⚛️ nuclear theory

Comparison of pair-source and correlation-function based Lévy analyses in Ar+Sc system at SPS energies

This study utilizes UrQMD simulations to compare pair-source and correlation-function based Lévy analyses of pion sources in central Ar+Sc collisions at SPS energies, establishing a hadronic baseline for NA61/SHINE femtoscopy research by examining non-Gaussian features as functions of transverse mass and collision energy.

Original authors: Barnabás Pórfy

Published 2026-10-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Barnabás Pórfy

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 high-energy world of particle physics, scientists smash atomic nuclei together at speeds close to the speed of light to recreate the conditions that existed just moments after the Big Bang. The goal is to understand the quark-gluon plasma, a state of matter where the building blocks of protons and neutrons melt into a hot, dense soup. To see what happens inside this fleeting fireball, researchers use a technique called femtoscopy. This method does not rely on taking a photograph with a camera, but rather on listening to the subtle statistical whispers of particles as they fly apart. By measuring how often pairs of identical particles, such as pions, emerge with similar speeds and directions, physicists can reconstruct the size and shape of the source that emitted them. For decades, scientists assumed this source was a simple, smooth blob, similar to a perfect sphere of light. However, recent observations suggest the reality is more complex, with the source exhibiting jagged, irregular edges that a simple sphere cannot describe.

To investigate this complexity, a researcher at Eötvös Loránd University in Budapest turned to computer simulations to study a specific collision system: argon nuclei striking scandium nuclei. This particular setup is a key part of the NA61/SHINE experiment, which operates at the Super Proton Synchrotron accelerator in Switzerland. The researcher used a sophisticated computer model known as UrQMD to simulate ten thousand of these collisions at various energy levels, ranging from 13 to 150 billion electron volts per nucleon. The simulation allowed the scientist to track the exact moment and location where every single particle stopped interacting with its neighbors, a point known as freeze-out. From these coordinates, the researcher could map out the three-dimensional shape of the particle-emitting source in a way that is impossible to do directly in a real experiment, where only the final momentum of the particles is visible.

The study focused on comparing two different ways of measuring the source's shape. The first method looked directly at the spatial distribution of the particles as they froze out, essentially creating a 3D map of where they were. The second method took that same data and transformed it into a correlation function, which is the type of measurement actually performed in real experiments. This transformation involves a mathematical process that converts spatial information into momentum information, similar to how a prism splits light into a rainbow, but here it splits the data to reveal how particles are correlated in their motion. By running both methods on the exact same simulated events, the researcher could see if the two approaches yielded the same picture of the source or if they told different stories.

The results revealed that while both methods generally agreed on the overall behavior of the source, they diverged in specific details, particularly at lower energies and for certain types of measurements. The simulations showed that the source is not a simple sphere but follows a more complex pattern known as a Lévy distribution, which allows for long, heavy tails rather than a sharp cutoff. When the researcher compared the direct spatial map with the reconstructed momentum correlation, they found that the correlation method consistently estimated the source to be slightly smaller in some directions and slightly larger in others compared to the direct map. Specifically, the parameter describing the stability of the source shape was found to be systematically lower when derived from the correlation function than from the direct spatial data. However, as the energy of the collision increased, these two different ways of looking at the data began to converge, suggesting that at higher energies, the distinction between the two methods becomes less significant.

A crucial finding of this work was the behavior of the source size as the mass of the particle pairs changed. In the simulations, the size of the emitting region shrank as the transverse mass of the particles increased, a trend that aligns with the idea of an expanding source. Yet, when the researcher compared these simulation results to actual experimental data collected by the NA61/SHINE collaboration, a notable difference emerged. The experimental data showed a non-monotonic trend, where the source size did not simply shrink in a straight line but fluctuated in a way the simulation did not capture. The researcher suggests that this discrepancy likely arises because the computer model used for the simulation lacks a hydrodynamic phase, a stage of fluid-like expansion that is believed to occur in real collisions. Without this fluid-like behavior, the simulation cannot reproduce the complex, non-linear changes seen in the real world.

Ultimately, this study provides a vital baseline for future research. By demonstrating that the method used to extract source parameters can influence the results, the work highlights the importance of understanding the limitations of different analysis techniques. The researcher concludes that while the computer model successfully reproduces many features of the collision, it falls short of fully describing the source because it misses the hydrodynamic expansion. This gap between simulation and reality points the way forward, indicating that future models must include these fluid dynamics to accurately describe the quark-gluon plasma. The work does not solve the mystery of the source's shape, but it clarifies how we measure it and where our current tools need to be sharpened to see the true nature of the subatomic fireball.

