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Quasi-two-body decays BcDhDπh in perturbative QCD

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Yan-Chao Zhao, Zhi-Qing Zhang, Zi-Yu Zhang, Zhi-Jie Sun and Qiu-Bo Meng. Quasi-two-body decays BcDhDπh in the perturbative QCD[J]. Chinese Physics C. doi: 10.1088/1674-1137/acd23e
Yan-Chao Zhao, Zhi-Qing Zhang, Zi-Yu Zhang, Zhi-Jie Sun and Qiu-Bo Meng. Quasi-two-body decays BcDhDπh in the perturbative QCD[J]. Chinese Physics C.  doi: 10.1088/1674-1137/acd23e shu
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Received: 2023-02-17
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Quasi-two-body decays BcDhDπh in perturbative QCD

    Corresponding author: Zhi-Qing Zhang, zhangzhiqing@haut.edu.cn
  • Institute of Theoretical Physics, School of Sciences, Henan University of Technology, Zhengzhou 450052, China

Abstract: In this work, we investigate the quasi-two-body decays BcDhDπh with h=(K0,π0,η,η) using the perturbative QCD (PQCD) approach. The description of final state interactions of the Dπ pair is achieved through the two-meson distribution amplitudes (DAs), which are normalized to the time-like form factor. The PQCD predictions on the branching ratios of the quasi-two-body decays BcDhDπh show an obvious hierarchy: Br(B+cD+K0D0π+K0)=(5.22+0.860.74)×106,Br(B+cD+π0D0π+π0)=(0.93±0.26)×107, Br(B+cD+ηD0π+η)=(2.83+0.590.52)×108 and Br(B+cD+ηD0π+η)=(1.89+0.400.36)×108. From the invariant mass mDπ-dependence of the decay spectrum for each channel, one can find that the branching fraction is concentrated in a narrow region around the D pole mass. Thus, one can obtain the branching ratios for the corresponding two-body decays BcD+h under the narrow-width approximation. We find that the branching ratios of the decays BcD+h are consistent with the previous PQCD calculations within errors. These predictions will be tested in future experiments.

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    I.   INTRODUCTION
    • In recent years, the Bc meson decays have attracted a large amount of research interest for experimental studies [18]. Compared with the Bu,d,s mesons, the Bc meson is unique, as it is composed of both heavy quarks with different flavors. It can decay only via weak interaction, because the two flavor asymmetric quarks (b and c) cannot be annihilated into gluons (photons) via strong (electromagnetic) interaction. The Bc meson has many rich decay modes, because it has a sufficiently large mass and its constituent quarks (b and c) can decay individually. It provides a very good platform to study the nonleptonic weak decays of heavy mesons, to test the standard model, and to search for new physics signals. With more and more Bc decay events being collected at the Large Hadron Collider (LHC), the Bc meson three-body decays will be an important research topic in both experiment and theory in the next few years.

      Many approaches based on symmetry principles and factorization theorems have been used to study the B(c) meson three-body decays. The former incude the U-spin [911], isospin and flavor SU(3) symmetry [1214], and factorization assisted topological diagram amplitude approaches [15]. The latter include the QCD-improved factorization approach [1619] and the PQCD approach [2030], and it has been proposed that the factorization theorem of three-body B decays is approximately valid when the two particles move collinearly and the bachelor particle recoils back in the final states. According to this quasi-two-body-decay mechanism, the two-hadron distribution amplitudes (DAs) are introduced into the PQCD approach, where the strong dynamics between the two final hadrons in the resonant regions are included.

      The corresponding two-body decays BcDh with h=(K0,π0,η,η) have been studied by the different theories, such as the PQCD approach [31] and the relativistic constituent quark model (RCQM) [32]. In this work, assuming that D is an internal resonance state, which further decays into D0π+, we will study the quasi-two-body decays BcDhD0π+h using the PQCD approach. After the new non-perturbative inputs are introduced, i.e., the two-meson Dπ distribution amplitudes, the factorization formulas for the BcDhD0π+h decay amplitudes can be written as [20, 33, 34]

      M=ΦBcHΦDπΦh,

      (1)

      where ΦBc(Φh) denotes the DAs of the initial (final bachelor) meson, ΦDπ is the Dπ two-meson DAs, and denotes the convolution integrations over the parton momenta. Similar to the two-body decay case, the evolution of the hard kernel H for the b quark decay starts with the diagrams of single hard gluon exchange.

      The remainder of this paper is organized as follows. In Section II, the kinematic variables for the Bc meson three-body decays are defined. The considered two-meson (Dπ) P-wave DAs are parametrized, whose normalization form factors are assumed to follow the relativistic Breit-Wigner (RBW) model. Then, the Feynman diagrams and the total amplitudes for these decays are given. In Section III, the numerical results are presented and discussed. The analytical formulas of the decay amplitudes for each Feynman diagram are presented in the Appendix.

    II.   FRAMEWORK

      A.   Distribution amplitudes

    • We begin with the parametrization of the kinematic variables involved in the decays BcDhDπh with h=(K0,π0,η,η). In the rest frame of the Bc meson, we define the Bc meson momentum PBc, the D meson momentum P, and the bachelor meson h momentum P3 in the light-cone coordinates as

      PBc=mBc2(1,1,0T),P=mBc2(1,η,0T),P3=mBc2(0,1η,0T).

      (2)

      With the Bc meson mass mBc. Here, η=ω2/m2Bc=s/m2Bc, with ω2=s=m2Dπ being the square of the invariant mass of the Dπ pair. The momenta of the light quarks in the Bc, D, and bachelor meson h are denoted as k1, k2, and k3, respectively:

      k1=(0,mBc2x1,k1T),k2=(mBc2z,0,k2T),k3=(0,mBc2(1η)x3,k3T),

      (3)

      where x1, z, and x3 are the momentum fractions.

