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