Study of the $\bar{B}_{q}^{\ast}$ ${\to}$ $DM$ decays with perturbative QCD approach
Junfeng Sun Jie Gao Yueling Yang Qin Chang Na Wang Gongru Lu Jinshu Huang

  
  
  Institute of Particle and Nuclear Physics,
              Henan Normal University, Xinxiang 453007, China
  
  Institute of Particle and Nuclear Physics,
              Henan Normal University, Xinxiang 453007, China
  
  Institute of Particle and Nuclear Physics,
              Henan Normal University, Xinxiang 453007, China
  
  Institute of Particle and Nuclear Physics,
              Henan Normal University, Xinxiang 453007, China
  
  Institute of Particle and Nuclear Physics,
              Henan Normal University, Xinxiang 453007, China
  
  Institute of Particle and Nuclear Physics,
              Henan Normal University, Xinxiang 453007, China
  
  College of Physics and Electronic Engineering,
              Nanyang Normal University, Nanyang 473061, China


  
  
  The B_q^∗ → DP, DV weak decays
  are studied with the perturbative QCD approach, where q =
  u, d and s; P and V denote the ground SU(3)
  pseudoscalar and vector meson nonet.
  It is found that the branching ratios for the color-allowed
  B_q^∗ → D_qρ^- decays
  can reach up to 10^-9 or more, and should be
  promisingly measurable at the running LHC and forthcoming
  SuperKEKB experiments in the near future.
  
  12.15.Ji 12.39.St 13.25.Hw 14.40.Nd
  
  
  
  
  
  Study of the B_q^∗ → DM decays with perturbative QCD approach
    Jinshu Huang
    Received August 8, 2016; accepted March 2, 2017
=============================================================



  
  

§ INTRODUCTION

  
  In accordance with the conventional quark model assignments, the ground
  spin-singlet pseudoscalar B_q mesons and spin-triplet vector
  B^∗_q mesons have the same flavor components, and consist
  of one valence heavy antiquark b̅ and one light quark q,
  i.e., b̅q, with q = u, d, s <cit.>.
  With the two e^+e^- B-factory BaBar and Belle experiments,
  there is a combined data sample of over 1 ab^-1 at the
  Υ(4S) resonance. The B_u,d meson weak decay modes with branching
  ratio of over 10^-6 have been well measured <cit.>.
  The B_s meson, which can be produced in hadron collisions or at/over
  the resonance Υ(5S) in e^+e^- collisions[1],
  [1]
  In hadron colliders, CDF and D0 each have accumulated about 10 fb^-1
  data, and LHCb has accumulated over 5 fb^-1 data up to the year
  of 2016 <cit.>. In e^+e^- colliders, Belle has
  accumulated more than 100 fb^-1 data at the resonance
  Υ(5S) <cit.>.
  is being carefully scrutinized.
  However, the study of the B_q^∗ mesons has not actually attracted
  much attention yet, subject to the relatively inadequate statistics.
  Because the mass of the B_q^∗ mesons is a bit larger than
  that of the B_q mesons, the B_q^∗ meson should
  be produced at the relatively higher energy rather than at the
  resonance Υ(4S) in e^+e^- collisions.
  With the high luminosities and large production cross section
  at the running LHC, the forthcoming SuperKEKB and future Super
  proton proton Collider (SppC, which is still in the preliminary
  discussion and research stage up to now), more and more B_q^∗
  mesons will be accumulated in the future,
  which makes the B_q^∗ mesons another research laboratory
  for testing the Cabibbo-Kobayashi-Maskawa (CKM) picture for
  CP-violating phenomena, examining our comprehension of the
  underlying dynamical mechanism for the weak decays of the heavy
  flavor hadrons.

  Having the same valence quark components and approximately an equal
  mass, both the B^∗_q and B_q mesons can decay via
  weak interactions into the same final states.
  On the one hand, the B^∗_q and B_q meson weak decays
  would provide each other with a spurious background;
  on the other hand, the interplay between the B_q^∗ and B_q
  weak decays could offer some potential useful information to constrain
  parameters within the standard model, and might shed some fresh light on
  various intriguing puzzles in the B_q meson decays.
  The B_q meson decays are well described by the bottom quark decay
  with the light spectator quark q in the spectator model.
  At the quark level, most of the hadronic B_q meson decays involve
  the b → c transition due to the hierarchy relation among
  the CKM matrix elements.
  As is well known, there is a more than 3 σ discrepancy between
  the value of |V_cb| obtained from inclusive determinations,
  |V_cb| = (42.2±0.8)×10^-3, and from exclusive
  ones, |V_cb| = (39.2±0.7)×10^-3 <cit.>.
  Besides the semileptonic B_q^(∗) → D^(∗)ℓν̅
  decays, the nonleptonic B_q^(∗) → DM
  decays, with M representing the ground SU(3) pseudoscalar P and the
  vector V meson nonet, are also induced by the b → c transition,
  and hence could be used to extract/constrain the CKM matrix element
  |V_cb|.

  From the dynamical point of view, the phenomenological models used for the
  B_q → DM decays might, in principle, be extended
  and applied to the B_q^∗ → DM decays.
  The practical applicability and reliability of these models could be
  reevaluated with the B_q^∗ → DM decays.
  Recently, some attractive QCD-inspired methods, such as the perturbative
  QCD (pQCD) approach <cit.>,
  the QCD factorization (QCDF) approach <cit.>,
  soft and collinear effective theory <cit.>
  and so on, have been developed vigorously and employed widely to
  explain measurements on the B_q meson decays.
  The B_q → DM decays have been studied with
  the QCDF <cit.> and pQCD <cit.>
  approaches, but there are few research works on the B_q^∗ meson
  weak decays.
  Recently, the B_q^∗ → D_qV decays have been
  investigated with the QCDF approach <cit.>, and it is shown
  that the B_q^∗0 → D_q^+ρ^- decays
  with branching ratios of O(10^-8) might be accessible to the
  existing and future heavy flavor experiments.
  In this paper, we will give a comprehensive investigation into the
  two-body nonleptonic B_q^∗ → DM decays
  with the pQCD approach in order to provide the future experimental
  research with an available reference.

  As is well known, the B^∗_q meson decays are dominated by
  the electromagnetic interactions rather than the weak interactions,
  which differs significantly from the B_q meson decays.
  One can easily expect that the branching ratios for the
  B_q^∗ → DM weak decays should be very
  small due to the short electromagnetic lifetimes of the B_q^∗
  mesons <cit.>, although these processes are
  favored by the CKM matrix element |V_cb|.
  Of course, an abnormal large branching ratio might be a possible hint
  of new physics beyond the standard model.
  There is still no experimental report on the B_q^∗
  → DM weak decays so far.
  Furthermore, the B_q^∗ → DM weak decays
  offer the unique opportunity of observing the weak decay of a vector
  meson, where polarization effects could be explored.

  This paper is organized as follows.
  In section <ref>, we present the theoretical framework, the
  conventions and notations, together with amplitudes for the
  B_q^∗ → DM decays.
  Section <ref> is devoted to the numerical results and discussion.
  The final section is a summary.

  
  

§ THEORETICAL FRAMEWORK

  
  
  

 §.§ The effective Hamiltonian

  
  As is well known, the weak decays of the B_q^(∗) mesons
  inevitably involve multiple length scales, including the mass of
  m_W for the virtual gauge boson W, the mass of m_b for
  the decaying bottom quark, the infrared confinement scale
  Λ_ QCD of the strong interactions, and
  m_W ≫ m_b ≫ Λ_ QCD.
  So, one usually has to resort to the effective theory approximation
  scheme. With the operator product expansion and the renormalization
  group (RG) method, the effective Hamiltonian for the
  B_q^∗ → DM decays can be written as
  <cit.>,
 
   
    H_ eff  = G_F/√(2) ∑_q^'=d,s V_cb V_uq^'^∗{ C_1(μ) Q_1(μ)+C_2(μ) Q_2(μ)}
      + h.c.,

 
  where G_F ≃ 1.166×10^-5  GeV^-2
  <cit.> is the Fermi coupling constant.

  Using the Wolfenstein parametrization, the CKM factor
  V_cbV_uq^'^∗ are expressed as a series expansion of
  the small Wolfenstein parameter λ ≈ 0.2 <cit.>.
  Up to the order of O(λ^7), they
  can be written as follows:
  
  
    V_cb V_ud^∗   =    A λ^2 ( 1 -λ^2/2-λ^4/8 )
          + O(λ^7)
      , 
    
      V_cb V_us^∗   =    A λ^3+ O(λ^7)
      .

  
  It is very clearly seen that the both V_cb V_ud^∗
  and V_cb V_us^∗ are real-valued, i.e.,
  there is no weak phase difference.
  However, nonzero weak phase difference is necessary and
  indispensable for the direct CP violation.
  Therefore, none of direct CP violation should be expected
  for the B_q^∗ → DM decays.

  The renormalization scale μ separates the physical
  contributions into the short- and long-distance parts.
  The Wilson coefficients C_1,2 summarize the physical
  contributions above the scale μ.
  They, in principle, are calculable order by order in the
  strong coupling α_s at the scale m_W with
  the ordinary perturbation theory,
  and then evolved with the RG equation to the characteristic
  scale μ ∼ O(m_b) for the bottom quark
  decay <cit.>.
  The Wilson coefficients at the scale m_W are determined at the
  quark level rather than the hadron level, so they are regarded
  as process-independent couplings of the local operators Q_i.
  Their explicit analytical expressions, including the next-to-leading
  order corrections, have been given in Ref.<cit.>.

  The physical contributions from the scales lower than μ are
  contained in the hadronic matrix elements (HME) where the local
  four-quark operators are sandwiched between the initial and final
  hadron states.
  The local six-dimension operators arising from the W-boson exchange
  are defined as follows:
 
  
    Q_1   =    [ c̅_α γ_μ (1-γ_5) b_α]             [ q̅^'_β γ^μ (1-γ_5) u_β]
      ,
      
    
      Q_2   =    [ c̅_α γ_μ (1-γ_5) b_β]             [ q̅^'_β γ^μ (1-γ_5) u_α]
      .

 
  where α and β are color indices, i.e., the gluonic
  corrections are included.
  The operator Q_1 (Q_2) consists of two color-singlet
  (color-octet) currents.
  The operators Q_1 and Q_2, called current-current operators
  or tree operators, have the same flavor form and a different color structure.
  It is obvious that the B_q^∗ → DM decays
  are uncontaminated by the contributions from the penguin operators,
  which is positive to extract the CKM matrix element |V_cb|.

  Because of the participation of the strong interaction, especially, the
  long-distance effects in the conversion from the quarks of the local
  operators to the initial and final hadrons, barricades are still erected
  on the approaches of nonleptonic B_q^(∗) weak
  decays, which complicates the calculation.
  HME of the local operators are the most intricate part for theoretical
  calculation, where the perturbative and nonperturbative contributions
  entangle with each other.
  To evaluate the HME amplitudes, one usually has to resort to some
  plausible approximations and assumptions, which results in the
  model-dependence of theoretical predictions.
  It is obvious that a large part of the uncertainties does come from
  the practical treatment of HME, due to our inadequate understanding
  of the hadronization mechanism and the low-energy QCD behavior.
  For the phenomenology of the B_q^∗ → DM
  decays, one of the main tasks at this stage is how to effectively
  factorize HME of the local operators into hard and soft parts, and
  how to evaluate HME properly.