      The P-wave Dπ two-meson DAs are defined as in Ref. [35]:

      ΦP-waveDπ=12Ncϵ̸L(+s)ϕDπ(z,b,s),

      (4)

      with the distribution amplitude

      ϕDπ(z,b,s)=FDπ(s)22Nc6z(1z)[1+aDπ(12z)]×exp(ω2Dπb2/2),

      (5)

      where the Gegenbauer moment is aDπ=0.50±0.10 and the shape parameter is ωDπ=0.10±0.02 GeV [35].

      The strong interactions between the resonance and the final-state meson pair can be factorized into the time-like form factor, which is guaranteed by the Watson theorem [36]. For the narrow resonances, the RBW function [37] is a convenient model for separation from any other resonant or nonresonant contributions with the same spin, and it has been widely used in experimental data analyses. Here, the time-like form factor FDπ(s) can be defined using the matrix element

      D(p1)π(p2)|ˉcγμ(1γ5)q|0=[(p1p2)μm2Dm2πp2pμ]×FDπ(s)+m2Dm2πp2pμF0(s),

      (6)

      where p=p1+p2; p1(p2) and mD(mπ) represent the D(π) meson momentum and mass, respectively. FDπ(s) and F0(s) are the P-wave and S-wave form factors, respectively, for the Dπ system. FDπ(s) is parameterized with the RBW line shape

      FDπ(s)=sfDgDDπm2DsimDΓ(s),

      (7)

      where fD and mD are the decay constant and the pole mass of the D meson, respectively. The coupling constant gDDπ can be determined using the decay width Γ(s). The invariant mass dependent decay width Γ(s) is defined as

      Γ(s)=Γ0(qq0)3(mDs)X2(qrBW),

      (8)

      where the Blatt-Weisskopf barrier factor is [38]

      X(qrBW)=1+(q0rBW)21+(qrBW)2,

      (9)

      with the barrier radius rBW=4.0 GeV1 [39, 40]. Here, q represents the momentum for the daughter meson D or π in the D meson rest frame, i.e.,

      q=12[s(mD+mπ)2][s(mDmπ)2]/s,

      (10)

      and q0 is the value of q at s=m2D.

      The twist-2 distribution amplitude ϕAh and the twist-3 ones ϕPh and ϕTh have been parameterized as follows [4143]:

      ϕAh(x)=fh22Nc6x(1x)[1+ah1C3/21(2x1)+ah2C3/22(2x1)+ah4C3/24(2x1)],

      (11)

      ϕPh(x)=fh22Nc[1+(30η352ρ2h)C1/22(2x1)3(η3ω3+920ρ2h(1+6ah2))C1/24(2x1)],

      (12)

      ϕTh(x)=fh22Nc(12x)[1+6(5η312η3ω3720ρ2h35ρ2haP2)(110x+10x2)],

      (13)

      where the subscript h represents the pseudoscalar mesons K,π and the flavor states ηq=uˉu+dˉd2,ηs=sˉs. Additionally, η3=0.015,ω3=3, and the mass ratios are ρK(π)=mK(π)/m0K(π),ρηq=2mq/mqq,ρηs=2ms/mss with m0K(π),mqq, and mss being the chiral enhancemnet scales. The Gegenbauer polynomials Cνn(t)

      C1/22(t)=12(3t21),C1/24(t)=18(330t2+35t4),

      (14)

      C3/21(t)=3t,C3/22(t)=32(5t21),C3/24(t)=158(114t2+21t4).

      (15)

      The parameters of the hadronic wave functions are taken from Refs. [44, 45]:

      aK1=0.108±0.053,aK2=0.170±0.046,aK4=0.073±0.022,aπ1=0,aπ2=0.258±0.087,

      (16)

      aπ4=0.122±0.055,aηq1=aηs1=0,aηq2=aηs2=0.115±0.115,aηq4=aηs4=0.015.

      (17)

      For the wave function of the heavy Bc meson, we take

      d4zeik1z0|ˉbα(0)cβ(z)|Bc(PBc)=i2Nc[(Bc+mBc)γ5ϕBc(k1)]βα,

      (18)

      where we only consider the contribution from the dominant Lorentz structure. In the coordinate space, the distribution amplitude ϕBc with an intrinsic b (the conjugate space coordinate to kT) dependence is adopted in a Gaussian form as [46]

      ϕBc(x,b)=NBcx(1x)exp[(1x)m2c+xm2b8ω2bx(1x)2ω2bb2x(1x)],

      (19)

      where the shape parameter ωb=1.0±0.1GeV is related to the factor NBc by the normalization 10ϕBc(x,0)dx=1.

    • B.   Analytical formulas

    • For the quasi-two-body decays BcDhDπh, the effective Hamiltonian relevant to the bD(D=d,s) transition is given in Ref. [47]:

      Heff=GF2[q=u,cVqbVqD{C1(μ)O(q)1(μ)+C2(μ)O(q)2(μ)}i=310VtbVtDCi(μ)Oi(μ)],

      (20)

      where the Fermi coupling constant is GF1.166×105 GeV2 [48], and VqbVqD and VtbVtD are the products of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The scale μ separates the effective Hamiltonian into two distinct parts: the Wilson coefficients Ci and the local four-quark operators Oi. The local four-quark operators are written as follows:

      O(q)1=(ˉDiqj)VA(ˉqjbi)VA,O(q)2=(ˉDiqi)VA(ˉqjbj)VA,O3=(ˉDibi)VAq(ˉqjqj)VA,O4=(ˉDibj)VAq(ˉqjqi)VA,O5=(ˉDibi)VAq(ˉqjqj)V+A,O6=(ˉDibj)VAq(ˉqjqi)V+A,O7=32(ˉDibi)VAqeq(ˉqjqj)V+A,

      O8=32(ˉDibj)VAqeq(ˉqjqi)V+A,O9=32(ˉDibi)VAqeq(ˉqjqj)VA,O10=32(ˉDibj)VAqeq(ˉqjqi)VA,

      (21)

      with the color indices i and j. Here V±A refer to the Lorentz structures γμ(1±γ5).

      The typical Feynman diagrams at the leading order for the quasi-two-body decays B+cD+hD0π+h are shown in Figs. 1 and 2, where we take the decay B+cD+K0D0π+K0 as an example. The analysis formulas for the decay amplitudes of the different Feynman diagrams are given in the appendix.

