Particle dispersion in the classical vector dark matter background

Interactions with a background medium modify in general the dispersion relation and canonical normalization of propagating particles. This can have an important phenomenological consequence when considering light dark matter coupling to quarks and leptons. In this paper, we address this issue in the vector dark matter background with the randomly distributed polarizations or a fixed polarization to the single direction. The observations associated with particle dispersion can give constraints on new light Abelian gauge boson models. Considering the solar neutrino transition and the electron mass measurement, stringent bounds can be put on the gauged $L_\mu - L_\tau$ model and the dark photon model. Moreover, the classical vector field turns out to induce drastic changes in the particle normalization, which rule out a significant parameter region of the generic vector dark matter model.