Quasi-two-body decays $B \to D K^*(892) \to D K \pi$ in the perturbative QCD approach

We study the quasi-two-body decays $B\to D K^*(892) \to D K\pi$ by employing the perturbative QCD approach. The two-meson distribution amplitudes $\Phi_{K\pi}^{\text{$P$-wave}}$ are adopted to describe the final state interactions of the kaon-pion pair in the resonance region. The resonance line shape for the $P$-wave $K\pi$ component $K^*(892)$ in the time-like form factor $F_{K\pi}(s)$ is parameterized by the relativistic Breit-Wigner function. For most considered decay modes, the theoretical predictions for their branching ratios are consistent with currently available experimental measurements within errors. We also disscuss some ratios of the branching fractions of the concerned decay processes. More precise data from LHCb and Belle-II are expected to test our predictions.


I. INTRODUCTION
Many three-body hadronic B meson decays including B → DKπ have been studied experimentally in recent years [1,2]. The B → DKπ decays have demonstrated the potential to determine the CKM angles precisely. Suggestions for the determination of the unitarity triangle angle γ through Dalitz plot [3] analyses of the decays B ± → DK ± π 0 and B 0 → DK + π − were proposed in [4] and [5,6], respectively. And the measurement has been performed by LHCb [7]. The BABAR Collaboration has presented a measurement of the weak phase 2β + γ from the time-dependent Dalitz plot analysis of the B 0 → D ∓ K 0 π ± decays [8]. In addition, the decay modes B → DKπ have provided rich opportunities to investigate the spectroscopy of excited charm mesons and the significant components originated from Kπ system, corresponding results have been acquired from Dalitz plot analyses of B 0 → D ( * )± K 0 π ∓ [9], B 0 s →D 0 K − π + [10,11], B − → D + K − π − [12] and B + → D + K + π − [13] decays. In the amplitude analyses of B → DKπ decays, contributions from the P -wave Kπ resonant state K * (892) 1 were found to be the largest proportion in most cases, and the B → DK * decays have been substantially studied in experiment by quasi-two-body approach [14][15][16][17][18][19][20][21][22][23].
On the theoretical side, the charmed hadronic B meson decays B → DK * have been studied by using rather different methods in Refs. [24][25][26][27][28][29][30][31][32]. The K * was treated as a stable particle in the framework of two-body decays while as a resonance with the cascade decay K * → Kπ in the three-body decays. Several approaches have been adopted to describe those three-body B decays involving Kπ systems. For instance, within the QCD factorization [33][34][35], the authors studied the CP violation and the contribution of the strong kaon-pion interactions in the three-body B → Kππ decays [36,37] where the K * (892) and K * 0 (1430) resonance effects were mainly taken into account. In Ref. [38], the calculation of the localized CP violation in B − → K − π + π − decays has been done with the Kπ channels including K * (892), K * 0 (1430), K * (1410), K * (1680) and K * 2 (1430). Using a simple model on the basis of the factorization approach, the branching ratios and direct CP violation for the charmless threebody hadronic decays B (s) → KKπ and B (s) → Kππ have been calculated in Refs. [39][40][41][42]. In the recent works, the Kπ contributions to the decay channels B → ψKπ [43] and B → P Kπ [44], with ψ = (J/ψ, ψ(2S)) and P = (K, π), were analyzed by employing the perturbative QCD (PQCD) factorization approach [45][46][47][48]. In addition, phenomenological studies of the processesB 0 → D 0 π + π − andB 0 s → D 0 K + π − based on results of the chiral unitary approach has been performed in Ref. [49]. Motivated by the abundant experimental data and theoretical studies, we shall analyse the contributions of the resonance K * in the B → DKπ decays in this work. Theoretically, the three-body hadronic B meson decays are much more complicated than the two-body cases because of their non-trivial kinematics and the different phase space distributions. While these three-body decay processes are known to be dominated by the low energy scalar, vector and tensor resonant states, which could be handled in the quasi-two-body framework by neglecting the three-body and rescattering effects [50][51][52][53][54]. In the quasi-two-body region of the phase space, the three final states are quasi-aligned in the rest frame of the B meson and two of them almost collimate to each other, the related processes can be denoted as B → h 1 R → h 1 h 2 h 3 where h 1 represents the bachelor particle and the h 2 h 3 pair proceeds by the intermediate state R. In our previous works, the S-, P -and D-wave ππ (Kπ) resonance contributions to a series of charmed or charmless three-body B (s) meson decays have been studied [43,44,53,[55][56][57][58][59][60][61][62][63][64] in the PQCD approach by introducing twomeson distribution amplitudes [65][66][67][68][69][70][71]. The consistency of the theoretical studies and experimental results indicates the PQCD factorization approach are applicable to the three-body and quasi-two-body hadronic B meson decays. More recently, several quasi-two-body decays involving D * 0 (2400) [72], D * (2007) 0 and D * (2010) ± [54] as the intermediate states have been studied. And the isovector scalar resonances a 0 (980) and a 0 (1450) in the B → ψ(KK, πη) decays were presented in [73]. In this work, we will extend the previous studies to the quasi-two-body decays B → DK * → DKπ.
This paper is organized as follows. In Sec. II, we give a brief introduction for the theoretical framework and perturbative calculations for the considered decays. Then, the numerical values and phenomenological analyses are given in Sec. III. Finally, the last section contains a short summary.