  
  

 §.§ Hadronic matrix elements

  
  One of the phenomenological schemes for the HME calculation is the
  factorization approximation based on Bjorken's a priori color
  transparency hypothesis, which says that the color singlet energetic
  hadron would have flown rapidly away from the color fields existing
  in the neighborhood of the interaction point before the soft gluons
  are exchanged among hadrons <cit.>.
  Modeled on the amplitudes for exclusive processes with the Lepage-Brodsky
  approach <cit.>, HME are usually written as the convolution integral
  of the hard kernels and the hadron distribution amplitudes (DAs).
  Hard kernels are expressed as the scattering amplitudes for the transition
  of the heavy bottom quark into light quarks.
  They are generally computable at the quark level with the perturbation
  theory as a series of expansion in the parameter 1/m_b and the strong
  coupling constant α_s in the heavy quark limit.
  It is assumed that the soft and nonperturbative contributions of HME
  could be absorbed into hadron DAs.
  The distribution amplitudes are functions of parton momentum fractions.
  They, although not calculable, are regarded as universal
  and can be determined by nonperturbative means or extracted from data.
  With the traits of universality and determinability of hadron DAs, HME
  have a sample structure and can be evaluated to make predictions.

  Besides the factorizable contributions to HME, the nonfactorizable
  corrections to HME also play an important role in commenting on
  the experimental measurements and solving the so-called puzzles and
  anomalies, and hence should be carefully considered, as commonly
  recognized by theoretical physicists.
  In order to regulate the endpoint singularities which appear
  in the spectator scattering and annihilation amplitudes with the
  QCDF approach and spoil the perturbative calculation with the
  collinear approximation <cit.>, it is suggested by the pQCD approach
  <cit.> that the transverse momentum
  of quarks should be conserved and, additionally, that a Sudakov factor
  should be introduced to DAs for all the participant
  hadrons to further suppress the long-distance and soft
  contributions.
  The basic pQCD formula for nonleptonic weak decay amplitudes
  could be factorized into three parts: the hard effects enclosed
  by the Wilson coefficients C_i, hard scattering kernels
  H_i, and the universal wave functions Φ_j.
  The general form is a multidimensional integral
  <cit.>,
  
  
    A_i ∝ ∫ ∏_jdx_j db_j 
      C_i(t)  H_i(t_i,x_j,b_j) Φ_j(x_j,b_j) e^-S_j,

  
  where x_j is the longitudinal momentum fraction of the valence
  quarks. b_j is the conjugate variable of the transverse momentum k_jT.
  The scale t_i is preferably chosen to be the maximum virtuality of
  all the internal particles. The Sudakov factor e^-S_j, together
  with the particular scale t_i, will ensure the perturbative
  calculation is feasible and reliable.

  
  

 §.§ Kinematic variables

  

  The B_q^(∗) weak decays are actually dominated
  by the b quark weak decay.
  In the heavy quark limit, the light quark originating from the heavy
  bottom quark decay is assumed to be energetic and race quickly away
  from the weak interaction point.
  If the velocity v ∼ c (the speed of light), the light
  quarks move near the light-cone line.
  The light-cone dynamics can be used to describe the relativistic
  system along the light-front direction.
  The light-cone coordinates (x^+,x^-,x_⊥) of space-time
  are defined as x^±
  = (x^0±x^3)/√(2) (or (t±x^3)/√(2)) and
  x_⊥ = x^i with i = 1 and 2.
  x^± = 0 is called the light-front.
  The scalar product of any two four-dimensional vectors is given by
  a·b = a_μb^μ = a^+b^- + a^-b^+
  - a_⊥·b_⊥.
  In the rest frame of the B_q^∗ meson, the final
  D and M mesons are back-to-back.
  The light-cone kinematic variables are defined as follows:
  
  
    p_B_q^∗  =  p_1  = m_1/√(2)(1,1,0)
      ,

  
  
    p_D  =  p_2  =  (p_2^+,p_2^-,0)
      ,

  
  
    p_M  =  p_3  =  (p_3^-,p_3^+,0)
      ,

  
  
    k_i  =  x_i p_i+(0,0,k_iT)
      ,

  
  
    p_i^±  =  (E_i ± p)/√(2),

  
  
    t  =  2 p_1·p_2  =  m_1^2+m_2^2-m_3^2  = 2 m_1 E_2,

  
  
    u  =  2 p_1·p_3  =  m_1^2-m_2^2+m_3^2  = 2 m_1 E_3,

  
  
    s  =  2 p_2·p_3  =  m_1^2-m_2^2-m_3^2,

  
  
    s t +s u-t u  =  4 m_1^2 p^2,

  
  where the subscripts i = 1, 2 and 3 of the variables (such as,
  four-dimensional momentum p_i, energy E_i, and mass m_i)
  correspond to the B_q^∗, D and M mesons,
  respectively.
  k_i is the momentum of the light antiquark carrying the longitudinal
  momentum fraction x_i. k_iT is the transverse momentum.
  t, u and s are the Lorentz scalar variables. p is the
  common momentum of the final states.
  These momenta are shown in Fig.<ref>(a), Fig.<ref>(a)
  and Fig.<ref>(a).

  
  

 §.§ Wave functions

  
  As aforementioned, wave functions are the essential input parameters
  in the master pQCD formula for the HME calculation.
  Following the notations in Refs. <cit.>, the wave functions
  of the participating meson are defined as the meson-to-vacuum HME.
  
  
    ⟨0|q̅_i(z)b_j(0)|B_q^∗(p,ϵ^∥)⟩ 
     = f_B_q^∗/4∫d^4k e^-ik·z{ϵ̸^∥ [
      m_B_q^∗ Φ_B_q^∗^v(k)  -
      p̸ Φ_B_q^∗^t(k) ] }_ji,

  
  
    ⟨0|q̅_i(z)b_j(0)|B_q^∗(p,ϵ^⊥)⟩ 
     = f_B_q^∗/4∫d^4k e^-ik·z{ϵ̸^⊥ [
      m_B_q^∗ Φ_B_q^∗^V(k)  -
      p̸ Φ_B_q^∗^T(k) ] }_ji,

  
  
    ⟨D_q(p)|c̅_i(0)q_j(z)|0⟩ 
     = i f_D_q/4∫d^4k e^+ik·z {γ_5[ p̸ Φ_D_q^a(k)
      +m_D_q Φ_D_q^p(k) ] }_ji,

  
  
    ⟨P(p)|q̅_i(0)q^'_j(z)|0⟩    =   1/4∫d^4k e^+ik·z {γ_5[ p̸ Φ_P^a(k) +
     μ_P Φ_P^p(k)
      
                       +
     μ_P (n̸_+n̸_--1) Φ_P^t(k)
      ] }_ji,

  
  
    ⟨V(p,ϵ^∥)|q̅_i(0)q^'_j(z)|0⟩ 
     = 1/4∫d^4k e^+ik·z {ϵ̸^∥ m_V Φ_V^v(k)
      +ϵ̸^∥p̸ Φ_V^t(k)
      -m_V Φ_V^s(k) }_ji,

  
  
    ⟨V(p,ϵ^⊥)|q̅_i(0)q^'_j(z)|0⟩    =   1/4∫d^4k e^+ik·z {ϵ̸^⊥ m_V Φ_V^V(k)
      +ϵ̸^⊥p̸ Φ_V^T(k)
      
               +
      i m_V/p·n_+ γ_5 ε_μναβγ^μ ϵ^⊥ν p^α 
      n_+^β Φ_V^A(k) }_ji,

  
  where f_B_q^∗ and f_D_q are the decay
  constants of the B_q^∗ meson and the D_q
  meson, respectively.
  ϵ^∥ and ϵ^⊥ are the
  longitudinal and transverse polarization vectors.
  n_+ = (1,0,0) and n_- = (0,1,0)
  are the positive and negative null vectors,
  i.e., n_±^2 = 0. The chiral factor μ_P
  relates the pseudoscalar meson mass to the quark mass through
  the following way <cit.>,
  
  
    μ_P  = m_π^2/m_u+m_d  = m_K^2/m_u,d+m_s ≈  (1.6±0.2) GeV.

  

  With the twist classification based on the power counting rule
  in the infinite momentum frame <cit.>,
  the wave functions Φ_B_q^∗,V^v,T and
  Φ_D_q,P^a are twist-2 (the leading twist),
  while the wave functions Φ_B_q^∗,V^t,V,s,A
  and Φ_D_q,P^p,t are twist-3.
  By integrating out the transverse momentum from the wave
  functions, one can obtain the corresponding distribution
  amplitudes.
  In our calculation, the expressions of the DAs for the
  heavy-flavored mesons are <cit.>
  
   
    ϕ_B_q^∗^v,T(x) = A  x x̅ exp{ -1/8 ω_B_q^∗^2 ( m_q^2/x+m_b^2/x̅) },

  
   
    ϕ_B_q^∗^t(x) = B  (x̅-x)^2 exp{ -1/8 ω_B_q^∗^2 ( m_q^2/x+m_b^2/x̅) },

  
   
    ϕ_B_q^∗^V(x) = C {1+(x̅-x)^2} exp{ -1/8 ω_B_q^∗^2 ( m_q^2/x+m_b^2/x̅) },

  
  
   
    ϕ_D_q^a(x) = D  x x̅ exp{ -1/8 ω_D_q^2 ( m_q^2/x+m_c^2/x̅) },

  
   
    ϕ_D_q^p(x) = E exp{ -1/8 ω_D_q^2 ( m_q^2/x+m_c^2/x̅) },

  
   where x and x̅ (≡ 1 - x) are the
   longitudinal momentum fractions of the light and heavy partons;
   m_b, m_c and m_q are the mass of the valence
   b, c and q quarks.
   The parameter ω_i determines the average transverse
   momentum of the partons, and
   ω_i ≈ m_i α_s(m_i).
   The parameters A, B, C, D and E are the normalization
   coefficients to satisfy the conditions,
  
   
    ∫_0^1dx ϕ_B_q^∗^v,t,V,T(x)=1
       ,

  
   
    ∫_0^1dx ϕ_D_q^a,p(x) =1
       .

  

  The main distinguishing feature of the above DAs in
  Eqs.(<ref>-<ref>) is the exponential
  functions, where the exponential factors are proportional
  to the ratio of the parton mass squared m_i^2 to the
  momentum fraction x_i, i.e., m_i^2/x_i.
  Hence, the DAs of Eqs.(<ref>-<ref>) are generally
  consistent with the ansatz that the momentum fractions are
  shared among the valence quarks according to the quark mass,
  i.e., a light quark will carry a smaller fraction of the parton
  momentum than a heavy quark in a heavy-light system.
  In addition, the exponential functions strongly suppress
  the contributions from the endpoint of x, x̅ →
  0, and naturally provide the effective truncation for the
  endpoint and soft contributions.

  As is well known, there are many phenomenological DA models for
  the charmed mesons. Some have been recited by Eq.(30) in
  Ref.<cit.>. One of the favorable DA models from
  the experimental data, without the distinction between the
  twist-2 and twist-3, has the common expression as below,
  
   
    ϕ_D_q(x) = 6 x x̅ {1+C_D_q (x̅-x) },

  
  where the parameter C_D_u,d = 0.5 for the D_u,d meson,
  and C_D_s = 0.4 for the D_s meson.