Technical Summary: Comparison of Pair-Source and Correlation-Function Based Lévy Analyses in Ar+Sc System at SPS Energies

Problem Statement
Femtoscopy in high-energy heavy-ion collisions aims to reconstruct the space-time structure of particle-emitting sources. While Gaussian source distributions have historically been the standard approximation, recent experimental data (e.g., from NA61/SHINE) indicate that non-Gaussian features are prevalent, necessitating the use of Lévy-stable source distributions. A critical methodological challenge exists in how these distributions are extracted: experimentalists measure momentum correlation functions ( $C_2$ ), whereas theoretical models often provide direct access to the pair-source distribution ( $D$ ). The origin of non-Gaussian features (attributed to mechanisms like Lévy walks, jet fragmentation, or critical phenomena) remains not fully understood. This study addresses the methodological discrepancy between extracting Lévy parameters directly from the pair-source distribution versus reconstructing them from the correlation function via Fast Fourier Transform (FFT).

Methodology
The study utilizes the UrQMD (Ultra-relativistic Quantum Molecular Dynamics) microscopic transport model to simulate central (0–10%) $^{40}\text{Ar}+^{45}\text{Sc}$ collisions at SPS energies ( $\sqrt{s_{NN}} \approx 5\text{--}17$ GeV, corresponding to beam momenta of 13–150 A GeV/c). The analysis focuses on like-charged pions ( $\pi^\pm\pi^\pm$ ).

Two distinct methodological approaches are applied to the same set of simulated events to extract Lévy parameters ( $\alpha$ , $\lambda$ , $R_{out}$ , $R_{side}$ , $R_{long}$ ):

Direct Pair-Source Analysis ( $D(\rho)$ ):
- The freeze-out coordinates of particles are used to construct the three-dimensional pair-source distribution $D(\vec{\rho})$ in the Longitudinally Co-Moving System (LCMS) using Bertsch–Pratt coordinates.
- To ensure statistical stability, distributions are constructed by summing over event blocks.
- One-dimensional projections of $D(\vec{\rho})$ are fitted directly with the 3D Lévy distribution formula (Eq. 3).
Correlation-Function Reconstruction ( $C(q)$ ):
- The three-dimensional pair-source distribution $D(\vec{\rho})$ is transformed into momentum space using FFT to generate the correlation function $C(q)$ .
- This reconstructed $C(q)$ is fitted with the Lévy-shaped correlation function (Eq. 4), restricted to a spherical momentum range $|\vec{q}| < q_{max}$ .
- This approach mimics the experimental procedure where only the correlation function is observable.

Both methods utilize the same kinematic cuts, centrality selections, and $K_T$ (transverse mass) binning. The parameters are studied as functions of transverse mass ( $m_T$ ) and collision energy.

Key Results
The comparison reveals systematic differences between the two extraction methods, particularly at lower transverse masses:

Lévy Index ( $\alpha$ ): Parameters extracted from $C(q)$ are systematically smaller than those from $D(\rho)$ at low $m_T$ . However, the two methods converge at higher $m_T$ . Notably, the UrQMD simulation shows a monotonic decrease in $\alpha$ with increasing $m_T$ , which contrasts with the non-monotonic trend observed in experimental data.
Scale Parameters ( $R_{out}, R_{side}, R_{long}$ ):
- $R_{out}$ and $R_{side}$ show a clear decrease with increasing $m_T$ , consistent with an expanding source. The $C(q)$ results for these parameters converge toward the $D(\rho)$ results at higher $m_T$ .
- $R_{long}$ exhibits a divergence between the two methods as $m_T$ increases, with $C(q)$ yielding systematically smaller values.
Intercept Parameter ( $\lambda^*$ ): The parameter extracted from $C(q)$ shows a rapid increase in the first few $m_T$ bins before stabilizing, whereas the $D(\rho)$ extraction follows a different trend. The values converge quantitatively at higher $m_T$ .
Averaged Radius ( $\bar{R}$ ): The directionally averaged radius $\bar{R} = \sqrt{(R^2_{out} + R^2_{side} + R^2_{long})/3}$ demonstrates strong convergence between the two methods, suggesting that directional averaging mitigates the discrepancies found in individual components.

Significance and Claims
The paper establishes a hadronic baseline for femtoscopic analyses in the Ar+Sc system, motivated by the NA61/SHINE experiment. Its primary contribution is a direct methodological comparison demonstrating that while the two approaches (direct source fitting vs. correlation function reconstruction) yield similar results at high transverse mass, they produce systematically different values at low transverse mass.

The author claims that the monotonic behavior of the Lévy index $\alpha$ in the UrQMD simulation, which lacks a hydrodynamic phase, stands in direct contrast to experimental data. This discrepancy highlights the limitations of the current UrQMD configuration and underscores the necessity of including a hydrodynamic phase to accurately describe the source evolution and the non-monotonic trends observed in real experiments. The study serves as a reference for future measurements, clarifying how methodological choices in analysis (source vs. correlation function) impact the extracted physical parameters.

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