Beam-energy dependence of the relativistic retardation effects of electrical fields on the ${\pi }^{-}/{\pi }^{+}$ ratio in heavy-ion collisions

In this article we investigate the beam-energy dependence of relativistic retardation effects of electrical fields on the single and double ${\pi }^{-}/{\pi }^{+}$ ratios in three heavy-ion reactions with an isospin- and momentum-dependent transport model IBUU11. With the beam energy increasing from 200 to 400 MeV/nucleon, effects of the relativistically retarded electrical fields on the ${\pi }^{-}/{\pi }^{+}$ ratio are found to increase gradually from negligible to considerably significant as expected; it is, however, an interesting observation that the relativistic retardation effects of electrical fields on the ${\pi }^{-}/{\pi }^{+}$ ratio become gradually insignificant as the beam energy further increases from 400 to 800 MeV/nucleon. That is to say, with the beam energy increasing, the competition of enhanced anisotropic features of retarded electrical fields and reduced duration time of the reactions gets effects of the relativistically retarded electrical fields on the ${\pi }^{-}/{\pi }^{+}$ ratio to be maximum around 400 MeV/nucleon. Therefore, the relativistic retardation effects of electrical fields should be carefully considered in heavy-ion collisions at intermediate energy especially around 400 MeV/nucleon when using the ${\pi }^{-}/{\pi }^{+}$ ratio as the probe of nuclear symmetry energy. Moreover, we also investigate the isospin dependence of relativistic retardation effects of electrical fields on the ${\pi }^{-}/{\pi }^{+}$ ratio in two isobar reaction systems of ${}^{96}\mathrm{Ru}+{}^{96}\mathrm{Ru}$ and ${}^{96}\mathrm{Zr}+{}^{96}\mathrm{Zr}$ at the beam energies from 200 to 800 MeV/nucleon. It is shown that the relativistic retardation effects of electrical fields on the ${\pi }^{-}/{\pi }^{+}$ ratio are independent of the isospin of reaction. Furthermore, we also examine the double ${\pi }^{-}/{\pi }^{+}$ ratio in reactions of ${}^{96}\mathrm{Zr}+{}^{96}\mathrm{Zr}$ over ${}^{96}\mathrm{Ru}+{}^{96}\mathrm{Ru}$ at the beam energies from 200 to 800 MeV/nucleon with the static field and retarded field, respectively. It is shown that the double ${\pi }^{-}/{\pi }^{+}$ ratio from two reactions is still an effective observable of symmetry energy without the interference of electrical field due to using the relativistic calculation compared to the nonrelativistic calculation.

Figure 1

Contours of the electric fields in the $X\text{−}o\text{−}Z$ reaction plane (upper) and $X\text{−}o\text{−}Y$ plane (lower) at the maximum compression stage in central $\text{Au}+\text{Au}$ collisions at the beam energies of 200, 400, and 800 MeV/nucleon. Panels (a)–(c) are for the static fields while panels (d)–(f) are for the retarded fields.

Figure 2

Upper: The final ${\pi }^{-}/{\pi }^{+}$ ratio in central $\text{Au}+\text{Au}$ collisions as a function of beam energy with three symmetry energy settings from the hard $x=-1$ to soft $x=1$. Lower: The relative effects of relativistically retarded electrical fields on the final ${\pi }^{-}/{\pi }^{+}$ ratio shown in the same reactions.

Figure 3

Evolution of dynamical multiplicities of ${\pi }^{+}$ (i.e., ${\pi }^{+}+{\mathrm{\Delta }}^{++}+\frac{1}{3}{\mathrm{\Delta }}^{+}$), ${\mathrm{\Delta }}^{++}$, and ${\mathrm{\Delta }}^{+}$ resonances in central $\text{Au}+\text{Au}$ collisions at the beam energies of 200, 400, and 800 MeV/nucleon, respectively.

Figure 4

Evolution of dynamical multiplicities of ${\pi }^{-}$ (i.e., ${\pi }^{-}+{\mathrm{\Delta }}^{-}+\frac{1}{3}{\mathrm{\Delta }}^0$), ${\mathrm{\Delta }}^{-}$, and ${\mathrm{\Delta }}^0$ resonances in central $\text{Au}+\text{Au}$ collisions at the beam energies of 200, 400, and 800 MeV/nucleon, respectively.

Figure 5

The ratio $R^n$ for neutrons and $R^p$ for protons at suprasaturation densities at the maximum compression stage as a function of nucleon kinetic energy in central $\text{Au}+\text{Au}$ collisions at the beam energies of 200, 400, and 800 MeV/nucleon, respectively.

Figure 6

The final ${\pi }^{-}/{\pi }^{+}$ ratio in two isobar reaction systems of ${}^{96}\mathrm{Ru}+{}^{96}\mathrm{Ru}$ (a) and ${}^{96}\mathrm{Zr}+{}^{96}\mathrm{Zr}$ (b) at the beam energies from 200 to 800 MeV/nucleon. The hard symmetry energy with parameter $x=-1$ and the soft one with parameter $x=1$ are used.

Figure 7

The double ${\pi }^{-}/{\pi }^{+}$ ratio [i.e., $\mathrm{DR}({\pi }^{-}/{\pi }^{+}$)] of reactions ${}^{96}\mathrm{Zr}+{}^{96}\mathrm{Zr}$ over ${}^{96}\mathrm{Ru}+{}^{96}\mathrm{Ru}$ at the beam energies from 200 to 800 MeV/nucleon with the static field and retarded field, respectively.