      Figure 1.  (color online) Factorizable (a)(b) and non-factorizable (c)(d) emission diagrams for the decay B+cD+K0D0π+K0.

      Figure 2.  (color online) Nonfactorizable (e)(f) and factorizable (g)(h) annihilation diagrams for the decay B+cD+K0D0π+K0.

      By combining the amplitudes from the different Feynman diagrams, the total decay amplitudes for these considered decays are given as follows:

      A(B+cD+K0D0π+K0)=VcsVcb[a1FLLa+C1MLLa]VtsVtb[(C312C9)MLLe+(C3+C9)MLLa+(C512C7)MLRe+(C5+C7)MLRa+(C4+13C3+C10+13C9)FLLa+(C4+13C312C1016C9)FLLe+(C6+13C512C816C7)FSPe+(C6+13C5+C8+13C7)FSPa],

      (22)

      2A(B+cD+π0D0π+π0)=VudVub[a2FLLe+C2MLLe]VcdVcb[a1FLLa+C1MLLa]VtdVtb[(32C10C3+12C9)MLLe(C3+C9)MLLa+(C5+12C7)MLRe+(C413C3C1013C9)FLLa+(C10+53C913C3C432C712C8)FLLe+(C613C5+12C8+16C7)FSPe(C5+C7)MLRa+(C613C5C813C7)FSPa],

      (23)

      2A(B+cD+ηqD0π+ηq)=VudVub[a2FLLe+C2MLLe]+VcdVcb[a1FLLa+C1MLLa]VtdVtb[(2C4+C3+12C1012C9)MLLe+(C3+C9)MLLa+(C512C7)MLRe+(C5+C7)MLRa+(C4+13C3+C10+13C9)FLLa+(73C3+53C4+13(C9C10))FLLe+(2C5+23C6+12C7+16C8)FLRe+(C6+13C512C816C7)FSPe+(C6+13C5+C8+13C7)FSPa],

      (24)

      A(B+cD+ηsD0π+ηs)=VtdVtb[(C412C10)MLLe+(C612C8)MSPe+(C3+13C412C916C10)FLLe+(C5+13C612C716C8)FLRe],

      (25)

      where the combinations of the Wilson coefficients are a1=C2+C1/3 and a2=C1+C2/3.

      It should be noticed that Eqs. (24) and (25) give the decay amplitudes corresponding to the flavor states ηq and ηs, respectively. For the physical states (η,η), the decay amplitudes are written as follows:

      A(B+cD+ηD0π+η)=A(B+cD+ηqD0π+ηq)cosϕA(B+cD+ηsD0π+ηs)sinϕ,

      (26)

      A(B+cD+ηD0π+η)=A(B+cD+ηqD0π+ηq)sinϕ+A(B+cD+ηsD0π+ηs)cosϕ,

      (27)

      where ϕ=39.3±1.0 [49] is the mixing angle between these two flavor states, and the physical states are defined as follows:

      (ηη)=(cosϕsinϕsinϕcosϕ)(ηqηs).

      (28)

      Then, the differential decay rate is described as [50, 51]

      dBds=τBcq3q3h48π3m7Bc¯|A|2,

      (29)

      where τBc represents the Bc meson mean lifetime, and the kinematic variable qh denotes the momentum magnitude of the bachelor meson h in the center-of-mass frame of the Dπ pair:

      qh=12[(m2Bcm2h)22(m2Bc+m2h)s+s2]/s.

      (30)
    III.   NUMERICAL RESULTS
    • The input parameters adopted in our numerical calculations are summarized as follows (the masses, decay constants, and QCD scale are in units of GeV, and the Bc meson lifetime is in units of ps) [31, 43, 48]:

      Λ(5)QCD=0.112,mB+c=6.27447,mb=4.18,mK0=0.498,m0K=1.7,mπ±=0.140,

      (31)

      mπ0=0.135,m0π=1.3,mη=0.548,mη=0.958,mqq=0.110,mss=0.707,

      (32)

      mD±=2.010,mD0=1.865,τB+c=0.510,fBc=0.489,fD=0.25,fK=0.16,

      (33)

      fπ=0.13,fηq=(1.07±0.02)fπ,fηs=(1.34±0.06)fπ.

      (34)

      For the CKM matrix elements, we employ the Wolfenstein parametrization with the inputs [48]

      λ=0.22453±0.00044,A=0.836±0.015,

      (35)

      ˉρ=0.122+0.0180.017,ˉη=0.355+0.0120.011.

      (36)

      Using the decay amplitudes given in the Appendix, the total amplitudes given by Eqs. (23)–(27), and the differential branching ratio given by Eq. (29), by integrating over the full Dπ invariant mass region (mD+mπ)s(mBcmh)2 with h=(K0,π0,η()), we obtain the branching ratios for the quasi-two-body decays:

      Br(B+cD+K0D0π+K0)=(5.22+0.53+0.47+0.21+0.09+0.15+0.410.380.450.000.210.150.37)×106,

      (37)

      Br(B+cD+π0D0π+π0)=(0.93+0.11+0.09+0.21+0.01+0.03+0.060.080.080.230.030.020.05)×107,

      (38)

      Br(B+cD+ηD0π+η)=(2.83+0.22+0.25+0.19+0.28+0.18+0.310.190.230.210.250.200.18)×108,

      (39)

      Br(B+cD+ηD0π+η)=(1.89+0.13+0.17+0.12+0.19+0.14+0.210.120.160.140.150.160.15)×108,

      (40)

      where the first error originates from the shape parameter in the Bc meson DAs, i.e., ωBc=1.0±0.1 GeV; the second error comes from the decay constant fD=(250±11) MeV; the third and the fourth errors are induced by the Gegenbauer coefficient aDπ=0.50±0.10 and the shape parameter ωDπ=0.10±0.02 GeV in the Dπ pair DAs, respectively; the fifth error is caused by the decay width of the resonance D+, i.e., ΓD+=83.3±2.6 keV; and the last error is induced by the next-to-leading-order effect in the PQCD approach, where the hard scale t is changed from 0.75t to 1.25t. Other errors, which come from the uncertainties of the parameters in the DAs of the bachelor meson h, the Wolfenstein parameters, etc., have been neglected because they are very small.

