Contributions of $K_0^*(1430)$ and $K_0^*(1950)$ in the charmed three-body $B$ meson decays

In this work, we investigate the resonant contributions of $K_0^*(1430)$ and $K_0^*(1950)$ in the three-body $B_{(s)}\to D_{(s)}K\pi$ within the perturbative QCD approach. The form factor $F_{k\pi}(s)$ are adopted to describe the nonperturbative dynamics of the S-wave $K\pi$ system. The branching ratios of all concerned decays are calculated and predicted to be in the order of $10^{-10}$ to $10^{-5}$. The ratio $R$ of branching fractions between $B^0\to \bar{D}^0 K_0^{*0}(1430) \to \bar{D}^0K^+\pi^-$ and $B_s^0 \to \bar{D}^0 \bar{K}_0^{*0}(1430)\to \bar{D}^0K^-\pi^+$ are predicted to be 0.0552, which implies the discrepancy for the LHCb measurements. We expect that the predictions in this work can be tested by the future experiments, especially, to resolve $R$ ratio discrepancy.

In this work, we investigate the resonant contributions of K * 0 (1430) and K * 0 (1950) in the threebody B (s) → D (s) Kπ within the perturbative QCD approach. The form factor F kπ (s) are adopted to describe the nonperturbative dynamics of the S-wave Kπ system. The branching ratios of all concerned decays are calculated and predicted to be in the order of 10 −10 to 10 −5 . The ratio R of branching fractions between B 0 →D 0 K * 0 0 (1430) →D 0 K + π − and B 0 s →D 0K * 0 0 (1430) →D 0 K − π + are predicted to be 0.0552, which implies the discrepancy for the LHCb measurements. We expect that the predictions in this work can be tested by the future experiments, especially, to resolve R ratio discrepancy.

I. INTRODUCTION
Decays of the type B → Dhh ′ , where a B meson decays to a charmed meson and two light pseudoscalar mesons, have attracted people's attention in recent years. On the one hand, the studies of these three-body processes have shown the potential to constrain the parameters of the unitarity triangle. For instance, the decay B 0 →D 0 π + π − is sensitive to measure the CKM angle β [1][2][3], while Dalitz plot analysis of the decays B 0 →D 0 K + π − and B 0 s →D 0 K + K − can further improve the determination of the CKM angle γ [4][5][6][7]. On the other hand, the B → Dhh ′ decays provide opportunities for probing the rich resonant structure in the final states, including the spectroscopy of charmed mesons and the components in two light mesons system. A series of results in this area have been acquired from the measurements performed by the Belle [8][9][10], BaBar [11][12][13][14] and LHCb [3,5,7,[15][16][17][18][19][20] Collaborations.
In theory, a direct analysis of the three-body B decays is particularly difficult on account of the entangled resonant and nonresonant contributions, the complex interplay between the weak processes and the low-energy strong interactions [21], and other possible final state interactions [22,23]. Fortunately, most of three-body hadronic B meson decay processes are considered to be dominated by the low-energy S-, P -and D-wave resonant states, which could be treated in the quasi-two-body framework. By neglecting the interactions between the meson pair originated from the resonant states and the bachelor particle in the final states, the factorization theorem is still valid as in the two-body case [24,25], and substantial theoretical efforts for different quasi-two-body B meson decays has been made within different theoretical approaches . As well, the contributions from various intermediate resonant state for the three-body decays B → Dhh ′ have been investigated in Ref. [48][49][50][51][52][53].
The understanding of the scalar mesons is a difficult and long-standing issue [54]. The scalar resonances usually have large decay widths which make them overlap strongly with the background. In the specific regions, such as the KK and ηη thresholds, cusps in the line shapes of the near-by resonances will appear due to the contraction of the phase space. Moreover, the inner natures of scalars are still not completely clear. Part of them, especially the ones below 1 GeV, have also been interpreted as glueballs, meson-meson bound states or multi-quark states, besides the traditional quark-antiquark configurations [55][56][57][58][59][60][61][62][63][64][65][66]. The K * 0 (1430) is perhaps the least controversial of the light scalar mesons and generally believed to be a qq state [67]. It predominantly couples to the Kπ channel and has been studied experimentally in many charmless three-body B meson decays [68][69][70][71][72][73][74][75]. Recently, measurements of the charmed three-body decays B 0 →D 0 K + π − and B 0 s →D 0 K + π − involving the resonant state K * 0 (1430) were also presented by LHCb [5,16]. In addition, the subprocess K * 0 (1950) → Kπ which often ignored in literatures has also been considered in Ref. [16].
In the framework of the PQCD approach [76][77][78], the investigation of S-wave Kπ contributions to the B 0 (s) → ψKπ decays was carried out in Ref. [79]. In a more recent work [80], contributions of the resonant states K * 0 (1430) and K * 0 (1950) in the three-body decays B → Kπh (h = K, π) were studied systematically within the same method. The K * 0 (1430) is treated as the lowest lying qq state in view of the controversy for K * 0 (700), and the scalar Kπ timelike form factor F Kπ (s) was also discussed in detail. Motivated by the related results measured by LHCb [5,16], we shall extent the previous work [80] to the study of the charmed three-body B decays and analyse the contributions of the resonances K * 0 (1430) and K * 0 (1950) in the B → DKπ decays in this work. The rest of this article is structured as follows. In Sec. II, we give a brief review of the framework of the PQCD approach. The numerical results and phenomenological discussions are presented in Sec. III and a short summary is given in Sec. IV, respectively. Finally, the relevant factorization formulae for the decay amplitudes are collected in the Appendix.

