Polarization, CP Asymmetry and Branching ratios in B \to K^* K^* with Perturbative QCD approach

We study the charmless rare decays $B\to K^{\ast }{K}^{\ast }$ within the Perturbative QCD picture. We calculate not only factorizable and non-factorizable diagrams, but also annihilation ones. Our predictions are the following: The longitudinal polarization fraction vary from 75% to 99% depending on channels, the branching ratios are of order ${10}^{-7}$ for $B^0({\bar{B}}^0)\to K^{\ast 0}{\bar{K}}^{\ast 0}$ and $B^{\pm} \to K^{\ast\pm} \bar{K}^{\ast0}(K^{\ast0})$,muchbiggerthanthatfor$B^0(\bar{B}^0) \to K^{\ast+}K^{\ast-}$($10^{-8}$)\mathrm{.}ThedirectCPasymmetryin$B^{\pm} \to K^{\ast\pm} \bar{K}^{\ast0}(K^{\ast0})$and$B^0(\bar{B}^0) \to K^{\ast+}K^{\ast-}$isabout-15$\alpha (\phi_2)$as$95^{\circ}$\mathrm{.}Ther{e}^{\mathrm{\prime }}snodirectCPVin$B^0(\bar{B}^0) \to K^{\ast0}\bar{K}^{\ast0}$decaysbecauseofthepure$b \to d$ penguin topology. Our predictions will be tested in the future B experiments.