. When the gauge group is SU(N), we show, by taking into account the effect of the would be Chern-Simons term, that the corrections are always integer multiples of ${1\over 4\pi}$, as they should for the theories to be quantum-mechanically consistent. In particular, the correction is vanishing for SU(2). The same method can also be applied to the case that the gauge group is SO(N). The result for SO(2) agrees with that found in the abelian Chern-Simons theories. Therefore, the calculation provides with us a unified understanding of the quantum correction to the Chern-Simons coefficient.