Investigating quarkonium production in proton-proton collisions with ALICE

Quarkonium production in high-energy proton-proton (pp) collisions provides a unique probe of both perturbative and non-perturbative aspects of quantum chromodynamics (QCD). Charmonium states, such as the J/$\psi$ and $\psi$(2S), are produced through a two-stage process involving hard parton-parton scatterings followed by non-perturbative hadronization. Precise measurements are essential to constrain theoretical models and improve our understanding of these mechanisms. In this proceeding, preliminary results on the J/$\psi$ production cross section at $\sqrt{s}$ = 13.6 TeV, measured by ALICE, will be shown. Thanks to the new Muon Forward Tracker (MFT), the separation of prompt and non-prompt J/$\psi$ production is now possible at forward rapidity (2.5 < $y$ < 3.6), and first measurements of their relative contributions will be presented. Additionally, the dependence of inclusive J/$\psi$ yields on the relative charged-particle multiplicity, measured at $\sqrt{s}$ = 13 TeV in the pseudorapidity range -3.7 < $\eta$ < -1.7, will be discussed. The observed steeper-than-linear trend highlights the role of multi-parton interactions. Finally, the $\psi$(2S)-to-J/$\psi$ ratio and its comparison with model predictions provides further insight into the interplay between soft and hard processes in charmonium production.