Effects of an induced electric field on the ${\pi }^{-}/{\pi }^{+}$ ratio in heavy-ion collisions

Using an isospin- and momentum-dependent transport model, we examine the effects of an electric field induced by a variable magnetic field on the ${\pi }^{-}/{\pi }^{+}$ ratio in central to peripheral heavy-ion collisions at beam energies of 400 and 1500 MeV/nucleon. It is shown that while the induced electric field does not affect the total multiplicities of both ${\pi }^{-}$ and ${\pi }^{+}$ mesons at both the lower beam energy of 400 MeV/nucleon and the higher beam energy of 1500 MeV/nucleon, it reduces (enhances) the emission of ${\pi }^{-}\left({\pi }^{+}\right)$ mesons in midrapidity, but enhances (reduces) the emission of ${\pi }^{-}\left({\pi }^{+}\right)$ mesons in forward and backward rapidities especially for the more peripheral collisions at the lower beam energy because of the rapidly transient variable magnetic field at more peripheral collisions and the longer reaction duration time at the lower beam energy. These findings indicate that the total ${\pi }^{-}/{\pi }^{+}$ ratio is still a precisely reliable probe of symmetry energy at both the lower and the higher beam energies, but one should consider the induced electric field when using the differential ${\pi }^{-}/{\pi }^{+}$ ratio to probe the symmetry energy especially for the lower beam energy and more peripheral collisions. Finally, the relative suppression factor based on the ratio of ${\pi }^{-}/{\pi }^{+}$ in different rapidities is proposed to be an effective probe of the induced electric field generated in heavy-ion collisions due to its maximizing effects of induced electric fields on the differential ${\pi }^{-}/{\pi }^{+}$ ratio but minimizing effects of some uncertainty factors in heavy-ion collisions.

Figure 1

The beam-energy dependence of the central magnetic field $e{B}_y\left(0\right)$ in Au + Au collisions at the impact parameter of 10 fm (a), and the impact-parameter dependence of the central magnetic field $e{B}_y\left(0\right)$ in Au + Au collisions with a beam energy of 500 MeV/nucleon (b). Data are taken from Ref. [20].

Figure 2

A sketch of the induced electric field ${\vec{E}}_F$ due to the variable magnetic field $\vec{B}$ in the direction perpendicular to the reaction plane. Modified from Fig. 1 of Ref. [22].

Figure 3

The time evolution of the impact-parameter-dependent dynamic ${({\pi }^{-}/{\pi }^{+})}_{\mathrm{like}}$ ratio in Au + Au collisions with (labeled as With) and without (labeled as W/O) the consideration of the induced electric field at two beam energies of 400 MeV/nucleon (left panels) and 1500 MeV/nucleon (right panels).

Figure 4

The impact-parameter-dependent percentage of ${\pi }^{-}$ (a) and ${\pi }^{+}$ (b) multiplicities of midrapidity in Au + Au collisions with (labeled as With) and without (labeled as W/O) the consideration of the induced electric field at a beam energy of 400 MeV/nucleon.

Figure 5

Same as Fig. 4 but for a beam energy of 1500 MeV/nucleon.

Figure 6

The impact-parameter-dependent percentage of ${\pi }^{-}$ (a) and ${\pi }^{+}$ (b) multiplicities of forward and backward rapidities in Au + Au collisions with (labeled as With) and without (labeled as W/O) the consideration of the induced electric field at a beam energy of 400 MeV/nucleon.

Figure 7

Same as Fig. 6 but for a beam energy of 1500 MeV/nucleon.

Figure 8

The rapidity distribution of the ${\pi }^{-}/{\pi }^{+}$ ratio in central to peripheral Au + Au collisions with (labeled as With) and without (labeled as W/O) the consideration of the induced electric field at a beam energy of 400 MeV/nucleon.

Figure 9

The impact-parameter-dependent relative suppression factor $F$ in Au + Au collisions with (labeled as With) and without (labeled as W/O) the consideration of the induced electric field at a beam energy of 400 MeV/nucleon.