$J\mathrm{/}\psi$ and $\psi$(2S) polarization in proton-proton collisions at energies available at the CERN Large Hadron Collider using PYTHIA8

The production mechanisms of charmonium states in both hadronic and heavy-ion collisions hold great significance for investigating the hot and dense QCD matter. Studying charmonium polarization in ultra-relativistic collisions can also provide insights into the underlying production mechanisms. With this motivation, we explore the $J\mathrm{/}\psi$ and $\psi$(2S) polarization in proton+proton collisions at $\sqrt{s}$ = 7, 8, and 13 TeV using a pQCD-inspired Monte-Carlo event generator called PYTHIA8. This work considers reconstructed quarkonia through their dimuons decay channel in the ALICE forward rapidity acceptance range of $2.5<y_{\mu \mu }<4$. Further, we calculate the polarization parameters ${\lambda }_{\theta }$, ${\lambda }_{\phi }$, ${\lambda }_{\theta \phi }$ from the polar and azimuthal angular distributions of the dimuons in helicity and Collins-Soper frames. This study presents a comprehensive measurement of the polarization parameters as a function of transverse momentum, charged-particle multiplicity, and rapidity at the LHC energies. Our findings of charmonium polarization are in qualitative agreement with the corresponding experimental data.