Charge-dependent directed flows in heavy-ion collisions by Boltzmann-Maxwell equations

We have calculated the directed flow $v_1$ and charge-dependent directed flow $\mathrm{\Delta }{v}_1$ for pions and protons in Au+Au collisions at $\sqrt{s_{NN}}=200$GeV by solving the coupled Boltzmann-Maxwell equations self-consistently. Our numerical results show that $v_1$ for pions and protons are all negative in the positive mid rapidity region and have similar behavior and magnitude. In contrast we find a quite different behavior in $\mathrm{\Delta }{v}_1$ for pions and protons. The difference lies in that $\Delta v_{1}$ for protons mainly comes from pressure gradients of the medium, while the dominant contribution to $\mathrm{\Delta }{v}_1$ for pions is from electromagnetic fields. Our results indicate that the effect of the electric field will slightly exceed that of the magnetic and lead to a small negative slope of $\mathrm{\Delta }{v}_1$ for pions