  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.
