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Anisotropic flows in heavy-ion collisions at HADES with Skyrme pseudopotential

Xin Li, Si-Pei Wang, Rui Wang, Zhen Zhang, Jie Pu, Chun-Wang Ma, Lie-Wen Chen

TL;DR

The paper investigates proton anisotropic flows in Au+Au collisions at $ tsnn=2.4$ GeV using a lattice BUU framework with the flexible N$5$LO Skyrme pseudopotential to separately vary momentum-dependent mean fields, symmetric-matter stiffness via $K_0$, high-density symmetry energy, and in-medium NN cross sections. By constructing multiple Skyrme-based interactions and running detailed LBUU simulations with realistic centrality sampling, the authors quantify how $v_1$–$v_4$ respond to these inputs and compare to HADES data. They find strong sensitivity to the momentum dependence of the mean field and to $K_0$, while high-density symmetry energy effects are limited and in-medium cross-section corrections mainly affect $v_1$; higher-order flows $v_2$–$v_4$ thus serve as robust probes of the symmetric EOS and momentum-dependent interactions. The work highlights the need for including momentum dependence and higher-order EOS parameters in transport-model analyses and points toward Bayesian frameworks to disentangle competing influences on nuclear matter properties at suprasaturation densities.

Abstract

Within the framework of the lattice Boltzmann-Uehling-Uhlenbeck transport model, we present a systematic study of proton anisotropic flow observables measured by the HADES collaboration, by utilizing the recently developed nuclear effective interaction based on the N$5$LO Skyrme pseudopotential. In particular, we investigate the impacts of the momentum dependence of nucleon mean-field potentials, the stiffness of symmetric nuclear matter (SNM) EOS, the high-density behaviors of the symmetry energy and the in-medium modification of nucleon-nucleon elastic cross sections on proton $v_{1}$, $v_{2}$, $v_{3}$, and $v_{4}$ in Au+Au collisions at $\sqrt{s_{\rm{NN}}} = 2.4\,\text{GeV}$. Our results show that the proton anisotropic flows are strongly sensitive to the momentum dependence of nucleon mean-field potential as well as the incompressibility coefficient $K_0$ of SNM. In addition, the transverse momentum dependence of the proton $v_2$ exhibits a modest sensitivity to the higher-order skewness coefficient $J_0$ and kurtosis coefficient $I_0$ of SNM as well as the momentum dependence of the symmetry potential, while the transverse momentum dependence of the proton $v_1$ is shown to modestly depend on the in-medium modification of nucleon-nucleon elastic cross sections. Moreover, the high-density symmetry energy seems to have limited effects on the proton anisotropic flows. These findings highlight the necessity of considering the momentum dependence of nucleon mean-field potentials including the symmetry potential, the higher-order characteristic parameters of SNM EOS beyond $K_0$, and the in-medium modification of nucleon-nucleon elastic cross sections, in future Bayesian transport model analyses on proton anisotropic flows in heavy-ion collisions at HADES energies, thereby to extract information on nuclear matter EOS as well as the associated underlying nuclear effective interactions.

Anisotropic flows in heavy-ion collisions at HADES with Skyrme pseudopotential

TL;DR

The paper investigates proton anisotropic flows in Au+Au collisions at GeV using a lattice BUU framework with the flexible NLO Skyrme pseudopotential to separately vary momentum-dependent mean fields, symmetric-matter stiffness via , high-density symmetry energy, and in-medium NN cross sections. By constructing multiple Skyrme-based interactions and running detailed LBUU simulations with realistic centrality sampling, the authors quantify how – respond to these inputs and compare to HADES data. They find strong sensitivity to the momentum dependence of the mean field and to , while high-density symmetry energy effects are limited and in-medium cross-section corrections mainly affect ; higher-order flows – thus serve as robust probes of the symmetric EOS and momentum-dependent interactions. The work highlights the need for including momentum dependence and higher-order EOS parameters in transport-model analyses and points toward Bayesian frameworks to disentangle competing influences on nuclear matter properties at suprasaturation densities.

Abstract

Within the framework of the lattice Boltzmann-Uehling-Uhlenbeck transport model, we present a systematic study of proton anisotropic flow observables measured by the HADES collaboration, by utilizing the recently developed nuclear effective interaction based on the NLO Skyrme pseudopotential. In particular, we investigate the impacts of the momentum dependence of nucleon mean-field potentials, the stiffness of symmetric nuclear matter (SNM) EOS, the high-density behaviors of the symmetry energy and the in-medium modification of nucleon-nucleon elastic cross sections on proton , , , and in Au+Au collisions at . Our results show that the proton anisotropic flows are strongly sensitive to the momentum dependence of nucleon mean-field potential as well as the incompressibility coefficient of SNM. In addition, the transverse momentum dependence of the proton exhibits a modest sensitivity to the higher-order skewness coefficient and kurtosis coefficient of SNM as well as the momentum dependence of the symmetry potential, while the transverse momentum dependence of the proton is shown to modestly depend on the in-medium modification of nucleon-nucleon elastic cross sections. Moreover, the high-density symmetry energy seems to have limited effects on the proton anisotropic flows. These findings highlight the necessity of considering the momentum dependence of nucleon mean-field potentials including the symmetry potential, the higher-order characteristic parameters of SNM EOS beyond , and the in-medium modification of nucleon-nucleon elastic cross sections, in future Bayesian transport model analyses on proton anisotropic flows in heavy-ion collisions at HADES energies, thereby to extract information on nuclear matter EOS as well as the associated underlying nuclear effective interactions.

