U (1)$_{X}$ $^{1}$, we can introduce a singlet scalar ξ with Yukawa couplings to the neutral singlets N$_{L,R}$ , which have mass matrices with a same texture, Here U is the MNS lepton flavor mixing matrix [20]. Through the above Yukawa interactions, the Majorana masses in Eq. (12) can be given by By integrating out the neutral singlets, the full neutrino masses would contain a Dirac mass term and two Majo- rana ones, with Here we have assumed and then have defined the left- and right-handed Majo- rana fermions, We thus integrate out the right-handed physical Higgs boson and then derive the following Lagrangian, Here the dimension-5 operator φ † $_{L}$φ$_{L}$ ¯ ν$_{R}$R and its CP con- jugate have been omitted since they are sufficiently sup- pressed in comparison with the couplings in (25). Clearly, Eq. (25) can accommodate the standard lep- togeneisis scenario, where the 2-body decays of the Ma- jorana fermions into the SM lepton and Higgs doublets can generate a lepton asymmetry if CP is not conserved. This lepton asymmetry can be partially converted to a baryon asymmetry through sphaleron [21]. So, the ab- sence of baryonic antimatter is well explained. The final baryon asymmetry can be described by by choosing the base where the Majorana mass matrix M M N is real and diagonal, Clearly, the right-handed neutrinos will give their left- handed partners an additional Majorana mass term through the seesaw since their Dirac masses are not van- ishing. This gift is negligible, Therefore, we can well define the left- and right-handed Majorana neutrinos, Here c = − 28 79 [22] is the sphaleron induced lepton-to- baryon conversion coefficient, g$_{∗}$ ≃ 112 is the relativistic degrees of freedom (the SM fields plus three right-handed neutrinos), κ$_{R}$$_{j}$ ( y$_{N}$, M ) and κ$_{L}$$_{j}$ ( y$_{N}$, M ) are wash out factors, ε$_{R}$$_{j}$ ( ε$_{L}$$_{j}$ ) is the lepton asymmetry induced by the decays of a R$_{j}$ ( L$_{j}$ ) and is calculated at one-loop order, 1 There is a kinetic mixing between the gauge bosons associated with U (1)$_{X}$ and U (1)$_{B}$$_{−}$$_{L}$. We simply assume this mixing is small enough to fulfill the experimental constraints. ε$_{R}$$_{j}$ ( y$_{N}$, M ) = Γ$_{R}$$_{j}$$_{→}$$_{l}$$_{L}$$_{+}$$_{φ}$$_{L}$ − Γ$_{R}$$_{j}$$_{→}$$_{l}$ c $_{L}$+ φ ∗ L Γ$_{R}$$_{j}$ , (27a) ε$_{L}$$_{j}$ ( y$_{N}$, M ) = Γ$_{L}$$_{j}$$_{→}$$_{l}$$_{L}$$_{+}$$_{φ}$$_{L}$ − Γ$_{L}$$_{j}$$_{→}$$_{l}$ c $_{L}$+ φ ∗ L Γ$_{L}$$_{j}$ , (27b) χ = ν$_{R}$ + ν $_{R}$, (22b) ν = ν$_{L}$ + ν $_{L}$, (22a) δm$_{L}$ = − m$_{D}$ 1 m † R m T D = O [( M D N M M N ) 2 ] m$_{L}$ ≪ m$_{L}$ . (21) n$_{b}$ s = c Σ$_{j}$ κ$_{R}$$_{j}$ε$_{R}$$_{j}$ + κ$_{L}$$_{j}$ε$_{L}$$_{j}$ g$_{∗}$ . (26) M M N = diag { M$_{1}$, M$_{2}$, M$_{3}$ } ≃ M. (20) R$_{i}$ = N$_{R}$$_{i}$ + N $_{R$_{i}$}$, (19b) L$_{i}$ = N$_{L}$$_{i}$ + N c $_{L$_{i}$}$, (19a) M M N ≫ M D $_{N}$, y$_{N}$ v$_{R}$, y$_{N}$v$_{L}$ , (18) L ⊃ − y$_{N}$ ¯ l$_{L}$ ˜ φ$_{L}$R + y$_{N}$ 1 M M D $_{N}$¯ l$_{L}$ ˜ φ$_{L}$L + λ$_{3}$ 4 λ$_{2}$v$_{R}$ y$_{N}$φ † $_{L}$φ$_{L}$ ¯ ν$_{R}$L + H.c. − 1 2 M ( ¯ LL + ¯ RR ) . (25) m$_{D}$ = y$_{N}$ 1 M M N ( M D $_{N}$) T 1 M M N y † $_{N}$v$_{L}$ v$_{R}$ . (17c) m$_{R}$ = − y$_{N}$ 1 M M N y T $_{N}$v 2 $_{R}$, (17b) m$_{L}$ = − y$_{N}$ 1 M M N y T $_{N}$v 2 $_{L}$, (17a) L ⊃ − 1 2 ¯ ν$_{L}$m$_{L}$ν c L − 1 2 ¯ ν$_{R}$m$_{R}$ν c R − ¯ ν$_{L}$m$_{D}$ν$_{R}$ + H.c. (16) M M N ≪ v$_{R}$ . (24) We now discuss the physics below the scale M M N , which is assumed much smaller than the left-right symmetry breaking scale v$_{R}$ , i.e. M M N = f$_{N}$ 〈 ξ 〉 . (15) L ⊃ − 1 2 f$_{N}$ ( ξ ¯ N c $_{L}$N$_{L}$ + ξ ∗ ¯ N c $_{R}$N$_{R}$ ) + H.c. . (14) m$_{ν}$ = m$_{L}$ = U $^{†}$diag { m$_{1}$, m$_{2}$, m$_{3}$ } U $^{∗}$, (23a) m$_{χ}$ = m$_{R}$ = U $^{†}$diag { m$_{1}$, m$_{2}$, m$_{3}$ } U $_{∗}$v R v 2 L . (23b) 3 III. BARYON ASYMMETRY