  We consider free massive matter fields in static scalar, electric and
gravitational backgrounds. Tuning these backgrounds to the brink of vacuum
decay, we identify a term in their effective action that is singular. This
singular term is universal, being independent of the features of the background
configuration. In the case of gravitational backgrounds, it can be interpreted
as a quantum mechanical analog of Choptuik scaling. If the background is tuned
slightly above the instability threshold, this singular term gives the leading
contribution to the vacuum decay rate.
