On Rayleigh scattering in the massless Nelson model

Asymptotic completeness of Rayleigh scattering in models of atoms and molecules of non-relativistic QED is expected, but for a proof we still lack sufficient control on the number of emitted soft photons. So far, this obstacle has only been overcome for the spin-boson model. In a general class of models asymptotic completeness holds provided the expectation value of the photon number $N$ remains bounded uniformly in time. This has been established by Faupin and Sigal. We review and simplify their work, and, more importantly, we replace the bound on $N$ by a weaker assumption on the distribution of $N$ that is both necessary and sufficient for asymptotic completeness.