  The processes of P-wave Quarkonium exclusive decays to two mesons are
investigated, in which the final state vector mesons with various polarizations
are considered separately. In the calculation, the initial heavy quarkonia are
treated in the framework of non-relativistic quantum chromodynamics, whereas
for light mesons, the light cone distribution amplitudes up to twist-3 are
employed. It turns out that the higher twist contribution is significant and
provides a possible explanation for the observation of the hadron helicity
selection rule violated processes $\chi_{c1}\rightarrow \phi\phi,\omega\omega$
by the BESIII collaboration in recently. We also evaluate the $\chi_{b1}\to
J/\psi J/\psi$ process and find that its branching ratio is big enough to be
measured at the B-factories.