  The expressions of the twist-2 quark-antiquark DAs for the
  light pseudoscalar and vector mesons have the expansion
  <cit.>,
  
   
    ϕ_P^a(x)  =  i f_P 6 x x̅ ∑_i=0 a^P_i C_i^3/2(ξ)
       ,

  
   
    ϕ_V^v(x)   =  f_V 6 x x̅ ∑_i=0 a^∥_i C_i^3/2(ξ)
      ,

  
  
    ϕ_V^T(x)   =  f_V^T 6 x x̅ ∑_i=0 a^⊥_i C_i^3/2(ξ)
      ,

  
  where f_P is the decay constant for the pseudoscalar meson P;
  f_V and f_V^T are the vector and tensor (also called the
  longitudinal and transverse) decay constants for the vector meson V.
  The nonperturbative parameters of a_i^P,∥,⊥
  are called the Gegenbauer moments, and a_0^P,∥,⊥
  = 1 for the asymptotic forms, a_ odd i^P,∥,⊥
  = 0 for the DAs of the G-parity eigenstates, such as the unflavored
  π, η, η^', ρ, ω, ϕ mesons.
  The short-hand notation ξ = x - x̅ = 2 x - 1.
  The analytical expressions of the Gegenbauer polynomials C_i^j(ξ)
  are as below,
  
   
    C_0^j(ξ)  =  1
       , 
           C_1^j(ξ)  =  2 j ξ, 
           C_2^j(ξ)  =  2 j (j+1) ξ^2-j
       , 
           ......

  

  As for the twist-3 DAs for the light pseudoscalar and vector mesons,
  their asymptotic forms will be employed in this paper for the
  simplification <cit.>, i.e.,
  
   
    ϕ_P^p(x)   =    +i f_P  C_0^1/2(ξ) 
       , 
    
      ϕ_P^t(x)   =    -i f_P  C_1^1/2(ξ)
       , 
    
     ϕ_V^t(x)    =    +3  f_V^T ξ^2, 
    
     ϕ_V^s(x)    =    -3  f_V^T ξ, 
    
     ϕ_V^V(x)    =    +3/4 f_V (1+ξ^2)
      , 
    
     ϕ_V^A(x)    =    -3/2 f_V ξ.

  

  
  

 §.§ Decay amplitudes

  
  As aforementioned, the B_q^∗ → DM weak decays
  are induced practically by the b quark decay at the quark level.
  There are three possible types of Feynman diagrams for the
  B_q^∗ → DM decays with the pQCD approach, i.e.,
  the color-allowed topologies of Fig.<ref> induced by
  the external W-emission interactions,
  the color-suppressed topologies of Fig.<ref> induced by
  the internal W-emission interactions,
  and the annihilation topologies of Fig.<ref> induced by
  the W-exchange interactions.
  In the emission topologies of Fig.<ref> (Fig.<ref>),
  the light spectator quark in the B_q^∗ meson is
  absorbed by the recoiled D_q (M_q) meson, and the exchanged
  gluons are space-like.
  In the annihilation topologies of Fig.<ref>, the exchanged
  gluons are time-like, which then split into the light quark-antiquark
  pair.

  
  
  
  
  
  
  
  
  

  The first two diagrams of Fig.<ref>, Fig.<ref>,
  and Fig.<ref> are usually called the factorizable topologies.
  In the color-allowed (color-suppressed) factorizable emission topologies,
  the gluons are exchanged only between the initial B_q^∗
  and the recoil D_q (M_q) meson pair, and the emission M (D^0)
  meson could be completely parted from the B_q^∗D_q
  (B_q^∗M_q) system.
  In the factorizable annihilation topologies, the gluons are exchanged
  only between the final DM meson pair, and the initial B_q^∗
  meson could be directly separated from the DM meson pair.
  Hence, in the factorizable emission (annihilation) topologies, the integral
  of the wave functions for the emission (initial) mesons reduces to the
  corresponding decay constant.
  For the factorizable topologies, the decay amplitudes will have the relatively
  simple structures, and can be written as the product of the decay constants
  and the hadron transition form factors.
  With the pQCD approach, the form factors can be written as the convolution
  integral of the hard scattering amplitudes and the hadron DAs.

  The last two diagrams of Fig.<ref>, Fig.<ref>, and
  Fig.<ref> are usually called the nonfactorizable topologies.
  In the nonfactorizable topologies, the emission meson is entangled
  with the gluons that radiated from the spectator quark, and hence
  on meson can be separated clearly from the other mesons.
  Hence, the decay amplitudes for the nonfactorizable topologies
  have quite complicated structures, and the amplitude convolution
  integral involve the wave functions of all the participating mesons.
  The nonfactorizable emission topologies within the pQCD framework
  are also called the spectator scattering topologies with the QCDF
  approach.
  Especially for the color-suppressed emission topologies, the factorizable
  contributions are proportional to the small parameter a_2, hence,
  the nonfactorizable contributions, being proportional to the large
  Wilson coefficient C_1, should be significant.
  As widely recognized, the nonfactorizable contributions play
  an important role in clarifying or reducing some discrepancies
  between the theoretical results and the experimental data on the
  nonleptonic B meson weak decays.

  Among the three possible types of Feynman diagrams (Fig.<ref>,
  Fig.<ref>, and Fig.<ref>), only one or two of them
  will contribute to the specific B_q^∗ → DM decays.
  The explicit amplitudes for the concrete B_q^∗ → DP, DV
  decays have been collected in the Appendixes <ref> and <ref>, and
  the building blocks in the Appendixes <ref>, <ref> and <ref>.
  According to the polarization relations between the initial and final vector
  mesons, the amplitudes for the B_q^∗ → DV decays
  can generally be decomposed into the following structures
  <cit.>,
  
   
    A(B_q^∗→DV)  =  A_L(ϵ_B_q^∗^∥,ϵ_V^∥)
     + A_N(ϵ_B_q^∗^⊥·ϵ_V^⊥)
     +i  A_T ε_μναβ ϵ_B_q^∗^μ ϵ_V^ν 
       p_B_q^∗^α p_V^β.

  
  which is conventionally written as the helicity amplitudes,
  
   
    H_0 =  A_L(ϵ_B_q^∗^∥,ϵ_V^∥)
       ,

  
   
    H_∥ = √(2)  A_N,

  
   
    H_⊥ = √(2) m_B_q^∗ p  A_T.

  

  As is well known, it is commonly assumed that the SU(3) symmetry breaking
  interactions mixes the isospin-singlet neutral members of the octet with
  the singlet states.
  The ideal mixing angle θ_V (with sinθ_V = 1/√(3))
  between the octet and the singlet states is almost true in practice for
  the physical ω and ϕ mesons, i.e.,
  the valence quark components are ω = (uu̅+dd̅)/√(2)
  and ϕ = ss̅.
  As for the mixing among the light pseudoscalar mesons, the notations known as
  the quark-flavor basis description <cit.> is adopted here, and for
  simplicity, the possible gluonium and charmonium compositions are neglected
  for the time being, i.e.,
  
   
    ([   η; η^' ]) =
       ([  cosθ_P -sinθ_P;  sinθ_P  cosθ_P ])
       ([ η_q; η_s ])
       ,

  
  where the flavor states η_q = (uu̅+dd̅)/√(2) and
  η_s = ss̅. The mixing angle determined from experimental
  data is θ_P = (39.3±1.0)^∘ <cit.>.
  The mass relations between the physical states (η
  and η^') and the flavor states (η_q
  and η_s) are
  
   
    m_η_q^2   =   
       m_η^2 cos^2θ_P
      +m_η^'^2 sin^2θ_P
      -√(2) f_η_s/f_η_q 
      (m_η^'^2- m_η^2) cosθ_P sinθ_P, 
    
       m_η_s^2   =   
       m_η^2 sin^2θ_P
      +m_η^'^2 cos^2θ_P
      -f_η_q/√(2) f_η_s
      (m_η^'^2- m_η^2) cosθ_P sinθ_P,

  
  where f_η_q and f_η_s are the decay constants.

  The amplitudes for the B_q^∗ → Dη,
  Dη^' decays can be written as
  
  
   
    A(B_q^∗→Dη)
          =   cosθ_P  A(B_q^∗→Dη_q)
      -sinθ_P  A(B_q^∗→Dη_s)
        , 
    
        A(B_q^∗→Dη^')
          =   sinθ_P  A(B_q^∗→Dη_q)
      +cosθ_P  A(B_q^∗→Dη_s)
        .

  

  
  

§ NUMERICAL RESULTS AND DISCUSSION

  
  In the rest frame of the B_q^∗ meson,
  the branching ratio is defined as
  
   
    Br(B_q^∗→DV)  = 1/24π p/m_B^∗_q^2Γ_B^∗_q {|H_0|^2+|H_∥|^2
     +|H_⊥|^2},

  
   
    Br(B_q^∗→DP)  = 1/24π p/m_B^∗_q^2Γ_B^∗_q | A(B_q^∗→DP)|^2,

  
  where Γ_B^∗_q is the full decay width
  of the B_q^∗ meson.

  Unfortunately, the experimental data on Γ_B^∗_q
  are still unavailable until now.
  As is generally known, the electromagnetic radiation processes
  B^∗_q → B_qγ dominate
  the B^∗_q meson decays, and the mass
  differences between the B^∗_q and B_q
  mesons are very small, m_B_q^∗ - m_B_q ≲
  50 MeV <cit.>, which results in the fact that the photons from
  the B^∗_q → B_qγ process
  are too soft to be easily identified by the detectors at the existing
  experiments.
  A good approximation for the decay width is Γ_B_q^∗
  ≈ Γ(B_q^∗→B_qγ).
  Theoretically, there is the close relation between the partial decay width
  for the B_q^∗ → B_qγ decay
  and the magnetic dipole (M1) moment of the B_q^∗ meson
  <cit.>, i.e.,
  
   
    Γ(B_q^∗→B_qγ)  = 4/3 α_ em  k_γ^3 μ^2_h,

  
  where α_ em is the fine structure constant;
  k_γ = (m_B_q^∗^2-m_B_q^2)/2m_B_q^∗
  is the photon momentum in the rest frame of the B_q^∗
  meson; μ_h is the M1 moment of the B_q^∗ meson.
  There are a large number of theoretical predictions on the partial decay
  width Γ(B_q^∗→B_qγ).
  Many of these have been collected into Table 7 of Ref.<cit.>
  and Tables 3 and 4 of Ref.<cit.>.
  However, there are big differences among these estimations with
  various models, due to our inaccurate information about the M1
  moments of mesons.
  In principle, the M1 moment of a hadron should be the sum of the
  M1 moments of its constituent quarks.
  As is well known, for an elementary particle, the M1 moment is
  proportional to the charge and inversely proportional to the mass.
  Hence, the M1 moment of the heavy-light B_q^∗
  meson should be mainly affected by the M1 moment of the light quark
  rather than the bottom quark.
  With the M1 moments of the light u, d and s quarks in the
  terms of the nuclear magnetons μ_N, i.e.,
  μ_u ≃ 1.85 μ_N,
  μ_d ≃ -0.97 μ_N, and
  μ_s ≃ -0.61 μ_N <cit.>,
  it is expected to have the relations
  Γ(B_u^∗→B_uγ) >
  Γ(B_d^∗→B_dγ) >
  Γ(B_s^∗→B_sγ), and therefore the relations
  Γ_B_u^∗ > Γ_B_d^∗
  > Γ_B_s^∗.
  It is far beyond the scope of this paper to elaborate more on the
  details of the decay width Γ_B_q^∗.
  In our calculation, in order to give a quantitative estimation of
  the branching ratios for the B_q^∗ → DM
  decays, we will use the following values of the decay widths,
  
   
    Γ_B_u^∗ ∼Γ(B_u^∗→B_uγ) ∼   450 eV,

  
   
    Γ_B_d^∗ ∼Γ(B_d^∗→B_dγ) ∼   150 eV,

  
   
    Γ_B_s^∗ ∼Γ(B_s^∗→B_sγ) ∼   100 eV,

  
  which is basically consistent with the recent results
  of Ref.<cit.>.

