]>PLB34269S03702693(18)30911010.1016/j.physletb.2018.11.054The Author(s)PhenomenologyFig. 1Typical Feynman diagrams for the quasitwobody decays B→D0⁎h→Dπh, h=(π,K). The symbol ⊗ stands for the weak vertex, × denotes possible attachments of hard gluons, and the grey rectangle represents the intermediate states D0⁎.Fig. 1Fig. 2The differential branching fractions for the decays B−→D0⁎0π−→D+π−π− and B¯0→D0⁎+π−→D0π+π− (left), B−→D0⁎0K−→D+π−K− and B¯0→D0⁎+K−→D0π+K− (right).Fig. 2Fig. 3Energy dependent ratios for the branching fractions between the decays B−→D0⁎0K−→D+π−K− and B−→D0⁎0π−→D+π−π− (the solid curve), B¯0→D0⁎+K−→D0π+K− and B¯0→D0⁎+π−→D0π+π− (the dash curve).Fig. 3Table 1Data for the quasitwobody hadronic B meson decays including the D0⁎ as the resonant state.Table 1ModeUnitBranching fractionRef.
B−→D0⁎0π−→D+π−π−(10−4)6.1 ± 0.6 ± 0.9 ± 1.6Belle [6]
(10−4)6.8 ± 0.3 ± 0.4 ± 2.0BaBar [7]
(10−4)5.78 ± 0.08 ± 0.06 ± 0.09 ± 0.391LHCb [42]
B¯0→D0⁎+π−→D0π+π−(10−4)0.60 ± 0.13 ± 0.15 ± 0.22Belle [43]
(10−5)7.7 ± 0.5 ± 0.3 ± 0.3 ± 0.42LHCb [44]
(10−5)8.0 ± 0.5 ± 0.8 ± 0.4 ± 0.43LHCb [44]
B−→D0⁎0K−→D+π−K−(10−6)6.1 ± 1.9 ± 0.5 ± 1.4 ± 0.4LHCb [45]
B0→D0⁎−K+→D¯0π−K+(10−5)1.77 ± 0.26 ± 0.19 ± 0.67 ± 0.20LHCb [46]
1Total Swave D+π− contribution.2Isobar model.3Kmatrix model.Table 2PQCD predictions for the concerned quasitwobody decays including the D0⁎ as the intermediate state.Table 2ModeUnitBranching fraction
B−→D0⁎0π−→D+π−π−(10−4)5.95−1.64+2.37(ωB)−1.55+1.97(ωDπ)−0.49+0.54(aDπ)−0.21+0.29(ΓD0⁎0)
B¯0→D0⁎+π−→D0π+π−(10−4)2.85−0.80+1.23(ωB)−0.81+1.05(ωDπ)−0.31+0.33(aDπ)−0.05+0.06(ΓD0⁎+)
B−→D0⁎0K−→D+π−K−(10−5)4.65−1.30+1.89(ωB)−1.24+1.51(ωDπ)−0.38+0.40(aDπ)−0.18+0.22(ΓD0⁎0)
B¯0→D0⁎+K−→D0π+K−(10−5)2.38−0.65+0.95(ωB)−0.68+0.85(ωDπ)−0.28+0.30(aDπ)−0.03+0.04(ΓD0⁎+)
Table 3Data for the concerned decays from Review of Particle Physics [5] and the ratios for the related branching fractions.Table 3ModeBModeBRD(⁎)
B− → D0K−(3.63 ± 0.12)×10−4B− → D0π−(4.68 ± 0.13)×10−30.078 ± 0.003
B− → D⁎(2007)0K−(3.97−0.28+0.31)×10−4B− → D⁎(2007)0π−(4.90 ± 0.17)×10−30.081−0.006+0.007

B¯0→D+K−(1.86 ± 0.20)×10−4B¯0→D+π−(2.52 ± 0.13)×10−30.074 ± 0.009
B¯0→D⁎(2010)+K−(2.12 ± 0.15)×10−4B¯0→D⁎(2010)+π−(2.74 ± 0.13)×10−30.077 ± 0.007
Resonant state D0⁎(2400) in the quasitwobody B meson decaysWenFeiWangab⁎wfwang@sxu.edu.cnaInstitute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, ChinaInstitute of Theoretical PhysicsShanxi UniversityTaiyuanShanxi030006ChinabState Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan, Shanxi 030006, ChinaState Key Laboratory of Quantum Optics and Quantum Optics DevicesShanxi UniversityTaiyuanShanxi030006China⁎Correspondence to: Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China.Institute of Theoretical PhysicsShanxi UniversityTaiyuanShanxi030006ChinaEditor: J. HisanoAbstractWe study four quasitwobody decay processes including D0⁎(2400) as the intermediate state in the perturbative QCD (PQCD) approach. The branching fraction predicted in this work for the decay mode B−→D0⁎(2400)0π−→D+π−π− agree with the data from Belle, BaBar and LHCb Collaborations. The PQCD prediction of the branching ratio for the decay B¯0→D0⁎+K−→D0π+K− is consistent with the value given by LHCb. For the decays B¯0→D0⁎+π−→D0π+π− and B−→D0⁎0K−→D+π−K−, the PQCD predicted