      If we assume the isospin conservation for the strong decay DDπ,

      Γ(D+D0π+)Γ(D+Dπ)=2/3.

      (41)

      Under the narrow width approximation relation, the branching ratios of these quasi-two-body decays can be written as

      Br(B+cD+hD0π+h)=Br(B+cD+h)Br(D+D0π+).

      (42)

      According to Eqs. (41) and (42), we can obtain the branching ratios of the corresponding two-body decays:

      Br(B+cD+K0)=(7.83+0.79+0.70+0.32+0.13+0.22+0.610.570.670.000.310.230.55)×106,

      (43)

      {\rm Br}(B_{c}^+\to D^{*+}\pi^0) = \left({1.40}_{-0.12-0.12-0.35-0.05-0.03-0.07}^{+0.16+0.13+0.32+0.01+0.04+0.09}\right)\times{10}^{-7},

      (44)

      {\rm Br}(B_{c}^+\to D^{*+}\eta) = \left({4.25}_{-0.29-0.35-0.32-0.38-0.30-0.27}^{+0.33+0.38+0.29+0.42+0.27+0.47}\right)\times{10}^{-8},

      (45)

      {\rm Br}(B_{c}^+\to D^{*+} \eta^{\prime}) =\left({2.84}_{-0.18-0.24-0.21-0.23-0.24-0.23}^{+0.20+0.26+0.18+0.29+0.21+0.32}\right)\times{10}^{-8},

      (46)

      where we assume the branching ratio of the decay D^{*+}\to D\pi to be 100\% , assume isospin conservation, and use the narrow width approximation.

      The branching ratios of the two-body decays B_c\to D^*h with h= (K^0, \pi^0, \eta, \eta^{\prime}) have been calculated in the PQCD approach [31], where the results were given as

      {\rm Br}(B_{c}^+\to D^{*+}K^0) = \left({7.78}_{-2.40-0.02-0.52}^{+2.54+0.02+0.72}\right)\times{10}^{-6},

      (47)

      {\rm Br}(B_{c}^+\to D^{*+}\pi^0) = \left({1.3}_{-0.3-0.3-0.0}^{+0.4+0.2+0.0}\right)\times{10}^{-7},

      (48)

      {\rm Br}(B_{c}^+\to D^{*+}\eta) = \left({3.4}_{-0.9-1.5-0.00}^{+1.4+1.9+0.4}\right)\times{10}^{-8},

      (49)

      {\rm Br}(B_{c}^+\to D^{*+} \eta^{\prime}) =\left({1.5}_{-0.5-0.6-0.1}^{+0.8+0.8+0.3}\right)\times{10}^{-8}.

      (50)

      From the upper formulas, one can find that the branching ratios of the decays B_{c}^+\to D^{*+}h calculated in two-body and three-body frameworks are consistent with each other within errors. The decay widths for the decays B_{c}^+\to D^{*+}K^0 and B_{c}^+\to D^{*+}\pi^0 were calculated by using the RCQM [32] and are given as \Gamma(B_{c}^+\to D^{*+}K^0)= 4.10\times10^{-19} GeV and \Gamma(B_{c}^+\to D^{*+}\pi^0)=9.83\times10^{-20} GeV, respectively. Taking \tau_{B_{c}^+}= 0.510 ps, we can obtain their branching ratios as {\rm BR} (B_{c}^+\to D^{*+}K^0)=3.18\times10^{-7} and {\rm Br}(B_{c}^+\to D^{*+}\pi^0)=0.76\times10^{-7}, respectively, which are far smaller than the PQCD predictions. From Eqs. (22) and (23), one can find that the dominant contributions for these two decays come from the factorization annihilation amplitude \mathcal{F}_{a}^{L L} asociated with the large Wilson coefficient a_1 . Unfortunately, this type of annihilation contribution is not calculable under the RCQM. We hope that it can be verified by future experiments. As we know, the annihilation diagram contributions associated with the CKM matrix elements V^*_{cb}V_{cs} are dominant in both the decays B^+_c\to D^{*0}K^+ and B^+_c\to D^{*+}K^0 . In fact, such contributions are the same for these two channels; thus, we argue that the branching ratios for these two decays B^+_c\to D^{*0}K^+ and B^+_c\to D^{*+}K^0 should be close to each other. For example, {\rm Br}(B^+_c\to D^{*0}K^+)=(7.35^{+3.28}_{-2.34})\times10^{-6} and {\rm Br}(B^+_c\to D^{*+}K^0)=(7.78^{+2.64}_{-2.46})\times10^{-6} were given by the PQCD approach [31]. In Ref. [52], the upper limit for R_{D^{*0}K^+} at the 95\% confidence level was given as

      R_{D^{*0}K^+}=\frac{f_c}{f_u}{\rm Br}(B^+_c\to D^{*0}K^+)<1.1\times 10^{-6},

      (51)

      where f_c/f_u is the ratio of the inclusive production cross-sections of B^+_c and B^+ mesons, which can be related to the decays B^+_c\to J/\Psi\pi^+ and B^+\to J/\Psi K^+ through the formula

      R_{c/u}=\frac{f_c}{f_u}\frac{{\rm Br}(B^+_c\to J/\Psi \pi^+)}{{\rm Br}(B^+\to J/\Psi K^+)},

      (52)

      and it was measured as R_{c/u}=(0.68\pm0.10)\% by the LHCb collaboration [9]. Unfortunately, {\rm Br}(B^+_c\to J/\Psi\pi^+) has not been well measured via experiments. According to different theoretical predictions, the LHCb collaboration gave a range of 0.004\sim0.012 for the f_c/f_u values [52]. Then, we can obtain the lowest upper limit for the branching ratio of the decay B^+_c\to D^{*0}K^+

      \begin{array}{*{20}{l}} {\rm Br}(B^+_c\to D^{*0}K^+)<9.2\times 10^{-5}. \end{array}

