Process dependence of the gluon Sivers function in p↑p→J/ψ+X within a TMD scheme in NRQCD

We consider the transverse single-spin asymmetry (SSA) for $J/\ensuremath{\psi}$ production in ${p}^{\ensuremath{\uparrow}}p\ensuremath{\rightarrow}J/\ensuremath{\psi}+X$ within a phenomenological transverse momentum dependent scheme in nonrelativistic QCD. Extending a previous study [U. D'Alesio et al., Eur. Phys. J. C 79, 1029 (2019)], 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 us, 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 $J/\ensuremath{\psi}$ formation [U. D'Alesio et al., Phys. Rev. D 96, 036011 (2017)], 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.