A gauged horizontal $SU(2)$ symmetry and $R_{K^{(\ast)}}$

One of the greatest challenges for models of $b \to s$ anomalies is the necessity to produce a large contribution to a quark times a lepton current, $J_q \times J_\ell$, and to avoid accordingly large contributions to flavour-changing $J_q \times J_q$ and $J_\ell \times J_\ell$ amplitudes, which are severely constrained by data. We consider a gauged horizontal symmetry involving the two heaviest generations of all left-handed fermions. In the limit of degenerate masses for the horizontal bosons, and in the absence of mixing between the two heavier generations and the lighter one, such symmetry would make $J_q \times J_q$ and $J_\ell \times J_\ell$ amplitudes exactly flavour-diagonal. Mixing with the first generation is however inescapable due to the CKM matrix, and the above mechanism turns out to be challenged by constraints such as $D^0 -\bar D^0$ mixing. Nonetheless, we show that a simultaneous description of all data can be accomplished by simply allowing for non-degenerate masses for the horizontal bosons. Such scenario predicts modifications in several processes that can be tested at present and upcoming facilities. In particular, it implies a lower and upper bound for $\mathcal B (B \to K \mu^\pm \tau^\mp)$, an asymmetry between its two charge conjugated modes, and well-defined correlations with LFV in $\tau$ decays.