content_image/366593.jpg content_image/366594.jpg content_image/366595.jpg
for the scales allowed by EWPT, but in GHU models we can actually test how good this approximation is because in this case it is possible to get a solution to all orders in the Higgs v.e.v.. These two types of corrections, which do not significantly modify the picture drawn above, are studied in the next two subsections. Finally, we have only included the LO A$_{4}$ × Z$_{8}$ breaking terms. Higher orders, although suppressed by extra powers of 1 / Λ $^{(}$′$^{)}$, could destabilize the TBM mixing pattern and introduce new sources of LFV. We will consider these higher order corrections in the following section. 

## B. Inclusion of massive KK modes 

The lepton mass Lagrangian contains a Dirac part that includes the Yukawa Lagrangian plus the KK mass terms, 

<img src='content_image/366593.jpg'>

where the SU (2)$_{L}$ doublets with hypercharges 1 2 and − 3 2 are denoted by ˜ l 1 2 and ˜ l − 3 $_{2}$, re- spectively, and the SM-like (hypercharge − 1 $_{2}$) doublets which participate from all SO (5) multiplets by l . Obviously, L Y also includes Yukawa couplings with the massive KK modes. 

The Dirac mass Lagrangian can be written in matrix form 

<img src='content_image/366594.jpg'>

where we have grouped together the charge − ν$_{L,R}$ . 5 The UV brane term, Eq. (36), induces a Majorana mass term that now involves all KK modes of the RH neutrinos 

<img src='content_image/366595.jpg'>

The mass Lagrangian is diagonal for the charge +1 and − 2 sectors but not for the charge − 1 and neutral ones. However, it is still true that it is family diagonal except for the terms involving the Majorana neutrino masses. Thus, although we have now to diagonalize the charged lepton mass term, this diagonalization does not mix different generations and then 

