The electromagnetic vertex of neutrinos in an electron background and a magnetic field

We study the electromagnetic vertex function of a neutrino that propagates in an electron background in the presence of a static magnetic field. The structure of the vertex function under the stated conditions is determined and it is written down in terms of a minimal and complete set of tensors. The one-loop expressions for all the form factors is given, up to terms that are linear in the magnetic field, and the approximate integral formulas that hold in the long wavelength limit are obtained. We discuss the physical interpretation of some of the form factors and their relation with the concept of the neutrino induced charge. The neutrino acquires a longitudinal and a transverse charge, due to the fact that the form factors depend on the transverse and longitudinal components of the photon momentum independently. We compute those form factors explicitly in various limiting cases and find that the longitudinal and transverse charge are the same for the case of a non-relativistic electron gas, but not otherwise.