branch ratios are 2.85−0.80+1.23(ωB)−0.81+1.05(ωDπ)−0.31+0.33(aDπ)−0.05+0.06(ΓD0⁎+)×10−4 and 4.65−1.30+1.89(ωB)−1.24+1.51(ωDπ)+0.40−0.38(aDπ)−0.18+0.22(ΓD0⁎0)×10−5, respectively. We analyze the experimental branching fractions using the ratios RD0⁎0 and RD0⁎+ which are related to the decays with the neutral and charged D0⁎(2400), respectively. The available experimental results for the quasitwobody decays including D0⁎(2400) are not in agreement with the isospin relation and SU(3) flavor symmetry.Many excited opencharm states have been discovered by various experiments in recent years, see Ref. [1] for a review. One of them, the pwave orbitally excited state D0⁎(2400), with the light degree of freedom jqp=12+ [2–4] and quantum number JP=0+ [5], was first discovered by Belle Collaboration in the threebody decays B−→D+π−π−, with the mass mD0⁎0=2308±17±15±28 MeV and width ΓD0⁎0=276±21±18±60 MeV [6]. For simplicity, we adopt D0⁎ to denote the D0⁎(2400) state and the inclusion of chargeconjugate processes is implied throughout this work. The neutral resonant state D0⁎0 has been confirmed by BaBar Collaboration in the same decay processes in [7], with the close but preciser values for its mass and width. While in the wideband photoproduction experiment, differ from those threebody decay processes, FOCUS Collaboration provided quite different values for the broad structure D0⁎0 in [8]. One has mD0⁎0=2407±21±35 MeV and ΓD0⁎0=240±55±59 MeV in company with mD0⁎+=2403±14±35 MeV and ΓD0⁎+=283±24±34 MeV for a charged state D0⁎+ from Ref. [8].Unlike the charmedstrange state Ds0⁎(2317) [9–11], which lies just below DK threshold and mainly decays into the isospin breaking channel Dsπ, the state D0⁎ is expected to decay rapidly through the swave pion emission, the conservation of its angular momentum implies this resonance primarily couple to Dπ and has a broad decay width [12] as revealed by experiments. While the discrepancy of its properties between the experimental results [6,7] and the predictions from the quark model [13,14] has triggered many studies on its true nature. The strong decays, radiative decays and/or the spectra have been studied extensively in Refs. [3,4,15–21] to explore the exact feature of the resonant state D0⁎. In Refs. [22,23], the possible fourquark structure of D0⁎ was investigated, the authors pointed out that the fourquark structure is acceptable for the resonant state observed by Belle [6] and BaBar [7], but not for the cases observed by FOCUS [8]. While in Refs. [24–26], it was claimed that there exist two poles in D0⁎ energy region. And a pole near the Dπ threshold was obtained from lattice QCD in [27], which was said to share the similarities with the experimental resonance D0⁎. The resonant state D0⁎ has also been explained as a mixture of two and fourquark state [28] or the bound state of Dπ [29].Semileptonic or hadronic B meson decays including a resonant state D0⁎ shall yield clues to its properties. Employing constituent quark model or the lightcone sum rules to evaluate the B→D⁎0 transition form factors, the decays of B→D0⁎lν have been studied in Refs. [30,31]. With the help of a chiral unitarity model, the ratio between the decay widths of B¯s0→Ds0⁎+(2317)lν¯l and B¯0→D0⁎+lν¯l was calculated in [32]. The model