Kinetic mixing, custodial symmetry and a lower bound on the dark $Z^{\prime }$ mass

Abstract

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 $M_W^2=M_{\tilde{Z}}^2{\cos }^2{\tilde{\theta }}_w$ 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 ${\rho }_0=M_W^2/{\hat{c}}_Z^2{M}_Z^2\hat{\rho }$, 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.

9 pages, notation changed to harmonize it with the PDG one, radiative corrections by SM particles explicitly included

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