$S$-wave $K\pi$ contributions to the hadronic charmonium $B$ decays in the perturbative QCD approach

We extend our recent works on the two-pion $S$-wave resonance contributions to the kaon-pion ones in the $B$ meson hadronic charmonium decay modes based on the perturbative QCD approach. The $S$-wave $K\pi$ time-like form factor in its distribution amplitudes is described by the LASS parametrization, which consists of the $K^*_0(1430)$ resonant state together with an effective range nonresonant component. The predictions for the decays $B\rightarrow J/\psi K\pi$ in this work agree well with the experimental results from $BABAR$ and Belle Collaborations. We also discuss theoretical uncertainties, indicating the results of this work, which can be tested by the LHCb and Belle-II experiments, are reasonably accurate.

On the theoretical side, several approaches have been used for describing charmless three-body B decays involving Kπ systems. For example, in Refs. [14][15][16][17][18], the authors predicted the branching ratios and direct CP violations in charmless three-body decays B → Kππ and B → KKπ using a model based on the factorization approach. The method was extended further in charmless three-body B s decays in Ref. [19]. In Ref. [20], the CP violation and the contributions of the strong kaon-pion interactions have been studied in B → Kππ decays using an approximate construction of relevant scalar and vector form factors. The K * resonance effects on direct CP violation have been taken into account based on the QCD factorization scheme [21], while, the three-body B meson decays with the charmonium mesons in the final states have not received much attention in the literature.
In our previous works, the decays B (s) → (J/ψ, η c )ππ [22,23], as well as the corresponding ψ(2S), η c (2S) modes [24][25][26], with the pion pair in S-wave resonant states, have been studied in the perturbative QCD (PQCD) [27][28][29] framework by introducing two-pion distribution amplitudes for the resonances [30,31]. These processes have been well described by a series of scalar resonances such as f 0 (500), f 0 (980), f 0 (1500), f 0 (1790), and so on. In the present paper, motivated by the recent detailed DP analyses of the Kπ invariant mass spectrum by the BABAR [4], Belle [8,11], and LHCb [12] collaborations, we will work on the decays of B (s) → (J/ψ, ψ(2S))Kπ, and we will focus on the Kπ pair originating from a scalar quark-antiquark state, while other partial wave and charmoniumlike resonances are beyond the scope of the present analysis. The S-wave contributions are parametrized into the timelike scalar form factors involved in the kaon-pion distribution amplitudes. For these form factors, we will adopt the LASS parametrization in Ref. [32], which consists of a linear combination of the K * 0 (1430) resonance and a nonresonant term coming from elastic scattering. By introducing the kaon-pion distribution amplitudes, the S-wave contributions of the related three-body B decays can be simplified into the quasi-two-body processes B → ψ(Kπ) S-wave → ψKπ. Following the steps of Refs. [22,26], the decay amplitude of B → ψ(Kπ) S-wave can be written as the convolution where the hard kernel H includes the leading-order contributions plus next-to-leading-order (NLO) vertex corrections.
) absorbs the nonperturbative dynamics in the hadronization processes.
The layout of this paper is as follows. In Sec. II, elementary kinematics, meson distribution amplitudes, and the required timelike scalar form factor are described. In Sec. III, we present a discussion following the presentation of the significant results on branching ratios. Finally, Sec. IV will be the conclusion of this work.
FIG. 1: The leading-order Feynman diagrams for the quasi-two-body decays B → ψK * 0 → ψKπ. The first two are factorizable and the last two are nonfactorizable, and K * 0 is the S-wave intermediate state.
In the light-cone coordinates, the kinematic variables of the decay B(p B ) → ψ(p 3 )(Kπ)(p) can be described in the B meson rest frame as with the mass ratio r = m/M , and where m(M ) is the mass of the charmonium (B) meson, the variable η = ω 2 /(M 2 − m 2 ), and the invariant mass squared ω 2 = p 2 for the kaon-pion pair. As usual we also define the kaon momentum p 1 and pion momentum p 2 as with ξ being the kaon momentum fraction. The momenta satisfy the momentum conservation p = p 1 + p 2 . The three-momenta of the kaon and charmonium in the Kπ center of mass are given by respectively, with m K (m π ) the kaon (pion) mass and the Källén function λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc).
