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Freeze-in and freeze-out generation of lepton asymmetries after baryogenesis in the $ν$MSM

The $ν$MSM -- an extension of the Standard Model by three relatively light singlet Majorana fermions $N_{1,2,3}$ -- allows for the generation of lepton asymmetry which is several orders of magnitude larger than the observed baryon asymmetry of the Universe. The lepton asymmetry is produced in interactions of $N_{2,3}$ (with masses in the GeV region) at temperatures below the sphaleron freeze out $ T \lesssim 130$ GeV and can enhance the cosmological production of dark matter (DM) sterile neutrinos $N_1$ (with the mass of the keV scale) happening at $T \sim 200$ MeV due to active-sterile neutrino mixing. This asymmetry can be generated in freeze-in, freeze-out, or later in decays of heavy neutral leptons. In this work, we address the question of the magnitude of the late-time asymmetry (LTA) generated by the heavy neutral leptons $N_{2,3}$ during their freeze-in and freeze-out, leaving the decays for later work. We study how much of this asymmetry can survive down to the lower temperatures relevant for the sterile neutrino DM creation. We find that this LTA could result in the production of a sizeable fraction of dark matter. We also examine a role played by magnetic fields and the Abelian chiral anomaly in the generation of LTA, not accounted for in the previous studies. We argue that the production of LTA can be increased significantly and make an estimate of the influence of this effect.