Effects of retarded electrical fields on observables sensitive to the high-density behavior of the nuclear symmetry energy in heavy-ion collisions at intermediate energies

Within the isospin- and momentum-dependent transport model IBUU11, we examine the relativistic retardation effects of electrical fields on the ${\pi }^{-}/{\pi }^{+}$ ratio and neutron-proton differential transverse flow in heavy-ion collisions at intermediate energies. Compared to the static Coulomb fields, the retarded electric fields of fast-moving charges are known to be anisotropic and the associated relativistic corrections can be significant. They are found to increase the number of energetic protons in the participant region at the maximum compression by as much as 25% but that of energetic neutrons by less than 10% in ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ reactions at a beam energy of 400 MeV/nucleon. Consequently, more ${\pi }^{+}$ and relatively fewer ${\pi }^{-}$ mesons are produced, leading to an appreciable reduction of the ${\pi }^{-}/{\pi }^{+}$ ratio compared to calculations with the static Coulomb fields. Also, the neutron-proton differential transverse flow, as another sensitive probe of high-density symmetry energy, is also decreased appreciably due to the stronger retarded electrical fields in directions perpendicular to the velocities of fast-moving charges compared to calculations using the isotropic static electrical fields. Moreover, the retardation effects on these observables are found to be approximately independent of the reaction impact parameter.

Figure 1

The density dependence of nuclear symmetry energy $E_{\mathrm{sym}}\left(\rho \right)$.

Figure 2

Contours of the electric fields in the $X$-o-$Z$ reaction plane (upper) and $X$-o-$Y$ plane (lower) at the initial compression ($t=10$ fm/$c$), maximum compression ($t=20$ fm/$c$) and expansion stage ($t=30$ fm/$c$) in central Au+Au collisions at a beam energy of 400 MeV/nucleon, respectively. The panels (a), (b), and (c) are for the static fields while the panels (d), (e), and (f) are for the retarded fields.

Figure 3

Upper: Contours of the $X$ and $Z$ components of the electric fields in the $X$-o-$Z$ plane at the maximum compression ($t=20$ fm/$c$) in central Au+Au collisions at a beam energy of 400 MeV/nucleon, respectively. Lower: Contours of the $X$ and $Y$ components of the electric fields in the $X$-o-$Y$ plane. The panels (a) and (c) are for the static fields while the panels (b) and (d) are for the retarded fields.

Figure 4

Density contours in the $X$-o-$Z$ plane (upper) and $X$-o-$Y$ plane (lower) at the initial compression ($t=10$ fm/$c$), maximum compression ($t=20$ fm/$c$), and expansion stage ($t=30$ fm/$c$) in central Au+Au collisions at a beam energy of 400 MeV/nucleon, respectively. The panels (a), (b), and (c) are for the static fields while the panels (d), (e), and (f) are for the retarded fields.

Figure 5

Time evolution of the (${\pi }^{-}/{\pi }^{+}$ )${}_{\mathrm{like}}$ ratio in central ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at 400 MeV/nucleon with symmetry energies ranging from superhard of $x=-2$ to supersoft of $x=2$. The panels (a) and (b) are results of calculations using the static and retarded electric fields, respectively.

Figure 6

Upper: The final ${\pi }^{-}/{\pi }^{+}$ ratio versus the symmetry energy parameter $x$ for the results shown in Fig. 5. Lower: Multiplicities of ${\pi }^{-}$ (a) and ${\pi }^{+}$ (b) in the same reactions.

Figure 7

The ratio $R^n$ (left) and $R^p$ (right) for neutrons (a) and protons (b), respectively, at supra-saturation densities at the maximum compression stage ($t=20$ fm/$c$) as a function of nucleon kinetic energy in central Au+Au collisions at a beam energy of 400 MeV/nucleon, respectively.

Figure 8

The impact parameter dependence of ${\pi }^{-}/{\pi }^{+}$ ratio in Au+Au collisions at 400 MeV/nucleon with different symmetry energy settings from superhard of $x=-2$ to supersoft of $x=2$. The impact parameters are set as $b/b_{\max }\le 0.15$, $b=5$ fm, and $b=7.2$ fm for central, midcentral, and peripheral collisions, respectively.

Figure 9

The neutron-proton differential flow in central (upper) and peripheral (lower) collisions for the same reactions and conditions as in Fig. 8.