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the n transverse dimensions [4, 6], 

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With the natural choice r = R and 〈 σ 〉 ≲ M$_{∗}$ , it is easy to read 

We will clarify later a heavy mass m$_{χ}$ is necessary for a successful leptogenesis. However, the modified Bessel function K n − 2 2 ( m$_{χ}$r ) will exponentially suppress 〈 χ 〉 if m$_{χ}$ ≫ 1 /r . This 〈 χ 〉 is too small to generate the desired neutrino masses. Therefore, we consider the brane-lattice crystallization scenario [18] where the bulk is populated with large numbers of branes. One finds the lepton num- ber violation in our brane would be [2] 

with the brane density [18] 

By taking the natural assumption m$_{χ}$ ≲ M$_{∗}$ , the VEV 〈 χ 〉 is power suppressed by the ratio of M$_{∗}$ over M$_{Pl}$ . In the following we will consider 

〈 χ 〉 ∼ 260 eV for n = 6 ,M$_{∗}$ = 3 TeV and m$_{χ}$ ≲ M$_{∗}$ . (9) 

Due to the VEV 〈 χ 〉 , the masses of the right-handed neutrinos N$_{R}$$_{1}$$_{,}$$_{2}$ would be 

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with 

We can diagonalize the above mass terms to be 

by taking the rotations as below, 

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Here we have take the following notations, 

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and 

Without loss of generality we will assume m$_{1}$ < m$_{2}$ so that 

Actually, the small and large masses (11), i.e. m$_{1}$$_{,}$$_{2}$ ≪ M$_{N}$ will induce 

It is convenient to define the Majorana fermions 

as the mass eigenstates, 

We further assume the doublet scalar η much heavier than the Majorana fermions X $^{±}$, say m 2 η ≫ M 2 $_{±}$. In this case, we can integrate out η and then simplify the Yukawa couplings given by the first line of Eq. (2) in a new form, 

with 

For M$_{±}$ ≫ M$_{+}$ − M$_{−}$ , we can follow the standard method [12] of the resonant leptogenesis 1 to compute 

1 The right-handed neutrinos for the type-I seesaw will induce the vertex loop besides the self-energy loop in their decays. At the same time, the Higgs triplet for the type-II seesaw, which will be introduced later, will mediate another vertex correction in the de- cays of the right-handed neutrinos. The right-handed neutrinos and the Higgs triplet are at the TeV scale so that the CP asym- metry constrained by the neutrino masses would be too small unless we take the resonant enhancement into account. How- ever, the resonant effect only exists in the self-energy loop. So, the vertex corrections induced by the Higgs triplet or the right- handed neutrinos would have no significant contributions to the leptogenesis. 

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