Induced electric current along the external magnetic field in electroweak matter

I start with the motivation for the study of an electric current flowing along the external magnetic field, which consists in the instability of this field. Then I consider one of the implementations of such a phenomenon, which is the chiral magnetic effect (CME) in the presence of the electroweakly interacting background matter. I discuss the derivation the current of massless particles using the relativistic quantum mechanics, which is based on the exact solution of the Dirac equation accounting for external fields. The qualitative derivation of the CME is also presented. Then the contribution to the current induced by the background matter is obtained by calculating the antisymmetric part of the polarization tensor in relativistic plasma. I also consider the situation of the parity violating self-interaction between particles in plasma. Finally, I discuss the new effect consisting in the appearance of the current along the magnetic field in the system of massive charged particles, which possess anomalous magnetic moments and electroweakly interact with background matter.