Searching for exclusive leptoquarks with the Nambu-Jona-Lasinio composite model at the LHC and HL-LHC

We present a detailed study concerning a new physics scenario involving four fermion operators of the Nambu-Jona-Lasinio type characterized by a strong-coupling ultraviolet fixed point where composite particles are formed as bound states of elementary fermions at the scale $\Lambda ={\cal O}(\text{TeV})$. After implementing the model in the Universal FeynRules Output format, we focus on the phenomenology of the scalar leptoquarks at the LHC and the High-Luminosity option. Leptoquark particles have undergone extensive scrutiny in the literature and experimental searches, primarily relying on pair production and, more recently, incorporating single, t-channel, and lepton-induced processes. This study marks, for the first time, the examination of these production modes at varying jet multiplicities. Novel mechanisms emerge, enhancing the total production cross-section, especially for leptoquarks couplings to higher fermion generations. A global strategy is devised to capture all final state particles produced in association with leptoquarks or originating from their decay, which we termed ``exclusive'', in an analogy to the nomenclature used in nuclear reactions. The assessment of the significance in current and future LHC runs, focusing on the case of leptoquark coupling to a muon - $\textit{c}$ quark pair, reveals superior sensitivity compared to ongoing searches. Given this heightened discovery potential, we advocate the incorporation of exclusive leptoquark searches in future investigations at the LHC.