# Studies of the resonance components in the \(B_s\) decays into charmonia plus kaon pair

## Abstract

In this work, the decays of \(B_s\) meson to a charmonium state and a \(K^+K^-\) pair are carefully investigated in the perturbative QCD approach. Following the latest fit from the LHCb experiment, we restrict ourselves to the case where the produced \(K^+K^-\) pair interact in isospin zero *S*, *P*, and *D* wave resonances in the kinematically allowed mass window. Besides the dominant contributions of the \(\phi (1020)\) resonance in the *P*-wave and \(f_2'(1525)\) in the *D*-wave, other resonant structures in the high mass region as well as the *S*-wave components are also included. The invariant mass spectra for most of the resonances in the \(B_s\rightarrow J/\psi K^+K^-\) decay are well reproduced. The obtained three-body decay branching ratios can reach the order of \(10^{-4}\), which seem to be accessible in the near future experiments. The associated polarization fractions of those vector-vector and vector-tensor modes are also predicted, which are compared with the existing data from the LHCb Collaboration.

## 1 Introduction

The three-body mode \(B_s\rightarrow J/\psi K^+K^-\) is of particular interest in searches for intermediate states in the \(B_s\) decay chain. Since the LHCb Collaboration [1] found no obvious structures in the \(J/\psi K^+\) invariant mass distribution, the \(B_s\rightarrow J/\psi K^+K^-\) decay proceeds predominantly via \(B_s\rightarrow J/\psi R\) with the quasi-two-body intermediate state *R* subsequently decaying into \(K^+K^-\). For the concerned \(B_s\) decay, the \(K^+K^-\) system arise from pure \(s{{\bar{s}}}\) source, and thus these resonances are isoscalar. Taking into account the conservation of *P*-parity and *C*-parity, the produced resonances are limited to quantum numbers \(J^{PC}=0^{++},1^{{-}{-}},2^{++},...\) with isospin \(I=0\). Among them, the largest component comes from the \(\phi (1020)\) in a *P*-wave configuration [2]. Several Collaborations [1, 3, 4] have presented a measurement of the \(B_s\rightarrow J/\psi \phi (1020)\) mode with \(\phi (1020)\) decays to \(K^+K^-\). The current world averages of the absolute branching ratio \({\mathcal {B}}(B_s\rightarrow J/\psi \phi (1020))\) can reach the order of \(10^{-3}\). Another *P*-wave resonance \(\phi (1680)\), whose contribution has more than 2 statistical standard deviation \((\sigma )\) significance, is also included by the LHCb Collaboration [1] in its best fit model. Two well known scalar resonances, the \(f_0(980)\) and \(f_0(1370)\), are observed in the \(K^+K^-\) mass spectrum by LHCb [1], which is the only data set available so far for the *S*-wave resonant structures.

Contributions from *D*-wave resonances are known to be non-negligible in this decay. The first observation of the decay sequence \(B_s\rightarrow J/\psi f_2'(1525), f_2'(1525)\rightarrow K^+K^-\), was recently reported by the LHCb Collaboration [5], and later confirmed by the D0 Collaboration [6]. Subsequently, the LHCb Collaboration [1] have determined the final state composition of the decay channel using a modified Dalitz plot analysis where the decay angular distributions are included. The best fit model includes a nonresonant component and eight resonance states, whose absolute branching ratios are measured relative to that of the normalization decay mode \(B^+\rightarrow J/\psi K^+\). In contrast to hadron collider experiments, the Belle Collaboration [4] normalize to the absolute number of \(B^0_s {\bar{B}}^0_s\) pairs produced and also present a measurement of the entire \(B_s\rightarrow J/\psi K^+K^-\) components including resonant and nonresonant decays. More recently, the LHCb Collaboration [7] improved their measurements, in which the fit fractions of six resonances including \(\phi (1020), \phi (1680), f_2(1270), f_2'(1525), f_2(1750), f_2(1950)\) together with a *S*-wave structure in \(B_s\rightarrow J/\psi K^+K^-\) are determined.

Above measurements have caught theoretical attention recently. The three-body decay \(B_s\rightarrow J/\psi K^+K^-\) including its dominant contributions of the resonances \(\phi (1020)\) and \(f_2'(1525)\) have been studied [8, 9] and the associated branching ratios have been obtained based on the framework of the factorization approach. Some recent analyses [10, 11, 12] had been carried out for \(B/B_s\) decays into \(J/\psi \) and the scalar, vector, and tensor resonances using chiral unitary theory, for which these states are shown to be generated from the meson-meson interaction. In Refs. [13, 14, 15], the \(K^+K^-\) *S*-wave contribution in the \(\phi (1020)\) resonance region is estimated to be of the order \(1-10\%\), in agreement with previous measurements from LHCb [1, 16], CDF [17], and ATLAS [18]. The significant *S*-wave effects may affect measurements of the *CP* violating phase \(\beta _s\) [13, 14, 19].

In this paper, we will consider the three-body \(B_s\) decays involving charmonia and kaon pair in the final state under the quasi-two-body approximation in the framework of perturbative QCD approach (PQCD) [20, 21]. The factorization formalism for the three-body decays can be simplified to that for the two-body cases with the introduction of two-kaon distribution amplitudes (DAs), which absorb the strong interaction related to the production of the two kaon system. For the detailed description of the three-body nonleptonic *B* decays in this approach, one can refer to [22, 23]. The PQCD approach so far, has been successfully applied to the studies of the resonance contributions to the three-body \(B/B_s\) decays in several recent papers [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. As advanced before, the decays under study are dominated by a series of resonances in *S*, *P* and *D* waves, while contributions from resonances with spin greater than two are not expected since they are well beyond the available phase space. Each partial wave contribution is parametrized into the corresponding timelike form factors involved in the two-kaon DAs. For each partial wave form factor, we adopt the form as a linear combination of those resonances with the same spin. In the present paper, we take into account the following resonances.^{1}: \(f_0(980)\), \(f_0(1370)\), \(\phi (1020)\), \(\phi (1680)\), \(f_2(1270)\), \(f'_2(1525)\), \(f_2(1750)\), \(f_2(1950)\), two scalar, two vector, and four tensor resonances in the context of the data presented in Refs. [1, 7] All resonances are commonly described by Breit Wigner (BW) distributions, except for the \(f_0(980)\) state, which is modelled by a Flatté function [35].

