⚛️ nuclear theory

Studying the QCD Matter produced in Heavy-Ion Collisions using the MUSES Calculation Engine

This paper presents the capabilities of the MUSES Calculation Engine's *Calliope* version to compute and thermodynamically merge diverse equations of state for heavy-ion collisions, demonstrating their application in relativistic viscous hydrodynamic simulations across various collision energies with movable critical points and critical scaling effects.

Original authors: Johannes Jahan (MUSES Collaboration), Kevin P. Pala (MUSES Collaboration), Yumu Yang (MUSES Collaboration), Isabella Danhoni (MUSES Collaboration), Prachi Garella (MUSES Collaboration), Jonathan Gonza

Published 2026-06-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Johannes Jahan (MUSES Collaboration), Kevin P. Pala (MUSES Collaboration), Yumu Yang (MUSES Collaboration), Isabella Danhoni (MUSES Collaboration), Prachi Garella (MUSES Collaboration), Jonathan Gonzales (MUSES Collaboration), Joaquin Grefa (MUSES Collaboration), Mauricio Hippert (MUSES Collaboration), Surkhab Kaur Virk (MUSES Collaboration), Micheal Kahangirwe (MUSES Collaboration), Musa R. Khan (MUSES Collaboration), Feyisola Nana (MUSES Collaboration), Mateus Reinke Pelicer (MUSES Collaboration), Tulio E. Restrepo (MUSES Collaboration), Hitansh Shah (MUSES Collaboration), T. Andrew Manning (MUSES Collaboration), Mark Alford (MUSES Collaboration), Dekrayat Almaalol (MUSES Collaboration), Ahmed Abuali (MUSES Collaboration), Alexander Clevinger (MUSES Collaboration), Nikolas Cruz-Camacho (MUSES Collaboration), Carlos Conde-Ocazionez (MUSES Collaboration), Francesco Di Clemente (MUSES Collaboration), David Friedenberg (MUSES Collaboration), Hosein Gholami (MUSES Collaboration), Marco Hofmann (MUSES Collaboration), Jeremy W. Holt (MUSES Collaboration), Isaac Legred (MUSES Collaboration), Jamie M. Karthein (MUSES Collaboration), Toru Kojo (MUSES Collaboration), Konstantin Maslov (MUSES Collaboration), Paolo Parotto (MUSES Collaboration), Leonardo Pena (MUSES Collaboration), Grégoire Pihan (MUSES Collaboration), Christopher Plumberg (MUSES Collaboration), Roman Poberezhniuk (MUSES Collaboration), Romulo Rougemont (MUSES Collaboration), Jordi Salinas San Martín (MUSES Collaboration), Rajesh Kumar (MUSES Collaboration), Volodymyr Vovchenko (MUSES Collaboration), Veronica Dexheimer (MUSES Collaboration), Jorge Noronha (MUSES Collaboration), Jacquelyn Noronha-Hostler (MUSES Collaboration), Claudia Ratti (MUSES Collaboration), Nicolás Yunes (MUSES Collaboration)

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 kitchen. In the very beginning, just after the Big Bang, everything was a super-hot, super-dense soup of tiny particles called quarks and gluons. This soup is known as Quark-Gluon Plasma (QGP). Today, scientists try to recreate this ancient soup in massive particle accelerators (like the Large Hadron Collider or RHIC) by smashing heavy atoms together at nearly the speed of light.

The problem is, we don't have a complete recipe for this soup. We know the ingredients (quarks and gluons), but we don't fully understand how they behave when mixed together under extreme heat and pressure. This "recipe" is called the Equation of State (EoS).

