Intermittency and fractal behaviour of charged particles generated using EPOS4 and PYTHIA8 at LHC energies

This paper investigates the intermittency and fractal behavior of charged particles in Pb-Pb collisions at 5.02 TeV using EPOS4 and PYTHIA8 simulations to analyze large density fluctuations as potential signatures of the QCD phase transition and critical point.

The Gist

The Big Picture: Smashing Atoms to Find the "Critical Point"

Imagine you are trying to understand how water turns into steam. If you heat water slowly, it bubbles gently. But if you hit a specific "critical point," the water doesn't just boil; it starts behaving strangely, with huge, chaotic bubbles forming and popping everywhere. Physicists believe that when they smash heavy atoms (like Lead) together at near-light speed, they create a similar "critical point" for the building blocks of matter (quarks and gluons). They call this state Quark-Gluon Plasma (QGP).

The goal of this paper is to see if the particles flying out of these collisions show signs of this "critical point." To do this, the authors use a mathematical tool called Intermittency.

The Analogy: The Grainy Photo vs. The Smooth Image

To understand "Intermittency," imagine taking a photo of a crowd of people.

• Random Crowd (No Critical Point): If you zoom in on the photo, the people are spread out evenly. Whether you look at the whole room or just a tiny square inch, the density of people looks roughly the same. It's "smooth."
• Critical Crowd (The Critical Point): If the crowd is at a "critical point," it's chaotic. If you zoom in, you might see huge clumps of people in some spots and empty spaces in others. The pattern looks the same no matter how much you zoom in (this is called fractal behavior). It's like a snowflake or a coastline: the more you zoom in, the more jagged and complex the edges look.

The authors are looking for that "jagged, clumpy" pattern in the particles created by the collisions. If they find it, it suggests the system is undergoing a phase transition (like water turning to steam).

The Tools: Two Different Simulators

Since we can't easily see the "critical point" in real life yet, the authors used computer simulations (Monte Carlo event generators) to predict what the data should look like. They used two different "simulators":

1. PYTHIA8: Think of this as a simulator that treats the collision like a game of billiards. It focuses on individual particles bouncing off each other and creating new ones based on standard rules. It's like simulating a crowd where everyone is just walking around randomly.
2. EPOS4: Think of this as a more complex simulator that includes "fluid dynamics." It assumes the particles form a hot, dense soup (like a liquid) that expands and cools down. It even has a switch (UrQMD) to see what happens if the particles crash into each other after the soup cools down (like people bumping into each other after a concert ends).

They ran these simulations for Lead-Lead collisions at the Large Hadron Collider (LHC) energy levels.

The Experiment: Counting the Clumps

The researchers took the simulated data and divided the space where the particles fly into a grid (like a chessboard). They then counted how many particles landed in each square.

• The Test: They kept making the squares on the chessboard smaller and smaller (increasing the resolution).
• The Expectation: If the system were at a "critical point," the number of clumps would grow in a very specific, predictable mathematical way (a power law) as the squares got smaller. This is the "Intermittency" signal.
• The Reality: They found no such signal.

The Results: Smooth, Not Jagged

Here is what they actually found:

1. No "Fractal" Pattern: When they zoomed in on the particle distribution, the pattern didn't get more complex. It stayed relatively smooth and random. It looked like a standard Poisson distribution (purely random noise), not a fractal structure.
2. No Critical Point Detected: The mathematical "scaling exponents" (the numbers that tell us if we are at a critical point) were way off from what theory predicts for a phase transition.
3. Both Simulators Agree: Both the "billiard ball" simulator (PYTHIA8) and the "fluid soup" simulator (EPOS4) produced similar results: no evidence of the critical point.

The Conclusion

The paper concludes that, within the rules and constraints of these two specific computer models, the production of particles in these collisions behaves like a statistical, random process.

• What this means: The models do not naturally produce the "clumpy, fractal" behavior that would indicate a phase transition or a critical point.
• The Takeaway: If scientists want to find the critical point in real experiments, they cannot rely on these specific models to show it to them. These models act as a "baseline" or a "control group." They tell us what the data looks like without the critical point. If real experimental data (from the ALICE detector) looks different from these simulations, then we might know we found something new. But based on these simulations alone, the "critical point" signal is missing.

In short: The authors tried to find a specific "fingerprint" of a phase transition in two popular computer simulations. They looked very closely, but the simulations showed a smooth, random pattern instead of the chaotic, fractal pattern they were hoping for. This suggests that, according to these models, the particle production is just a standard statistical event, not a sign of a critical phase transition.

Technical Summary

Technical Summary: Intermittency and Fractal Behaviour of Charged Particles in Pb–Pb Collisions

Problem Statement
The investigation of the Quark-Gluon Plasma (QGP) and the Quantum Chromodynamics (QCD) phase transition is a central goal in high-energy physics. Large density fluctuations in charged particle production are considered promising signatures for exploring the QCD phase transition and the critical point in heavy-ion collisions. These fluctuations are expected to exhibit fractal and scale-invariant behavior, which can be probed using intermittency methodology. While empirical evidence from heavy-ion collisions suggests multifractal behavior, and theories like the Ginzburg-Landau (GL) model predict specific anomalous scaling for second-order phase transitions, there is a need to quantify these properties within the constraints of current Monte Carlo (MC) event generators. This study aims to analyze the scaling behavior of normalized factorial moments (NFM) and the fractal nature of charged particles generated by the PYTHIA8 and EPOS4 models at LHC energies ($\sqrt{s_{NN}} = 5.02$ TeV) to determine if these models exhibit the critical fluctuations associated with a phase transition.

