  Dark Matter models that employ a vector portal to a dark sector are usually
treated as an effective theory that incorporates kinetic mixing of the photon
with a new U(1) gauge boson, with the $Z$ boson integrated out. However, a more
complete theory must employ the full SU(2)$_L\times $U(1)$_Y \times
$U(1)$_{Y^\prime}$ gauge group, in which kinetic mixing of the $Z$ boson with
the new U(1) gauge boson is taken into account. The importance of the more
complete analysis is demonstrated by an example where the parameter space of
the effective theory that yields the observed dark matter relic density is in
conflict with a suitably defined electroweak $\rho$-parameter that is deduced
from a global fit to $Z$ physics data.
