Neutrino propagation in an electron background with an inhomogeneous magnetic field

Abstract

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 \emph{normal} electron background plus another electron \emph{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 ($\vec{B}$), 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 $\vec{B}$, such as the gradient of $\hat{k}\cdot (\vec{v}\times \vec{B})$, which involve the stream background velocity, and a term of the form $\hat{k}\cdot (\nabla \times \vec{B})$ that can be present in the absence of the stream background, in addition to a term of the form $\hat{k}\cdot \vec{v}$ and the well known term $\hat{k}\cdot \vec{B}$ that arises in the constant $\vec{B}$ 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.

Comments: Title changes, 39 pages, 2 figures