Geometry of particle emission in UrQMD Ar+Sc collisions at SPS energies

This paper utilizes Ultra-Relativistic Quantum Molecular Dynamics (UrQMD) simulations to demonstrate that the three-dimensional two-pion emission source in intermediate-mass $^{40}$Ar+$^{45}$Sc collisions at SPS energies is well-described by Lévy-stable distributions, thereby establishing a theoretical baseline for future femtoscopy measurements in such systems.

The Gist

Imagine you are at a massive, chaotic concert. The band (the colliding atomic nuclei) has just finished a song, and the crowd (the particles) is rushing toward the exits. You want to know: How big was the crowd? How long did they linger? And how did they move?

This paper is about using a special kind of "crowd control camera" called femtoscopy to answer those questions, but instead of a concert, the event is a high-energy collision between Argon and Scandium atoms.

Here is the breakdown of what the scientists did, using simple analogies:

1. The Setup: The "Smash"

The researchers used a supercomputer simulation called UrQMD (Ultra-Relativistic Quantum Molecular Dynamics). Think of this as a incredibly detailed video game engine that simulates what happens when you smash two heavy atomic nuclei together at nearly the speed of light.

• The Experiment: They simulated smashing Argon and Scandium atoms together at various speeds (energies) found at the SPS (Super Proton Synchrotron) accelerator.
• The Goal: They wanted to see the "shape" of the explosion. In the past, scientists assumed the explosion was a perfect, smooth ball (like a Gaussian distribution). But recent evidence suggests the explosion is actually lumpy and has "tails" that stretch out further than expected.

2. The New Lens: The "Levy" Shape

Instead of assuming the crowd spreads out in a perfect circle, the scientists used a mathematical shape called a Lévy distribution.

• The Analogy: Imagine throwing a handful of confetti.
  • A Gaussian (old model) is like throwing confetti that forms a neat, round pile in the middle.
  • A Lévy (new model) is like throwing confetti where most falls in the middle, but a few pieces fly very far away, creating long, wispy tails.
• Why it matters: The paper confirms that in these Argon+Scandium collisions, the particles behave like the Lévy confetti. They have a "power-law tail," meaning some particles travel much further than a simple ball model would predict.

3. The Investigation: Measuring the "Ghost"

You can't take a photo of the explosion because it happens too fast (in a femtosecond, which is a quadrillionth of a second). Instead, the scientists look at pairs of particles (specifically pions, which are like the "ghosts" of the collision).

• The Trick: They measure how often two particles are born close together in space and time. If they are born close, they "bunch up" due to quantum mechanics.
• The Result: By analyzing these bunches, they reconstructed the 3D shape of the source. They found that the "Lévy" shape fits the data perfectly.

4. The Findings: What the "Shape" Tells Us

The scientists measured three main things about the shape of the explosion:

• The "Stiffness" (The Alpha Index):

  • What it is: This measures how "pointy" or "lumpy" the distribution is.
  • The Finding: As the particles move faster (higher momentum), the shape gets slightly "stiffer" (closer to a normal ball). But as the collision energy gets higher, the shape gets "lumpier" (more Lévy-like).
  • The Metaphor: Think of it like a crowd running. If they are running slowly, they stay in a tight group. If they are running at a frantic, high-energy sprint, they scatter in weird, long trails. The simulation shows that at higher energies, the "scattering" becomes more extreme.

• The "Size" (The Radii):

  • What it is: How big is the explosion in different directions?
  • The Finding: The explosion gets smaller for faster particles (because fast particles escape the "fireball" quickly) and larger for higher collision energies (because the fireball lasts longer and expands more).
  • The Metaphor: Imagine a balloon popping. If you look at the slow-moving air molecules, they seem to come from a large, expanding cloud. If you look at the fast-moving ones, they seem to come from a tiny, tight spot where the pop happened.

• The "Strength" (The Lambda Parameter):

  • What it is: How strong is the "bunching" effect?
  • The Finding: The bunching gets slightly stronger for faster particles. However, the authors note that their simulation is missing some "long-distance" particles (from the decay of heavier, unstable particles like Kaons). If they included those, the bunching would look slightly weaker, but the overall trend remains the same.

5. Why This Matters

This paper is a baseline.

• The Problem: We have experimental data from real machines (like the NA61/SHINE experiment), but we need to know if our computer models (UrQMD) are telling the truth.
• The Solution: This paper says, "Here is what our computer model predicts for Argon+Scandium collisions."
• The Future: Now, when real experimentalists measure these collisions, they can compare their real data against this "baseline."
  • If the real data matches the simulation, our understanding of how matter behaves is correct.
  • If they don't match (and the paper hints at some differences, like a "dip" in the data that the simulation missed), it tells us there is something new and mysterious happening—perhaps a phase transition or a critical point in the universe's matter that our current models can't explain yet.

Summary

In short, the authors built a high-speed simulation of an atomic collision. They found that the particles don't explode in a simple, round ball, but in a complex, stretched-out shape called a Lévy distribution. This shape changes depending on how fast the particles are and how hard the nuclei were smashed. This work provides a crucial "control group" for scientists to test their theories against real-world experiments, helping us understand the fundamental geometry of the universe's most energetic events.

Technical Summary

1. Problem Statement and Motivation

The primary goal of high-energy heavy-ion physics is to understand the space-time geometry of the particle-emitting source created in collisions. While femtoscopy (quantum-statistical correlation analysis) has traditionally assumed a Gaussian source shape, recent experimental data (from RHIC, LHC, and SPS) indicate that the two-pion source function exhibits power-law tails, better described by Lévy-stable distributions.

