J an 2 02 0 The contributions of K ∗ 0 ( 1430 ) and K ∗ 0 ( 1950 ) in the three-body decays B → Kπh

Abstract We study the contributions of the resonant states K∗ 0 (1430) and K ∗ 0 (1950) in the three-body decays B → Kπh (with h = π,K) in the perturbative QCD approach in this wok. The crucial nonperturbative factor FKπ(s) in the distribution amplitudes of the S-wave Kπ system is derived from the matrix element of vacuum to Kπ pair. The CP averaged branching fraction of a quasitwo-body decay process B → K∗ 0 (1950)h → Kπh is about one order smaller than the corresponding decay B → K∗ 0 (1430)h → Kπh. We compare our predictions with the results in literature. And the perturbative QCD predictions in this work for the relevant decays agree well with the existing experimental data.

The total decay amplitude for the B meson decays into three light mesons K, π and h as the final states can be described as the coherent sum of the nonresonant and resonant contributions in the isobar formalism [62][63][64]. The nonresonant contributions are spread all over the phase space and play an important role in the corresponding decay processes [65][66][67]. The resonant contributions from low energy scalar, vector and tensor resonances are known experimentally, in most cases, to be the dominated portion of the related decays and could be studied in the quasi-two-body framework [68][69][70] when the rescattering effects [71] and three-body effects [72,73] are neglected. For the B → Kπh decays, we have the resonant contributions from the Kπ, πh and Kh pairs which are originated from different intermediate states and as well containing the two-body final state interactions, while the J P = 0 + component of the Kπ spectrum, denoted as (Kπ) * 0 , are always found very important for the relevant physical observables.
In this work, we will focus on the contributions of the resonant state K * 0 (1430) in the B → Kπh decay processes in the PQCD approach based on the k T factorization theorem [100][101][102][103]. The contributions of the resonant state K * 0 (1950) in the three-body B decays involving Kπ pair have been ignored in the relevant theoretical studies and only be noticed by LHCb Collaboration very recently in the works [104,105]. We will systematically estimate, for the first time, the contributions from the state K * 0 (1950) for the B → Kπh decays in this work. As for the resonance K * 0 (700), we shall leave to the future studies in view of its ambiguous internal structure and the accompanying complicated results for the three-body B decays [106], in addition, the corresponding contributions have been covered up by the effective range part of LASS line shape for the experimental results [38-40, 42, 43].
This work is organized as follows. In Sec. II, we give a brief introduction for the theoretical framework. In Sec. III, we show the numerical results and give some discussions. Conclusions are presented in Sec. IV. The factorization formulas and functions for the related quasi-twobody decay amplitudes are collected in the Appendix.

II. FRAMEWORK
In the rest frame of B meson, we define its momentum p B and its light spectator quark momentum k B as in the light-cone coordinates, where x B is the momentum fraction and m B is the mass. For the resonant states K * 0 and the Kπ pair generated from it by the strong interaction as revealed in the Fig. 1, we define their momentum p = m B √ 2 (ζ, 1, 0). It's easy to validate ζ = s/m 2 B , where the invariant mass square s = p 2 = m 2 Kπ for the Kπ pair. The light spectator quark comes from B meson and goes into intermediate states in the hadronization of K * 0 as shown in Fig. 1 (a) has the momentum k = (0, m B √ 2 z, k T ). For the bachelor final state h and its spectator quark, their momenta p 3 and k 3 have the definitions as Where x 3 and z, which run from 0 to 1, are the corresponding momentum fractions. The matrix element from the vacuum to the K + π − final state is given by [120] with the p 1 (p 2 ) is the momentum for kaon(pion) in the Kπ system, ∆ Kπ = (m 2 K − m 2 π ) and m K (m π ) is the mass of K(π) meson. The F Kπ + (s) is the vector form factor which has been discussed in detail in the Refs. [80,[121][122][123][124][125][126][127]. The the scalar form factor F Kπ 0 (s) is defined as [128][129][130] where q is the light quark u or d, the isospin factor C X = 1 for X = {K + π − , K 0 π + } and The form factor F Kπ 0 (s) above is suppose to be one when s is zero. When the K + π − pair originated from the resonant state K * 0 (1430) 0 , we have [129] K + π − |ds|0 ≈ K + π − |K * 0 and [106], and the mass m K * 0 could be replaced by the invariant mass √ s for the off-shell K * 0 . One can find different values of f K * 0 for K * 0 (1430) in [131], we employ f K * 0 (1430) m 2 K * 0 (1430) = 0.0842 ± 0.0045 GeV 3 [132] and f K * 0 (1950) m 2 K * 0 (1950) = 0.0414 GeV 3 [133] in this work. The Breit-Wigner formula for the denominator , with the mass-dependent decay width Γ(s) = Γ 0 q q 0 m K * 0 √ s and Γ 0 is the full width for resonant state K * 0 . In the rest frame of the resonance K * 0 , its daughter kaon or pion has the magnitude of the momentum as The q 0 in Γ(s) is the value for q at s = m 2 where the Γ K * 0 →Kπ is the partial width for K * 0 → Kπ. The S-wave Kπ system distribution amplitudes are collected into [106,130,134,135] with the v = (0, 1, 0 T ) and n = (1, 0, 0 T ) being the dimensionless vectors. The twist-2 light-cone distribution amplitude has the form [106,130,134] with C 3/2 m the Gegenbauer polynomials, a 0 = (m s (µ) − m q (µ))/ √ s for (K * − 0 ,K * 0 0 ) and a 0 = (m q (µ) − m s (µ))/ √ s for (K * + 0 , K * 0 0 ) according to Ref. [134]. The a m are scale-dependent Gegenbauer moments, with a 1 = −0.57 ± 0.13 and a 3 = −0.42 ± 0.22 at the scale µ = 1 GeV for the resonance K * 0 (1430), and the contributions from the even terms could be neglected [106]. There is no available Gegenbauer moments for the state K * 0 (1950), we employ the scale-dependent a 1 and a 3 of K * 0 (1430) for the entire S-wave Kπ system in the numerical calculation. For the twist-3 light-cone distribution amplitudes in this work, we take the asymptotic forms as The factor F Kπ (s) is related to scalar form factor F Kπ The distribution amplitudes for B meson and the bachelor final state h in this work are the same as those widely employed in the studies of the hadronic B meson decays in the PQCD approach, one can find their expressions and parameters in the Appendix.

III. RESULTS AND DISCUSSIONS
In the numerical calculation, we adopt the decay constants f B = 0.189, f Bs = 0.231 GeV [136], the mean lifetimes τ B 0 = (1.520±0.004)×10 −12 s, τ B + = (1.638±0.004)×10 −12 s and τ B 0 s = (1.509 ± 0.004) × 10 −12 s [86] for the B 0 , B + and B 0 s mesons, respectively. The masses and the decay constants for the relevant particles in the numerical calculation in this work, the full widths for K * 0 (1430) and K * 0 (1950), and the Wolfenstein parameters of the CKM matrix are presented in Table I.  [86]. Utilizing the differential branching fraction Eq. (A8) and the decay amplitudes collected in Appendix A, we obtain the CP averaged branching fractions (B) and the direct CP asymmetries (A CP ) in Table II and Table III for the concerned quasi-two-body decay processes including the resonances K * 0 (1430) and K * 0 (1950), respectively, as the intermediate states.
The results for those quasi-two-body decays with one daughter of the K * 0 is the neutral pion are omitted. One will get a half value of the B and the same value of the A CP of the corresponding result in Tables II, III for a decay with the subprocesses K * 0 → Kπ 0 considering the isospin relation. For example, we have while these two decay processes have the same direct CP asymmetry. For the PQCD predictions in Tables II, III, Table I result in quite small errors for these quasi-two-body predictions because the most variation effect of decay width in the denominator D K * 0 of Eq. (6) is offset by the Γ in its numerator g K * 0 Kπ . For instance, the corresponding errors for the decay process B + → K * 0 (1430) 0 π + → K + π − π + are 0.04×10 −5 and 0.2% for its branching fraction and direct CP asymmetry, respectively, while for B + → K * 0 (1950) 0 π + → K + π − π + , the two errors are 0.01 × 10 −6 and 0.1%. There are other errors, which come from the uncertainties of the Wolfenstein parameters of the CKM matrix, the parameters in the distribution amplitudes for bachelor pion or kaon, the masses and the decay constants of the initial and final states, etc. are small and have been neglected. One can find that for those decay modes with the main contributions come from the annihilation diagrams of Fig. 1, their branching fraction errors generated from the variations of the a 1 and a 3 could be larger than the corresponding errors from ω B or ω Bs , because there is no shape parameter for B meson in the factorizable annihilation diagrams.