  
  
  

  
  
  

  
  
  

  The numerical values of other input parameters are collected in
  Table <ref>, where their central values will be fixed
  as the default inputs unless otherwise specified.
  In addition, in order to investigate the effects from different
  DA models, we explore three scenarios,
  
  
  * Scenario I:
   Eqs.(<ref>-<ref>)
   for the DAs of ϕ_B^∗^v,t,V,T and ϕ_D_q^a,p;
  
  * Scenario II:
  ϕ_B^∗^v,t,V,T = Eq.(<ref>),
  and ϕ_D_q^a,p = Eq.(<ref>);
  
  * Scenario III:
  ϕ_B^∗^v,t,V,T = Eq.(<ref>),
  and ϕ_D_q^a,p = Eq.(<ref>).
  

  Our numerical results on the branching ratios
  are presented in Tables <ref>
  and <ref>, where the uncertainties come from the
  typical scale (1±0.1)t_i, the mass m_c and m_b,
  and the hadronic parameters (including the decay constants,
  Gegenbauer moments, and so on), respectively.
  The following are some comments.

  (1)
  Generally, the B_q^∗ → DP decay modes
  could be divided into three categories, i.e. the “T”, “C”, and “A”
  types are dominated by contributions from the color-allowed emission
  topologies of Fig.<ref>, the color-suppressed emission
  topologies of Fig.<ref>, and the pure annihilation topologies
  of Fig.<ref>, respectively.
  And each category could be further divided into two classes,
  i.e., the decay amplitudes of the classes “I” and “II” are
  proportional to the CKM factors of V_cb V_ud^∗ ∼
  Aλ^2 and V_cb V_us^∗ ∼
  Aλ^3, respectively.
  There are many hierarchical relations among the branching ratios,
  such as,
  
   
    Br(class T-I) >
       Br(class C-I) >
       Br(class A-I)
       ,

  
   
    Br(class T-II) >
       Br(class C-II) >
       Br(class A-II)
       ,

  
   
    Br(class X-I) >
       Br(class X-II),   for X = T, C, A.

  
  These categories and relations also happen to hold true
  for the B_q^∗ → DV decays.

  For the “T” and “C” types of the B_q^∗
  → DM decays, the annihilation contributions have a
  negligible impact on the branching ratios, and they are
  strongly suppressed relative to the emission contributions,
  as is stated by the QCDF approach <cit.>.

  For the “T” types of the B_q^∗ →
  DM decays, the factorizable contributions from the emission
  topologies to the branching ratios are dominant over other
  contributions.
  However, for the “C” types of the B_q^∗ →
  DM decays, the nonfactorizable contributions to the branching
  ratios become very important, and sometimes even dominant.

  (2)
  With the law of conservation of angular momentum,
  three partial wave amplitudes, including the s-, p-, and d-wave
  amplitudes, all will contribute to the B_q^∗
  → DV decays, while only the p-wave amplitude will
  contribute to the B_q^∗ → DP decays.
  Besides, the branching ratios are proportional to the squares
  of the decay constants with the pQCD approach.
  With the magnitude relations between the decay constants
  f_V > f_P, one should expect to have the general
  relation of the branching ratios,
  
   
    Br(B_q^∗→DV) >
       Br(B_q^∗→DP)
       ,

  
  for the final vector V and pseudoscalar P mesons
  carrying the same flavor, azimuthal and magnetic isospin
  quantum numbers.
  And due to the relations between the decay constants
  f_B_s^∗ > f_B_u,d^∗ and
  f_D_s > f_D_u,d,
  and the relations between the decay widths
  Γ_B_s^∗ < Γ_B_u,d^∗,
  the color-allowed B_s^∗0 →
  D_s^+ρ^- decay has a relatively large
  branching ratio.

  Furthermore, our study results show that for the “T”
  types of the B_q^∗ → DV decays,
  the contributions of the longitudinal polarization
  part are dominant. Take the B_s^∗0
  → D_s^+ρ^- decay for example,
  the longitudinal polarization fraction f_0 ≡
  |H_0|^2/|H_0|^2
  +|H_∥|^2+|H_⊥|^2
  ≈ 90% (85%),
  the parallel polarization fraction f_∥ ≡
  |H_0|^2/|H_∥|^2
  +|H_∥|^2+|H_⊥|^2
  ≈ 9% (12%),
  and the perpendicular polarization fraction f_⊥ ≡
  |H_0|^2/|H_⊥|^2
  +|H_∥|^2+|H_⊥|^2
  ≈ 1% (3%) with the DA scenarios I (II and III),
  which generally agree with those obtained by the QCDF approach <cit.>.

  (3)
  As is well known, the theoretical results depend on the values
  of the input parameters.
  From the numbers in Tables <ref> and <ref>,
  it is clearly seen that the main uncertainty is due to the
  limited knowledge of the hadron DAs,
  for example, the large discrepancy among the different DA scenarios.
  Besides the theoretical uncertainties listed in Tables <ref>
  and <ref>, the CKM parameters will bring some 6%
  uncertainties.
  With a different value of the decay width Γ_B_q^∗,
  the branching ratios in Tables <ref> and <ref>
  should be multiplied by the factors of
  450  eV/Γ_B_u^∗,
  150  eV/Γ_B_d^∗,
  100  eV/Γ_B_s^∗
  for the B_u^∗, B_d^∗, B_s^∗
  weak decays, respectively.
  To reduce the theoretical uncertainties, one of the
  commonly used methods is to exploit the rate of the
  branching ratios, such as,
  
   
    Br(B_u^∗-→D_u^0π^-)/ Br(B_u^∗-→D_u^0K^-) ≈ f_π^2/λ^2 f_K^2,

  
   
    Br(B_u^∗-→D_u^0ρ^-)/ Br(B_u^∗-→D_u^0K^∗-) ≈ f_ρ^2/λ^2 f_K^∗^2,

  
   
    Br(B_s^∗0→D_u^0ϕ)/ Br(B_s^∗0→D_u^0K^∗0) ≈ λ^2 f_ϕ^2/f_K^∗^2.

  

  (4)
  The branching ratios for the B_q^∗ →
  DM decays are smaller by at least five orders of magnitude
  than the branching ratios for the B_q →
  DM decays <cit.>.
  This fact implies that the possible background from the
  B_q^∗ → DM decays could be safely
  neglected when the B_q → DM decays
  were analyzed, but not vice versa, i.e., one of main pollution
  backgrounds for the B_q^∗ → DM
  decays would come from the B_q → DM
  decays, even if the invariant mass of the DM meson pair
  could be used to distinguish the B_q^∗
  meson from the B_q meson experimentally.

  
   
  

  (5)
  The event numbers of the B_q^∗ meson in a data sample
  can be calculated by the following formula,
  
   
    N(B_q^∗) =  L_ int × σ_bb̅ × f_B_q ×  f_B_q^∗/ f_B_q.

  
   
    f_B_q^∗ = 2 × f_B_q^∗B_q^∗
       +2 × f_B_q^∗B_q^∗π
       +f_B_q^∗B_q+ c.c.
       +f_B_q^∗B_qπ+ c.c
       +⋯,

  
  where L_ int is the integrated luminosity,
  σ_bb̅ denotes the bb̅ pair production cross section,
  f_B_q, f_B_q^∗B_q^∗, ⋯ refer
  to the production fraction of all the B_q meson, the
  B_q^∗B_q^∗ meson pair, ⋯.
  The production fractions of specific modes at the center-of-mass
  of the Υ(5S) resonance <cit.> are listed
  in Table <ref>.
  With a large production cross section of the process e^+e^- →
  bb̅ at the Υ(5S) peak σ_bb̅ =
  (0.340±0.016)  nb <cit.>, it is expected that
  some 3.3×10^9 B_u,d^∗ and 1.2×10^9
  B_s^∗ mesons could be available per 10  ab^-1
  Υ(5S) dataset.
  The branching ratios of the color-allowed “T-I” class
  B_q^∗ → DM decays can reach up
  to O(10^-9) or more, which are essentially coincident
  with those obtained by the QCDF approach <cit.>.
  Hence, a few events of the B_q^∗ →
  D_qπ^- and B_u,d^∗ →
  D_u,dρ^- decays, and dozens of the
  B_s^∗0 → D_s^+ρ^- decay,
  might be available at the forthcoming SuperKEKB.
  At high energy hadron colliders, for example, given with the
  cross section at the LHCb σ_bb̅ ≈
  100 μb <cit.>,
  with a similar ratio f_B_u = f_B_d = 0.344±0.021
  and f_B_s = 0.115±0.013 at Tevatron <cit.>
  and a similar ratio f_B_q^∗/f_B_q at the
  Υ(5S) meson <cit.>,
  some 9.8×10^13 B_u,d^∗ events and 2.2×10^13
  B_s^∗ events per ab^-1 dataset could be available at
  the LHCb, corresponding to more than 10^5 of the
  B_s^∗0 → D_s^+ρ^- decay
  events and over 10^4 of the B_q^∗ →
  D_qπ^- and B_u,d^∗ →
  D_u,dρ^- decay events,
  which should be easily measured by the future LHCb experiments.

  
  

§ SUMMARY

  
  Besides the dominant electromagnetic decay mode, the ground vector
  B_q^∗ meson (q = u, d and s) can also decay via
  the weak interactions within the standard model.
  A large amount of the B_q^∗ mesons are expected to be
  accumulated with the running LHC and the forthcoming SuperKEKB,
  which makes it seemingly possible to explore the B_q^∗
  meson weak decays experimentally.
  The theoretical study is necessary to offer a ready reference.
  In this paper, we investigated the B_q^∗ →
  DP, DV decays with the phenomenological pQCD approach.
  It is found that the color-allowed B_q^∗ →
  D_qρ^- decays have branching ratios ≳ 10^-9,
  and should be promisingly accessible at the high luminosity experiments
  in the future.

  
  

§ ACKNOWLEDGMENTS

  The work is supported by the National Natural Science Foundation
  of China (Grant Nos. U1632109, 11547014 and 11475055).

