Kinetic mixing, custodial symmetry and a lower bound on the dark Z^(prime) mass

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.