parameter to be consistent with the current observations. We also calculate the scattering cross section between the DM particle and nucleon and discuss the implication for the direct DM search experiments. We summarize our results in the section IV. Our notations and the formulas used in our analysis are listed in Appendix. The model is based on the gauge group SU (3)$_{C}$ × SU (2)$_{L}$ × U (1)$_{Y}$ × U (1)$_{B}$ − $_{L}$. Additional fields besides the standard model fields are a gauge field Z ′ µ of the U (1)$_{B}$ − $_{L}$, a SM singlet B − L Higgs boson Ψ with two U (1)$_{B}$ − L charge, and three RH neutrinos N$_{i}$ which are necessary for the gauge and gravitational anomaly cancellations. In describing the RH neutrinos, we use the four component representation of RH neutrino constructed from the Weyl spinor ν$_{R}$$_{i}$ , For the two RH neutrinos, N$_{1}$ and N$_{2}$ , we assign Z$_{2}$ parity even, while odd for N$_{3}$ , so that the RH neutrino N$_{3}$ is stable and, hence, the DM candidate. Due to the additional gauge symmetry U (1)$_{B}$ − $_{L}$, the covariant derivative for each fields is given by where D ( SM ) µ is the covariant derivative in the SM, and q$_{B}$ − L is the charge of each fields under the U (1)$_{B}$ − L with its gauge coupling g$_{B}$ − $_{L}$. Yukawa interactions relevant for the neutrino masses are given by where ˜ Φ = − iτ$_{2}$ Φ ∗ for Φ being the SM Higgs doublet, and without loss of generality we have worked out in the basis where the second term in the right-hand-side is in flavor diagonal for RH neutrinos. Because of the Z$_{2}$ parity, the DM candidate N$_{3}$ has no Yukawa couplings with the left-handed lepton doublets. The general Higgs potential for the SU (2)$_{L}$ doublet Φ and a singlet B − L Higgs Ψ is generally given by II. THE MINIMAL GAUGED U $^{(1)$_{B}$}$− L MODEL WITH Z$_{2}$ PARITY N$_{i}$ ≡   ν$_{R}$$_{i}$ ϵ ν ∗ R$_{i}$   , (1) D$_{µ}$ = D ( SM ) µ − iq$_{B}$ − $_{L}$g$_{B}$ − $_{L}$Z ′ $_{µ}$, (2) L int = 3 ∑ α =1 2 ∑ i =1 y$_{αi}$ ¯ L$_{α}$ ˜ Φ N$_{i}$ − 1 2 3 ∑ i =1 λ$_{R}$$_{i}$ ¯ N$_{i}$ Ψ P$_{R}$N$_{i}$ + h . c ., (3) V (Φ , Ψ) = m 2 1 | Φ | 2 + m 2 | Ψ | 2 + λ$_{1}$ | Φ | 4 + λ$_{2}$ | Ψ | 4 + λ$_{3}$ | Φ | $^{2}$| Ψ | $^{2}$. (4) 3