Bayesian Analyses of Proton Multiple Flow Components in Intermediate Heavy Ion Collisions with Momentum-Dependent Interactions

TL;DR

This study tackles the challenge of disentangling the dense-matter equation of state from in-medium nucleon-nucleon interactions in heavy-ion collisions. By applying a Bayesian framework with a Gaussian Process emulator to the momentum-dependent IBUU transport model and constraining it with HADES proton-flow observables from Au+Au at $1.23$ GeV/nucleon, the authors jointly infer the incompressibility $K_0$ and the in-medium cross-section factor $X$. The results favor a soft to moderately soft EoS with $K_0 \\lesssim 250$ MeV and an in-medium cross-section suppression around $X \\sim 0.9$–$1.0$, with momentum dependence reducing the need for large cross-section modifications compared to momentum-independent analyses. The work demonstrates the critical role of momentum-dependent mean fields in interpreting heavy-ion data and provides a framework for incorporating more microscopic, density- and momentum-dependent physics and additional observables across energies to further constrain dense-matter properties.

Abstract

We perform a comprehensive Bayesian analyses of Au + Au collision data at 1.23 GeV/nucleon using an isospin-dependent Boltzmann-Uehling-Uhlenbeck transport model that incorporates a momentum-dependent mean field and medium-modified baryon-baryon cross sections. The model parameters are calibrated to empirical properties of nuclear matter at saturation density, with particular attention to variations in the incompressibility $K_0$. Within a Bayesian statistical framework and using a Gaussian Process emulator, we simultaneously extract constraints on the incompressibility $K_0$ and the in-medium baryon-baryon scattering modification factor $X$ by systematically comparing model predictions with HADES measurements of proton collective flow, including the slopes ($F_1$ and $F_3$) of directed and triangular flow, as well as elliptic ($v_2$) and quadrupole ($v_4$) flow observables. We find that the extracted incompressibility favors relatively small values, indicating a soft nuclear equation of state, while the inferred average $X$ values fall at $0.9$-$1.0$, suggesting mild suppression of baryon-baryon cross sections in the medium. Furthermore, we demonstrate that transport models employing momentum-independent mean fields require stiffer equations of state and stronger in-medium corrections to reproduce the same observables. These results highlight the critical role of momentum dependence in the mean field and its interplay with in-medium scattering in constraining the properties of dense nuclear matter from heavy-ion collisions.

Figures (5)

• Figure 1: Time evolution of central density in Au + Au collisions at 1.23 GeV for two centrality $10$--$20\%$ and $20$--$30\%$.
• Figure 2: Predicted slope $F_1$ and $F_3$ of the directed flow $v_1$ and triangular flow $v_3$ at mid-rapidity, as well as the elliptic flow $v2$ and quadrupole flow $v_4$ for the free proton, respectively, as functions of the incompressibility $K_0$ and in-medium cross section modification factor $X$ generated by using the IBUU transport model for the Au + Au reactions for two centrality $10$--$20\%$ (blue) and $20$--$30\%$ (red). The dashed line is experimental data from the HADES Collaboration 2023EPJA...59...80A.
• Figure 3: Validation of Gaussian Process (GP) emulations against IBUU predictions for $F_1$, $v_2$, $F_3$, and $v_4$ in mid-central Au+Au collisions. Perfect agreement corresponds to points lying on the $y = x$ line.
• Figure 4: Posterior PDFs of parameters $X$ and $K_0$, from the collective flow observables $F_1$, $v_2$, $F_3$, and $v_4$, for the centrality $10$--$20\%$ (upper panels) and $20$--$30\%$ (lower panels) of Au + Au collisions, respectively.
• Figure 5: Posterior PDFs of $X$ and $K_0$ as well as their correlation, for the centrality $10$--$20\%$ (left panels) and $20$--$30\%$ (right panels) of Au + Au collisions, respectively.