⚛️ phenomenology

Two-pion source analysis in small to large systems at SPS energies

This study utilizes the UrQMD model to simulate two-pion source distributions in central $^7$ Be+ $^9$ Be and $^{129}$ Xe+ $^{139}$ La collisions at SPS energies, demonstrating that Lévy-stable functions effectively describe the emission source across varying system sizes and providing a purely hadronic reference for NA61/SHINE femtoscopic measurements.

Original authors: Barnabás Pórfy, Zsombor Marcell Rácz

Published 2026-10-14

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CC BY 4.0

Original authors: Barnabás Pórfy, Zsombor Marcell Rácz

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 nearly the speed of light to study the behavior of matter under extreme conditions. These collisions are so violent that they produce a hot, dense environment of fundamental particles. To understand how this matter behaves, how long it lasts, and how it cools down into ordinary matter, researchers need to measure the size and shape of the fireball created in the crash. They do this using a technique called femtoscopy, which acts like a microscopic camera. Instead of taking a photograph with light, which would be too slow and blurry for such tiny scales, scientists look at how pairs of identical particles, specifically pions, fly apart from the collision. Because of the strange rules of quantum mechanics, these identical particles tend to clump together or avoid each other depending on how far apart they were when they were born. By analyzing the patterns in how these particles separate, physicists can reconstruct the three-dimensional shape and the lifetime of the source that emitted them, measuring distances that are smaller than an atomic nucleus.

A team of researchers recently turned their attention to this problem by simulating collisions between different sizes of atomic nuclei to see how the source of these particles changes. They focused on two specific scenarios: a collision between a small beryllium nucleus and a beryllium target, and a much larger collision between a xenon nucleus and a lanthanum target. These events were modeled using a powerful computer program that tracks every single particle interaction as if it were a cascade of billiard balls, operating in a purely hadronic mode without assuming the formation of a fluid-like plasma. The goal was to create a pure reference point for real-world experiments, specifically for the NA61/SHINE experiment at CERN, which is currently scanning a wide range of collision energies and system sizes. By simulating these events, the researchers could look directly at the space and time coordinates of the particles as they last interacted, a detail that real experiments can only guess at by looking at the final momentum of the particles.

The researchers discovered that the shape of the particle-emitting source in these simulations fits a specific mathematical pattern known as a Lévy distribution. For a long time, scientists assumed these sources were simple, smooth blobs, similar to a bell curve. However, this new work confirms that the reality is more complex, featuring "heavy tails" where particles are more likely to be found far from the center than a simple model would predict. This shape suggests that the particles are not just moving in a straight line but are undergoing a kind of erratic, wandering motion through the debris of the collision, a process physicists call a Lévy walk. This behavior appears consistently across both the small beryllium collisions and the large xenon collisions, suggesting a common underlying structure to how pions are emitted, regardless of the size of the system.

When the team looked at how the size of this source changed, they found clear patterns related to the energy of the collision and the mass of the nuclei involved. As the energy of the collision increased, the source size increased, and the particles seemed to wander further from the center before stopping. More importantly, the size of the source itself grew with the energy of the crash. The large xenon and lanthanum collisions produced a significantly larger source than the small beryllium collisions, which makes intuitive sense given the larger initial size of the nuclei. However, the researchers also observed that as the particles moved faster, the apparent size of the source shrank, a signature of the rapid expansion and flow of the material created in the crash. This shrinking effect was visible even in the simulation that did not include a fluid stage, indicating that the expansion is a fundamental feature of the particle interactions themselves.

The study also examined how strongly the particles correlated with one another. The strength of this connection remained relatively steady across different collision energies and system sizes, though the researchers noted that their simulated numbers were likely slightly higher than what would be seen in a real experiment. This is because the simulation did not include certain long-lived particles that decay into pions much later, which would dilute the correlation in a real detector. Despite this small difference, the overall picture is robust: the simulations provide a solid baseline of what to expect from purely hadronic interactions. The results show that even without a fluid phase, the complex dance of particles creates a source with a distinct, non-Gaussian shape that evolves predictably with energy and size. This work serves as a crucial reference for experimentalists, allowing them to separate the effects of simple particle collisions from the more exotic signals of a quark-gluon plasma in their upcoming measurements.

Technical Summary: Two-pion source analysis in small to large systems at SPS energies

Problem and Motivation
Characterizing the space-time structure of particle emission in high-energy heavy-ion collisions is central to understanding the properties of the quark–gluon plasma (QGP) and the hadronic phase. While femtoscopy (the study of quantum-statistical correlations between identical particles) provides access to femtometer-scale emission sources, experimental measurements only access the pair source indirectly via momentum correlations. To complement these measurements, event generators can reconstruct the pair source directly from the space-time coordinates of final-state particles.

