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U (1)$_{X}$ $^{1}$, we can introduce a singlet scalar ξ with Yukawa couplings to the neutral singlets N$_{L,R}$ , 

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, 

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with 

Here we have assumed 

and then have defined the left- and right-handed Majo- rana fermions, 

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, 

<img src='content_image/424134.jpg'>

Therefore, we can well define the left- and right-handed Majorana neutrinos, 

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. 

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which have mass matrices with a same texture, 

Here U is the MNS lepton flavor mixing matrix [20]. 

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. 

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 

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, 

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## III. BARYON ASYMMETRY 

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