⚛️ nuclear theory

Hadronic rescattering effects on net-proton cumulants from functional renormalization group calculations

This study demonstrates that hadronic rescattering significantly suppresses higher-order net-proton cumulant signals, particularly the $C_4/C_2$ ratio, during the late-stage evolution of heavy-ion collisions, necessitating its inclusion as a non-negligible background when extracting QCD critical endpoint signatures from experimental data.

Original authors: Qianru Lin, Shi Yin, Jianing Li, Hannah Elfner, Fabian Rennecke, Long-Gang Pang, Jan M. Pawlowski

Published 2026-08-31

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CC BY 4.0

Original authors: Qianru Lin, Shi Yin, Jianing Li, Hannah Elfner, Fabian Rennecke, Long-Gang Pang, Jan M. Pawlowski

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

In the subatomic world, matter is not a static solid but a seething soup of particles that can transform into one another. Physicists believe that under extreme heat and pressure, such as those that existed a fraction of a second after the Big Bang, protons and neutrons melt away to form a state of matter called the quark-gluon plasma. As the universe cooled, this plasma froze back into the particles we see today. A major goal of modern physics is to map the exact conditions where this transition happens, specifically looking for a "critical endpoint." This is a unique point on the map where the transition changes character, much like the specific temperature and pressure where water turns from a liquid into a gas. To find this point, scientists smash heavy atomic nuclei together at incredible speeds, recreating the hot, dense conditions of the early universe. By measuring how the number of protons fluctuates from one collision to the next, they hope to spot the tell-tale signs of this critical point.

However, the story does not end the moment the plasma freezes. The particles that emerge continue to interact with one another as they fly apart, a phase known as hadronic rescattering. For years, researchers have focused on the initial conditions of the collision or the moment of the phase transition itself, often assuming that what happens afterward is a minor detail. A new study challenges this assumption, suggesting that these final interactions are actually a significant background noise that can distort the very signals scientists are trying to detect. The researchers, led by Qianru Lin and colleagues, set out to quantify exactly how much these late-stage interactions change the measurements of proton fluctuations, using a combination of advanced theoretical calculations and computer simulations.

The team began with a sophisticated theoretical framework called the functional renormalization group, which provided a detailed prediction of what the proton fluctuations should look like at the moment the particles stop interacting chemically. To make these abstract numbers usable for a simulation, the researchers used a statistical method to reconstruct the likely distribution of protons and antiprotons for millions of individual collision events. They then fed these particles into a computer model called SMASH, which acts as a virtual laboratory. In this digital environment, the particles were allowed to fly apart and collide with one another, mimicking the real-world expansion of the fireball created in a heavy-ion collision. The simulation covered five different collision energies, ranging from 3.0 to 7.7 giga-electronvolts, a region where the search for the critical endpoint is most intense.

The results revealed that the hadronic stage is far from passive. As the particles bounced off one another, the fluctuations in their numbers changed significantly. The most striking effect was observed at a collision energy of 4.9 giga-electronvolts, a specific point where the theoretical models predicted a strong signal for the critical endpoint. In the simulations, the ratio of the fourth-order fluctuation to the second-order fluctuation—a key indicator used to identify the critical point—dropped by approximately 20 percent during the early, dense phase of the particle cascade. This suppression was most severe right after the initial freeze-out and then partially recovered as the system expanded and the particles became too far apart to interact frequently. The study showed that while the overall shape of the signal remained, its height was substantially reduced by these final collisions.

This finding has profound implications for how experimental data is interpreted. The researchers demonstrated that the non-monotonic pattern of the signal, which rises and falls with energy, survives the chaotic aftermath of the collision, but its magnitude is altered. This means that if experimentalists do not account for these rescattering effects, they might underestimate the strength of the critical signal or misjudge its location. The study confirms that the laws of conservation, specifically the strict conservation of the total number of baryons in the system, play a crucial role in dampening these fluctuations, particularly at the lower energies studied. The microscopic mechanism driving this change was identified as the formation of short-lived particle resonances, which dominate the interactions in the early stages of the expansion.

Ultimately, this work provides a necessary correction for the ongoing search for the critical endpoint. It establishes that hadronic rescattering is not a negligible background but a dynamic process that reshapes the observables. The study does not rule out the existence of the critical endpoint, nor does it erase the signal; rather, it clarifies that the signal seen in experiments will be a modified version of the original theoretical prediction. By quantifying this modification, the researchers have provided a clearer path for future experiments, such as those at the Facility for Antiproton and Ion Research and the Relativistic Heavy Ion Collider, to distinguish between the true signatures of the critical point and the distortions introduced by the final moments of the collision. The path to understanding the fundamental structure of matter now requires looking not just at the birth of the fireball, but also at its quiet, complex death.

Technical Summary: Hadronic Rescattering Effects on Net-Proton Cumulants

Problem Statement
Net-proton cumulants measured in the Beam Energy Scan (BES) of heavy-ion collisions are primary observables for probing the conjectured critical endpoint (CEP) of the Quantum Chromodynamics (QCD) phase diagram. While theoretical frameworks, particularly functional renormalization group (fRG) calculations, provide equilibrium cumulants at chemical freeze-out, the impact of subsequent hadronic rescattering on these observables remains insufficiently quantified. Existing studies often focus on initial-state or phase-transition contributions, neglecting how the hadronic cascade modifies fluctuations before kinetic freeze-out. Furthermore, exact conservation laws (e.g., net-baryon number) and the finite lifetime of the fireball introduce non-trivial background effects that complicate the extraction of critical signals.

