Third Family Hypercharge Model for $R_{K^{(\ast )}}$ and Aspects of the Fermion Mass Problem

We present a model to explain LHCb’s recent measurements of $R_K$ and $R_{K^{\ast }}$ based on an anomaly-free, spontaneously-broken $U(1{)}_F$ gauge symmetry, without any fermionic fields beyond those of the Standard Model (SM). The model explains the hierarchical heaviness of the third family and the smallness of quark mixing. The $U(1{)}_F$ charges of the third family of SM fields and the Higgs doublet are set equal to their respective hypercharges. A heavy $Z^{\mathrm{\prime }}$ particle with flavour-dependent couplings can modify the $[\overline{b_L}{\gamma }^{\rho }{s}_L][\overline{{\mu }_L}{\gamma }_{\rho }{\mu }_L]$ effective vertex in the desired way. The $Z^{\mathrm{\prime }}$ contribution to $B_s-\overline{B_s}$ mixing is suppressed by a small mixing angle connected to $V_{ts}$, making the constraint coming from its measurement easier to satisfy. The model can explain $R_K$ and $R_{K^{(\ast )}}$ whilst simultaneously passing other constraints, including measurements of the lepton flavour universality of $Z$ couplings.