Real-time analysis of thermal activation via sphaleron transitions.

We study the process of thermal activation mediated by sphaleron transitions by analyzing the real-time dynamics of the decay out of equilibrium in a 1+1 dimensional field theory with a metastable state. The situation considered is that of a rapid supercooling in which the system is trapped in a metastable state at a temperature larger than the mass of the quanta, but smaller than the energy to create a critical droplet. The initial density matrix is evolved in time and the nucleation rate (probability current at the saddle point) is computed. The nucleation rate is time dependent, vanishing at early times, reaching a maximum at a time t ≈ 1/m with m the mass of quanta in the metastable state, and decreasing at long times as a consequence of unitarity. An estimate for the average number of particles of “true vacuum” produced as a function of time during the nucleation process is obtained.