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.
Categories: hep-ph