Electromagnetic fields with electric and chiral magnetic conductivities in heavy ion collisions

Abstract

We derive an analytic formula for electric and magnetic fields produced by a moving charged particle in a conducting medium with the electric conductivity $\ensuremath{\sigma}$ and the chiral magnetic conductivity ${\ensuremath{\sigma}}_{\ensuremath{\chi}}$. We use the Green's function method and assume that ${\ensuremath{\sigma}}_{\ensuremath{\chi}}$ is much smaller than $\ensuremath{\sigma}$. The compact algebraic expressions for electric and magnetic fields without any integrals are obtained. They recover the Lienard--Wiechert formula at vanishing conductivities. Exact numerical solutions are also found for any values of $\ensuremath{\sigma}$ and ${\ensuremath{\sigma}}_{\ensuremath{\chi}}$ and are compared with analytic results. Both numerical and analytic results agree very well for the scale of high-energy heavy ion collisions. The spacetime profiles of electromagnetic fields in noncentral $\text{Au}+\text{Au}$ collisions have been calculated based on these analytic formula as well as exact numerical solutions.