Exploring QGP-like phenomena with Charmonia in $p+p$ collisions at $\sqrt{s} = 13$ TeV

In ultrarelativistic collisions of nuclei at the Large Hadron Collider, the created QCD environment rapidly changes, leading to a non-adiabatic evolution of the quantum states involved. Considering this, we first examine the pre-equilibrium state of QCD matter and its effect on the initially produced charmonium using a temperature-independent Hamiltonian. As the QCD matter reaches local thermal equilibrium, this Hamiltonian transforms to its finite temperature counterpart. To model the pre-equilibrium stage, we use the bottom-up thermalization approach to determine the effective temperature of the QCD matter, followed by a Gubser-type expansion for the thermalized medium. Additionally, we consider collisional damping, gluonic dissociation, and regeneration mechanisms, which specifically modify the charmonium yield in the thermalized medium. Mainly, the gluonic dissociation and collisional damping cause a reduction in the yield conversely, regeneration through gluonic deexcitation enhances the yield of charmonium. Further, we explore the combined effects of these mechanisms on the collective yield of charmonium states with transverse momentum ($p_{\rm T}$) and event multiplicity in the proton-proton collisions at $\sqrt{s} = 13$ TeV. Based on our findings, we contend that the combined effects of these mechanisms can serve as a robust probe for determining the possible existence of a thermalized QCD medium in such a small collision system.