For the valence quarks, momenta k B , k 3 , k, whose notations are displayed in Fig. 1, are chosen as where k iT , x i represent the transverse momentum and longitudinal momentum fraction of the quark inside the meson, respectively. The B meson can be treated as a heavy-light system, whose wave function in impact coordinate space can be expressed by [27] where b is the conjugate variable of the transverse momentum of the valence quark of the meson, and N c is the color factor. The distribution amplitude φ B (x, b) is adopted in the same form as it was in Refs. [27,33] with shape parameter ω b = 0.40 ± 0.04 GeV for the B u,d mesons and ω b = 0.50 ± 0.05 GeV for the B s meson. The normalization constant N is related to the decay constant f B through For the considered decays, the vector charmonium meson is longitudinally polarized. The longitudinal polarized component of the wave function is defined as [34,35] with the longitudinal polarization vector ǫ L = 1 √ 2r (−r 2 , 1 − η, 0 T ). For the twist-2 (twist-3) distribution amplitudes φ L (φ t ), the same form and parameters are adopted as in Refs. [34,35].
The S-wave kaon-pion distribution amplitudes are introduced in analogy with the case of two-pion ones [22,30], which are organized into where n = (1, 0, 0 T ) and v = (0, 1, 0 T ) are two dimensionless vectors. For vµ=− contributes at twist-2, while φ I=1/2 s and φ I=1/2 tµ=+ contribute at twist-3. It is worthwhile to stress that this kaon-pion system has similar asymptotic distribution amplitudes (DAs) as the ones for a light scalar meson [36], but we replace the scalar decay constants with the timelike form factor: where µ S = mS m2−m1 and m S and m 1,2 are the scalar meson mass and running current quark masses, respectively; their values can be found in Refs. [38][39][40]. B 1 is the first odd Gegenbauer moment for the light scalar mesons. According to Refs. [38,39,41], there are two scenarios for the scalar mesons. In scenarios I, all scalar mesons are viewed as the conventional two-quark states. In scenarios II, the light scalar mesons below or near 1 GeV are treated as the four-quark states, while those above 1 GeV scalar mesons such as f 0 (1370), K * 0 (1430), a 0 (1450), and so on are regarded as the ground states of qq. As noticed, scenario II is more favored for explaining the B + → K * 0 (1430)π + data measured by both BaBar [42] and Belle [43]. Besides, scenario II is also supported by a lattice calculation [44] and the recent Regge trajectory calculation [45]. Hence, we prefer to use the Gegenbauer moments B 1 = −0.57 ± 0.13 at the 1 GeV scale in scenario II obtained using the QCD sum rule method [38,39].
As is known, the relativity Breit-Wigner (RBW) model is unsuitable for describing the Kπ S-wave contributions because the broad κ and K * 0 (1430) resonance interferes strongly with a slowly varying nonresonant (NR) component. Detailed discussions of the S-wave Kπ systems in the isobar model, K-matrix model, and model-independent partial wave analysis method can be found in Refs. [46][47][48][49]. In this work, we parametrize the timelike scalar form factor F s (ω 2 ) for the S-wave Kπ systems by the LASS line shape [32], which has been widely adopted in the experimental data analysis, its expression is given as [4,50] where the first term is an empirical term from inelastic scattering and the second term is the resonant contribution with a phase factor to retain unitarity. m 0 = 1.435 GeV and Γ 0 = 0.279 GeV [4] are the pole mass and width of the K * 0 (1430) resonance state. | p 10 | is | p 1 | evaluated at the Kπ pole mass. The phase factor cot(δ B ) is defined as with the shape parameters a = 1.94 and r = 1.76 [4]. The differential branching ratio for the B → ψ(Kπ) S−wave decay takes the explicit form Since the S-wave kaon-pion distribution amplitude in Eq. (10) has the same Lorentz structure as that of two-pion ones in Ref. [22], the decay amplitude A here can be straightforwardly obtained just by replacing the twist-2 or twist-3 DAs of the ππ system with the corresponding twists of the Kπ one in Eq. (11). In addition, we also consider the NLO vertex corrections to the factorizable diagrams in Fig. 1, whose effects are included by the modifications to the Wilson coefficients as usual [51][52][53].

