One-loop expressions for $H^{\pm} \rightarrow W^{\pm} Z$ and their implications at muon--TeV colliders

One-loop contributions for decay process $H^{\pm} \rightarrow W^{\pm}Z$ within the Two-Higgs-Doublet Model are computed in the general $\mathcal{R}_{\xi}$ gauge, and its phenomenological applications at future muon--TeV colliders are studied in this paper. The analytic results are confirmed by several consistency tests, for example, the $\xi$-independence, the renormalization-scale stability and the ultraviolet finiteness of the one-loop amplitude. We first perform an updated parameter scan of the Type-X THDM in the phenomenological studies. The production of charged Higgs boson pairs at future muon--TeV colliders is investigated through the two processes $\mu^+\mu^- \rightarrow H^+H^- \rightarrow W^{\pm}W^{\mp}Zh$ and $\mu^+\mu^- \rightarrow \gamma\gamma \rightarrow H^+H^- \rightarrow W^{\pm}W^{\mp}Zh$. Both signal events and their significances are evaluated with taking into account the corresponding Standard Model backgrounds. We find that the signal significances can exceed $5\sigma$ at several benchmark points in the viable parameter space of the Type-X THDM.