## 2 Kinematics and the two-kaon distribution amplitudes

*M*. The momenta of the decay products will be denoted as \(p_1,p_2\) for the two kaons, and \(p_3\) for the charmonia, with the specific charge assignment according to

*m*. The kaon transverse momenta is expressed as \(\mathbf{P }_{\text {T}}=(\omega \sqrt{\zeta (1-\zeta )},0)\). The factor \(\eta =p^2/(M^2-m^2)\) is defined in terms of the invariant mass squared of the kaon pair \(p^2=\omega ^2\), which satisfies the momentum conservation \(p=p_1+p_2=p_B-p_3\). The valence quark momenta labeled by \(k_B\), \(k_3\), and

*k*, as indicated in Fig. 1a, are parametrized as

*z*denote the longitudinal momentum fractions, and \(k_{iT}\) represent the transverse momenta. Since the light spectator quark momentum

*k*moves with the kaon pair in the plus direction, the minus component of its parton momentum should be very small, thus it can be neglected in the hard kernel, and then integrated out in the definition of the two-kaon distribution amplitudes. We also dropped \(k_B^+\) because it vanishes in the hard amplitudes.

Since the \(B_s\) meson wave function and the charmonium distribution amplitudes have successfully described various hadronic two-body and three-body charmonium *B* decays [27, 28, 29, 30, 36], we use the same ansatz as them. For the sake of brevity, their explicit expressions are not shown here and can be found in Refs. [36, 37, 38]. Below, we briefly describe the two-kaon DAs in three partial waves and the associated form factors.

### 2.1 *S*-wave two-kaon DAs

*S*-wave two-kaon DAs are introduced in analogy with the case of two-pion ones [22, 24], which are organized into

*S*,

*P*, and

*D*always associate with the corresponding partial waves. Above various twists DAs have similar forms as the corresponding twists for a scalar meson by replacing the scalar decay constant with the scalar form factor [39], we adopt their asymptotic models as shown below [24, 30]:

*R*, to be determined by data. Their values are given in the next section. In what follows, all resonances

*R*with different quantum numbers will be labeled by the single letter

*R*, without pointing to its quantum numbers. The two phase-space factors are \(\rho _{\pi \pi }=2q_{\pi }/\omega \), \(\rho _{KK}=2q_{K}/\omega \), where \(q_{\pi (K)}\) is the pion (kaon) momentum in the dipion (dikaon) rest frame. The exponential factor \(F_{KK}=e^{-\alpha q_K^2}\) with \(\alpha =2.0\pm 0.25\) \(\hbox {GeV}^{-2}\) [41, 42] is introduced above the

*KK*threshold and serves to reduce the \(\rho _{KK}\) factor as the invariant mass increases. The constants \(g_{\pi \pi }\) and \(g_{KK}\) are the \(f_0(980)\) couplings to \(\pi \pi \) and \(K{{\bar{K}}}\) final states respectively. We use \(g_{\pi \pi }=167\) MeV and \(g_{KK}/g_{\pi \pi }=3.47\) as determined by LHCb [43]. The BW amplitude in generic form is

*R*because kaons have spin 0. The \(L_R=0,1,2,...\) correspond to the

*S*,

*P*,

*D*, ... partial wave resonances. The Blatt-Weisskopf barrier factors \(F_R\) [44] for scalar, vector and tensor states are

*z*when \(\omega =m_R\). The meson radius parameters

*r*are dependent on the momentum of the decay particles in the parent rest frame. Modifying the

*r*changes slightly our results, as discussed in the next section. Hence, we set this parameter to be 1.5 \(\text {GeV}^{-1}\) (corresponding to 0.3 fm) for all the considered resonances, as is obtained in [1].

### 2.2 *P*-wave two-kaon DAs

*P*-wave DAs including both longitudinal and transverse polarizations for the pion pair. Naively, the

*P*-wave two-kaon ones can be obtained by replacing the pion vector form factors by the corresponding kaon ones. The explicit expressions read

*L*and

*T*on the left-hand side denote the longitudinal polarization and transverse polarization, respectively. Here \(\epsilon ^{\mu \nu \rho \sigma }\) is the totally antisymmetric unit Levi-Civita tensor with the convention \(\epsilon ^{0123}=1\). The transverse polarization vector \(\epsilon _{T}\) for the dikaon system has the same form as that of dipion [29]. The various twists DAs in Eq. (10) can be expanded in terms of the Gegenbauer polynomials:

*P*-wave form factors \(F_P^{\parallel }\) and \(F_P^{\perp }\) serve as the normalization of the two-kaon DAs. They play a similar role with the vector and tensor decay constants in the definition of the vector meson DAs [45]. The Gegenbauer moments \(a_2^i\) will be regarded as free parameters and determined in this work.

### 2.3 *D*-wave two-kaon DAs

*D*-wave two-kaon DAs associated with longitudinal and transverse polarizations into

*p*and \(m_T\) are momentum and mass of the tensor meson, respectively. The two interpolating currents \(j_{\mu \nu }(0)\) and \(j_{\mu \nu \rho }(0)\) are defined in [46, 47]. Let us begin with the local matrix element \(\langle K^+(p_1)K^-(p_2)|j_{\mu \nu }(0)/j_{\mu \nu \rho }(0)|0\rangle \) associated with the

*D*-wave form factors. Under the tensor-meson-dominant hypothesis [47], inserting the tensor intermediate in above matrix element, we get

*D*-wave amplitudes, the helicity angle \(\theta \) is encoded into the Wigner-d functions, schematically [2]:

### 2.4 The differential branching ratio

^{2}Note that interference between different partial wave vanishes because the \(\zeta \) functions in Eqs. (5), (11), and (23), corresponding to

*S*,

*P*, and

*D*partial waves, are orthogonal. In the \(J/\psi \) and \(\psi (2S)\) cases, the decay amplitudes \({\mathcal {A}}_S\) and \({\mathcal {A}}_P\) here can be straightforwardly obtained from the previous publications [24, 29] by replacing the two-pion form factors and all pion masses and momenta with the respective kaon quantities. For \({\mathcal {A}}_D\), its factorization formula can be related to \({\mathcal {A}}_P\) by making the following replacement,

## 3 Numerical results

The relevant resonance parameters in the \(B_s\rightarrow X K^+K^-\) decays

Resonance | \(J^{PC}\) | Resonance formalism | Mass (MeV) | Width (MeV) | Source |
---|---|---|---|---|---|

