Gauge invariance of the one-loop effective action of the Higgs field in the SU(2) Higgs model

The one-loop effective action of the abelian and nonabelian Higgs models has been studied in various gauges, in the context of instanton and sphaleron transition, bubble nucleation and most recently in nonequilibrium dynamics. Gauge invariance is expected on account of Nielsen' s theorem, if the classical background field is an extremum of the classical action, i.e., a solution of the classical equation of motion. We substantiate this general statement for the one-loop effective action, as computed using mode functions. We show that in the gauge-Higgs sector there are two types of modes that satisfy the same equation of motion as the Faddeev-Popov modes. We apply the general analysis to the computation of the fluctuation determinant for bubble nucleation in the SU(2) Higgs model in the 't Hooft gauge with general gauge parameter $\xi$. We show that due to the cancellation of the modes mentioned above the fluctuation determinant is independent of $\xi$.