      (53)

      Certainly, the upper limit here is strongly dependent on the branching ratio of the decay B^+_c\to J/\Psi\pi^+ . As a rough estimate, this upper limit can also be applied to the branching ratio of the decay B^+_c\to D^{*+}K^0 . Our prediction is found to satisfy this limit, which can be tested by the present LHCb experiments. There exists constructive (destrutive) interference between the amplitudes \mathcal{A}\left(B_{c}^+ \to D^{*+} \eta_{q}\to D^0\pi^+ \eta_{q}\right) and \mathcal{A} (B_{c}^+\to D^{*+} \eta_{s}\to D^0\pi^+ \eta_{s}) in the decay B_{c}^+\to D^{*+}\eta\to D^0\pi^+ \eta (B_{c}^+\to D^{*+}\eta^\prime\to D^0\pi^+ \eta^\prime) , which increases (reduces) the branching ratio of the corresponding decay.

      Last, we discuss the invariant mass m_{D\pi} -dependence of the decay spectrum. Here, we take the decay B_{c}^+ \to D^{*+} \pi^{0}\to D^0\pi^+ \pi^{0} as an example and plot the m_{D\pi} -dependence of the differential branching fraction in Fig. 3, where a maximum in the D\pi pair invariant mass is observed at approximately 2.010 GeV. It is easy to see that the main contribution to the branching ratio comes from the region around the pole mass of the D^{*+} resonance, as we expected. The peak located at the D^{*+} mass is very sharp, because the decay width is tiny, i.e., \Gamma_{D^{*+}}=83.3 keV [53, 54]; at the same time, the D^0\pi^+ threshold is too close to the resonance mass m_{D^{*+}} . This differs significantly from the case of the K^{*} resonance [22], whose mass is far larger than the K\pi threshold. If we integrate over m_{D\pi} by limiting the range of m_{D\pi}=[m_{D^{*+}}-\delta_m,m_{D^{*+}}+\delta_m] with \delta_m=2\Gamma_{D^{*+}}, 3\Gamma_{D^{*+}}, 4\Gamma_{D^{*+}} , we can find that the corresponding branching factions are 93.6\%, ~94.3\%,~ 95.0\% of the total branching ratio {\rm Br}(B_{c}^+ \rightarrow D^{*+} \pi^{0}\rightarrow D^0\pi^+ \pi^{0})= 0.93\times10^{-7}. In contrast, if we consider the virtual contribution from the region 2.1\sim3.5 GeV of the invariant mass m_{D\pi} shown in the right panel of Fig. 3, the corresponding branching fraction only amounts to 7.1\% of the total. The situation is simliar for the other decay channels.

      Figure 3.  (color online) Invariant mass m_{D\pi} dependence of the differential branching fraction for the decay B_c\to D^{*+}\pi^{0}\to D^{0}\pi^{+} \pi^{0} (left panel) and the corresponding virtual contribution in the m_{D\pi} region 2.1\sim3.5 GeV (right panel).

    IV.   SUMMARY
    • In this paper, we studied the quasi-two-body decays B_{c}^+\to D^{*+}h\to D^0\pi^+ h with h=(K^0,\pi^0,\eta^{(\prime)}) in the PQCD approach. The di-meson distribution amplitude for the D\pi system with the P-wave time-like form factor F_{D\pi}(s) is employed to describe the D^{*+} resonance and its interactions with the D^0\pi^+ pair. We predict the branching ratios of the concerned decays and find the following:

      1. The branching ratios of the quasi-two-body decays B_{c}^+\to D^{*+}h\to D^0\pi^+ h exhibit an obvious hierarchy

      \begin{aligned}[b]& {\rm Br}(B_{c}^+\to D^{*+}K^0\to D^0\pi^+ K^0)\\&> {\rm Br}(B_{c}^+\to D^{*+}\pi^0\to D^0\pi^+ \pi^0)\\&> {\rm Br}(B_{c}^+\to D^{*+}K^0\to D^0\pi^+ \eta^{(\prime)}), \end{aligned}

      (54)

      where {\rm Br}(B_{c}^+\to D^{*+}K^0\to D^0\pi^+ K^0) is the largest one and reaches the order of 10^{-6} . Thus, the decay B_{c}^+\to D^{*+}K^0\to D^0\pi^+ K^0 can be observed by future LHCb experiments.

      2. Using the narrow width approximation relation and the isospin conservation \Gamma(D^{*+}\to D^0\pi^+)/\Gamma(D^{*+}\to D\pi)= 2/3 , we can relate the branching ratios of these quasi-two-body decays B_{c}^+\to D^{*+}h\to D^0\pi^+ h with those of the corresponding two-body channels B_{c}^+\to D^{*+}h . Our results for the branching ratios of the decays B_{c}^+\to D^{*+}h are consistent with the previous PQCD calculations within errors, while there is considerable tension between the predictions of the PQCD approach and the RCQM for the decay B_{c}^+\to D^{*+}K^0 . The former is one order of magnitude larger than the latter. This is because the annihilation type contribution is dominant for the decay B_{c}^+\to D^{*+}K^0 , which is not calculable under the RCQM.

      3. The branching ratio of the decay B_{c}^+\to D^{*+}\eta\to D^{0}\pi^+\eta is larger than that of B_{c}^+\to D^{*+}\eta^\prime\to D^{0}\pi^+\eta^\prime , which is induced by the opposite interferences between the amplitudes \mathcal{A}\left(B_{c}^+ \rightarrow D^{*+} \eta_{q}\rightarrow D^0\pi^+ \eta_{q}\right) and \mathcal{A} (B_{c}^+\to D^{*+} \eta_{s}\to D^0\pi^+ \eta_{s}) .

