content_image/356010.jpg content_image/356013.jpg content_image/356014.jpg content_image/356011.jpg content_image/356012.jpg
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 

<img src='content_image/356010.jpg'>

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 SM LH doublets in ζ$_{1}$ and ζ$_{3}$ , and 

<img src='content_image/356013.jpg'>

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 

<img src='content_image/356014.jpg'>

<img src='content_image/356011.jpg'>

<img src='content_image/356012.jpg'>