\(f_0(980)\) | \(0^{++}\) | Flatté | 990 | \(\cdots \) | PDG [2] |

\(f_0(1370)\) | \(0^{++}\) | BW | 1475 | 113 | LHCb [52] |

\(\phi (1020)\) | \(1^{{-}{-}}\) | BW | 1019 | 4.25 | PDG [2] |

\(\phi (1680)\) | \(1^{{-}{-}}\) | BW | 1689 | 211 | Belle [53] |

\(f_2(1270)\) | \(2^{++}\) | BW | 1276 | 187 | PDG [2] |

\(f'_2(1525)\) | \(2^{++}\) | BW | 1525 | 73 | PDG [2] |

\(f_2(1750)\) | \(2^{++}\) | BW | 1737 | 151 | Belle [54] |

\(f_2(1950)\) | \(2^{++}\) | BW | 1980 | 297 | Belle [54] |

*S*-wave sector, the two experimental results of the \(f_0(980)\) and \(f_0(1370)\) components in Ref. [1] are used to fit out \(a_1\) and \(c_{f_0(1370)}\). In Fig. 2a, b, we show the dependence of the branching ratios of the \(f_0(980)\) and \(f_0(1370)\) components as well as their combination in the \(B_s\rightarrow J/\psi K^+K^-\) decay on the Gegenbauer moment \(a_1\) and the phase of \(c_{f_0(1370)}\), respectively. The module \(|c_{f_0(1370)}|\) is chosen as 0.12 to maximize the overlap between the predicted curves and the experimental range. Apparently, both the \(f_0(980)\) and \(f_0(1370)\) modes can meet the data as setting \(a_1\sim 0.8\) in Fig. 2a. This value is much larger than the corresponding \(a_1=0.2\) that obtained in Ref. [24]. The discrepancy is understandable with respect to the different nonperturbative dynamics of \(f_0(980)\) decaying to the

*KK*and \(\pi \pi \) pairs. In Fig. 2b, it is reflected that the constructive or destructive interference pattern between the two resonances vary with the phase. Here its value is taken to be \(-\pi /2\) since the LHCb’s data [1] favor the destructive interference. Of course, considering the sizeable experimental uncertainties, these parameters are difficult to be restricted precisely at this moment. As a case study with rough estimation, the related treatment about these parameters in this work is just a try. A convincible research should be performed through a global fit to more rich measurements in the future.

For the *P*-wave ones, we first use the experimental branching ratios of three decay channels \(B_s\rightarrow J/\psi \phi (1020)(\rightarrow K^+K^-)\) (longitudinal) [7], \(B_s\rightarrow \psi (2S) \phi (1020)(\rightarrow K^+K^-)\) (longitudinal) [2], and \(B_s\rightarrow \eta _c \phi (1020)(\rightarrow K^+K^-)\) [55] to fit the three longitudinal Gegenbauer moments \(a_2^0\), \(a_2^s\), and \(a_2^t\), then the three transverse ones can be constrained by the transverse polarization fractions of the former two modes. Finally, according to the fit fraction and polarizations of the \(\phi (1680)\) component in the \(B_s\rightarrow J/\psi K^+K^-\) decay from the LHCb [7], one can determine the values of \(r^T(\phi (1680))\) and \(c_{\phi (1680)}\).

*D*-wave resonances by the LHCb Collaboration [7], we can exactly determine its weight coefficient \(c_{f_2'(1525)}\) and two Gegenbauer moments \(a_1^0\) and \(a_1^T\) based on its fit fraction and three polarizations. Following a similar procedure as above, we can determine \(r^T(R)\) and the module of \(c_R\) for other tensor resonances. As pointed out in [56], the form factor

*F*(

*s*) with the time-like momentum transfer squared \(s>4m_K^2\) could be analytically continued to the space-like region \(s<0\). It has been known that a form factor is normalized to unity at \(s=0\), because a soft probe cannot reveal the structure of a bound state. Therefore, we postulate that the kaon form factors should be constrained by such normalization condition. According to our fitted modules of the

*D*-wave weight coefficients in Eq. (34), the phases of \(c_{f_2(1270)}\) and \(c_{f_2(1750)}\) are set to \(\pi \) to ensure the normalization of the form factor \(F^{\parallel }_D(0)=1\). Strictly speaking, the phases of the various coefficient \(c_i\) in Eq. (34) should be determined from the interference fit fractions. However, the current available data are not yet sufficiently precise to extract them. Furthermore, the \(f'_2(1525)\) dominates over the

*D*-wave contributions as shown below, the relative phases among these

*c*parameters have little effect on the total

*D*-wave decay branching ratios, and our choice of these phases does not affect the magnitude estimation of either the individual resonance or the total contribution.

Branching ratios of *S*-wave resonance contributions to the \(B_s\rightarrow (J/\psi ,\psi (2S),\eta _c,\eta _c(2S)) K^+K^-\) decays. Theoretical errors correspond to the uncertainties of Gegenbauer moments and hard scales, respectively

Modes | \({\mathcal {B}}(R=f_0(980))\) | \({\mathcal {B}}(R=f_0(1370))\) | \(\text {S-wave}^{a}\) |
---|---|---|---|

\(J/\psi K^+K^- \) | \((4.3_{-1.3-0.3}^{+1.3+0.7})\times 10^{-5}\) | \((9.4^{+3.1+1.5}_{-3.4-0.4})\times 10^{-6}\) | \((3.4_{-0.9-0.4}^{+1.0+0.5})\times 10^{-5}\) |

Data [1] | \((3.7\sim 9.2)\times 10^{-5}\,^{b}\) | \((9.2^{+3.3}_{-9.2})\times 10^{-6}\,^{c}\) | \(\cdots \) |

\(\eta _c K^+K^- \) | \((4.6_{-1.1-0.5}^{+0.8+0.8})\times 10^{-5}\) | \((1.1^{+0.2+0.2}_{-0.3-0.1})\times 10^{-5}\) | \((3.5_{-0.7-0.3}^{+0.8+0.6})\times 10^{-5}\) |

\(\psi (2S) K^+K^- \) | \((8.0_{-2.6-1.2}^{+2.2+1.8})\times 10^{-6}\) | \((8.3^{+3.0+1.9}_{-3.0-0.8})\times 10^{-7}\) | \((6.2_{-1.7-0.8}^{+1.7+1.4})\times 10^{-6}\) |

\(\eta _c(2S) K^+K^-\) | \((9.3_{-1.9-1.1}^{+1.7+1.9})\times 10^{-6}\) | \((1.4^{+0.3+0.3}_{-0.3-0.1})\times 10^{-6}\) | \((7.2^{+1.5+1.4}_{-1.3-0.9})\times 10^{-6}\) |

Branching ratios of *P*-wave resonance contributions to the \(B_s\rightarrow (J/\psi ,\psi (2S),\eta _c,\eta _c(2S)) K^+K^-\) decays. Theoretical errors are attributed to the Gegenbauer moments, form factor ratios, and hard scales, respectively. The statistical and systematic uncertainties from data [2, 7, 55] are combined in quadrature

Modes | \({\mathcal {B}}(R=\phi (1020))\) | \({\mathcal {B}}(R=\phi (1680))\) | P-wave |
---|---|---|---|

\(J/\psi K^+K^-\) | \(5.7^{+0.2+0.9+0.2}_{-0.2-0.7-0.0}\times 10^{-4}\) | \(3.6^{+0.1+0.4+0.3}_{-0.1-0.3-0.3}\times 10^{-5}\) | \(5.9_{-0.1-0.7-0.0}^{+0.1+0.9+0.3}\times 10^{-4}\) |

