Process dependence of the gluon Sivers function in $p^\uparrow p \to J/\psi + X$ within a TMD approach in NRQCD

We consider the transverse single-spin asymmetry (SSA) for $J/\psi$ production in $p^\uparrow p \to J/\psi + X$ within a TMD approach in non-relativistic QCD. Extending a previous study [1], we employ here the color-gauge invariant generalized parton model (CGI-GPM), in which spin and intrinsic transverse momentum effects are taken into account, together with leading-order initial- and final-state interactions (ISIs and FSIs). We find that, even when the heavy-quark pair is produced in a color-octet state, ISIs and FSIs lead to a nonvanishing SSA, allowing, in principle, to test the process dependence of the gluon Sivers function (GSF). We show that of the two independent contributions, due to the so-called $f$- and $d$-type GSFs, appearing in the CGI-GPM, the $d$-type one turns out to be dynamically suppressed. Therefore, as already found adopting the Color-Singlet Model approach for the $J/\psi$ formation [2], only the $f$-type GSF could play a role in phenomenology. A comparison with the corresponding results obtained in the generalized parton model, without the inclusion of ISIs and FSIs, is also carried out.