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Abstract We extend the Standard Model gauge group by $$U(1)_{L_e-L_\mu }$$ U ( 1 ) L e - L μ and introduce two scalars, a doublet and a singlet, that are charged under this new group and have lepton flavour violating couplings. Since in this model $$\mu \rightarrow e$$ μ → e processes can only be mediated by $$\mu \rightarrow \tau \times \tau \rightarrow e$$ μ → τ × τ → e interactions, bounds from $$\mu \rightarrow e$$ μ → e transitions can be avoided while allowing for accessible new physics. We consider the case of a $$Z'$$ Z ′ boson with a mass of $$M_{Z'}\simeq 10~$$ M Z ′ ≃ 10 GeV and a gauge coupling $$g'\simeq 10^{-4}$$ g ′ ≃ 10 - 4 , which is in reach of Belle-II, and a long-lived $$Z'$$ Z ′ boson with a mass of $$\text {MeV}\lesssim M_{Z'}\lesssim m_\mu -m_e$$ MeV ≲ M Z ′ ≲ m μ - m e which can be probed by searching for $$\ell \rightarrow \ell ' +\mathrm {inv.}$$ ℓ → ℓ ′ + inv . . Neutrino masses and mixing angles can also be accounted for if sterile neutrinos are added to the spectrum.