      4. From the D^0\pi^+ invariant mass m_{D^0\pi^+} -dependences of these concerned decay spectrums, one can find that the main portions of the branching fractions are concentrated in a very small region of the m_{D^0\pi^+} . For example, approximately 94\% of the branching ratio of the decay B_{c}^+\to D^{*+}\pi^0\to D^{0}\pi^+\pi^0 comes from the realm of 2.1 MeV around the D^{*+} pole mass.

    APPENDIX: DECAY AMPLITUDES
    • In this appendix, we present the PQCD factorization formulas for the amplitudes of the decays B\to D^{*+}h\to D^0\pi^+ h

      \begin{aligned}[b] \mathcal{F}_{e}^{LL} =&4 \sqrt{2}G_F C_F \pi f_h (\eta -1) m_{B_c}^4 \int_0^1 {\rm d} x_1 {\rm d} z\int_0^{\infty}b_1{\rm d}b_1b_2{\rm d}b_2\phi_{B_c}(x_1,b_1)\phi_{D \pi}(z,b_2,\omega)\Big\{ [\sqrt{\eta } (r_b+2 z-2)\\{} &-2 r_b-z+1]h(\alpha_e,\beta_a,b_1,b_2)E_a(t_a)-\eta h(\alpha_e,\beta_b,b_2,b_1)E_b(t_b)\Big\}, \end{aligned}\tag{A1}

      \begin{aligned}[b] \mathcal{F}_{e}^{SP} =&-8 \sqrt{2} G_F C_F \pi f_h r_h m_{B_c}^4 \int_0^1{\rm d}x_1{\rm d}z\int_0^{\infty}b_1{\rm d}b_1b_2{\rm d}b_2\phi_B(x_1,b_1)\phi_{D \pi}(z,b_2,\omega)\Big\{[(\eta-1)rb-\sqrt{\eta}z+2 \eta(z-1) \\{} &+2]h(\alpha_e,\beta_a,b_1,b_2)E_a(t_a)+x_1h(\alpha_e,\beta_b,b_2,b_1)E_b(t_b)\Big\}, \end{aligned}\tag{A2}

      \begin{aligned}[b] \mathcal{M}_{e}^{LL} =&16 \frac{\sqrt{3}}{3} G_F C_F \pi (\eta -1) m_{B_c}^4\int_0^1{\rm d}x_1{\rm d}z{\rm d}x_3\int_0^{\infty}b_1{\rm d}b_1b_3{\rm d}b_3\phi_{B_c}(x_1,b_1)\phi_{D \pi}(z,b_1,\omega)\phi_h^{A}(x_3)(x_3)\Big\{ (\eta -x_1- \eta x_3 \\{} &+x_3+ \eta z - \sqrt{\eta} z -1)h(\beta_c,\alpha_e,b_1,b_3)E_{cd}(t_c) -(x_1+\eta x_3-x_3+\sqrt{\eta } z-z)h(\beta_d,\alpha_e,b_1,b_3)E_{cd}(t_d)\Big\}, \end{aligned}\tag{A3}

      \begin{aligned}[b] \mathcal{M}_{e}^{LR} =&-16 \frac{\sqrt{3}}{3} G_F C_F \pi r_h (\sqrt{\eta} -1) m_{B_c}^4\int_0^1{\rm d}x_1{\rm d}z{\rm d}x_3\int_0^{\infty}b_1{\rm d}b_1b_3{\rm d}b_3\phi_{B_c}(x_1,b_1)\phi_{D \pi}(z,b_1,\omega)\Big\{[\phi_h^{P}(x_3) (\eta +x_1\\{} &-\eta x_3+x_3-\sqrt{\eta } z-1)+\phi_h^{T}(x_3) \left(\eta +x_1-\eta x_3+x_3+\sqrt{\eta } z-1\right)]h(\beta_c,\alpha_e,b_1,b_3)E_{cd}(t_c)\\{} &-[x_1 (\phi_h^{P}(x_3)-\phi_h^{T}(x_3))+(\eta -1) x_3 (\phi_h^{P}(x_3)-\phi_h^{T}(x_3))-\sqrt{\eta } z (\phi_h^{P}(x_3)+\phi_h^{T}(x_3))]\\{} &\times h(\beta_d,\alpha_e,b_1,b_3)E_{cd}(t_d)\Big\}, \end{aligned}\tag{A4}

      \begin{aligned}[b] \mathcal{M}_{e}^{SP} =&-16 \frac{\sqrt{3}}{3} G_F C_F \pi (\eta-1) m_{B_c}^4 \int_0^1 {\rm d}x_1{\rm d}z{\rm d}x_3 \int_0^{\infty}b_1{\rm d}b_1b_3{\rm d}b_3\phi_{B_c}(x_1,b_1)\phi_{D \pi}(z,b_1, \omega) \Big\{[(\eta+x_1-\eta x_3+x_3 \\{} &+\eta z-\sqrt{\eta} z-1) \phi_{h}^{A}(x_3)]h(\beta_c,\alpha_e,b_1,b_3)E_{ef}(t_e)-[(x_1+\eta x_3-x_3+\sqrt{\eta}z-z) \phi_{h}^{A}(x_3)]\\{} &\times h(\beta_d,\alpha_e,b_1,b_3)E_{ef}(t_f)\Big\}, \end{aligned}\tag{A5}

      \begin{aligned}[b] \mathcal{M}_{a}^{LL} =&16 \frac{\sqrt{3}}{3} G_F C_F \pi m_{B_c}^4\int_0^1{\rm d}x_1{\rm d}z{\rm d}x_3\int_0^{\infty}b_1{\rm d}b_1b_2{\rm d}b_2\phi_{B_c}(x_1,b_1)\phi_{D \pi}(z,b_2,\omega)\Big\{[(\eta -1) \phi_h^{A}(x_3) (\eta +r_b\\{} &-\eta x_3+x_3+\eta z -1)+\sqrt{\eta } r_h (\phi_h^{T}(x_3) (-\eta +(\eta-1) x_3+ z+1)+\phi_h^{P}(x_3) (\eta -\eta x_3+x_3+z-1))] \\{} &h(\beta_e,\alpha_a,b_1,b_2)E_{ef}(t_e)-[(\eta -1) \phi_h^{A}(x_3) (r_c+(\eta -1) (z-1))+\sqrt{\eta } r_h (\phi_h^{P}(x_3) (\eta -x_1-\eta x_3+x_3+z-1)\\{} &+\phi_h^{T}(x_3) (x_1+\eta (x_3-1)-x_3+z-1))]h(\beta_f,\alpha_a,b_1,b_2)E_{ef}(t_f)\Big\}, \end{aligned}\tag{A6}

