Contributions of S-, P-, and D-wave resonances to the quasi-two-body decays Bs0→ψ(3686,3770)Kπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{0}_{s} \rightarrow \psi (3686,3770)K \pi $$\end{document} in the perturbative QCD approach

Based on the perturbative quantum chromodynamics (pQCD) approach and the quasi-two-body approximation, we have studied the three-body decays Bs0→ψ(3686,3770)Kπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$\end{document}, which include the contributions of the intermediate resonances K¯0∗(1430)0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{K }^{*}_{0}(1430)^{0}$$\end{document}, K¯∗(892)0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{K }^{*}(892)^{0}$$\end{document}, K¯∗(1410)0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{K }^{*}(1410)^{0}$$\end{document}, K¯∗(1680)0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{K }^{*}(1680)^{0}$$\end{document}, and K¯2∗(1430)0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{K }^{*}_{2}(1430)^{0}$$\end{document}. The time-like form factors corresponding to the distribution amplitudes of the S-, P-, and D-wave of the kaon–pion pair were adopted in parameterized form, and describe the interactions between K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K $$\end{document} and π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document} in the resonance region. First, the decays Bs0→ψ(2S,1D)K-π+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{0}_{s}\rightarrow \psi (2S,1D)K ^{-}\pi ^{+}$$\end{document} were calculated, followed by the calculation of the branching ratios of the decays Bs0→ψ(3686,3770)K-π+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{0}_{s}\rightarrow \psi (3686,3770)K ^{-}\pi ^{+}$$\end{document} using the 2S–1D mixing scheme. In addition, the pQCD predictions for the decays Bs0→ψ(2S,1D)Kπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{0}_{s}\rightarrow \psi (2S,1D)K \pi $$\end{document} and Bs0→ψ(3686,3770)Kπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$\end{document} were obtained using the narrow-width approximation relation given by the Clebsch–Gordan coefficients. Our work shows that the K¯∗(892)0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{K }^{*}(892)^{0}$$\end{document} resonance is the main contributor to the total decay, and the branching ratio and the longitudinal polarization fraction of the ψ(2S)K¯∗(892)0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (2S)\overline{K }^{*}(892)^{0}$$\end{document} decay mode agree well with the currently available data within errors. Furthermore, the theoretical predictions of the ψ(2S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (2S)$$\end{document} and ψ(3686)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (3686)$$\end{document} decay modes are very close, indicating that they can be regarded as the same meson state. Finally, the pQCD predictions for branching ratios of decays Bs0→ψ(3686,3770)Kπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$\end{document} are of the order of 10-5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-5}$$\end{document} and 10-6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-6}$$\end{document}, respectively, which can be verified using the ongoing LHCb and Belle II experiments.


I. INTRODUCTION
In recent years, studies on B-meson decays have attracted increasing attention since they enable the testing of the standard model (SM) and enrich the field of quantum chromodynamics (QCD).The three-body decays of the B meson involve resonant as well as for non-resonant contributions.Thus their calculations are more complicated than those for two-body decays.There are mutual interferences between the resonant and non-resonant states, and thus it is difficult to calculate them separately [1].Based on the symmetry principles and the factorization theorems, a few theoretical models for calculating the three-body decay have been developed.In this study, we have adopted the widely used perturbative QCD (pQCD) factorization approach [2][3][4][5].The color-suppressed phenomenon occurs when a B 0 s meson decays into a kaon-pion pair and a charmonium.Thus, it is meaningful to study the B 0 s → ψ(2S, 1D)Kπ decays.Recently, significant advances have been made in the research on heavy quarkonium generation mechanism [6].The LHCb collaboration has detected the B 0 s → ψ(2S)K − π + decay [7] and found that the main source of the decay branching ratio is the K * (892) 0 resonance.These advances have allowed us to reliably calculate and test the B 0 s → ψ(2S, 1D)Kπ decays.The pQCD factorization approach was proposed based on the k T factorization theorem [8][9][10].In this approach, a three-body problem can be simplified to a quasi-two-body problem by introducing two-hadron distribution amplitudes(DAs) [11,12].The predominant contributions in the decay process are from the parallel motion range, where the invariant mass of the double light meson pair is lower than O( ΛM B ), and Λ = M B − m b represents the mass difference between the B meson and the b-quark.Thus, the pQCD factorization formula for the three-body decay of the B 0 s meson can be generally described as [9,10] where the hard decay kernel, H, represents the contribution of the Feynman diagram with only one gluon exchange in the leading order, which can be calculated using the perturbation theory.The terms φ B 0 s , φ h3 , and φ h1h2 represent the wave functions of B 0 s , h 3 , and h 1 h 2 pair, respectively.They are considered as non-perturbative inputs, which can be constructed by extracting the relevant experimentally measured quantities or calculating them using the non-perturbative model.
Though the decay B 0 s → ψ(3770)Kπ has not been observed experimentally, the mixing structure of ψ(3770) can be investigated by making a theoretical prediction for this decay channel.Since the charmonium mesons ψ(3686) and ψ(3770) are regarded as the 2S-1D mixed states, the decays B 0 s → ψ(2S)Kπ and B 0 s → ψ(1D)Kπ should first be calculated, and then the fitting should be performed based on the 2S-1D mixing scheme to obtain the branching ratios of the decays B 0 s → ψ(3686, 3770)Kπ.The ψ(1D) state denotes the orbital quantum number l = 2 and the principal quantum number n = 1, and ψ(2S) is the first radially excited state of the charmonium meson.

