  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 $\xi$.
