No net charge separation in hot QCD in a magnetic field

We study the realization of axion electrodynamics in QCD in the presence of a background magnetic field at temperatures high enough for the occurrence of topological charge transitions that are reflected in the presence of a θ -vacuum term in the action. We show that in this system, the Maxwell equations contain two equal and opposite electric currents that are proportional to the time derivative of the axion field θ . One of these currents comes directly from the Abelian chiral anomaly term in the action and can be interpreted as a polarization current due to the magnetoelectricity of the system with C P -broken symmetry. The other current is obtained from the regular tadpole diagrams and can be understood as produced by the medium chiral imbalance and the single spin projection of the quarks in the lowest Landau level. Since the two currents cancel out, the net electric charge separation along the magnetic field, a phenomenon known as the chiral magnetic effect, does not take place. Thus, if chirally imbalanced QCD is correctly represented by the model under consideration, then the chiral magnetic effect is absent. We discuss the similarities and differences with Weyl semimetals in a magnetic field.