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Dark Photon Dark Matter in the minimal $B-L$ Model

The extension of the Standard model (SM) with three heavy right handed neutrinos, a complex scalar and the gauged $U(1)_{\rm B-L}$ symmetry (the minimal $B-L$ model) is considered the most compelling minimal one: the presence and the out-of-equilibrium decay of the heavy right handed neutrinos can account for the small masses of the active neutrinos and the baryon asymmetry of the universe. A natural accompanying question concerns whether the minimal $B-L$ model can naturally accommodate an interesting dark matter (DM) candidate. We study the possibility where the current DM population is explained by the gauge boson of $U(1)_{\rm B-L}$ symmetry. We discuss how the minimal set-up originally aimed at the seesaw mechanism and the leptogenesis is connected to conditions making the gauge boson promoted to a DM candidate.