Widening the $U(1)_{L_\mu-L_\tau}$ $Z^\prime$ mass range for resolving the muon $g-2$ anomaly

Exchanging a $Z^\prime$ gauge boson is a favored mechanism to solve the muon $(g-2)_\mu$ anomaly. Among such models the $Z^\prime$ from $U(1)_{L_\mu - L_\tau}$ gauge group has been extensively studied. In this model the same interaction addressing $(g-2)_\mu$, leads to an enhanced muon neutrino trident (MNT) process $\nu_\mu N \to \nu_\mu \mu \bar \mu N$ constraining the $Z^\prime$ mass to be less than a few hundred MeV. Many other $Z^\prime$ models face the same problem. It has long been realized that the coupling of $Z^\prime$ in the model can admit $(\bar \mu \gamma^\mu \tau + \bar \nu_\mu \gamma^\mu L \nu_\tau)Z^\prime_\mu$ interaction which does not contribute to the MNT process. It can solve $(g-2)_\mu$ anomaly for a much wider $Z^\prime$ mass range. However this new interaction induces $\tau \to \mu \bar\nu_\mu \nu_\tau$ which rules out it as a solution to $(g-2)_\mu$ anomaly. Here we propose a mechanism by introducing type-II seesaw $SU(2)_L$ triplet scalars to evade constraints from all known data to allow a wide $Z^\prime$ mass range to solve the $(g-2)_\mu$ anomaly. This mechanism opens a new window for $Z^\prime$ physics.