For a quasi-two-body decay process with the resonance K * 0 (1950) involved, its branching fraction is predicted to be roughly one order smaller than the corresponding decay mode including the resonant state K * 0 (1430). Or rather, the decays in Table III with the factorizable emission diagrams of Fig. 1 (c), for their CP averaged branching ratios, will be about 12%-15% (R 1 ) of the corresponding values in Table II, and the others will be about 6%-9% (R 2 ) for the corresponding branching fractions in Table II. The difference between R 1 and R 2 mainly originated from the (S − P )(S + P ) current amplitude the Eq. (A43), which possess the intermediate state invariant mass dependent factor m B √ ζ (≡ √ s). Take the decays B + → K * 0 0 π + → K + π − π + as an example, if we neglect the factorizable contributions from Fig. 1 (c), the ratio between two branching fractions of B + → K * 0 (1950) 0 π + → K + π − π + and B + → K * 0 (1430) 0 π + → K + π − π + will be 0.08, which is in the range of R 2 . The diagram of the differential branching fractions for B + → K * 0 (1950) 0 π + → K + π − π + and B + → K * 0 (1430) 0 π + → K + π − π + is shown in the Fig. 2. From which one can find that the main portion of the branching fractions lies in the region around the corresponding pole mass of the intermediate states. This feature makes that the proportion of the contribution of the branching ratio from Eq. (A43) will be larger for a decay mode including the K * 0 (1950) than the corresponding decay process including a K * 0 (1430) as the intermediate state.
We must stress that the ratios R 1 , R 2 , and also the branching fractions in Table III for the quasi-two-body decays involving K * 0 (1950) are strongly dependent on the relation [133]. If the value 0.0414 becomes two times larger, the R 1 , R 2 and the branching fractions in Table III will become four times larger than their current values. In Ref. [105], there are two branching fractions measured by LHCb to be The two central values above give us the ratio R = 0.15 which is in the range of R 1 for these two branching factions, but there is no diagrams like Fig. 1 (c) for B 0 → η c K * 0 0 decays. Because of the large errors for B 0 → η c K * 0 (1950) 0 , we can not extract the decay constant f K * 0 (1950) from this measurement. While from the data of the fit fractions for η c → K 0 S K ± π ∓ in [79] and η c → K + K − π 0 in [85] both from BaBar, one can expect a larger value than 0.0414 GeV 3 for the f K * 0 (1950) m 2  Table II, the second error comes from the uncertainty of Belle [31] The two-body results for the branching fractions of B → K * 0 h can be extracted from the quasi-two-body predictions in this work with the relation In Ref. [137], a parameter η was defined to measure the violation of the factorization relation the Eq.
where for the decays B + → K * 0 (1430) 0 π + , which means the violation of the factorization relation is not large when neglecting the effect of the invariant mass s in the decay amplitudes of the quasi-two-body decays. In order to check this conclusion, we calculate the decay B + → K * 0 (1430) 0 π + in the two-body framework of the PQCD approach, and we have B(B + → K * 0 (1430) 0 π + ) = 35.2 × 10 −6 , which is about 96.2% of the result in Table IV extracted with Eq (15), and Table II.
The comparison of the PQCD branching fractions with the experimental measurements for the two-body decays Table IV, with the first error is added in quadrature from the errors in Table II and the second error comes from the uncertainty of B(K * 0 (1430) → Kπ) = 0.93±0.10 [86] for these theoretical results. The branching fraction and direct CP asymmetry for B + → K * 0 (1430) 0 π + in Review of Particle Physics [86] averaged from the results in [30,38,43] are 39 +6 −5 × 10 −6 and 0.061 ± 0.032, respectively, which are consistent with the predictions (36.6 ± 11.3 ± 3.9) × 10 −6 in Table IV and (−1.3 ± 0.5)% in Table II. Because of the large uncertainty of the A CP = 0.26 +0. 18 −0.14 for B + → K * 0 (1430) + π 0 in [86], we can not evaluate the significance of the prediction (1.5 ± 1.0)%, but our branching fraction agree very well with BaBar's result in [43] for this decay mode. For the decay B 0 → K * 0 (1430) + π − , one has two results as listed in Table IV from BaBar and Belle Collaborations, its average B is presented to be (33 ± 7) × 10 −6 in Review of Particle Physics [86], this value agree well with the PQCD prediction (33.4 ± 10.2 ± 3.6) × 10 −6 . There is an upper limit of 2.2 × 10 −6 for the decay B + →K * 0 (1430) 0 K + , which is below the expectation. Our predictions will be tested by future experiments. In the very recent work, LHCb Collaboration presented the branching fractions for the combined decays B 0 which are in agreement with the PQCD predictions in Table V.
