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In this paper we analyze azimuthal asymmetries in the processes of unpolarized and polarized $$J/\psi \,(\Upsilon )$$ production at an Electron-Ion Collider. Apart from giving access to various unknown gluon transverse momentum distributions, we suggest to use them as a new method to extract specific color-octet NRQCD long-distance matrix elements, i.e. $$\langle 0\vert \mathcal{O}_{8}^{J/\psi }(^{1}S_{0})\vert 0\rangle $$ and $$\langle 0\vert \mathcal{O}_{8}^{J/\psi }(^{3}P_{0})\vert 0\rangle $$, whose values are still quite uncertain and for which lattice calculations are unavailable. The new method is based on combining measurements of analogous asymmetries in open heavy-quark pair production which can be performed at the same energy. We also study for the first time the effects of transverse-momentum smearing in the quarkonium formation process. To enhance the gluon contribution one can consider smaller values of x and, in order to assess the impact of small-x evolution, we perform a numerical study using the MV model as a starting input and evolve it with the JIMWLK equations.

Gluon TMDs and NRQCD matrix elements in $${J}{/}{\psi }$$ production at an EIC