Neutrino propagation in an electron background with an inhomogeneous magnetic field

We study the electromagnetic coupling of a neutrino that propagates in a two-stream electron background medium. Specifically, we calculate the electromagnetic vertex function for a medium that consists of a normal electron background plus another electron stream background that is moving with a velocity four-vector v(mu) relative to the normal background. The results can be used as the basis for studying the neutrino electromagnetic properties and various processes in such a medium. As an application, we calculate the neutrino dispersion relation in the presence of an external magnetic field (B), focused in the case in which B is inhomogeneous, keeping only the terms of the lowest order in 1/m(W)(2) and linear in the B and its gradient. We show that the dispersion relation contains additional anisotropic terms involving the derivatives of B, such as the gradient of (k) over cap . (v x B), which involve the stream background velocity, and a term of the form (k) over cap . (del x B) that can be present in the absence of the stream background, in addition to a term of the form (k) over cap . v and the well known term (k) over cap . B that arises in the constant B case. The derivative-dependent terms are even under a CP transformation. As a result, in contrast to the latter two just mentioned, they depend on the sum of the particle and antiparticle densities and therefore can be non-zero in a CP-symmetric medium in which the particle and antiparticle densities are equal.