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We study the electromagnetic coupling of a neutrino that propagates in atwo-stream electron background medium. Specifically, we calculate theelectromagnetic vertex function for a medium that consists of a \emph{normal}electron background plus another electron \emph{stream} background that ismoving with a velocity four-vector $v^\mu$ relative to the normal background.The results can be used as the basis for studying the neutrino electromagneticproperties and various processes in such a medium. As an application, wecalculate the neutrino dispersion relation in the presence of an externalmagnetic field ($\vec B$), focused in the case in which $B$ is inhomogeneous,keeping only the terms of the lowest order in $1/m^2_W$ and linear in the $B$and its gradient. We show that the dispersion relation contains additionalanisotropic terms involving the derivatives of $\vec B$, such as the gradientof $\hat k\cdot(\vec v\times\vec B)$, which involve the stream backgroundvelocity, and a term of the form $\hat k\cdot(\nabla\times \vec B)$ that can bepresent in the absence of the stream background, in addition to a term of theform $\hat k\cdot\vec v$ and the well known term $\hat k\cdot\vec B$ thatarises in the constant $\vec B$ case. The derivative-dependent terms are evenunder a $CP$ transformation. As a result, in contrast to the latter two justmentioned, they depend on the sum of the particle and antiparticle densitiesand therefore can be non-zero in a $CP$-symmetric medium in which the particleand antiparticle densities are equal.

Neutrino propagation in an electron background with an inhomogeneous magnetic field