On the experimental side, the LASS parametrization [36,74] are employed in most cases to describe the S-wave Kπ system, where m 0 and Γ 0 are now the pole mass and full width for K * 0 (1430), and cotδ B = 1 aq + 1 2 rq with the parameters a = 2.07 ± 0.10 GeV −1 and r = 3.32 ± 0.34 GeV −1 [36]. The relativistic Breit-Wigner term of Eq. (19) is different from the Eq. (6). Before the F Kπ (s) in Eqs. (10)(11) be replaced by the LASS expression, a coefficient is needed for R(s). We have the replacement on the theoretical side. WithR(s) in the concerned quasi-two-body decay amplitudes, one will have the results for the decays B → (Kπ) * 0 h, including the contributions from the nonresonant effective range term, and the contributions from the resonance K * 0 (1430) which are the same as in the Table II. As the examples, we listed the results for B + → (Kπ) * 0  Table II and are added in quadrature. The percentages of the branching ratios in the column NERT are about 49% of the total results with the cutoff at 1.8 GeV, which are close to the percentages for the nonresonant effective range term in Refs. [38,40,43], while the total branching fractions from the LASS formula in Table VI are smaller than those values in [38,40,43]. We argue that, it's not really good for the effective range part of Eq. (19) to be studied in the quasi-two-body framework with the same expressions of the decay amplitudes in Appendix A, because the nonresonant term of a three-body decay amplitude should not be included in Eq. (9) and the effective range term hides the possible contributions from the exotic K * 0 (700).

Two-body decays
This work Theory Ref.  There is no direct CP asymmetries for the decays B 0 (s) → K * 0 0K 0 and B 0 (s) →K * 0 0 K 0 in Tables II, III, because these processes have only contributions from the penguin operators for their decay amplitudes. For the processes B 0 → K * 0 (1430) + π − → K 0 π + π − and B 0 → K * 0 (1430) 0 π 0 → K + π − π 0 via the b → sqq transition at quark level, the very small fraction of the total branching ratio for the contributions from the current-current operators led to the small direct CP asymmetries for these two decays as shown in Table II. The same pattern will appear for the decays B + → K * 0 (1430) 0 π + → K + π − π + and B + → K * 0 (1430) + π 0 → K 0 π + π 0 , and also for the relevant decays with the K * 0 (1950) replace K * 0 (1430) as the intermediate, but not for the decays B 0 s → K * 0 (1430) − π + →K 0 π − π + and B 0 s →K * 0 (1430) 0 π 0 → K − π + π 0 via the b → dqq transition. The interference between weak and strong phase of the decay amplitudes from current-current and penguin operators results in the large direct CP asymmetries for the B 0 s → K * 0 (1430) − π + →K 0 π − π + and B 0 s →K * 0 (1430) 0 π 0 → K − π + π 0 decays. The differential distribution curve of the A CP in m Kπ for the decay process B 0 s → K * 0 (1430) − K + →K 0 π − K + is displayed in Fig. 3. For the decays B + → K * 0 (1430) 0 π + and B 0 → K * 0 (1430) + π − , they both receive the contributions from the Fig. 1 (c, d). With the isospin limit we have [138] With the predictions in Table VII, one has R = 1.017 ± 0.003 from PQCD. The small error of R is because of the cancellation between the errors of two branching ratios, which means the increase or decrease of the parameters that caused the errors will result in nearly identical change of the weight for the numerator and denominator of R. For the decays B + → K * 0 (1430) + π 0 and B 0 → K * 0 (1430) 0 π 0 , the diagrams of Fig. 1 (a, c, d) will contribute to the branching fractions, the decay amplitudes from Fig. 1 (a) are same for both B + → K * 0 (1430) + π 0 and B 0 → K * 0 (1430) 0 π 0 , but the decay amplitudes from Fig. 1 (c, d) have the opposite sign considering the difference forūu anddd to form a neutral pion. It is not strange for the ratio between branching fractions of B + → K * 0 (1430) + π 0 and B 0 → K * 0 (1430) 0 π 0 away from unity. Because of the amplitude pollution from Fig. 1 (a) with the tree operators, the ratio for the branching fractions between B + → K * 0 (1430) + π 0 and B + → K * 0 (1430) 0 π + , and the ratio for B 0 → K * 0 (1430) 0 π 0 and B 0 → K * 0 (1430) + π − could deviate from the isospin limit.