Methodology
The study utilizes simulated event samples of Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV. Two distinct MC generators are employed:

1. PYTHIA8 with Angantyr: A general-purpose generator extended for nucleus-nucleus collisions via a Glauber-inspired framework. It models multiplicity fluctuations arising from geometric variations in nucleon-nucleon sub-collisions and internal multi-parton interactions (MPI).
2. EPOS4: A framework featuring a parallel scattering approach and core-corona separation. It includes two configurations: "UrQMD ON" (incorporating late-stage hadronic interactions via the UrQMD transport model) and "UrQMD OFF" (excluding the hadronic phase).

Centrality classes (0–5%, 5–10%, 10–20%, etc.) are defined using impact parameter ($b$) for EPOS4 and summed transverse energy ($P_{\Sigma}E_T$) for PYTHIA8 to ensure a consistent comparison framework. The analysis focuses on charged particles ($\pi^\pm, K^\pm, p, \bar{p}$) within the pseudorapidity acceptance $|\eta| \leq 0.8$ and full azimuth ($0 \leq \phi \leq 2\pi$) across three transverse momentum ($p_T$) intervals: $0.4 \leq p_T \leq 2.0$, $0.4 \leq p_T \leq 1.5$, and $0.4 \leq p_T \leq 1.0$ GeV/c.

The core analysis involves calculating the normalized factorial moments, $F_q(M)$, defined as:
$$F_q(M) = \frac{1}{N} \sum_{e=1}^{N} \frac{1}{M^2} \sum_{i=1}^{M^2} f_q(n_{ie}) \left( \frac{1}{N} \sum_{e=1}^{N} \frac{1}{M^2} \sum_{i=1}^{M^2} f_1(n_{ie}) \right)^{-q}$$
where $n_{ie}$ is the number of particles in the $i$-th bin of the $e$-th event, $M$ is the number of bins per dimension, and $f_q$ represents the factorial moment of the bin multiplicity.

The study examines:

• M-scaling: The power-law growth $F_q(M) \propto M^{\phi_q}$ as the bin size decreases (increasing $M$).
• F-scaling: The relationship between higher-order moments and the second-order moment, $F_q(M) \propto (F_2(M))^{\beta_q}$.
• Fractal Dimensions: The generalized (Rényi) dimensions $D_q$ are derived from the intermittency indices $\phi_q$ using the relation $D_q = D_T (1 - \frac{\phi_q}{q-1})$, where $D_T=2$.
• Scaling Exponent ($\nu$): Determined from the power-law relation $\beta_q \propto (q-1)^\nu$.

Key Results

• Absence of Power-Law Scaling: The analysis reveals that for both PYTHIA8 and EPOS4 (in both UrQMD modes), the normalized factorial moments $F_q(M)$ do not exhibit a linear dependence on $\ln M$ in the log-log plots. Instead, $F_q(M)$ shows an initial rise at small $M$ followed by saturation. This indicates a lack of robust M-scaling and, consequently, the absence of intermittency.
• Monofractal Character: The calculated generalized dimensions $D_q$ exhibit negligible dependence on the moment order $q$ within statistical uncertainties. This suggests a monofractal character of particle production rather than the multifractal behavior expected in systems undergoing a second-order phase transition.
• Scaling Exponent ($\nu$): The extracted scaling exponent $\nu$ ranges between 1.6 and 1.9 across various centralities and $p_T$ intervals. These values are significantly larger than those predicted by Ginzburg-Landau theory or the SCR model for a system at criticality (where $\nu$ would be related to specific critical exponents, e.g., $1/8$ for the Ising model).
• Model Comparison: Both PYTHIA8 and EPOS4 follow similar trends, showing no significant differences in scaling behavior between the hydrodynamic (EPOS4 UrQMD ON) and non-hydrodynamic (EPOS4 UrQMD OFF / PYTHIA8) configurations regarding intermittency.

Significance and Claims
The paper concludes that the particle generation processes simulated by PYTHIA8 and EPOS4 under the default constraints at $\sqrt{s_{NN}} = 5.02$ TeV do not display the scale-invariant, self-similar, or multifractal behaviors indicative of a critical point or a second-order phase transition. The observed fluctuations are characterized as statistical in nature rather than dynamical critical fluctuations.

The significance of this work lies in establishing a quantitative baseline for understanding multiplicity fluctuations. By demonstrating that these standard MC models produce monofractal, non-intermittent distributions, the study provides a reference point for distinguishing between statistical noise and genuine critical phenomena in future experimental data. The absence of the predicted scaling behavior in these models suggests that if critical fluctuations exist in real heavy-ion collisions, they are not captured by the current default configurations of these generators, or that the specific observables (intermittency in soft $p_T$) are not sensitive enough to the underlying dynamics in these simulations.