Quasi-two-body decays $B_{(s)}\to K^*(892)h\to K\pi h$ in perturbative QCD approach

We extend our recent works on the $P$-wave two-pion resonant contributions to the kaon-pion cases in the hadronic charmless $B$ meson decays by employing the perturbative QCD approach. The concerned decay modes are analysed in the quasi-two-body framework by parameterizing the kaon-pion distribution amplitude $\Phi_{K \pi}^{\rm P}$, which contains the final state interactions between the kaon and pion in the resonant region. The relativistic Breit-Wigner formula for the $P$-wave resonant state $K^*(892)$ is adopted to parameterize the time-like form factor $F_{K\pi}$. We calculate the $CP$-averaged branching ratios and direct $CP$-violating asymmetries of the quasi-two-body decays $B_{(s)}\to K^*(892) h\to K\pi h$, with $h=(\pi, K)$, in this work. It is shown that the agreement of theoretical results with the experimental data can be achieved, through which Gegenbauer moments of the $P$-wave kaon-pion distribution amplitudes are determined. The predictions in this work will be tested by the precise data from the LHCb and the future Belle II experiments.


I. INTRODUCTION
Three-body hadronic B meson decays are a rich field for the experiments and theoretical studies. These decay processes offer one of the best tools for the analyses of direct CP violation and also provide a testing ground for the dynamical models of the strong interaction. Strong dynamics in a three-body B meson decay is much more complicated than that in a two-body case, the three-body processes receive nonresonant and resonant contributions, as well as the significant final-state interactions [1][2][3]. The nonresonant contributions have been studied with the method of heavy meson chiral perturbation theory [4][5][6]. In addition to the nonresonant background, it is urgent to study the resonant contributions which are, in most cases, the dominant part of a threebody decay process. Analyses of the three-body decays utilizing the Dalitz plots [7,8] enable us to investigate the properties of various scalar, vector and tensor resonant states with the isobar model [9,10] in terms of the usual Breit-Wigner model [11].
On the theoretical side, no proof of factorization has been done for the decays of the B meson into three final state mesons. As a first step, however, we can restrict ourselves to the specific kinematical configurations, in which two energetic final state mesons almost collimating to each other. For such topologies, the three-body interactions are expected to be suppressed strongly due to power counting rules [12]. It's reasonable for us to assume the validity of the factorization for these quasi-two-body B decays. In the "quasi-two-body" mechanism, the two-body scattering and all possible interactions between the two involved particles are included but the interactions between the bachelor particle and the daughter mesons from the resonance are neglected. Substantial progress on three-body hadronic B meson decays by means of symmetry principles has been made for example in Refs. [13][14][15][16][17][18][19][20]. The QCD-improved factorization [21] has also been widely applied to the studies of the three-body hadronic B meson decays in Refs. [12,[22][23][24][25][26][27][28][29][30][31]. The detailed factorization properties of the B + → π + π + π − in different regions of phase space were investigated in Ref. [22]. The CP violations and the contributions of the strong kaon-pion interactions have been investigated in the B → Kππ decays utilizing an approximate construction of relevant scalar and vector form factors in Ref. [12]. In Ref. [25], the authors studied the decays of B ± → π ± π ∓ π ± within a quasi-two-body QCD factorization approach and introducing the scalar and vector form factors for the S and P waves, as well as a relativistic Breit-Wigner (RBW) formula for the D wave to describe the meson-meson final state interactions.
The relevant Feynman diagrams are the same as Fig. 1 in the Ref. [50]. The P -wave contributions are parameterized into the time-like vector form factors involved in the kaon-pion DAs. We adopt the RBW line shape for the P -wave resonance K * (892) to parameterize the time-like form factors [62]. Throughout the remainder of the paper, the symbol K * is used to denote the K * (892) resonance. By employing the kaon-pion DAs, the P -wave contributions to the related three-body B meson decays can be simplified into quasi-two-body processes B → K * h → Kπh. As a result, one can describe the typical PQCD factorization formula for a B → h 1 h 2 h 3 decay amplitude as the form of [32,33], The hard kernel H describes the dynamics of the strong and electroweak interactions in three-body hadronic decays in a similar way as the one for the two-body B → h 1 h 2 decays, the Φ B and Φ h3 are the wave functions for the B meson and the bachelor paticle h 3 , which absorb the non-perturbative dynamics in the process. The Φ h1h2 is the two-hadron (K plus π in this work) distribution amplitude, which describes the structure of the final-state K-π pair. This paper is organized as follows. In Sec. II, we give a brief introduction for the theoretical framework. The numerical results, some discussions and the conclusions will be given in last two sections. The factorization formulas for the relevant three-body decay amplitudes are collected in the Appendix.

