A new approach to semi-leptonic tags in $B$-meson semi-invisible decays

Kinematic variables designed for pairwise decays to partly undetected final states -- a prominent example being $M_{T2}$ and its Lorentz-invariant version $M_2$ -- have been extensively deployed in high-$p_T$ collider searches. A new range of potential applications at flavour facilities -- where $B$ mesons or $\tau$ leptons are also pairwise produced -- was recently proposed. One general challenge in these decays arises if both the signal parent and the 'other' parent, often used as a tag, decay semi-invisibly. In such cases, which notably include semi-leptonic tags, signal identification is generally hindered by the ensuing lack of knowledge of the signal-parent boost. $M_2$ helps precisely to overcome this challenge, and allows to leverage the otherwise superior efficiency of semi-leptonic decays. Our strategy rests on two novel constraints that can be imposed on $M_2$. The first is that of the known mass of the decaying-parent mass squared which, in connection with other constraints, gives rise to $M_{2sB}$. The second is on the flight direction of the signal parent, often well reconstructed at facilities with high vertexing capabilities such as Belle II and LHCb. This constraint gives rise to the $M_{2V}$ variable, that can be used even at facilities where the collision energy is not known. We test these ideas in a decay of great current interest in the context of the persistent discrepancies in $B$ decays, namely $B \to K \tau \mu$. We find that a bare-bones application of $M_{2sB}$ leads, alone, to an improvement that is already halfway between the current approach and the "truth-level" semi-leptonic case. Ceteris paribus -- in particular statistics -- our approach thus makes semi-leptonic tags competitive with fully reconstructed hadronic tags.