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Baselines for Abelian Charge Fluctuations in Nuclear Collisions:Theory and Comparison with Experimental Data

This paper establishes a theoretical framework for analyzing Abelian charge fluctuations in nuclear collisions by deriving analytical expressions for cumulants in the canonical ensemble and incorporating local multi-particle interactions, which successfully reproduce STAR and HADES experimental data by revealing the decisive role of repulsive two-proton interactions at high energies and attractive three-particle interactions at lower energies.

Original authors: Bengt Friman, Krzysztof Redlich, Anar Rustamov

Published 2026-09-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bengt Friman, Krzysztof Redlich, Anar Rustamov

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 heart of matter, where protons and neutrons collide at speeds approaching the speed of light, physicists search for the fundamental rules that govern how the universe is built. When heavy atomic nuclei smash together in particle accelerators, they create a fleeting, super-hot soup of particles that mimics the conditions of the early universe. To understand what happens in these collisions, scientists do not just count how many particles are produced; they also measure how those numbers fluctuate from one crash to the next. These tiny variations, or fluctuations, act as a sensitive probe, potentially revealing whether the matter inside the collision undergoes a dramatic phase change, similar to water turning into steam, or if it hints at a deeper, critical point in the laws of physics. However, interpreting these fluctuations is tricky because the total number of particles is strictly conserved by the laws of physics, and this conservation alone can create patterns that look like something more exotic is happening.

A team of researchers from Germany and Poland has developed a new way to untangle these patterns, focusing specifically on the fluctuations of baryon number, a property that distinguishes ordinary matter from antimatter. Their work, which combines rigorous mathematical theory with computer simulations, aims to establish a reliable baseline for what these fluctuations should look like if only the standard laws of conservation were at play. By comparing their theoretical predictions with real data from major experiments, they discovered that the simple rules of conservation are not enough to explain what is seen in the laboratory. Instead, the data reveals that the particles themselves are interacting with one another in complex ways that change depending on how hard the nuclei are smashed together.

The researchers began by building a detailed mathematical model of a system where the total number of baryons is fixed, a scenario known as the canonical ensemble. In this framework, they calculated how the numbers of protons and antiprotons should fluctuate if they were simply moving around without influencing each other, other than being bound by the rule that the total count must remain constant. They derived precise formulas for these fluctuations, extending previous work to cover any level of detail, or order, of measurement. This allowed them to predict exactly what the data should look like if the only force at work was the global conservation of charge. When they compared these predictions to the actual measurements taken by the STAR experiment at the Relativistic Heavy Ion Collider and the HADES experiment at GSI, a clear mismatch emerged. The simple model of conservation alone failed to reproduce the specific patterns of fluctuation observed in the experiments, particularly the ratios between different types of measurements.

To solve this puzzle, the team introduced the concept of local interactions, where particles influence their neighbors directly. They simulated two types of forces: repulsive forces that push particles apart and attractive forces that pull them together. By adjusting these forces in their computer models, they found that the nature of the interaction changes dramatically with the energy of the collision. At the highest energies, where the collisions are most violent, the data is best explained by a repulsive force between protons, which keeps them from clustering too closely. This repulsion acts like a pressure that smooths out the fluctuations. However, as the collision energy drops to lower levels, the picture flips. The experimental data at these lower energies cannot be explained by repulsion; instead, it requires an attractive force that pulls particles together. Remarkably, the best fit for the low-energy data came from a model where groups of three protons attract one another, rather than just pairs.

This shift from repulsion at high energies to attraction at low energies suggests that the behavior of matter in these collisions is governed by a dynamic interplay of forces, much like the competition between pressure and cohesion in everyday fluids. The researchers found that at high energies, the repulsive interactions between pairs of protons dominate, successfully reproducing the data from the STAR experiment. In contrast, at the lower energies studied by HADES, the inclusion of attractive interactions involving three particles was essential to match the observations. Without accounting for these specific local correlations, the theoretical models consistently missed the mark, either predicting too much fluctuation or the wrong direction of change. The study demonstrates that to truly understand the signals of a potential phase transition in nuclear matter, scientists must first accurately account for these mundane but crucial local interactions.

The findings provide a more realistic foundation for future searches for the critical endpoint of the strong interaction, a hypothetical point in the phase diagram of matter that has long been sought by physicists. By showing that the observed fluctuations are largely driven by the interplay of attractive and repulsive forces among protons, the study clarifies what the "background noise" looks like. This clarity is vital because it allows researchers to distinguish between fluctuations caused by ordinary particle interactions and those that might signal a genuine phase transition. The work confirms that the nature of baryon interactions is not static but evolves with the energy of the collision, moving from a regime dominated by repulsion to one where attraction, particularly among three-particle groups, becomes the governing factor. This insight refines the tools physicists use to explore the deepest secrets of the nuclear world.

Technical Summary: Baselines for Abelian Charge Fluctuations in Nuclear Collisions

Problem Statement
The primary objective of this study is to establish robust theoretical baselines for fluctuations in conserved Abelian charges, specifically baryon number, within the context of nucleus-nucleus collisions. While fluctuations of conserved charges are key observables for mapping the QCD phase diagram and searching for a chiral critical endpoint, experimental measurements are significantly influenced by non-critical effects. Two dominant non-critical effects are volume fluctuations and exact conservation laws. Previous analyses demonstrated that global conservation of net-baryon number suppresses net-proton cumulants relative to a non-critical (Skellam) baseline. However, recent updated measurements from the STAR BES-II program and the HADES experiment reveal substantial deviations from these global conservation baselines, particularly in the ratios of factorial cumulants. The central problem addressed is that existing theoretical frameworks, which account only for global baryon number conservation, fail to reproduce the experimental data for proton factorial cumulants across a wide range of collision energies.

