The ρ (770 , 1450) → ωπ contributions for three-body decays B → ¯ D (

: The decays B → ¯ D ( ∗ ) ωπ are very important for the investigation of ρ excitations and the test of factorization hypothesis for B meson decays. The B + → ¯ D ( ∗ )0 ωπ + and B 0 → D ( ∗ ) − ωπ + have been measured by different collaborations but without any predictions for their observables on theoretical side. In this work, we study the contributions of ρ (770 , 1450) → ωπ for the cascade decays B + → ¯ D ( ∗ )0 ρ + → ¯ D ( ∗ )0 ωπ + , B 0 → D ( ∗ ) − ρ + → D ( ∗ ) − ωπ + and B 0 s → D ( ∗ ) − s ρ + → D ( ∗ ) − ωπ + . We introduce ρ (770 , 1450) → ωπ subpro-cesses into the distribution amplitudes for ωπ system via the vector form factor F ωπ ( s ) and then predict the branching fractions for the first time for concerned quasi-two-body decays with ρ (770 , 1450) → ωπ , as well as the corresponding longitudinal polarization fractions Γ L / Γ for the cases with the vector ¯ D ∗ 0 or D ∗− ( s ) in their final states. The branching

ρ + → D ( * )− ωπ + .We introduce ρ(770, 1450) → ωπ subprocesses into the distribution amplitudes for ωπ system via the vector form factor F ωπ (s) and then predict the branching fractions for the first time for concerned quasi-two-body decays with ρ(770, 1450) → ωπ, as well as the corresponding longitudinal polarization fractions Γ L /Γ for the cases with the vector D * 0 or D * − (s) in their final states.The branching fractions of these quasi-two-body decays are predicted at the order of 10 −3 , which can be detected at the LHCb and Belle-II experiments.The predictions for the decays B 0 → D * − ρ(770) + → D * − ωπ + and B 0 → D * − ρ(1450) + → D * − ωπ + agree well with the measurements from Belle Collaboration.In order to avoid the pollution from annihilation Feynman diagrams, we recommend to take the B 0 s → D * − s ρ(770, 1450) + decays, which have only emission diagrams at quark level, to test the factorization hypothesis for B decays.

Introduction
Three-body hadronic B meson decay processes always provide us a rich field to investigate various aspects of the strong and weak interactions.We may rely on them to study dynamical models for the strong interaction, to analyse hadron spectroscopy and explore the properties and substructures of resonant states, to determine the fundamental parameters for quark mixing and to understand the essence of CP asymmetries.In recent years, experimental efforts on these decay processes by employing Dalitz plot technique [1] have revealed valuable insights into the involved strong and weak dynamics.But on the theoretical side, it is complicated to describe the strong dynamics in these decays because of the rescattering processes [2][3][4][5], hadron-hadron interactions and three-body effects [6,7] in the final states.The resonance contributions in relevant decay channels, which are associated with the scalar, vector and tensor intermediate states, could be isolated from the total decay amplitudes and can be studied within the quasi-two-body framework [8][9][10].
In this paper, we shall concentrate on the cascade decays , where ρ + in this work stands for the intermediate states ρ(770) + and ρ(1450) + decaying into ωπ + .In the very recent study performed by SND Collaboration for e + e − → ωπ 0 → π + π − π 0 π 0 process in the energy range 1.05-2.00GeV, four isovector vector resonances covering ρ(770), ρ(1450), ρ(1700) and ρ(2150) have been employed to parametrize the related form factor for the ρ → ωπ transition [31].But we noticed from the Born cross section in Ref. [31] that the contributions for ωπ from ρ(1700) and the so called ρ(2150) state are not large and not important when comparing with those from ρ(770) and ρ(1450).In addition, the excited ρ states around 2 GeV are not well understood [30,50].In this context we will leave the contributions for ωπ from ρ(1700) and ρ(2150) in the concerned decays to future studies.
This paper is organized as follows.In Sec. 2, we give a brief introduction of the theoretical framework for the quasi-two-body decays B → D( * ) ρ(770, 1450) → D( * ) ωπ within PQCD approach.In Sec. 3, we present our numerical results of the branching fractions for with some necessary discussions.Summary of this work is given in Sec. 4. The factorization formulae for the related decay amplitudes are collected in the Appendix.