IV. CONCLUSION
In this work, we studied the contributions of the resonant state K * 0 (1430) and, for the first time, the resonance K * 0 (1950) in the three-body decays B → Kπh in the PQCD approach.
The crucial nonperturbative factor F Kπ (s) in the distribution amplitudes of the S-wave Kπ system was derived from the matrix element of the vacuum to Kπ final state and was related to the time-like scalar form factor F Kπ 0 (s) by the relation F Kπ (s) = B 0 /m K * 0 F Kπ 0 (s). This relation also means that the LASS parametrization for the (Kπ) * 0 system which frequently appeared in the experimental works cannot be adopted directly for the Kπ system distribution amplitudes in the PQCD approach.
With The Lorentz invariant decay amplitude A for the quasi-two-body decay processes B → K * 0 h → Kπh in the PQCD approach, according to Fig. 1, is given by [26,68] The symbol ⊗ here means convolutions in parton momenta, the hard kernel H contains one hard gluon exchange at the leading order in strong coupling α s as in the two-body formalism. The distribution amplitudes Φ B , Φ h and Φ Kπ absorb the nonperturbative dynamics in the relevant decay processes. The B meson light-cone matrix element can be decomposed as [141][142][143] where the distribution amplitude φ B is of the form with N B the normalization factor. The shape parameters ω B = 0.40 ± 0.04 GeV for B 0 and B ± , ω Bs = 0.50 ± 0.05 for B 0 s , respectively. The light-cone wave functions for pion and kaon are written as [144][145][146][147] The distribution amplitudes of φ A (x 3 ), φ P (x 3 ) and φ T (x 3 ) are 2,4 (t) and C The magnitude momentum for the bachelor h is in the center-of-mass frame of the K * 0 , where m h is the mass of the bachelor state. The direct CP asymmetry A CP is defined as The errors induced by the parameter P ± ∆P for the B and A CP in this work, we employ the formulas With the subprocesses K * + 2K 0 π 0 }, and the K * 0 is K * 0 (1430) or K * 0 (1950), the concerned quasi-two-body decay amplitudes are given as follows: in which G F is the Fermi coupling constant, V 's are the CKM matrix elements. The combinations a i of Wilson coefficients are defined as 3 , a 5 = C 5 + C 6 3 , (A28) a 6 = C 6 + C 5 3 , a 7 = C 7 + C 8 3 , a 8 = C 8 + C 7 3 , a 9 = C 9 + C 10 3 , a 10 = C 10 + C 9 3 . (A29) It should be understood that the Wilson coefficients C and the amplitudes F and M for the factorizable and nonfactorizable contributions, respectively, appear in convolutions in momentum fractions and impact parameters b.
With the ratio r 0 = m h 0 /m B , the amplitudes from Fig. 1 (a) are written as with the color factor C F = 4/3. The amplitudes from Fig. 1 (b) are written as The amplitudes from Fig. 1 (c) are The amplitudes from Fig. 1 (d) are The hard functions are written as where H (1) 0 (χ) = J 0 (χ) + iY 0 (χ). The factor S t (χ) with the expression [149] S t (χ) = 2 1+2c Γ(3/2 + c) resums the threshold logarithms ln 2 χ appearing in the hard kernels to all orders, and the parameter c has its expression as c = 0.04Q 2 − 0.51Q + 1.87 with Q 2 the invariant mass square of the final state f in the B → f transition [135,150]. The evolution factors in the factorization expressions are given by in which the Sudakov exponents are defined as with the quark anomalous dimension γ q = −α s /π. The explicit form for the function s(Q, b) is [143] s(Q, b) = and the coefficients A (i) and β i are where n f is the number of the quark flavors and γ E is the Euler constant.