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We investigate the quarkonium production mechanisms in jets at the LHC, usingthe Fragmenting Jet Functions (FJF) approach. Specifically, we discuss the jetenergy dependence of the $J/\psi$ production cross section at the LHC. Bycomparing the cross sections for the different NRQCD production channels($^1S_0^{[8]}$,$^3S_1^{[8]}$,$^3P_J^{[8]}$, and $^3S_1^{[1]}$), we find that atfixed values of energy fraction $z$ carried by the $J/\psi$, if the normalizedcross section is a decreasing function of the jet energy, in particular for $z> 0.5$, then the depolarizing $^1S_0^{[8]}$ must be the dominant channel. Thismakes the prediction made in [Baumgart et al., JHEP 1411, 003 (2014)] for theFJF's also true for the cross section. We also make comparisons between thelong distance matrix elements extracted by various groups. This analysis couldpotentially shed light on the polarization properties of the $J/\psi$production in high $p_T$ region.

Quarkonium Polarization and the Long Distance Matrix Elements Hierarchies using Jet Substructure