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In $N+1$ dimensions, false vacuum decay at zero temperature is dominated bythe $O(N+1)$ symmetric instanton, a sphere of radius $R_0$, whereas attemperatures $T>>R_0^{-1}$, the decay is dominated by a `cylindrical' (static)$O(N)$ symmetric instanton. We study the transition between these two regimesin the thin wall approximation. Taking an $O(N)$ symmetric ansatz for theinstantons, we show that for $N=2$ and $N=3$ new periodic solutions exist in afinite 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 thederivative of the nucleation rate at a certain temperature $T_*$, when thestatic instanton starts dominating. For $N=1$, on the other hand, the newsolutions are dominant and they smoothly interpolate between the zerotemperature instanton and the high temperature one, so the transition is ofsecond order. The determinantal prefactors corresponding to the `cylindrical'instantons are discussed, and it is pointed out that the entropic contributionsfrom massless excitations corresponding to deformations of the domain wall giverise to an exponential enhancement of the nucleation rate for $T>>R_0^{-1}$.

Instantons for vacuum decay at finite temperature in the thin wall limit