Hard thermal loops in a magnetic field and the chiral anomaly

The fermionic dispersion relation in the presence of a background magnetic field and a high temperature QED plasma is calculated exactly in the external field, using the Hard Thermal Loop effective action. As the field strength increases there is a smooth transition from the weak-field ($qB\ll q^2T^2$) thermal dispersion relations to the vacuum Landau levels when the background field is much stronger than any thermal effects ($qB\gg q^2T^2$). The self-energy at finite field strength acquires an imaginary part. The spectral width becomes important for critical field strengths ($qB \sim q^2T^2$), necessitating the use of the full spectral function. It is shown that the spectral function satisfies the usual condition of normalization and causality. Using the exact spectral function I also show that the production of chirality in an external electromagnetic field at high temperature is unaffected by the presence of the thermal masses of the fermions.