⚛️ nuclear theory

Femtoscopic parameters of pion emission sources in EPOS4 Au+Au collisions from 7.7 to 200 GeV

This paper presents a direct three-dimensional femtoscopic analysis of pion emission sources in EPOS4 Au+Au collisions across the RHIC Beam Energy Scan range, utilizing Fast Fourier Transforms to extract Lévy parameters and reduce projection biases while comparing out-side observables with UrQMD calculations to explore QCD phase diagram signatures.

Original authors: Yan Huang, Máté Csanád

Published 2026-10-14

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Yan Huang, Máté Csanád

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 most extreme laboratories on Earth, scientists smash heavy atomic nuclei together at speeds approaching the speed of light. These collisions create a fleeting, super-hot soup of particles that existed only microseconds after the Big Bang. To understand what happens inside this fireball, researchers do not look at the particles as they fly apart, but rather at how they clump together. Because identical particles, like two pions, behave like waves, they interfere with one another in a way that reveals the size and shape of the space they came from. This technique, known as femtoscopy, acts like a microscopic ruler, allowing physicists to measure distances smaller than an atomic nucleus. By studying how these particles correlate, scientists can map the three-dimensional structure of the explosion and the timing of when the particles stopped interacting, a moment called kinetic freeze-out.

A team of researchers has now refined this measuring technique to look at the source of these particles in a new, more complete way. Instead of flattening the complex, three-dimensional shape of the explosion into simple, one-dimensional slices, they analyzed the entire volume at once. Using powerful computer simulations of gold nuclei colliding at energies ranging from 7.7 to 200 billion electron volts, the team reconstructed the full three-dimensional map of where the particles were emitted. They then transformed this spatial map into a momentum map, which is what detectors actually measure, and fitted the data with a flexible mathematical shape that can describe both smooth, round sources and those with long, heavy tails. This direct approach avoids the distortions that can occur when scientists try to guess the full 3D shape based only on its 2D shadows.

The study focused on collisions of gold nuclei, a standard testbed for exploring the behavior of matter under extreme heat and pressure. The researchers compared their new, direct three-dimensional method against their previous technique, which relied on analyzing one-dimensional projections. They found that while both methods agreed on the general trends of how the source size changes with energy, the new method provided slightly different numerical values for the shape of the source. Specifically, the new analysis suggested the source was slightly less "round" and more elongated than the old method indicated, particularly at the lowest collision energies. However, the overall size of the source and how long the particles took to emerge remained consistent between the two approaches. This confirmed that the previous findings were robust, even if the new method offered a more precise view of the source's geometry.

To understand what these shapes meant for the physics of the collision, the team compared their results from a model that includes fluid-like behavior with another model that treats the collision as a series of individual particle crashes. The two models agreed on the general shape of the source, but they told very different stories about how long the explosion lasted. In the fluid-like model, the time it took for particles to stop interacting changed very little as the collision energy increased. In the particle-crash model, however, this duration grew significantly longer as the energy went up. This difference is crucial because it shows that the way particles spread out depends heavily on the underlying rules of the collision. If the source behaves like a fluid, the expansion is collective and uniform; if it behaves like a swarm of individual particles, the duration of the event stretches out as the collision becomes more violent.

The researchers also examined specific measurements that compare the size of the source in the direction of the particle's motion versus the direction perpendicular to it. These measurements are sensitive to how long the particles are being emitted. In the fluid model, this ratio decreased as energy increased, while in the particle-crash model, it increased. This divergence highlights that the expansion of the fireball is not a simple, universal process but is deeply tied to the specific dynamics of the matter involved. The study did not find any sudden, non-smooth changes in these measurements that would signal a critical point in the phase diagram of matter, but it established a clear baseline for what to expect from standard physics. By providing a more direct and less biased way to measure these sources, the work offers a sharper tool for future experiments. When real-world data from particle accelerators is eventually compared against these refined simulations, scientists will be better equipped to distinguish between ordinary changes in the collision dynamics and the rare, exotic signals that might reveal new phases of matter.

Problem Statement
Three-dimensional femtoscopy is a critical tool for investigating the space-time structure of particle-emitting sources in relativistic heavy-ion collisions, offering insights into the evolution of strongly interacting matter and the QCD phase diagram. While Lévy-stable distributions have been established as a more flexible description of non-Gaussian source structures than traditional Gaussian parameterizations, previous analyses of EPOS4 simulations relied on fitting one-dimensional projections of the three-dimensional pair-distance distribution $D(\rho)$ . The authors identify a potential limitation in this approach: reducing a three-dimensional source to one-dimensional projections implicitly assumes that the projections contain sufficient information to characterize the underlying 3D structure. However, 3D $D(\rho)$ distributions can exhibit non-elliptical structures where different 3D shapes yield similar 1D projections, potentially introducing bias when extracting source parameters. Furthermore, there is a need to understand how different collision dynamics (hydrodynamic vs. transport) influence the energy dependence of femtoscopic observables, particularly out-side parameters like $R^2_{diff}$ and $R_{out}/R_{side}$ , which are sensitive to emission duration.