II. COMPUTATIONAL FRAMEWORK
The weak-effective Hamiltonian of the B 0 s → ψ(2S, 1D)K * 0 (→ K − π + ) decays is expressed as [25] H where V * cb V cd and V * tb V td are the CKM factors, O i is the localized four-quark operator, and C i is the Wilson coefficient corresponding to the quark operator.
A form similar to the two-pion DA has been adopted for the S-wave of the kaon-pion pair DA [33]: The subscripts S, P, and D denote the corresponding sub-waves, respectively, in the following description.
Using the description given by Wang et al. [34], the twist-2 DAs have been described in a form similar to the scalar meson [35,36], whereas asymptotic forms for the twist-3 DAs have been adopted in this work.They can be expressed as follows: The Gegenbauer polynomials are C  [35,37,38].
For the time-like scalar form factor, F S (ω 2 ), we have adopted the parameterized fitting results of an improved LASS line type presented by Aston et al. [39].F S (ω 2 ) is expressed as [34] In Eq. ( 17), the first term contains the resonant contribution with a phase factor to maintain unitarity, and the second term is an empirical term of the elastic Kπ scattering.According to the Li et al. [40], the P-wave kaon-pion DAs related to the longitudinal and transverse polarizations can be expressed as The different twists in Eq. ( 19) when expanded using the Gegenbauer polynomial have the specific forms as follows: The SU (3)

and the Gegenbauer moments
3 and a 1v = 0.3 [40] have been adopted in this work.The time-like shape factor, F P (ω 2 ), of the P-wave is expressed as [41] .
The mass-related width is given by where Γ i and m i denote the width and the pole mass, respectively, of the corresponding resonance, L R represents the orbital angular momentum, with values of 0, 1 and 2 for the S, P, and D-wave, respectively.According to the study by Wang and Li [19], the following relation can be obtained where f T K * = 0.185 ± 0.010 GeV and f K * = 0.217 ± 0.005 GeV [29].We have adopted the procedure from the work by Li et al. [40]: studies on the decay constants of K * (1410) 0 and K * (1680) 0 are limited, and thus we have used the two decay constants of K * (892) 0 to determine the ratio f T K * /f K * .A form similar to the two-kaon DAs has also been considered in the D-wave kaon-pion DAs [24]: where the coefficient 2 3 ( 1 2 ) comes from the different definitions of the polarization vector between the vector and tensor mesons in the longitudinal(transverse) polarization.
The different twists in the D-wave DAs are [24,[42][43][44] The Gegenbauer moments are a 0 1 = 0.4 ± 0.1 and a T 1 = 0.8 ± 0.2, and a form similar to Eq. ( 26) has been adopted for the time-like shape factor, F D (ω 2 ).Furthermore, the approximate relation ) can also be found, with f T K * 2 (1430) = 0.077 ± 0.014 GeV and f K * 2 (1430) = 0.118 ± 0.005 GeV [42].The differential decay ratios for the B 0 s → ψ(2S, 1D)K − π + decays in the B 0 s meson rest frame can be written as where the three-momenta of K − and ψ(2S, 1D) in the kaon-pion center-of-mass system are expressed as The terms A 0 , A , and A ⊥ represent the longitudinal, parallel, and perpendicular polarization amplitudes, respectively.The related expressions are where the subscripts L, N, and T denote the longitudinal, normal, and transverse polarizations, respectively.The polarization fraction is defined as with the normalization relation