Framework
The relevant effective weak Hamiltonian H eff for the decays B → D( * ) ρ(770, 1450) with subprocesses ρ(770, 1450) → ωπ via the b → c transition is written as [11] where G F = 1.1663788(6)×10 −5 GeV −2 [16] is the Fermi coupling constant, V cb and V ud are the Cabibbo-Kobayashi-Maskawa (CKM) matrix [96,97] elements.The Wilson coefficients C 1,2 (µ) at scale µ are always combined as a 1 = C 1 +C 2 /3 and a 2 = C 2 +C 1 /3.The detailed discussion of the evaluation for C 1,2 (µ) in PQCD approach is found in Ref. [55], where one will also find the values  [11].In light cone coordinates the momentum p B is equal to m B √ 2 (1, 1, 0 T ) in the rest frame of B meson, where the mass m B stands for initial state B + , B 0 or B 0 s .In the same coordinates, the resonance ρ(770), its excited state ρ(1450) and the ωπ system generated from resonances by strong interaction have the same momentum p = m B √ 2 (ζ, 1 − r 2 , 0 T ), with the squared invariant mass p 2 = s for ωπ system.For the bachelor state D( * ) in the related processes, its momentum is defined as Figure 2. Typical Feynman diagrams for the quasi-two-body decays B → D( * ) ρ → D( * ) ωπ at quark level, where q ∈ {u, d} and q ′ ∈ {u, d, s}, the symbol ⊗ stands for the weak interaction vertex.
for the intermediate state and the D * meson, respectively, are where the parameter r will be satisfied by the relation p where the distribution amplitude ϕ B is of the form with two shape parameters ω B = 0.40 ± 0.04 GeV for B ±,0 and ω Bs = 0.50 ± 0.05 for B 0 s , respectively, the N B is a normalization factor.
The wave functions for D * 0 and D * − (s) have been discussed in detail in Ref. [103].Up to twist-3 accuracy, their two-particle light-cone distribution amplitudes are defined as ) with the normalization conditions ) ) The distribution amplitude for the D0 and D − (s) mesons is [103,104] ϕ (2.12) We adopt the same structure of the distribution amplitude for both D (s) and D * (s) in view the detailed discussions in Ref. [103] for them, but we employ different Gegenbauer moments C D = 0.6 ± 0.15 and C D * = 0.5 ± 0.10 for D (s) and D * (s) , respectively, in order to cater to the existing experimental data and also taking into account the different decay constants for them as they in this work and in Ref. [103].
For the P -wave ωπ system along with the subprocess ρ → ωπ, the distribution amplitudes hold the same structure of the vector mesons and could be organized into [42,105,106] with two dimensionless lightlike vectors n = (1, 0, 0 T ) and v = (0, 1, 0 T ), and N c is the number of colors for QCD.We adopt the convention ϵ 0123 = 1 for the Levi-Civita tensor ϵ µναβ .The twist-2 distribution amplitude for a longitudinally polarized ρ state can be parametrized as [105] where the Gegenbauer polynomial . The twist-2 transversely polarized distribution amplitude ϕ T (x, s) has a similar form as the longitudinally polarized one, we have [105] The forms of the twist-3 distribution amplitudes are [42,105,106] We adopt the same Gegenbauer moments for the P -wave ωπ system in this work as they were in Refs.[8,42,43] for the pion pair or kaon pair in view of the fact that these parameters are employed to describe the formation rather than the decay for the intermediate states.
And the value of Gegenbauer moment a T R in twist-2 transversely polarized distribution amplitude ϕ T (x, s) is set to be the same as it for a 0 R in this work.The form factor F T ωπ for the twist-3 distribution amplitudes of ϕ P -wave ωπ,L (x, s) and the twist-2 of ϕ P -wave ωπ,T (x, s) are deduced from the relation The factor f ωπ (s) in Eq. (2.15) is employed as the abbreviation of the transition form factor for ρ(770, 1450) → ωπ decays in the concerned processes.The related effective Lagrangian is written as [108][109][110] With the help of this Lagrangian, we can define the form factor F ωπ (s) from the matrix element [111][112][113] where j µ is the isovector part of the electromagnetic current, λ and ε is the polarization and polarization vector for ω meson, p a and p b are the momenta for ω and pion, respectively, and p = p a + p b .We need to stress that, in order to make the expression of differential branching fraction the Eq.(2.29) brief and concise, we employ f ωπ = f 2 ρ /m ρ F ωπ to describe the distribution amplitudes above for the P -wave ωπ system in Eqs.(2.15)-(2.20).