      \begin{aligned}[b] \mathcal{M}_{a}^{LR} =&-16 \frac{\sqrt{3}}{3} G_F C_F \pi m_{B_c}^4\int_0^1{\rm d}x_1{\rm d}z{\rm d}x_3\int_0^{\infty}b_1{\rm d}b_1b_2{\rm d}b_2\phi_{B_c}(x_1,b_1)\phi_{D \pi}(z,b_2,\omega)\Big\{[(\eta -1) r_b (\sqrt{\eta } \phi_h^{A}(x_3)\\{} &-r_h \phi_h^{P}(x_3)) +(\eta+1) r_b r_h \phi_h^{T}(x_3)+(\eta-1) r_P (x_3-1) (\phi_h^{P}(x_3)+\phi_h^{T}(x_3))+z (\eta^{3/2} \phi_h^{A}(x_3)-\sqrt{\eta} \phi_h^{P}(x_3)\\{} &+\eta r_h (\phi_h^{T}(x_3)-\Phi_h^{P}(x_3)))]h(\beta_e,\alpha_a,b_1,b_2)E_{ef}(t_e) -[r_h (\phi_h^{P}(x_3)+\phi_h^{T}(x_3)) (r_c+x_1-x_3)\\{} &+\eta^{3/2} \phi_h^{A}(x_3) (r_c+z-1)-\sqrt{\eta} \phi_h^{A}(x_3) (r_c+z-1) ]h(\beta_f,\alpha_a,b_1,b_2)E_{ef}(t_f)\Big\}, \end{aligned}\tag{A7}

      \begin{aligned}[b] \mathcal{F}_{a}^{LL} =&-4 \sqrt{2}G_F C_F \pi f_{B_c} m_{B_c}^4 \int_0^1{\rm d}z{\rm d}x_3\int_0^{\infty}b_2{\rm d}b_2b_3{\rm d}b_3\phi_{D \pi}(z,b_2,\omega)\Big\{[(\eta -1) (z-1) \phi_h^{A}(x_3)\\{} &-2 \sqrt{\eta } r_h z \phi_h^{P}(x_3)]h(\alpha_a,\beta_g,b_2,b_3)E_{g}(t_g)-[-(\eta -1) (\eta \phi_h^{A}(x_3)+r_c r_h \phi_h^{P}(x_3))-(\eta +1) r_c r_h \phi_h^{T}(x_3)\\{} &+(\eta-1)^{2} x_3 \phi_h^{A}(x_3))]h(\alpha_a,\beta_h,b_3,b_2)E_{h}(t_h)\Big\}, \end{aligned}\tag{A8}

      \begin{aligned}[b] \mathcal{F}_{a}^{SP} =&8 \sqrt{2}G_F C_F \pi f_{B_c} C_{D \pi} m_{B_c}^4 \int_0^1{\rm d}z{\rm d}x_3\int_0^{\infty}b_2{\rm d}b_2b_3{\rm d}b_3\phi_{D \pi}(z,b_2,\omega)\Big\{[2 r_h \phi_h^{P}(x_3) (\eta-z \eta -1)\\{} &+(\eta-1) \sqrt{\eta} (z-1) \phi_h^{A}(x_3) ]h(\alpha_a,\beta_g,b_2,b_3)E_{g}(t_g)+[-(\eta -1) (\eta \phi_h^{A}(x_3)+r_c r_h \phi_h^{P}(x_3))\\{} &-(\eta+1) r_c r_h \phi_h^{T}(x_3)+(\eta-1)^{2} x_3 \phi_h^{A}(x_3)]h(\alpha_a,\beta_h,b_3,b_2)E_{h}(t_h)\Big\}, \end{aligned}\tag{A9}

      where the mass ratio is r_h/m_{B_c} (r_b=m_b/m_{B_c}, r_c=m_c/m_{B_c}) , with r_h (r_b, r_c) being the final bachelor meson ( b, c quark) mass, and f_{B_c} and f_h are the decay constants of B_c and the final bachelor meson h=(K, \pi, \eta^{(\prime)}) , respectively. The hard scales are chosen as follows:

      \begin{aligned}[b] t_a &= \max\{\sqrt{|\beta_a|},1/b_1,1/b_2\},\;\;\;\;\;\;\;\;\;\;\;\;\;\; t_b = \max\{\sqrt{|\beta_b|},1/b_1,1/b_2\},\\ t_c &= \max\{\sqrt{|\beta_c|},\sqrt{|\alpha_e|},1/b_1,1/b_3\},\;\;\; t_d = \max\{\sqrt{|\beta_d|},\sqrt{|\alpha_e|},1/b_1,1/b_3\},\\ t_e &= \max\{\sqrt{|\beta_e|},\sqrt{|\alpha_a|},1/b_1,1/b_2\},\;\;\; t_f = \max\{\sqrt{|\beta_f|},\sqrt{|\alpha_a|},1/b_1,1/b_2\},\\ t_g &= \max\{\sqrt{|\beta_g|},\sqrt{|\alpha_a|},1/b_2,1/b_3\},\;\;\; t_h = \max\{\sqrt{|\beta_h|},\sqrt{|\alpha_a|},1/b_2,1/b_3\}, \end{aligned}\tag{A10}

      where

      \begin{aligned}[b] \alpha_e =& x_1 z m_{B_c}^2, \;\;\;\alpha_a = (z-1)[x_3+(1-x_3) \eta] m_{B_c}^2,\\ \beta_a =& (z+r_b^2-1)m_{B_c}^2,\;\;\; \beta_b = x_1m_B^2,\\ \beta_c =& [x_1 z -z (1-x_3) (1-\eta)]m_{B_c}^2,\;\;\;\beta_d = [x_1 z -z x_3 (1-\eta)]m_{B_c}^2,\\ \beta_e =& [r_b^2-z(1-x_3)(1-\eta)]m_{B_c}^2,\;\;\; \beta_f = [r_c^2-(z-1)(x_1-x_3(1-\eta)-\eta)]m_{B_c}^2,\\ \beta_g =& (z-1)m_{B_c}^2,\;\;\; \beta_h = [r_c^2-(\eta+x_3(1-\eta))]m_{B_c}^2. \end{aligned}\tag{A11}

