Direct searches motivated by recent $B$-anomalies

We discuss the physics case for direct searches at the LHC motivated by the $B$-physics anomalies. After correlating semi-tauonic $B$ decays to di-tau production at the LHC, and discussing the possible models solving the $B$-anomalies, we show how current LHC data in $\tau\bar\tau$ tails exclude most beyond the SM scenarios except for a handful of leptoquark (LQ) models. We analyze in detail the impact of LHC searches for some of these LQ solutions using current data. In particular, we focus on the well known $U_1\sim({\bf3},{\bf1})_{2/3}$ vector LQ solution as well as the GUT-inspired scalar LQ solutions, $R_2\sim({\bf3},{\bf2})_{7/6}$ and $S_3\sim({\bf\bar3},{\bf3})_{1/3}$. By exploiting the complementarity between high-$p_T$ searches in di-tau tails and the lepton flavor violating decays $B\to K\mu\tau$ and $\tau\to\mu\phi$ we argue that these model can be cornered by the LHC, Belle II and LHCb in the near future.