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Investigating the $Z^\prime$ gauge boson at the future lepton colliders

$Z^\prime$ boson as a new gauge boson has been proposed in many new physics models. The interactions of $Z^\prime$ coupling to fermions are detailed studied at the large hadron collider. A $Z^\prime$ with the mass of a few TeV has been excluded in some special models. The future lepton colliders will focus on the studies of Higgs physics which provide the advantage to investigate the interactions of Higgs boson with the new gauge bosons. We investigate the $Z^\prime ZH$ interaction via the process of $e^+e^- \to Z^\prime/Z \to ZH \to l^+l^- b \bar{b}$. The angular distribution of the final leptons decaying from the $Z$-boson is related to the mixing of $Z^\prime$-$Z$ and the mass of $Z^\prime$. The forward-backward asymmetry has been proposed as an observable to investigate the $Z^\prime$-$Z$ mixing. The angular distributions change significantly with some special beam polarization comparing to the unpolarized condition.