Study of quasi-two-body B^{0}\rightarrow T(\pi\pi, K\bar{K},\pi\eta) decays in perturbative QCD approach

In this study, we calculate the CP-averaged branching ratios and the direct CP-violating asymmetries of the three body decays B^{0}\rightarrow T(\pi\pi, K\bar{K},\pi\eta) in the perturbative QCD (PQCD) approach, where T denotes tensor mesons a_{2}(1320), K^{*}_{2}(1430), f_{2}(1270) and f^{'}_{2}(1525), the daughter branching is f_{0}(980)\rightarrow \pi\pi, K\bar{K},f_{0}(500)\rightarrow \pi\pi, a_{0}(980)\rightarrow K\bar{K}, \pi\eta. By introducing two-meson distribution amplitudes parameterized by the time-like form factors, the three-body decay is simplified to quasi-two-body decay. Based on the two-quark structure, considering the effect of mixing angle \theta between f_{0}(980) and f_{0}(500), \phi between f_{2}(1270) and f^{'}_{2}(1525) on our calculations. We found that (a) Taking \theta=135^{\circ}, under the narrow-width approximation we extract the decays B^{0}\rightarrow K^{*}_{2}(1430)^{0}f_{0}(980) branching fractions is agree with the current experimental data well. (b) The decay rates for the considered decay modes are generally in the order of 10^{-8}$ to $10^{-5}. (c)The branching fractions are sensitive to the \theta, and the opposite is true for \phi. The \phi is really small, so the decay branching ratio only has little change, except for some decays involving f^{'}_{2}(1525). (d)The \theta and \phi can bring remarkable change to the direct CP asymmetries of pure penguin processes so that it is not 0. (e)We calculate the relative partial decay widths {\Gamma}(a_{0} \rightarrow K\overline{K})/{\Gamma}(a_{0} \rightarrow \pi\eta) and the ratio {\cal B}(f_{0} \rightarrow K^{+}K^{-})/{\cal B}(f_{0} \rightarrow \pi^{+}\pi^{-}), which are in agreement with the existing experimental values. Our results can help to understand the internal structure of scalar mesons and the nature of tensor mesons and be tested by future experiments.