  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.
