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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 

<img src='content_image/328055.jpg'>

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 

<img src='content_image/328054.jpg'>

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 

<img src='content_image/328053.jpg'>

<img src='content_image/328052.jpg'>

In a basis where both charged lepton and heavy Majorana neutrino mass matrices are diagonal, the Yukawa interactions in Eq. (3) are replaced by 

<img src='content_image/328051.jpg'>

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 

<img src='content_image/328050.jpg'>

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 

