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

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