III. DECAY AMPLITUDES
Based on the pQCD approach, the decay amplitude of where F and M represent the factorization and non-factorization contributions, respectively.The superscripts LL and LR denote the weak vertices of the operators, and SP is the Fierz transformation of LR.For the S-wave, the amplitude is only a longitudinal polarization.The total decay amplitudes of the P-wave and the D-wave are decomposed into The decay amplitudes of the longitudinal polarization are as follows: A L (P) and A L (D) can be expressed by the following replacement: The decay amplitudes of normal polarization are as follows: A N (D) can be expressed by the following replacement: The decay amplitudes of transverse polarization are as follows: A T (D) can be expressed by the following replacement: The mass ratio r c = mc M B 0 s and the group factor C F = 4 3 .The expressions for the Sudakov exponents S B 0 s (t), S M (t), and S ψ (t), the threshold resummation factor S t (x), the scattering kernel functions h i (i = a, b, c, d), and the hard scales t i have been given in the APPENDIX.
Vertex correction has been performed on the factorization diagrams in this work.According to the NDR scheme [45][46][47], the relevant Wilson coefficients are expressed as The renormalization scale, µ, has been selected to be of the order of m b .The Wilson coefficients a 1,2,3 (S) were applied to the decay amplitude A(S) with only longitudinal polarization, and the hard scattering functions, f I and g I , are given in Ref. [48].Meanwhile, the Wilson coefficients a 1,2,3 (P,D) were applied to the decay amplitudes A(P,D) with both longitudinal and transverse polarizations, the hard scattering function, f h , comes from the vertex corrections, and the superscript h denotes the polarization state: h = 0 for the helicity 0 state, whereas h = ± for the helicity ± states.The expressions for f 0 and f ± can be found in Ref. [49].
Masses The parameters used in the calculation have been presented in Table I, which include the masses of the involved mesons, their decay constants, the lifetime of the B 0 s meson, and the Wolfenstein parameters.The pole masses of the quarks were adopted in this study [52].
The data in Table II have been taken from Ref. [41], the relevant information that should be considered in the study for the S, P, and D-wave resonances are contained in the table.In this work, the dynamic limit of the invariant mass of the resonance In addition, although the mass of the K * (1680) 0 resonance exceeds the upper limit, its decay channels should be considered in the study because of its large width(Γ K * (1680) 0 = 322 ± 110 MeV).
The decay branching ratios of the K * 0 (1430) 0 resonance of the S-wave were first calculated and the results obtained have been given in Table III.The errors were derived from the shape parameter, ω Bs , in the wave function of the B 0 s meson, the Gegenbauer moments in the DAs of the kaon-pion pair, and the hard scale t(0.9t ∼ 1.1t), respectively.The errors in the following tables were analyzed in the same order.
Next, the resonances of the P-wave were calculated considering K * (892) 0 , K * (1410) 0 , and K * (1680) 0 , and the results thus obtained have been given in Table IV.The experimental measurement data 3) × 10 −5 was taken from the article of Zyla et al. [51].Our pQCD prediction agrees well with it within errors.Finally, the contributions of the K * 2 (1430) 0 intermediate resonance of the D-wave were considered and the calculation results have been presented in Table V. TABLE III.Branching ratios of the S-wave resonance in the quasi-two-body decays B 0 s → ψ(2S, 1D)K * 0 (→ K − π + ) calculated using the pQCD factorization approach.

Decay mode pQCD prediction
Experimental data The theoretical prediction for the branching ratio of the B 0 s → ψ(2S)K − π + decay is 3.67 +1.56+1.42+0.15−1.12−1.15−0.10× 10 −5 in this work, which includes contributions from the intermediate resonances of the S, P, and D-wave.This result is consistent with the latest experimental data (3.1 ± 0.4) × 10 −5 [51] within errors.From the numerical results, it has been observed that K TABLE IV.Branching ratios of the P-wave resonances in the quasi-two-body decays B 0 s → ψ(2S, 1D)K * 0 (→ K − π + ) calculated using the pQCD factorization approach.

