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.
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