Anomalous $Z^{\mathrm{\prime }}$ bosons for anomalous $B$ decays

Motivated by the intriguing discrepancies in $b\to s\mathrm{\ell }\mathrm{\ell }$ transitions, the fermion mass problem, and a desire to preserve the accidental symmetries of the Standard Model (SM), we extend the SM by an anomalous $U(1{)}_X$ gauge symmetry where $X=Y_3+a(L_{\mu }-L_{\tau })\mathrm{/}6$. The heavy $Z^{\mathrm{\prime }}$ boson associated with spontaneously breaking $U(1{)}_X$ at the TeV scale mediates the $b\to s\ell\ell$anomaliesvia$$\mathcal{O}$^\mu_9 \sim\frac{1}{\Lambda^2}(\bar{s}\gamma_\rho P_L b)(\bar{\mu} \gamma^\rho \mu)$. We show that this model, which features mixed gauge anomalies involving $U(1{)}_X$ and hypercharge, can be made anomaly-free for any $a\in \mathbb{Z}$ by integrating in a pair of charged fermions whose masses naturally reside somewhere between 1 and 30 TeV. The gauge symmetry permits only the third family Yukawas at the renormalisable level, and so the light quark masses and mixings are controlled by accidental $U(2{)}^3$ flavour symmetries which we assume are minimally broken alongside $U(1{)}_X$. The lepton sector is not governed by $U(2)$ symmetries, but rather one expects a nearly diagonal charged lepton Yukawa with $m_{e,\mu }\ll m_{\tau }$. The model does not explain the hierarchy $m_e\ll m_{\mu }$, but it does possess high quality lepton flavour symmetries that are robust to the heavy physics responsible for generating $m_{e,\mu }$. We establish the viability of these models by checking agreement with the most important experimental constraints. We comment on how the model could also explain neutrino masses and the muon $g-2$.