parameter to be consistent with the current observations. We als o calculate the scattering
cross section between the DM particle and nucleon and discuss the im plication 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.
II. THE MINIMAL GAUGEDU(1)B−L MODEL WITH Z2 PARITY
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 Ni which are
necessary for the gauge and gravitational anomaly cancellations. In describing the RH
neutrinos, we use the four component representation of RH neut rino constructed from the
Weyl spinor νRi ,
Ni ≡

νRi
ǫν∗
Ri

, (1)
For the two RH neutrinos, N1 and N2, we assign Z2 parity even, while odd for N3, so that
the RH neutrino N3 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
Dµ = D(SM )
µ − iqB−LgB−LZ′
µ , (2)
where D(SM )
µ is the covariant derivative in the SM, and qB−L is the charge of each ﬁelds
under the U(1)B−L with its gauge coupling gB−L.
Yukawa interactions relevant for the neutrino masses are given by
Lint =
3∑
α =1
2∑
i=1
yαi ¯Lα ˜Φ Ni − 1
2
3∑
i=1
λRi
¯NiΨ PRNi + h.c., (3)
where ˜Φ = −iτ2Φ ∗ for Φ being the SM Higgs doublet, and without loss of generality we hav e
worked out in the basis where the second term in the right-hand-sid e is in ﬂavor diagonal
for RH neutrinos. Because of the Z2 parity, the DM candidate N3 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
V(Φ ,Ψ) = m2
1|Φ |2 + m2
2|Ψ |2 + λ1|Φ |4 + λ2|Ψ |4 + λ3|Φ |2|Ψ |2. (4)
3