Symmetry of Bounce Solutions at Finite Temperature
Yutaro Shoji, Masahide Yamaguchi
TL;DR
This work extends the Coleman–Glaser–Martin analysis to finite temperature by reformulating the search for least-action saddle points as a genuine minimization problem of a scale-invariant functional $\mathcal{R}[\phi]$. Employing Steiner symmetrization and its refined equality conditions—extended to functions vanishing at spatial infinity—the authors prove that all least-action saddles are $O(D-1)$-symmetric and monotone in the spatial directions for admissible potentials with $D>3$. The main result provides a rigorous foundation for the common finite-temperature assumptions used in thermal vacuum decay and cosmological phase transitions, and it recovers the zero-temperature symmetry in the high-$\beta$ limit. These findings have direct implications for bubble nucleation rates and gravitational-wave predictions in early-universe cosmology, while suggesting future work including gravity, multi-component fields, and potential relaxations at finite temperature.
Abstract
The seminal work of Coleman, Glaser, and Martin established that, at zero temperature, any non-trivial solution to the equations of motion with the least Euclidean action is $O(D)$-symmetric. This paper extends their foundational analysis to finite temperature. We rigorously prove that for a broad class of scalar potentials, any saddle-point configuration with the least action is necessarily $O(D\!-\!1)$-symmetric and monotonic in the spatial directions. This result provides a firm mathematical justification for the symmetry properties widely assumed in studies of thermal vacuum decay and cosmological phase transitions.