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In leptonic flavour models with discrete flavour symmetries, couplingsbetween flavons and leptons can result in special flavour structures after theygain vacuum expectation values. At the same time, they can also contribute tothe other lepton-flavour-violating processes. We study the flavon-induced LFV3-body charged lepton decays and radiative decays and we take as example the$A_4$ discrete symmetry. In $A_4$ models, a $Z_3$ residual symmetry roughlyholds in the charged lepton sector for the realisation of tri-bimaximal mixingat leading order. The only processes allowed by this symmetry are $\tau^-\to\mu^+ e^- e^-, e^+ \mu^- \mu^-$, and the other 3-body and all radiative decaysare suppressed by small $Z_3$-breaking effects. These processes also depend onthe representation the flavon is in, whether pseudo-real (case i) or complex(case ii). We calculate the decay rates for all processes for each case andderive their strong connection with lepton flavour mixing. In case i, sum rulesfor the branching ratios of these processes are obtained, with typical examples$\text{Br}(\tau^-\to \mu^+ e^- e^-) \approx \text{Br}(\tau^-\to e^+ \mu^-\mu^-)$ and $\text{Br}(\tau^-\to e^- \gamma) \approx \text{Br}(\tau^-\to \mu^-\gamma)$. In case ii, we observe that the mixing between two $Z_3$-covariantflavons plays an important role. All processes are suppressed by charged leptonmasses and current experimental constraints allow the electroweak scale and theflavon masses to be around hundreds of GeV. Our discussion can be generalisedin other flavour models with different flavour symmetries.

Flavon-induced connections between lepton flavour mixing and charged lepton flavour violation processes