Energy conservation and the chiral magnetic effect

We analyze the chiral magnetic effect in a homogeneous neutral plasma from the point of view of energy conservation, and construct an effective potential for the growth of maximally helical perturbations of the electromagnetic field. We show that a negative curvature at the origin of the potential, indicating instability of the plasma, is induced by a chiral asymmetry in electron Fermi energy, as opposed to number density, while the potential grows at large field value. It follows that the ground state for a plasma has zero magnetic helicity; a nonzero electron mass will allow an excited state of a plasma with nonzero helicity to relax to that ground state quickly. We conclude that a chiral plasma instability triggered by weak interactions is not a viable mechanism for explaining magnetic fields in stars except possibly when dynamics drives the system far from equilibrium.

Figure 1

The electron energy spectrum at zero temperature in a magnetic field $\overrightarrow{B}=B\widehat{z}$, plotted versus $p_z$ for positive (blue) and negative (red) chirality states. The single particle energy levels are shifted vertically by ${\varphi }_{\pm }$ respectively in the presence of a background potential due to neutral current interactions with the ambient nucleons. Thick lines indicate occupied states, which are filled to Fermi energies ${\mu }_{\pm }$.

Figure 2

The potential $V({\mathcal{A}}_k)$ in Eq. (28) governing the time evolution of ${\mathcal{A}}_k$ for a particular set of initial conditions and $k=k_{\star }$, defined in Eq. (29).