On different approaches to freeze-in and freeze-out leptogenesis with quasi-degenerate neutrinos

We compare two approaches for determining the generation of lepton asymmetry during production and decay of quasi-degenerate neutrinos, namely the density matrix formalism and a recent proposal which does not involve any counting of neutrino number densities and is based on plugging the resummed propagator in a quantum field theory model for neutrino oscillations. We show numerically and analytically that they are almost equivalent except in the limit of very large mass splittings. The comparison, performed in a simple scalar toy model, helps to understand several issues that have been discussed in the literature.