Methodology
The authors employ a statistical approach based on the canonical ensemble, where the net-baryon number is exactly conserved in the full phase space.

Analytical Derivation: The study begins by formulating the canonical partition function for an ideal gas of baryons and antibaryons (in the Boltzmann approximation) with exact net-baryon number conservation. Using the S-matrix approach, the authors derive analytical expressions for the cumulants and factorial cumulants of baryon ( $N_B$ ) and antibaryon ( $N_{\bar{B}}$ ) multiplicity distributions.
Subsystem Fluctuations: A key theoretical development involves deriving closed-form expressions for factorial cumulants within a subsystem (a limited acceptance window) in terms of the cumulants of the total system. This is achieved using the generalized Faà di Bruno formula and multivariate Bell polynomials. The authors establish explicit relations connecting the factorial cumulants of the net-baryon number in a subsystem to the factorial cumulants of the baryon and antibaryon numbers in the full system.
Incorporation of Local Interactions: To account for dynamics beyond global conservation, the authors introduce a phenomenological framework incorporating local multi-particle interactions. They model both repulsive and attractive correlations using a probabilistic approach inspired by equilibrium statistical mechanics.
- Repulsive Interactions: Modeled via a short-range Buckingham-type potential.
- Attractive Interactions: Modeled via a phenomenological confining power-law potential.
- Multi-particle Clusters: The framework extends to clusters of three particles, summing pairwise interactions to generate higher-order correlations.
Event Generation: The authors utilize a Metropolis algorithm with simulated annealing to generate event configurations that satisfy the target correlation coefficients defined by the interaction potentials. This allows for the generation of realistic event samples that can be directly compared with experimental acceptance criteria.

Key Contributions

Analytical Framework: The paper provides the first derivation of closed-form analytical expressions for factorial cumulants of arbitrary order for baryon, antibaryon, and net-baryon number distributions in a subsystem, explicitly accounting for exact global conservation in the total system. These results generalize previous work limited to lower orders.
Correlation Linkage: A rigorous correlation is established between net-baryon fluctuations in a subsystem and the fluctuations of baryon and antibaryon numbers in the total system.
Phenomenological Extension: The study introduces a unified framework that simultaneously incorporates both attractive and repulsive multi-particle interactions within the canonical ensemble, a feature not previously addressed in fluctuation analyses.
Realistic Event Generation: The methodology enables the generation of events with specific local correlation structures, facilitating a direct, differential comparison with experimental data (e.g., rapidity dependence).

Results
The study compares the calculated cumulants and factorial cumulants against experimental data from the STAR collaboration (BES-I, BES-II, and Fixed-Target Mode) and the HADES collaboration.

Failure of Global Conservation Baselines: The authors confirm that baselines considering only global baryon number conservation (the "canonical baseline") fail to reproduce the experimental ratios of proton factorial cumulants ( $F^{(n,0)}/F^{(1,0)}$ ).
High Energies (STAR BES-II): At higher collision energies (e.g., $\sqrt{s_{NN}} = 17.3$ GeV and above), the experimental data are well-reproduced by incorporating repulsive two-proton interactions. The inclusion of these repulsive correlations significantly improves the agreement with STAR data for ratios such as $F^{(2,0)}/F^{(1,0)}$ and $F^{(3,0)}/F^{(1,0)}$ .
Low Energies (HADES and STAR FXT): At lower energies (e.g., $\sqrt{s_{NN}} = 2.55$ GeV for HADES Ag-Ag collisions and $\sqrt{s_{NN}} \approx 3$ GeV for STAR FXT), repulsive interactions alone drive the theoretical predictions further away from the data. Instead, attractive interactions are required. Specifically, the data at low energies are best described by attractive three-particle interactions, rather than simple two-particle attraction.
Energy Dependence: The results indicate a transition in the nature of baryon interactions: from predominantly repulsive at high energies to predominantly attractive at low energies. This trend is visualized in the evolution of factorial cumulant ratios across the beam energy scan.

Significance and Claims
The paper claims that the inclusion of local multi-particle correlations is decisive for describing observed fluctuation patterns in heavy-ion collisions. The authors assert that:

Necessity of Interactions: Global conservation laws alone are insufficient to explain the energy dependence of proton factorial cumulants. The interplay between attractive and repulsive interactions is essential for a quantitative description.
Physical Interpretation: The observed transition from repulsive to attractive dominance with decreasing collision energy is reminiscent of the dynamics in classical systems described by the Van der Waals equation of state (competition between short-range repulsion and long-range attraction).
Connection to QCD: While the authors note that the temperatures probed are above 60 MeV, they suggest that these fluctuation patterns may be connected to the conjectured chiral transition of QCD rather than a simple gas-liquid transition of nuclear matter.
Methodological Utility: The derived analytical expressions and the event generation framework provide a necessary tool for future fluctuation analyses, allowing researchers to disentangle non-critical effects (conservation laws and local interactions) from potential critical signals associated with the QCD phase transition.

The study concludes that a realistic baseline for fluctuation observables must account for both exact global conservation and local dynamical correlations, the nature of which evolves with collision energy.

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