For the differential branching fraction, one has the formula [16] for the quasi-two-body decays B → D( * ) ρ → D( * ) ωπ, where τ B is the mean lifetime for B meson, s is the squared invariant mass for ωπ system.One should note that the phase space factor in Eq. (2.29) is different from that for the decays with subprocesses of ρ → ππ and ρ → K K as a result of the definition of F ωπ (s) in Eq. (2.22); the relations are employed for the derivation of Eq. (2.29), where θ is the angle between the threemomenta of ω and bachelor state D( * ) .In rest frame of intermediate states, the magnitude of the momenta are written as ) for pion and the bachelor meson D( * ) , where m π , m ω and m D are the masses for pion, ω and the bachelor meson, respectively.The Lorentz invariant decay amplitudes according to Fig. 2 for the concerned decays are given in the Appendix.
Table 1.Masses, decay constants and full widths (in units of GeV) for relevant states as well as the Wolfenstein parameters for CKM matrix elements from Review of Particle Physics [16], the f D * and f D * s are cited from [118].
Utilizing differential branching fractions the Eq.(2.29) and the decay amplitudes collected in Appendix, we obtain the branching fractions in Tables 3-4 for the concerned quasi-two-body decays with ρ(770) + and (1450) + decaying into ωπ + .For these PQCD branching fractions in Tables 3-4, their first error comes from the uncertainties of the shape parameter ω B = 0.40 ± 0.04 or ω Bs = 0.50 ± 0.05 for the B ±,0 or B 0 s meson; the Gegenbauer moments C D = 0.6 ± 0.15 or C D * = 0.5 ± 0.10 for D (s) or D * (s) mesons contribute the second error; the third one is induced by the Gegenbauer moments a 0 R = 0.25 ± 0.10, a t R = −0.60 ± 0.20 and a s R = 0.75 ± 0.25 [8] for the intermediate states; the fourth one for the decay results with ρ → ωπ comes from the uncertainties of the coupling g ρωπ or A 1 in Eq. (2.23).The uncertainties of the PQCD results in Table 2 are obtained by adding the individual theoretical errors in quadrature which induced by the uncertainties of ω B (s) , C D ( * ) and a 0,t,s R , respectively.There are other errors for the PQCD predictions in this work, which come from the uncertainties of the masses and the decay constants of the initial and final states, from the uncertainties of the Wolfenstein parameters, etc., are small and have been neglected.Table 5. Experimental data for the relevant three-body branching fractions from Review of Particle Physics [16].

Decay mode B [16]
The four decay channels B + → D( * )0 ωπ + and B 0 → D ( * )− ωπ + have been observed by CLEO Collaboration in Ref. [13], the updated studies for the decay B 0 → D * − ωπ + were presented later by BABAR and Belle Collaborations in Refs.[14,15].In these measurements, the ωπ + system in the final states showed a preference for the 1 − resonances.The relevant data from Review of Particle Physics [16] are found in Table 5.In addition to the total branching fraction for B 0 → D * − ωπ + decay, one finds the fitted branching fractions in Table 4 for the corresponding two quasi-two-body decays are in agreement with these two branching fractions presented by Belle Collaboration in [15].In consideration of the fitted branching fractions for B 0 → D * − ρ(770) + → D * − ωπ + and B 0 → D * − ρ(1450) + → D * − ωπ + in [15] and the data in Table 5 for the three-body decay B 0 → D * − ωπ + , one finds that the contributions from subprocesses ρ(770) + → ωπ + and ρ(1450) + → ωπ + are dominant for this three-body process.By examining the fraction of the longitudinal polarization Γ L /Γ at a fixed value of the momentum transfer, the decays B 0 → D * − ρ(770, 1450) + → D * − ωπ + can be employed to test the factorization hypothesis for B meson decays [125,126].The measurement of the fraction of longitudinal polarization in Ref. [126] for the decays B 0 → D * − ρ(770) + and B + → D * 0 ρ(770) + confirmed the validity of the factorization assumption at relatively low region of the momentum transfer.In Ref. [127], the authors proposed that if the ωπ + system in the B → D * ωπ + decays is composed of two or more particles not dominated by a single narrow resonance, factorization can be tested in different kinematic regions.In Table 4, we list PQCD predictions for the corresponding longitudinal polarization fractions Γ L /Γ for the relevant decays.The errors, which are added in quadrature, for these longitudinal polarization fractions are quite small from the uncertainties of ω B (s) , C D ( * ) , Gegenbauer moments for resonances, coupling g ρωπ or the weight parameter A 1 .The explanation is that the increase or decrease for the relevant numerical results from the uncertainties of these parameters will result in nearly identical change of the weight for the numerator and denominator of the corresponding Γ L /Γ predictions.In Ref. [13], the longitudinal polarization fraction for B 0 → D * − ωπ + was measured to be Γ L /Γ = 0.63 ± 0.09; for the same decay channel in mass region of 1.1-1.9GeV for ωπ + , the result Γ L /Γ = 0.654 ± 0.042(stat.)± 0.016(syst.)was provided by BABAR in Ref. [14].These two measurements agree well with the corresponding predictions in Table 4.
