content only contains a left-handed doublet and its right- handed partner, one [18] need assume the Higgs poten- tial to be O (2) invariant for giving an acceptable sym- metry breaking pattern. Alternatively we shall extend the original model [17] with spontaneous D-parity viola- tion [19]. For generating the masses of the SM fermions, we introduce some heavy charged singlet fermions with Dirac masses and then integrating them out. For the neutrinos, we introduce left- and right-handed neutral singlet fermions with not only Dirac but also Majo- rana masses. The right(left)-handed neutral singlets can dominate the seesaw for the left(right)-handed neutrino masses. Their decays can produce the observed baryon asymmetry through the leptogenesis and can give the right-handed neutrinos a desired relic density for the dark matter. In this left-right symmetric scenario, the proper- ties of the dark matter are related to the parameters for generating the neutrino masses and the baryon asymme- try. grating out these singlets [17], where the SM Yukawa couplings are given by Here we have chosen the base where the mass matrices M$_{D,U,E}$ are real and diagonal. Note the mass matrices M$_{D,U,E}$ now are not hermitian as a result of the sponta- neous D-parity violation, i.e. The most general scalar potential of our model is with M 0 D,U,E = M 0 † D,U,E while h † D,U,E = − h$_{D,U,E}$ . If the Yukawa couplings h$_{D,U,E}$ vanish, the left-handed yukawa couplings y L D,U,E should be equal to the right-handed ones y R D,U,E as M 0 D,U,E are hermitian matrices. Since the Yukawa coupling of top quark is very close to 1, Eq. (10b) will constrain M$_{T}$ ≲ v$_{R}$ for y$_{T}$ < √$_{4}$$_{π}$$_{. For simplic-}$ ity, we shall conveniently assume M$_{D}$ = M$_{U}$ = M$_{E}$ = v$_{R}$ to make the heavy charged singlets decoupled from the following discussions. In the neutrino sector, we consider the left- and right- handed neutral singlets with the Yukawa couplings and the masses as below, with σ → − σ and φ$_{L}$ → φ$_{R}$ under the parity symmetry. Here λ$_{1}$$_{,}$$_{2}$ > 0, λ$_{3}$ > − 2 λ$_{2}$ and λ$_{4}$ > − 2 √ λ$_{1}$λ$_{2}$ so that the potential is bounded from below. For appropriate parameter choice, it is easy to give the desired vacuum expectation values (VEVs), In the present model without the Higgs bi-doublets, the charged gauge bosons W ± L and W ± R are mass eigenstates [18], In the presence of the spontaneous D-parity violation, the Dirac mass matric M D N is not hermitian, i.e. with M 0 N = M 0 † N while h † N = − h$_{N}$ . Furthermore, we have forbidden the Yukawa couplings ¯ l$_{L}$ ˜ φ$_{L}$N c $_{L}$, ¯ l$_{R}$ ˜ φ$_{R}$N c R and their CP conjugates. This can be achieved by im- posing a discrete, global or local symmetry. For exam- ple, we consider a U (1)$_{X}$ local symmetry under which D$_{L,R}$, U c $_{L,R}$, E$_{L,R}$ , N c $_{L,R}$, φ ∗ L,R carry a quantum number X = 1. Clearly, this U (1)$_{X}$ is free of gauge anomaly. In this context, the Yukawa couplings and the Dirac mass terms in Eqs. (9) and (12) are allowed while the Majo- rana mass terms in Eq. (12) are forbidden. To break this The left- and right-handed fermions in the SM with the right-handed neutrinos are placed in doublets for each family. For generating the masses of the SM fermions, we introduce some heavy charged singlets to construct the Yukawa couplings to the Higgs and fermion doublets so that we can derive the SM yukawa couplings by inte- M D N = M 0 N + h$_{N}$ 〈 σ 〉 (13) L ⊃ − y$_{N}$ ( ¯ l$_{L}$ ˜ φ$_{L}$N$_{R}$ + ¯ l$_{R}$ ˜ φ$_{R}$N$_{L}$ ) − M D N ¯ N$_{L}$N$_{R}$ − 1 2 M M N ( ¯ N c $_{L}$N$_{L}$ + ¯ N c $_{R}$N$_{R}$ + H.c. . (12) v$_{R}$ = 〈 φ$_{R}$ 〉 ≫ v$_{L}$ = 〈 φ$_{L}$ 〉 ≃ 174 GeV . (7) m$_{W}$$_{L}$ = 1 √$_{2}$ gv$_{L}$ , m$_{W}$$_{R}$ = 1 √$_{2}$ gv$_{R}$ . (8) V = − µ 2 $_{1}$σ 2 − µ 2 2 ( φ † $_{L}$φ$_{L}$ + φ † $_{R}$φ$_{R}$ ) + µ$_{3}$σ ( φ † $_{L}$φ$_{L}$ − φ † $_{R}$φ$_{R}$ ) + λ$_{1}$σ 4 + λ$_{2}$ [ ( φ † $_{L}$φ$_{L}$ ) 2 + φ † $_{R}$φ$_{R}$ ) 2 ] + λ$_{3}$φ † $_{L}$φ$_{L}$ φ † $_{R}$φ$_{R}$ + λ$_{4}$σ 2 φ † $_{L}$φ$_{L}$ + φ † $_{R}$φ$_{R}$ ) , (6) M$_{D,U,E}$ = M 0 D,U,E + h$_{D,U,E}$ 〈 σ 〉 (11) y$_{e}$ = − y L E v$_{R}$ M$_{E}$ y R † E . (10c) y$_{u}$ = − y L U v$_{R}$ M$_{U}$ y R † U , (10b) y$_{d}$ = − y L D v$_{R}$ M$_{D}$ y R † D , (10a) L ⊃ − y$_{D}$ (¯ q$_{L}$φ$_{L}$D$_{R}$ + ¯ q$_{R}$φ$_{R}$D$_{L}$ ) − M$_{D}$ ¯ D$_{L}$D$_{R}$ − y$_{U}$ ( ¯ q$_{L}$ ˜ φ$_{L}$U$_{R}$ + ¯ q$_{R}$ ˜ φ$_{R}$U$_{L}$ ) − M$_{U}$ ¯ U$_{L}$U$_{R}$ − y$_{E}$ ¯ l$_{L}$φ$_{L}$E$_{R}$ + ¯ l$_{R}$φ$_{R}$E$_{L}$ − M$_{E}$ ¯ E$_{L}$E$_{R}$ +H.c. ⇒ − y$_{d}$ ¯ q$_{L}$φ$_{L}$d$_{R}$ − y$_{u}$ ¯ q$_{L}$ ˜ φ$_{L}$u$_{R}$ − y$_{e}$ ¯ l$_{L}$φ$_{L}$e$_{R}$ +H.c. , (9) 2 II. THE MODEL