Excitation function of femtoscopic Lévy source parameters of pion pairs in EPOS4

This study utilizes the EPOS4 model to systematically investigate the transverse mass and collision-energy dependence of three-dimensional femtoscopic Lévy source parameters for pion pairs across the STAR Beam Energy Scan range, revealing distinct trends in radii, the Lévy index, and correlation strength while highlighting a systematic reduction in the sideward radius compared to EPOS3 results.

The Gist

Imagine you are trying to figure out what a giant, invisible balloon looks like just by watching how the air molecules inside it bounce off each other. That is essentially what this paper is doing, but instead of a balloon, it's studying the tiny, super-hot "soup" of particles created when heavy gold atoms smash into each other at nearly the speed of light.

Here is a breakdown of the research using simple analogies:

The Big Picture: The "Particle Soup"

When scientists smash gold atoms together (like in the STAR experiment at the Relativistic Heavy Ion Collider), they create a state of matter called a Quark-Gluon Plasma. Think of this as a microscopic, super-hot explosion. For a split second, it's a chaotic soup of particles. Then, it cools down and freezes into a cloud of pions (a type of particle).

The scientists want to know: How big was this explosion? How long did it last? And what shape did it have?

The Tool: "Femtoscopy" (The Microscopic Ruler)

To measure something this small (smaller than an atom), they use a technique called Femtoscopy.

• The Analogy: Imagine you are in a dark room with a bunch of people shouting. If two people shout at the exact same time and from the exact same spot, their voices blend together in a specific way. If they are far apart, the voices sound different.
• In Physics: By looking at pairs of pions that are created very close together, scientists can measure how their "voices" (momentum) interfere with each other. This interference tells them how far apart the particles were when they were born. It's like using sound waves to map the shape of a room you can't see.

The New Twist: The "Levy" Shape

Traditionally, scientists assumed these particle clouds were perfect spheres (like a smooth, round balloon). This is called a Gaussian shape.

• The Problem: Real life is messy. Sometimes particles get stuck in long-lived "ghosts" (resonances) or move in weird, long-distance jumps.
• The Solution: This paper uses a Lévy distribution. Think of this as a balloon that isn't perfectly round; it has fuzzy edges and long, wispy tails. It's a more realistic way to describe a chaotic explosion. The paper introduces a "shape parameter" (called $\alpha$) to measure how "fuzzy" or "tail-heavy" the explosion is.

The Experiment: Changing the Volume and the Speed

The researchers used a supercomputer simulation called EPOS4 to run thousands of these collisions. They changed two main things:

1. The Energy ($\sqrt{s_{NN}}$): They smashed the atoms together at different speeds, from a "gentle" 7.7 GeV up to a "hard" 200 GeV.
2. The Transverse Mass ($m_T$): They looked at particles moving at different speeds sideways.

What Did They Find? (The Results)

1. The Size of the Explosion (Radii)

• The Finding: As the collision energy gets higher, the "soup" gets bigger, especially in the direction the beam is traveling (longitudinal).
• The Analogy: If you pop a balloon gently, it's small. If you pop it with a hammer, it expands further. The paper found that the "long" part of the explosion stretches out more at higher energies, while the "sideways" part grows a bit, and the "outward" part stays roughly the same size.

2. The Shape (The $\alpha$ parameter)

• The Finding: The shape of the explosion is fairly consistent. It's not a perfect sphere, but it doesn't change drastically with energy.
• The Analogy: Whether you pop the balloon gently or hard, it still has those same "fuzzy, wispy" edges. It doesn't suddenly turn into a perfect cube or a flat pancake.

3. The "Core" vs. The "Halo" (Correlation Strength $\lambda$)

• The Finding: At higher energies, fewer particles seem to come from the "core" (the immediate explosion), and more seem to come from the "halo" (long-lived ghosts that decay later).
• The Analogy: Imagine a firework. At low energy, you see mostly the bright center burst. At high energy, you see more of the lingering sparks flying off far away. The paper found that as energy goes up, the "core" signal gets a bit weaker relative to the "halo."

The Comparison: EPOS4 vs. EPOS3

The scientists compared their new simulation (EPOS4) with an older one (EPOS3).

• The Result: They mostly agreed, which is good news! It means the physics is working correctly.
• The Exception: There was one difference in the "sideways" size. The new model (EPOS4) predicts the explosion is slightly smaller sideways than the old one. This is like two different architects drawing the same house; they agree on the roof and the front door, but one thinks the side wall is a few inches shorter. This small difference helps scientists refine their models.