This paper introduces a new, powerful digital kitchen tool called MUSES (Modular Unified Solver of the Equation of State). Here is a simple breakdown of what the authors did and why it matters:

1. The Problem: A Puzzle with Missing Pieces

Scientists have different ways to calculate the recipe for the soup:

Lattice QCD: This is like trying to calculate the recipe using pure math and supercomputers. It's very accurate, but it only works well when the soup is hot and not too heavy (low pressure). It hits a "wall" when things get too dense.
Phenomenological Models: These are like "best guess" recipes based on how we think the soup should behave. They work well for dense, heavy soup but aren't as grounded in pure math.
The Gap: Until now, these two methods didn't talk to each other well. If you tried to use the math recipe for the whole simulation, it would break. If you used the guess recipe, it might not match the real physics.

2. The Solution: The MUSES "Lego" System

The authors built a new software engine called Calliope. Think of MUSES as a set of high-tech Lego blocks.

Each block is a different "recipe" (EoS module). Some blocks are for the hot, light soup (Lattice QCD). Some are for the cold, heavy soup (Hadronic models). Some even include a special "Critical Point" block, which is a theoretical spot where the soup changes its nature dramatically (like water turning to steam, but for subatomic particles).
The Synthesis Module is the special tool that snaps these different Lego blocks together. It takes the math recipe and the guess recipe and blends them smoothly so they don't have a jagged edge where they meet. This creates one long, continuous recipe that works from the hottest, lightest soup to the densest, heaviest soup.

3. The "Inverter" Tool: Changing the Language

When scientists run computer simulations of these collisions, their programs speak a specific language: they need to know the density of the soup (how much stuff is in a box) and the entropy (how messy it is).
However, the MUSES recipes are written in a different language: Temperature and Chemical Potential (a measure of how much "push" the particles have).

The EoS Inverter is a translator. It takes the MUSES recipes and flips them around so the simulation computer can understand them. It's like taking a menu written in French and instantly translating it into English so the chef can cook.

4. The Simulation: Testing the Recipes

The authors used this new system to simulate heavy-ion collisions at three different energy levels (7.7, 19.6, and 39 GeV). They wanted to see:

Where does the soup go? They tracked tiny "fluid particles" (like tracking individual drops of water in a storm) to see which parts of the recipe they passed through.
Does the Critical Point matter? They tested recipes with the "Critical Point" in different locations. They found that if the Critical Point is in a certain spot, the fluid particles tend to "bunch up" or slow down as they pass through it, much like cars slowing down when they see a traffic jam ahead.
The Results:
- The new "blended" recipes (Holography + HRG) covered a much wider area of the phase diagram than the old math-only recipes.
- They found that for lower energy collisions, the fluid spends a lot of time in the "dense" part of the recipe, which is where the old math recipes often failed.
- They confirmed that the location of the Critical Point changes how the fluid moves, but they didn't see a massive "traffic jam" (critical lensing) effect in their specific simulations, suggesting the Critical Point might be oriented in a way that doesn't trap the fluid as much as some theories predicted.

5. Why This Matters

This paper doesn't claim to have found the Critical Point yet. Instead, it built the infrastructure to look for it.

Before this, scientists had to patch together different recipes manually, which was messy and prone to errors.
Now, with MUSES, they can mix and match different theoretical models, ensure they fit together perfectly, and run complex simulations to see which recipe matches the real data from particle accelerators best.

In a nutshell: The authors built a universal translator and a modular construction kit for the "recipe of the universe." This allows scientists to simulate heavy-ion collisions more accurately, helping them figure out if and where the mysterious "Critical Point" exists in the subatomic world.

Technical Summary: Studying QCD Matter in Heavy-Ion Collisions using the MUSES Calculation Engine

Problem Statement

The equation of state (EoS) of hot and dense Quantum Chromodynamics (QCD) matter is fundamental for describing heavy-ion collisions across all collision energies. However, determining the EoS presents two primary challenges:

