A possible symmetry of the $\nu$MSM

To explain the dark matter and the baryon asymmetry of the Universe, the parameters of the $\nu$MSM (an extension of the Minimal Standard Model by three singlet neutrinos with masses smaller than the electroweak scale) must be fine-tuned: one of the masses should be in the ${\cal O} (10)$ keV region to provide a candidate for the dark-matter particle, while two other masses must be almost the same to enhance the CP-violating effects in the sterile neutrino oscillations leading to the baryon asymmetry. We show that a specifically defined global lepton-number symmetry, broken on the level of ${\cal O} (10^{-3})$ leads to the required pattern of sterile neutrino masses being consistent with the data on neutrino oscillations.