Single Leptoquark Solutions to the $B$-physics Anomalies

We examine various scenarios in which the Standard Model is extended by a light leptoquark state to explain one or both $B$-physics anomalies. Combining low-energy constraints and direct searches at the LHC, we confirm that the only single leptoquark model that can explain both anomalies at the same time is a vector leptoquark, known as $U_1$. Focusing on $U_1$, we highlight the complementarity between LHC and low--energy constraints, and argue that improving the experimental bound on $\mathcal{B}(B\to K\mu\tau)$ by two orders of magnitude could compromise its viability as a solution to the $B$-physics anomalies.