the n transverse dimensions [4, 6], With the natural choice r = R and 〈 σ 〉 ≲ M$_{∗}$ , it is easy to read Here we have take the following notations, 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 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}$$_{1}$$_{,}$$_{2}$ would be with We can diagonalize the above mass terms to be by taking the rotations as below, It is convenient to define the Majorana fermions as the mass eigenstates, We further assume the doublet scalar η much heavier than the Majorana fermions X $^{±}$, say m 2 η ≫ M 2 $_{±}$. In this case, we can integrate out η and then simplify the Yukawa couplings given by the first line of Eq. (2) in a new form, with For M$_{±}$ ≫ M$_{+}$ − M$_{−}$ , we can follow the standard method [12] of the resonant leptogenesis 1 to compute Δ$_{n}$( r ) = 1 (2 π ) n 2 ( m$_{χ}$ M$_{∗}$ ) n − 2 ( m$_{χ}$r ) − n − 2 2 K n − 2 2 ( m$_{χ}$r ) . (5) 〈 χ 〉 ∼ M$_{∗}$ Δ$_{n}$( R ) . (6) 〈 χ 〉 ∼ M$_{∗}$ ∫ d $^{n}$rn$_{brane}$ Δ$_{n}$( r ) = M$_{∗}$n$_{brane}$ ( m$_{χ}$ M$_{∗}$ ) n − 2 1 m n χ = M$_{∗}$ ( M$_{∗}$ m$_{χ}$ ) $^{2}$( M$_{∗}$ M$_{Pl}$ ) 4 n (7) n$_{brane}$ ∼ M n ∗ ( M$_{∗}$ M$_{Pl}$ ) 4 n . (8) L ⊃ − M$_{N}$ ¯ N c $_{R$_{1}$}$N$_{R}$$_{2}$ − 1 2 m$_{1}$ ¯ N c $_{R$_{1}$}$N$_{R}$$_{1}$ − 1 2 m$_{2}$ ¯ N c $_{R$_{2}$}$N$_{R}$$_{2}$ +H.c. (10) m$_{1}$$_{,}$$_{2}$ = h$_{1}$$_{,}$$_{2}$ 〈 χ 〉 ≪ M$_{N}$ . (11) L ⊃ − 1 2 M$_{±}$ ¯ X ± c R X ± R + H.c. (12) N$_{R}$$_{1}$ = cX + R − isX − R , (13a) N$_{R}$$_{2}$ = sX + R + icX − R . (13b) y ′ α 2 = − y$_{α}$$_{2}$ ρ 〈 χ 〉 m 2 η . (21) L ⊃ − ( cy$_{α}$$_{1}$ − sy ′ α $_{2}$) ¯ ψ$_{L}$$_{α}$φX + − i ( sy$_{α}$$_{1}$ + cy ′ α $_{2}$) ¯ ψ$_{L}$$_{α}$φX − + H.c. (20) L ⊃ − 1 2 M$_{±}$ ¯ X $^{±}$X $^{±}$. (19) X − = X − R + X − c R , (18b) X + = X + R + X + c R , (18a) M$_{±}$ ≃ M$_{N}$ ± 1 $_{2}$( m$_{1}$ + m$_{2}$ ) ≫ M$_{+}$ − M$_{−}$ . (17b) ϑ ≃ π 4 , (17a) M$_{+}$ > M$_{−}$ > 0 . (16) M$_{−}$ = 2 scM$_{N}$ − s $^{2}$m$_{1}$ − c $^{2}$m$_{2}$ . (15b) M$_{+}$ = 2 scM$_{N}$ + c $^{2}$m$_{1}$ + s $^{2}$m$_{2}$ , (15a) c ≡ cos ϑ, s ≡ sin ϑ for ϑ = 1 2 arctan 2 M$_{N}$ m$_{2}$ − m$_{1}$ (14) 2 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.