When employing B 0 → D * − ρ(770, 1450) + → D * − ωπ + to test of the factorization hypothesis, we should keep in mind that there are contributions from the annihilation Feynman diagrams as shown in Fig. 2-(c) for these two decay processes.By comparing the data B = (3.2+1.5 −1.3 ) × 10 −5 for the pure annihilation decay B 0 → D * − s K * + [16] with the results in Table 2 for B 0 → D * − ρ(770) + , one can roughly take the annihilation diagram contributions to be around a few percent at the decay amplitude level.In order to avoid the pollution from annihilation Feynman diagrams, we recommend to take the decays B 0 s → D * − s ρ(770, 1450) + with ρ(770, 1450) + decay into π + π 0 or ωπ + to test of the factorization hypothesis, in view of these decay channels have only emission diagrams the Fig. 2-(b) at quark level.We plot the invariant mass √ s dependent Γ L /Γ in Fig. 3 for the decay B 0 s → D * − s ρ(770) + with the subprocess ρ(770) + → ωπ + .One finds that the Γ L /Γ for B 0 s → D * − s ρ(770) + is going down as the increase of the invariant mass √ s for ωπ + system.Since the subprocesses ρ(770) + → π + π 0 and ρ(770) + → ωπ + are described by the electromagnetic form factors F π and F ωπ , respectively, in the quasi-two-body decay amplitudes, they are independence from the weak interaction in the related decay processes and wouldn't disturb the measurement results of Γ L /Γ for relevant channels.
Because the threshold for ωπ + is larger than the mass of ρ(770) + , we don't see a typical BW shape for the curve with subprocess ρ(770) + → ωπ + in Fig. 4, the bump of the curve is attributed to the kinematic characteristics in corresponding decay process rather than the properties of the involved resonant state ρ(770) + .The resonance ρ(770) + as a virtual bound state [17,18] in the process ρ(770) + → ωπ + can not completely present its properties in the concerned processes because of the phase space of the relevant decay processes.But the quantum number of the involved resonance could be fixed from its decay daughters the ωπ + system.The exact resonant source for ωπ + makes the cascade decay like B 0 → D * − ρ(770) + → D * − ωπ + to be a quasi-two-body process, although the invariant mass region for the ωπ + system is excluded from the region around pole mass of ρ(770).The resonance ρ(1450) + with the mass larger than the threshold of ωπ + contribute a normal BW shape for the curve of the differential branching fraction in Fig. 4 for the decay B 0 → D * − ρ(1450) + → D * − ωπ + .But in the decay D + s → ωπ + η which has been measured by BESIII recently [128], since the initial decaying state D + s does not have enough energy to make ρ(1450) demonstrate its intact properties, it will provide only the virtual contribution for ωπ + system in this three-body D + s decay process, we shall leave the detailed discussion of it to future study.
With one open charm meson in the final state of each decay channel, the decay amplitudes of these processes were described well by effective Hamiltonian H eff with the treelevel W exchange operators O 1 and O 2 in the quasi-two-body framework.The subprocesses ρ(770, 1450) + → ωπ + , which are related to the processes e + e − → ωπ 0 and τ → ωπν τ and can not be calculated in PQCD, were introduced into the distribution amplitudes for ωπ system in this work via the vector form factor F ωπ (s) which has measured by different collaborations recently.
