content_image/330577.jpg content_image/330576.jpg
where the dots denote massive modes. (We have chosen the normalization constant to obtain a canonically normalized scalar, and in the second equality we have used R ≪ R ′ .) These four scalars transform as a 4 of SO (4) and are identified with the SM Higgs. 5D gauge invariance guarantees that any potential generated for these scalars has to arise from non-local contributions and therefore, it is finite to all orders in perturbation theory [36]. 

Regarding the matter content of the model, there are several possibilities. We consider here all fermions to be in fundamental representations of SO (5). Thus, four multiplets per family are required in order to have independent localizations for left and right-handed zero modes. This construction is parallel to the one giving rise to realistic composite Higgs models in the quark sector [17, 37]; and as we will show, a similar matter content transforming non- trivially under a global A$_{4}$ symmetry generates the observed leptonic spectrum in a natural way, without conflict with present experimental data. Hence, there are four 5D fermion representations per generation transforming as the fundamental SO (5) representation 5 , with boundary conditions 

where ζ$_{1}$$_{,}$$_{2}$ and ζ$_{3}$$_{,α}$ have U (1)$_{X}$ charge 0 and − 1, respectively. Note that there are three copies for each ζ$_{1}$$_{,}$$_{2}$$_{,}$$_{3}$ because there are three families, but only one ζ$_{α}$ set with α running over the three lepton flavors e, µ, τ . We explicitly show the decomposition under SU (2)$_{L}$ × SU (2)$_{R}$, 5 = ( 2 , 2 ) ⊕ ( 1 , 1 ). The bi-doublet is represented by a 2 × 2 matrix with the SU (2)$_{L}$ rotation acting vertically and the SU (2)$_{R}$ one horizontally ( i.e. the left and right columns correspond to fields with T 3 R = ± 1 / 2, whereas the upper and lower components have T 3 L = ± 1 / 2, respectively). The bi-doublets in ζ$_{1}$$_{,}$$_{2}$ contain two SU (2)$_{L}$ doublets of hypercharge 1 2 and − 1 $_{2}$, and those in ζ$_{3}$$_{,α}$ two SU (2)$_{L}$ doublets of hypercharge − 1 2 and − 3 $_{2}$, respectively. The corresponding electric charges read 

<img src='content_image/330577.jpg'>

<img src='content_image/330576.jpg'>

where the dot denotes all possible values of the corresponding subscript. The signs in square brackets are a shorthand for the boundary conditions. A Dirichlet boundary condition 