      The hard functions are written as follows:

      \begin{aligned}[b] h(\alpha, \beta, b_1, b_2) =& h_1(\alpha,b_1)\times h_2(\beta,b_1,b_2), \\ h_1(\alpha,b_1) =&\Bigg\{\begin{array}{cc} K_0(\sqrt{\alpha}b_1), & \alpha > 0 \\ K_0({\rm i} \sqrt{-\alpha}b_1), & \alpha < 0 \end{array}\\ h_{2}\left(\beta, b_{1}, b_{2}\right)=&\left\{\begin{array}{ll} \theta\left(b_{1}-b_{2}\right) I_{0}\left(\sqrt{\beta} b_{2}\right) K_{0}\left(\sqrt{\beta} b_{1}\right)+\left(b_{1} \leftrightarrow b_{2}\right), & \beta>0 \\ \theta\left(b_{1}-b_{2}\right) J_{0}\left(\sqrt{-\beta} b_{2}\right) K_{0}\left({\rm i} \sqrt{-\beta} b_{1}\right)+\left(b_{1} \leftrightarrow b_{2}\right), & \beta<0 \end{array}\right. \end{aligned}\tag{A12}

      with

      K_0(ix) = \frac{\pi}{2}(-N_0(x)+{\rm i}J_0(x)). \tag{A13}

      The Sudakov factor S_t(x) from the threshold resummation is given as

      S_t(x) = \frac{2^{1+2a}\Gamma(3/2+a)}{\sqrt{\pi}\Gamma(1+a)}[x(1-x)]^a, \tag{A14}

      with the parameter a = 0.4 . The evolution factors are given as follows:

      E_{a}(t) = \alpha_s(t)\exp[-S_{B_c}(t)-S_{D^*}(t)]S_t(z), \;\;\;E_{b}(t) = \alpha_s(t)\exp[-S_{B_c}(t)-S_{D^*}(t)]S_t(x_1), \tag{A15}

      E_{g}(t) = \alpha_s(t)\exp[-S_{D^*}(t)-S_h(t)]S_t(x_3), \;\;\;E_{h}(t) = \alpha_s(t)\exp[-S_{D^*}(t)-S_h(t)]S_t(z), \tag{A16}

      E_{cd}(t) = \alpha_s(t)\exp[-S_{B_c}(t)-S_{D^*}(t)-S_h(t)]|_{b_2=b_1}, \tag{A17}

      E_{ef}(t) = \alpha_s(t)\exp[-S_{B_c}(t)-S_{D^*}(t)-S_h(t)]|_{b_3=b_2}, \tag{A18}

      where the Sudakov exponents are expressed as

      \begin{aligned}[b] S_{B_c}(t) =& s\left(x_1\frac{m_{B_c}}{\sqrt{2}},b_1\right)+\frac{5}{3}\int_{1/b_1}^t\frac{{\rm d} \bar{\mu}}{\bar{\mu}}\gamma_q(\alpha_s(\bar{\mu})),\\ S_{D^*}(t) =& s\left(z\frac{m_{B_c}}{\sqrt{2}},b_2\right)+s\left((1-z)\frac{m_{B_c}}{\sqrt{2}},b_2\right)+2\int_{1/b_2}^t\frac{{\rm d}\bar{\mu}}{\bar{\mu}}\gamma_q(\alpha_s(\bar{\mu})),\\ S_{h}(t)=& s\left(x_{3} \frac{m_{B_c}}{\sqrt{2}}, b_{3}\right)+s\left(\left(1-x_{3}\right) \frac{m_{B_c}}{\sqrt{2}}, b_{3}\right)+2 \int_{1 / b_{3}}^{t} \frac{{\rm d} \bar{\mu}}{\bar{\mu}} \gamma_{q}\left(\alpha_{s}(\bar{\mu})\right), \end{aligned}\tag{A19}

      with \gamma_q = -\alpha_s/\pi being the quark anomalous dimension. The function s(Q,b) is expressed as

      s(Q,b) = \frac{A^{(1)}}{2\beta_1}\hat{q}\ln\left(\frac{\hat{q}}{\hat{b}}\right)-\frac{A^{(1)}}{2\beta_1}(\hat{q}-\hat{b})+\frac{A^{(2)}}{4\beta_1^2}\hat{q}\ln\left(\frac{\hat{q}}{\hat{b}}-1\right)-\left[\frac{A^{(2)}}{4\beta_1^2}-\frac{A^{(1)}}{4\beta_1}\ln\left(\frac{{\rm e}^{2\gamma_E-1}}{2}\right)\right]ln\left(\frac{\hat{q}}{\hat{b}}\right), \tag{A20}

      where

      \begin{aligned}[b] \hat{q} = \ln\frac{Q}{\sqrt{2}\Lambda_{\rm QCD}},\quad\hat{b} = \ln\frac{1}{b\Lambda_{\rm QCD}},\quad \beta_1 = \frac{33-2n_f}{12},\quad\beta_2 = \frac{153-19n_f}{24},\quad A^{(1)} = \frac{4}{3}, \quad A^{(2)} = \frac{67}{9}-\frac{\pi^2}{3}-\frac{10}{27}n_f+\frac{8}{3}\beta_1\ln(\frac{1}{2} {\rm e}^{\gamma_E}), \end{aligned}\tag{A21}

      with n_f being the number of the quark flavor and \gamma_E being Euler's constant.

Reference (54)

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