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. The lepton mass Lagrangian contains a Dirac part that includes the Yukawa Lagrangian plus the KK mass terms, 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 where we have grouped together the charge − 1 leptons into e$_{L,R}$ and the neutral ones into 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 B. Inclusion of massive KK modes 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 5 In this subsection we use the same calligraphic notation to denote matrices although they have a larger size here because they also include massive KK modes. 15 L D = L Y − ∑ $^{n}$≥ 1 [ m $^{l}$$_{n}$l ( n ) L l ( n ) R + m ˜ l 1 / 2 n ˜ l ( n ) 1 $_{$_{2}$L}$˜ l ( n ) 1 $_{2}$R + m ˜ l$_{−}$ 3 / 2 n ˜ l ( n ) − 3 $_{$_{2}$L}$˜ l ( n ) − 3 $_{$_{2}$R}$+ m ν ′ n ν ′ ( n ) L ν ′ ( n ) R + m e ′ $_{n}$e ′ ( n ) L e ′ ( n ) R + h . c . ] , (48) − L D = e$_{L}$ M e $_{D}$e$_{R}$ + ν$_{L}$ M ν $_{D}$ν$_{R}$ + ∑ $^{n}$≥ 1 m ˜ l 1 / 2 n ˜ X ( n ) L ˜ X ( n ) R + ∑ $^{n}$≥ 1 m ˜ l$_{−}$ 3 / 2 n ˜ Y ( n ) L ˜ Y ( n ) R + h . c ., (49) − L M = 1 2 ν c R M ν $_{M}$ν$_{R}$ + h . c . . (50)