II. FRAMEWORK
In the light-cone coordinate system, the B meson momentum p B , the total momentum of the Kπ pair and the D meson momentum p 3 under the rest frame of B meson can be written as with m B being the B meson mass and the mass ratio r = m D /m B . The variable η equals to s/(m 2 B − m 2 D ) where s is the invariant mass squared of Kπ pair in the range from (m K + m π ) 2 to (m B − m D ) 2 . We also set the momenta of the light quarks in the B meson, the Kπ pair and the D meson as K B , K and K 3 , and have the definitions as follow where x B , z and x 3 are the momentum fractions and run from zero to unity. In the PQCD approach, the decay amplitude for the quasi-two-body decay B (s) → D (s) K * 0 (1430, 1950) → D (s) Kπ can be expressed as the convolution [81] where the symbol H represents the hard kernel with single hard gluon exchange. φ B and φ D are the distribution amplitudes for the B and D mesons, respectively. φ Kπ denotes the distribution amplitude for the Kπ pair with certain spin in the resonant region. In this work, we use the same distribution amplitudes for the B (s) and D (s) mesons as in Ref [50] where one can easily find their expressions and the relevant parameters. Inspired by generalized distribution amplitude [82][83][84][85], the generalized LCDA for two-meson system are introduced [81,86] for three-body B-meson decay in the framework of PQCD approach and the heavy-to-light transition form factor in light-cone sum rules, respectively. The nonlocal matrix elements of vacuum to Kπ with various spin projector can be written as Kπ|s(x) q(0)|0 = √ s 1 0 dz e i z p·x φ s (z, s) , the Kπ S-wave distribution amplitude is chosen as [80] Φ Kπ (z, s) where n = (1, 0, 0 T ) and v = (0, 1, 0 T ) are the dimensionless lightlike unit vectors. The twist-2 and twist-3 light-cone distribution amplitudes have the form Here, C 3/2 m are the Gegenbauer polynomials, a m (µ) are the Gegenbauer moments and F Kπ (s) is the scalar form factor for the Kπ pair. In this work, we adopt the same formulae and parameters for the Kπ S-wave distribution amplitude as them in Ref. [80].
According to the typical Feynman diagrams as shown in Fig. 1 and the quark currents for each decays, the decay amplitudes for the considered quasi-two-body decays B → DK * 0 (1430, 1950) → DKπ are given as where G F is the Fermi constant, V ij is the CKM matrix element, and the combinations of the Wilson coefficients a 1,2 are defined as a 1 = C 1 /3 + C 2 and a 2 = C 2 /3 + C 1 . The expressions of individual amplitudes At last, we give the definition of the differential branching ratio for the considered quasi-two-body decays In the center-of-mass frame of Kπ system, the magnitudes of the momenta | p 1 | and | p 3 | can be expressed as

III. RESULTS
In the numerical calculations, the masses of the involved mesons (GeV), the lifetime of the B mesons (ps), the resonance decay widths (GeV) and the Wolfenstein parameters are taken from the Review of P article P hysics [54] The decay constants of the B (s) and D (s) mesons are set to the values f B (s) = 0.190 (0.230) GeV and f D (s) = 0.212 (0.250) GeV [87]. By integrating the differential branching ratio in Eq. (23), we obtain the branching ratios for the considered quasitwo-body processes with the intermediate resonances K * 0 (1430) and K * 0 (1950) in Tables I and II,

