What should be the limits of integration for euclidean action $S(\phi)$ in 3d and 4d? This action is negatively exponentiated to calculate the decay rate. I suspect that it is variable limit problem.
The limits of integration are usually all space, when you want the vacuum correlation functions, or around a circle in time, if you want thermal correlation functions. A circle in time means a finite 0 to T time window, with periodic boundary conditions for the bosonic fields and anti-periodic boundary conditions for the fermions.