Data [7] | \((5.6\pm 0.5)\times 10^{-4}\,^{a}\) | \((3.2\pm 0.4)\times 10^{-5}\,^{a}\) | \(\cdots \) |

\( \eta _c K^+K^- \) | \(2.4^{+0.2+0.5+0.2}_{-0.2-0.4-0.0}\times 10^{-4}\) | \(1.1^{+0.1+0.2+0.0}_{-0.1-0.2-0.0}\times 10^{-5}\) | \(2.4^{+0.2+0.6+0.4}_{-0.1-0.3-0.0}\times 10^{-4}\) |

Data [55] | \((2.5\pm 0.4)\times 10^{-4}\,^{b}\) | \(\cdots \) | \(\cdots \) |

\(\psi (2S) K^+K^- \) | \(2.4^{+0.0+0.3+0.0}_{-0.1-0.4-0.0}\times 10^{-4}\) | \(3.2^{+0.1+0.7+0.4}_{-0.1-0.4-0.0}\times 10^{-6}\) | \(2.3^{+0.1+0.3+0.1}_{-0.1-0.4-0.0}\times 10^{-4}\) |

Data [2] | \((2.6\pm 0.3)\times 10^{-4}\,^{c}\) | \(\cdots \) | \(\cdots \) |

\( \eta _c(2S) K^+K^- \) | \(8.0^{+0.6+2.2+0.9}_{-0.6-1.8-0.5}\times 10^{-5}\) | \(1.1^{+0.1+0.2+0.2}_{-0.1-0.2-0.0}\times 10^{-6}\) | \(8.0^{+0.6+2.0+1.0}_{-0.7-1.8-0.5}\times 10^{-5}\) |

Modes | \({\mathcal {B}}(R=f'_2(1525))\) | \({\mathcal {B}}(R=f_2(1270))\) | \({\mathcal {B}}(R=f_2(1750))\) | \({\mathcal {B}}(R=f_2(1950))\) | D-wave |
---|---|---|---|---|---|

\(J/\psi K^+K^-\) | \(8.9^{+2.8+1.8+0.9}_{-2.3-1.3-0.1}\times 10^{-5}\) | \(2.3^{+0.8+0.5+0.3}_{-0.5-0.4-0.1}\times 10^{-7}\) | \(5.0^{+1.5+0.6+0.1}_{-1.3-0.4-0.0}\times 10^{-6}\) | \(4.1^{+1.3+0.8+0.1}_{-1.0-0.7-0.0}\times 10^{-6}\) | \(9.2^{+2.8+1.9+0.9}_{-2.5-1.2-0.0}\times 10^{-5}\) |

Data [7] | \((8.5\pm 1.2)\times 10^{-5}\) | \((1.3\pm 0.3)\times 10^{-5}\) | \(4.7^{+2.4}_{-2.1}\times 10^{-6}\) | \(3.5^{+1.7}_{-1.4}\times 10^{-6}\) | \(\cdots \) |

\(\eta _c K^+K^-\) | \(4.9^{+2.1+1.6+0.5}_{-1.8-1.3-0.2}\times 10^{-5}\) | \(1.5^{+0.6+0.3+0.1}_{-0.5-0.3-0.1}\times 10^{-7}\) | \(2.3^{+1.0+0.8+0.3}_{-0.8-0.5-0.1}\times 10^{-6}\) | \(2.6^{+1.1+0.4+0.1}_{-0.9-0.4-0.0}\times 10^{-6}\) | \(4.9^{+2.2+1.6+0.4}_{-1.7-1.2-0.1}\times 10^{-5}\) |

\( \psi (2S)K^+K^- \) | \(1.3^{+0.4+0.2+0.0}_{-0.4-0.2-0.1}\times 10^{-5}\) | \(5.9^{+1.8+1.4+0.3}_{-1.6-1.2-0.3}\times 10^{-8}\) | \(\cdots \) | \(\cdots \) | \(1.3^{+0.3+0.2+0.0}_{-0.4-0.2-0.1}\times 10^{-5}\) |

\(\eta _c(2S)K^+K^- \) | \(0.7^{+0.4+0.3+0.1}_{-0.2-0.2-0.1}\times 10^{-5}\) | \(3.2^{+1.4+0.8+0.1}_{-1.1-0.7-0.1}\times 10^{-8}\) | \(\cdots \) | \(\cdots \) | \(0.7^{+0.3+0.3+0.1}_{-0.2-0.2-0.0}\times 10^{-5}\) |

The calculated branching ratios of *S*, *P*, and *D*-wave resonance contributions to the \(B_s\rightarrow (J/\psi ,\psi (2S),\eta _c,\eta _c(2S))K^+K^-\) decays are collected in Tables 2, 3, and 4, respectively. The last column of each Table are the corresponding total partial wave branching ratios. The theoretical errors stem from the uncertainties for fitted values of Gegenbauer moments \(a_i\), the form factor ratios \(r^T(R)\), and the hard scales *t*, respectively. For Gegenbauer moments in the twist-2 DAs, we vary their values within a \(20\%\) range for the error estimation. The uncertainty of the ratio \(r^T(R)\) in Eqs. (33) and (34) are general assigned to be \(\delta r= \pm 0.2\). The hard scales vary from 0.75*t* to 1.25*t* to characterize the energy release in decay process. It is necessary to stress that the second uncertainty from \(r^T(R)\) is absent for the *S*-wave resonance contributions in Table 2. The uncertainties stemming from the weight coefficients \(c_R\) are not shown explicitly in these Tables, whose effect on the branching ratios via the relation of \({\mathcal {B}}\propto |c_R|^2\). For the *S* and *D*-waves resonance contributions, the twist-3 in the two-kaon DAs are taken as the asymptotic forms for lack of better results from nonperturbative methods, which may give significant uncertainties. We have checked the sensitivity of our results to the choice of the meson radius parameter *r* [ see Eq. (9)] in the BW parametrization. The variation of its value from 0 to 3.0 \(\text {GeV}^{-1}\) results in the change of the branching ratios and polarizations by only a few percents. In general, our results are more sensitive to those hadronic parameters.

