where ρ$_{i}$ ∼ v $^{2}$/ Λ 2 and γ$_{i}$ ∼ v$_{η}$v/ Λ $^{2}$. Let us discuss now the terms on the IR brane. The leading term mixing ζ$_{1}$ with ζ$_{2}$ contains no power of φ ′ , so the only new structure corresponds to one factor of φ ′ . The first such term comes at order 1 / Λ ′ $^{3}$, due to the Z$_{8}$ symmetry. At this order we have where ϵ$_{1}$ ∼ v ′ $_{η}$v ′ $^{2}$/ Λ ′ 3 and ϵ$_{2}$$_{,}$$_{3}$ get contributions ∼ v ′ $_{η}$v ′ $^{2}$/ Λ ′ 3 and ∼ v ′ 2 η v ′ / Λ ′ $^{3}$, and similarly for ν ′ 1 $_{L}$ν ′ 2 $_{R}$. Finally, the coupling between ζ$_{3}$ and ζ$_{α}$ is not modified by higher order terms, because we already have a term with one power of φ ′ and the singlet contribution cannot result from a singlet structure under A$_{4}$ . No further structures are generated at higher orders. Therefore the higher order effects in the A$_{4}$ breaking can be summarized, after the rotation ζ$_{k}$ → Ω ζ$_{k}$ in Eq. (14), by the following replacements for the Majorana masses, with for the mixing between the bi-doublets or the singlets in ζ$_{1}$ and ζ$_{2}$ . The IR terms remain diagonal whereas the UV terms receive non-diagonal corrections. All three effects are a [ η ′ ∗ $^{2}$φ ′ Λ ′ 3 + η $^{′}$| φ $^{′}$| 2 Λ ′ 3 + η ′ ∗ φ ′ † 2 Λ ′ 3 ] ( l$_{1}$$_{L}$l$_{2}$$_{R}$ + ˜ l$_{1}$$_{L}$ ˜ l$_{2}$$_{R}$ ) +h . c . → l$_{1}$$_{L}$   $^{ϵ$_{1}$}$ ϵ$_{2}$ ϵ$_{3}$ ϵ$_{3}$ ϵ$_{1}$ ϵ$_{2}$ ϵ$_{2}$ ϵ$_{3}$ ϵ$_{1}$    l$_{2}$$_{R}$ +[ l$_{1}$$_{,}$$_{2}$ → ˜ l$_{1}$$_{,}$$_{2}$ ]+h . c ., (57) ˆ θ$_{M}$ → ˆ θ$_{M}$ $_{+}$  $^{δ$_{1}$}$ δ$_{3}$ δ$_{2}$ δ$_{3}$ δ$_{2}$ δ$_{1}$ δ$_{2}$ δ$_{1}$ δ$_{3}$    (58) x$_{l}$ → x$_{l}$ + X$_{l}$, (59) y$_{b,s}$ v ′ Λ ′ → y$_{b,s}$ v ′ Λ ′ $_{+}$  $^{ϵ}$$^{b,s}$ 1 + ϵ b,s 2 + ϵ b,s 3 0 0 0 ϵ b,s 1 + ωϵ b,s 2 + ω $^{2}$ϵ b,s 3 0 0 ϵ b,s 1 + ω $^{2}$ϵ b,s 2 + ωϵ b,s 3    (61) X$_{l}$ = Ω †   $^{ρ$_{1}$}$ + ρ$_{2}$ + ρ$_{3}$ 0 0 0 ρ$_{1}$ + ωρ$_{2}$ + ω $^{2}$ρ$_{3}$ γ$_{1}$ 0 γ$_{2}$ ρ$_{1}$ + ω $^{2}$ρ$_{2}$ + ωρ$_{3}$    Ω (60) for the mixing between the SM LH doublets in ζ$_{1}$ and ζ$_{3}$ , and 19