Elliptic and quadrangular flow of protons in the high baryon density region

The collective flow provides valuable insights into the anisotropic expansion of particles produced in heavy-ion collisions and is sensitive to the equation of the state of nuclear matter in high-baryon-density regions. In this paper, we use the hadronic transport model SMASH to investigate the elliptic flow ($v_2$), quadrangular flow ($v_4$), and their ratio ($v_{4}/v_{2}^{2}$) in Au+Au collisions at high baryon density. Our results show that the inclusion of baryonic mean-field potential in the model successfully reproduces experimental data from the HADES experiment, indicating that baryonic interactions play an important role in shaping anisotropic flow. In addition to comparing the transverse momentum ($p_T$), rapidity, and centrality dependence of $v_{4}/v_{2}^{2}$ between HADES data and model calculations, we also explore its time evolution and energy dependence across $\sqrt{s_{NN}} =$ 2.4 to 4.5 GeV. While the ratio $v_{4}/v_{2}^{2}$ for high-$p_{T}$ particles approaches 0.5, which aligns with expectations from hydrodynamic behavior, we emphasize that this result primarily reflects agreement with the HADES measurements rather than a definitive indication of ideal fluid behavior. These findings contribute to understanding the early-stage dynamics in heavy-ion collisions at high baryon density.