Symmetry potential of the $\mathrm{\Delta }\left(1232\right)$ resonance and its effects on the ${\pi }^{-}/{\pi }^{+}$ ratio in heavy-ion collisions near the pion-production threshold

Effects of the completely unknown symmetry (isovector) potential of the $\mathrm{\Delta }\left(1232\right)$ resonance on the total and differential ${\pi }^{-}/{\pi }^{+}$ ratio in heavy-ion collisions at beam energies from 100 to 1000 MeV/nucleon are explored within an isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model. The effects are found to be negligible at beam energies above the pion-production threshold owing to the very short lifetimes of less than 2 fm/$c$ for $\mathrm{\Delta }$ resonances with masses around $m_{\mathrm{\Delta }}=1232$ MeV, leaving the ${\pi }^{-}/{\pi }^{+}$ ratios of especially the energetic pions still a reliable probe of the high-density behavior of nuclear symmetry energy $E_{\mathrm{sym}}\left(\rho \right)$. However, as the beam energy becomes deeply subthreshold for pion production, effects of the $\mathrm{\Delta }$ symmetry potential become appreciable, especially on the ${\pi }^{-}/{\pi }^{+}$ ratio of low-energy pions from the decays of low-mass $\mathrm{\Delta }$ resonances which have lived long enough to be affected by their mean-field potentials, providing a useful tool to study the symmetry potential and spectroscopy of $\mathrm{\Delta }$ resonances in neutron-rich nuclear matter. Interestingly, though, even at the deeply subthreshold beam energies, the differential ${\pi }^{-}/{\pi }^{+}$ ratio of energetic pions remains sensitive to the $E_{\mathrm{sym}}\left(\rho \right)$ at suprasaturation densities with little influence from the uncertain symmetry potential of the $\mathrm{\Delta }$ resonance.

Figure 1

The density dependence of the potential symmetry energy used in this study.

Figure 2

The mean lifetime (black line) of $\mathrm{\Delta }\left(1232\right)$ resonance as a function of its mass. The dashed line at 1.7 fm/$c$ indicates the lifetime of the $\mathrm{\Delta }$ as its centroid mass of 1232 MeV. The blue lines are the $\mathrm{\Delta }$ mass distributions in $NN\to N\mathrm{\Delta }$ collisions at nucleon-nucleon $\left(NN\right)$ center-of-mass energy of $\sqrt{s}=2.11$ and 2.32 GeV corresponding to first-chance $NN$ collisions in nucleus-nucleus reactions at a beam energy of 500 and 1000 MeV/nucleon, respectively.

Figure 3

The ${({\pi }^{-}/{\pi }^{+})}_{\mathrm{like}}$ ratio as a function of time in head-on $\text{Au}+\text{Au}$ reactions at a beam energy of 100, 200, 500, and 1000 MeV/nucleon with the same $F\left(u\right)=u$ but three different $f_{\mathrm{\Delta }}$ values.

Figure 4

The ${\pi }^{-}/{\pi }^{+}$ ratio as a function of pion kinetic energy in head-on $\text{Au}+\text{Au}$ reactions at a beam energy of 200 MeV/nucleon with the same $F\left(u\right)=u$ but three different isovector potentials for the $\mathrm{\Delta }$ resonance.

Figure 5

The ${({\pi }^{-}/{\pi }^{+})}_{\mathrm{like}}$ ratio as a function of time in head-on $\text{Au}+\text{Au}$ reactions at a beam energy of 200 MeV/nucleon with $f_{\mathrm{\Delta }}=1$ but two different forms for the $F\left(u\right)$.

Figure 6

The ${({\pi }^{-}/{\pi }^{+})}_{\mathrm{like}}$ ratio as a function of pion kinetic energy in head-on $\text{Au}+\text{Au}$ reactions at a beam energy of 200 MeV/nucleon with $f_{\mathrm{\Delta }}=1$ but two different forms for the $F\left(u\right)$.