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 ${\cal O} (10^{-4})$ leads to the required pattern of sterile neutrino masses being consistent with the data on neutrino oscillations. Moreover, the existence of this symmetry allows to fix the flavour structure of couplings of singlet fermions to the particles of the Standard Model and indicates that their masses are likely to be smaller than ${\cal O} (1)$ GeV, opening a possibility of their search in decays of charmed, beauty and even $K$ or $\pi$-mesons.