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We study the thermalization of a scalar field $\Phi$ coupled to two otherscalar fields $\chi_{1,2}$ that constitute a bath in thermal equilibrium. For arange of masses the $\Phi$ propagator features threshold and infrareddivergences, a vanishing residue at the (quasi) particle pole and vanishing\emph{on-shell} decay rates thereby preventing the equilibration of $\Phi$ withthe bath via on-shell processes. Inspired by the theory of quantum open systemswe obtain a quantum master equation for the reduced density matrix of $\Phi$that includes the time dependence of bath correlations, yielding time dependentrates in the dynamics of relaxation and allowing virtual processes of smallvirtuality $\propto 1/t$ at long time $t$. These \emph{off-shell} processeslead to thermalization despite vanishing S-matrix rates. In the case ofthreshold divergences we find that a thermal fixed point is approached as$e^{-\sqrt{t/t^*}}$ with the relaxation time $t^*$ becoming shorter at hightemperature as a consequence of stimulated emission and absorption. In theinfrared 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 inrelaxational dynamics in this case is strikingly reminiscent of theorthogonality catastrophe in heavy impurity systems. The results yield moregeneral lessons on thermalization via virtual processes.

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