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Methodology for Analyzing Proton Multiplicity Fluctuations with Azimuthal Partitions in Heavy-Ion Collisions

This paper proposes and validates a novel methodology using the $\Delta\sigma^2$ observable to analyze proton multiplicity fluctuations in azimuthal partitions, demonstrating its potential to quantify clustering tendencies and extract interaction parameters relevant to the first-order phase transition in heavy-ion collisions through simulations with AMPT and MUSIC+FIST models.

Original authors: Dylan Neff, Zhongling Ji, Roli Esha, Gang Wang, Huan Huang

Published 2026-10-27

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CC BY 4.0

Original authors: Dylan Neff, Zhongling Ji, Roli Esha, Gang Wang, Huan Huang

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 most energetic collisions between heavy atomic nuclei, scientists are searching for a fundamental shift in the nature of matter itself. Under normal conditions, protons and neutrons are locked inside atomic nuclei, held together by the strong force. However, when these nuclei smash into each other at nearly the speed of light, the energy is so intense that the protons and neutrons melt into a hot, dense soup of their constituent parts, known as quarks and gluons. This state of matter, called the quark-gluon plasma, is believed to have existed just microseconds after the Big Bang. As this fireball cools and expands, it undergoes a phase transition, snapping back into the familiar protons and neutrons that make up our world today. Physicists are eager to understand the details of this transition: does it happen smoothly, or does it occur abruptly, like water boiling into steam? If it is abrupt, the cooling matter might form clumps of different densities, leaving behind subtle fingerprints in the particles that fly out of the collision.

To find these fingerprints, a team of researchers has proposed a new way to look at the debris from these collisions. Instead of counting all the particles produced in an entire event, they decided to look at how protons are distributed in specific slices of the circle around the collision point. They introduced a method to measure whether protons tend to group together or spread apart in these angular slices. By analyzing the fluctuations in the number of protons found in these slices, they developed a new tool to detect if the protons are interacting with each other in a way that suggests a sudden phase change occurred. This approach offers a fresh perspective on an old problem, allowing scientists to peer into the microscopic behavior of matter during its most extreme transformation.

The researchers focused on a specific observable they call the variance of proton multiplicity in azimuthal partitions. In simpler terms, imagine taking a snapshot of a collision and drawing a wedge-shaped slice around the center, like a piece of a pie. They count how many protons fall inside that slice. They then repeat this process many times, moving the slice to different angles around the circle for the same collision event. If the protons were completely random, the number of protons in each slice would follow a predictable pattern, similar to flipping a coin many times. However, if the protons are influenced by a phase transition, they might clump together, causing some slices to have many more protons than expected and others to have very few. The team created a mathematical measure to quantify how much the actual distribution of protons deviates from this random baseline. A positive result would indicate that protons are clustering together, while a negative result would suggest they are avoiding each other.

To test whether this new measure works, the scientists ran detailed computer simulations using two different models of heavy-ion collisions. One model, known as AMPT, simulates the complex microscopic interactions of particles, while the other, a combination of MUSIC and FIST, uses fluid dynamics to describe the expansion of the collision fireball. They applied their new method to simulated collisions of gold nuclei at various energy levels, ranging from 7.7 to 62.4 billion electron volts. The results showed that the new measure was robust and reliable. In the AMPT simulations, the protons appeared to repel each other, spreading out more evenly than random chance would predict, which resulted in a negative value for their measure. In contrast, the fluid dynamics models showed a much weaker effect, with the values hovering close to zero. This difference confirmed that the method is sensitive enough to detect the specific ways protons interact with one another, such as the repulsive forces that keep them from occupying the same space.

The researchers also investigated how the size of the slice they used affected the results. They found that the strength of the signal depended on the width of the angular slice, and they developed a way to describe this relationship using a simple curve. By fitting their data to this curve, they could extract two key numbers: one that represented the overall strength of the interaction between protons, and another that described the range over which this interaction occurred. When they tested their method on a simplified model where they manually programmed protons to either attract or repel each other, the numbers they extracted matched the input settings perfectly. This demonstrated that their technique could accurately measure both the strength and the distance of the forces acting between protons in the collision.

A significant part of their work involved ensuring that the results were not distorted by the limitations of real-world detectors or the natural flow of particles in the collision. They showed that by comparing the data from the same collision events with data from mixed events—where protons from different collisions are combined—they could cancel out detector errors. They also accounted for the elliptic flow, a natural tendency of particles to fly out in an oval shape due to the geometry of the collision, which could otherwise mimic the clustering signal. After applying these corrections, the method remained stable and consistent. They discovered that the repulsive behavior seen in the AMPT simulations became stronger at lower collision energies, a trend they attributed to the conservation of momentum in events with fewer particles.

The study concludes that this new approach provides a powerful and sensitive way to study the clustering of protons in heavy-ion collisions. By measuring how protons fluctuate in angular slices, scientists can now probe the nature of the phase transition with greater precision. While the current results are based on simulations, the method is ready to be applied to real experimental data from facilities like the Relativistic Heavy Ion Collider. If applied to real collisions, this technique could reveal whether the quark-gluon plasma undergoes a first-order phase transition, a scenario where matter changes state abruptly and forms distinct clumps. Such a discovery would provide crucial insights into the fundamental structure of the universe and the forces that govern the behavior of matter at its most extreme.