independent studies of B→D⁎⁎lν have been performed within the standard model in [33] and beyond the standard model in [34] based on heavy quark symmetry. In [35], the branching fraction for the semileptonic decay B→D0⁎lν was predicted assuming the conventional quarkantiquark configuration for D0⁎ state. And the hadronic matrix elements were evaluated in the Bethe–Salpeter approach for Bc→D0⁎μμ¯ decays in [36]. Since Belle's announcement [6], many works have been done on twobody hadronic B meson decays including the state D0⁎. For example, within the covariant lightfront approach, the branching ratio of B−→D0⁎0π− was predicted to be 7.3×10−4 in [37]. In [38], the information on the Isgur–Wise functions at zero recoil was extracted from the B¯0→D0⁎+π− decay process. Using the improved version of the Isgur–Scora–Grinstein–Wise quark model for the B→D0⁎ transition form factors, the decays B−→D0⁎0π− and B0→D0⁎−π+ have their branching ratios as 7.7×10−4 and 2.6×10−4, respectively, in Ref. [39]. Twobody decays B→D0⁎π have also been discussed within factorization framework [40] and the perturbative QCD (PQCD) approach [41].In Table 1, we have the data for the four quasitwobody hadronic B meson decay modes including D0⁎ from Belle, BaBar and LHCb Collaborations. In the processes B→D0⁎h→Dπh, where h is a charged pion or kaon, the weak interaction point accompany the birth of the bachelor particle h, the intermediate state D0⁎0(+) generated from the hadronization of cquark plus u¯(d¯)quark as demonstrated in Fig. 1. We stress that the resonance D0⁎ is not necessary to be conventional quartantiquark structure. In this letter, we shall analyze those four decay processes in a quasitwobody framework based on the PQCD factorization approach [47–50]. In Refs. [51–55], the PQCD approach has been employed in the studies of the threebody B meson decays. With the help of the twopion distribution amplitudes [56–58] and the experimental inputs for the timelike pion form factors, in Ref. [59], we calculated the decays B→Kρ(770),Kρ′(1450)→Kππ in the quasitwobody framework. The method used in [59] have been adopted for some other quasitwobody B decays in Refs. [60–62]. In this work, we extend the previous studies to the B¯0→D0⁎+h−→D0π+h− and B−→D0⁎0h−→D+π−h− decays.Refer to the Kπ system in Refs. [63,64], we define the scalar form factor F0Dπ(s) for the final state D+π− decays from D0⁎0 as(1)〈D+π−c¯u0〉=2B0F0Dπ(s), with the constant(2)B0=mD2−mπ22(mc−mu)≈1.93GeV, where the mD(mπ) is the mass of D(π) meson, the mc=1.275 GeV and mu=2.2 MeV (which could be neglected safely) for the mass of c and u quarks are adopted from [5]. Then we have(3)〈D+π−c¯u0〉≈〈D+π−D0⁎0〉1DBW〈D0⁎0c¯u0〉=ΠD0⁎DπBW〈D0⁎0c¯u0〉, and(4)ΠD0⁎DπBW=gD0⁎DπDBW=2B0F0Dπ(s)〈D0⁎0c¯u0〉=2B0f¯D0⁎m0F0Dπ(s), with f¯D0⁎=m0mc(μ)−mu(μ)⋅fD0⁎, and fD0⁎ is the decay constant of D0⁎. One has different values from 78 MeV [65] to 148−46+40 MeV [40] in different works for this decay constant, see [37,40,41,65–68], we support the moderate one fD0⁎=0.13 GeV which was adopted in the PQCD approach in [41]. The denominator DBW=m02−s−im0Γ(s), the massdependent decay width Γ(s) has its definition as Γ(s)=Γ0qq0m0s, m0 and Γ0 are the pole mass and width of the resonant state D0⁎ and s is the invariant mass square for the Dπ pair in the