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

We derive an analytic formula for electric and magnetic fields produced by a moving charged particle in a conducting medium with the electric conductivity $\sigma$ and the chiral magnetic conductivity ${\sigma }_{\chi }$. We use the Green's function method and assume that ${\sigma }_{\chi }$ is much smaller than $\sigma$. 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 $\sigma$ and ${\sigma }_{\chi }$ 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.

Figure 1

The electromagnetic fields at $\mathbf{x}=(0,0,0)$ fm produced by a point charge (proton) of 100 GeV which are located at $\mathbf{x}=(6,0,0)$ fm and moving along ${\mathbf{e}}_z$. We choose the following values of conductivities: $\sigma =5.8$ MeV and ${\sigma }_{\chi }=1.5$ MeV.

Figure 2

The geometry of two colliding nuclei in the transverse plane at $z=0$. One nucleus at $(b/2,0)$ in the transverse plane is moving along ${\mathbf{e}}_z$, while another nucleus at $(-b/2,0)$ is moving along $-{\mathbf{e}}_z$. The points $P$ and $Q$ in the transverse plane are two typical points at which $\mathbf{B}$ and $\mathbf{E}$ will be calculated.

Figure 3

Time evolution of $B_y$ and $E_y$ in $\text{Au}+\text{Au}$ collisions at $\sqrt{s}=200$ GeV and $\mathbf{x}=(0,6,0)$ fm for three cases: (a) Lienard–Wiechert potential ($\sigma ={\sigma }_{\chi }=0$); (b) with $\sigma$ ($\sigma \ne 0$ and ${\sigma }_{\chi }=0$); (c) with $\sigma$ and ${\sigma }_{\chi }$ ($\sigma \ne 0$ and ${\sigma }_{\chi }\ne 0$). The $x$ and $z$ components are vanishing, $B_{x,z}\approx 0$ and $E_{x,z}\approx 0$.

Figure 4

The time evolution of $\mathbf{B}$ and $\mathbf{E}$ in $\text{Au}+\text{Au}$ collisions at $\sqrt{s}=200$ GeV and $\mathbf{x}=(6,0,0)$ fm for three cases: (a) Lienard–Wiechert potential (L-W, $\sigma ={\sigma }_{\chi }=0$); (b) with $\sigma$ ($\sigma \ne 0$ and ${\sigma }_{\chi }=0$); (c) with $\sigma$ and ${\sigma }_{\chi }$ ($\sigma \ne 0$ and ${\sigma }_{\chi }\ne 0$).

Figure 5

Contour plots for electric (upper panel) and magnetic (lower panel) fields in the transverse plane of $z=0$ at $t=2$ fm/$c$ and $\sqrt{s}=200$ GeV in $\text{Au}+\text{Au}$ collisions. The two colliding nuclei are shown as two red dashed circles.

Figure 6

The two-dimensional vector fields for transverse components ${\mathbf{B}}_T$ and ${\mathbf{E}}_T$ in the transverse plane of $z=0$ at $t=2$ fm/$c$ and $\sqrt{s}=200$ GeV in $\text{Au}+\text{Au}$ collisions.

Figure 7

Illustration of the asymmetry of the magnetic fields at nonvanishing ${\sigma }_{\chi }$. Two positive point charges at $(\pm a,0,0)$ move in the $\pm {\mathbf{e}}_z$ directions: (a) azimuthal components, (b) radial components. The azimuthal components are symmetric at symmetric points $(0,\pm b,0)$, while the radial components have opposite signs.