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Calculation of the $B\to K_{0,2}^*(1430)f_0(980)/σ$ decays in the Perturbative QCD Approach

Motivated by the observations of the decays $B^0 \to K_0^{*}(1430)^0 f_0(980)$ and $ B^0 \to K_2^{*}(1430)^0 f_0(980)$ from BaBar collaboration, we study the $B^{0(+)} \to K_{0,2}^{*}(1430)^{0(+)} f_0(980)/σ$ decays in the perturbative QCD approach for the first time. In the absence of reliable nonperturbative wave functions we only assume the scalar meson $f_0(980)$ and $σ$ are two-quark ground states. In our calculations, these decays are all dominated by the hard-scattering emission and annihilation diagrams, while the factorizable emission diagrams are forbidden or suppressed heavily by the vector decay constants. Furthermore, the branching fractions are sensitive to the mixing between $f_0(980)$ and $σ$. Comparing our results with the experimental data, a large mixing angle $θ$ is favored. Taking $θ=145^\circ$, the orders of branching fractions of $B \to K_0^{*}(1430)^0 σ$, $B \to K_{2}^{*}(1430)^0 σ$ and $B \to K_{0,2}^{*}(1430)^0 f_0(980)$ are predicted to be $10^{-4}$, $10^{-5}$ and $10^{-6}$, respectively, which can be measured in the current experiments such as LHCb and Belle-2. In addition, although these decays are penguin dominant, the mixing also leads to large direct $CP$ asymmetries in these decays. With the precise data in future, our results could shed light on the inner structure of the scalar mesons and can be used to determine the mixing angle of the $σ-f_0(980)$ system.