Kinetic mixing, custodial symmetry, and a lower bound on the mass of a dark gauge boson

Abstract

We consider the extension of the standard model by dark fields with an Abelian spontaneously broken gauge symmetry in a hidden dark matter scenario. The dimension-four gauge-invariant terms include a kinetic mixing term and a Higgs mixing term, and we show that, after spontaneous symmetry breaking, the tree-level relation $M^{2}_{W}=M^{2}_{\tilde{Z}} \cos ^{2} \tilde{\theta }_{w}$ holds and permits us to write the mixing angle induced by the kinetic mixing in the neutral massive gauge boson sector, θ_ζ, in terms of the values of M_Z, the weak mixing angle, and of the mass of the physical dark gauge boson Z_D. 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 the lower bound $M_{Z_{D}}\gt M_{Z}$ for the dark gauge boson 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.