content_image/329404.jpg content_image/329406.jpg content_image/329405.jpg
## Appendix A: Higgs Potential and vacuum alignment 

Since it is not trivial to ensure that the different vacuum alignments of 〈 Φ $^{0}$〉 = ( υ, υ, υ ), 〈 ϑ 〉 = υ$_{ϑ}$ and 〈 χ 〉 , 〈 ψ 〉 in Eq. (7) are preserved, or at least approximately preserved, we shall briefly discuss these vacuum alignments. In order for the different vacuum alignments of 〈 Φ 〉 , 〈 χ 〉 , 〈 ϑ 〉 and 〈 ψ 〉 to be ensured, let us consider the most general renormalizable scalar potential of Φ , χ, ϑ, ψ and η invariant under SU (2) × U (1) × A$_{4}$ with the Z$_{2}$ × Z$_{4}$ symmetry is given as 

<img src='content_image/329404.jpg'>

where the bilinears B , trilinears T and quartic terms Q are given as 

<img src='content_image/329406.jpg'>

<img src='content_image/329405.jpg'>

