The Higgs fields φ and ψ are obtained by expanding Φ and Ψ as Φ =   0 1 √$_{2}$ ( v + φ )   , (5) Ψ = 1 √$_{2}$ ( v ′ + ψ ) , (6)   h H   =   cos θ − sin θ sin θ cos θ     φ ψ   , (7) around the true vacuum with the vacuum expectation values v and v ′ . These are related with the mass eigenstates h and H through with θ being the mixing angle. Their masses are given by The mass of the new neutral gauge boson Z ′ arises by the U (1)$_{B}$ − L gauge symmetry breaking, Associated with the U (1)$_{B}$ − L gauge symmetry breaking, the RH neutrinos N$_{i}$ acquire masses From LEP experiment, the current lower bound on the Z ′ boson mass has been found to be [10, 11] Two Z$_{2}$ -even RH neutrinos N$_{1}$ and N$_{2}$ are responsible for light neutrino masses via the seesaw mechanism, Note that the rank of this mass matrix is two, so that the lightest neutrino is massless. Due to the Z$_{2}$ parity, one of RH neutrino N$_{3}$ (we denote it as N hereafter) in our model can be the DM candidate. We first estimate its relic abundance and identify the model M 2 h = 2 λ$_{1}$v $^{2}$cos2 θ + 2 λ$_{2}$v ′ $^{2}$sin2 θ − 2 λ$_{3}$vv ′ sin θ cos θ, (8) M 2 H = 2 λ$_{1}$v $^{2}$sin2 θ + 2 λ$_{2}$v ′ $^{2}$cos2 θ + 2 λ$_{3}$vv ′ sin θ cos θ. (9) M 2 Z ′ = 4 g 2 $^{B}$−$_{L}$v ′ $^{2}$. (10) M$_{N}$$_{i}$ = − λ$_{R}$$_{i}$ v ′ √$_{2}$ . (11) M$_{Z}$ ′ g$_{B}$ − L = 2 v ′ ≳ 6 − 7 TeV . (12) m$_{ν}$$_{αβ}$ = − ∑ i =1 , 2 y$_{αi}$y$_{iβ}$ v 2 2 M$_{N}$$_{i}$ . (13) III. RIGHT-HANDED NEUTRINO DARK MATTER 4