Sterile neutrinos and B–L symmetry

We revisit the relation between the neutrino masses and the spontaneous breaking of the $B–L$ gauge symmetry. We discuss the main scenarios for Dirac and Majorana neutrinos and point out two simple mechanisms for neutrino masses. In this context the neutrino masses can be generated either at tree level or at quantum level and one predicts the existence of very light sterile neutrinos with masses below the eV scale. The predictions for lepton number violating processes such as $\mu \to e$ and $\mu \to e\gamma$ are discussed in detail. The impact from the cosmological constraints on the effective number of relativistic degree of freedom is investigated.