Decay mode pQCD prediction
Experimental data −12.9−7.9−0.6 2.33 +0.80+0.92+0.10−0.58−0.56−0.05The polarization fractions are defined by Eq. ( 39), and they have been listed in Tables IV and V.For the P-wave ψ(2S) decay mode, the longitudinal polarization fraction is approximately 43%, whereas in the ψ(1D) mode, it is about 10%, with parallel and vertical fractions being approximately equal in both modes.For the D-wave ψ(2S) decay mode, the three polarization fractions are roughly at the same level of approximately 33% but they are distinctly different in the ψ(1D) mode.We expect additional abundant and detailed data to be obtained from future experiments so that our theoretical predictions can be accurately verified and more systematic analysis for B 0 s → ψ(2S, 1D)K * (→ K − π + ) decays can be performed.
From the experimental data, the relative fraction between the branching ratios has been obtained to be [7] B(B By comparing the branching ratio of the B 0 s → ψ(2S)K * (892) 0 (→ K − π + ) decay, calculated using the pQCD factorization approach, with the pQCD prediction for the B 0 → ψ(2S)K * (892) 0 (→ K + π − ) decay [40], we obtain the relative fraction of the theoretical calculation as The discrepancy in the values comes from the vertex correction and the selection of different values for some of the parameters.However, this discrepancy is still within the acceptable limit.The relative fraction results predicted by the theory agree somewhat with the experimental data, which support the pQCD factorization approach and also contribute to the further studies on resonance mesons.
Figs. 2− 4 depict the function images of the ω dependence of the differential branching ratios of the S, P, and D-wave of the B 0 s → ψ(2S, 1D)K − π + decays, respectively.Fig. 2 shows that a small peak can always be detected near the invariant mass ω = 0.892GeV, which can be attributed to the interference effect of the K * (892) 0 resonance on the S-wave.On the other hand, the function images of ψ(1D) mode drop faster at the end than of the ψ(2S) mode due to the difference in the upper limit of their invariant masses ω of Kπ.Obviously, the peak values of all function images appear at the pole mass of the corresponding resonance.Therefore, the main part of the branching ratios is in the region around the resonance and almost in the range of the branching ratios of S, P, and D-wave decay modes in this range account for 43.91%, 74.73%, and 78.68% of the total branching ratios, respectively.The value of 43.91% can be interpreted as the interference effect of the K * (892) 0 resonance on the S-wave that is not included.
Using Eqs. ( 59) and (60),the branching ratios of the B 0 s → ψ(3686, 3770)K − π + decays were obtained using the fitting scheme based on the S-D mixing mechanism.The calculation results are presented in Tables VI and VII, respectively.
Considering the Clebsch-Gorden coefficients, we can write the following relation In our calculation, for the quasi-two-body decay B 0 s → ψK * 0 → ψK − π + , isospin conservation was assumed for the strong decays of an I = 1/2 intermediate resonance K * 0 to Kπ, which can be expressed as follows: Therefore, the branching ratios of B 0 s → ψ(2S, 1D)K * 0 (→ Kπ) and B 0 s → ψ(3686, 3770)K * 0 (→ Kπ) decays can be extracted directly under the narrow-width approximation relation A comparison of the branching ratios for ψ(3770) decay modes when the mixing angle is set to θ = −12 • and θ = 27 • reveal a significant difference between the two choices, which can be attributed to the visibly small decay constant of ψ(1D) compared to that of ψ(2S).These results are in accordance with the analyses presented in other studies [16,17,50,53,54].In addition, when the 2S-1D mixing scheme is considered for the B 0 s → ψ(3686)K − π + decay, the numerical result changes slightly compared to that of the B 0 s → ψ(2S)K − π + decay, indicating that the ψ(3686) state might be deemed as the ψ(2S) state.Further, according the Eqs.( 59) and (60), the reason for the ψ(3686) and ψ(3770) decay modes having markedly different sensitivities to the change in the mixing angle under the 2S-1D mixing scheme could be provided.Numerically, A(B 0 s → ψ(2S)K − π + ) is much larger than A(B 0 s → ψ(1D)K − π + ), and thus the former dominates the decay amplitudes of the ψ(3686) as well as ψ(3770) decay modes.The value of the amplitude sin θA(B 0 s → ψ(2S)K − π + ) is greatly changed when the mixing angle is switched between θ = −12 • and θ = 27 • .On the contrary, the amplitude cos θA(B 0 s → ψ(2S)K − π + ) is relatively stable under this of the ψ(2S) decay modes agree well with the existing experimental data within acceptable errors.Our calculations show that the branching ratios of the ψ(3686) and ψ(2S) decay modes are very similar, suggesting that they can be regarded as the same state.Theoretical predictions for the branching ratios of ψ(3686) and ψ(3770) decay channels are of the order of 10 −5 and 10 −6 , respectively, which will be verified using the data from future experimental measurements.The detected data will help us to gain further understanding about the internal structures of the ψ(3686) and ψ(3770) mesons.