II. THE THEORETICAL FRAMEWORK
In the framework of the PQCD approach for the quasi-two-body decays, the nonperturbative dynamics associated with the pair of the mesons are absorbed into two-meson distribution amplitudes, then the relevant decay amplitude A for the quasi-two-body decays B → DK * → DKπ can be written as the convolution [65,66] where the symbol ⊗ means the convolution integrations over the parton momenta and the hard kernel H includes the leadingorder contributions. The B meson (D meson, P -wave Kπ pair) distribution amplitude Φ B (Φ D ,Φ P -wave

Kπ
) absorbs the nonperturbative dynamics in the decay processes.

A. Coordinates and wave functions
In the rest frame of the B meson, we define the B meson momentum p B , the kaon momentum p 1 , the pion momentum p 2 , the K * meson momentum p = p 1 + p 2 and the D meson momentum p 3 in the light-cone coordinates as where the corresponding momentum fractions x B , z and x 3 run between zero and unity. The P -wave kaon-pion distribution amplitudes are defined in the same way as in Ref. [65,66], with the functions [44] where the Legendre polynomial P 1 (2ζ − 1) = 2ζ − 1 and the variable s = ω 2 = m 2 (Kπ). For the Gegenbauer moments, we adopt a || 1K * = 0.05 ± 0.02 and a || 2K * = 0.15 ± 0.05 determined in Ref. [44]. The relativistic Breit-Wigner (RBW) function is an appropriate model for narrow resonances which are well separated from any other resonant or nonresonant contributions with the same spin, and it is widely used in the experimental data analyses. Here, the time-like form factor F Kπ (s) is parameterized with the RBW line shape and can be expressed as the following form [7,11,12] with the mass-dependent decay width Γ(s) The | − → p 1 | is the momentum of one of the resonance daughters evaluated in the Kπ rest frame and . The parameter r BW is the barrier radius which is set to 4.0 GeV −1 as in Ref. [7,11,12]. Following Ref. [53], we also assume that [74]. In this work, we use the same distribution amplitudes for the B and D meson as in Ref. [59,62] where one can easily find their expressions and the relevant parameters.

