Metastability of quantum fields in a thermal bath

We investigate the metastability of scalar fields in quantum field theories at finite temperature, focusing on a detailed understanding of the bounce solution. At finite temperature, the bounce solution depends on two variables: the Euclidean time $\tau$ and the spatial radial distance $r$, and it is periodic in the $\tau$ direction. We propose a novel method to determine the bounce that describes transitions in a thermal bath, suitable for numerical calculations. Two types of bounces exist for transitions in a thermal bath: $\tau$-dependent and $\tau$-independent bounces. We apply our method to compute these bounces in several models, including both thin-wall and thick-wall scenarios, to examine their properties. Specifically, we evaluate the critical temperature below which the $\tau$-independent bounce becomes destabilized due to fluctuations, rendering it irrelevant. We demonstrate that in the thick-wall case, the $\tau$-dependent bounce smoothly transitions into the $\tau$-independent one as temperature increases, whereas in the thin-wall case, the transition between the two types of bounces is discontinuous.