\begin{document}$ CP $\end{document} violation may play an important role in baryogenesis in the early universe and should be examined comprehensively at colliders. We study the \begin{document}$ CP $\end{document} properties of \begin{document}$ HVV $\end{document} vertexes between Higgs and gauge boson pairs by defining a \begin{document}$ CP $\end{document} violation phase angle ξ, which indicates the mixture of \begin{document}$ CP $\end{document}-even and \begin{document}$ CP $\end{document}-odd Higgs states in \begin{document}$ HVV $\end{document} in new physics. A series of \begin{document}$ HVV $\end{document} amplitudes, \begin{document}$ H\to\gamma\gamma, H\to\gamma V\to \gamma \ell\ell $\end{document}, and \begin{document}$ H\to VV\to 4\ell $\end{document}, with a \begin{document}$ CP $\end{document} phase angle are studied systematically to explicitly explain why \begin{document}$ CP $\end{document} violation can only be probed independently in the \begin{document}$ 4\ell $\end{document} process. We obtain a novel amplitude decomposition relation that illustrates that if two preconditions (multilinear momentum dependent vertexes, and the current \begin{document}$ J_\mu $\end{document} of \begin{document}$ V\to \ell^+ \ell^- $\end{document} is formally proportional to a photon's polarization vector) are satisfied, a higher-point amplitude can be decomposed into a summation of a series of lower-point amplitudes. As a practical example, the amplitude of the \begin{document}$ H\to\gamma V\to \gamma \ell\ell $\end{document} and \begin{document}$ H\to VV\to 4\ell $\end{document} processes can be decomposed into a summation of many \begin{document}$ H\to\gamma\gamma $\end{document} amplitudes. We calculate these amplitudes in the framework of the on-shell scattering amplitude method, considering both massless and massive vector gauge bosons with the \begin{document}$ CP $\end{document} violation phase angle. The above two approaches provide consistent results and clearly reveal the \begin{document}$ CP $\end{document} violation ξ dependence in the amplitudes."> Exploring <i>HVV</i> amplitudes with <i>CP</i> violation using decomposition and the on-shell scattering amplitude method -
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