Neutrino dispersion relation in a magnetized multi-stream matter background

We study the propagation of a neutrino in a medium that consists of two or more thermal backgrounds of electrons and nucleons moving with some relative velocity, in the presence of a static and homogeneous electromagnetic field. We calculate the neutrino self-energy and dispersion relation using the linear thermal Schwinger propagator, we give the formulas for the dispersion relation and discuss general features of the results obtained, in particular the effects of the stream contributions. As a specific example we discuss in some detail the case of a magnetized two-stream electron, i.e., two electron backgrounds with a relative velocity $\vec v$ in the presence of a magnetic field. For a neutrino propagating with momentum $\vec k$, in the presence of the stream the neutrino dispersion relation acquires an anisotropic contribution of the form $\hat k\cdot\vec v$ in addition to the well known term $\hat k\cdot\vec B$, as well as an additional contribution proportional to $\vec B\cdot\vec v$. We consider the contribution from a nucleon stream background as an example of other possible stream backgrounds, and comment on possible generalizations to take into account the effects of inhomogeneous fields. We explain why a term of the form $\hat k\cdot(\vec v\times\vec B)$ does not appear in the dispersion relation in the constant field case, while a term of similar form can appear in the presence of an inhomogeneous field involving its gradient.