Fermion mass and width in QED in a magnetic field

We revisit the calculation of the fermion self-energy in QED in the presence of a magnetic field. We show that, after carrying out the renormalization procedure and identifying the most general tensor structure for the modified fermion propagator, when working in the large field limit, the mass function develops an imaginary part. This happens when account is made of the sub-leading contributions associated to Landau levels other than the lowest one. The imaginary part is associated to a spectral density describing the spread of the mass function in momentum. The center of the distribution corresponds to the magnetic-field modified mass. The width becomes small as the field intensity increases in such a way that for asymptotically large values of the field, when the separation between Landau levels becomes also large, the mass function describes a stable particle occupying only the lowest Landay level. For large but finite values of the magnetic field, the spectral density represents a finite probability for the fermion to occupy Landau levels other than the LLL.