NLO contributions to $B\to K{K}^{\ast }$ Decays in the pQCD approach

We calculate the important next-to-leading-order (NLO) contributions to the $B\to K{K}^{\ast }$ decays from the vertex corrections, the quark loops, and the magnetic penguins in the perturbative QCD (pQCD) factorization approach. The pQCD predictions for the CP-averaged branching ratios are $Br(B^+ \to K^+ \bar{K}^{0}) \approx 3.2\times 10^{-7}$,$Br(B^+ \to \bar{K}^0 {K}^{+}) \approx 2.1\times 10^{-7}$,$Br(B^0/\ov{B}^0 \to K^0\bar{K}^{0}+\bar{K}^0 K^{0}) \approx 8.5\times 10^{-7}$,$Br(B^0/\ov{B}^0 \to K^+K^{-} + K^-K^{+}) \approx 1.3\times 10^{-7}$, which agree well with both the experimental upper limits and the predictions based on the QCD factorization approach. Furthermore, the CP-violating asymmetries of the considered decay modes are also evaluated. The NLO pQCD predictions for $\text{acp}(B^{+}\to K^{+}{\bar{K}}^{\ast 0})$ and $\text{acp}(B^{+}\to K^{\ast +}{\bar{K}}^0)$ are $\acp^{dir}(K^+\bar{K}^{0})\approx -6.9 %$and$\acp^{dir}(K^{+}\bar{K}^0)\approx 6.5 %$.