Why Does This Matter?

The ultimate goal of this research is to find the QCD Critical Point.

• The Analogy: Think of water. It can be ice, liquid, or steam. There is a specific point where the transition between liquid and gas becomes "critical" (fuzzy and wild). Physicists think nuclear matter has a similar critical point.
• The Hope: If they smash atoms at just the right energy, the "shape" of the explosion (the Lévy parameters) might suddenly change in a weird, non-smooth way. This paper provides a "baseline" (a map of what happens without a critical point) so that when real experiments see a weird blip, they know, "Aha! That's the critical point!"

Summary

This paper is a detailed "dress rehearsal" using a supercomputer. The scientists simulated heavy-ion collisions to see how the "shape" and "size" of the particle explosion change with energy. They found that the explosion gets bigger and more stretched out at higher energies, but its fundamental "fuzzy" shape stays mostly the same. This gives them a solid reference point to help future real-world experiments hunt for the elusive "Critical Point" of the universe.

Technical Summary

1. Problem Statement and Motivation

High-energy heavy-ion collisions create extreme conditions of temperature and density, allowing for the study of strongly interacting matter and the Quantum Chromodynamics (QCD) phase diagram. A primary goal is to locate the QCD critical point, where the transition between hadronic matter and quark-gluon plasma changes from a crossover to a first-order phase transition.

Femtoscopic measurements (Hanbury Brown-Twiss or HBT interferometry) of identical particle correlations are sensitive probes of the space-time structure of the particle-emitting source. While traditional analyses assumed Gaussian source distributions, recent experimental data suggests that Lévy-stable distributions (characterized by long tails) provide a better description, potentially arising from critical phenomena, jet fragmentation, or anomalous diffusion.

The specific problem addressed in this work is the lack of a comprehensive theoretical baseline for three-dimensional (3D) Lévy femtoscopy across the Beam Energy Scan (BES) range. The authors aim to:

• Systematically investigate the excitation function (collision energy dependence) of 3D Lévy source parameters using the EPOS4 event generator.
• Establish a non-critical theoretical baseline to interpret future experimental data (e.g., from STAR) and identify potential signatures of critical behavior.
• Compare EPOS4 results with the previous EPOS3 model to assess improvements in the description of collective source characteristics.

2. Methodology

The study utilizes the EPOS4 Monte Carlo event generator (version 4.0.3), which implements a fully self-consistent, energy-momentum conserving parallel scattering scheme with sub-scattering dependent saturation scales. This represents a paradigm shift from EPOS3, leading to harder high-multiplicity events.

Simulation Setup:

• System: Central (0–10%) Au+Au collisions.
• Energy Range: $\sqrt{s_{NN}} = 7.7$ to $200$ GeV (covering the STAR BES range).
• Particles: Identical charged pion pairs ($\pi^+\pi^+$ and $\pi^-\pi^-$).
• Kinematic Cuts: $0.15 < p_T < 1.0$ GeV/c and $|\eta| < 1$.
• Equation of State (EoS): Default X3FF EoS with a crossover transition and three-flavor conservation.

Analysis Procedure:

1. Distance Distribution: Instead of measuring momentum correlations directly (as in experiments), the simulation calculates the 3D pair distance distribution $D(\vec{\rho})$ in the longitudinally co-moving system (LCMS) using the Bertsch-Pratt coordinate system (out, side, long).
2. Lévy Parameterization: The spatial distribution is fitted using a 3D Lévy-stable distribution function:
   $$D(\vec{\rho}) \propto \exp\left(-\left|\frac{\vec{\rho}}{R}\right|^\alpha\right)$$
   where $\alpha$ is the Lévy index (shape), $R$ represents the radii ($R_{out}, R_{side}, R_{long}$), and $\lambda$ is the correlation strength.
3. Fitting: The parameters $\alpha, R_{out}, R_{side}, R_{long},$ and $\lambda$ are extracted by fitting the simulated distance distributions.
4. Systematic Uncertainties: Uncertainties were estimated by varying:
   • Number of averaged events ($N_{events}$).
   • Maximum momentum difference ($Q_{max}$).
   • Fit range in $\rho$.
   • Centrality definition (impact parameter vs. multiplicity).
   • Note: Uncertainties are generally low (<3%) for $m_T < 0.7$ GeV/c² but increase at higher $m_T$ due to reduced statistics.

