Probing Saturon-like Limits in QCD Systems

TL;DR

The paper investigates saturon-like limits in QCD by solving the BK equation to obtain the gluon occupancy $N_g(x)$ and a thermodynamic entropy $S(x)$ derived from an Unruh-inspired temperature $T=Q_s/(2\pi)$ and an emergent gluon mass $M_g\sim Q_s$. It contrasts proton and nuclear targets, finding that protons approach but do not reach the bound $S_{ m max}\sim 1/\alpha(Q_s)$ within the explored small-$x$ range, while a lead nucleus can, in a narrow small-$x$ window, reach this bound, suggesting nuclei as the natural environment to search saturon-like behavior. The results highlight the potential of high-occupancy gluon states in heavy-ion systems to realize saturon-like entropy and motivate precision small-$x$ measurements and high-occupancy $pA$ and $AA$ collisions, as well as future Electron-Ion Collider studies. The work also outlines theoretical directions to sharpen the saturon criterion, including higher-order QCD corrections and entanglement-based entropy constructions, linking saturation physics to gravity-inspired entropy bounds.

Abstract

High-occupancy QCD matter enters a saturated regime when its entropy or occupancy approaches the unitarity bound $\sim 1/α$, the ``saturon" criterion. We test this criterion for protons and nuclei at small $x$ using analytic and numerical solutions of the BK equation. From these solutions we construct the gluon occupancy $N_g(x)$ and a thermodynamic entropy $S(x)$ via an Unruh-like temperature $T = Q_s/(2π)$ and an emergent gluon mass $M_g \sim Q_s$. For protons, both $N_g$ and $S$ rise toward small $x$ yet stay below $1/α_s$ in our baseline setup. For nuclei, by contrast, the nuclear entropy $S_A$ attains the $1/α_s$ benchmark in a small-$x$ window where the proton does not. This singles out nuclei as the natural environment to search for saturon-like behavior and motivates precision small-$x$ measurements and high-occupancy $pA$ and $AA$ collisions.

Figures (5)

• Figure 1: The gluon density inside the proton increases from low to high energies (from left to right), eventually reaching a state of gluon saturation as a result of overlap and gluon recombination.
• Figure 2: From left to right and top to bottom, the panels correspond to the analytical and numerical solutions of the BK equation in the fixed and running coupling cases, respectively. The solution methods follow Wang:2020stjCai:2023izaEnberg:2005cb. In each subfigure, the four curves from left to right represent $x = 10^{-1},\ 10^{-3},\ 10^{-5},\ 10^{-7}$, respectively.
• Figure 3: The variation of the extracted saturation scale $Q_s^2$ with respect to $x$ under different schemes is shown. The solid line, dashed line, dash-dotted line, and dotted line correspond to the four respective solutions.
• Figure 4: The four subplots, arranged from left to right and top to bottom, represent respectively: the analytical solution of the BK equation with fixed coupling, the analytical solution with running coupling, the numerical solution with fixed coupling, and the numerical solution with running coupling. Each panel displays the resulting gluon occupation number and the associated entropy.
• Figure 5: Gluon occupancy $N_g(x)$ and thermodynamic entropy $S(x)$ for the nucleus ($A=208$) at fixed coupling; the black horizontal line denotes the criterion ($1/\alpha_s$).