constraints to realize the vacuum alignments. Then, we rewrite the right-handed Majorana neutrino mass term and neutrino Yukawa coupling matrix, which are given as where κ = | λ$_{χ}$ | υ$_{χ}$/M and x ≡ υ$_{ψ}$ | f$_{ν}$ | / (Λ g$_{ν}$ ). As will be shown later, the size of x is restricted by the unknown mixing angle θ$_{13}$ . Diagonalizing M$_{R}$ in order to go into the physical basis (mass basis) of the right-handed neutrino, the diagonalization of M$_{R}$ is given as where a = √ 1 + κ 2 + 2 κ cos ξ, b = √ 1 + κ 2 − 2 κ cos ξ , with real and positive mass eigenval- ues, M$_{1}$ = Ma, M$_{2}$ = M, M$_{3}$ = Mb , and the diagonalizing matrix V$_{R}$ is with the phases In a basis where both charged lepton and heavy Majorana neutrino mass matrices are diagonal, the Yukawa interactions in Eq. (3) are replaced by where M d R = diag( M$_{1}$, M$_{2}$, M$_{3}$ ) and the couplings of N$_{i}$ with leptons and scalar η , ˜ Y$_{ν}$ ≡ $_{V}$† $_{L}$Y$_{ν}$ V$_{R}$ , is given as Concerned with CP violation, we notice that the CP phases ϕ$_{1}$, ϕ$_{2}$ coming from M$_{R}$ as well as the CP phase φ from Y$_{ν}$ obviously take part in low-energy CP violation, as you can 7 ˜ Y$_{ν}$ = V e † L Y$_{ν}$V$_{R}$ = $_{g$_{ν}$}$   − 2+ e $^{iφ}$x √$_{6}$ 1+ e $^{iφ}$x √$_{3}$ e $^{iφ}$x √$_{6}$ 1 − e $^{iφ}$x √$_{6}$ 1+ ωe $^{iφ}$x √$_{3}$ $_{i}$√$_{3+}$$_{e}$ $^{iφ}$x √$_{6}$ 1 − e $^{iφ}$x √$_{6}$ 1 − ωe $^{iφ}$x √$_{3}$ − $_{i}$√$_{3+}$$_{e}$ $^{iφ}$x √$_{6}$       e i ϕ$_{1}$ 2 0 0 0 1 0 0 0 e i ϕ$_{2}$ 2    . (13) L Yuk = ( ˜ Y$_{ν}$ )$_{ji}$ ¯ ℓ$_{Lj}$ηN$_{i}$ + M d R ¯ N$_{i}$ ( N$_{i}$ ) c + h.c (12) ϕ$_{1}$ $_{= tan}$− 1 ( κ sin ξ 1 + κ cos ξ ) and ϕ$_{2}$ $_{= tan}$− 1 ( κ sin ξ κ cos ξ − 1 ) . (11) V$_{R}$ = 1 $_{√$_{2}$}$   0 √$_{2}$ 0 − 1 0 − 1 − 1 0 1       e i ϕ$_{1}$ 2 0 0 0 1 0 0 0 e i ϕ$_{2}$ 2    , (10) M d R = $_{V}$† $_{R}$M$_{R}$ $_{V}$∗ R = M Diag . ( a, 1 , b ) , (9) M$_{R}$ = $_{M}$   1 0 0 0 1 κe iξ 0 κe iξ 1    , Y$_{ν}$ = $_{g$_{ν}$}$   1 0 xe iφ 0 1 0 xe iφ 0 1    , (8)