  
  
  

§ THE AMPLITUDE FOR THE B_Q^∗ → DP DECAYS

  
  
   
    A(B_u^∗-→D_u^0π^-)  =  F  V_cb V_ud^∗ {∑_i  M^T_i,P
      +∑_j  M^C_j,P},

  
   
    A(B_u^∗-→D_u^0K^-)   =  F  V_cb V_us^∗ {∑_i  M^T_i,P
      +∑_j  M^C_j,P},

  
   
    A(B_d^∗0→D_d^+π^-)   =  F  V_cb V_ud^∗ {∑_i  M^T_i,P
      +∑_j  M^A_j,P},

  
   
    A(B_d^∗0→D_d^+K^-)   =  F  V_cb V_us^∗ ∑_i  M^T_i,P,

  
   
    √(2)  A(B_d^∗0→D_u^0π^0)   =  F  V_cb V_ud^∗ {
      -∑_i  M^C_i,P
      +∑_j  M^A_j,P},

  
   
    √(2)  A(B_d^∗0→D_u^0η_q)   =  F  V_cb V_ud^∗ {∑_i  M^C_i,P
      +∑_j  M^A_j,P},

  
   
    A(B_d^∗0→D_u^0K^0)   =  F  V_cb V_us^∗ ∑_i  M^C_i,P,

  
   
    A(B_d^∗0→D_s^+K^-)   =  F  V_cb V_ud^∗ ∑_i  M^A_i,P,

  
   
    A(B_s^∗0→D_s^+π^-)   =  F  V_cb V_ud^∗ ∑_i  M^T_i,P,

  
   
    A(B_s^∗0→D_s^+K^-)   =  F  V_cb V_us^∗ {∑_i  M^T_i,P
      +∑_j  M^A_j,P},

  
   
    A(B_s^∗0→D_d^+π^-)   =  F  V_cb V_us^∗ ∑_i  M^A_i,P,

  
   
    √(2)  A(B_s^∗0→D_u^0π^0)   =  F  V_cb V_us^∗ ∑_i  M^A_i,P,

  
   
    √(2)  A(B_s^∗0→D_u^0η_q)   =  F  V_cb V_us^∗ ∑_i  M^A_i,P,

  
   
    A(B_s^∗0→D_u^0η_s)   =  F  V_cb V_us^∗ ∑_i  M^C_i,P,

  
   
    A(B_s^∗0→D_u^0K^0)   =  F  V_cb V_ud^∗ ∑_i  M^C_i,P,

  
   
    F  = G_F/√(2) π C_F/N_c 
      f_B_q^∗  f_D,

  
  where M^k_i,j is the amplitude building blocks.
  The superscripts k = T, C, A correspond to the
  color-allowed emission topologies of Fig.<ref>,
  the color-suppressed emission topologies of Fig.<ref>,
  the annihilation topologies of Fig.<ref>.
  The subscripts i = a, b, c, d correspond to the
  diagram indices.
  The subscripts j = P, L, N, T correspond to
  the different helicity amplitudes.
  The analytical expressions of the amplitude building blocks
  M^k_i,j are given in the Appendix <ref>,
  <ref>, <ref>.

  
  

§ THE AMPLITUDE FOR THE B_Q^∗ → DV DECAYS

  
  
   
    i  A_λ(B_u^∗-→D_u^0ρ^-)  =  F  V_cb V_ud^∗ {∑_i  M^T_i,λ
      +∑_j  M^C_j,λ},

  
   
    i  A_λ(B_u^∗-→D_u^0K^∗-)  =  F  V_cb V_us^∗ {∑_i  M^T_i,λ
      +∑_j  M^C_j,λ},

  
   
    i  A_λ(B_d^∗0→D_d^+ρ^-)   =  F  V_cb V_ud^∗ {∑_i  M^T_i,λ
      +∑_j  M^A_j,λ},

  
   
    i A_λ(B_d^∗0→D_d^+K^∗-)   =  F  V_cb V_us^∗ ∑_i  M^T_i,λ,

  
   
    i √(2)  A_λ(B_d^∗0→D_u^0ρ^0)   =  F  V_cb V_ud^∗ {
      -∑_i  M^C_i,λ
      +∑_j  M^A_j,λ},

  
   
    i √(2)  A_λ(B_d^∗0→D_u^0ω)   =  F  V_cb V_ud^∗ {∑_i  M^C_i,λ
      +∑_j  M^A_j,λ},

  
   
    i A_λ(B_d^∗0→D_u^0K^∗0)   =  F  V_cb V_us^∗ ∑_i  M^C_i,λ,

  
   
    i A_λ(B_d^∗0→D_s^+K^∗-)   =  F  V_cb V_ud^∗ ∑_i  M^A_i,λ,

  
   
    i A_λ(B_s^∗0→D_s^+ρ^-)   =  F  V_cb V_ud^∗ ∑_i  M^T_i,λ,

  
   
    i A_λ(B_s^∗0→D_s^+K^∗-)   =  F  V_cb V_us^∗ {∑_i  M^T_i,λ
      +∑_j  M^A_j,λ},

  
   
    i A_λ(B_s^∗0→D_d^+ρ^-)   =  F  V_cb V_us^∗ ∑_i  M^A_i,λ,

  
   
    i √(2)  A_λ(B_s^∗0→D_u^0ρ^0)   =  F  V_cb V_us^∗ ∑_i  M^A_i,λ,

  
   
    i √(2)  A_λ(B_s^∗0→D_u^0ω)   =  F  V_cb V_us^∗ ∑_i  M^A_i,λ,

  
   
    i A_λ(B_s^∗0→D_u^0ϕ)   =  F  V_cb V_us^∗ ∑_i  M^C_i,λ,

  
   
    i A_λ(B_s^∗0→D_u^0K^∗0)   =  F  V_cb V_ud^∗ ∑_i  M^C_i,λ,

  
  where the index λ corresponds to three different
  helicity amplitudes, i.e., λ = L, N, T.


  
  

§ AMPLITUDE BUILDING BLOCKS FOR THE COLOR-ALLOWED
   B_Q^∗ → D_QM DECAYS

  
  The expressions of the amplitude building blocks M^T_i,j
  for the color-allowed topologies are presented as follows, where
  the subscript i corresponds to the diagram indices of Fig.<ref>;
  and j corresponds to the different helicity amplitudes.
  
  
   
    M^T_a,P   =   
      2 m_1 p ∫_0^1dx_1∫_0^1dx_2∫_0^∞b_1db_1∫_0^∞b_2db_2 
      H^T_f(α^T,β^T_a,b_1,b_2) 
      E^T_f(t^T_a)
       
       ×   α_s(t^T_a)  a_1(t^T_a) ϕ_B_q^∗^v(x_1) {ϕ_D^a(x_2) 
       ( m_1^2 x̅_2+m_3^2 x_2 )
       + ϕ_D^p(x_2)  m_2 m_b},

  
  
   
    M^T_a,L   =   ∫_0^1dx_1∫_0^1dx_2∫_0^∞b_1db_1∫_0^∞b_2db_2 
      H^T_f(α^T,β^T_a,b_1,b_2) 
      E^T_f(t^T_a) α_s(t^T_a)
       
       ×   
      a_1(t^T_a) ϕ_B_q^∗^v(x_1) {ϕ_D^a(x_2) 
       ( m_1^2 s x̅_2+m_3^2 t x_2 )
      + ϕ_D^p(x_2)  m_2 m_b u },

  
  
   
    M^T_a,N   =    m_1 m_3 ∫_0^1dx_1∫_0^1dx_2∫_0^∞b_1db_1∫_0^∞b_2db_2 
      H^T_f(α^T,β^T_a,b_1,b_2) 
      E^T_f(t^T_a)
       
       ×   α_s(t^T_a)  a_1(t^T_a) ϕ_B_q^∗^V(x_1) {ϕ_D^a(x_2)  (2 m_2^2 x_2 -t)
      -2  m_2 m_b ϕ_D^p(x_2) },

  
  
   
    M^T_a,T   =    2 m_1 m_3 ∫_0^1dx_1∫_0^1dx_2∫_0^∞b_1db_1∫_0^∞b_2db_2 
      H^T_f(α^T,β^T_a,b_1,b_2) 
      E^T_f(t^T_a)
       
       ×   α_s(t^T_a)  a_1(t^T_a) ϕ_B_q^∗^V(x_1) ϕ_D^a(x_2)
       ,

  
  
  
   
    M^T_b,P   =   
      2 m_1 p ∫_0^1dx_1∫_0^1dx_2∫_0^∞b_1db_1∫_0^∞b_2db_2 
      H^T_f(α^T,β^T_b,b_2,b_1) 
      E^T_f(t^T_b)
       
       ×   α_s(t^T_b) {ϕ_B_q^∗^v(x_1) [
      2 m_2 m_c ϕ_D^p(x_2)
     -ϕ_D^a(x_2)  (m_2^2 x̅_1+m_3^2 x_1) ]
       
       +   ϕ_B_q^∗^t(x_1) [
       2 m_1 m_2 ϕ_D^p(x_2) x̅_1
      -m_1  m_c ϕ_D^a(x_2) ] }  a_1(t^T_b)
       ,

  
  
   
    M^T_b,L   =   ∫_0^1dx_1∫_0^1dx_2∫_0^∞b_1db_1∫_0^∞b_2db_2 
      H^T_f(α^T,β^T_b,b_2,b_1) 
      E^T_f(t^T_b) α_s(t^T_b)
       
       ×   
       a_1(t^T_b) {ϕ_B_q^∗^t(x_1) [
      2 m_1 m_2 ϕ_D^p(x_2) (s-u x_1)
      -m_1 m_c s ϕ_D^a(x_2) ]
       
       +   ϕ_B_q^∗^v(x_1) [
      ϕ_D^a(x_2) ( m_3^2 t  x_1
      -m_2^2 u x̅_1 )
      +2 m_2 m_c u ϕ_D^p(x_2) ] },

  
  
   
    M^T_b,N   =    m_3 ∫_0^1dx_1∫_0^1dx_2∫_0^∞b_1db_1∫_0^∞b_2db_2 
      H^T_f(α^T,β^T_b,b_2,b_1) 
      E^T_f(t^T_b)
       
       ×   α_s(t^T_b) {ϕ_B_q^∗^V(x_1)  m_1 [
      ϕ_D^a(x_2) (2 m_2^2-t x_1)
      -4 m_2 m_c ϕ_D^p(x_2) ]
       
       +   ϕ_B_q^∗^T(x_1) [
      ϕ_D^a(x_2) t m_c
     +ϕ_D^p(x_2) 2 m_2 (2 m_1^2 x_1-t)
       ] }  a_1(t^T_b)
       ,

  
  
   
    M^T_b,T   =    2 m_3 ∫_0^1dx_1∫_0^1dx_2∫_0^∞b_1db_1∫_0^∞b_2db_2 
      H^T_f(α^T,β^T_b,b_2,b_1) 
      E^T_f(t^T_b) α_s(t^T_b)
       
       ×   
      a_1(t^T_b) {ϕ_B_q^∗^T(x_1) [
      ϕ_D^p(x_2) 2 m_2
     -ϕ_D^a(x_2) m_c]
     -ϕ_B_q^∗^V(x_1) ϕ_D^a(x_2) m_1 x_1},

  
  
  
   
    M^T_c,P   =   2 m_1 p/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 δ(b_1-b_2)
       
