Extraction of Effective Parameters From Transverse Momentum Spectra of Heavy Quarkonia in Proton–Proton Collisions at the LHC

The effective string tension ( κ ) 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 ( Q G P ) 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 ( κ and T ) are obtained by fitting the experimental p T spectra of J / ψ and Y ( n S ) ( n = 1, 2, and 3) within various rapidity intervals, produced in p + p collisions at center‐of‐mass energies of 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 κ and T increase, indicating a directly proportional relationship between them. Based on κ , the average minimum strong force radius of participant quarks is determined.