Theoretical Limitations: First-principles Lattice QCD (LQCD) calculations are restricted to vanishing or small real baryon chemical potentials ( $\mu_B$ ) due to the fermion sign problem. While LQCD provides reliable data at high temperatures and low $\mu_B$ , phenomenological models are required for higher densities. Ensuring these models match LQCD constraints in a thermodynamically consistent manner remains a significant hurdle.
Dynamical Complexity: Heavy-ion collisions are inherently dynamical, out-of-equilibrium systems. Comparing experimental data to theory requires hydrodynamic simulations that evolve multiple conserved charges (baryon number, strangeness, and electric charge). These simulations require EoS inputs that are not only thermodynamically consistent but also invertible (converting from $(T, \mu_X)$ to natural hydrodynamic variables like entropy density $s$ and charge densities $n_X$ ) and capable of handling phase transitions, including potential critical points (CP) and first-order transitions.

Furthermore, connecting constraints from heavy-ion collisions to neutron stars requires a unified framework capable of handling the four-dimensional phase diagram $(T, \mu_B, \mu_S, \mu_Q)$ , as the systems explore different regions of this space despite potentially similar baryon densities.

Methodology

The authors utilize the MUSES (Modular Unified Solver of the Equation of State) Calculation Engine, specifically the new Calliope release. This framework provides a composable workflow management system that orchestrates a collection of software modules to compute EoSs and observables.

Key Modules and Workflows

The paper focuses on modules designed for heavy-ion phenomenology, expanding the previous set of 10 modules to 17:

Hadronic Phase: The HRG (Hadron Resonance Gas) module, based on Thermal-FIST, computes thermodynamics in the full 4D space $(T, \mu_B, \mu_S, \mu_Q)$ . It supports ideal, excluded volume, and Quantum van der Waals (QvdW) interactions.
Lattice QCD Expansions:
- 4D-Taylor (BQS): Expands pressure in powers of $\hat{\mu}_X = \mu_X/T$ using continuum-estimated susceptibilities.
- 4D-TExS: Generalizes the $T'$ -Expansion Scheme to 4D, mapping Cartesian chemical potentials to spherical coordinates to extend the validity range beyond standard Taylor expansions.
Phenomenological Models:
- Ising 2D-TExS: Combines 2D lattice $T'$ -expansion with 3D-Ising critical scaling to model a critical point and first-order line in the $(T, \mu_B)$ plane.
- Holography (NumRelHolo): Uses an Einstein-Maxwell-Dilaton (EMD) gauge/gravity correspondence model to describe strongly coupled matter, calibrated to LQCD at $\mu_B=0$ and extended to finite density. It includes transport coefficients.
Synthesis Module: A new procedure based on statistical mixture and stability criteria allows merging two different EoSs (e.g., Holography and HRG) into a single, thermodynamically consistent EoS covering a broader range of $(T, \mu_X)$ . This procedure can generate crossover transitions, first-order transitions, or critical points.
EoS Inverter: A module that numerically inverts EoS tables from $(T, \mu_X)$ coordinates to natural hydrodynamic variables $(s, n_X)$ or $(\epsilon, n_X)$ , essential for Lagrangian hydrodynamic solvers like CCAKE.

Simulation Framework

The authors perform relativistic viscous hydrodynamic simulations using the NuclearConfectionary framework (specifically AMPT for initial conditions, CCAKE 2.0 for hydrodynamics, and SMASH for the hadronic afterburner).

Collisions: Central Au+Au collisions at $\sqrt{s_{NN}} = 7.7, 19.6,$ and $39$ GeV.
Variables: Simulations track individual SPH (Smoothed Particle Hydrodynamics) particles through the phase diagram, allowing for the analysis of dynamical trajectories, heat maps, and freeze-out distributions.
Transport Coefficients: Shear viscosity is set to a constant $\eta T/(\epsilon+P) = 0.08$ . Bulk viscosity includes a critical scaling component mimicked via the baryon susceptibility $\chi_B^2$ .