II. FRAMEWORK
In the light-cone coordinates, we let the kaon-pion pair and the final-state h move along the direction of n = (1, 0, 0 T ) and v = (0, 1, 0 T ), respectively, in the rest frame of the B meson. The kinematic variables of the decay B(p B ) → (Kπ)(p)h(p 3 ) can be chosen as where m B is the mass of B meson, the variable η is defined as η = ω 2 /m 2 B , the invariant mass squared ω 2 = p 2 for the kaon-pion pair. If we choose ζ = p + 1 /p + as kaon momentum fraction, the kaon momentum p 1 and pion momentum p 2 can be written as We employ x B , z, x 3 to denote the momentum fraction of the positive quark in each meson, k BT , k T , k 3T is assigned to the transverse momentum of the positive quark, respectively. The momentum k B of the spectator quark in the B meson, the momentum k for the resonant state K * (892) and k 3 for the final-state h are of the form of The momentum fractions x B , z and x 3 run from zero to unity.
The P -wave kaon-pion DAs are introduced in analogy with the case of two-pion ones [49], Following the steps of S-wave kaon-pion resonance [48,88], it is worthwhile to stress that the P -wave kaon-pion system has similar DAs as the ones for a light vector meson, but we replace the vector decay constants with the time-like form factor: where the Legendre polynomial P 1 (2ζ − 1) = 2ζ − 1. The Gegenbauer moments a || 1K * and a || 2K * will be regarded as free parameters and determined in this work.
For the narrow resonance K * , we adopt the RBW line shape for the P -wave resonance K * to parameterize the time-like form factors F Kπ (s), which is widely adopted in the experimental data analyses. The explicit expressions are in the following form [62], with the kaon-pion invariant mass squared s = ω 2 = m 2 (Kπ).
Here, the mass-dependent width Γ(s) is defined by where | − → p 1 | is the momentum vector of the resonance decay product measured in the resonance rest frame, and | − → p 0 | is the value of | − → p 1 | when √ s = m K * . The m K * and Γ K * are the pole mass and width of resonance state K * , respectively. Following Ref. [49], we also assume that with f K * = 0.217 ± 0.005GeV, f T K * = 0.185 ± 0.010GeV [89].

III. NUMERICAL RESULTS
The following input parameters (the masses, decay constants and QCD scale are in units of GeV) will be used [90] in numerical calculations, The values of the Wolfenstein parameters are adopted as given in the Ref. [90]: A = 0.836 ± 0.015, λ = 0.22453 ± 0.00044, ρ = 0.122 +0.018 −0.017 ,η = 0.355 +0.012 −0.011 . While the B meson and kaon (pion) DAs are the same as widely adopted in the PQCD approach [50].
For the decay B → K * (892)h → Kπh, the differential branching ratio is described as [90], with τ B the mean lifetime of B-meson, and s = ω 2 the invariant mass squared. The kinematic variables | − → p 1 | and | − → p 3 | denote the kaon momentum and h's momentum in the center-of-mass frame of the K-π pair, with the kaon (pion) mass m K (m π ) and the Källén function λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc). By using the differential branching fraction in Eq. (14), and the decay amplitudes in the Appendix, we calculate the CP averaged branching ratios (B) and the direct CP -violating asymmetries (A CP ) for the concerned decays B → K * h → Kπh, which are shown in the Table I and Table II together with some currently available experimental measurements. The Gegenbauer moments a || 1K * = 0.05 ± 0.02, a || 2K * = 0.15 ± 0.05 are determined to cater to the data [90,91], which differ from those in the DAs for a longitudinally polarized K * meson [89]. The first theoretical error from the variation of the hard scale t from 0.75t to 1.25t (without changing 1/b i ) and the QCD scale Λ QCD = 0.25 ± 0.05 GeV, which characterizes the effect of the NLO QCD contributions. The second error comes from the variations of the shape parameter of the B (s) meson distribution amplitude ω B = 0.40 ± 0.04 GeV or ω Bs = 0.50 ± 0.05 GeV. The last one is caused by the Gegenbauer moments a || 1K * = 0.05 ± 0.02, a || 2K * = 0.15 ± 0.05. The first two errors are comparable and contribute the main uncertainties in our approach, while the last one is less than 15%. The errors from τ B ± , τ B 0 , τ Bs and the Wolfenstein parameters in [90] are very small and have been neglected.
