Naturally small Dirac neutrino mass and $B-L$ dark matter

In the conventional gauged ${B-L}$ extension of the standard model, the $B-L$ charge of the singlet scalar $\chi$, responsible for the breaking of $U(1)_{B-L}$ symmetry, is taken to be 2 such that it can anchor type-I seesaw by giving Majorana masses to the right-handed neutrinos, $\nu_R$. In this paper, we consider instead the cases $\chi \sim 3$ or 4 under $B-L$, so that $\nu_R$ may not acquire any Majorana mass and neutrinos are Dirac fermions. We then consider a vector-like fermion $S$ with 2 units of $B-L$ charge, which becomes a good candidate for dark matter, either Dirac for $\chi \sim 3$ or Majorana for $\chi \sim 4$. In both cases, spontaneous $B-L$ breaking can induce a strong first-order phase transition, producing stochastic gravitational waves (GW) which can be tested at GW experiments. Moreover, the presence of light $\nu_R$s gives rise to an additional contribution to the effective number of relativistic degrees of freedom, $\Delta{N}_{\rm eff}$, providing complementary constraints from current and upcoming CMB observations.