Paper Structure

This paper contains 12 sections, 25 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical framework
  3. Skyrme pseudopotential and nuclear matter equation of state
  4. The lattice BUU transport model
  5. Description of the interactions employed
  6. Anisotropic flows in heavy-ion collisions and centrality selection
  7. Results and discussion
  8. Evolutions of ground state properties and the centre density of collision system
  9. Effects of the momentum-dependent nuclear mean-field potentials
  10. Effects of the stiffness of symmetric nuclear matter EOS
  11. Effects of the symmetry energy and in-medium nucleon-nucleon elastic cross sections
  12. Summary

Figures (10)

  • Figure 1: The total kinetic energy $T_{\rm{c.m.}}$ dependence of the medium correction for nucleon-nucleon elastic cross sections at different local density $\rho$ from Eq. (\ref{['CS-form']}) with $\alpha = 1.8$. The vertical dashed line indicates the $T_{\rm{c.m.}}$ of two scattering nucleons corresponding to the HADES energy ($\sqrt{s_{\rm NN}}=2.4~\mathrm{GeV}$).
  • Figure 2: Panel (a): The kinetic energy dependence of the single-nucleon potential in cold symmetric nuclear matter predicted by the interactions L$45$D$03$, L$45$D$03$FL and L$45$MID. The nucleon optical potential (Schrödinger equivalent potential) in symmetric nuclear matter at saturation density $\rho_{0}$ obtained by Hama $et$$al.$Hama:1990vr is shown as black stars. The black squares show the extrapolation of Hama’s data. The black circles named FL are the optical potential obtained by Feldmeier and Lindner in Ref. Feldmeier:1991ey. Panel (b): Same as Panel (a) but for the density $\rho=2.5\rho_{0}$. Panel (c): The momentum dependence of the symmetry potential $U_{\rm sym}(\rho_{0},p)$ given by the interactions with distinct $U_{\rm sym}(\rho_{0},p)$, namely, L$45$D$03$, L$45$Dm$03$ and L$45$MID. The results from the global optical model analyses Li:2014qtaXu:2010fh are shown in light blue band. Panel (d): Same as Panel (c) but for the density $\rho=2.5\rho_{0}$. Panel (e): The pressure of symmetric nuclear matter ($P_{\text{SNM}}$) as a function of nucleon density given by the interactions: L$45$D$03$, L$45$D$03$K$_{0}380$, L45D$03$J$_{0}$m$800$ and L45D$03$I$_{0}$$4000$. The gray band shows the constrains obtained by Danielewize et al. Danielewicz:2002pu. Panel (f): The density dependence of the symmetry energy ($E_{\rm sym}$) for interactions L45D03 and L35D03.
  • Figure 3: Time evolution of (a) nucleon root-mean-square radius, (b) the fraction of bound nucleons, and (c) binding energy $E_{\rm B}$ of ground state $^{197}$Au with interactions: L$45$D$03$, L$45$D$03$FL, L$45$Dm$03$, L$45$MID, L$35$D$03$, L$45$D$03$K$_{0}380$, L$45$D$03$J$_{0}$m$800$ and L$45$D$03$I$_{0}4000$ up to $1000~{\rm fm}/c$. Calculations are performed with time step $\Delta t = 0.2~{\rm fm}/c$ and $100,000$ test particles.
  • Figure 4: Time evolution of the central baryon density $\rho_{\rm{B}}$ in Au+Au collisions at $\sqrt{s_{\rm NN}}=2.4~\mathrm{GeV}$ and $20$--$30\%$ centrality predicted by the LBUU transport model with interactions (a) L$45$D$03$, L$45$D$03$FL, L$45$Dm$03$ and L$45$MID as well as (b) L$35$D$03$, L$45$D$03$K$_{0}380$, L$45$D$03$J$_{0}$m$800$, L$45$D$03$I$_{0}4000$ and L$45$D$03$ with CS.
  • Figure 5: Directed ($v_1$) (a), elliptic ($v_2$) (b), triangular ($v_3$) (c) and quadrangular ($v_4$) (d) flows as functions of the center-of-mass rapidity $y_{\mathrm{cm}}$ for free protons in Au+Au collisions at $\sqrt{s_{\rm NN}}=2.4 \,\mathrm{GeV}$ (corresponding to $E_\mathrm{beam}$$=$$1.23A~\rm GeV$) predicted by the lattice BUU transport model with interactions L$45$D$03$, L$45$D$03$FL, L$45$Dm$03$ and L$45$MID. The corresponding HADES data HADES:2020lobHADES:2022osk are also included for comparison.
  • ...and 5 more figures