Before discussing the results of our calculations in detail, we wish to explain the quoted experimental values that appear in these Tables. The measured branching ratio for each resonant component in the concerned decays are calculated by multiplying its fit fraction and the total three-body decay branching ratio.^{3} The fit fractions of *P* and *D*-wave resonances are taken from the most recent LHCb experiment [7], which superseded the earlier one from [1]. However, in Ref. [7], the *S*-wave component is described in a model-independent pattern, making no assumptions of any \(f_0\) resonant structures. Therefore, we use the *S*-wave \(f_0(980)\) and \(f_0(1370)\) fractions from [1]. Note that the \(f_0(980)\) fraction is strongly parametrization dependent. For instance, the parameter set by *BABAR* gives a smaller fit fraction \((4.8\pm 1.0)\%\), while the parameter set by LHCb gives a larger value \((12.0\pm 1.8)\%\) [see Table VI of Ref. [1]]. Therefore, we quote the central values in a wide range according to the two models rather than a central value plus or minus its statistical and systematic uncertainties for the \(f_0(980)\) resonance in Table 2. For other charmonium channels, the detailed partial wave analysis for determining various resonance fractions are still missing due to a limited number of events. The quasi-two-body branching ratios can be built from product of two two-body branching ratios when available in the narrow-width limit, namely, \({\mathcal {B}}(B_s\rightarrow X R(\rightarrow K^+K^-))\approx {\mathcal {B}}(B_s\rightarrow X R)\times {\mathcal {B}}(R\rightarrow K^+K^-)\). For example, we have used the experimental numbers \({\mathcal {B}}(B_s\rightarrow \eta _c\phi (1020))=(5.0\pm 0.9)\times 10^{-4}\) [2, 55] and \({\mathcal {B}}(\phi (1020)\rightarrow K^+K^-)=(49.2\pm 0.5)\%\) [2] to obtain the experimental branching ratio for \({\mathcal {B}}(B_s\rightarrow \eta _c\phi (1020)(\rightarrow K^+K^-))=(2.5\pm 0.4)\times 10^{-4}\), which is shown in Table 3.

^{4}For illustration we have explicitly written the relative ratio of \({\mathcal {B}}(B_s\rightarrow J/\psi f_2(1270)(\rightarrow K^+K^-))\) compared to \({\mathcal {B}}(B_s\rightarrow J/\psi f_2(1270)(\rightarrow \pi ^+\pi ^-))\) in the narrow-width limit as

- (1)Using our values from Table 2, we expect thatwhere the value of \({\mathcal {B}}(B_s\rightarrow J/\psi f_0(980)(\rightarrow \pi ^+\pi ^-))=1.15^{+0.52}_{-0.41}\times 10^{-4}\) is read from the previous PQCD calculations [24]. On the experimental side,$$\begin{aligned}&\frac{{\mathcal {B}}(B_s\rightarrow J/\psi f_0(980)(\rightarrow K^+K^-))}{{\mathcal {B}}(B_s\rightarrow J/\psi f_0(980)(\rightarrow \pi ^+\pi ^-))} \nonumber \\&\quad \approx \frac{{\mathcal {B}}( f_0(980)\rightarrow K^+K^-)}{{\mathcal {B}}(f_0(980)\rightarrow \pi ^+\pi ^-)}=0.37^{+0.23}_{-0.13}, \end{aligned}$$(36)
*BABAR*measures the ratio of the partial decay width of \(f_0(980)\rightarrow K^+K^-\) to \(f_0(980)\rightarrow \pi ^+\pi ^-\) of \(0.69\pm 0.32\) using \(B\rightarrow KKK\) and \(B\rightarrow K\pi \pi \) decays [58]. While BES performs a partial wave analysis of \(\chi _{c0}\rightarrow f_0(980)f_0(980)\rightarrow \pi ^+\pi ^-\pi ^+\pi ^-\) and \(\chi _{c0}\rightarrow f_0(980)f_0(980)\rightarrow \pi ^+\pi ^-K^+K^-\) in \(\psi (2S)\rightarrow \gamma \chi _{c0}\) decay and extracts the ratio as \(0.25^{+0.17}_{-0.11}\) [59, 60]. Their weighted average yields \(0.35^{+0.15}_{-0.14}\). It can be seen that our estimate in Eq. (36) is consistent with this experimental average value. - (2)Combining Tables 2, 3 and the number in Eq. (36) , we obtain the ratio:comply with the latest average of Heavy Flavor Averaging Group (HFAVG) \({\mathcal {R}}_{f_0/\phi }=0.207\pm 0.016\) [61] from the measurements [62, 63, 64, 65]$$\begin{aligned} {\mathcal {R}}_{f_0/\phi }= & {} \frac{{\mathcal {B}}(B_s\rightarrow J/\psi f_0(980)(\rightarrow \pi ^+\pi ^-))}{{\mathcal {B}}(B_s\rightarrow J/\psi \phi (1020)(\rightarrow K^+K^-))}\nonumber \\= & {} 0.203^{+0.126}_{-0.095}, \end{aligned}$$(37)In comparison to previous theoretical estimation \(0.122^{+0.081}_{-0.058}\) obtained in [66], our value turns out to be larger.$$\begin{aligned} {\mathcal {R}}_{f_0/\phi }=\left\{ \begin{aligned}&0.252^{+0.046}_{-0.032}(\text {stat})^{+0.027}_{-0.033}(\text {syst}) \quad \quad \quad&\text {LHCb}, \\&0.275 \pm 0.041 (\text {stat})\pm 0.061 (\text {syst}) \quad \quad \quad&\text {D0}, \\&0.140 \pm 0.008 (\text {stat})\pm 0.023 (\text {syst}) \quad \quad \quad&\text {CMS}, \\&0.257 \pm 0.020 (\text {stat})\pm 0.014 (\text {syst}) \quad \quad \quad&\text {CDF}. \\ \end{aligned}\right. \end{aligned}$$
- (3)
Evidence of the \(f_0(1370)\) resonance in \(B_s\rightarrow J/\psi \pi ^+\pi ^-\) decay is reported by Belle [67] with a significance of \(4.2\sigma \). The corresponding product branching fraction is measured to \({\mathcal {B}}(B_s\rightarrow J/\psi f_0(1370), f_0(1370)\rightarrow \pi ^+\pi ^-)= 3.4^{+1.4}_{-1.5}\times 10^{-5}\) [67].