Methodology
The authors construct a framework to propagate equilibrium fRG cumulants through a hadronic transport model to quantify rescattering effects. The methodology involves three primary steps:

Reconstruction of Initial Distributions:
- Input: The study utilizes equilibrium cumulants of the net-baryon number calculated via fRG in the grand canonical ensemble (GCE) [10].
- Mapping: Net-baryon cumulants are approximated as net-proton cumulants ( $C_n^B \approx C_n^P$ ), a standard assumption that becomes less certain at lower collision energies.
- Conservation Laws: To account for exact net-baryon number conservation in finite systems, the subensemble acceptance method (SAM) is applied to convert GCE susceptibilities into canonical ensemble (CE) cumulants.
- Maximum Entropy Principle: Since only the first four cumulants are available, the probability distribution function for net-proton multiplicity is reconstructed using the maximum entropy principle to minimize bias.
- Event-by-Event Generation: Separate proton ( $N_p$ ) and antiproton ( $N_{\bar{p}}$ ) multiplicities are generated for each event. $N_p$ is sampled, and $N_{\bar{p}}$ is derived via $N_{\bar{p}} = N_p - N_{net}^p$ , ensuring $N_{\bar{p}} \geq 0$ . Kinematics are sampled from thermal distributions with radial flow, initialized in a spherical volume ( $R=5.89$ fm).
Hadronic Transport Simulation:
- The generated particle lists are propagated through the SMASH (Simulating Many Accelerated Strongly-interacting Hadrons) transport model (version 3.2).
- The system evolves up to $t = 101$ fm/c until scattering rates become negligible (kinetic freeze-out).
- Simulations cover collision energies $\sqrt{s_{NN}} = 3.0, 3.9, 4.9, 7.2,$ and $7.7$ GeV.
- No experimental acceptance cuts (e.g., $p_T$ or rapidity) are applied in the spherical setup; instead, a total momentum cut ( $0.4 < p < 2$ GeV/c) is used.
Analysis:
- Net-proton cumulants ( $C_1$ through $C_4$ ) and their ratios (specifically $C_4/C_2$ ) are recalculated after the hadronic evolution.
- Collision rates and center-of-mass energy distributions for elastic, resonance, and inelastic channels are analyzed to identify microscopic drivers of cumulant modification.

Key Results

Suppression of Higher-Order Cumulants: Hadronic rescattering significantly modifies the initial equilibrium signals. In the canonical ensemble (enforcing exact conservation), the ratio $C_4/C_2$ at $\sqrt{s_{NN}} = 4.9$ GeV is suppressed by approximately 20% during the early stage of the cascade.
Non-Monotonicity Survival: The non-monotonic energy dependence of $C_4/C_2$ (the peak structure associated with the fRG input and the assumed CEP) survives the hadronic evolution in both GCE and CE frameworks. However, the magnitude of this peak is substantially reduced.
Time Evolution Dynamics:
- The suppression of $C_4$ and $C_2$ is most pronounced in the early, dense stage of the hadronic cascade (first few fm/c).
- Following the initial drop, a partial recovery occurs as the system expands and dilutes, indicating that the hadronic evolution does not act as a simple monotonic damping mechanism.
- The suppression of $C_4$ is more severe than that of $C_2$ , leading to the net reduction in the $C_4/C_2$ ratio.
Microscopic Mechanisms: The modification is driven primarily by resonance formation, which dominates the collision rate, followed by elastic scattering. The effect is strongest at 4.9 GeV, where the initial fluctuations are largest, suggesting that enhanced fourth-order fluctuations inherited from the fRG input are particularly susceptible to hadronic interactions.
Canonical vs. Grand Canonical: The CE results show stronger suppression than GCE results, highlighting the critical role of exact net-baryon number conservation in the intermediate-energy regime.

Significance and Claims
The paper claims that hadronic rescattering constitutes a non-negligible background effect that must be accounted for when extracting QCD critical-point signals from experimental data. Specifically:

The magnitude of the critical signal (peak height of $C_4/C_2$ ) is significantly modified (reduced) by the hadronic stage, even if the qualitative non-monotonic structure remains.
The study provides the first quantitative assessment of these effects using direct fRG inputs (rather than model parametrizations) across the BES energy range.
The results imply that the location of the CEP may be encoded in the peak position (which survives), but the peak height (which is suppressed) requires careful correction for hadronic afterburner effects to be correctly interpreted.

Limitations and Future Work
The authors note several systematic uncertainties:

Baryon-to-Proton Mapping: The approximation $C_n^B \approx C_n^P$ introduces uncertainty at lower energies where hyperon contributions are non-negligible.
Acceptance Fraction: The phenomenological fixing of the acceptance fraction $\alpha$ in SAM requires further dedicated study.
Geometry: The use of a uniform spherical expansion simplifies the realistic hydrodynamic evolution where freeze-out occurs gradually from the surface inward. Future work aims to couple this framework to realistic fluctuating initial states and hybrid hydrodynamic simulations.

Technical Summary: Hadronic Rescattering Effects on Net-Proton Cumulants

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