Key Contributions

Release of MUSES Calliope: The paper introduces a new set of modules and workflows specifically tailored for heavy-ion collisions, complementing existing neutron-star modules.
Extended EoS Coverage: The authors demonstrate the construction of EoSs that cover the relevant phase space for the RHIC Beam Energy Scan (BES-II), including regions with high $\mu_B$ and low $T$ where LQCD expansions typically fail.
Thermodynamic Merging: The implementation of the Synthesis module allows for the seamless merging of disparate EoS models (e.g., Holography and HRG) with tunable critical points, ensuring thermodynamic consistency across phase boundaries.
Inversion and Stability: The EoS Inverter module is tested for its ability to handle the non-linear mapping between $(T, \mu_X)$ and $(s, n_X)$ , including strategies for handling first-order phase transitions and metastable/unstable regimes.
Dynamical Trajectory Analysis: By utilizing a Lagrangian hydrodynamic solver, the authors map the specific trajectories of fluid cells through the QCD phase diagram, identifying which regions are probed at different beam energies and rapidities.

Results

EoS Validity and Coverage:
- The 4D-TExS EoS provides significantly better coverage in the $(s, n_B)$ plane compared to the standard 4D-Taylor expansion, particularly at lower temperatures and higher baryon densities.
- The Holography+QvdW-HRG merged EoS offers the broadest coverage, effectively spanning the hadronic and deconfined phases.
- Simulations at $\sqrt{s_{NN}} = 19.6$ GeV using the 4D-Taylor EoS show that only ~50-70% of fluid cells at mid-rapidity remain within the valid EoS table, necessitating the use of backup conformal EoSs. In contrast, the 4D-TExS and Holography+HRG models maintain >90% coverage at mid-rapidity for $\sqrt{s_{NN}} \ge 19.6$ GeV.
Critical Point Sensitivity:
- The location of the critical point significantly influences which beam energies are sensitive to its presence.
- For a critical point at $\mu_B^c = 598$ MeV (consistent with recent theoretical predictions), only the lowest energy ( $\sqrt{s_{NN}} = 7.7$ GeV) shows a significant fraction of fluid cells ( $\sim$ 40%) passing near the critical region. Higher energies probe the critical region primarily at large forward/backward rapidities.
- No significant "critical lensing" (bunching of isentropic trajectories) was observed in the simulations, suggesting the critical region may be oriented horizontally along the $\mu_B$ axis rather than the $T$ axis, or that out-of-equilibrium effects smear the signal.
First-Order Transitions:
- Simulations including metastable and unstable regimes across a first-order transition did not cause numerical crashes in the CCAKE solver, as the region of negative squared speed of sound ( $c_s^2 < 0$ ) was extremely small and short-lived ( $\langle \Delta \tau \rangle \approx 1.4 \times 10^{-5}$ fm/c).
- However, the authors note that the current treatment of the first-order line is a preliminary exploration and not a complete microscopic dynamical theory of phase conversion.
Observables:
- The choice of EoS (and critical point location) had a minimal effect on the total pion multiplicity ($dN/dy$), suggesting that normalization parameters dominate these observables.
- The width of the rapidity distribution ( $dN/d\eta$ ) was found to be strongly dependent on the initial state smearing length in AMPT rather than the EoS details.

Significance and Outlook

The paper establishes the infrastructure necessary for systematic studies of dense QCD matter within a unified, open-source framework. By providing modular, thermodynamically consistent EoSs that can be merged and inverted, the MUSES Calliope engine enables more robust comparisons between heavy-ion collision data and theoretical models.

The authors emphasize that their work is a "first step" and "preliminary exploration," particularly regarding the handling of first-order phase transitions and the inclusion of critical scaling in transport coefficients. They explicitly state that a systematic treatment of first-order transitions in dynamical simulations requires further theoretical and numerical work. The primary significance lies in the capability to track which regions of the QCD phase diagram are sampled during fireball evolution and to quantify how the limited validity of expansion-based EoSs affects hydrodynamic predictions. This framework is positioned to facilitate future joint analyses connecting constraints from heavy-ion collisions and neutron stars.