3. Key Contributions

• First 3D Lévy Analysis in EPOS4: This work provides the first systematic 3D Lévy femtoscopic analysis using the updated EPOS4 model across the full STAR BES energy range.
• Excitation Functions: It establishes detailed excitation functions (dependence on $\sqrt{s_{NN}}$) for all key Lévy parameters ($\alpha, \lambda, R_{out}, R_{side}, R_{long}$) and derived quantities ($R_{diff}, R_{out}/R_{side}$).
• Model Comparison: It offers a critical comparison between EPOS4 and EPOS3, highlighting specific improvements and discrepancies, particularly in the $R_{side}$ radius.
• Baseline for Critical Point Search: By providing a "non-critical" baseline, the study enables future experimentalists to distinguish between standard hydrodynamic evolution and signals of a critical point (which would manifest as non-monotonic behaviors in these parameters).

4. Key Results

A. Source Shape (Lévy Index $\alpha$)

• $m_T$ Dependence: $\alpha$ shows a mild increase with transverse mass ($m_T$) across all energies. This is consistent with a reduced relative contribution from long-lived resonances at higher $p_T$.
• Energy Dependence: $\alpha$ is largely independent of collision energy. A slight localized decrease is observed around $\sqrt{s_{NN}} = 11.5$ GeV, but no clear non-monotonic behavior (a signature of the critical point) is present in the EPOS4 baseline.
• Reliability Note: The authors note that EPOS4 is less reliable below 19.6 GeV; results in this low-energy region should be interpreted with caution.

B. Source Size (Radii $R_{out}, R_{side}, R_{long}$)

• $m_T$ Dependence: All radii decrease as $m_T$ increases, a standard hydrodynamic effect (Hubble-like expansion). The decrease is most pronounced in the longitudinal direction ($R_{long}$).
• Energy Dependence:
  • $R_{side}$ and $R_{long}$ increase gradually with collision energy, reflecting the longer system lifetime and larger spatial expansion at higher energies.
  • $R_{out}$ shows little to no dependence on collision energy, a behavior consistent with STAR experimental data.
• Derived Quantities:
  • $R_{diff} = R_{out}^2 - R_{side}^2$: Shows a non-monotonic pattern with a drop at 14.5 GeV, but no clear global trend.
  • $R_{out}/R_{side}$: Decreases with energy, with a notable enhancement in the 14.5–39 GeV range. The deviation from unity at low energies suggests limitations in modeling pre-thermalized acceleration or viscous corrections.

C. Correlation Strength ($\lambda$)

• $m_T$ Dependence: $\lambda$ increases with $m_T$, showing a "hole" at low $m_T$. This is attributed to the increasing purity of the core (primordial pions) relative to the halo (long-lived resonance decays) as $m_T$ rises.
• Energy Dependence: $\lambda$ generally decreases with increasing collision energy, likely due to the enhanced role of resonance decays at higher energies. Anomalies (deviations from the trend) are observed at 9.2 and 11.5 GeV.

D. Comparison with EPOS3

• General Agreement: Most parameters agree with EPOS3 results within $\sim 2\sigma$.
• Significant Discrepancy: The $R_{side}$ radius in EPOS4 is systematically smaller than in EPOS3, with deviations ranging from $2.5\sigma$ to $5.8\sigma$. This suggests that the new scattering scheme in EPOS4 modifies the transverse source geometry significantly.
• Data Compatibility: Despite the shift in $R_{side}$, EPOS4 remains compatible with existing one-dimensional experimental data (e.g., PHENIX), though 3D experimental comparisons are needed for full validation.

5. Significance and Outlook

This study provides a crucial theoretical baseline for the interpretation of femtoscopic data in the search for the QCD critical point.

• Critical Point Signatures: The absence of strong non-monotonic features in the EPOS4 baseline (which assumes a crossover EoS) implies that any non-monotonic behavior observed in future experimental data could be a genuine signature of criticality or a first-order phase transition.
• Future Directions: The authors emphasize the need to:
  1. Investigate the dependence of these observables on different Equations of State (EoS), specifically those describing a first-order phase transition or a second-order critical point.
  2. Compare these 3D Lévy predictions with upcoming 3D experimental measurements from the STAR experiment.
  3. Refine the modeling of low-energy collisions ($\sqrt{s_{NN}} < 19.6$ GeV) where EPOS4 currently shows limitations.

In summary, the paper successfully maps the expected behavior of Lévy source parameters in a standard hydrodynamic scenario, setting the stage for identifying anomalous signals in experimental data that may point to new physics in the QCD phase diagram.