⚛️ nuclear theory

Searching for the QCD critical point with relativistic viscous hydrodynamics

This paper presents the latest results from the MUSES collaboration using the 3+1D relativistic viscous hydrodynamics code CCAKE, enhanced with numerical improvements like "glass" initial conditions and adaptive smoothed particle hydrodynamics, to dynamically probe the QCD critical point and quantify its interplay with beam energy and rapidity range within the context of STAR's Beam Energy Scan II data.

Original authors: Kevin P. Pala, Surkhab K. Virk, Isabella Danhoni, Fernando Gardim, Christopher Plumberg, Jacquelyn Noronha-Hostler

Published 2026-10-01

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CC BY 4.0

Original authors: Kevin P. Pala, Surkhab K. Virk, Isabella Danhoni, Fernando Gardim, Christopher Plumberg, Jacquelyn Noronha-Hostler

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

Deep within the heart of every atom lies a world of intense energy and pressure, a realm where the fundamental building blocks of matter behave in ways that defy our everyday experience. Under normal conditions, protons and neutrons are locked together, but scientists believe that if enough heat and pressure are applied, these particles melt into a soupy, free-flowing state known as the quark-gluon plasma. This state of matter existed for a fleeting moment just after the Big Bang, before the universe cooled enough for particles to clump back together. For decades, physicists have been trying to map the boundaries of this exotic state, specifically looking for a mysterious spot on the map called the critical point. This is a unique location where the transition between the free-flowing plasma and the clumped-up particles changes from a smooth blend into a sharp, sudden jump. Finding this point would be a monumental discovery, helping us understand how the universe evolved and revealing the deep rules that govern all matter.

To hunt for this elusive spot, researchers at the Relativistic Heavy Ion Collider smash heavy atomic nuclei together at incredible speeds. By adjusting the energy of these collisions, they can create environments with different densities of matter, effectively scanning the map of the quark-gluon plasma. However, the critical point cannot be seen directly; it is hidden inside a system that expands and cools in a fraction of a second, often far from a state of balance. To find it, scientists must build a digital model of these collisions that is precise enough to track how the fluid behaves as it passes through different conditions. This is exactly what a team of researchers, led by Kevin Pala and Jacquelyn Noronha-Hostler, has achieved by refining the tools used to simulate these cosmic events.

The team focused on a specific energy level where the conditions are just right to potentially reach the critical point, using data from the Beam Energy Scan II experiment. Their work relies on a sophisticated computer code called CCAKE, which acts as a virtual laboratory for these high-energy crashes. In the past, these simulations faced a significant hurdle: they often struggled to maintain accuracy while running fast enough to be useful. The researchers solved this by upgrading the way the simulation starts and how it tracks the fluid. Instead of placing the initial particles on a rigid, fixed grid, they allowed the particles to settle into a smooth, glass-like arrangement that has no preferred direction. This "glass" starting point, combined with a new method that automatically adjusts the resolution of the simulation—zooming in where the action is complex and zooming out where it is simple—allowed the computer to run much faster while using far fewer virtual particles. This improvement meant the team could conserve energy in their models with much greater precision, a crucial step for getting reliable results.

With these new tools in hand, the team ran simulations of collisions at an energy of 7.7 billion electron volts, a setting chosen because it creates a high enough density of matter to potentially touch the critical point. They tested two different scenarios: one where the critical point exists at a specific location predicted by other theories, and another where it is located far away, beyond the reach of this energy. The results were revealing. In the scenario where the critical point was present, the simulated fluid spent a significant amount of time lingering near that special spot as it expanded and cooled. This lingering effect, which the researchers describe as a form of dynamical lensing, suggests that the system is sensitive to the presence of the critical point. In contrast, when the critical point was placed far away, the fluid passed through that region much more quickly, spending very little time there. Interestingly, the researchers found that the basic properties of the fluid, such as its density and entropy, looked almost identical in both cases, meaning the difference was subtle and required looking at how the system evolved over time to detect.

The team then compared their simulations to real data collected by the STAR experiment at the collider. They looked at how many particles were produced, their average speed, and how they moved in different directions. Their new model, which included the effects of a critical point, matched the real-world data remarkably well. It successfully reproduced the number of particles created and the patterns of their flow across a wide range of collision energies. This agreement gives the researchers confidence that their model is capturing the essential physics of the collision. While the study does not definitively prove that the critical point has been found, it demonstrates that the current tools are now sensitive enough to detect its influence if it is there. The work provides a robust framework for future experiments, showing that by refining how we simulate these tiny, fleeting explosions, we can get closer to uncovering one of the most important secrets of the subatomic world.

Technical Summary: Searching for the QCD Critical Point with Relativistic Viscous Hydrodynamics

Problem Statement
The primary objective of this work is to address the challenge of identifying the Quantum Chromodynamics (QCD) critical point using data from the STAR collaboration's Beam Energy Scan II (BESII) at RHIC. While the phase transition from quark-gluon plasma (QGP) to hadrons is established as a crossover at vanishing net-baryon density, the existence of a critical point at large baryon chemical potentials ( $\mu_B$ ) followed by a first-order phase transition remains a central open question. A significant theoretical hurdle is that the critical point cannot be directly extracted from experimental data; rather, it must be probed dynamically within an expanding, cooling system that may be far from equilibrium. The paper aims to provide the necessary theoretical input to interpret BESII data, specifically net-proton fluctuations and transverse momentum fluctuations, by simulating the system's passage through the QCD phase diagram.

