On the single leptoquark solutions to the $B$-physics anomalies

We revisit the possibilities of accommodating the experimental indications of the lepton flavor universality violation in $b$-hadron decays in the minimal scenarios in which the Standard Model is extended by the presence of a single $\mathcal{O}(1\,\mathrm{TeV})$ leptoquark state. To do so we combine the most recent low energy flavor physics constraints, including $R_{K^{(\ast)}}^\mathrm{exp}$ and $R_{D^{(\ast)}}^\mathrm{exp}$, and combine them with the bounds on the leptoquark masses and their couplings to quarks and leptons as inferred from the direct searches at the LHC and the studies of the large $p_T$ tails of the $pp\to \ell\ell$ differential cross section. We find that none of the scalar leptoquarks of $m_\mathrm{LQ} \simeq 1\div 2$~TeV can accommodate the $B$-anomalies alone. Only the vector leptoquark, known as $U_1$, can provide a viable solution which, in the minimal setup, provides an interesting prediction, i.e. a lower bound to the lepton flavor violating $b\to s\mu^\pm\tau^\mp$ decay modes, such as $\mathcal{B}(B\to K\mu\tau) \gtrsim 0.7\times 10^{-7}$.