Technical Summary: Methodology for Analyzing Proton Multiplicity Fluctuations with Azimuthal Partitions in Heavy-Ion Collisions

Problem Statement
High-energy heavy-ion collisions aim to investigate the phase transition between confined hadronic matter and deconfined quark-gluon plasma. While higher-order cumulants of conserved charges (such as net-baryon number) are theoretically sensitive to the order of this phase transition, their experimental measurement faces significant challenges. These include susceptibility to detector anomalies, rising statistical uncertainties with higher orders, and the need to disentangle physical signals from effects like finite volume, initial volume fluctuations, and baryon transport. Furthermore, mapping coordinate-space clustering (expected during a first-order phase transition due to surface tension) to momentum-space correlations is non-trivial. This paper addresses the need for an alternative, robust experimental approach to probe proton clustering and potential phase transition signals.

Methodology
The authors propose a novel observable, $\Delta\sigma^2$ , designed to quantify proton multiplicity fluctuations within azimuthal partitions. The methodology proceeds as follows:

Azimuthal Partitions: For each collision event, a partition of width $w$ (in degrees) is randomly placed in the azimuthal angle. The number of protons ( $N_w$ ) falling within this partition is recorded. This process is repeated for multiple random orientations per event ("event resampling") to optimize statistical usage.
The $\Delta\sigma^2$ Observable: The observable is defined as the normalized difference between the variance of the measured $N_w$ $N_{w}$ distribution and a binomial baseline:
$\Delta\sigma^2 \equiv \frac{\sigma^2(N_w) - \sigma^2_{\text{binomial}}}{N(N-1)}$
where $\sigma^2_{\text{binomial}} = NP_0(1-P_0)$ $σ_{binomial}^{2} = N P_{0} (1 - P_{0})$ represents the variance expected for uncorrelated protons with $P_0 = w/2\pi$ $P_{0} = w /2 π$ .
- Physical Interpretation: A positive $\Delta\sigma^2$ indicates proton clustering (attractive interactions), while a negative value suggests repulsion (more uniform distribution).
Corrections:
- Event Mixing: To mitigate azimuthal-dependent detector inefficiencies, a mixed-event baseline is subtracted from same-event data.
- Elliptic Flow Correction: Collective motion (elliptic flow, $v_2$ ) mimics clustering. The authors derive an analytical correction term based on Fourier coefficients ( $v_n$ ) to subtract this background, noting that the correction vanishes for $w=180^\circ$ .
- Normalization: The normalization by $N(N-1)$ ensures the observable is independent of the total proton multiplicity $N$ and insensitive to homogeneous detector inefficiencies.

Key Contributions and Results
The paper validates this methodology through simulations using the AMPT (A Multi-Phase Transport) and MUSIC+FIST (Hydrodynamics + FIST sampler) models for Au+Au collisions at $\sqrt{s_{NN}} = 7.7$ –$62.4$ GeV.

Model Dependence on $N$ : Simulations confirm that $\Delta\sigma^2$ is largely independent of the total proton multiplicity $N$ , validating the proposed normalization scheme.
Model Comparisons:
- AMPT: Results show significantly negative $\Delta\sigma^2$ values, indicating effective repulsive interactions among protons. The magnitude of this repulsion increases at lower collision energies and lower multiplicities, a trend the authors attribute to global momentum conservation constraints in events with fewer particles.
- MUSIC+FIST: The default model yields values close to zero (weak correlation). However, a version incorporating an excluded-volume effect (preventing baryons from occupying the same volume) produces markedly more negative $\Delta\sigma^2$ , demonstrating the observable's sensitivity to repulsive interactions.
Dependence on Partition Width ( $w$ ): The authors analyze the dependence of the average $\langle \Delta\sigma^2 \rangle$ $⟨ Δ σ^{2} ⟩$ on the partition width $w$ $w$ . They parameterize this dependence using a quadratic function:
$\langle \Delta\sigma^2 \rangle = b \left[ 1 - \left( \frac{w - 180}{z} \right)^2 \right]$
- Parameter $b$ : Represents the general scale (magnitude) of the correlation. In Gaussian correlation toy models, $b$ correlates linearly with the interaction strength ( $A$ ) and its sign indicates attraction (positive) or repulsion (negative).
- Parameter $z$ : Represents the inverse curvature. In the Gaussian model, $z$ is found to be solely dependent on the interaction range ( $\sigma$ ), following the relation $z \approx 12/\sigma + 131$ .

Significance and Claims
The paper claims that the $\Delta\sigma^2$ observable offers a robust method to probe proton clustering that is:

Resilient: Insensitive to homogeneous detector inefficiencies and correctable for elliptic flow and detector-specific azimuthal biases.
Sensitive: Capable of distinguishing between attractive (clustering) and repulsive interactions, as demonstrated by the distinct results from AMPT, MUSIC+FIST, and the excluded-volume model.
Quantitative: The parameters $b$ and $z$ , extracted from the $w$ -dependence, can theoretically map to the strength and range of proton interactions, respectively.

The authors conclude that applying this approach to RHIC Beam Energy Scan (BES) data could provide new insights into the nature of the quark-hadron phase transition. Specifically, the beam-energy dependence of the correlation width (reflected in $z$ ) is highlighted as a potential experimental signature of the underlying physics of the phase transition, distinct from the overall scale effects driven by momentum conservation.

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