Pion emission source shape in UrQMD Au + Au collisions at STAR energies

Technical Summary: Pion Emission Source Shape in UrQMD Au + Au Collisions at STAR Energies

Problem and Motivation
Femtoscopy, specifically the measurement of two-pion Bose–Einstein correlations, serves as a primary tool for probing the space-time geometry of particle-emitting sources in heavy-ion collisions. While Gaussian source distributions were historically assumed, recent experimental data across SPS, RHIC, and LHC energies suggest that Lévy-stable distributions provide a superior description of correlation functions, particularly for sources exhibiting long tails. A critical objective in this field is to identify signatures of the Quantum Chromodynamics (QCD) critical point, which may manifest as a non-monotonic energy dependence of the Lévy index of stability, $\alpha$. To interpret experimental data from the STAR Collaboration's Beam Energy Scan (BES-II), a theoretical baseline representing a system without critical behavior or hydrodynamic evolution is required. This work addresses the need for such a baseline by simulating pion emission sources using the UrQMD transport model, which lacks explicit hydrodynamic evolution and a QCD phase transition.

Methodology
The study utilizes the UrQMD (Ultra-relativistic Quantum Molecular Dynamics) event generator, version 3.4, in its default hadronic cascade mode. This mode employs stochastic binary scatterings, color string formation, and resonance decays but excludes explicit hydrodynamic evolution or an equation of state.

• Simulation Setup: The authors simulated 100,000 events for lower energies ($\sqrt{s_{NN}} = 3.0$ to $7.2$ GeV) and 10,000 events for higher energies ($7.7$ to $27.0$ GeV) for central (0–10%) Au+Au collisions.
• Data Extraction: The analysis focuses on the freeze-out coordinates of identical charged pions ($\pi^+\pi^+$ and $\pi^-\pi^-$). The three-dimensional spatial separation vector $\vec{\rho}$ is calculated in the Longitudinal Co-Moving System (LCMS) using Bertsch–Pratt coordinates ("out", "side", "long").
• Fitting Procedure: Unlike experimental analyses that measure momentum correlation functions, this simulation directly accesses the pair distance distribution $D(\vec{\rho})$. The one-dimensional projections of this distribution ($D(\rho_{out})$, $D(\rho_{side})$, $D(\rho_{long})$) are fitted simultaneously using three-dimensional Lévy distributions.
• Parameters Extracted: The fit yields the Lévy index $\alpha$, the source radii ($R_{out}, R_{side}, R_{long}$), and the pair multiplicity parameter $\lambda^*$. The parameter $\lambda^*$ is used instead of the standard $\lambda$ because UrQMD does not simulate weak decays of long-lived resonances (e.g., $K^0_S, \eta$), meaning the source consists almost entirely of the "core" component.
• Systematic Uncertainties: Uncertainties were evaluated by varying the relative momentum selection ($Q_{max}^{LCMS}$), the fit range ($\rho_{max}^{fit}$), and the number of merged events ($N_{events}$) used for averaging.

Key Results
The analysis covers the collision energy range $\sqrt{s_{NN}} = 3–27$ GeV and investigates dependencies on transverse mass ($m_T$) and collision energy.

1. Source Radii ($R_{out}, R_{side}, R_{long}$):

   • All three radii exhibit a monotonic decrease with increasing $m_T$, consistent with collective expansion dynamics.
   • Conversely, all radii increase monotonically with collision energy. $R_{long}$ shows the strongest energy dependence, indicating a longer system lifetime at higher energies, while $R_{side}$ shows the weakest dependence.
   • The derived radius difference $R^2_{diff} = R^2_{out} - R^2_{side}$ and the ratio $R_{out}/R_{side}$ both increase monotonically with $\sqrt{s_{NN}}$. This contrasts with EPOS4 model results in the overlapping energy range, where these quantities decrease, highlighting differences in space-time evolution between transport-only and hydrodynamic models.

2. Lévy Index ($\alpha$):

   • The index $\alpha$ decreases monotonically with increasing collision energy.
   • At the lowest energies, $\alpha$ is consistent with a constant; at higher energies, a stronger dependence on $m_T$ is observed.
   • Values of $\alpha$ remain well below 2 across all energies and $m_T$ bins, confirming that Gaussian descriptions are inadequate. The decrease in $\alpha$ suggests an increasing contribution of Lévy walk behavior (anomalous diffusion via scattering and decays) at higher energies and lower $m_T$.

3. Correlation Strength ($\lambda^*$):

   • The parameter $\lambda^*$ remains consistent with a constant value close to unity across all energies. This is expected given the absence of long-lived resonance contributions in the simulation, which would otherwise reduce the intercept below unity.

Significance and Claims
The primary contribution of this work is the establishment of a "hadronic transport baseline" for femtoscopic observables in the RHIC BES-II energy range. Because the UrQMD model operates without a QCD phase transition or critical point dynamics, the resulting smooth, monotonic trends in $\alpha(\sqrt{s_{NN}})$ and source radii represent the expected behavior of a non-critical hadronic system.

The authors claim that these results provide a necessary reference for future comparisons with experimental measurements from the STAR Collaboration. Specifically:

• A statistically significant non-monotonic deviation in the experimental $\alpha(\sqrt{s_{NN}})$ or $R^2_{diff}(\sqrt{s_{NN}})$ from the smooth baselines established here would serve as a strong indicator of dynamics beyond pure hadronic transport, potentially signaling the proximity of the QCD critical point.
• The comparison with EPOS4 results highlights that differences in the energy dependence of $R_{out}/R_{side}$ and $R^2_{diff}$ may reflect the varying influence of hydrodynamic phases versus hadronic rescattering.
• The study underscores that while UrQMD provides a robust baseline for hadronic transport, it does not capture partonic degrees of freedom or critical fluctuations; thus, deviations from this baseline in experimental data are crucial for constraining the QCD phase diagram.