On the behaviour of composite resonances breaking lepton flavour universality

Within the context of composite Higgs models, recent hints on lepton-flavour non-universality in $B$ decays can be explained by a vector resonance $V$ with sizeable couplings to the Standard Model leptons ($\ell$). We argue that, in such a case, spin-$1/2$ leptonic resonances ($L$) are most probably light enough to open the decay mode $V \rightarrow L\ell$. This implies, in combination with the fact that couplings between composite resonances are much larger than those between composite and elementary fields, that this new decay can be important. In this paper, we explore under which conditions it dominates over other decay modes. Its discovery, however, requires a dedicated search strategy. Employing jet substructure techniques, we analyse the final state with largest branching ratio, namely $\mu^+\mu^- Z/h, Z/h \rightarrow$ jets. We show that (i) parameter space regions that were believed excluded by di-muon searches are still allowed, (ii) these regions can already be tested with the dedicated search we propose and (iii) $V$ masses as large as $\sim 3.5$ TeV can be probed at the LHC during the high-luminosity phase.