Searching for new physics effects in future $W$ mass and $\sin^2\theta_W (Q^2)$ determinations

We investigate the phenomenology of the dark $Z$ boson, $Z_d$, which is associated with a new Abelian gauge symmetry and couples to the standard model particles via kinetic mixing $\varepsilon$ and mass mixing $\varepsilon_Z^{}$. We examine two cases: (i) $Z_d$ is lighter than the $Z$ boson, and (ii) $Z_d$ is heavier than that. In the first case, it is known that $Z_d$ causes a deviation in the weak mixing angle at low energies from the standard model prediction. We study the prediction in the model and compare it with the latest experimental data. In the second case, the $Z$-$Z_d$ mixing enhances the $W$ boson mass. We investigate the effect of $Z_d$ on various electroweak observables including the $W$ boson mass using the $S$, $T$, and $U$ parameters. We point out an interesting feature: in the limit $\varepsilon \to 0$, the equation $S = - U$ holds independently of the mass of $Z_d$ and the size of $\varepsilon_Z^{}$, while $|S|\gg |U|$ in many new physics models. We find that the dark $Z$ boson with a mass of $O(100)~\mathrm{GeV}$ with a relatively large mass mixing can reproduce the CDF result within $2\sigma$ while avoiding all other experimental constraints. Such dark $Z$ bosons are expected to be tested at future high-energy colliders.