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

Abstract

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 negligibly to considerably significant as expectedly; it is however, the interesting observation is the relativistic retardation effects of electrical fields on the ${\pi }^{-}/{\pi }^{+}$ ratio are becoming gradually insignificant as the beam energy further increasing from 400 to 800 MeV/nucleon. 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 ${}^{96}$Ru+${}^{96}$Ru and ${}^{96}$Zr+${}^{96}$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 ${}^{96}$Zr+${}^{96}$Zr over ${}^{96}$Ru+${}^{96}$Ru at the beam energies from 200 to 800 MeV/nucleon with the static field and retarded field, respectively. It is shown 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.

Comments: 8 pages, 7 figures. Abbreviated abstract to meet the criterion of arXiv platform. Accepted for publication in Physical Review C. An extended but necessary complementary study to arXiv:1709.09127