Gauge fixing and renormalisation scale independence of tunneling rate in abelian Higgs model and in the Standard Model

We explicitly show perturbative gauge fixing independence of the tunneling rate to a stable radiatively induced vacuum in the abelian Higgs model. We work with a class of $R_\xi$ gauges in the presence of both dimensionless and dimensionful gauge fixing parameters. We show that Nielsen identities survive the inclusion of higher order oparators and compute the tunnelling rate to the vacua modified by the nonrenormalisable operators in a gauge invariant manner. We also discuss implications of this method for the complete Standard Model.