Matching high- and low-temperature regimes of massive scalar fields

We analyze the matching of high- and low-temperature expansions of the effective action of massive scalar fields confined between two infinite walls with different boundary conditions. One remarkable effect in the low-temperature regime is the exponential decay of the vacuum energy with the distance of the boundary walls. This exponential decay depends on the boundary conditions. When the conditions involve any kind of interconnection between the boundary values of the fields at the two walls the rate of the exponential decay is half than when the two boundaries are not connected. In particular, this rate for periodic boundary conditions is half of that for Dirichlet boundary conditions.