⚛️ nuclear theory

Critical net-proton number fluctuations with hydrodynamics

This study computes net-proton number fluctuations across nine collision energies using a hydrodynamic model incorporating critical fluctuations from the functional renormalization group, revealing that while low-order cumulant ratios show minimal deviation from non-critical baselines, higher-order ratios like $C_4/C_2$ exhibit a distinct non-monotonic energy dependence driven by the critical end point.

Original authors: Rui-zhe Zhao, Shi Yin, Shanjin Wu, Lipei Du, Xiaofeng Luo, Wei-jie Fu

Published 2026-07-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Rui-zhe Zhao, Shi Yin, Shanjin Wu, Lipei Du, Xiaofeng Luo, Wei-jie Fu

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

Imagine the universe as a giant, cosmic soup. In the very first moments after the Big Bang, this soup was so hot and dense that the basic building blocks of matter—quarks and gluons—were swimming freely, unable to stick together to form protons or neutrons. This state of matter is called the Quark-Gluon Plasma (QGP). As the universe cooled, these free-floating particles "froze" into the solid matter we see today, much like water turning into ice. But physicists suspect that if you could squeeze this soup with enough pressure (baryon density) while keeping it hot, it wouldn't just freeze smoothly. Instead, it might hit a "critical point," a special spot on the map of the universe where the rules of the game change dramatically. Finding this Critical End Point (CEP) is like finding the exact temperature and pressure where water suddenly decides to boil or freeze in a weird, chaotic way. Scientists care about this because it helps us understand the fundamental laws of nature, the history of our universe, and even the mysterious insides of neutron stars.

To hunt for this elusive point, researchers smash heavy atoms together at nearly the speed of light, creating tiny, fleeting fireballs of QGP. They then look for "fluctuations"—random wiggles in the number of particles produced. If the collision hits the critical point, these wiggles should get huge and behave in a very specific, non-monotonic way (going up and down instead of just steadily increasing or decreasing). However, reading these signals is incredibly hard because the detectors only see a slice of the explosion, and the background noise from normal physics is loud.

This paper is a sophisticated attempt to clean up that noise and see if the signal is real. The authors, a team of physicists, combined two powerful tools: a method called the Functional Renormalization Group (fRG) to predict what happens near the critical point, and a complex computer simulation of fluid dynamics (hydrodynamics) to model how the fireball expands and cools in a real collision. Instead of using a simplified, one-size-fits-all model, they simulated the entire messy, expanding fireball at nine different collision energies, ranging from 7.7 to 200 GeV. They then applied the same "filters" that real-world detectors use, such as only counting particles moving at certain speeds or angles, and accounted for the fact that the total number of particles must be conserved.

The results of their simulation offer a nuanced picture. For simple, low-order measurements (like the basic variance of particle counts), the difference between a world with a critical point and one without is tiny; the background noise drowns out the signal. However, as they looked at more complex, higher-order measurements (specifically the ratio of the fourth-order fluctuation to the second, known as $C_4/C_2$ ), the story changed. In the low-energy collisions, which probe the high-density region of the phase diagram, the simulation including the critical point showed a distinct "non-monotonic" behavior—a bump and dip in the data as energy changes. This wiggly pattern was completely absent in the simulations that ignored the critical point.

Crucially, the paper suggests that previous, simpler models that assumed the fireball froze at a single, uniform temperature and pressure might have overestimated how dramatic this signal would be. By using a realistic, fluid-dynamic simulation that accounts for the messy, uneven nature of the collision, the authors found a signal that aligns much better with the actual experimental data collected by the STAR collaboration at the Relativistic Heavy Ion Collider (RHIC). While the paper does not claim to have definitively proven the existence of the Critical End Point, it demonstrates that when you model the collision with high fidelity, the theoretical predictions for critical fluctuations become much more consistent with what we actually see in the lab. This gives scientists more confidence that the strange, non-monotonic patterns they are hunting for are indeed the fingerprints of the critical point, rather than just a trick of the math.

Technical Summary: Critical Net-Proton Number Fluctuations with Hydrodynamics

Problem Statement
The location and existence of the QCD Critical End Point (CEP) remain an outstanding open question in the study of the QCD phase diagram. While theoretical approaches such as the functional renormalization group (fRG), Dyson-Schwinger equations (DSE), and lattice QCD extrapolations suggest the CEP exists at high baryon chemical potential ( $\mu_B/T \gtrsim 4-5$ ), direct experimental verification is challenging. Fluctuation observables, particularly net-proton number fluctuations, serve as proxies for baryon number fluctuations to search for the CEP. However, reliable theoretical predictions for experimental signals are difficult to achieve because previous simplified models often assumed a single baryon chemical potential and temperature per collision energy (parametrized chemical freeze-out curves), which fails to capture the realistic spacetime evolution of heavy-ion collisions.