Methodology
The paper introduces a direct three-dimensional analysis framework to address the biases of projection-based methods. The methodology proceeds as follows:

Data Generation: The authors utilize EPOS4 simulations for Au+Au collisions across the RHIC Beam Energy Scan range ( $\sqrt{s_{NN}} = 7.7$ to $200$ GeV). Unlike previous analyses that merged events, individual events are analyzed to maximize statistical precision.
3D Histogram Construction: The relative space-time coordinates of pion pairs are transformed into the longitudinally co-moving system (LCMS) using Bertsch–Pratt decomposition ( $\rho_{out}, \rho_{side}, \rho_{long}$ ) to construct a three-dimensional pair-distance distribution $D(\rho)$ as a $TH3F$ histogram.
FFT Transformation: The normalized 3D $D(\rho)$ histogram is transformed into the momentum-space correlation function $C(q_{out}, q_{side}, q_{long})$ using a Fast Fourier Transform (FFT) on a cubic grid. This preserves the full multidimensional structure without reducing the data to 1D projections.
Direct Fitting: The resulting 3D correlation function is fitted directly with an analytic Lévy formula: $C(q) = 1 + \lambda \exp[-|q^T R^2 q|^{\alpha/2}]$ . The fit extracts five parameters: the Lévy index $\alpha$ , correlation strength $\lambda$ , and the radii $R_{out}, R_{side}, R_{long}$ .
Comparative Analysis: The results from this direct 3D FFT method are compared against the previous projection-based $D(\rho)$ analysis. Additionally, the EPOS4 results are compared with recent UrQMD calculations (using the $D(\rho)$ method) to investigate differences in collision dynamics.

Key Contributions

Development of a Direct 3D Framework: The paper establishes a robust method for Lévy femtoscopy that transforms the full 3D source distribution to momentum space via FFT and fits the correlation function directly, avoiding potential biases associated with 1D projections of non-elliptical sources.
Systematic Comparison of Methods: It provides a quantitative comparison between the new direct 3D FFT method and the traditional projection-based method, validating the stability of energy dependence trends while quantifying systematic shifts in parameter values.
Dynamical Model Comparison: The study offers a detailed comparison between EPOS4 (hydrodynamic-based) and UrQMD (hadronic transport-based) models, specifically focusing on the excitation functions of out-side observables to elucidate the role of emission duration and collective evolution.

Results

Method Comparison (EPOS4):
- Lévy Index ( $\alpha$ ): The direct $C(q)$ fits yield systematically lower values of $\alpha$ (by approximately 10–15%) compared to the $D(\rho)$ -based method, with the difference being most pronounced at the lowest energies (7.7 and 9.2 GeV). However, the energy and transverse mass ( $m_T$ ) dependence trends remain consistent.
- Radii ( $R_{out}, R_{side}, R_{long}$ ): $R_{out}$ values show close agreement (<5% difference). $R_{side}$ is systematically larger (5–20%) and $R_{long}$ is systematically smaller (5–15%) in the $C(q)$ method compared to the projection method. Despite these absolute shifts, the collision-energy dependence trends for all radii are consistent between the two methods.
- Correlation Strength ( $\lambda$ ): Values from the $C(q)$ fits are generally 7% larger, but their energy and $m_T$ dependences are preserved.
- Derived Observables: The derived quantities $R^2_{diff} = R^2_{out} - R^2_{side}$ and $R_{out}/R_{side}$ exhibit energy dependences consistent with the previous analysis.
Model Comparison (EPOS4 vs. UrQMD):
- Lévy Index: Both models show good agreement in $\alpha$ values and energy dependence.
- Radii: While $R_{side}$ shows a similar moderate increase with energy in both models, $R_{out}$ exhibits a much stronger energy dependence in UrQMD compared to the weak dependence in EPOS4.
- Out-Side Observables: Consequently, $R^2_{diff}$ and $R_{out}/R_{side}$ show opposite energy dependences: they decrease with collision energy in EPOS4 but increase in UrQMD.
- Emission Duration: Using the approximation $R^2_{out} - R^2_{side} \simeq \beta_T^2 \Delta\tau^2$ , the effective emission duration $\Delta\tau_{eff}$ shows a weak energy dependence in EPOS4 but a pronounced, approximately linear increase with $\sqrt{s_{NN}}$ in UrQMD. This suggests that the different excitation functions arise from distinct space-time evolutions: hydrodynamic expansion in EPOS4 versus hadronic rescattering in UrQMD.

Significance
The paper claims that this work establishes a more direct and robust three-dimensional framework for femtoscopic studies, which is essential for future comparisons with experimental measurements. By demonstrating that the excitation functions of out-side observables are highly sensitive to the underlying collision dynamics (hydrodynamics vs. transport), the authors argue that non-monotonic structures in experimental data cannot be attributed solely to critical-point dynamics without first accounting for these baseline model differences. The EPOS4 and UrQMD results serve as complementary baselines, providing a necessary reference for distinguishing generic dynamical effects from potential signatures of the QCD phase diagram or critical point. The study emphasizes that understanding the space-time evolution of the source is a prerequisite for interpreting femtoscopic observables in the search for QCD phase transitions.

More like this