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We present a novel study on the inclusive production of a heavy quarkonium($J/\psi$ or $\Upsilon$), in association with a light-flavored jet, as a testfield of the high-energy QCD dynamics. The large transverse momenta at whichthe two final-state objects are detected permits us to perform an analysis inthe spirit of the variable-flavor number scheme (VFNS), in which the crosssection for the hadroproduction of a light parton is convoluted with aperturbative fragmentation function that describes the transition from a lightquark to a heavy hadron. The quarkonium collinear fragmentation function isbuilt as a product between a short-distance coefficient function, which encodesthe resummation of DGLAP type logarithms, and a non-perturbative long-distancematrix element (LDME), calculated in the non-relativistic QCD (NRQCD)framework. Our theoretical setup is the hybrid high-energy and collinearfactorization, where the standard collinear approach is supplemented by theresummation of leading and next-to-leading energy-type logarithms \emph{\`a la}BFKL. We propose this reaction as a suitable channel to probe the productionmechanisms of quarkonia at high energies and large transverse momenta and topossibly unveil the transition region from the heavy-quark pair productionmechanism to the single-parton fragmentation one.

Inclusive $J/\psi$ and $\Upsilon$ emissions from single-parton fragmentation in hybrid high-energy and collinear factorization