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 zero-temperature threedimensional ones. Our approach is based on the notion of a coarse-grained potential Uk that incorporates the effect of fluctuations with characteristic momenta above a given scale k. Uk is non-convex and becomes equal to the convex effective potential for k → 0. We demonstrate that a consistent calculation of the nucleation rate must be performed at non-zero values of k, larger than the typical scale of the saddlepoint configuration that dominates tunnelling. The nucleation rate is exponentially suppressed by the action Sk of this saddle point. The pre-exponential factor Ak, which includes the fluctuation determinant around the saddle-point configuration, is well-defined and finite. Both Sk and Ak are k-dependent, but this dependence cancels in the expression for the nucleation rate. This picture breaks down in the limit of very weakly first-order phase transitions, for which the pre-exponential factor compensates the exponential suppression.