Mode
Unit B Data  Tables I and II, we have the following comments: For the branching fractions of two-body decays with K * 0 (1430) and K * 0 (1950), we shall apply Combined with results listed in Tables I and II,  (2) The PQCD prediction for the branching fraction B(B 0 s →D 0K * 0 0 (1430) →D 0 K − π + ) agrees with LHCb's data (3.00 ± 0.24 ± 0.11 ± 0.50 ± 0.44) × 10 −4 [16] within errors, while the PQCD predicted B(B 0 →D 0 K * 0 0 (1430) → D 0 K + π − ) is much larger than the value (0.71 ± 0.27 ± 0.33 ± 0.47 ± 0.08) × 10 −5 measured by LHCb [5] with significant uncertainties. By comparison, one can find that the decay modes B 0 s →D 0K * 0 0 (1430) →D 0 K − π + and B 0 →D 0 K * 0 0 (1430) →D 0 K + π − contain the same decay topology when neglecting the differences of hadronic parameters between B 0 and B 0 s . Then, we evaluate the ratio which is close to the PQCD prediction 0.0593 by using the results listed in Table I, but different from the value 0.0237 acquired from the central values of the measured branching ratio by LHCb [5,16]. One can find that in the Ref [16], the K * 0 (1430) component receive 20% fit fraction of total B(B 0 s →D 0 K − π + ), but in Ref [5,16], K * 0 (1430) component receive only 5.1% of total B(B 0 →D 0 K + π − ). The K * 0 (1430) component playing a such different role in two different process, however, on the theoretical side, the decay amplitudes are exactly same for B 0 →D 0 K * 0 0 (1430) →D 0 K + π − and B 0 s →D 0K * 0 0 (1430) →D 0 K − π + if we neglect the SU(3) symmetry breaking effect, R ratio will independent of theoretical framework. More precise measurements and more proper partial wave analysis are needed to resolve the discrepancy.
(3) For the CKM suppressed decay modes B 0 s → D 0K * 0 0 (1430) → D 0 K − π + , their branching ratios are much smaller than the corresponding results of B 0 s → D 0K * 0 0 (1430) → D 0 K − π + decays as predicted by PQCD in this work. The major reason comes from the strong CKM suppression factor as discussed in Ref [89]. The non-vanishing charm quark mass in the fermion propagator generates the main differences between the Similarly, for the B + → D 0 K * + 0 (1430) → D 0 K 0 π + decay and B + →D 0 K * + 0 (1430) →D 0 K 0 π + decay, there still exist the CKM suppression but much moderate than the previous cases: from Table I we have The main differences between the R s CKM and R s1,s2 CKM comes from the nonvanishing charm quark mass contributions in the non-factorizable B → K * 0 (1430) emission diagram. We also suggest more study on the decay mode B 0 s → D + s K * − 0 (1430) → D + sK 0 π − because it has a large branching ratio and can be found in future experiments.
(4) K * 0 (1430) was often parameterized by LASS lineshape [90] in partial wave analysis, which incorporate both cusps resonance and slowly varying nonresonance contribution, and it was applied in LHCb measurements [5,16]. However, rigorous theoretical calculation for nonresonance contribution in the context of PQCD framework is still absent [80], the comparison between theoretical calculations and experiment measurements focus only on the S-wave K * 0 (1430) contribution. More attempts can be make in future study to parameterize the nonresonance contribution for sake of giving a more reliable result.
(5) The CP -averaged branching fraction of the charmless quasi-two-body decay involving the intermediate state K * 0 (1950) is predicted to be about one magnitude smaller than the corresponding process containing K * 0 (1430) in [80]. In quasi-two-body charmed decays, the ratio of branching fractions between Table II and I are about few percentage, which are smaller than that of charmless cases mainly due to the absence of (S − P )(S + P ) amplitude, which receive resonance pole mass enhancement as discussed in [80]. And the more compact phase space can also reduce the branching fractions for the decay mode involving K * 0 (1950). From the partial wave analysis in [16], the K * 0 (1950) mode is measured to be about 1.5% than that of K * 0 (1430) mode , which is about one third of our prediction, ie. 4.6%, more precise measurements and more reliable theoretical predictions are needed in the future study. (6) In Fig. 2, we show the Kπ invariant mass-dependent differential branching fraction for the quasi-two-body decays B 0 s →D * 0 . One can easily find that the main portion of the branching fraction comes from the region around the pole mass of the corresponding resonant states, the contributions from the m Kπ mass region greater than 3 GeV is evaluated about 0.4% compared with the whole kinematic region (i.e. [m K + m π , m B − m D ]) in this work and can be safely neglected. can be obtained by using the quasi-two-body approximation relation in Eq (26). Under SU(3) flavor symmetry, we found the theoretical framework independent ratio R = by neglecting the differences of hadronic parameters between B 0 and B 0 s , this result is consistent with our PQCD prediction, but inconsistent with LHCb measurements. For the decays B 0 s → D 0K * 0 0 (1430) → D 0 K − π + and B 0 s → D 0K * 0 0 (1430) → D 0 K − π + , the great difference in their corresponding branching fractions can be understood by a strong CKM suppression factor R CKM ≈ λ 4 (ρ 2 +η 2 ) ≈ 3 × 10 −4 , while the moderate difference between B + → D 0 K * + 0 (1430) → D 0 K 0 π + and B + →D 0 K * + 0 (1430) →D 0 K 0 π + as well as B 0 → D 0 K * 0 0 (1430) → D 0 K + π − and B 0 →D 0 K * 0 0 (1430) →D 0 K + π − are mainly due to the R s CKM ≈ (ρ 2 +η 2 ) ≈ 0.147. More reliable theoretical predictions are needed in the future study for the nonresonance contribution and S-wave K * 0 (1950) contribution. We hope the predictions in this work can be tested by the future experiments, especially, to resolve R ratio discrepancy.

Acknowledgments
We are grateful to Ai-jun Ma for helpful comments. This work was supported by the National Natural Science Foundation of China under Grant No. 11947040.

Appendix: Decay Amplitudes
The factorization formulae for the individual amplitudes from different subdiagrams in Fig. 1 are where the hard functions are written as where E 1mn , E 2mn (m = a, c, e, g and n = b, d, f, h) are the evolution factors, which given by where we have (A.14)