Study of $B\to K^* \rho, K^*\omega$ Decays with Polarization in Perturbative QCD Approach

The $B \to K^{*}\rho$, $ K^{*}\omega$ decays are useful to determine the CKM angle $\phi_3=\gamma$. Their polarization fractions are also interesting since the polarization puzzle of the $B\to \phi K^*$ decay. We study these decays in the perturbative QCD approach based on $k_T$ factorization. After calculating of the non-factorizable and annihilation type contributions, in addition to the conventional factorizable contributions, we find that the contributions from the annihilation diagrams are crucial. They give dominant contribution to the strong phases and suppress the longitudinal polarizations. Our results agree with the current existing data. We also predict a sizable direct CP asymmetries in $B^+ \to K^{*+}\rho^0$, $B^0 \to K^{*+}\rho^-$, and $B^+ \to K^{*+}\omega$ decays, which can be tested by the oncoming measurements in the B factory experiments.