The pion production mechanism is explored based on the ultrarelativistic quantum molecular dynamics model (UrQMD) in which the $\mathrm{\Delta }$-mass dependence of the $\mathcal{M}$ matrix and $\mathrm{\Delta }$-mass dependence of the momentum $p_{N\mathrm{\Delta }}$ in $N\mathrm{\Delta }\to NN$ cross sections are taken into account. By analyzing the time evolution of the pion production rate and the density in the overlapped region for the reaction of Au+Au at the beam energy of $0.4A$ GeV, we find that characteristic density of pion observable is in the region of 1–2 times normal density. The process of pion production in the reaction is tracked, including the loops of $NN\leftrightarrow N\mathrm{\Delta }$ and $\mathrm{\Delta }\leftrightarrow N\pi$, and our calculations show that the sensitivity of ${\pi }^{-}/{\pi }^{+}$ to symmetry energy is weakened after 4–5 $N\text{−}\mathrm{\Delta }\text{−}\pi$ loops in the pion production path. The ${\pi }^{-}/{\pi }^{+}$ ratio in the reaction near the threshold energies retains its sensitivity to the symmetry energy, and it is insensitive to the nuclear incompressibility $K_0$ and effective mass when their values are selected in the commonly accepted range. By comparing the UrQMD calculations to the FOPI data at $0.4A$ GeV and considering the constraint of symmetry energy from neutron star properties, the slope of symmetry energy $L=54–91$ MeV and the symmetry energy at two times normal density $S\left(2{\rho }_0\right)=48–59$ MeV are deduced, and they are also consistent with the constraint from the ASY-EOS flow data.

• Received 16 March 2020
• Revised 24 June 2020
• Accepted 7 January 2021

DOI:https://doi.org/10.1103/PhysRevC.103.014616