Generalised $\mu$-$\tau$ symmetries and calculable gauge kinetic and mass mixing in $U(1)_{L_\mu-L_\tau}$ models

Extensions of the standard model with a $U(1)$ gauge symmetry contain gauge invariant kinetic mixing, $\sin\chi$, and gauge non-invariant mass mixing, $\delta M^2$, between the hypercharge and the new gauge boson $Z'$. These represent a priori incalculable but phenomenologically important parameters of the theory. They become calculable if there exist spontaneously or softly broken symmetries which forbid them at tree level but allow their generation at the loop level. We discuss various symmetries falling in this category in the context of the gauged $L_\mu-L_\tau$ models and their interplay with lepton mixing. It is shown that one gets phenomenologically inconsistent lepton mixing parameters if these symmetries are exact. Spontaneous breaking of these symmetries can lead to consistent lepton mixing and also generates finite and calculable values of these parameters at one or two loop order depending on the underlying symmetry. We calculate these parameters in two specific cases: (i) the standard seesaw model with $\mu$-$\tau$ symmetry broken by the masses of the right-handed neutrinos and (ii) in a model containing a pair of vectorlike charged leptons which break $\mu$-$\tau$ symmetry. In case (i), the right-handed neutrinos are the only source of gauge mixing. The kinetic mixing parameters are suppressed and vanish if the right-handed neutrinos decouple from the theory. In contrast, there exists a finite gauge mixing in case (ii) which survives even when the masses of vectorlike leptons are taken to infinity, exhibiting non-decoupling behaviour. The seesaw model discussed here represents a complete framework with practically no kinetic mixing and hence can survive a large number of experimental probes used to rule out specific ranges in the coupling $g^\prime$ and mass $M_{Z^\prime}$. The model can generate non-universality in tau decays, which can be tested in future experiments.