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

TL;DR

This paper explores the implications of kinetic mixing and custodial symmetry in models with an extra dark $Z'$ gauge boson, deriving a lower mass bound from electroweak precision data.

Contribution

It derives a relation between the dark $Z'$ mass, mixing angle, and electroweak parameters, establishing a lower mass bound under custodial symmetry assumptions.

Findings

Lower bound on $Z'$ mass: $M_{Z'} > M_Z$ at 94% confidence level.

Relation between mixing angle and physical parameters at tree and loop levels.

Custodial symmetry ensures the validity of the derived bounds in extended models.

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 $M^{2}_{W}=M^{2}_{\tilde Z} \cos^{2} \tilde \theta_{w}$ 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.