Estimates for the single-spin asymmetries in the p↑p→J/ψX process at PHENIX RHIC and SPD NICA

We study the transverse single-spin asymmetry (TSSA) in $p^{\uparrow}p \to J/\psi X$ reaction, incorporating both transverse-momentum and spin effects. To predict production cross section of prompt $J/\psi$ we use two different approaches, non-relativistic QCD (NRQCD) factorization approach Improved Color Evaporation Model (ICEM), show how predicted results for TSSAs depend on choice hadronization model. For initial-state consider models: standard Generalized Parton (GPM) Colour Gauge-Invariant version it (CGI-GPM). demonstrate that PHENIX collaboration data TSSA process constrain gluon Sivers function proton rule-out one existing parameterizations. Estimates conditions future SPD NICA experiment are presented first time.