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https://doi.org/10.1140/epjc/s10052-024-13130-9

Regular Article - Theoretical Physics

## Branching ratios and CP asymmetries of the quasi-two-body decays ${B}_{c}\to \phantom{\rule{4pt}{0ex}}K_{0}^{\ast}(1430,1950){D}_{(s)}\to K\pi {D}_{(s)}$ in the PQCD approach

^{1}
Institute of Theoretical Physics, School of Sciences, Henan University of Technology, 450001, Zhengzhou, Henan, China

^{2}
School of Mathematics and Statistics, Zhengzhou University, 450001, Zhengzhou, Henan, China

Received:
14
May
2024

Accepted:
15
July
2024

Published online:
29
August
2024

In this paper, we investigate the quasi-two-body decays ${B}_{c}\to K_{0}^{\ast}(1430,1950){D}_{(s)}\to K\pi {D}_{(s)}$ within the perturbative QCD (PQCD) framework. The S-wave two-meson distribution amplitudes (DAs) are introduced to describe the final state interactions of the $K\pi $ 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 ${B}_{c}\to K_{0}^{\ast}(1430){D}_{(s)}$ obtained from those of the quasi-two-body decays ${B}_{c}\to K_{0}^{\ast}(1430){D}_{(s)}\to K\pi {D}_{(s)}$ under the narrow width approximation (NWA) can be consistent well with the previous PQCD results calculated in the two-body framework by assuming that ${K}_{0}^{\ast}(1430)$ is the lowest lying $\overline{q}s$ state, which is the so-called scenario II (SII). We conclude that the LASS parameterization is more suitable to describe the $K_{0}^{\ast}(1430)$ than the RBW formula, and the nonresonant components play an important role in the branching ratios of the decays ${B}_{c}\to K_{0}^{\ast}(1430){D}_{(s)}\to K\pi {D}_{(s)}$. In view of the large difference between the decay width measurements for the $K_{0}^{\ast}(1950)$ given by BaBar and LASS collaborations, we calculate the branching ratios and the CP violations for the quasi-two-body decays ${B}_{c}\to K_{0}^{\ast}(1950){D}_{(s)}\to K\pi {D}_{(s)}$ by using two values, ${\mathrm{\Gamma}}_{{K}_{0}^{\ast}(1950)}=0.100\pm 0.04$ GeV and ${\mathrm{\Gamma}}_{{K}_{0}^{\ast}(1950)}=0.201\pm 0.034$ GeV, besides the two kinds of parameterizations for the resonance ${K}_{0}^{\ast}(1950)$. We find that the branching ratios and the direct CP violations for the decays ${B}_{c}\to K_{0}^{\ast}(1950){D}_{(s)}\to K\pi {D}_{(s)}$ have not as large difference between the two parameterizations as the case of decays involving the ${K}_{0}^{\ast}(1430)$, especially for the results with ${\mathrm{\Gamma}}_{{K}_{0}^{\ast}(1950)}=0.201\pm 0.034$ GeV. The effect of the nonresonant component in the ${K}_{0}^{\ast}(1950)$ may be not so serious as that in the ${K}_{0}^{\ast}(1430)$. The quasi-two-body decays ${B}_{c}^{+}\to K_{0}^{\ast +}(1430){D}^{0}\to {K}^{0}{\pi}^{+}{D}^{0}$ and ${B}_{c}^{+}\to K_{0}^{\ast 0}(1430){D}^{+}\to {K}^{+}{\pi}^{-}{D}^{+}$ have large branching ratios, which can reach to the order of ${10}^{-4}$ and are most likely to be observed in the future LHCb experiments. Furthermore, the branching ratios of the quasi-two-body decays ${B}_{c}\to K_{0}^{\ast}(1950){D}_{(s)}\to K\pi {D}_{(s)}$ are about one order smaller than those of the corresponding decays ${B}_{c}\to K_{0}^{\ast}(1430){D}_{(s)}\to K\pi {D}_{(s)}$.

*© The Author(s) 2024*

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Funded by SCOAP^{3}.