A viable $L_e-L_{\mu }$ model with $\mu \to e$ violation

We extend the Standard Model gauge group by $U{\left(1\right)}_{L_e-L_{\mu }}$ 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 $\mu \to e$ processes can only be mediated by $\mu \to \tau \times \tau \to e$ interactions, bounds from $\mu \to e$ transitions can be avoided while allowing for accessible new physics. We consider the case of a $Z^{\text{'}}$ boson with a mass of $M_{Z^{\text{'}}}\simeq 10$GeV and a gauge coupling $g^{\text{'}}\simeq {10}^{-4}$, which is in reach of Belle-II, and a long-lived $Z^{\text{'}}$ boson with a mass of $\text{MeV}\lesssim M_{Z^{\text{'}}}\lesssim m_{\mu }-m_e$ which can be probed by searching for $\ell \to {\ell }^{\text{'}}+\mathrm{inv}.$. Neutrino masses and mixing angles can also be accounted for if sterile neutrinos are added to the spectrum.