Extraction of Effective Parameters from Transverse Momentum Spectra of Heavy Quarkonia in Proton-Proton Collisions at the LHC
The effective string tension ($\kappa$) in the Schwinger mechanism and the effective temperature ($T$) in Bose-Einstein statistics are extracted from the transverse momentum ($p_T$) spectra of heavy quarkonia produced in proton-proton (p+p) collisions at the Large Hadron Collider (LHC). Here, $T$ derived from the heavy quarkonium $p_T$ spectra also serves as the initial effective temperature (effective temperature at the initial stage) of small collision systems. This is because, despite the absence of quark-gluon plasma (QGP) formation during the collisions, which leaves $T$ largely unaffected by QGP-related effects, the initial geometric asymmetry and local partonic thermalization still induce radial and transverse flows, thereby contributing to an increase in $T$. The effective parameters ($\kappa$ and $T$) are obtained by fitting the experimental $p_T$ spectra of $J/\psi$ and $\Upsilon(nS)$ ($n=1$, 2, and 3) within various rapidity intervals, produced in p+p collisions at center-of-mass energies of $\sqrt{s}=13$ and 8 TeV, as measured by the LHCb Collaboration. It is found that the multi-component distribution structured within the framework of the Schwinger mechanism or Bose-Einstein statistics can effectively describe the heavy quarkonium $p_T$ spectra in small collision systems. With decreasing rapidity in the forward region, both $\kappa$ and $T$ increase, indicating a directly proportional relationship between them. Based on $\kappa$, the average minimum strong force radius of participant quarks is determined.