       ×   ϕ_P^a(x_3) α_s(t^T_c) 
      C_2(t^T_c) {ϕ_B_q^∗^v(x_1) ϕ_D^a(x_2) 
      (2 m_2^2 x_2+s x̅_3-t x_1)
       
       +   ϕ_B_q^∗^t(x_1) ϕ_D^p(x_2) 
       m_1 m_2 (x_1-x_2) } 
       H^T_n(α^T,β^T_c,b_3,b_2) 
       E_n(t^T_c)
       ,

  
  
   
    M^T_c,L   =   1/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^T_n(α^T,β^T_c,b_3,b_2)
       
       ×   δ(b_1-b_2) 
       E_n(t^T_c) ϕ_V^v(x_3) {ϕ_B_q^∗^v(x_1) ϕ_D^a(x_2)  u 
       (2 m_2^2 x_2+s x̅_3-t x_1)
       
       +   ϕ_B_q^∗^t(x_1) ϕ_D^p(x_2)  m_1 m_2 
       (u x_1-s x_2-2 m_3^2 x̅_3)
       } α_s(t^T_c) 
       C_2(t^T_c)
       ,

  
  
   
    M^T_c,N   =   m_3/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3
       ×   
      H^T_n(α^T,β^T_c,b_3,b_2) 
      E_n(t^T_c) α_s(t^T_c) 
      C_2(t^T_c) δ(b_1-b_2)
       
       ×   {ϕ_B_q^∗^V(x_1) ϕ_D^a(x_2) ϕ_V^V(x_3)  2 m_1 
      (t x_1-2 m_2^2 x_2-s x̅_3)
       
           +
      ϕ_B_q^∗^T(x_1) ϕ_D^p(x_2) ϕ_V^V(x_3)  m_2 
      (t x_2+u x̅_3-2 m_1^2 x_1)
       
           +
      ϕ_B_q^∗^T(x_1) ϕ_D^p(x_2) ϕ_V^A(x_3)  2 m_1 m_2 p 
      (x_2-x̅_3) },

  
  
   
    M^T_c,T   =   m_3/N_c p ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3
       ×   
      H^T_n(α^T,β^T_c,b_3,b_2) 
      E_n(t^T_c) α_s(t^T_c) 
      C_2(t^T_c) δ(b_1-b_2)
       
       ×   {ϕ_B_q^∗^V(x_1) ϕ_D^a(x_2) ϕ_V^A(x_3)  2  (2 m_2^2 x_2+s x̅_3-t x_1)
       
           +
      ϕ_B_q^∗^T(x_1) ϕ_D^p(x_2) ϕ_V^A(x_3)  r_2 
      (2 m_1^2 x_1-t x_2-u x̅_3)
       
           +
      ϕ_B_q^∗^T(x_1) ϕ_D^p(x_2) ϕ_V^V(x_3)  2 m_2 p  (x̅_3-x_2) },

  
  
  
   
    M^T_d,P   =   2 m_1 p/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 ϕ_P^a(x_3)
       
       ×   δ(b_1-b_2) α_s(t^T_d) 
      C_2(t^T_d)  E_n(t^T_d) {ϕ_B_q^∗^v(x_1) ϕ_D^a(x_2)  s (x_2-x_3)
      
       +   ϕ_B_q^∗^t(x_1) ϕ_D^p(x_2) 
       m_1 m_2 (x_1-x_2) } 
      H^T_n(α^T,β^T_d,b_3,b_2)
       ,

  
  
   
    M^T_d,L   =   1/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 δ(b_1-b_2)
       
       ×   
      E_n(t^T_d) α_s(t^T_d) 
      C_2(t^T_d) ϕ_V^v(x_3) {ϕ_B_q^∗^v(x_1) ϕ_D^a(x_2) 
       4 m_1^2 p^2 (x_2-x_3)
       
       +   ϕ_B_q^∗^t(x_1) ϕ_D^p(x_2) 
       m_1 m_2 ( u x_1-s x_2-2 m_3^2 x_3) } 
      H^T_n(α^T,β^T_d,b_3,b_2)
       ,

  
  
   
    M^T_d,N   =   m_2 m_3/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 δ(b_1-b_2)
       
       ×   ϕ_B_q^∗^T(x_1) ϕ_D^p(x_2) α_s(t^T_d) 
      C_2(t^T_d) {ϕ_V^V(x_3) 
      ( t x_2+u x_3-2 m_1^2 x_1)
       
       +   ϕ_V^A(x_3) 2 m_1 p (x_2-x_3)
       }  H^T_n(α^T,β^T_d,b_3,b_2) 
       E_n(t^T_d)
       ,

  
  
   
    M^T_d,T   =   m_2 m_3/N_c m_1 p ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 δ(b_1-b_2)
       
       ×   ϕ_B_q^∗^T(x_1) ϕ_D^p(x_2) α_s(t^T_d) 
      C_2(t^T_d) {ϕ_V^A(x_3) 
      ( 2 m_1^2 x_1-t x_2-u x_3)
       
       +   ϕ_V^V(x_3) 2 m_1 p (x_3-x_2)
       }  H^T_n(α^T,β^T_d,b_3,b_2) 
       E_n(t^T_d)
       ,

  
  
  where N_c = 3 is the color number.
  α_s is the strong coupling constant.
  C_1,2 are the Wilson coefficients.
  The parameter a_i is defined as
  
   
    a_1   =    C_1+1/N_c C_2, 
    
       a_2   =    C_2+1/N_c C_1.

  

  The functions H_f,n^T and the Sudakov factors E_f,n^T
  are defined as follows, where the subscripts f and n correspond
  to the factorizable and nonfactorizable topologies.
  
   
    H_f^T(α,β,b_i,b_j)  = 
       K_0(b_i√(-α)) {θ(b_i-b_j) K_0(b_i√(-β)) 
       I_0(b_j√(-β))
       + (b_i↔ b_j) },

  
   
    H_n^T(α,β,b_i,b_j)    =   {θ(b_i-b_j)  K_0(b_i√(-α)) 
       I_0(b_j√(-α)) + (b_i↔ b_j)
       }
          × {θ(-β)  K_0(b_i√(-β))
      +π/2 θ(+β) [ i J_0(b_i√(β))
       - Y_0(b_i√(β)) ] },

  
   
    E_f^T(t) = exp{ -S_B_q^∗(t)-S_D(t) },

  
   
    E_n(t) = exp{ -S_B_q^∗(t)-S_D(t)-S_M(t) },

  
   
    S_B_q^∗(t)  =  s(x_1,b_1,p_1^+)
      +2∫_1/b_1^tdμ/μγ_q,

  
   
    S_D(t)  =  s(x_2,b_2,p_2^+) + s(x̅_2,b_2,p_2^+)
      +2∫_1/b_2^tdμ/μγ_q,

  
   
    S_M(t)  =  s(x_3,b_3,p_3^+) + s(x̅_3,b_3,p_3^+)
      +2∫_1/b_3^tdμ/μγ_q,

  
  where I_0, J_0, K_0 and Y_0 are the Bessel
  functions; γ_q = -α_s/π is the
  quark anomalous dimension; the expression of s(x,b,Q)
  can be found in the appendix of Ref.<cit.>;
  α^T and β_i^T are the virtualities
  of the gluon and quark propagators;
  the subscripts of the quark virtuality β_i^T
  and the typical scale t_i^T correspond to the diagram
  indices of Fig.<ref>.
  
   
    α^T   =    x_1^2 m_1^2+x_2^2 m_2^2-x_1 x_2 t
       , 
    
      β_a^T   =    x_2^2 m_2^2-x_2 t+m_1^2-m_b^2, 
    
      β_b^T   =    x_1^2 m_1^2-x_1 t+m_2^2-m_c^2, 
    
      β_c^T   =   α^T+x̅_3^2 m_3^2
                         -x_1 x̅_3 u+x_2 x̅_3 s
       , 
    
      β_d^T   =   α^T+x_3^2 m_3^2
                         -x_1 x_3 u+x_2 x_3 s
       , 
    
       t_a(b)^T   =   max(√(-α^T),√(|β_a(b)^T|),1/b_1,1/b_2)
       , 
    
       t_c(d)^T   =   max(√(-α^T),√(|β_c(d)^T|),1/b_2,1/b_3)
       .

  

  

§ AMPLITUDE BUILDING BLOCKS FOR THE COLOR-SUPPRESSED
    B_Q^∗ → DM_Q DECAYS

  
  The expressions of the amplitude building blocks M^C_i,j
  for the color-suppressed topologies are displayed as follows, where
  the subscript i corresponds to the diagram indices of Fig.<ref>;
  and j corresponds to the different helicity amplitudes.
  
  
   
    M^C_a,P   =   
      -∫_0^1dx_1∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_3db_3 ϕ_B_q^∗^v(x_1) α_s(t^C_a)  a_2(t^C_a)
       
       ×   
       H^C_f(α^C,β^C_a,b_1,b_3) { 2 m_1 p ϕ_P^a(x_3) 
       (m_1^2 x̅_3+m_2^2 x_3)
       
       +   
       2 m_1 p μ_P m_b ϕ_P^p(x_3)
       + μ_P m_b  t ϕ_P^t(x_3) } 
       E^C_f(t^C_a)
       ,

  
  
   
    M^C_a,L   =   
     -∫_0^1dx_1∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_3db_3 
      H^C_f(α^C,β^C_a,b_1,b_3)
       
       ×   α_s(t^C_a)  a_2(t^C_a) ϕ_B_q^∗^v(x_1) {ϕ_V^v(x_3) 
      (m_1^2 s x̅_3+m_2^2 u x_3)
       
       +   
       m_3 m_b t ϕ_V^t(x_3)
       + 2 m_1 p  m_3 m_b ϕ_V^s(x_3)
       }   E^C_f(t^C_a)
       ,

  
  
   
    M^C_a,N   =   ∫_0^1dx_1∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_3db_3 
      H^C_f(α^C,β^C_a,b_1,b_3)
       
       ×   α_s(t^C_a)  a_2(t^C_a) ϕ_B_q^∗^V(x_1) {ϕ_V^V(x_3) m_1 m_3 (t-s x_3)
       
       +   
       m_1 m_b s ϕ_V^T(x_3)
       + 2 m_3 p  m_1^2 x̅_3 ϕ_V^A(x_3)
       }   E^C_f(t^C_a)
       ,

  
  
   
    M^C_a,T   =   
     -∫_0^1dx_1∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_3db_3 
      H^C_f(α^C,β^C_a,b_1,b_3)
       
       ×   α_s(t^C_a)  a_2(t^C_a) ϕ_B_q^∗^V(x_1) { (m_3/p) ϕ_V^A(x_3) (t-s x_3)
       
       +   ϕ_V^V(x_3) 2 m_1 m_3 x̅_3
      + ϕ_V^T(x_3) 2 m_1 m_b}   E^C_f(t^C_a)
       ,

  
  
  
   
    M^C_b,P   =   
      2 m_1 p ∫_0^1dx_1∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_3db_3 
      H^C_f(α^C,β^C_b,b_3,b_1) 
      E^C_f(t^C_b) α_s(t^C_b)
       
