Leptoquarks facing flavour tests and $b\to s\ell\ell$ after Moriond 2021

In view of the emerging hints for the violation of lepton flavour universality in several $B$-meson decays, we conduct a model-independent study (effective field theory approach) of several well-motivated new physics scenarios. Taking into account the most recent LHCb data, we provide updates to Wilson coefficient fits for numerous popular new physics hypotheses. We also consider a promising model of vector leptoquarks, which in addition to explaining the $B$-meson decay anomalies ($R_{K^{(*)}}$ and $R_{D^{(*)}}$) would have an extensive impact for numerous flavour observables. We identify promising decay modes allowing to (indirectly) probe such an extension: these include positive signals (at Belle II or LHCb) for $\tau\to \phi \mu$, $B_{(s)}$-meson decays to $\tau^+ \tau^-$ and $\tau^+ \mu^-$ ($\tau^+ e^-$) final states, as well as an observation of certain charged lepton flavour violation transitions at COMET and Mu2e. We also argue how the evolution of the experimental determination of $R_{D^{(*)}}$ can prove instrumental in falsifying a vector leptoquark explanation of the anomalous $B$-meson decay data.