  In this work we consider the extension of the standard model by dark fields
with an Abelian $U(1)_{d}$ spontaneously broken gauge symmetry in a hidden dark
matter scenario. Considering all the dimension four gauge invariant terms we
show that the tree-level relation $M^{2}_{W}=M^{2}_{\tilde Z} \cos^{2} \tilde
\theta_{w}$ holds and permits to write the mixing angle induced by the kinetic
mixing in the neutral massive gauge boson sector, $\theta_{\zeta}$, in terms of
the values of $M_{Z}$, the weak mixing angle and of the mass of the physical
dark gauge $Z^{\prime}$ boson. At the loop level, a similar relation is
obtained in the $\overline{MS}$ scheme. Using the result extracted from the
global fit to electroweak precision data for the ratio
$\rho_{0}=M^{2}_{W}/\hat{c}^{2}_{Z} M^{2}_{Z}\hat{\rho}$, we obtain a lower
bound $M_{Z^{\prime}}> M_{Z}$ for the dark $Z^{\prime}$ mass at the $94\%$
confidence level. We argue that this lower bound holds in the general case of
theories for physics beyond the standard model with an extra $U(1)$ gauge
factor subgroup, whenever the extended Higgs potential respects custodial
symmetry.
