and on the square of the SM lepton mixing with heavier vector-like fermions [46]. where α = e, µ, τ . In terms of this effective Lagrangian, the branching ratios for these decays read For the µ − e conversion rate we have applied the usual expression where g µe L,R are the corresponding off-diagonal Z ¯ eµ couplings, G$_{F}$ is the Fermi constant and α the QED coupling strength, while the other terms are atomic physics constants defined in [48]. We shall use the current PDG [45] bounds for the tri-lepton decays and the titanium bound B$_{conv}$ < 6 . 1 × 10 − 13 from the SINDRUM II experiment [49] for µ − e conversion. 7 Higgs mediated contributions [47] are suppressed by the A 4 symmetry and the SM lepton masses, and then very small in this class of models. 23 − L eff = 4 G$_{F}$ √$_{2}$ [ g αβ 3 ( ¯ l β $_{R}$γ$^{µ}$l α R ) ( ¯ l β $_{R}$γ$^{µ}$l β R ) + g αβ 4 ( ¯ l β $_{L}$γ$^{µ}$l α L ) ( ¯ l β $_{L}$γ$^{µ}$l β L ) + g αβ 5 ( ¯ l β $_{R}$γ$^{µ}$l α R ) ( ¯ l β $_{L}$γ$^{µ}$l β L ) + g αβ 6 ( ¯ l β $_{L}$γ$^{µ}$l α L ) ( ¯ l β $_{R}$γ$^{µ}$l β R )] + h . c ., (66) B ( µ → eee ) = 2 ( | g µe 3 | 2 + | g µe 4 | 2 ) + | g µe 5 | 2 + | g µe 6 | $^{2}$, B ( τ → µµµ ) = { 2 ( | g τ µ 3 | 2 + | g τ µ 4 | 2 ) + | g τ µ 5 | 2 + | g τ µ 6 | 2 } B ( τ → eνν ) , B ( τ → eee ) = 2 | g τ e 3 | 2 + | g τ e 4 | 2 + | g τ e 5 | 2 + | g τ e 6 | 2 B ( τ → eνν ) , B ( τ → eeµ ) = { | g τ µ 3 | 2 + | g τ µ 4 | 2 + | g τ µ 5 | 2 + | g τ µ 6 | 2 } B ( τ → eνν ) , B ( τ → eµµ ) = | g τ e 3 | 2 + | g τ e 4 | 2 + | g τ e 5 | 2 + | g τ e 6 | 2 B ( τ → eνν ) . (67) B$_{conv}$ = 2 p$_{e}$E$_{e}$G 2 $_{F}$m 3 $_{µ}$α $^{3}$| F$_{q}$ | $^{2}$Z4 $_{eff}$Q 2 N π $^{2}$Z Γ$_{capt}$ [ | g µe R | 2 + | g µe L | 2 ] , (68) • Tree-level LFV . We have included the most relevant constraints following [42]. Ex- plicitly, we have studied the decays µ → e − e $^{+}$e − , τ → µ − µ $^{+}$µ − , τ → e − e $^{+}$e − , τ → µ − e $^{+}$e − , τ → µ − e $^{+}$µ − and the µ − e nuclear conversion rate. The tri-lepton decays l → l$_{1}$ ¯ l$_{2}$l$_{3}$ are mediated by LFV tree-level couplings to the physical Z gauge boson and its KK excitations. (The effects due to fermion mixing are negligible). 7 At low energies, these contributions can be parameterized by the following effective Lagrangian,