Extraction of Effective Parameters from Transverse Momentum Spectra of Heavy Quarkonia in Proton-Proton Collisions at the LHC

TL;DR

This work addresses whether a temperature concept is applicable in small collision systems by extracting two effective parameters, the Schwinger string tension $\kappa$ and the initial effective temperature $T$, from heavy quarkonium $p_T$ spectra in p+p collisions at the LHC. It introduces a multi-component framework combining Schwinger mechanism and Bose-Einstein statistics to fit the spectra of $J/\psi$ and $\Upsilon(nS)$ measured by LHCb at $\sqrt{s}=13$ and 8 TeV, yielding a consistent $\kappa$–$T$ correlation and a derived $R_{\min}$ that characterizes the initial color field overlap. The extracted $\kappa$ values are large (tens to hundreds of GeV/fm) and $T$ values (roughly 0.7–2.1 GeV) exceed typical QGP freeze-out scales, consistent with initial-state dynamics and possible collectivity in small systems. The positive $\kappa$–$T$ correlation and the very small $R_{\min}$ imply intense initial energy density and potential early-time collective behavior, offering a microscopic lens for comparing small- and large-system dynamics and guiding future investigations across collision systems and energies.

Abstract

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 (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 ($κ$ and $T$) are obtained by fitting the experimental $p_T$ spectra of $J/ψ$ and $Υ(nS)$ ($n=1$, 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 $κ$ and $T$ increase, indicating a directly proportional relationship between them. Based on $κ$, the average minimum strong force radius of participant quarks is determined.

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