QED vertex and anomalous magnetic moment in the presence of a magnetic field

Abstract

We compute the fermion-photon vertex in QED in the presence of a constant and uniform magnetic background up to one-loop order. We show that even at tree-level, the vertex is modified due to the loss of Lorentz invariance induced by the magnetic field, thus breaking into longitudinal and transverse pieces. Moreover, the radiative corrections induce the emergence of a rich tensor structure that includes the anomalous magnetic moments in the transverse, parallel, and mixed transverse/parallel directions. We concentrate on studying one of these anomalous magnetic moment components, the one in the purely transverse direction. We find the selection rules for transitions between a few low-lying Landau levels and show that the amplitudes for transitions from an initial to a final Landau level differ by a sign from the reverse process due to the loss of time reversal invariance induced by the presence of the field. Contrary to the vacuum case, the amplitudes are, in general, complex, and the phase factor can be interpreted in terms of a finite life-time of the decaying state. For the anomalous magnetic moment in the purely transverse direction, transitions between states occupying both the lowest Landau levels are forbidden. Moreover, for the computation of the allowed transitions, we find that it is not necessary to include a photon mass since the magnetic field acts as an infrared regulator.