Methodology
This work combines functional renormalization group (fRG) calculations with state-of-the-art (3+1)-dimensional viscous hydrodynamic simulations to compute net-proton number fluctuations across nine collision energies ( $\sqrt{s_{NN}} = 7.7 - 200$ GeV).

Hydrodynamic Evolution: The spacetime evolution of Au+Au collisions is simulated using the MUSIC viscous hydrodynamics framework with rapidity-dependent 3D initial conditions, followed by iS3D particlization on a constant energy-density hypersurface ( $\epsilon_{fo} = 0.26$ GeV/fm $^3$ ) and an UrQMD hadronic afterburner. The model is calibrated to hadronic observables (yields, spectra, etc.) at the specified energies.
Encoding Critical Fluctuations: The freeze-out hypersurface is treated as a collection of fluid elements, each characterized by local temperature $T(x_i)$ and baryon chemical potential $\mu_B(x_i)$ . The net-baryon number fluctuations, including critical contributions from the CEP calculated via fRG, are mapped onto these fluid elements.
From Net-Baryon to Proton Fluctuations: Since fRG provides net-baryon fluctuations ( $\chi_n^B$ ) rather than separate baryon/antibaryon fluctuations, the authors infer (anti)baryon fluctuations using a relation based on the Hadron Resonance Gas (HRG) model (Eq. 4). Proton fluctuations are then derived from baryon fluctuations via isospin randomization (binomial filtering), assuming a probability $q \approx 1/2$ .
Experimental Constraints: The calculations implement several experimental realities:
- Acceptance Cuts: Binomial acceptance corrections are applied to match STAR detector kinematics ( $0.4 < p_T < 2.0$ GeV, $|y| < 0.5$ ).
- Global Conservation: The subensemble acceptance method (SAM) is used to correct for global baryon number conservation effects.
Comparison Baselines: Results are compared against:
- A non-critical baseline using HRG susceptibilities within the same hydrodynamic framework.
- Previous fRG results calculated along a parametrized chemical freeze-out curve without hydrodynamic evolution.

Key Contributions

Realistic Framework: The paper establishes a method to encode fRG-derived critical fluctuations into a realistic hydrodynamic freeze-out hypersurface, moving beyond the oversimplified assumption of a single freeze-out point per collision energy.
Comprehensive Implementation: It integrates critical fluctuations with experimental acceptance cuts, isospin randomization, and global baryon conservation simultaneously within a dynamical background.
Systematic Energy Scan: The study covers the full Beam Energy Scan (BES) range ( $\sqrt{s_{NN}} = 7.7 - 200$ GeV), allowing for a detailed investigation of collision energy dependence.

Results
The study computes the cumulant ratios $C_2/C_1$ , $C_3/C_2$ , and $C_4/C_2$ for net-proton distributions:

Low-Order Cumulants ( $C_2/C_1$ ): The difference between results with critical fluctuations (fRG+hydro) and non-critical results (HRG+hydro) is negligible across the entire energy range.
High-Order Cumulants at High Energy: For $\sqrt{s_{NN}} \gtrsim 14.5$ GeV, the difference between critical and non-critical results for $C_3/C_2$ and $C_4/C_2$ is small, as these energies probe the low $\mu_B$ region where critical fluctuations are suppressed.
High-Order Cumulants at Low Energy: In the low collision energy regime (e.g., $\sqrt{s_{NN}} = 7.7$ GeV), which probes high $\mu_B$ , the difference between critical and non-critical results increases with the order of the cumulant.
Non-Monotonic Behavior: A distinct non-monotonic dependence on collision energy is observed in $C_4/C_2$ when critical fluctuations are included. This feature is absent in the non-critical (HRG) baseline.
Comparison with Experiment: The fRG+hydro results for $C_3/C_2$ and $C_4/C_2$ show better agreement with STAR BES-II experimental data than the simplified fRG results calculated on a parametrized freeze-out curve, which tend to overestimate the non-monotonicity.

Significance and Claims
The paper claims that combining fRG calculations with hydrodynamic simulations provides a promising and more realistic approach to describing fluctuation observables in heavy-ion collisions compared to parametrized freeze-out models. The authors emphasize that while low-order cumulants are dominated by non-critical physics and experimental constraints, higher-order cumulants in the low-energy regime are sensitive to the CEP. The observation of non-monotonic behavior in $C_4/C_2$ specifically with critical fluctuations suggests that this observable could be a signature of the CEP, provided the realistic hydrodynamic evolution is accounted for. The authors modestly note that future work should extend these calculations to the fixed-target energy range ( $\sqrt{s_{NN}} < 7.7$ GeV) to further explore the CEP region.

Technical Summary: Critical Net-Proton Number Fluctuations with Hydrodynamics

More like this