B. Analytic formulae
For the quasi-two-body decays B → DK * → DKπ, the effective Hamiltonian is defined as where the Fermi coupling constant G F = 1.16638 × 10 −5 GeV −2 , C 1,2 (µ) denote the Wilson coefficients at the renormalization scale µ, O 1,2 (µ) represent the effective four quark operators and V ij are the CKM matrix elements. The typical Feynman diagrams at the leading order for the quasi-two-body decays Fig. 1 and 2, respectively. By making analytical evaluations for those Feynman diagrams in Fig. 1 and Fig. 2, we can obtain the total decay amplitudes of these concerned decays.
For the B (s) →D (s) K * →D (s) Kπ decays, their total decay amplitudes can be written explicitly in the following form FIG. 1: The leading-order Feynman diagrams for the quasi-two-body decays . The h1h2 denotes the Kπ pair and the blue ellipse represents the intermediate state while the total decay amplitudes for B (s) → D (s) K * → D (s) Kπ decays can be written as where the individual amplitude F LL eK * , M LL eK * , F LL eD , M LL eD , F LL aK * and M LL aK * are the amplitudes from different sub-diagrams in Fig. 1 and Fig. 2. Since the P -wave kaon-pion distribution amplitudes in Eq. (4) have the same Lorentz structure as that of two-pion ones in Ref. [53,59], the concerned expressions of those individual amplitudes in Ref. [59] can be employed in this work directly by replacing the distribution amplitudes (φ 0 , φ s , φ t ) of the ππ system with the corresponding twists of the Kπ ones in Eq. (5)- (7). The parameter c in the Eq. (33) of [59] is adopted to be 0.4 in this work according to the Refs. [75,76].
For the B → DK * → DKπ decays, the differential decay rate can be described as where τ B is the mean lifetime of B meson, the kinematic variables | p 1 | and | p 3 | denote the magnitudes of the K and D momenta in the center-of-mass frame of the kaon-pion pair,  Table I and Table II. The first error of these PQCD predictions comes from the B(B s ) meson shape parameter uncertainty ω B = 0.40 ± 0.04 (ω Bs = 0.50 ± 0.05) GeV, the following two errors are from the Gegenbauer coefficients in the kaon-pion distribution amplitudes: a || 1K * = 0.05 ± 0.02, a || 2K * = 0.15 ± 0.05 and the last one is induced by C D = 0.5 ± 0.1 (C Ds = 0.4 ± 0.1) for D(D s ) meson wave function. The errors come from the uncertainties of the parameters, for instance, the Wolfenstein parameters, the pole mass m K * and width Γ K * , are very small and have been neglected. Although the three-body B meson decay offers an ideal ground to study the distribution of CP asymmetry, there are no direct CP violations for these decays in this work since only the tree diagrams contribute to the considered decay processes.  From the calculations and the numerical results as listed in Table I and Table II, one can find the following points: (1) By assuming the B(K * → Kπ) ≈ 100 % [1] and accepting a simple relation between the branching ratio of the same kind of decay evaluated in the quasi-two-body and the two-body framework, it is easy to have One can extract the PQCD predictions for the decay rates of the related quasi-two-body decays from the results in Table I  and Table II. Take the quasi-two-body decay B + →D 0 K * + →D 0 K + π 0 for example, the relation between B(B + → D 0 K * + →D 0 K + π 0 ) and B(B + →D 0 K * + ) can be described as where the isospin relation B(K * + → K + π 0 ) = 1 3 . Combining with the central value of B(B + →D 0 K * + →D 0 Kπ) = 5.38 × 10 −4 in Table I, one can obtain the PQCD prediction for B(B + →D 0 K * + →D 0 K + π 0 ) = 1.79 × 10 −4 easily.
(3) For the branching ratios of eight CKM suppressed B (s) → D (s) K * → D (s) Kπ decays as listed in Table II, the PQCD predictions are in the order of 10 −8 to 10 −5 and we have some comments as follows: • Since no enough significant signals have been observed, there were hardly any specific data for the branching fractions of those decays but the upper limits. As shown in Table II, the experimental collaborations have determined the upper limits for three of the considered decays at 90(95)% confidence level and it is easy to see that all the PQCD predictions for the branching ratios are consistent with the corresponding experimental ones.
• We suggest more studies for those CKM suppressed B (s) → D (s) K * → D (s) Kπ decays in which the decay mode Kπ has a large branching ratio, and could be measured in the LHCb and Belle-II experiments. (4) Different from the fixed kinematics of the two-body B meson decays, the decay amplitudes of the quasi-two-body B meson decays show a strong dependence on the Kπ invariant mass ω. In Fig. 3, we plot the differential decay branching ratio of the decay mode B + →D 0 K * + →D 0 Kπ versus the invariant mass ω in the range of [m K + m π , 2GeV]. The main portion of the branching ratio lies obviously in the region around the pole mass of the resonant state which presented as a narrow peak in the plot, the contributions from the energy region ω > 1.2 GeV can be omitted safely.

IV. SUMMARY
Motivated by the abundant experimental data, we studied the contributions of the P -wave resonant states K * to the decays B (s) →D (s) Kπ and the CKM suppressed decays B (s) → D (s) Kπ by employing the PQCD factorization approach. The final-state interactions between the Kπ pair are factorized into the two-meson distribution amplitudes in which the resonant line shape for the resonance K * in the time-like form factor F Kπ is described by the RBW function.
By the numerical evaluations and the phenomenological analyses, we found the following points: (1) By adopting the B(K * → Kπ) ≈ 100 %, one can obtain B(B → DK * → DKπ) ≈ B(B → DK * ) easily, and it provides us a new way to study those two-body B meson decays in the framework for the quasi-two-body cases.
(2) The PQCD predictions for the branching fractions of the decay processes B (s) →D (s) K * →D (s) Kπ do agree with the experimental data, except for the cases of the color-suppressed decay B 0 →D 0 K * 0 →D 0 Kπ and the pure annihilation decay B 0 → D − s K * + → D − s Kπ. More precise data from the LHCb and the Belle-II experiments can help us to test our predictions and to improve the theoretical framework itself.
(3) For the CKM suppressed B (s) → D (s) K * → D (s) Kπ decays, all the PQCD predictions for the branching ratios are consistent with currently available experimental measurements. Experimentally, the upper limit for the branching fraction of B + → D + K * 0 → DKπ given by Ref. [13] is much less than that in the Ref. [22] and Ref. [23], and it is to be verified further.
(4) Unlike the fixed kinematics of the two-body B meson decays, the decay amplitudes of the quasi-two-body B meson decays have a strong dependence on the Kπ invariant mass ω and the main portion lies in the region around the pole mass of the resonant state.