One-loop off-shell decay $H^{\ast }\to ZZ$ at future colliders

We present one-loop formulas for contributing to the $HZZ$ vertex in 't Hooft-Veltman gauge within Standard Model framework. One-loop off-shell Higgs decay rates to $Z$-pair are investigated in both unpolarized and longitudinal polarization for $Z$ bosons in final state. The corrections are range of $7\mathrm{\%}$ to $8.4\mathrm{\%}$ when we vary the off-shell Higgs mass from $200$ GeV to $500$ GeV. In applications, we study off-shell Higgs decay $H^{\ast }\to ZZ$ in the Higgs productions at future colliders such as the signal processes ${\gamma }^{\ast }(Q^2)\gamma \to H^{\ast }\to ZZ$ and $e^{-}\gamma \to e^{-}{H}^{\ast }\to e^{-}ZZ$ are analyzed.