       ×   
      a_2(t^C_b) {ϕ_B_q^∗^v(x_1) ϕ_P^a(x_3)  (m_3^2 x̅_1+m_2^2 x_1)
      -ϕ_B_q^∗^t(x_1) ϕ_P^p(x_3) 
      2 m_1 μ_P x̅_1},

  
  
   
    M^C_b,L   =   ∫_0^1dx_1∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_3db_3 
      H^C_f(α^C,β^C_b,b_3,b_1) 
      E^C_f(t^C_b) α_s(t^C_b) 
      a_2(t^C_b)
       
       ×   {ϕ_B_q^∗^v(x_1) ϕ_V^v(x_3)  (m_2^2 u x_1-m_3^2 t x̅_1)
      -ϕ_B_q^∗^t(x_1) ϕ_V^s(x_3) 
      4 m_1^2 m_3 p x̅_1},

  
  
   
    M^C_b,N   =   
      m_1 m_3 ∫_0^1dx_1∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_3db_3 
      H^C_f(α^C,β^C_b,b_3,b_1) 
      E^C_f(t^C_b)
       
       ×   α_s(t^C_b)  a_2(t^C_b) ϕ_B_q^∗^V(x_1) {ϕ_V^V(x_3)  (s-t x_1)
     +ϕ_V^A(x_3)  2 m_1 p x̅_1},

  
  
   
    M^C_b,T   =   -m_3/p ∫_0^1dx_1∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_3db_3 
      H^C_f(α^C,β^C_b,b_3,b_1) 
      E^C_f(t^C_b)
       
       ×   α_s(t^C_b)  a_2(t^C_b) ϕ_B_q^∗^V(x_1) {ϕ_V^A(x_3)  (s-t x_1)
     +ϕ_V^V(x_3)  2 m_1 p x̅_1},

  
  
  
   
    M^C_c,P   =   ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^C_n(α^C,β^C_c,b_2,b_3)
       
       ×   δ(b_1-b_3) {ϕ_B_q^∗^t(x_1) ϕ_D^a(x_2) 
       m_1 μ_P [ ϕ_P^t(x_3) 
      (t x_1-2 m_2^2 x̅_2-s x_3)
      
       +   ϕ_P^p(x_3)  2 m_1 p (x_3-x_1)
      ] -
      ϕ_B_q^∗^v(x_1) ϕ_P^a(x_3) 
      2 m_1 p [ ϕ_D^p(x_2)  m_2 m_c
       +   ϕ_D^a(x_2)  (s x̅_2+2 m_3^2 x_3-u x_1)
       ] }  E_n(t^C_c) α_s(t^C_c) 
       C_1(t^C_c)/N_c,

  
  
   
    M^C_c,L   =   ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^C_n(α^C,β^C_c,b_2,b_3)
       
       ×   δ(b_1-b_3) {ϕ_B_q^∗^t(x_1) ϕ_D^a(x_2) 
       m_1  m_3 [ ϕ_V^t(x_3) 
      (t x_1-2 m_2^2 x̅_2-s x_3)
      
       +   ϕ_V^s(x_3)  2 m_1 p (x_3-x_1)
       ] + ϕ_B_q^∗^v(x_1) ϕ_V^v(x_3) [ -ϕ_D^p(x_2)  m_2 m_c u
       
       +   ϕ_D^a(x_2)  4 m_1^2 p^2 
      (x_1-x̅_2)
       ] }  E_n(t^C_c) α_s(t^C_c) 
      C_1(t^C_c)/N_c,

  
  
   
    M^C_c,N   =   1/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^C_n(α^C,β^C_c,b_2,b_3)
       
       ×   δ(b_1-b_3)  E_n(t^C_c) α_s(t^C_c)  {ϕ_B_q^∗^V(x_1) ϕ_D^p(x_2) ϕ_V^V(x_3) 2 m_1 m_2 m_3 m_c
       +   ϕ_B_q^∗^T(x_1) ϕ_D^a(x_2) ϕ_V^T(x_3) [
      m_1^2 s (x̅_2-x_1)
      +m_3^2 t (x_3-x̅_2) ] } 
      C_1(t^C_c)
       ,

  
  
   
    M^C_c,T   =   2/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^C_n(α^C,β^C_c,b_2,b_3)
       
       ×   
      C_1(t^C_c) {ϕ_B_q^∗^T(x_1) ϕ_D^a(x_2) ϕ_V^T(x_3) [ m_1^2 (x_1-x̅_2)
      + m_3^2 (x̅_2-x_3) ]
       
       -   ϕ_B_q^∗^V(x_1) ϕ_D^p(x_2) ϕ_V^A(x_3) m_2 m_3 m_c/p
       }  E_n(t^C_c) α_s(t^C_c) δ(b_1-b_3)
       ,

  
  
  
   
    M^C_d,P   =   1/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^C_n(α^C,β^C_d,b_2,b_3) 
      E_n(t^C_d)
       
       ×   δ(b_1-b_3) α_s(t^C_d) 
      C_1(t^C_d) ϕ_D^a(x_2) {ϕ_B_q^∗^v(x_1) ϕ_P^a(x_3) 
      2 m_1 p s (x_2-x_3)
       
       +   ϕ_B_q^∗^t(x_1) m_1 μ_P [ ϕ_P^p(x_3)  2 m_1 p  (x_3-x_1)
       + ϕ_P^t(x_3) (2 m_2^2 x_2+s x_3-t x_1)
       ] },

  
  
   
    M^C_d,L   =   1/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^C_n(α^C,β^C_d,b_2,b_3) 
      E_n(t^C_d)
       
       ×   δ(b_1-b_3) α_s(t^C_d) 
      C_1(t^C_d) ϕ_D^a(x_2) {ϕ_B_q^∗^v(x_1) ϕ_V^v(x_3) 
      4 m_1^2 p^2 (x_2-x_3)
       
       +   ϕ_B_q^∗^t(x_1) m_1 m_3 [ ϕ_V^s(x_3)  2 m_1 p  (x_3-x_1)
       + ϕ_V^t(x_3) (2 m_2^2 x_2+s x_3-t x_1)
       ] },

  
  
   
    M^C_d,N   =   1/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^C_n(α^C,β^C_d,b_2,b_3) 
      E_n(t^C_d)
       
          × δ(b_1-b_3) α_s(t^C_d) 
      C_1(t^C_d) ϕ_B_q^∗^T(x_1) ϕ_D^a(x_2) ϕ_V^T(x_3) { m_1^2 s (x_1-x_2)
       +m_3^2 t (x_2-x_3) },

  
  
   
    M^C_d,T   =   2/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞db_1∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^C_n(α^C,β^C_d,b_2,b_3) 
      E_n(t^C_d)
       
          × δ(b_1-b_3) α_s(t^C_d) 
      C_1(t^C_d) ϕ_B_q^∗^T(x_1) ϕ_D^a(x_2) ϕ_V^T(x_3) { m_1^2 (x_2-x_1)
       +m_3^2 (x_3-x_2) }.

  
  

  The functions H_f,n^C have the similar expressions for
  H_f,n^T, i.e.,
  
   
    H_f^C(α,β,b_i,b_j)  = 
       H_f^T(α,β,b_i,b_j)
       ,

  
   
    H_n^C(α,β,b_i,b_j)  = 
       H_n^T(α,β,b_i,b_j)
       .

  

  The Sudakov factor E_f^C are defined as
  
   
    E_f^C(t) = exp{ -S_B_q^∗(t)-S_M(t) },

  
  and the expressions for E_n(t), S_B_q^∗(t),
  S_D(t) and S_M(t) are the same as those given in
  the Appendix <ref>.
  α^C and β_i^C are the gluon and quark
  virtualities; the subscripts of β_i^C and t_i^C
  correspond to the diagram indices of Fig.<ref>.
  
   
    α^C   =    x_1^2 m_1^2+x_3^2 m_3^2-x_1 x_3 u
       , 
    
      β_a^C   =    x_3^2 m_3^2-x_3 u+m_1^2-m_b^2, 
    
      β_b^C   =    x_1^2 m_1^2-x_1 u+m_3^2, 
    
      β_c^C   =   α^C+x̅_2^2 m_2^2
           -x_1 x̅_2 t+x_3 x̅_2 s-m_c^2, 
    
      β_d^C   =   α^C+x_2^2 m_2^2
           -x_1 x_2 t+x_2 x_3 s
       , 
    
       t_a(b)^C   =   max(√(-α^C),√(|β_a(b)^C|),1/b_1,1/b_3)
       , 
    
       t_c(d)^C   =   max(√(-α^C),√(|β_c(d)^C|),1/b_2,1/b_3)
       .

  

  

§ AMPLITUDE BUILDING BLOCKS FOR THE ANNIHILATION
   B^∗0 → DM DECAYS

  
  The expressions of the amplitude building blocks M^A_i,j
  for the annihilation topologies are listed as follows, where the
  subscript i corresponds to the diagram indices of Fig.<ref>;
  and j corresponds to different helicity amplitudes.
  
  
   
    M^A_a,P   =   ∫_0^1dx_2∫_0^1dx_3∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^A_f(α^A,β^A_a,b_2,b_3) 
      E^A_f(t^A_a) α_s(t^A_a)
       
       ×   
       a_2(t^A_a) {ϕ_D^p(x_2) [
      ϕ_P^a(x_3) 4 m_1 m_2 m_c p
     +ϕ_P^p(x_3) 4 m_1 m_2 μ_P p x_3
       +   ϕ_P^t(x_3) 2 m_2 μ_P (t+u x̅_3)
       ] - ϕ_D^a(x_2) [
      ϕ_P^p(x_3)  2 m_1 m_c μ_P p
       
       +   ϕ_P^a(x_3) 2 m_1 p 
      (m_1^2 x̅_3+m_2^2 x_3)
     +ϕ_P^t(x_3) m_c μ_P t ] },

  
  
   
    M^A_a,L   =   ∫_0^1dx_2∫_0^1dx_3∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^A_f(α^A,β^A_a,b_2,b_3) 
      E^A_f(t^A_a) α_s(t^A_a)
       
       ×   
       a_2(t^A_a) {ϕ_D^p(x_2) [
      ϕ_V^v(x_3) 2 m_2 m_c u
     -ϕ_V^t(x_3) 2 m_2 m_3 (t+u x̅_3)
       
       -   ϕ_V^s(x_3) 4 m_1  m_2 m_3 p x_3] + ϕ_D^a(x_2) [
      ϕ_V^s(x_3) 2 m_1 m_3 m_c p
       
       -   ϕ_V^v(x_3) 
      (m_2^2 u x_3+m_1^2 s x̅_3)
     +ϕ_V^t(x_3) m_3 m_c t ] },

  
  
   
    M^A_a,N   =   ∫_0^1dx_2∫_0^1dx_3∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^A_f(α^A,β^A_a,b_2,b_3) 
      E^A_f(t^A_a)
       
       ×   {ϕ_D^a(x_2) [
      ϕ_V^V(x_3) m_1 m_3 (s x̅_3+2 m_2^2)
     -ϕ_V^T(x_3) m_1 m_c s
       
       +   ϕ_V^A(x_3) 2 m_1^2  m_3 p x̅_3] - ϕ_D^p(x_2) [
      ϕ_V^V(x_3) 4 m_1 m_2 m_3 m_c
       -   ϕ_V^T(x_3) 2 m_1 m_2 (s+2 m_3^2 x̅_3)
       ] } α_s(t^A_a)  a_2(t^A_a)
       ,