With the parameters g ρωπ = 16.0 ± 2.0 GeV −1 and A 1 = 0.171 ± 0.036 for form factor F ωπ (s), we predicted the branching fractions for the first time on theoretical side for 12 quasi-two-body decays with ρ(770, 1450) + → ωπ + , as well as the corresponding longitudinal polarization fractions Γ L /Γ for the cases with the vector D( * )0 or D The decay B 0 → D * − ωπ + has been employed in literature to test the factorization hypothesis for B meson decays by examining the fraction of the longitudinal polarization Γ L /Γ at a fixed value of the momentum transfer.But we should take care about contributions from the annihilation Feynman diagrams for this decay process.In order to avoid the pollution from annihilation Feynman diagrams, we recommend to take the decays B 0 s → D * − s ρ(770, 1450) + with ρ(770, 1450) + decay into π + π 0 or ωπ + to test the factorization hypothesis for B decays.These decay channels have only emission diagrams with B s → D * − s transition at quark level, and the subprocesses which can be described with the corresponding electromagnetic form factors would not disturb the measurement results for Γ L /Γ.
The resonance ρ(770) + in the concerned quasi-two-decays of this work decaying to ωπ + system in the final states can not completely present its properties and contribute only the virtual contribution for the total branching fraction for corresponding three-body decay channels, because of the threshold for ωπ + and phase space limitation.But the quantum number of the involved resonance could be fixed from its decay daughters the ωπ + system.We want to stress here that the virtual contributions from specific known intermediate states are different from the nonresonant contributions demarcated in the experimental studies.
A Decay amplitudes for B → D( * ) ρ → D( * ) ωπ decays With the effective weak Hamiltonian H eff in Eq. (2.1), the total decay amplitudes for the concerned quasi-two-body decays are then written as According to the polarized decay amplitudes, one has

and
), the amplitudes A L , A ∥ and A ⊥ are related to the M L , M N and M T , respectively.For the detailed discussion, one is referred to Refs.[105,[129][130][131][132].
With the subprocesses ρ + → ωπ + , where ρ is ρ(770) or ρ(1450), the specific expressions in PQCD approach for the Lorentz invariant decay amplitudes of these general amplitudes F 's and M 's for B → D( * ) ρ → D( * ) ωπ decays are given as follows: The amplitudes from Fig. 2-(a) for the decays with a pseudoscalar D0 or D − (s) meson in the final state are written as The amplitudes from Fig. 2-(c) the annihilation diagrams are written as Where the T ρ, T D and Aρ in the subscript of above expressions stand for B → ρ, B → D transitions and the annihilation Feynman diagrams, respectively.The F 's stand for those factorizable diagrams and M 's for the nonfactorizable diagrams in Fig. 2.
The longitudinal polarization amplitudes from Fig. 2-(a) for the decays with a vector D * 0 or D * − (s) meson in the final state are written as vector D * 0 or D * − (s) are written as where n f is the number of the quark flavors and γ E is the Euler constant.
The hard scale, denoted as t i , are determined by selecting the maximum value of the virtuality associated with the internal momentum transition in the hard amplitudes, the specific expressions for the hard scales are given by : The threshold resummation factor S t (x) is of the form [133]: with the parameter c adopted to be 0.3.
The expressions of the hard functions h i with i ∈ {a, b, c, d, e, f, g, h, m, n, o, p} are obtained through the Fourier transform of the hard kernel: where K 0 , I 0 are modified Bessel function with K 0 (ix) = π 2 (−N 0 (x) + iJ 0 (x)) and J 0 is the Bessel function, α and β are the factors i 1 , i 2 .
27034 and C 2 = 1.11879 at m b scale.The local four-quark operators O c 1,2 are the products of two V − A currents, and one has

A. 3 )
by combining various of contributions from the related Feynman diagrams in Fig. 2.Where ρ + stands for the ρ(770) + or ρ(1450) + in the relevant decays.The other three decay amplitudes for the corresponding B + , B 0 and B 0 s decays with D * 0 , D * − and D * − s , respectively, can be obtained from Eq. (A.1)-(A.3)with the replacements of D (s) meson wave function by the D * (s) wave function.As has been done in two-body decays of B to two vector mesons as the final state, the decay amplitudes for B → D * ρ + → D * ωπ + in this work can be decomposed as x 3 , b B , b 3 ) , (A.6) with the symbol ζ = 1−ζ, the mass ratios r = m D ( * ) /m B and r c = m c /m B .The amplitudes from Fig. 2-(b) are written as

Table 4 .
Same as in Table3but with the different bachelor mesons D * 0 and D * − (s) ; the results in column Γ L /Γ are the predictions for the corresponding longitudinal polarization fraction.