The domain of soft hadrons in transverse momentum space in high energy collisions

TL;DR

This paper addresses how soft and hard processes shape hadron transverse-momentum spectra across $pp$, $p$-Pb, and AA collisions by identifying two curvature-based points, $p_0$ and $p_1$, in the second derivative of the log-spectrum, enabling a three-region momentum-space classification. Using Levy-Tsallis fits (and its variants) plus local smoothing, the authors extract $p_0$ and $p_1$ for multiple identified hadrons, examining their dependence on hadron species, collision energy, system size, and event activity, and revealing a notable $p_0$–$\langle p_T\rangle$ correlation with baryon/meson distinctions. They further relate the observed patterns to hadronization mechanisms via an equal-velocity combination (EVC) model, linking pre-hadronization quark spectra to hadron $p_T$ distributions and reproducing qualitative baryon–meson differences. The work provides a data-driven, quantitative criterion for the soft–hard boundary in high-energy collisions and highlights how soft expansion and hadronization shape the spectra across collision systems and centralities.

Abstract

By studying experimental data for invariant transverse momentum distribution $f(p_{T})$ of hadrons in high energy $pp$, $p$A and AA collisions, we find two characteristic $p_{T}$ points relating to the behavior of $\left[\ln f(p_{T})\right]^{''}$, i.e., the second derivative of the logarithm of hadronic $p_{T}$ spectrum with respect to $p_{T}$. One point is $p_{0}$ at which $\left[\ln f(p_{T})\right]^{''}$ is zero and another point is $p_{1}$ at which $\left[\ln f(p_{T})\right]^{''}$ reaches maximum. \bz{The hadronic distribution can be classified into three kinetic regions, i.e., soft region $0<p_{T}<p_{0}$ dominated by hadrons from soft parton system, hard region $p_{T}>p_{1}$ dominated by hadrons from high energy partonic jet, and the transition region $p_{0}<p_{T}<p_{1}$ of above two sources of hadron production.} Using rich data of hadronic $p_{T}$ spectra at RHIC and LHC, we carry out a systematical analysis for $p_{0}$ and $p_{1}$ of kaon, $K^{*}$, $φ$ and baryons such as $p$, $Λ$, $Ξ$, $Ω$ in $pp$, $p$-Pb and heavy-ion collisions, and show their dependence on hadron species, collision energy, collision centrality and/or charged-particle multiplicity. We also study the correlation between $p_{0}$ of hadrons and their average transverse momentum $\left\langle p_{T}\right\rangle $, and find a difference between $p_{0}$-$\left\langle p_{T}\right\rangle $ correlation of baryons and that of mesons which can be understood by quark combination mechanism at hadronization. The systematic comparison for $p_{0}$-$\left\langle p_{T}\right\rangle $ correlation of kaon, $φ$, $p$ and $Λ$ in $pp$, $p$-Pb, Pb-Pb and Au+Au collisions at RHIC and LHC energies indicates that $p_{0}$ is a sensitive physical quantity of reflecting the production property of soft hadrons in high energy collisions.

Figures (14)

• Figure 1: (a) $p_{T}$ spectrum of proton at mid-rapidity in 60-80% centrality in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV ALICE:2014juv and three fitting lines; (b) the second derivative of the logarithm of the fitting functions with respect to $p_{T}$.
• Figure 2: (a) $p_{T}$ spectrum of proton at mid-rapidity in 0-5% centrality in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV ALICE:2014juv and two fitting lines; (b) the second derivative of the logarithm of the spectrum and two fitting functions with respect to $p_{T}$.
• Figure 3: (a) $p_{T}$ spectrum of $\Lambda$ at mid-rapidity in 0-5% centrality in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV ALICE:2013cdo and three global fittings with Eq. (\ref{['eq:fpt_levy_v0']}) labeled “ Levy fit”, Eq. (\ref{['eq:fpt_levy_v2']}) labeled “ Levy_v2 fit” and Eq. (\ref{['eq:fpt_levy_v3']}) labeled “ Levy_v3 fit” ; (b) Ratios of data to fittings.
• Figure 4: (a) Results (lines) of Levy-Tsallis fittings for experimental data of $p_{T}$ spectra of identified hadrons at mid-rapidity in inelastic $pp$ collisions at $\sqrt{s}=13$ TeV ALICE:2020jshALICE:2023egx and (b) the obtained $p_{0}$ (open circles) and $p_{1}$ (open squares).
• Figure 5: $p_{0}$ and $p_{1}$ of identified hadrons at mid-rapidity in inelastic $pp$ collisions at $\sqrt{s}=13,7,5.02,2.76$ TeV and those in NSD events in $pp$ collisions at $\sqrt{s}=200$ GeV and those at 900 GeV ($|y|<2$).