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We investigate the inclusive hadroproduction of a heavy quarkonium ($J/\psi$or $\Upsilon$), in association with a light-flavored jet, as a testing groundfor the semi-hard regime of QCD. Our theoretical setup is the hybridhigh-energy and collinear factorization, where the standard collinear approachis supplemented by the $t$-channel resummation of leading and next-to-leadingenergy logarithms \`a la BFKL. We present predictions for rapidity andazimuthal-angle differential distributions, hunting for stabilizing effects ofthe high-energy series under higher-order corrections. Our reaction representsan additional channel to test the production mechanisms of quarkonia at highenergies and large transverse momenta and to possibly shed light on thetransition region from heavy-quark pair production to single-partonfragmentation of $J/\psi$ and $\Upsilon$ states.

Diffractive semi-hard production of a $J/\psi$ or a $\Upsilon$ from single-parton fragmentation plus a jet in hybrid factorization