for the right-handed (RH) component is denoted by [+], whereas [ − ] denotes a Dirichlet boundary condition for the left-handed (LH) chirality. Finally, the first sign corresponds to the boundary condition at the UV brane and the second one at the IR brane. The chosen boundary conditions allow for a LH zero mode transforming as an SU (2)$_{L}$ doublet with hypercharge − 1 / 2 in ζ$_{1}$ , a RH singlet of charge − 1 in ζ$_{α}$ , and a RH neutral singlet in ζ$_{2}$ . As generally in A$_{4}$ models, an extra global symmetry must be imposed to forbid dangerous operators. A discrete Z$_{8}$ group does the job in our case. Both global symmetries will be broken at the two branes by localized scalars transforming as gauge singlets, φ and η at the UV brane and φ ′ and η ′ at the IR one. The fermion and scalar transformation properties under A$_{4}$ × Z$_{8}$ are gathered in Table I. The three copies of ζ$_{1}$$_{,}$$_{2}$$_{,}$$_{3}$ span the A$_{4}$ triplet TABLE I: Bulk fermion (left) and localized scalar (right) quantum number assignments under the discrete group A$_{4}$ × Z$_{8}$ . representation, whereas each ζ$_{α}$ transforms as one the three different A$_{4}$ one-dimensional representations (see Appendix A 1 for a summary of the A$_{4}$ representations). Once the matter content is fixed, we can write down the most general Lagrangian com- patible with the symmetries. The bulk Lagrangian reads where summation on repeated indices k ∈ { 1 , 2 , 3 } , α ∈ { e, µ, τ } is understood. D$_{µ,z}$ are the gauge covariant derivatives and the bulk Dirac masses are conventionally parametrized in terms of the fundamental scale R , Note that the A$_{4}$ symmetry implies a family independent bulk mass for ζ$_{k}$ . The most general localized Lagrangians, excluding kinetic terms (discussed below), compatible with 6 A$_{4}$ Z$_{8}$ ζ$_{1}$ 3 1 ζ$_{2}$ 3 2 ζ$_{3}$ 3 1 ζ$_{α}$ 1 , 1 ′ , 1 ′′ 4 A$_{4}$ Z$_{8}$ φ (UV) 3 4 η (UV) 1 4 φ ′ (IR) 3 5 η ′ (IR) 1 7 L = ∫ R ′ R d z a 4 { ζ$_{k}$ [ i GLYPH<a0> D + ( D$_{z}$ + 2 a ′ a ) γ 5 − aM$_{k}$ ] ζ$_{k}$ + ζ$_{α}$ [ i GLYPH<a0> D + ( D$_{z}$ + 2 a ′ a ) γ 5 − aM$_{α}$ ] ζ$_{α}$ } , (7) M$_{k,α}$ = c$_{k,α}$ R . (8)