Branching ratios and CP asymmetries of the quasi-two-body decays Bc → \ K*₀(1430,1950) D₍ₛ₎ → K π D₍ₛ₎ in the PQCD approach

In this paper, we investigate the quasi-two-body decays Bc → K₀*(1430,1950) D₍ₛ₎ → K π D₍ₛ₎ B c → K 0 ∗ ( 1430 , 1950 ) D ( s ) → K π D ( s ) within the perturbative QCD (PQCD) framework. The S-wave two-meson distribution amplitudes (DAs) are introduced to describe the final state interactions of the Kπ K π 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 suitable 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₍ₛ₎ B c → K 0 ∗ ( 1430 ) D ( s ) obtained from those of the quasi-two-body decays Bc → K₀*(1430) D₍ₛ₎ → K π D₍ₛ₎ B c → K 0 ∗ ( 1430 ) D ( s ) → K π D ( s ) 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) K 0 ∗ ( 1430 ) is the lowest lying q̄ s q ¯ s state, which is the so-called scenario II (SII). We conclude that the LASS parameterization is more suitable to describe the K₀*(1430) K 0 ∗ ( 1430 ) than the RBW formula, and the nonresonant components play an important role in the branching ratios of the decays Bc → K₀*(1430) D₍ₛ₎ → K π D₍ₛ₎ B c → K 0 ∗ ( 1430 ) D ( s ) → K π D ( s ) . In view of the large difference between the decay width measurements for the K₀*(1950) K 0 ∗ ( 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₍ₛ₎ → K π D₍ₛ₎ B c → K 0 ∗ ( 1950 ) D ( s ) → K π D ( s ) by using two values, Γ K^*₀(1950)=0.100± 0.04 Γ K 0 ∗ ( 1950 ) = 0.100 ± 0.04 GeV and Γ K^*₀(1950)=0.201± 0.034 Γ K 0 ∗ ( 1950 ) = 0.201 ± 0.034 GeV, besides the two kinds of parameterizations for the resonance K^*₀(1950) K 0 ∗ ( 1950 ) . We find that the branching ratios and the direct CP violations for the decays Bc → K₀*(1950) D₍ₛ₎ → K π D₍ₛ₎ B c → K 0 ∗ ( 1950 ) D ( s ) → K π D ( s ) have not as large difference between the two parameterizations as the case of decays involving the K^*₀(1430) K 0 ∗ ( 1430 ) , especially for the results with Γ K^*₀(1950)=0.201± 0.034 Γ K 0 ∗ ( 1950 ) = 0.201 ± 0.034 GeV. The effect of the nonresonant component in the K^*₀(1950) K 0 ∗ ( 1950 ) may be not so serious as that in the K^*₀(1430) K 0 ∗ ( 1430 ) . The quasi-two-body decays B⁺c → K*+₀(1430) D⁰ → K⁰ π ⁺ D⁰ B c + → K 0 ∗ + ( 1430 ) D 0 → K 0 π + D 0 and B⁺c → K*0₀(1430) D⁺ → K⁺ π ⁻ D⁺ B c + → K 0 ∗ 0 ( 1430 ) D + → K + π - D + have large branching ratios, which can reach to the order of 10⁻⁴ 10 - 4 and are most likely to be observed in the future LHCb experiments. Furthermore, the branching ratios of the quasi-two-body decays Bc → K₀*(1950) D₍ₛ₎ → K π D₍ₛ₎ B c → K 0 ∗ ( 1950 ) D ( s ) → K π D ( s ) are about one order smaller than those of the corresponding decays Bc → K₀*(1430) D₍ₛ₎ → K π D₍ₛ₎ B c → K 0 ∗ ( 1430 ) D ( s ) → K π D ( s ) .