⚛️ nuclear experiments

Proton Collectivity in Au+Au Collisions at $\sqrt{s_{\rm NN}}=2.4-4.5$ ~GeV from a Unified Purely Hadronic EOS without QCD Phase Transition

Using a unified purely hadronic equation of state with an incompressibility of 230 MeV, the study successfully reproduces proton flow data in Au+Au collisions up to 4.3 GeV but fails at 4.5 GeV, providing circumstantial evidence for the onset of partonic degrees of freedom and the hadron-quark phase transition.

Original authors: Gao-Feng Wei, Shuang-Jie Liu, Yu-Liang Zhao, Qi-Jun Zhi, Zhigang Xiao

Published 2026-07-28

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CC0 1.0

Original authors: Gao-Feng Wei, Shuang-Jie Liu, Yu-Liang Zhao, Qi-Jun Zhi, Zhigang Xiao

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic Lego set. For decades, scientists have been trying to figure out what happens when you smash two of these massive Lego structures together at incredible speeds. This is the world of heavy-ion collisions, where physicists crash heavy atoms (like gold) into each other to recreate the conditions that existed just moments after the Big Bang. The goal is to understand the "nuclear equation of state" (EOS), which is basically the rulebook for how matter behaves when you squeeze it tighter and tighter.

Think of this rulebook like a recipe for a super-dense soup. At normal densities, the soup is made of distinct ingredients: protons and neutrons (the "hadrons"). But as you squeeze the pot harder, the question is: does the soup just get thicker and harder to stir, or does it suddenly melt into a completely different kind of fluid made of its tiny building blocks (quarks and gluons)? This transition from "solid" atomic matter to a "liquid" of free-floating quarks is one of the biggest mysteries in physics. Scientists are looking for the exact moment this switch happens, but it's tricky because the forces involved are incredibly complex, and the experiments happen in a flash.

Now, enter a team of researchers who decided to play the role of cosmic chefs using a computer simulation. They wanted to test a specific theory: could the behavior of this ultra-dense soup be explained entirely by the rules of normal atomic matter, without needing to assume it turns into a quark soup? They simulated crashing gold nuclei together at energies between 2.4 and 4.5 GeV (a unit of energy that tells us how hard the smash is).

Here is what they found. When they smashed the atoms at energies up to 4.3 GeV, their "purely atomic" recipe worked perfectly. By using a specific type of force field that changes depending on how fast the particles are moving (a "momentum-dependent" field) and a specific stiffness value of 230 MeV, their simulation matched the real-world data from experiments like HADES, E895, and STAR. In this range, the matter behaved exactly as if it were still just a super-dense collection of protons and neutrons. There was no need to invent a new phase of matter; the atomic rules held up even when the density reached about five times the normal density of an atomic nucleus.

However, the story takes a sharp turn at 4.5 GeV. When the researchers cranked the energy up just a little bit more, their "purely atomic" simulation suddenly failed. No matter how they tweaked the atomic rules, the computer model could no longer predict how the protons moved after the crash. The particles started behaving in a way that the atomic recipe simply couldn't explain. This failure happened right at the same energy where other experiments had noticed a strange new pattern emerging (known as NCQ scaling), which is often seen as a sign that the matter has started to break apart into its fundamental quark pieces.

So, what does this mean? The paper suggests that up to 4.3 GeV, the universe's "soup" is still just a very thick, atomic broth, and we don't need to worry about it melting into quarks yet. But the moment we cross the 4.5 GeV threshold, the atomic rules break down. This isn't a proof that quarks are definitely there, but it is a strong hint—a "smoking gun"—that something new and fundamental is starting to happen in that tiny window between 4.3 and 4.5 GeV. It's as if the chef tried to cook the soup using only the old recipe, and at a specific temperature, the pot suddenly started bubbling in a way that recipe couldn't predict, signaling that a new ingredient (the quark phase) has finally entered the kitchen.

Technical Summary: Proton Collectivity in Au+Au Collisions at $\sqrt{s_{NN}} = 2.4 - 4.5$ GeV

Problem Statement
The nuclear equation of state (EOS) is widely hypothesized to soften in the density range of $2-5$ times the saturation density ( $\rho_0$ ). However, establishing a rigorous, unified hadronic baseline to test this hypothesis across a wide energy range has remained a challenge. Previous studies, such as those by the E895 Collaboration, suggested a transition from a hard to a soft EOS around $4\rho_0$ , while other analyses reported a wide range of incompressibility values ( $K_0 = 167-380$ MeV). Furthermore, existing transport models often exhibit strong energy-dependent inconsistencies: a soft momentum-dependent EOS may describe low-energy data well, while a hard one is required for high energies. Additionally, different transport codes frequently yield divergent predictions even with similar EOS inputs. A decisive test requires a purely hadronic transport model capable of reproducing collective flow data across the $(2-5)\rho_0$ density range using a global, beam-energy-independent EOS.

