Quark-Lepton Connections in $Z^\prime$ Mediated FCNC Processes: Gauge Anomaly Cancellations at Work

We consider scenarios with a heavy $Z^\prime$ gauge boson with flavour non-universal quark and lepton couplings with the goal to illustrate how the cancellation of gauge anomalies generated by the additional U(1)$^\prime$ gauge symmetry would imply correlations between FCNC processes in the quark sector, in the lepton sector and most interestingly between quark flavour and lepton flavour violating processes. We present scenarios with only left-handed flavour-violating $Z^\prime$ couplings and those with also right-handed flavour-violating couplings. These are characterized by a small number of free parameters but in contrast to gauge anomaly cancellation in the Standard Model, that takes place separately within each generation, in our scenarios anomaly cancellation involves simultaneously quarks and leptons of all three generations. Our models involve, beyond the ordinary quarks and leptons, three heavy right-handed neutrinos. The models with only left-handed FCNCs of $Z^\prime$ involve beyond $g_{Z^\prime}$ and $M_{Z^\prime}$ two real parameters characterizing the charges of all fermions under the U(1)$^\prime$ gauge symmetry and the CKM and PMNS ones in the quark and lepton sectors, respectively. The models with the right-handed FCNCs of $Z^\prime$ involve few additional parameters. Imposing constraints from well measured $\Delta F=2$ observables we identify interesting correlations that involve e.g. $\varepsilon'/\varepsilon$, $B_{s,d}\to\mu^+\mu^-$, $B\to K(K^*) \ell^+\ell^-$, $K^+\rightarrow\pi^+\nu\bar\nu$, $K_{L}\rightarrow\pi^0\nu\bar\nu$ and purely lepton flavour violating decays like $\mu\to e\gamma$, $\mu\to 3 e$, $\tau\to 3\mu$ and $\mu-e$ conversion} among others. Also $(g-2)_{\mu,e}$ are considered. The impact of the experimental $\mu\to e\gamma$, $\mu\to 3 e$ and in particular $\mu-e$ conversion bounds on rare $K$ and $B$ decays is emphasized.