Scaling behaviour of charged particles generated in Xe$-$Xe collisions at $\sqrt{s_{\rm{NN}}}$ = 5.44 TeV using the AMPT model

This paper utilizes the String Melting mode of the AMPT model to investigate the scaling behavior and intermittency of charged particle multiplicity fluctuations in Xe–Xe collisions at $\sqrt{s_{\rm{NN}}}$ = 5.44 TeV, determining key parameters such as anomalous fractal dimensions and scaling exponents to characterize the system's self-similar dynamics and provide baseline predictions.

The Gist

Imagine two giant, slightly squashed water balloons (representing Xenon nuclei) smashing into each other at nearly the speed of light. When they collide, they don't just splash; they create a tiny, super-hot fireball of energy that explodes into thousands of tiny particles.

This paper is like a detective story. The authors want to know: Is this explosion random chaos, or is there a hidden, repeating pattern?

Here is the story of their investigation, broken down into simple parts:

1. The Detective's Tool: The "Pixelated" Lens

To see if there is a pattern, the researchers used a computer model called AMPT (think of it as a highly sophisticated video game engine that simulates these crashes).

They looked at the spray of particles coming out of the crash. To analyze it, they imagined placing a grid over the explosion, like a sheet of graph paper.

• The Experiment: They started with a coarse grid (big squares). Then, they made the squares smaller and smaller (higher resolution), like zooming in with a camera.
• The Goal: They were looking for something called "Intermittency." In everyday terms, this is like looking at a cloud. If you zoom in, do you see the same fluffy shapes repeating over and over? If you see the same patterns at every level of zoom, that's a "fractal" pattern. In physics, finding this specific kind of pattern is a huge clue that the system went through a special "phase transition" (like water turning to steam, but for subatomic particles).

2. The Search for the "Critical Point"

In the world of heavy-ion physics, scientists are hunting for a "Critical End Point." Imagine a map of weather. There's a specific spot where rain turns into snow, and the air gets very turbulent and unpredictable. Scientists think a similar "turbulent zone" exists in the subatomic world.

If the particles in the collision show fractal patterns (self-similarity), it suggests the system hit this turbulent, critical zone. If the patterns are just random noise, it means the system behaved smoothly, like a calm river.

3. What They Found: The "Smooth River"

The researchers ran their simulation with the Xenon nuclei and analyzed the particle spray using their "pixelated lens." Here is what they discovered:

• No Magic Patterns: As they zoomed in (made the grid squares smaller), they did not see the repeating, self-similar fractal patterns they were hoping for. The fluctuations in the number of particles were just random noise.
• One Type of Fractal: They found that the particles behaved like a "monofractal." Think of this like a simple, smooth sheet of paper. No matter how you look at it, it's just a flat sheet. They did not find a "multifractal" (which would be like a crumpled piece of paper with complex, repeating wrinkles at every scale).
• The "Scaling" Number: They calculated a specific number (called $\nu$) that describes how the particles fluctuate. Their number came out to be around 1.78.
  • If the system had hit that "critical turbulent zone," theory says this number should be around 1.3.
  • Because 1.78 is different from 1.3, it confirms that the simulation did not produce critical fluctuations.

4. Why This Matters (The "Baseline")

You might wonder, "If they didn't find the special pattern, is the paper useless?" Not at all.

Think of this like a chef trying to bake a perfect soufflé. Before they can say, "My soufflé failed because I didn't use enough eggs," they need to know what a perfect soufflé looks like in a textbook.

• This paper provides the "textbook expectation" for what happens when you smash Xenon nuclei together using the AMPT model.
• It tells us: "If you use this specific computer model, you will get a smooth, non-critical result."
• This is crucial because when real scientists look at data from the Large Hadron Collider (LHC), they can compare their real-world results against this "baseline." If the real data looks different from this paper's results, it might mean the real world is doing something special (like hitting that critical point) that the computer model isn't capturing yet.

Summary

The authors simulated a high-speed crash between Xenon atoms. They looked for hidden, repeating patterns in the debris that would signal a major change in the state of matter. They found no such patterns. The debris behaved smoothly and randomly, without the complex "fractal" structure associated with critical points.

This result is valuable because it sets a standard expectation. It tells future researchers: "If you see something different in real experiments, it's not just the computer model acting up; it might be something new and exciting happening in the real universe."

Technical Summary

Technical Summary: Scaling Behaviour of Charged Particles in Xe–Xe Collisions at $\sqrt{s_{NN}} = 5.44$ TeV using the AMPT Model

Problem Statement and Motivation
The primary objective of contemporary high-energy physics is to identify the nature of phase transitions in strongly interacting matter and understand the mechanisms driving multiparticle production in heavy-ion collisions. A critical area of investigation involves the search for the Critical End Point (CEP) in the Quantum Chromodynamics (QCD) phase diagram. Near a second-order phase transition or the CEP, systems are expected to exhibit large-scale fluctuations characterized by a divergence in correlation length, manifesting as "critical opalescence." In QCD matter, this is hypothesized to result in self-similar spatial patterns and critical clustering in particle distributions.

Intermittency analysis, which studies the scaling behavior of multiplicity fluctuations in phase space, serves as a powerful tool to detect these critical phenomena. While intermittent patterns have been observed in various interactions (hadron-nucleon, muon-hadron, etc.), conclusive evidence regarding the specific mechanism of multiparticle production and critical fluctuations remains elusive. Recent proposals suggest utilizing high-multiplicity events from the Large Hadron Collider (LHC) to investigate these phenomena. However, establishing a baseline for these fluctuations is essential to distinguish critical effects from non-critical dynamical contributions. This study addresses the gap in understanding the scaling behavior of charged particle generation in Xenon-Xenon (Xe–Xe) collisions, an intermediate-sized system with deformed nuclear geometry, using the AMPT (A Multi-Phase Transport) model.

