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

We derive analytic formula for electric and magnetic fields produced by a moving charged particle in a conducting medium with the electric conductivity $σ$ and the chiral magnetic conductivity $σ_χ$. We use the Green function method and assume that $σ_χ$ is much smaller than $σ$. 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 $σ$ and $σ_χ$ and are compared to analytic results. Both numerical and analytic results agree very well for the scale of high energy heavy ion collisions. The space-time profiles of electromagnetic fields in non-central Au+Au collisions have been calculated based on these analytic formula as well as exact numerical solutions.