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In this paper, we investigate the quasi-two-body decays Bc K₀^* (1430, 1950) D (ₒ) K D (ₒ) within the perterbative QCD (PQCD) framework. The S-wave two-meson distribution amplitudes (DAs) are introduced to describe the final state interactions of the K pair, which involve the time-like form factors and the Gegenbauer polynomials. In the calculations, we adopt two kinds of parameterization schemes to describe the time-like form factors: One is the relativistic Breit-Wigner (RBW) formula, which is usually more siutable for the narrow resonances, and the other is the LASS line shape proposed by the LASS Collaboration, which includes both the resonant and nonresonant components. We find that the branching ratios and the direct CP violations for the decays Bc K₀^* (1430) D (ₒ) obtained from those of the quasi-two-body decays Bc K₀^* (1430) D (ₒ) K D (ₒ) under the narrow width approximation (NWA) can be consistent well with the previous PQCD results calculated in the two-body framework by assuming that K^*₀ (1430) being the lowest lying q s state, which is so-called scenario II (SII). We conclude that the LASS parameterization is more siutable to describe the K₀^* (1430) than the RBW formula, and the nonresonant components play an important role in the branching ratios of the decays Bc K₀^* (1430) D (ₒ) K D (ₒ). In view of the large difference between the decay width measurements for the K₀^* (1950) given by BaBar and LASS collaborations, we calculate the branching ratios and the CP violations for the quasi-two-body decays Bc K₀^* (1950) D (ₒ) K D (ₒ) by using two values, ₊^*䃐 (₁₉₅₀) =0. 1000. 04 GeV and ₊^*䃐 (₁₉₅₀) =0. 2010. 034 GeV, besides the two kinds of parameterizations for the resonance K^*₀ (1950).
Zhang et al. (Sat,) studied this question.