Precise limits on the charge-$2/3$ $U_1$ vector leptoquark

The $U_1$ leptoquark is known to be a suitable candidate for explaining the semileptonic $B$-decay anomalies. We derive precise limits on its parameter space relevant for the anomalies from the current LHC high-$p_{\rm T}$ dilepton data. We consider an exhaustive list of possible $B$-anomalies-motivated simple scenarios with one or two new couplings that can also be used as templates for obtaining bounds on more complicated scenarios. To obtain precise limits, we systematically consider all possible $U_1$ production processes that can contribute to the dilepton searches, including the resonant pair and single productions, nonresonant $t$-channel $U_1$ exchange, as well as its large interference with the Standard Model background. We demonstrate how the inclusion of resonant production contributions in the dilepton signal can lead to appreciably improved exclusion limits. We point out new search channels of $U_1$ that can act as unique tests of the flavour-motivated models. The template scenarios can also be used for future $U_1$ searches at the LHC. We compare the LHC limits with other relevant flavour bounds and find that a TeV-scale $U_1$ can accommodate both $R_{D^{(*)}}$ and $R_{K^{(*)}}$ anomalies while satisfying all the bounds.