J/ψ production and polarization with the inclusion of transverse momentum effects

J/ψ mesons, and quarkonia in general, are remarkable objects, since they are an astonishing tool to probe both the perturbative and non-perturbative aspects of QCD, as well as their interplay. At the same time, they are also sources of uncertainties since their hadronization mechanism is not yet fully understood. Hence, the study of observables which allow to discern among different hadronization models is extremely valuable. One remarkable example is given by the analysis of J/ψ polarization states. In particular, we consider the J/ψ yield in semi-inclusive deep-inelastic scattering (SIDIS) processes and its subsequent decay into a lepton pair. The corresponding cross section can be parameterized in terms of polarization parameters, which may be accessible at the future Electron-Ion Collider, located in the US. Hence, in this thesis we will present their theoretical estimates in kinematic regions potentially explored by this facility, highlighting the possibility to discern between two hadronization models: the non-relativistic QCD (NRQCD) approach and the Color-Singlet Model (CSM). Quite remarkably, in the small transverse momentum region these parameters are sensitive to the transverse momentum dependent (TMD) distributions of gluons inside the proton. We then propose a revised factorized formula valid for SIDIS-related observables evaluated at small transverse momentum. This involves a new TMD object related to quarkonia, called TMD shape function, which describes the transverse momentum smearing due to soft gluon interaction in their hadronization phase. In this thesis we will provide a systematic method to derive the (NLO) perturbative tail of the TMD shape function. Moreover, the potential of quarkonia as outstanding tools to access gluon TMD distributions can be shown by considering single-spin asymmetries (SSAs) in hadronic collisions for J/ψ production. Indeed, for processes like p p^↑ → J/ψ+X, this SSA is sensitive, within a phenomenological framework, to the gluon Sivers function, still poorly known. Thus, adopting the Generalized Parton Model (GPM) and its Color-Gauge Invariant extension (CGI-GPM), we will provide SSA estimates for PHENIX@RHIC and other upcoming experiments (SMOG@LHC, NICA-SPD, SpinQuest@Fermilab).