^{5}Combined with our prediction on the*KK*channel in Table 2, one can estimate the relative branching ratios of \(f_0(1370)\rightarrow K^+K^-/\pi ^+\pi ^-\) lie in the range (\(0.2\sim 0.5\)). Since the situation of the knowledge of the \(f_0(1370)\) decaying into*KK*or \(\pi \pi \) is rather unclear, above expected values should be investigated further in the future with more precise data. - (4)From Tables 3 and 4, we get another interesting ratioin which the two known branching ratios \({\mathcal {B}}(\phi (1020)\rightarrow K^+K^-)=(49.2\pm 0.5)\%\) and \({\mathcal {B}}(f'_2(1525)\rightarrow K^+K^-)=\frac{1}{2}(88.7\pm 2.2)\%\) [2] are used. Our central value is in accordance with the previous theoretical estimation of \(0.154^{+0.090}_{-0.070}\) [9]. Experimentally, different Collaborations reported their measurements \({\mathcal {R}}_{f'_2/\phi }=0.215\pm 0.049(\text {stat})\pm 0.026(\text {syst})\) (Belle [4]), \({\mathcal {R}}_{f'_2/\phi }=0.264\pm 0.027(\text {stat})\pm 0.024(\text {syst})\) (LHCb [5]), and \({\mathcal {R}}_{f'_2/\phi }=0.19\pm 0.05(\text {stat}) \pm 0.04(\text {syst})\) (D0 [6]). It seems that theoretical predictions are generally smaller than the experimental measurements. None the less, including the errors, both the theoretical predictions and experimental data can still agree with each other.$$\begin{aligned} {\mathcal {R}}_{f'_2/\phi }=\frac{{\mathcal {B}}(B_s\rightarrow J/\psi f'_2(1525))}{{\mathcal {B}}(B_s\rightarrow J/\psi \phi (1020))}=0.173^{+0.070}_{-0.059}, \end{aligned}$$(38)
- (5)Finally, we estimate the relative branching ratios between two tensor modesThe current PDG values of$$\begin{aligned} {\mathcal {R}}_{f_2/f'_2}=\frac{{\mathcal {B}}(B_s\rightarrow J/\psi f_2(1270))}{{\mathcal {B}}(B_s\rightarrow J/\psi f'_2(1525))}=0.050^{+0.005}_{-0.003}. \end{aligned}$$(39)are dominated by the LHCb measurement [1, 52]. Combined with the experiment value \({\mathcal {B}}(f_2(1270)\rightarrow \pi ^+\pi ^-)=\frac{2}{3}(84.2^{+2.9}_{-0.9})\%\), one obtains the measured ratio \({\mathcal {R}}_{f_2/f'_2}=0.02\pm 0.01\), which is only half of our prediction in Eq. (39). However, the datum for \(f_2(1270)\) mode has been further reviewed in Ref. [57], the updated branching ratio is \({\mathcal {B}}(B_s\rightarrow J/\psi f_2(1270)(\rightarrow \pi ^+\pi ^-))=(6.8 \pm 1.0)\times 10^{-6}\) with statistical uncertainty only, corresponding to \({\mathcal {R}}_{f_2/f'_2}=0.05\pm 0.01\). It is clear that our prediction on this ratio is marginally consistent with the updated experiment. In addition, based on the chiral unitary approach for mesons, the authors of Ref. [10] present a larger value \({\mathcal {R}}_{f_2/f'_2}=0.084\pm 0.046\). Recalling that the theoretical errors are relatively large, so within a \(1\sigma \) tolerance, one still can count them as being consistent.$$\begin{aligned}&{\mathcal {B}}(B_s\rightarrow J/\psi f_2(1270)(\rightarrow \pi ^+\pi ^-))\nonumber \\&\quad =(3.14\pm 0.9)\times 10^{-6},\nonumber \\&{\mathcal {B}}(B_s\rightarrow J/\psi f'_2(1525))\nonumber \\&\quad =(2.6\pm 0.6)\times 10^{-4}, \end{aligned}$$(40)

*S*-wave total contribution due to the destructive interference between the two resonances. In fact, the best fit model from the LHCb experiment [1] also shows that the destructive interference between \(f_0(980)\) and \(f_0(1370)\) resonances in the \(B_s\rightarrow J/\psi K^+K^-\) channel. The interference between the two

*P*-wave resonances \(\phi (1020)\) and \(\phi (1680)\) is rather small due to the relatively narrow width of the former (\(\Gamma _{\phi (1020)}=\)4.25 MeV). Since the contribution of the latter is an order of magnitude smaller, the

*P*-wave resonance contribution is almost equal to the \(\phi (1020)\) one. By the same token, the

*D*-wave resonance contribution mainly come from the \(f_2'(1525)\) component, while other tensor resonance contributions are at least one order smaller. The peak of the high-mass vector resonance \(\phi (1680)\) lie almost on the upper limit of the allowed phase space for the 2

*S*charmonium modes, their rates suffer a strong suppression and are smaller than that of ground state charmonium channels by almost a factor of 10. Higher-mass \(K^+K^-\) resonances like \(f_2(1750)\) and \(f_2(1950)\) are beyond the invariant mass spectra for the 2

*S*charmonium modes, their contributions are absent in the last two rows of Table 4. As stated above, any interference contribution between different spin-

*J*states integrates to zero. Therefore, summing over the contributions of the various partial waves, we can obtain the total three-body decay branching ratios

*S*-wave contribution in our calculations. The small gap might be offset by the nonresonant term and its interference with the resonant components. For other modes, their branching ratios can also reach the order of \(10^{-4}\), which is large enough to permit a measurement.

Modes | \(f_0(\%)\) | \(f_{\parallel }(\%)\) | \(f_{\perp }(\%)\) |
---|---|---|---|

\(B_s\rightarrow J/\psi \phi (1020)(\rightarrow K^+K^-)\) | \(50.6^{+1.2+5.9+2.5}_{-1.5-4.4-1.7}\) | \(24.4^{+1.1+1.8+1.0}_{-0.6-2.1-1.3}\) | \(24.9^{+0.6+2.7+0.9}_{-0.5-3.6-1.1}\) |

Data [7] | \(50.9 \pm 0.4 \) | \(23.1\pm 0.5\) | \(26.0\pm 0.6\) |

\(B_s\rightarrow J/\psi \phi (1680)(\rightarrow K^+K^-)\) | \(49.1^{+1.5+3.8+0.6}_{-1.1-3.6-0.9}\) | \(20.1^{+0.2+0.3+0.7}_{-0.5-0.2-0.3}\) | \(30.8^{+0.9+3.3+0.1}_{-1.1-3.6-0.3}\) |

Data [7] | \(44.0\pm 3.9 \) | \(32.7\pm 3.6\) | \(23.3\pm 3.6\) |

\(B_s\rightarrow \psi (2S) \phi (1020)(\rightarrow K^+K^-)\) | \(44.1^{+1.3+3.8+0.6}_{-2.4-5.0-1.8}\) | \(23.2^{+1.2+1.8+1.1}_{-0.8-1.3-0.4}\) | \(32.7^{+1.1+3.1+0.7}_{-0.5-2.4-0.2}\) |

Data [69] | \(42.2\pm 1.4\) | \(\cdots \) | \(26.4 \pm 2.4\) |

\(B_s\rightarrow \psi (2S) \phi (1680)(\rightarrow K^+K^-)\) | \(45.3^{+0.3+3.5+0.2}_{-0.4-3.3-1.3}\) | \(15.1^{+0.2+0.3+0.9}_{-0.1-0.2-0.3}\) | \(39.6^{+0.1+3.5+0.4}_{-0.1-3.8-0.0}\) |