final state. In the rest frame of the resonance D0⁎, its daughter D+ or π− has the magnitude of the momentum as(5)q=12[s−(mD+mπ)2][s−(mD−mπ)2]/s, and q0 is the value of q at s=m02. The coupling constant gD0⁎Dπ has its value from the relation [32,39](6)gD0⁎Dπ=8πm02Γ0q0. We define(7)FDπ(s)=m02m02−s−im0Γ(s), then we have F0Dπ(s)=CDπ⋅FDπ(s), with the parameter(8)CDπ=gD0⁎Dπf¯D0⁎2B0m0In the rest frame of the B meson, with mB being its mass, we define the momentum p=mB2(1,η,0) in the lightcone coordinates for the resonant state D0⁎ and the Dπ pair coming out from the resonance. Its easy to see η=s/mB2 with s=p2. The light spectator quark comes from B meson and goes into D0⁎ in the hadronization processes in Fig. 1 (a) got the momentum k=(mB2z,0,kT), z is the momentum fraction. The momenta pB,p3,kB and k3 for the B meson, bachelor meson h and the associated spectator quarks for B and h have their definitions as(9)pB=mB2(1,1,0T),p3=mB2(0,1−η,0T),kB=(0,mB2xB,kBT),k3=(0,mB2(1−η)x3,k3T), where xB and x3 are the corresponding momentum fractions.The Swave Dπ system distribution amplitude could be collected into [41,54,59](10)ΦDπS−wave=12Nc(p/+s)CDπϕDπ(z,b,s), and the distribution amplitude(11)ϕDπ(z,b,s)=FDπ(s)22Nc{6z(1−z)×[mc(s)−mu(s)s+aDπ(1−2z)]}exp(−ωDπ2b2/2), the aDπ=0.40±0.10 and ωDπ=0.40±0.10 GeV are adopted in the calculation in this work by catering to our numerical results to the data in Table 1 and considering the related parameters for the D and D⁎ mesons in the literature. The numbers for ωDπ and aDπ in Ref. [41] have been considered as the references in this work, but we don't use the same values because of the different framework of the twobody and quasitwobody decays and the different definitions of the distribution amplitudes. The distribution amplitudes for the pion, kaon and B meson are the same as those widely adopted in the PQCD approach to hadronic B meson decays, one can find their expressions and the relevant parameters in Ref. [69].The decay amplitude A for the quasitwobody decay processes B¯0→D0⁎+h−→D0π+h− and B−→D0⁎0h−→D+π−h− in the PQCD approach is given by [51,52,59](12)A=ϕB⊗H⊗ϕh⊗ϕDπ, where the symbol ⊗ means convolutions in parton momenta, the hard kernel H contains only one hard gluon exchange at leading order in the strong coupling αs as in the twobody formalism and the distribution amplitude ϕB (ϕh, ϕDπ) absorbs nonperturbative dynamics in the decay processes. We then have the differential branching fraction (B) [5](13)dBdη=τBqhqB02C2Dπ32π3mBm02A2‾, τB being the B meson mean lifetime, the magnitude momentum for bachelor h, in the centerofmass frame of the Dπ pair, as(14)qh=12[(mB2−mh2)2−2(mB2+mh2)s+s2]/s. The mh is the mass of the bachelor meson pion or kaon.In the numerical calculation, we adopt ΛMS‾(f=4)=0.25 GeV. The decay constant fB=0.19 GeV for B meson comes from lattice QCD [70]. The masses and the mean lifetimes for the neutral and charged B meson, the pole masses and the widths of the neutral and charged D0⁎ state, the Wolfenstein parameters, the masses of pion, kaon and D meson are all come from the Particle Data Group [5]. Utilizing the differential branching fraction Eq. (13) and the decay amplitudes collected in Appendix A, we obtain the branching fractions in Table 2 for the concerned quasitwobody decay