Branching ratios and CP asymmetries of the quasi-two-body decays in the PQCD approach

Zhi-Qing Zhang; Zi-Yu Zhang; Ming-Xuan Xie; Ming-Yang Li; Hong-Xia Guo

doi:10.1140/epjc/s10052-024-13130-9

2024 Impact factor 4.8

Particles and Fields

Open Access

Eur. Phys. J. C (2024) 84: 864
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 $B_{c} \to K_{0}^{} (1430, 1950) D_{(s)} \to K π D_{(s)}$ $B_c \rightarrow \ K^{}_0(1430,1950) D_{(s)} \rightarrow K \pi D_{(s)}$ in the PQCD approach

Zhi-Qing Zhang¹, Zi-Yu Zhang¹, Ming-Xuan Xie¹, Ming-Yang Li¹ and Hong-Xia Guo²^e

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

^e guohongxia@zzu.edu.cn

Received: 14 May 2024
Accepted: 15 July 2024
Published online: 29 August 2024

Abstract

In this paper, we investigate the quasi-two-body decays $B_{c} \to K_{0}^{*} (1430, 1950) D_{(s)} \to K π D_{(s)}$ $B_c \rightarrow K_0^{*}(1430,1950) D_{(s)} \rightarrow K \pi D_{(s)}$ within the perturbative QCD (PQCD) framework. The S-wave two-meson distribution amplitudes (DAs) are introduced to describe the final state interactions of the $K π$ $K\pi$ 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 $B_{c} \to K_{0}^{*} (1430) D_{(s)}$ $B_c \rightarrow K_0^{*}(1430) D_{(s)}$ obtained from those of the quasi-two-body decays $B_{c} \to K_{0}^{*} (1430) D_{(s)} \to K π D_{(s)}$ $B_c \rightarrow K_0^{*}(1430) D_{(s)} \rightarrow K \pi D_{(s)}$ under the narrow width approximation (NWA) can be consistent well with the previous PQCD results calculated in the two-body framework by assuming that $K_{0}^{*} (1430)$ $$K^*_0(1430)$$ is the lowest lying $\bar{q} s$ $\bar{q} s$ state, which is the so-called scenario II (SII). We conclude that the LASS parameterization is more suitable to describe the $K_{0}^{*} (1430)$ $K_0^{*}(1430)$ than the RBW formula, and the nonresonant components play an important role in the branching ratios of the decays $B_{c} \to K_{0}^{*} (1430) D_{(s)} \to K π D_{(s)}$ $B_c \rightarrow K_0^{*}(1430) D_{(s)} \rightarrow K \pi D_{(s)}$ . In view of the large difference between the decay width measurements for the $K_{0}^{*} (1950)$ $K_0^{*}(1950)$ given by BaBar and LASS collaborations, we calculate the branching ratios and the CP violations for the quasi-two-body decays $B_{c} \to K_{0}^{*} (1950) D_{(s)} \to K π D_{(s)}$ $B_c \rightarrow K_0^{*}(1950) D_{(s)} \rightarrow K \pi D_{(s)}$ by using two values, $Γ_{K_{0}^{*} (1950)} = 0.100 \pm 0.04$ $\Gamma _{K^*_0(1950)}=0.100\pm 0.04$ GeV and $Γ_{K_{0}^{*} (1950)} = 0.201 \pm 0.034$ $\Gamma _{K^*_0(1950)}=0.201\pm 0.034$ 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 $B_{c} \to K_{0}^{*} (1950) D_{(s)} \to K π D_{(s)}$ $B_c \rightarrow K_0^{*}(1950) D_{(s)} \rightarrow K \pi D_{(s)}$ 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 $Γ_{K_{0}^{*} (1950)} = 0.201 \pm 0.034$ $\Gamma _{K^*_0(1950)}=0.201\pm 0.034$ GeV. The effect of the nonresonant component in the $K_{0}^{*} (1950)$ $$K^*_0(1950)$$ may be not so serious as that in the $K_{0}^{*} (1430)$ . The quasi-two-body decays $B_{c}^{+} \to K_{0}^{* +} (1430) D^{0} \to K^{0} π^{+} D^{0}$ $B^+_c \rightarrow K^{*+}_0(1430) D^{0} \rightarrow K^0 \pi ^+ D^{0}$ and $B_{c}^{+} \to K_{0}^{* 0} (1430) D^{+} \to K^{+} π^{-} D^{+}$ $B^+_c \rightarrow K^{*0}_0(1430) D^{+} \rightarrow K^+ \pi ^- D^{+}$ 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 $B_{c} \to K_{0}^{*} (1950) D_{(s)} \to K π D_{(s)}$ $B_c \rightarrow K_0^{*}(1950) D_{(s)} \rightarrow K \pi D_{(s)}$ are about one order smaller than those of the corresponding decays $B_{c} \to K_{0}^{*} (1430) D_{(s)} \to K π D_{(s)}$ $B_c \rightarrow K_0^{*}(1430) D_{(s)} \rightarrow K \pi D_{(s)}$ .

© The Author(s) 2024

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.