A consistent calculation of bubble-nucleation rates

We present a consistent picture of tunnelling in field theory. Our results apply both to high-temperature field theories in four dimensions and to zer o-temperature three-dimensional ones. Our approach is based on the notion o f a coarse-grained potential U-k that incorporates the effect of fluctuatio ns with characteristic momenta above a given scale k. U-k is non-convex and becomes equal to the convex effective potential for k --> 0. We demonstrat e that a consistent calculation of the nucleation rate must be performed at non-zero values of k, larger than the typical scale of the saddle-point co nfiguration that dominates tunnelling. The nucleation rate is exponentially suppressed by the action S-k Of this Saddle point. The pre-exponential fac tor A(k), which includes the fluctuation determinant around the saddle-poin t configuration, is well-defined and finite. Both S-k and A(k) are k-depend ent, but this dependence cancels in the expression for the nucleation rate. This picture breaks down in the limit of very weakly first-order phase tra nsitions, for which the pre-exponential factor compensates the exponential suppression. (C) 1999 Elsevier Science B.V.