processes. The shape parameter uncertainty of the B meson, ωB=0.40±0.04 GeV, contributes the largest error for the branching fractions in Table 2, the ωDπ=0.40±0.10 GeV for Dπ system takes the second place and the aDπ=0.40±0.10 in the Eq. (11) generates the third one. For the decay width of the resonance D0⁎, the charged state got ΓD0⁎+=230±17 MeV and neutral one has ΓD0⁎0=267±40 MeV [5], then we have the quite different weight of the error from decay width for those processes including charged and neutral D0⁎ state, as shown in Table 2. There are other errors, which come from the uncertainties of the parameters in the distribution amplitudes for bachelor pion(kaon) [69] and the Wolfenstein parameters [5], are small and have been neglected.The distributions of those four branching ratios in Table 2 in the Dπ pair invariant mass mDπ are shown in Fig. 2, with the curves for B−→D0⁎0π−→D+π−π− (the dash line) and B¯0→D⁎+0π−→D0π+π− (the solid line) on the left, and the curves for B−→D0⁎0K−→D+π−K− (the dash line) and B¯0→D0⁎+K−→D0π+K− (the solid line) at the right. The small mass difference of the charged and the neutral D0⁎ exhibit the different peaks of the mDπ dependence for the different decay modes. The main portion of each branching ratio lies obviously in the region around the pole mass of the resonant state D0⁎ in the Fig. 2, the contributions from the energy region mDπ>3 GeV can be safely omitted.Assuming the D0⁎ state decays essentially into twobody modes, from the Clebsch–Gordan coefficients, we have B(D⁎00→D+π−)=B(D0⁎+→D0π+)=23. Then we have the twobody results as B(B−→D0⁎0π−)=8.93−3.48+4.71×10−4 and B(B¯0→D0⁎+π−)=4.28−1.77+2.48×10−4 from Table 2. The twobody value for the B−→D0⁎0π− decay agree well with the results in Refs. [37–39,41]. The result 4.28−1.77+2.48×10−4 for the decay B−→D0⁎0π− is consistent with the results 2.6×10−4 in [39] and 3.1×10−4 in [40] within errors, but it's smaller than the corresponding results in Ref. [41].Compare our numerical results in Table 2 with the corresponding data in Table 1, we find that the PQCD value of the branching fraction for the quasitwobody decay process B−→D0⁎0π−→D+π−π− in this work agree well with the values (6.1±0.6±0.9±1.6)×10−4 taken from Belle [6] and (6.8±0.3±0.4±2.0)×10−4 picked up from BaBar [7]. In Ref. [42], LHCb Collaboration presented a result (5.78±0.08±0.06±0.09±0.39)×10−4 for the total Swave Dπ system, which should be supposed to mainly contributed by the D0⁎ state, in the B−→D+π−π− decays. For the decay B¯0→D0⁎+K−→D0π+K−, the result 2.38−0.65−0.68−0.28−0.03+0.95+0.85+0.30+0.04×10−5 in Table 2 is consistent with the data (1.77±0.26±0.19±0.67±0.20)×10−5 given by LHCb [46]. While for the other two decay modes, there are apparent inconsistencies for the branching ratios between the PQCD predictions and the results from Belle and LHCb Collaborations. The Belle's branching fraction [43] for the decay B¯0→D0⁎+π−→D0π+π− is only about 21% of the PQCD prediction in this work, other two values from LHCb [44] in Table 1 for this process are some larger, but still less than 30% of our result when considering only the central values. The data for the decay B−→D0⁎0K−→D+π−K− selected from LHCb [45] is probably worse than the B¯0→D0⁎+π−→D0π+π− case, the branching fraction B=(6.1±1.9±0.5±1.4±0.4)×10−6 