Low scale Dirac leptogenesis and dark matter with observable $\Delta N_{\rm eff}$

We propose a gauged $B-L$ extension of the standard model (SM) where light neutrinos are of Dirac type by virtue of tiny Yukawa couplings with the SM Higgs. To achieve leptogenesis, we include additional heavy Majorana fermions without introducing any $B-L$ violation by two units. An additional scalar doublet with appropriate $B-L$ charge can allow heavy fermion coupling with the SM leptons so that out of equilibrium decay of the former can lead to generation of lepton asymmetry. Due to the $B-L$ gauge interactions of the decaying fermion, the criteria of successful Dirac leptogenesis can also constrain the gauge sector couplings so as to keep the corresponding washout processes under control. The same $B-L$ gauge sector parameter space can also be constrained from dark matter requirements if the latter is assumed to be a SM singlet particle with non-zero $B-L$ charge. The same $B-L$ gauge interactions also lead to additional thermalised relativistic degrees of freedom $\Delta N_{\rm eff}$ from light Dirac neutrinos which are tightly constrained by Planck 2018 data. While there exists parameter space from the criteria of successful low scale Dirac leptogenesis, dark matter and $\Delta N_{\rm eff}$ even after incorporating the latest collider bounds, all the currently allowed parameters can be probed by future measurements of $\Delta N_{\rm eff}$.