Minimal Dirac seesaw accompanied by Dirac fermionic dark matter

The $SU(3)_c^{}\times SU(2)_L^{} \times U(1)_Y^{}$ standard model is extended by a $U(1)_{B-L}^{}$ gauge symmetry with four right-handed neutrinos. Because of their Yukawa couplings to a Higgs singlet for spontaneously breaking the $U(1)_{B-L}^{}$ symmetry, two right-handed neutrinos can form a Dirac fermion to become a stable dark matter particle. Meanwhile, mediated by additionally heavy Higgs doublet(s), fermion singlet(s) and/or fermion doublet(s), the other two right-handed neutrinos can have a dimension-5 operator with the standard model lepton and Higgs doublets as well as the $U(1)_{B-L}^{}$ Higgs singlet. This context can realize a minimal Dirac neutrino mass matrix only with two nonzero eigenvalues. In association with the sphaleron processes, the interactions for generating the Dirac neutrino masses can also produce the observed baryon asymmetry in the universe.