From the numerical results as shown in those two tables, one can address some issues as follows: • The isospin conservation is assumed for the strong decays of an I = 1/2 resonance K * to Kπ when we compute the branching fractions of the quasi-two-body process B → K * h → Kπh, namely, Taking B 0 → π 0 (K * 0 →)Kπ decay as an example, we can obtain the quasi-two-body branching fraction where we assume the K * → Kπ branching fraction to be 100%.
• We calculated the branching ratios and CP violations of the quasi-two-body decays B → KK * → KKπ as shown in Table I. Moreover, there is no CP violation for the decays B 0 → K 0 (K * 0 →)Kπ, B 0 →K 0 (K * 0 →)Kπ, B 0 s → K 0 (K * 0 →)Kπ and B 0 s →K 0 (K * 0 →)Kπ within the standard model, since there is only one kind of penguin operator involved in the decay amplitudes of the considered decays, which can be seen from Eqs. (A7-A10). The PQCD predictions of the sum of branching ratios of B 0 s → K + (K * − →)Kπ and B 0 s → K − (K * + →)Kπ decays, as well as the sum of branching ratios of B 0 s → K 0 (K * 0 →)Kπ and B 0 s →K 0 (K * 0 →)Kπ are in consistent with the LHCb measurements [86,87] and support their first observations of B 0 s meson decays to K * ± K ∓ and K 0 S K * 0 . The LHCb Collaboration reported that there is no evidence for the decay B 0 → K 0 S K * 0 and an upper limit is set on the branching ratio. Our result for the B(B 0 → K 0 (K * 0 →)Kπ) plus B(B 0 →K 0 (K * 0 →)Kπ) is around 0.54 × 10 −6 , which can be examined in the forthcoming experiments.
• For the B 0 s → π + (K * − →)Kπ decay process, our prediction is B = (8.08 +2.88 −2.09 ) × 10 −6 at leading-order in the quasitwo-body framework in this work, such a branching ratio is a bit larger than the value (3.3 ± 1.2) × 10 −6 in [90]. For the corresponding two-body modes, the previous theoretical predictions as given in Refs. [21,89,92] are larger than the data as well. In Ref. [93], the authors considered the next-to-leading-order (NLO) corrections and found that the NLO contribution will result in a 37% reduction of the leading order PQCD prediction for the "tree" dominated decay B 0 s → π + K * − . The authors confirmed that the branching ratios of the quasi-two-body modes in the three-body and two-body frameworks are close to each other in Ref. [49], since the B(ρ → ππ) ≈ 100%. Compared with the previous calculations of the two-body decays B (s) → P K * from PQCD [89,[93][94][95], we can obtain the consistency between the two-body and three-body modes. Maybe we can assume that the PQCD prediction of the branching ratio of the quasi-tobody decay B 0 s → π + (K * − →)Kπ will accommodate to data if we take the NLO contributions into consideration in the three-body framework. However, how to evaluate the NLO corrections to the three-body decays in the PQCD framework is a big task and will be left for the future studies.