III. RESULTS
In the numerical calculations, parameters such as the meson mass, the Wolfenstein parameters, the decay constants, and the lifetime of B s mesons are presented in Table I. Other parameters relevant to the kaon-pion DAs have been given in the second section.
By using Eq. (14), integrating separately for the K * 0 (1430) resonant and nonresonant components as well as their coherent sum, we obtained the CP -averaged branching ratios for the considered decays, which are shown in Table II  TABLE II: The PQCD predictions for the CP -averaged branching ratios from various components together with the S-wave contribution for the considered decays. The theoretical errors correspond to the uncertainties due to the shape parameters ω b in the wave function of the B (s) meson, the heavy quark masses m b and mc, the Gegenbauer moment B1, and the hard scale t, respectively. The experimental results are obtained by multiplying the fit fractions by the measured three-body branching ratios, where all errors are combined in quadrature.
This The fit fraction is obtained from a weighted average of three measurements by Belle [6,11] and LHCb [12], while the measured value for B(B 0 → ψ(2S)K + π − ) is given in PDG [37].
together with some of the experimental measurements. Since the charged and neutral decay modes differ only in the lifetimes of B 0 and B + in our formalism, we can obtain the branching ratios of charged decay modes by multiplying the neutral ones by the lifetime ratio τ B + /τ B 0 . Some dominant uncertainties are considered in our calculations. The first error is caused by the shape parameter ω b in the B (s) meson wave function. The second error comes from the uncertainty of the heavy quark masses. In the evaluation, we vary the values of m c(b) within a 20% range. The third error is induced by the Gegenbauer moment B 1 = −0.57 ± 0.13 [38,39]. The last one is caused by the variation of the hard scale from 0.75t to 1.25t, which characterizes the size of the NLO QCD contributions. The first three errors are comparable, and contribute the main uncertainties in our approach. While the last scale-dependent uncertainty is less than 20% due to the inclusion of the NLO vertex corrections. The errors from the uncertainty of the CKM matrix elements and the decay constants of charmonia are very small and have been neglected. We have checked the sensitivity of our results to the choice of the shape parameters a and r [see Eq. (13)] in the LASS parametrization. Some experimental groups [42,50] prefer to choose another set of solutions with a = 2.07 and r = 3.32. Using the above values, we find that the branching ratios displayed in Table II decrease by only a few percent. From Table II, we find that the K * 0 (1430) resonance accounts for 41% of the branching fraction and the LASS NR term accounts for 49%. The constructive interference between them is responsible for the remaining 10% in the B → J/ψKπ decays. For the corresponding ψ(2S) modes, since the K * 0 (1430) resonance region is very close to the upper limits of the Kπ invariant mass spectra, the resonance contribution is suppressed to 25% of the total S-wave decay fraction. A similar situation also exists in the Cabbibo suppressed B 0 s decay modes. All these channels receive a relatively large contribution from the LASS NR, which involved the component of κ resonance as mentioned in [12]. In fact, the κ fit fractions from both the Belle [6,8,11] and LHCb [12] measurements are larger than that of K * 0 (1430) resonance.
As for the data, the fit fractions determined from the Dalitz plot analyses can be converted into quasi-two-body branching fractions by multiplying the corresponding branching fractions of the three-body decays. Taking the B 0 → J/ψK + π − decay as an example, based on the fit fraction of the K * 0 (1430) component, which was measured to be f K * 0 (1430) = (5.9 +0.6 −0.4 )% with a significance of 22.0σ by the Belle Collaboration [8], we have the center value of the quasi-two-body branching fraction Other available fit fractions are also converted into branching fraction measurements which are listed in Table II. It is shown that the model calculations presented here are described reasonably well for the J/ψ mode, but less so for the case of ψ(2S), especially for the S-wave contributions, which fall short by a large factor. It is worth to noting that the fit fractions for the ψ(2S) modes have much larger relative errors because of limited statistics. For instance, the previous Belle Collaboration gives the fit f K * 0 (1430) = (5.3 ± 2.6)% [6], while the subsequent measurements from the Belle and LHCb collaborations are (1.1 ± 1.4)% [11] and (3.6 ± 1.1)% [12], respectively. Including the errors, all The ω dependence of the differential decay rates dB/dω for the decay modes (a) B 0 → J/ψK + π − , (b) B 0 → ψ(2S)K + π − , (c) B 0 s → J/ψK − π + , and (d) B 0 s → ψ(2S)K − π + with a logarithmic y-axis scale. The resonance K * 0 (1430) and LASS nonresonant components are shown by the dotted blue and dashed green curves, respectively, while the solid red curves represent the total S-wave contributions.