Methodology
The authors employed an isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model to simulate proton collective flow in Au+Au collisions at center-of-mass energies $\sqrt{s_{NN}} = 2.4 - 4.5$ GeV. Key methodological features include:

Inelastic Channels: The model was extended to include critical inelastic channels, specifically $\Delta$ and $N^*$ resonances, strange baryons, and multi-pion production (including $\rho$ and $\omega$ channels), which significantly affect nucleon-nucleon collisions and flow behavior.
EOS Scenarios: Two distinct mean-field scenarios were compared:
1. Momentum-Dependent Interaction (MDI): A unified mean field with incompressibility $K_0 = 230$ MeV and a slope parameter $L = 62.5$ MeV. This scenario does not include density-dependent softening.
2. Momentum-Independent Interaction (MID): Two parameter sets were tested: a "stiff" EOS ( $K_0 = 380$ MeV) and a "soft" EOS ( $K_0 = 210$ MeV), both with $L = 62.5$ MeV.
Observables: The study calculated proton directed flow ( $v_1$ ), sideward flow, and elliptic flow ( $v_2$ ). Results were filtered with acceptance cuts and centrality conditions matching experimental data from the HADES, E895, and STAR collaborations.

Key Results

Success up to 4.3 GeV: The MDI scenario with $K_0 = 230$ $K_{0} = 230$ MeV quantitatively reproduced experimental data for directed, sideward, and elliptic flows across the entire range of $\sqrt{s_{NN}} = 2.4 - 4.3$ $s_{N N} = 2.4 - 4.3$ GeV. This energy range corresponds to a maximum achieved density of approximately $5\rho_0$ $5 ρ_{0}$ .
- The MDI-230 model successfully described the excitation functions of directed and sideward flows, whereas the MID-stiff model failed to reproduce sideward flow, and the MID-soft model underestimated both.
- For elliptic flow, the MDI-230 model accurately captured the transverse momentum ( $p_t$ ) dependence and the sign change of $v_2$ occurring around $\sqrt{s_{NN}} = 3.3$ GeV. In contrast, both MID scenarios failed to reproduce the $p_t$ dependence or the correct transition energy.
- Crucially, this success was achieved without introducing density-dependent softening of the EOS.
Failure at 4.5 GeV: At $\sqrt{s_{NN}} = 4.5$ GeV, the same purely hadronic MDI-230 model failed to reproduce the proton directed and elliptic flow data. The $\chi^2$ /DoF for the elliptic flow increased significantly from 0.98 (at lower energies) to 5.2.
Comparison with NCQ Scaling: The failure of the hadronic model at 4.5 GeV coincides with the energy at which the STAR Collaboration observed the restoration of Number-of-Constituent-Quark (NCQ) scaling, a signature of partonic collectivity.

Significance and Claims
The paper claims to provide a robust, data-driven hadronic baseline for characterizing high-density nuclear matter and mapping the hadron-quark phase transition. The primary implications are twofold:

Uniform EOS: The results suggest that the nuclear EOS does not undergo softening in the density range $(2-5)\rho_0$ when described by a momentum-dependent interaction. This uniform hadronic EOS supports the existence of massive neutron stars ( $\gtrsim 2M_\odot$ ).
Onset of Partonic Degrees of Freedom: The sudden breakdown of the purely hadronic description at 4.5 GeV, despite its success just 0.2 GeV lower, provides circumstantial evidence for the onset of partonic degrees of freedom. The authors narrow the window for a possible first-order phase transition to $4.3 < \sqrt{s_{NN}} \le 4.5$ GeV, where the maximum density slightly exceeds $5\rho_0$ .

The study concludes that careful treatment of inelastic channels and the inclusion of momentum-dependent interactions are essential for modeling the pressure generated by compressing nuclear matter in heavy-ion collisions, resolving previous theoretical inconsistencies regarding the stiffness of the EOS.

Technical Summary: Proton Collectivity in Au+Au Collisions at sNN=2.4−4.5\sqrt{s_{NN}} = 2.4 - 4.5sNN​​=2.4−4.5 GeV

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Technical Summary: Proton Collectivity in Au+Au Collisions at $\sqrt{s_{NN}} = 2.4 - 4.5$ GeV