  
  
   
    M^A_a,T   =   ∫_0^1dx_2∫_0^1dx_3∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^A_f(α^A,β^A_a,b_2,b_3) 
      E^A_f(t^A_a)
       
       ×   {ϕ_D^p(x_2)  4 m_2[
      ϕ_V^T(x_3) m_1
     +ϕ_V^A(x_3) m_3 m_c/p ]
       
       -   ϕ_D^a(x_2) [
      ϕ_V^V(x_3) 2 m_1 m_3 x̅_3
     +ϕ_V^T(x_3) 2 m_1 m_c
       +   ϕ_V^A(x_3) (m_3/p) (s x̅_3+2 m_2^2)
       ] } α_s(t^A_a)   a_2(t^A_a)
       ,

  
  
  
   
    M^A_b,P   =   
      2 m_1 p ∫_0^1dx_2∫_0^1dx_3∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^A_f(α^A,β^A_b,b_3,b_2) 
      E^A_f(t^A_b) α_s(t^A_b)
       
       ×   
       a_2(t^A_b) {ϕ_D^p(x_2) ϕ_P^p(x_3) 
       2 m_2 μ_P x̅_2
       -ϕ_D^a(x_2) ϕ_P^a(x_3) 
        (m_1^2 x_2+m_3^2 x̅_2) },

  
  
   
    M^A_b,L   =   
      -∫_0^1dx_2∫_0^1dx_3∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^A_f(α^A,β^A_b,b_3,b_2) 
      E^A_f(t^A_b) α_s(t^A_b)  a_2(t^A_b)
       
       ×   {ϕ_D^p(x_2) ϕ_V^s(x_3) 
       4 m_1 m_2 m_3 p x̅_2
       +ϕ_D^a(x_2) ϕ_V^v(x_3) 
        (m_1^2 s x_2+m_3^2 t x̅_2) },

  
  
   
    M^A_b,N   =   
      m_1 m_3 ∫_0^1dx_2∫_0^1dx_3∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^A_f(α^A,β^A_b,b_3,b_2) 
      E^A_f(t^A_b)
       
       ×   α_s(t^A_b)  a_2(t^A_b) ϕ_D^a(x_2) {ϕ_V^V(x_3) 
      (s+2 m_2^2 x_2)- ϕ_V^A(x_3) 2 m_1 p },

  
  
   
    M^A_b,T   =   ∫_0^1dx_2∫_0^1dx_3∫_0^∞b_2db_2∫_0^∞b_3db_3 
      H^A_f(α^A,β^A_b,b_3,b_2) 
      E^A_f(t^A_b) α_s(t^A_b)
       
       ×   
      a_2(t^A_b) ϕ_D^a(x_2) {ϕ_V^V(x_3) 
      2 m_1 m_3- ϕ_V^A(x_3) (m_3/p) 
       (s+2 m_2^2 x_2) },

  
  
  
   
    M^A_c,P   =   ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_2db_2∫_0^∞db_3 
      H^A_n(α^A,β^A_c,b_1,b_2)
       
       ×   δ(b_2-b_3)  {ϕ_D^a(x_2) ϕ_P^a(x_3)  2 m_1 p [
      ϕ_B_q^∗^v(x_1) 
      (s x_2+2 m_3^2 x̅_3-u x̅_1)
       
       +   ϕ_B_q^∗^t(x_1) m_1 m_b] +
      ϕ_B_q^∗^v(x_1) ϕ_D^p(x_2) 
      m_2 μ_p [ ϕ_P^p(x_3) 
      2 m_1 p (x_2-x̅_3)
      
       +   ϕ_P^t(x_3) (2 m_1^2 x̅_1
      -t x_2-u x̅_3) ] }  E_n(t^A_c) α_s(t^A_c)  C_1(t^A_c)/N_c,

  
  
   
    M^A_c,L   =   ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_2db_2∫_0^∞db_3 
      H^A_n(α^A,β^A_c,b_1,b_2)
       
       ×   δ(b_2-b_3) {ϕ_B_q^∗^v(x_1) ϕ_D^p(x_2)  m_2 m_3[
      ϕ_V^t(x_3) 
      (t x_2+u x̅_3-2 m_1^2 x̅_1)
       
       +   ϕ_V^s(x_3)  2 m_1 p (x̅_3-x_2)
       ] + ϕ_D^a(x_2) ϕ_V^v(x_3) [ ϕ_B_q^∗^t(x_1)  m_1 m_b s
       
       +   ϕ_B_q^∗^v(x_1) 
       4 m_1^2 p^2 (x_2-x̅_1) ]
       }  E_n(t^A_c) α_s(t^A_c)  C_1(t^A_c)/N_c,

  
  
   
    M^A_c,N   =   ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_2db_2∫_0^∞db_3 
      H^A_n(α^A,β^A_c,b_1,b_2) 
      E_n(t^A_c)
       
       ×   δ(b_2-b_3) α_s(t^A_c) {ϕ_B_q^∗^V(x_1) ϕ_D^p(x_2) ϕ_V^T(x_3) 
      m_1 m_2 (u x̅_1-s x_2-2 m_3^2 x̅_3)
       
       +   ϕ_B_q^∗^T(x_1) ϕ_D^a(x_2) 
      m_3 m_b [ ϕ_V^A(x_3) 2 m_1 p
      -ϕ_V^V(x_3) t  ]
       }  C_1(t^A_c)/N_c,

  
  
   
    M^A_c,T   =   ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_2db_2∫_0^∞db_3 
      H^A_n(α^A,β^A_c,b_1,b_2)
       
       ×   δ(b_2-b_3)  
      E_n(t^A_c) α_s(t^A_c) {ϕ_B_q^∗^V(x_1) ϕ_D^p(x_2) ϕ_V^T(x_3) 
      2 m_1 m_2 (x̅_1-x_2)
       
       +   ϕ_B_q^∗^T(x_1) ϕ_D^a(x_2) 
      m_3 m_b [ ϕ_V^A(x_3) t/(m_1 p)
      -2 ϕ_V^V(x_3)  ]
       }  C_1(t^A_c)/N_c,

  
  
  
   
    M^A_d,P   =   1/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_2db_2∫_0^∞db_3 
      H^A_n(α^A,β^A_d,b_1,b_2)  E_n(t^A_d)
       
       ×   δ(b_2-b_3) α_s(t^A_d) 
      C_1(t^A_d) ϕ_B_q^∗^v(x_1) {ϕ_D^a(x_2) ϕ_P^a(x_3) 
      2 m_1 p ( 2 m_2^2 x_2+s x̅_3-t x_1)
       
       +   ϕ_D^p(x_2) 
      m_2 μ_P [ ϕ_P^t(x_3) 
      (2 m_1^2 x_1-t x_2-u x̅_3)
      +ϕ_P^p(x_3) 2 m_1 p (x̅_3-x_2)
       ] },

  
  
   
    M^A_d,L   =   1/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_2db_2∫_0^∞db_3 
      H^A_n(α^A,β^A_d,b_1,b_2)  E_n(t^A_d)
       
       ×   δ(b_2-b_3) α_s(t^A_d) 
      C_1(t^A_d) ϕ_B_q^∗^v(x_1) {ϕ_D^a(x_2) ϕ_V^v(x_3) 
      u ( 2 m_2^2 x_2+s x̅_3-t x_1)
       
       -   ϕ_D^p(x_2) 
      m_2 m_3 [ ϕ_V^t(x_3) 
      (2 m_1^2 x_1-t x_2-u x̅_3)
      +ϕ_V^s(x_3) 2 m_1 p (x̅_3-x_2)
       ] },

  
  
   
    M^A_d,N   =   1/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_2db_2∫_0^∞db_3 
      H^A_n(α^A,β^A_d,b_1,b_2)
       
       ×   
      E_n(t^A_d) α_s(t^A_d) ϕ_B_q^∗^V(x_1) {ϕ_D^a(x_2) ϕ_V^V(x_3) 
      2 m_1 m_3 ( t x_1-2 m_2^2 x_2-s x̅_3)
       
       +   ϕ_D^p(x_2) ϕ_V^T(x_3) 
      m_1 m_2 (u x_1-s x_2-2 m_3^2 x̅_3)
       ] }  C_1(t^A_d) δ(b_2-b_3)
       ,

  
  
   
    M^A_d,T   =   2/N_c ∫_0^1dx_1∫_0^1dx_2∫_0^1dx_3∫_0^∞b_1db_1∫_0^∞b_2db_2∫_0^∞db_3 
      H^A_n(α^A,β^A_d,b_1,b_2)
       
       ×   
      E_n(t^A_d) α_s(t^A_d) ϕ_B_q^∗^V(x_1) {ϕ_D^a(x_2) ϕ_V^A(x_3) 
      (m_3/p) ( 2 m_2^2 x_2+s x̅_3-t x_1)
       
       +   ϕ_D^p(x_2) ϕ_V^T(x_3) 
      m_1 m_2 (x_1-x_2) ] } 
      C_1(t^A_d) δ(b_2-b_3)
       .

  
  

  The functions H_f,n^A and the Sudakov factor E_f^A
  are defined as follows.
  
   
    H_f^A(α,β,b_i,b_j)    =   π^2/4 { i J_0(b_i√(α))
       - Y_0(b_i√(α)) }
          × {θ(b_i-b_j) [ i J_0(b_i√(β))
       - Y_0(b_i√(β)) ] J_0(b_j√(β))
      +  (b_i↔ b_j) },

  
   
    H_n^A(α,β,b_i,b_j)    =   {θ(-β)  K_0(b_i√(-β))
      + π/2θ(+β) [ i J_0(b_i√(β))
       - Y_0(b_i√(β)) ] }
          × π/2 {θ(b_i-b_j) [ i J_0(b_i√(α))
       - Y_0(b_i√(α)) ]
       J_0(b_j√(α))
       + (b_i↔ b_j) },

  
   
    E_f^A(t) = exp{ -S_D(t)-S_M(t) },

  
  and the expressions for E_n(t), S_B_q^∗(t),
  S_D(t) and S_M(t) are the same as those given in
  the Appendix <ref>.
  α^A and β_i^A are the gluon and quark
  virtualities; the subscripts of β_i^A and t_i^A
  correspond to the diagram indices of Fig.<ref>.
  
   
    α^A   =    x_2^2 m_2^2+x̅_3^2 m_3^2+x_2 x̅_3 s
       , 
    
      β_a^A   =   x̅_3^2 m_3^2+x̅_3 s+m_2^2-m_c^2, 
    
      β_b^A   =    x_2^2 m_2^2+x_2 s+m_3^2, 
    
      β_c^A   =   α^A+x̅_1^2 m_1^2
                          -x̅_1 x_2 t-x̅_1 x̅_3 u-m_b^2, 
    
      β_d^A   =   α^A+x_1^2 m_1^2
                          -x_1 x_2 t-x_1 x̅_3 u
       , 
    
       t_a(b)^A   =   max(√(α^A),√(|β_a(b)^A|),1/b_2,1/b_3)
       , 
    
       t_c(d)^A   =   max(√(α^A),√(|β_c(d)^A|),1/b_1,1/b_2)
       .

  
   

  
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