Recent phenomenological studies across SPS, RHIC, and LHC energies have indicated that particle-emitting sources are better described by Lévy-stable distributions featuring power-law tails, rather than the traditionally assumed Gaussian shapes. These tails are attributed to mechanisms such as jet fragmentation, critical behavior, and anomalous diffusion (Lévy walks) in the hadronic stage. This paper addresses the need for a purely hadronic reference for femtoscopic measurements by investigating two-pion pair-source distributions in small ( $^7$ Be+ $^9$ Be) and large ( $^{129}$ Xe+ $^{139}$ La) collision systems at SPS energies, simulated without a hydrodynamic stage.

Methodology
The study utilizes the UrQMD (Ultra-relativistic Quantum Molecular Dynamics) model version 3.4 in cascade mode, meaning the simulation proceeds without a hydrodynamic phase. Events were generated for central (0–10%) collisions of $^7$ Be+ $^9$ Be and $^{129}$ Xe+ $^{139}$ La at beam momenta ranging from 13 $A$ to 150 $A$ GeV/c ( $\sqrt{s_{NN}} \approx 5–17$ GeV).

Key methodological steps include:

Event Generation: The code was modified to allow $\eta$ meson decays, while treating kaons and $\Sigma, \Xi, \Omega$ baryons as stable.
Source Reconstruction: Spatial pair distributions were analyzed in the longitudinally comoving system (LCMS). The analysis focused on the angular average of the pair separation vector, $\rho = |\vec{\rho}|$ , to derive a one-dimensional source distribution $D(\rho)$ for the Be+Be and Xe+La systems.
Kinematic Selections: To ensure comparability with experimental data (specifically NA61/SHINE), strict kinematic cuts were applied, including rapidity ( $|y| < 1$ ), transverse momentum ( $p_T < 1$ GeV/c), and relative momentum cuts dependent on transverse mass ( $m_T$ ).
Fitting Procedure: The reconstructed source distributions were fitted with Lévy distributions. For the Be+Be and Xe+La systems, one-dimensional fits were performed to extract the Lévy stability index ( $\alpha$ ), the scale parameter ( $R$ ), and the intercept parameter ( $\lambda$ ). The intermediate Ar+Sc system, included for comparison from previous work, was analyzed using three-dimensional fits.
Uncertainty Analysis: Statistical uncertainties from event-block averaging were found to be negligible. Systematic uncertainties were estimated by varying the relative-momentum selection cuts (strict: $q_{LCMS} < \sqrt{m_T} \cdot 100$ MeV/c $^2$ ; loose: $q_{LCMS} < \sqrt{m_T} \cdot 200$ MeV/c $^2$ ).

Key Contributions and Results
The paper provides a systematic study of Lévy parameters as functions of pair transverse mass ( $m_T$ ), collision energy ( $\sqrt{s_{NN}}$ ), and system size, comparing the results with an intermediate system ( $^{40}$ Ar+ $^{45}$ Sc) from previous work.

Lévy Shape Compatibility: The simulated sources in both small and large systems are well-described by Lévy distributions, supporting a common phenomenological picture of pion emission space-time structure across different system sizes.
Stability Index ( $\alpha$ ):
- $\alpha$ increases with $m_T$ in all systems.
- $\alpha$ systematically decreases as collision energy rises. This trend is attributed to a stronger Lévy-walk phenomenon characteristic of hadronic rescattering at higher energies and the production of heavier resonances that decay within the simulation.
- At the top SPS energy, $\alpha$ values are ordered by system size: smallest in Be+Be, intermediate in Ar+Sc, and largest in Xe+La.
- The simulated Be+Be results differ from existing NA61/SHINE measurements, which yield $\alpha \approx 1$ .
Scale Parameter ( $R$ ):
- $R$ falls steeply with increasing $m_T$ , a signature typically associated with expanding sources and collective flow. The authors note this behavior is present even in the purely hadronic cascade of UrQMD.
- $R$ increases with collision energy.
- A clear system-size dependence is observed: the Be+Be source is considerably smaller than the Xe+La source, attributed to differences in lifetime, radial expansion velocity, and initial geometric size.
Correlation Strength ( $\lambda^*$ ):
- The parameter shows a slight increase at low $m_T$ before flattening, with almost no dependence on collision energy or system size.
- The authors note that the simulated $\lambda^*$ is likely overestimated because UrQMD does not include weak decays that populate the "halo" of the source. Based on branching ratios, the physical $\lambda$ would be approximately 92–93% of the simulated values.

Significance and Outlook
The primary significance of this work is establishing a purely hadronic reference for femtoscopic measurements at the NA61/SHINE experiment. By demonstrating that Lévy distributions describe the source shape even in the absence of a hydrodynamic stage, the results provide a baseline for understanding the contributions of hadronic rescattering.

The authors conclude that the observed system-size and energy dependencies of $\alpha$ and $R$ offer a consistent picture of the space-time structure of pion emission in hadronic collisions. The paper positions these results as a necessary foundation for future work, specifically extending the model to include a hydrodynamic stage, against which such extensions and upcoming experimental measurements can be compared.

Technical Summary: Two-pion source analysis in small to large systems at SPS energies

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