Insights into the pion production mechanism and the symmetry energy at high density

The pion production mechanism is explored based on the ultrarelativistic quantum molecular dynamics model (UrQMD) in which the $\mathrm{\ensuremath{\Delta}}$-mass dependence of the $\mathcal{M}$ matrix and $\mathrm{\ensuremath{\Delta}}$-mass dependence of the momentum ${p}_{N\mathrm{\ensuremath{\Delta}}}$ in $N\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}NN$ 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 $NN\ensuremath{\leftrightarrow}N\mathrm{\ensuremath{\Delta}}$ and $\mathrm{\ensuremath{\Delta}}\ensuremath{\leftrightarrow}N\ensuremath{\pi}$, and our calculations show that the sensitivity of ${\ensuremath{\pi}}^{\ensuremath{-}}/{\ensuremath{\pi}}^{+}$ to symmetry energy is weakened after 4--5 $N\text{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}\text{\ensuremath{-}}\ensuremath{\pi}$ loops in the pion production path. The ${\ensuremath{\pi}}^{\ensuremath{-}}/{\ensuremath{\pi}}^{+}$ 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 $S(2{\ensuremath{\rho}}_{0})=48--59$ MeV are deduced, and they are also consistent with the constraint from the ASY-EOS flow data.