three measurements agree with one another. In Table II, we calculate a weighted average and error from them as f K * 0 (1430) = (2.9 ± 0.8)%, which is closer to the LHCb data [12]. On the other hand, comparing with the ground state J/ψ modes, the branching ratio of the radially excited charmonium modes should be relatively small, owing to the phase space suppression and smaller decay constants. In Table II, our prediction of the S-wave branching ratio for B 0 → ψ(2S)K + π − is a few times smaller than that of B 0 → J/ψK + π − . However, the data from BABAR show the same order of magnitude between the two channels. Such a difference should be clarified in the forthcoming experiments based on much larger data samples.
The B 0 s decay modes can be theoretically related to the counterpart B 0 decays since they have identical topology and similar kinematic properties in the limit of SU (3) flavor symmetry. The relative ratios of the branching fractions for B 0 s and B 0 decay modes are dominated by a Cabibbo suppression factor of |V cd | 2 /|V cs | 2 ∼ λ 2 under the naive factorization approximation. From Table II, one can see that the B s channels have relatively small branching ratios (10 −6 ). Experimentally, the fraction of B 0 s → ψ(2S)K − π + decay proceeding via an S-wave is measured to be f S-wave = 0.339 ± 0.052 by the LHCb experiment [13], with statistical uncertainty only. Although the signal B 0 s → J/ψK − π + is found with a 4.7σ significance by the LHCb experiment [54] using a mass window of ±150 MeV around the nominal K * 0 mass, the small size of the data sample does not permit the determination of K * 0 (1430) and the S-wave fraction itself.
In Fig. 2, we plot the differential branching ratios as functions of the Kπ invariant mass ω for the considered decays. The red (solid) curve denotes the total S-wave contribution, while individual terms are given by the blue (dotted) curve for K * 0 (1430) resonance and green (dashed) curve for LASS NR contributions. Note that the J/ψ − ψ(2S) mass difference causes significant differences in the range spanned in the respective decay modes. As expected, the contributions from LASS NR and the K * 0 (1430) resonance are of comparable size. For the J/ψ modes, the dip region near 1.6 GeV is caused by strongly destructive interference between the resonance and nonresonant part of the LASS parametrization. From Eq. (12), one can estimate that the magnitude of these two terms is approximately equal, and the phase difference is roughly π around the 1.6 GeV regions. Experimentally, it is usually interpreted as resulting from interference between the K * 0 (1430) and its first radial excitation [4]. However, the dip is not seen in Figs 2 (b) and 2 (d) because its region is beyond the Kπ invariant mass spectra for the ψ(2S) modes. Comparing with the Kπ mass distributions obtained by BABAR (Fig. 11 of Ref. [4]) and LHCb (Fig. 2 of Ref. [55]), our distribution for the S-wave contribution agrees fairly well, showing a similar behavior.

IV. CONCLUSION
Motivated by the phenomenological importance of the hadronic charmonium B decays, in the present work we have carried out analyses of the B 0 (s) → ψKπ decays within the framework of the PQCD factorization approach by introducing the kaon-pion distribution amplitudes. Both the S-wave resonant and nonresonant components are parametrized into the timelike scalar form factors, which can be described by the LASS line shape. It is worth noting that fractions of the resonant and nonresonant components in these decays are comparable in size. Our predicted S-wave decay spectrum in the kaon-pion pair invariant mass show a similar behavior as the experiment. In particular, the K * 0 (1430) production in the B 0 → J/ψKπ decay agrees well with the results of a recent Dalitz plot analysis by the Belle Collaboration. Nevertheless, for the case of ψ(2S) modes, our results for the S-wave branching ratios turn out to be lower than the data. For the B s decays, an amplitude analysis to determine the fraction of decays proceeding via an intermediate K * 0 (1430) meson is still missing. We expect the relevant results could be tested by future experimental measurements.