  Searches for new physics (NP) at particle colliders typically involve
multivariate analysis of kinematic distributions of final state particles
produced in a decay of a hypothetical NP resonance. Since the pair-production
cross-sections mediated by such resonances are strongly suppressed by the NP
scale, this analysis becomes less relevant for NP searches for masses of the
BSM resonance above 1 TeV. On the other hand, $t$-channel processes are less
sensitive to the mass of the virtual mediator and therefore larger phase-space
can be potentially probed as well as the couplings between the NP particles and
the Standard Model fields. The fact that transitions between different
generations of quarks and leptons may exist, the potential of the search
presented in this article can be used, as a reference guide, to enlarge
significantly the scope of searches performed at the LHC to flavour
off-diagonal channels, in a theoretically consistent approach. In this work, we
study non-resonant production of scalar leptoquarks which have been proposed in
the literature to provide a potential avenue for radiative generation of
neutrino masses, accommodating as well the existing flavour physics data. Final
states involving just two muons at the LHC ($\mu^+, \mu^-$), are used as a
well-motivated case study.
