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Large mass expansion of the one-loop effective action induced by a scalar field on the two-dimensional Minkowski background with non-trivial $(1+1)$ splitting

A large mass expansion of the one-loop effective action of a scalar field on the two-dimensional Minkowski spacetime is found in the system of coordinates, where the metric $g_{μν}(t,x)\neqη_{μν}=diag(1,-1)$, and $g_{μν}(t,x)$ tends to $η_{μν}$ at the spatial and temporal infinities. It is shown that, apart from the Coleman-Weinberg potential, this expansion contains the terms both analytic and non-analytic in $m^{-2}$, where $m$ is the mass of a scalar field. A general unambiguous expression for the one-loop correction to the effective action on non-stationary backgrounds is derived.