is about one order of magnitude smaller than the predicted value in Table 2.The B→D0⁎(→Dπ)π processes can be decomposed in terms of two isospin amplitudes, A1/2 and A3/2, as have been done for the B¯0→D+π− and B−→D0π− decays in Ref. [72]. With the absolute value A1/2/2A3/2 and the relative strong phase between A1/2 and A3/2 in [72], we have the ratio R≈0.59, which is close to the result R≈0.54 from Table 3, between the branching fractions of decays B¯0→D+π− and B−→D0π−. It is reasonable to expect that the ratio between the branching fractions of the decays B¯0→D0⁎+(→D0π+)π− and B−→D0⁎0(→D+π−)π− is not far from the 0.54. Our results in this work provide R≈0.48 for the corresponding two decays with pion as the bachelor particle, while the value for R from the data in Table 1 is just slightly larger than 0.1. So one could conclude that the data in Table 1 for the decays B¯0→D0⁎+(→D0π+)π− and B−→D0⁎0(→D+π−)π− are probably inconsistent with the isospin relation.For the decay processes B−→D0⁎0π−→D+π−π− and B−→D0⁎0K−→D+π−K−, we have an identical step D0⁎0→D+π−, the difference of the two decay modes originated from the bachelor particles pion and kaon. Within the SU(3) flavor symmetry, we have a straightforward ratio RD0⁎0 for the branching fractions of these two decays as(15)RD0⁎0=B(B−→D0⁎0K−→D+π−K−)B(B−→D0⁎0π−→D+π−π−)≈VusVud2⋅fK2fπ2, with(16)VusVudfK+fπ+=0.276 from Review of Particle Physics [5], then we have RD0⁎0≈0.076. It's easy to obtain a similar ratio RD0⁎+≈RD0⁎0,(17)RD0⁎+=B(B¯0→D0⁎+K−→D0π+K−)B(B¯0→D0⁎+π−→D0π+π−)≈VusVud2⋅fK2fπ2 for the decay modes B¯0→D0⁎+K−→D0π+K− and B¯0→D0⁎+π−→D0π+π−. The energy dependent curves of the RD0⁎0 and RD0⁎+ predicted by PQCD are shown in Fig. 3, from which one can find that there is little variation for the RD0⁎0 or RD0⁎+ as mDπ runs from its threshold to 3.5 GeV. There are similar patterns for the ratios of the related branching fractions for the decay modes including a pseudoscalar D or a vector D⁎ rather than D0⁎ as listed in the Table 3. If we accept the average value B(B−→D0⁎0π−)×B(D0⁎0→D+π−)=(6.4±1.4)×10−4 in Ref. [5], the branching fraction B=(4.86±1.06)×10−5, which agree well with the PQCD prediction in Table 2, for the decay process B−→D0⁎0K−→D+π−K− could be derived from Eq. (15). If we believe the result B=(1.77±0.77)×10−5 given by LHCb [46] for the decay B0→D0⁎−K+→D¯0π−K+, then the three values listed in Table 1 for the decay B¯0→D0⁎+π−→D0π+π− announced by Belle and LHCb are simply not credible when considering the Eq (17). In fact, there is a preliminary result from the Dalitz plot analysis of the B0→D¯0π+π− decay processes in Ref. [71] announced by BaBar as(18)B(B0→D0⁎−π+)×B(D⁎−0→D¯0π−)=(2.18±0.23±0.33±1.15±0.03)×10−4. This result is consistent with the prediction 2.85+1.23+1.05+0.33+0.06−0.80−0.81−0.31−0.05×10−4 within errors.To sum up, we studied the quasitwobody decays B−→D0⁎0π−→D+π−π−, B¯0→D0⁎+π−→D0π+π−, B−→D0⁎0K−→D+π−K− and B¯0→D0⁎+K−→D0π+K− in the PQCD approach in this work. The branching fraction predicted by PQCD for the decay process B−→D0⁎0π−→D+π−π− agree well with the data from Belle, BaBar and LHCb Collaborations. The result for the B¯0→D0⁎+K−→D0π+K− in this work is consistent with the data (1.77±0.26±0.19±0.67±0.20)×10−5 given by LHCb. For the other two decays, we analyzed