  $CP$ violation may play an important role in Baryogenesis in early universe,
and should be examined at colliders comprehensively. We study $CP$ properties
of $HVV$ vertexes between Higgs and gauge boson pairs with defining a $CP$
violation phase angle $\xi$, which indicates the mixture of $CP$-even and
$CP$-odd Higgs states in $HVV$ in new physics. A series of $HVV$ amplitudes
$H\to\gamma\gamma, H\to\gamma V\to \gamma \ell\ell$, and $H\to VV\to 4\ell$
with $CP$ phase angle are studied systematically, which explains explicitly why
$CP$ violation could only be probed in $4\ell$ process independently. We get a
novel amplitude decomposition relation which illustrates if two preconditions
(multilinear momentum dependent vertexes and current $J_\mu$ of $V\to \ell^+
\ell^-$ is formally proportional to a photon's polarization vector) are
satisfied, an high-point amplitude can be decomposed into a summation of a
series of low-point amplitudes. As a practical example, the amplitude of
$H\to\gamma V\to \gamma \ell\ell$, and $H\to VV\to 4\ell$ processes can be
decomposed into summation of many $H\to\gamma\gamma$ amplitudes. Meanwhile, we
calculate these amplitudes in the framework of on-shell scattering amplitude
method, with considering both massless and massive vector gauge bosons with
$CP$ violation phase angle. The above two approaches provides consistent
results and exhibit clearly the $CP$ violation $\xi$ dependence in the
amplitudes.