\(B_s\rightarrow J/\psi f_2'(1525)(\rightarrow K^+K^-)\) | \(51.1^{+14.1+4.3+1.6}_{-14.2-3.6-0.3}\) | \(26.2^{+7.6+1.9+0.0}_{-7.6-2.1-0.4}\) | \(22.7^{+6.7+1.7+0.6}_{-6.5-2.2-1.4}\) |

Data [7] | \(46.8\pm 1.9\) | \(33.8\pm 2.3\) | \(19.4\pm 2.8\) |

\(B_s\rightarrow J/\psi f_2(1270)(\rightarrow K^+K^-)\) | \(42.9^{+12.5+3.0+2.3}_{-14.2-2.7-2.4}\) | \(29.5^{+7.4+1.3+0.6}_{-6.4-1.5-0.6}\) | \(27.6^{+6.7+1.4+1.8}_{-6.1-1.6-1.7}\) |

Data [7] | \(76.9\pm 5.5\) | \(6.0\pm 4.2\) | \(17.1\pm 5.0\) |

\(B_s\rightarrow J/\psi f_2(1750)(\rightarrow K^+K^-)\) | \(53.7^{+13.9+1.8+2.1}_{-14.4-3.3-1.7}\) | \(25.3^{+7.8+1.6+0.8}_{-7.6-0.7-0.3}\) | \(21.0^{+6.5+1.7+0.9}_{-6.3-1.0-1.9}\) |

Data [7] | \(58.2\pm 13.9\) | \(31.7\pm 12.4\) | \(10.1^{+16.8}_{-6.1}\) |

\(B_s\rightarrow J/\psi f_2(1950)(\rightarrow K^+K^-)\) | \(30.2^{+13.1+3.1+1.8}_{-11.1-1.2-0.0}\) | \(36.9^{+5.8+0.6+0.6}_{-6.9-1.5-1.4}\) | \(32.9^{+5.2+0.6+0.8}_{-6.2-1.6-2.5}\) |

Data [7] | \(2.2^{+6.7}_{-1.5}\) | \(38.3\pm 13.8\) | \(59.5\pm 14.2\) |

\(B_s\rightarrow \psi (2S) f_2'(1525)(\rightarrow K^+K^-)\) | \(41.9^{+14.2+1.8+1.0}_{-13.5-2.7-1.9}\) | \(34.7^{+8.0+2.8+0.6}_{-8.5-2.2-0.0}\) | \(23.4^{+5.5+0.5+1.5}_{-5.7-0.2-1.5}\) |

\(B_s\rightarrow \psi (2S) f_2(1270)(\rightarrow K^+K^-)\) | \(36.4^{+13.9+3.2+0.9}_{-12.4-2.1-1.0}\) | \(37.0^{+7.3+1.1+0.8}_{-8.1-1.7-0.6}\) | \(26.5^{+5.2+1.0+1.7}_{-5.8-1.4-1.6}\) |

In the literatures, most of the theory studies concentrate on several dominant resonant components. For example, the authors of Ref. [9] considered two dominant \(\phi (1020)\) and \(f'_2(1525)\) resonances in the \(B_s\rightarrow J/\psi K^+K^-\) decay. The predicted resonance contributions as well as the total three-body decay branching ration are \((5.6\pm 0.7)\times 10^{-4}\), \(1.8^{+1.1}_{-0.8}\times 10^{-4}\),^{6} and \(9.3^{+1.3}_{-1.1}\times 10^{-4}\), respectively. Another earlier paper [8] also discuss the concerned decays in the QCD factorization approach. The three-body branching ratio was obtained by applying Dalitz plot analysis to be \({\mathcal {B}}(B_s\rightarrow J/\psi K^+K^-)=(10.3\pm 0.9)\times 10^{-4}\). In a recent paper [68], the authors have performed phenomenological studies on the \(B_s\rightarrow J/\psi f_0(980)\) decay in the two-body PQCD formalism. With the mixing angle between the \(f_0(500)\) and \(f_0(980)\) in the quark-flavor basis adopting as \(25^{\circ }\), the calculated branching ratio for the two-body channel \(B_s\rightarrow J/\psi f_0(980)\), was converted into quasi-two-body one as \({\mathcal {B}}(B_s\rightarrow J/\psi f_0(980)(\rightarrow K^+K^-))=4.6^{+2.6}_{-2.3}\times 10^{-5}\). Overall, our results are comparable with these theoretical predictions within the error bars.

The differential branching ratios of the considered decays are plotted on \(\omega \) in Fig. 3, in which the green, purple, red, blue, orange, cyan, wine, and black lines show the \(f_0(980)\), \(\phi (1020)\), \(f_2(1270)\), \(f_0(1370)\), \(f'_2(1525)\), \(\phi (1680)\), \(f_2(1750)\), and \(f_2(1950)\) resonance contributions, respectively. To see more clearly all the resonance peaks, especially in the region of the \(f_2(1270)\) resonance, we draw them in both linear (left panels) and logarithmic (right panels) scales for each decay channel. It is clear that an appreciable peak arising from the \(\phi (1020)\) resonance, accompanied by \(f_2'(1525)\). Another three resonance peaks of \(f_0(980)\), \(f_0(1370)\), and \(\phi (1680)\) have relatively smaller strengths than the \(f_2'(1525)\) one, but their broader widths compensate the integrated strengths over the entire phase space. Therefore, the branching ratios of the four components are predicted to be of a comparable size. Apart from above obvious signal peak, there are two visible structures at about 1750 MeV and 1950 MeV in Fig. 3a, c, but not in Fig. 3e, g because the two higher mass regions are beyond the *KK* invariant mass spectra for the 2*S* charmonium state modes. The contributions of the tensor \(f_2(1270)\), however, can hardly be seen since its strength is found to be compatible with zero and its peak almost overlap with the tail of those higher mass states like \(f_2(1750)\) and \(f_2(1950)\). The obtained distribution for the most of resonance contributions to the \(B_s\rightarrow J/\psi K^+K^-\) decay agrees fairly well with the LHCb data shown in Fig. 7 of Ref. [7], while other predictions could be tested by future experimental measurements.

The PQCD results for the polarization fractions together with the LHCb data, are listed in Table 5. The sources of the errors in the numerical estimates have the same origin as in the discussion of the branching ratios in Table 3. For most modes, the transverse polarization fraction \(f_T=f_{\parallel }+f_{\perp }\) and the longitudinal one are roughly equal. Nevertheless, for the \(f_2(1950)\) mode, the longitudinal polarization fraction is suppressed to \(30\%\) owing to a larger \(r^T(f_2(1950))\) in Eq. (34) enhances its transverse polarization contribution. Even so, the longitudinal polarization fraction is still larger than the experimental value. Of course, taking into account both the theoretical and experimental errors, the deviation is less than \(3\sigma \).

