A combined explanation of the $B$-meson decay anomalies with a single vector leptoquark

Motivated by the recent experimental progress on the $R_{D^{(\ast)}}$ and $R_{K^{(\ast)}}$ anomalies in $B$-decays, we consider an extension of the Standard Model by a single vector leptoquark field. We study how one can achieve the required lepton flavour non-universality, starting from a priori universal gauge couplings. While the unitary coupling flavour structure, induced by the mass misalignment of quarks and leptons after $SU(2)_L$ breaking, does not allow to comply with stringent bounds from lepton flavour violating processes, we find that effectively non-unitary couplings, due to mixings with heavy vector-like fermions, hold the key to simultaneously address the $R_{K^{(\ast)}}$ and $R_{D^{(\ast)}}$ anomalies. Furthermore, in the near future, the expected progress in the sensitivity on charged lepton flavour violating observables should allow to probe a large region of the preferred parameter space in this class of vector leptoquark models.