In Fig. 1(a), we show the ω-dependence of differential decay rate dB(B + → K +K * 0 → K + Kπ)/dω. TheK * 0 is visible as a narrow peak near 0.89 GeV. We find that the main portion of the branching ratios lies in the region around the pole mass of the K * resonance as expected by examining the distribution of the branching ratios in the kaon-pion invariant mass ω. The central values of B are 0.25×10 −7 and 0.37×10 −7 when the integration over ω is limited in the range of ω = [m K * −0.5Γ K * , m K * +0.5Γ K * ] or ω = [m K * − Γ K * , m K * + Γ K * ] respectively, which amount to 50% and 74% of the total branching ratio B = 0.50 × 10 −7 as listed in Table I. In two-body B decays, the measured CP violation is just a number due to the fixed kinematics. While in threebody decays, the decay amplitudes depend on the Kπ invariant mass, which resulting in the differential distribution of direct CP asymmetries. In Fig. 1(b), we display the differential distributions of A CP for the four decay modes B + → K +K * 0 → K + Kπ (black solid line), B + →K 0 K * + →K 0 Kπ (purple dotted line), B + → π 0 K * + → π 0 Kπ (red dashed line) and B 0 → π − K * + → π − Kπ (blue dash-dotted line), respectively. One can find a falloff of A CP with ω for B + → K +K * 0 → K + Kπ, B + → π 0 K * + → π 0 Kπ and B 0 → π − K * + → π − Kπ. It implies that the direct CP asymmetries in the above three quasitwo-body decays, if calculated as the two-body decays with the K * resonance mass being fixed to m K * , may be overestimated. The ascent of the differential distribution of A CP with ω for B + →K 0 K * + →K 0 Kπ suggests that its direct CP asymmetry, if calculated in the two-body formalism, may be underestimated.

IV. SUMMARY
In this paper, we calculated the quasi-two-body decays B (s) → K * (892)h → Kπh by using the PQCD factorization approach. The relativistic Breit-Wigner formula for the P -wave narrow resonance K * (892) was adopted to parameterize the time-like form factor F Kπ . The kaon-pion distribution amplitude Φ P Kπ with the P -wave time-like form factor F Kπ was employed to describe the resonant state K * and its interactions with the kaon-pion pair. We predicted the branching ratios and the direct CP asymmetries of the concerned decay channels, and compared their differential branching ratios with currently available data. General agreements between the PQCD predictions and the data can be achieved by tuning the Gegenbauer moments of the P -wave kaon-pion DAs. The direct CP asymmetry of the B (s) → K * (892)h → Kπh modes are not numbers but depend on the kaon-pion invariant mass. Owing to the isospin conservation in K * (892) → Kπ decays, we can obtain the separate branching ratios of the corresponding quasi-two-body decays. More precise data from the LHCb and the future Belle II will test our predictions. The CP averaged branching ratios and direct CP -violating asymmetries of B (s) → K(K * →)Kπ decays calculated in PQCD approach together with experimental data [90,91]. The theoretical errors corresponding to the uncertainties due to the next-to-leading-order effects (the hard scale t and the QCD scale ΛQCD), the shape parameters ωB (s) in the wave function of B (s) meson and the Gegenbauer moments (a || 1K * and a || 2K * ), respectively.

Modes
Quasi-two-body results Experiment  The CP averaged branching ratios and direct CP -violating asymmetries of B (s) → π(K * →)Kπ decays calculated in PQCD approach together with experimental data [90,91]. The theoretical errors corresponding to the uncertainties due to the next-to-leading-order effects (the hard scale t and the QCD scale ΛQCD), the shape parameters ωB The decay amplitudes for considered quasi-two-body decay modes in this work are given as follows: 3 + C 10 )F LL eK * + ( C 5 3 + C 6 + C 7 3 + C 8 )F SP eK * + (C 3 + C 9 )M LL eK * + (C 5 + C 7 )M LR eK * + ( C 3 3 + C 4 − C 9 6 − C 10 2 )F LL aK * + ( A(B 0 s → π 0 (K * 0 →)Kπ) = where G F is the Fermi coupling constant. The V ij 's are the Cabibbo-Kobayashi-Maskawa matrix elements. The functions (F LL eK * , F LL aK * , M LL eK * , M LL aK * , · · · ) appeared in above equations are the individual decay amplitudes corresponding to different currents. Since the P -wave kaon-pion distribution amplitude in Eq. (5) has the same Lorentz structure as that of two-pion ones in Ref. [49], the explicit expressions of the individual decay amplitudes can be obtained straightforwardly just by replacing the twist-2 or twist-3 DAs of the ππ system with the corresponding twists of the Kπ ones in Eq. (6)- (8).