Despite this experimental confirmation, theoretical simulations validating these findings in intermediate-mass systems (like Argon+Scandium) have been lacking. Most existing phenomenological studies focus on large systems (Au+Au, Pb+Pb) at RHIC and LHC energies. There is a need to:

• Investigate whether Lévy distributions apply to intermediate systems (Ar+Sc) at SPS energies ($\sqrt{s_{NN}} \approx 5-17$ GeV).
• Understand the physical mechanisms (e.g., resonance decays, Lévy walks) driving these distributions in a hadronic transport model.
• Provide a theoretical baseline for upcoming experimental measurements by the NA61/SHINE collaboration.

2. Methodology

The study utilizes the Ultra-Relativistic Quantum Molecular Dynamics (UrQMD) Monte Carlo event generator (version 3.4) to simulate central (0-10%) $^{40}$Ar+$^{45}$Sc collisions.

• Simulation Setup:

  • Beam Energies: 13, 19, 30, 40, 75, and 150 A GeV/c ($\sqrt{s_{NN}} \approx 5-17$ GeV).
  • Event Generation: 10,000 events per energy in the 0-10% centrality range.
  • Modifications: The code was modified to include the decay of $\eta$ mesons to improve the connection to real data, though weak decays of kaons and hyperons ($\Sigma, \Omega, \Xi$) remain excluded.
  • Validation: The simulated hadronic spectra (transverse momentum $p_T$ and rapidity $y$) were validated against NA61/SHINE experimental data. UrQMD was found to describe pion and proton spectra well, though it slightly underestimates kaon multiplicities.

• Femtoscopy Analysis:

  • Frame of Reference: Analysis was performed in the Longitudinally Comoving System (LCMS) using Bertsch-Pratt coordinates ($out, side, long$).
  • Source Reconstruction: The three-dimensional pair source distribution $D(\vec{\rho})$ was calculated from the freeze-out coordinates of like-charged pion pairs.
  • Statistical Convergence: Due to limited statistics in single events, distributions from blocks of events (2,500–5,000 events depending on energy) were summed to ensure convergence.
  • Fitting: The projected 1D source distributions were fitted with 3D Lévy-stable distributions to extract parameters:
    • $\alpha$: Lévy stability index (shape, $1 < \alpha \le 2$).
    • $R_{out}, R_{side}, R_{long}$: Lévy scale parameters (radii).
    • $\lambda^*$: Correlation strength (intercept).

• Systematic Uncertainties: A rigorous error analysis was performed by varying three key parameters: the number of summed events ($N_{events}$), the maximum momentum difference ($Q_{max}$), and the fit range.

3. Key Contributions

• First Intermediate-System Simulation: This is one of the first detailed theoretical investigations of Lévy-stable source parameters in intermediate-mass systems (Ar+Sc) using a microscopic transport model.
• 3D Lévy Analysis in UrQMD: The study demonstrates that UrQMD-generated sources in intermediate systems are well-described by 3D Lévy distributions, validating the use of this parametrization beyond large collision systems.
• Baseline for NA61/SHINE: The results provide a crucial theoretical baseline for interpreting future experimental data from the NA61/SHINE experiment, specifically regarding the excitation function of femtoscopic parameters.

4. Key Results

The analysis yielded the following trends for the extracted Lévy parameters as functions of transverse mass ($m_T$) and collision energy ($\sqrt{s_{NN}}$):

• Lévy Index ($\alpha$):

  • $m_T$ Dependence: $\alpha$ shows a moderate increase with increasing $m_T$. This is attributed to a relative weakening of the contribution from resonance decays at higher momenta.
  • Energy Dependence: $\alpha$ shows a clear decrease with increasing collision energy. The authors attribute this to a denser environment at higher energies, which enhances the "Lévy walk" phenomenon (anomalous diffusion) caused by multiple scattering and resonance decays.
  • Discrepancy with Data: The simulated decrease in $\alpha$ with energy contrasts with preliminary NA61/SHINE data, which shows an increase. The authors suggest this discrepancy arises because UrQMD lacks a hydrodynamic medium and critical point dynamics, which are likely present in the experimental data.

• Correlation Strength ($\lambda^*$):

  • Shows a small increase with $m_T$ and no clear dependence on collision energy.
  • Note: The extracted values are systematically higher than experimental values because UrQMD excludes weak decays (e.g., $K_S^0, \Lambda$). The authors estimate the "true" $\lambda$ would be ~92-93% of the simulated value if these decays were included.

• Lévy Radii ($R_{out}, R_{side}, R_{long}$):

  • $m_T$ Dependence: All radii decrease with increasing $m_T$, consistent with the expectation of an expanding source with collective flow.
  • Energy Dependence: Radii generally increase with collision energy.
  • Ordering: At low $m_T$, the ordering is $R_{long} > R_{out} > R_{side}$. At higher $m_T$, $R_{long}$ becomes significantly smaller than $R_{out}$. This behavior matches findings in EPOS models and experimental data.
  • The study confirms that $R_{long}$ is more sensitive to collision energy than the other radii due to its connection with the system's lifetime.

5. Significance and Conclusion

The paper establishes that UrQMD successfully reproduces the Lévy-stable nature of pion emission sources in intermediate-mass Ar+Sc collisions. The extracted parameters follow physical expectations regarding source expansion and collective flow.

However, the study highlights a critical limitation: UrQMD is a hadronic transport model and does not include hydrodynamic evolution or critical point dynamics. The discrepancy between the simulated energy dependence of $\alpha$ and experimental data suggests that the experimental observations in the SPS energy range may be driven by the onset of a hydrodynamic phase or critical phenomena, which are absent in the current simulation.

Conclusion: These results serve as a vital baseline. Future comparisons between these UrQMD results and experimental data will help isolate the specific contributions of hadronic rescattering versus hydrodynamic flow and critical dynamics in the QCD phase diagram.