the experimental results using the ratio relations between the decay branching fractions including D0⁎0 or D0⁎+. From RD0⁎0 and RD0⁎+, we argued that the experimental results for the decays B¯0→D0⁎+π−→D0π+π− and B−→D0⁎0K−→D+π−K− are probably questionable. The PQCD predictions for these two decay modes, in this work, are 2.85−0.80+1.23(ωB)−0.81+1.05(ωDπ)−0.31+0.33(aDπ)−0.05+0.06(ΓD0⁎+)×10−4 and 4.65−1.30+1.89(ωB)−1.24+1.51(ωDπ)−0.38+0.40(aDπ)−0.18+0.22(ΓD0⁎0)×10−5, respectively. We concluded that the available experimental results for the four decays including D0⁎(2400) are not in agreement with the isospin relation and SU(3) flavor symmetry.AcknowledgementsThis work was supported in part by National Science Foundation of China under Grant No. 11547038.Appendix ADecay amplitudesThe concerned quasitwobody decay amplitudes are given, in the PQCD approach, by(A.1)A(B−→π−[D0⁎0→]D+π−)=GF2VcbVud⁎[(c13+c2)FTD0⁎+c1MTD0⁎+(c1+c23)FTπ+c2MTπ],(A.2)A(B−→K−[D0⁎0→]D+π−)=GF2VcbVus⁎[(c13+c2)FTD0⁎+c1MTD0⁎+(c1+c23)FTK+c2MTK],(A.3)A(B¯0→π−[D0⁎+→]D0π+)=GF2VcbVud⁎[(c13+c2)FTD0⁎+c1MTD0⁎+(c1+c23)FAπ+c2MAπ],(A.4)A(B¯0→K−[D0⁎+→]D0π+)=GF2VcbVus⁎[(c13+c2)FTD0⁎+c1MTD0⁎], in which GF is the Fermi coupling constant, V's are the CKM matrix elements. And it should be understood that the Wilson coefficients c1 and c2 appear in convolutions in momentum fractions and impact parameters b.The amplitudes from Fig. 1 are written as(A.5)FTD0⁎=8πCFmB4fπ(K)(η−1)×∫dxBdz∫bBdbBbdbϕB(xB,bB)ϕDπ(z,b,s)×{[η(2z−1)−1−z]Ea(1)(t(1)e)h(xB,z,bB,b)+(2η(rc−1)+η−rc)Ea(2)(te(2))h(z,xB,b,bB)},(A.6)MTD0⁎=32πCFmB4/2Nc(η−1)×∫dxBdzdx3∫bBdbBb3db3ϕB(xB,bB)ϕDπ(z,b,s)ϕA×{[η(1−z−x3)+zη+(xB+x3−1)]×Eb(tb(1))hb(1)(xi,bi)+[x3(1−η)+z(1−η)−xB]×Eb(tb(2))hb(2)(xi,bi)},(A.7)FTπ(K)=8πCFmB4FDπ(s)∫dxBdx3∫bBdbBb3db3ϕB(xB,bB)×{[ϕA(1−η)(x3(η−1)−1)−r0[ϕP(η+1+2(η−1)x3)+ϕT(η−1)(2x3−1)]]×E(1)c(ti(1))h(xB,x3(1−η),bB,b3)+[2r0ϕP(η(1+xB)−1)+η(η−1)xBϕA]×Ec(2)(ti(2))h(x3,xB(1−η),b3,bB)},(A.8)MTπ(K)=32πCFmB4/2Nc×∫dxBdzdx3∫bBdbBbdbϕB(xB,bB)ϕDπ(z,b,s)×{[ϕA(1−η)(ηrc+(1+η)(1−xB−z))+r0ϕP(η(xB+z+x3−2)−4ηrc−x3)+r0ϕT(η(xB+z−x3)+x3)]Ed(td(1))hd(1)(xi,bi)+[(η−1)[z−xB+(1−η)x3]ϕA+r0x3(1−η)(ϕP+ϕT)+r0η(xB−z)(ϕT−ϕP)]×Ed(td(2))hd(2)(xi,bi)},(A.9)FAπ=8πCFmB4fB∫dzdx3∫bdbb3db3ϕDπ(z,b,s))×{[(η−1)[(1+2ηrc+(η−1)x3)ϕA+r0(rc+2x3η)ϕT]−r0[(1+η)rc+2η(x3(η−1)+2)]ϕP]×Ee(1)(ta(1))ha(z,x3(1−η),b,b3)+[(1−η)zϕA+2r0η(1−η+z)ϕP]×Ee(2)(ta(2))ha(x3,z(1−η),b3,b)},(A.10)MAπ=32πCFmB4/2Nc×∫dxBdzdx3∫bBdbBb3db3ϕB(xB,bB)ϕDπ(z,b,s)×{[(η−1)(η(xB+z−1)+z+xB)ϕA+r0η[(η−1)(1−x3)(ϕP+ϕT)+(z+xB)(ϕT−ϕP)−2ϕP]]×Ef(tf(1))hf(1)(xi,bi)+[(1−η)[η(z−xB)+(1−x3)(1−η)]ϕA+r0η[(η−1)(x3−1)×(ϕP−ϕT)+(z−xB)(ϕP+ϕT)]]Ef(tf(2))hf(2)(xi,bi)}.The evolution factors in the above factorization formulas are given by(A.11)Ea(1)(t)=αs(t)exp[−SB(t)−SC(t)]St(z),Ea(2)(t)=αs(t)exp[−SB(t)−SC(t)]St(xB),(A.12)Eb(t)=αs(t)exp[−SB(t)−SC(t)−SP(t)]b=bB,(A.13)Ec(1)(t)=αs(t)exp[−SB(t)−SP(t)]St(x3),Ec(2)(t)=αs(t)exp[−SB(t)−SP(t)]St(xB),(A.14)Ed(t)=αs(t)exp[−SB(t)−SC(t)−SP(t)]b3=bB,(A.15)Ee(1)(t)=αs(t)exp[−SC(t)−SP(t)]St(x3),Ee(2)(t)=αs(t)exp[−SC(t)−SP(t)]St(z),(A.16)Ef(t)=αs(t)exp[−SB(t)−SC(t)−SP(t)]b3=b, in which S(B,C,P)(t) are in the Appendix of [73], the hard functions h,ha,h(1,2)(b,d,f) and the hard scales t(e,b,i,d,a,f)(1,2) have their explicit expressions in the Ref. 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