Self-interacting scalar field theory in general curved spacetimes at zero and finite temperature revisited

We revisit the problem of spontaneous symmetry breaking (SSB), its restoration, and phase transition for a self-interacting quantum scalar field in a general curved background, at zero and finite temperature. To the best of our knowledge, most of the earlier computations in this context have been done in the linear order in curvature, which may not be very suitable for the Ricci flat spacetimes. One of our objectives is to see whether the higher-order terms can bring in qualitatively new physical effects, and thereby attempting to fill in this gap in the literature. We use Bunch and Parker's local momentum space representation of the Schwinger-DeWitt expansion of the Feynman propagator. Such expansion, being based upon the local Lorentz symmetry of spacetime, essentially probes the leading curvature correction to short scale, ultraviolet quantum processes. We compute the renormalized, background spacetime curvature (up to quadratic order) and temperature-dependent one-loop effective potential for ${\ensuremath{\phi}}^{4}$ plus ${\ensuremath{\phi}}^{3}$ self-interaction. In particular for the de Sitter spacetime, we have shown for the ${\ensuremath{\phi}}^{4}$-theory that we can have SSB even with a positive rest mass squared and positive nonminimal coupling, at zero temperature. This cannot be achieved by the linear curvature term alone and the result remains valid for a very large range of renormalization scale. Such SSB will generate a field mass that depends upon the spacetime curvature as well as the nonminimal coupling. For a phase transition, we have computed the leading curvature correction to the critical temperature. At finite temperature, symmetry restoration is demonstrated. We also extend some of the above results to two-loop level. The symmetry breaking in de Sitter at two loop remains present. We have further motivated the necessity of treating this problem nonperturbatively in some instances.