For the *P*-wave resonant channels, the parallel polarization fractions are slightly smaller than the corresponding perpendicular one in our calculations, while the LHCb’s data show an opposite behavior for the \(\phi (1680)\) mode. As pointed out in Ref. [29], the relative importance of the parallel and perpendicular polarization amplitudes in the \(\rho \) channels are sensitive to the two Gegenbauer moments \(a_2^{a}\) and \(a_2^{v}\). The similar situation also exist in this work. Strictly speaking, the Gegenbauer moments in two-hadron DAs are not constants, but depend on the dihadron invariant mass \(\omega \). However, the explicit behaviors of those Gegenbauer moments with the \(\omega \) are still unknown and the available data are not yet sufficiently precise to control their dependence. Here, we do not consider the \(\omega \) dependence and assume the Gegenbauer moments for the resonances with same spin are universal. That is to say it is unlikely to accommodate the measured \(B_s\rightarrow J/\psi \phi (1020), J/\psi \phi (1680)\) parallel and perpendicular polarization simultaneously with the same set of Gegenbauer moments in PQCD. A further theoretical study of the \(\omega \) dependence of the Gegenbauer moments will clarify this issue.

For the *D*-wave mode \(B_s\rightarrow J/\psi f_2(1270)\), compared with the data from the LHCb, our predicted longitudinal polarization is smaller while the two transverse ones are larger [see Table 5]. As stressed before, the \(f_2(1270)\) fit fraction in the *KK* mode is unexpected, so its polarizations may have a similar situation. In fact, the best fit model from LHCb [42, 57] on the \(B_s\rightarrow J/\psi \pi ^+\pi ^-\) decay showing the longitudinal polarization for the \(f_2(1270)\) component is obviously smaller than the transverse ones. As it is hard to understand why the polarization patterns of \(f_2(1270)\) resonance decaying into \(\pi \pi \) and *KK* pairs are so different, a refined measurement of the \(f_2(1270)\) contribution to the \(J/\psi K^+K^-\) mode is urgently needed in order to clarify such issue.

## 4 Conclusion

In this paper we carry out an systematic analysis of the \(B_s\) meson decaying into charmonia and \(K^+K^-\) pair by using the PQCD approach. This type of process is expected to receive dominant contributions from intermediate resonances, such as the vector \(\phi (1020)\), tensor \(f'_2(1525)\), and scalar \(f_0(980)\), thus can be considered as quasi-two-body decays. In addition to the three prominent components mentioned above, some significant excitations in the entire \(K^+K^-\) mass spectrum, which have been well established in the \(B_s\rightarrow J/\psi K^+K^-\) decay, are also included. These resonances fall into three partial waves according to their spin, namely, *S*, *P*, and *D*-wave states. Each partial wave contribution is parametrized into the corresponding timelike form factor involved in the two-kaon DAs, which can be described by the coherent sum over resonances sharing the same spin. The \(f_0(980)\) component is described by a Flatté line shape, while other resonances are modeled by the Breit-Wigner function.

After determining the hadronic parameters involved in the two-kaon DAs by fitting our formalism to the available data, we have calculated each resonance contribution in the processes under consideration. It is found that the largest component is the \(\phi (1020)\), followed by \(f'_2(1525)\), with others being almost an order of magnitude smaller. The resultant invariant mass distributions for most resonances in the \(B_s\rightarrow J/\psi K^+K^-\) decay show a similar qualitative behavior as the LHCb experiment. Since the interference contributions between any two different spin resonances are zero, summing over various partial wave contributions, we can estimate the total three-body decay branching ratios. The obtained branching ratio of the \(B_s\rightarrow J/\psi K^+K^-\) decay is in accordance with available experimental data and numbers from other approaches. The modes involving 2S charmonium have sizable three-body branching ratios, of order \(10^{-4}\), which seem to be in the reach of future experiments. As a cross-check, we have discussed some interesting relative branching ratios and compared with available experimental data and other theoretical predictions.

Three polarization contributions were also investigated in detail for the vector-vector and vector-tensor modes. For most of channels, the transverse polarization is found to be of the same size as the longitudinal one and the parallel and perpendicular polarizations are also roughly equal, while for some higher resonance modes, the polarization patterns can be different. The obtained results can be confronted to the experimental data in the future.

Finally, we emphasize that further experimental investigations on the \(f_2(1270)\) component in the \(B_s \rightarrow J/\psi K^+K^-\) decay based on much larger data samples are urgently necessary.

## Footnotes

- 1.
In the following, we also use the abbreviation \(f_0\), \(\phi \), and \(f_2\) to denote the

*S*,*P*, and*D*-wave resonances for simplicity. - 2.
The last two terms do not appear for those modes involving spinless \(\eta _c/\eta _c(2S)\) in the final state.

- 3.
So far, only the \(B_s\rightarrow J/\psi K^+K^-\) mode is well measured. Its weighted average branching ratio, given by the Particle Data Group (PDG), is \({\mathcal {B}}(B_s\rightarrow J/\psi K^+K^-)=(7.9\pm 0.7)\times 10^{-4}\) [2], where the statistical and systematic uncertainties are combined in quadrature.

- 4.
From discussions with Liming Zhang and Xuesong Liu, the \(f_2(1270)\) fraction in \(B_s\rightarrow J/\psi K^+K^-\) [7] could be too high because the misidentified background from \(B_d\rightarrow J/\psi K^+\pi ^-\) in the \(f_2(1270)\) region may give some systematic uncertainties.

- 5.
The PDG also present a value of \({\mathcal {B}}(B_s\rightarrow J/\psi f_0(1370)(\rightarrow \pi ^+\pi ^-)) =4.5^{+0.7}_{-4.0}\times 10^{-5}\) measured by the LHCb Collaboration [52], which is obtained by multiplying the corresponding normalized fit fraction and the branching ratio of the normalization mode \(B_s\rightarrow J/\psi \phi (1020)\). Although its central value is consistent with former measurements from Belle, but suffers from sizeable systematic uncertainties. We do not use its result for further calculations.

- 6.
From discussion with Néstor Quintero, there is a typo for the \(f'_2(1525)\) contribution in the Table IV of [9], its value should be 1.8 rather than 0.8, such that the sum in the last column is 9.3.

## Notes

### Acknowledgements

We acknowledge Hsiang-nan Li for helpful discussions and Liming Zhang for enlightening discussions concerning the experiments. This work was supported in part by the National Natural Science Foundation of China under Grants No.11605060 and No.11547020, in part by Natural Science Foundation of Hebei Province under Grant no. A2019209449, and in part by the Program for the Top Young Innovative Talents of Higher Learning Institutions of Hebei Educational Committee under Grant No. BJ2016041. Ya Li is also supported by the Natural Science Foundation of Jiangsu under Grant No. BK20190508.

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