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 where the bilinears B , trilinears T and quartic terms Q are given as 27 V = µ 2 $_{Φ}$B$_{1}$ + λ Φ 1 Q Φ 1 + λ Φ 2 Q Φ 2 + λ Φ 3 Q Φ 3 + λ Φ 4 Q Φ 4 + iλ Φ 5 Q Φ 5 + µ 2 $_{χ}$B$_{2}$ + λ χ $_{1}$Q χ 1 + λ χ $_{2}$Q χ 2 + λ χ $_{3}$Q χ 3 + µ 2 $_{ψ}$B$_{3}$ + δ $^{ψ}$T$_{1}$ + λ ψ $_{1}$Q ψ 1 + λ ψ $_{2}$Q ψ 2 + λ ψ $_{3}$Q ψ 3 + µ 2 $_{ϑ}$B$_{4}$ + λ $^{ϑ}$Qϑ 1 + λ Φ χ 1 Q Φ χ 1 + λ Φ χ 2 Q Φ χ 2 + λ Φ χ ∗ 2 Q Φ χ 3 + λ Φ χ 3 Q Φ χ 4 + iλ Φ χ 4 Q Φ χ 5 + δ Φ ψ s T$_{2}$ + iδ Φ ψ a T$_{3}$ + λ Φ ψ 1 Q Φ ψ 1 + λ Φ ψ 2 Q Φ ψ 2 + λ Φ ψ ∗ 2 Q Φ ψ 3 + λ Φ ψ 3 Q Φ ψ 4 + iλ Φ ψ 4 Q Φ ψ 5 + λ Φ $^{ϑ}$Q Φ ϑ 1 + δ $^{ψχ}$T$_{4}$ + λ ψχ 1 Q ψχ 1 + λ ψχ 2 Q ψχ 2 + λ ψχ ∗ 2 Q ψχ 3 + λ ψχ 3 Q ψχ 4 + λ $^{ψϑ}$Q ψϑ 1 + δ $^{χϑψ}$T$_{5}$ + µ 2 $_{η}$B$_{5}$ + λ $^{η}$Q η 1 + λ Φ η 1 Q Φ η 1 + λ Φ η 2 Q Φ η 2 + λ Φ η 3 Q Φ η 3 + λ Φ η ∗ 3 Q Φ η 4 + λ $^{ηχ}$Q ηχ 1 + λ $^{ηψ}$Q ηψ 1 + λ $^{ηϑ}$Q ηϑ 1 , (A1) Q Φ 1 $_{= (Φ}$†$_{Φ)}$$_{1}$$_{(Φ}$ †$_{Φ)}$$_{1}$$_{,}$ Q Φ 2 $_{= (Φ}$†$_{Φ)}$$_{1}$ ′ $_{(Φ}$†$_{Φ)}$$_{1}$ ′′ , Q Φ 3 $_{= (Φ}$†$_{Φ)}$$_{3}$$_{s}$$_{(Φ}$ †$_{Φ)}$$_{3}$$_{s}$$_{,}$ Q Φ 4 $_{= (Φ}$†$_{Φ)}$$_{3}$$_{a}$$_{(Φ}$ †$_{Φ)}$$_{3}$$_{a}$$_{,}$ Q Φ 5 $_{= (Φ}$†$_{Φ)}$$_{3}$$_{s}$$_{(Φ}$ †$_{Φ)}$$_{3}$$_{a}$$_{,}$ Q χ 1 = ( χχ ) 1 ( χχ ) 1 , Q χ 2 = ( χχ ) 1 ′ ( χχ ) 1 ′′ , Q χ 3 = ( χχ ) 3 ( χχ ) 3 , Q ψ 1 = ( ψψ ) 1 ( ψψ ) 1 , Q ψ 2 = ( ψψ ) 1 ′ ( ψψ ) 1 ′′ , Q ψ 3 = ( ψψ ) 3 ( ψψ ) 3 , Q ϑ 1 = ( ϑϑ ) $^{2}$, Q Φ χ 1 $_{= (Φ}$†$_{Φ)}$$_{1}$$_{(}$$_{χχ}$$_{)}$$_{1}$$_{,}$ Q Φ χ 2 $_{= (Φ}$†$_{Φ)}$$_{1}$ ′ ( χχ ) 1 ′′ , Q Φ χ 3 $_{= (Φ}$†$_{Φ)}$$_{1}$ ′′ ( χχ ) 1 ′ , Q Φ χ 4 $_{= (Φ}$†$_{Φ)}$$_{3}$$_{s}$$_{(}$$_{χχ}$$_{)}$$_{3}$$_{,}$ Q Φ χ 5 $_{= (Φ}$†$_{Φ)}$$_{3}$$_{a}$$_{(}$$_{χχ}$$_{)}$$_{3}$$_{,}$ (A3) B$_{1}$ $_{= (Φ}$†$_{Φ)}$$_{1}$$_{,}$ B$_{2}$ = ( χχ ) 1 , B$_{3}$ = ( ψψ ) 1 , B$_{4}$ = ϑϑ, B$_{5}$ = $_{η}$†$_{η,}$ T$_{1}$ = ( ψψψ ) 1 , T$_{2}$ $_{= (Φ}$†$_{Φ)}$$_{3}$$_{s}$$_{ψ,}$ T$_{3}$ $_{= (Φ}$†$_{Φ)}$$_{3}$$_{a}$$_{ψ,}$ T$_{4}$ = ( ψχχ ) 1 , T$_{5}$ = ( χψ ) 1 ϑ, (A2)