Methodology
The study utilizes the String Melting (SM) mode of the AMPT model (version v1.26t9b-v2.26t9b) to simulate Xe–Xe collisions at a center-of-mass energy per nucleon pair of $\sqrt{s_{NN}} = 5.44$ TeV. The SM mode was selected over the Default (DF) mode because it offers an improved description of collective behavior and multi-particle correlations by converting all excited strings into partons before the parton cascade, followed by hadronization via a quark coalescence model.

• Event Generation: Approximately 500,000 minimum bias events were generated, with a subset of $3 \times 10^5$ central events (impact parameter $b \leq 3.5$ fm) selected for analysis. The model parameters were tuned to reproduce experimental data from the ALICE collaboration, specifically regarding charged-particle multiplicities and pseudorapidity density distributions.
• Phase Space Partitioning: The analysis focuses on charged particles (protons, pions, and kaons) within the kinematic acceptance of $|\eta| \leq 0.8$ and full azimuthal angle, restricted to the soft transverse momentum region ($p_T \leq 2.0$ GeV/c). The phase space is partitioned into a two-dimensional lattice of $M^2$ cells ($M_\eta \times M_\phi$), where $M$ varies from 4 to 80.
• Intermittency Analysis: The study calculates Normalized Factorial Moments (NFM), denoted as $F_q(M)$, for moment orders $q = 2, 3, 4, 5$. The scaling behavior is examined through:
  1. Resolution Scaling: Investigating the power-law dependence of $F_q$ on the number of bins ($M$) to determine the intermittency index ($\phi_q$).
  2. Fractal Dimensions: Calculating the anomalous fractal dimension ($d_q$) and the generalized fractal dimension ($D_q$) to distinguish between monofractal and multifractal nature.
  3. Order Scaling (F-scaling): Analyzing the relationship between higher-order moments and the second-order moment ($F_q \propto F_2^{\beta_q}$) to extract the scaling exponent ($\nu$), defined by $\beta_q = (q-1)^\nu$.
  4. Phase Transition Coefficient: Examining the coefficient $\lambda_q$ to identify potential minima indicative of non-thermal phase transitions.

Key Results
The analysis of the AMPT-generated Xe–Xe events yields the following observations:

• Absence of Intermittency: The normalized factorial moments $F_q(M)$ do not exhibit a significant power-law growth with increasing phase space resolution ($M$). Instead, $F_q$ values grow for small $M$ and subsequently saturate. This indicates a lack of large bin-to-bin fluctuations and the absence of scale invariance in the multiplicity distributions.
• Monofractal Nature: The intermittency indices ($\phi_q$) are found to be independent of the moment order $q$ within statistical errors. Consequently, the generalized fractal dimension ($D_q$) is independent of $q$, revealing a monofractal nature of particle generation. This contrasts with the multifractal behavior expected in systems exhibiting critical fluctuations.
• Scaling Exponent ($\nu$): A weak linear dependence of $\ln F_q$ on $\ln F_2$ is observed, allowing for the extraction of the scaling exponent $\nu$. The average value obtained across all transverse momentum bins is $\nu \simeq 1.78 \pm 0.04$. This value is significantly higher than the theoretical predictions for second-order phase transitions (e.g., Ginzburg-Landau theory predicts $\nu \approx 1.304$; Ising model with critical fluctuations predicts $\nu \approx 1.04$).
• Coefficient $\lambda_q$: The parameter $\lambda_q$ shows a clear dependence on $q$ but exhibits no minimum for $q$ up to 5. This absence of a minimum suggests the formation of a single-phase system rather than a system undergoing a non-thermal phase transition.
• Independence from $p_T$: The observed scaling parameters ($\nu$, $D_q$, $\lambda_q$) are largely independent of the specific transverse momentum bins studied, suggesting a uniform particle generation mechanism across the studied $p_T$ range in the SM AMPT framework.

Significance and Claims
The paper posits that the observed scaling behavior in the SM AMPT model for Xe–Xe collisions serves as a baseline expectation for particle production mechanisms in the absence of critical fluctuations.

• Validation of Methodology: The results confirm that the AMPT model, which incorporates smooth transport dynamics but does not explicitly include critical fluctuation physics, produces monofractal, non-intermittent patterns. This validates the methodology's utility: since the model fails to reproduce critical signatures (as expected), the same methodology can be effectively applied to experimental data to search for deviations that might signal the presence of a Critical End Point.
• System Size and Geometry: The study specifically addresses the Xe–Xe system, characterized by intermediate system size and geometric deformation. The findings indicate that neither the intermediate size nor the initial-state geometric deformation of the Xe nuclei is sufficient to generate critical fluctuation patterns within the current AMPT transport dynamics.
• Future Outlook: The authors conclude that while the SM AMPT framework provides a robust baseline, a direct comparison with experimental data from the LHC is necessary. Such a comparison would allow researchers to assess the extent to which observed scaling behaviors in real data can be reproduced by the model, thereby helping to distinguish genuine critical effects from non-critical dynamical contributions.

In summary, the paper establishes that within the string melting mode of the AMPT model, Xe–Xe collisions at 5.44 TeV do not exhibit signatures of intermittency, multifractality, or phase-transition-like scaling. These results provide a crucial reference point for interpreting fluctuation phenomena in future experimental analyses of deformed nuclear systems.