Thermalization by off-shell processes: the virtues of small virtuality

We study the thermalization of a scalar field $\Phi$ coupled to two other scalar fields $\chi_{1,2}$ that constitute a bath in thermal equilibrium. For a range of masses the $\Phi$ propagator features threshold and infrared divergences, a vanishing residue at the (quasi) particle pole and vanishing \emph{on-shell} decay rates thereby preventing the equilibration of $\Phi$ with the bath via on-shell processes. Inspired by the theory of quantum open systems we obtain a quantum master equation for the reduced density matrix of $\Phi$ that includes the time dependence of bath correlations, yielding time dependent rates in the dynamics of relaxation and allowing virtual processes of small virtuality $\propto 1/t$ at long time $t$. These \emph{off-shell} processes lead to thermalization despite vanishing S-matrix rates. In the case of threshold divergences we find that a thermal fixed point is approached as $e^{-\sqrt{t/t^*}}$ with the relaxation time $t^*$ becoming shorter at high temperature as a consequence of stimulated emission and absorption. In the infrared case, the thermal fixed point is approached as $e^{-\gamma(t)}$, where $\gamma(t)$ features a crossover between a $\propto \ln(t)$ and a $\propto t$ behavior for $t \gg 1/T$. The vanishing of the residue and the crossover in relaxational dynamics in this case is strikingly reminiscent of the orthogonality catastrophe in heavy impurity systems. The results yield more general lessons on thermalization via virtual processes.