Low-energy theorem for electron hyperfine interactions, and the special case of magnetic poles

Abstract

The $S$-wave scattering length for a Dirac electron passing through a nonsingular magnetic field is shown to be given exactly by lowest-order perturbation theory. (Fermi contact term). In the presence of a pair of Dirac magnetic poles, the result is still correct in the limit of pole separation small compared to the electron Compton wavelength. In the limit of large pole separation, the scattering length takes the form appropriate to the case of a small magnet (whose internal structure is irrelevant) probing the field of the pole pair. In units of the magnetic dipole moment, the effect is half as big and opposite in sign to that in the small-separation limit. The passage from one limit to the other is qualitatively determined, and depends on boundary conditions at the poles. All of these statements depend on the neglect of electric fields, and of higher-order quantum-electrodynamics corrections like the anomalous magnetic moment of the electron.