Instantons for Vacuum Decay at Finite Temperature in the Thin Wall Limit

TL;DR

This paper investigates the transition between different instanton solutions governing vacuum decay at finite temperature, revealing a first order transition for N=2,3 and a smooth second order transition for N=1, with implications for nucleation rates.

Contribution

It introduces a detailed analysis of instanton solutions in the thin wall limit across various dimensions, identifying the nature of phase transitions in vacuum decay.

Findings

New periodic instanton solutions exist near T ~ R_0^{-1} for N=2,3.

A first order transition occurs at T_* for N=2,3, with a sudden change in nucleation rate behavior.

For N=1, the transition is smooth (second order), with solutions interpolating between zero and high temperature regimes.

Abstract

In $N+1$ dimensions, false vacuum decay at zero temperature is dominated by the $O(N+1)$ symmetric instanton, a sphere of radius $R_0$, whereas at temperatures $T>>R_0^{-1}$, the decay is dominated by a `cylindrical' (static) $O(N)$ symmetric instanton. We study the transition between these two regimes in the thin wall approximation. Taking an $O(N)$ symmetric ansatz for the instantons, we show that for $N=2$ and $N=3$ new periodic solutions exist in a finite temperature range in the neighborhood of $T\sim R_0^{-1}$. However, these solutions have higher action than the spherical or the cylindrical one. This suggests that there is a sudden change (a first order transition) in the derivative of the nucleation rate at a certain temperature $T_*$, when the static instanton starts dominating. For $N=1$, on the other hand, the new solutions are dominant and they smoothly interpolate between the zero temperature instanton and the high temperature one, so the transition is of second order. The determinantal prefactors corresponding to the `cylindrical' instantons are discussed, and it is pointed out that the entropic contributions from massless excitations corresponding to deformations of the domain wall give rise to an exponential enhancement of the nucleation rate for $T>>R_0^{-1}$.