Electroweak Precision Measurements of a Nearly-Degenerate $Z^{\mathrm{\prime }}$-$Z$ System

In this paper, we discuss the possibility to probe a nearly-degenerate $Z^{\mathrm{\prime }}$-$Z$ system by analyzing the $Z$-lineshape at an electron-positron collider. Compared with the usual $Z^{\mathrm{\prime }}$ in the literature well separated with the standard model (SM) $Z$ boson in mass, the nearly-degenerate $Z^{\mathrm{\prime }}$-$Z$ mixing affects the observed effective oblique parameters'' $\tilde{S}$, $\tilde{T}$, $\tilde{U}$, and the effective deviation of number of neutrino species’’ $\delta {\tilde{N}}_{\nu }$ in a more complicated way and cannot be simply computed perturbatively up to a particular order. Aiming at solving this problem, we write down a general simplified effective Lagrangian and enumerate some parameter spaces corresponding to some typical models, and suggest a method to extract the constraints by looking into the line-shape of the $Z$-like resonance at an electron-positron collider.