Methodology
The authors employ the NuclearConfectionery framework, a modular open-source software suite for heavy-ion collisions, to perform 3+1D relativistic viscous hydrodynamic simulations. Key components of the methodology include:

Equation of State (EOS): The study utilizes the MUSES EOS (specifically the merged Holography+QvdW-HRG table), which features a movable critical point. This EOS covers a wide range of temperature ( $T$ ) and baryon chemical potential ( $\mu_B$ ), extending up to $\mu_B/T \sim 3.5$ . Since the hydrodynamic code requires entropy ( $s$ ) and net-baryon density ( $n_B$ ) as natural variables, while the MUSES EOS is defined in the grand canonical ensemble ( $T, \mu_B$ ), an "EOS Inverter" module is used to perform the necessary conversion. The simulations include metastable and unstable regions to account for jumps across the first-order phase transition, ensuring thermodynamic consistency.
Numerical Upgrades (CCAKE): The hydrodynamic evolution is solved using the CCAKE code, which employs Smoothed Particle Hydrodynamics (SPH). The authors introduce two critical numerical improvements to enhance accuracy and efficiency:
1. "Glass" Initial Conditions: Instead of initializing fluid particles on a fixed grid, particles are sampled from the local energy density and allowed to settle into an optimal configuration via a fictitious repulsive force (WVT-relaxation). This results in an amorphous, isotropic distribution ("glass") that better resolves fine features of the simulation.
2. Adaptive SPH: The smoothing scale ( $h$ ), which dictates the resolution, is no longer constant. It adapts based on the local particle density, providing high resolution in the dense core of the fluid and lower resolution at the edges.
Simulation Setup: The framework couples the initial state (AMPT), hydrodynamic evolution (CCAKE), particlization (BQSSampler), and hadronic afterburner (SMASH). Simulations are conducted for O+O collisions at 5.36 TeV (for convergence tests) and Au+Au collisions across the BESII energy range ( $\sqrt{s_{NN}} = 7.7$ to 200 GeV).

Key Contributions

Numerical Efficiency and Accuracy: The integration of glass initial conditions and adaptive SPH significantly improves energy conservation in CCAKE by an order of magnitude. This allows for high numerical accuracy with fewer SPH particles, reducing run times (e.g., approximately one minute less for O-O collisions) while maintaining resolution of rapid expansions.
Dynamic Critical Point Probing: The study demonstrates the ability to track the fluid's trajectory through the QCD phase diagram with a movable critical point. By comparing scenarios where the critical point is within reach ( $\mu_c^B = 598$ MeV) versus out of reach ( $\mu_c^B = 1000$ MeV), the authors quantify the "dynamical critical lensing" effect.
Model-to-Data Comparison: The authors present a comprehensive comparison of their framework against STAR BESII data, including pion multiplicity, mean transverse momentum ( $\langle p_T \rangle$ ), and azimuthal anisotropies ( $v_n\{2\}$ ).

Results

Phase Diagram Trajectory: For Au+Au collisions at $\sqrt{s_{NN}} = 7.7$ GeV, the simulations show that when the critical point is located at $\mu_c^B = 598$ MeV, the fluid spends a significantly larger amount of time in the vicinity of the critical point compared to when the critical point is at 1000 MeV. This is visualized as a higher density of "hits" (time spent) in the $T, \mu_B$ plane near the critical point.
Variable Insensitivity: The authors observe that the natural hydrodynamic variables ( $s, n_B$ ) are nearly insensitive to the location of the critical point; the distribution of fluid cells in the $s, n_B$ plane shows almost no difference between the two scenarios.
Experimental Agreement: The simulations, incorporating a shear viscosity ( $\eta/s$ ) that increases with $\mu_B$ , are able to reproduce STAR BESII data reasonably well. The results match experimental measurements for charged particle multiplicity, mean transverse momentum, and flow harmonics ( $v_2, v_3$ ) across the beam energy scan.

Significance and Claims
The paper claims that the upgraded NuclearConfectionery framework, combined with the MUSES EOS, provides a robust tool for direct model-to-data comparisons at RHIC beam energies. The authors assert that their setup can:

Quantify the interplay between beam energy, the location of the critical point, and the rapidity range.
Provide a reasonable description of multiplicity and flow observables, validating the framework for future calculations of experimental observables.
Offer a dynamic perspective on how the system interacts with a potential critical point, specifically highlighting that the time spent near the critical point is a function of the beam energy and the critical point's location, a phenomenon described as dynamical critical lensing.

The work emphasizes that while the framework successfully reproduces bulk observables, further work is required to fully incorporate critical fluctuations within these hydrodynamic frameworks.

Technical Summary: Searching for the QCD Critical Point with Relativistic Viscous Hydrodynamics

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