Bubble nucleation incphi4models at all temperatures

One possible way in which phase transitions in the early universe may have occurred is via nucleation of bubbles of the new phase (true vacuum) in the old phase (false vacuum). The technique most widely used to compute the probability of bubble nucleation is based on instanton methods in the context of the semiclassical approximation. At zero temperature in 3+1 dimensions the nucleation rate is dominated by the O(4) symmetric instanton, a sphere of radius R, while at temperatures T≫${\mathit{R}}^{\mathrm{-}1}$ the decay is dominated by a ‘‘cyclindrical’’ (static) instanton with O(3) invariance. There has been discussion in the literature as to whether the transition between these two regimens would be first order (discontinuity in the first derivative of the nucleation rate at the transition temperature ${\mathit{T}}_{\mathit{c}}$), or second order (continuity of the first derivative, but discontinuity of the second derivative at ${\mathit{T}}_{\mathit{c}}$). In this paper we obtain the finite temperature solutions corresponding to the quantum and the thermal regimes, and compute their action as a function of the temperature for different values of the wall thickness in a ${\mathit{cphi}}^4$ potential. Our results indicate that only for the cases of very large wall thickness a second-order transition takes place, while for all the other cases a first-order transition occurs. © 1995 The American Physical Society.