#  $B^{\ast}$ ${\to}$ $\overline{D}D$ decays with perturbative QCD approach

Junfeng Sun  Institute of Particle and Nuclear Physics, Henan Normal
University, Xinxiang 453007, China  Haiyan Li  Institute of Particle and
Nuclear Physics, Henan Normal University, Xinxiang 453007, China  Yueling Yang
Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang
453007, China  Na Wang  Institute of Particle and Nuclear Physics, Henan
Normal University, Xinxiang 453007, China  Qin Chang  Institute of Particle
and Nuclear Physics, Henan Normal University, Xinxiang 453007, China  Gongru
Lu  Institute of Particle and Nuclear Physics, Henan Normal University,
Xinxiang 453007, China

###### Abstract

The nonleptonic two-body $B^{\ast}$ ${\to}$ $\overline{D}D$ weak decays are
studied phenomenologically with the perturbative QCD factorization approach.
It is found that the $B_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{-}D_{s}^{+}$,
$B_{d}^{{\ast}0}$ ${\to}$ $D_{d}^{-}D_{s}^{+}$, and $B_{u}^{{\ast}+}$ ${\to}$
$\overline{D}_{u}^{0}D_{s}^{+}$ decays have branching ratios ${\gtrsim}$
$10^{-9}$, and might be promisingly measurable at the running LHC and
forthcoming SuperKEKB experiments in the future.

###### pacs:

12.15.Ji 12.39.St 13.25.Hw 14.40.Nd

##  I Introduction

The $B_{q}^{\ast}$ mesons, consisting of $\bar{b}q$ pair with $q$ $=$ $u$, $d$
and $s$, are spin-triplet ground vector states with definite spin-parity
quantum numbers of $J^{P}$ $=$ $1^{-}$ pdg . Because the mass splittings
$m_{B_{q}^{\ast}}$ $-$ $m_{B_{q}}$ ${\lesssim}$ 50 MeV pdg  are much smaller
than the mass of the lightest pion meson, the $B_{q}^{\ast}$ meson decays
dominantly into the ground pseudoscalar $B_{q}$ meson through the
electromagnetic interaction. Besides, the $B_{q}^{\ast}$ mesons can also decay
via the bottom-changing transition induced by the weak interaction within the
standard model (SM). Because of the strong phase-space suppression from their
dominant magnetic dipole (M1) transition $B_{q}^{\ast}$ ${\to}$
$B_{q}{\gamma}$, the lifetime of the $B_{q}^{\ast}$ meson is of the order of
$10^{-17}$ second or less, which, in general, is too short to enable the
$B_{q}^{\ast}$ meson to experience the weak disintegration 1509.05049 . The
$B_{q}^{\ast}$ weak decays have not actually attracted much attention yet.
Until now, there has been no experimental report and few theoretical works
concentrating on the $B_{q}^{\ast}$ weak decay, subject to the relatively
inadequate statistics on the $B_{q}^{\ast}$ mesons. Fortunately, the high
luminosities and large production rates at LHC and the forthcoming SuperKEKB
are promising, and the rapid accumulation of more and more $B_{q}^{\ast}$ data
samples is expected to be possible. Some $B_{q}^{\ast}$ weak decay modes might
be detected and investigated in the future, which undoubtedly makes the
$B_{q}^{\ast}$ mesons another a vibrant arena for testing the Cabibbo-
Kabayashi-Maskawa (CKM) picture for $CP$-violating phenomena, examining our
comprehension of the underlying dynamical factorization mechanism, and so on.
In addition, heavy quark symmetry relates hadronic transition matrix elements
(HTME) of the $B_{q}^{\ast}$ and $B_{q}$ weak decays. The interplay between
the $B_{q}^{\ast}$ and $B_{q}$ weak decays could prove useful information to
overconstraint parameters in the SM, and might shed some fresh light on
various anomalies in $B$ decays.

The purely leptonic decays $B_{q}^{\ast}$ ${\to}$ ${\ell}^{+}{\ell}^{-}$
induced by the flavor-changing neutral currents have been studied recently in
the SM 1509.05049 ; 1601.03386 . The semileptonic and nonleptonic
$B_{q}^{\ast}$ decays have been investigated also in the SM 1605.01630 ;
1605.01629 ; 1605.01631 ; 1606.09071 , where the transition form factors are
evaluated with the Wirbel-Stech-Bauer approach zpc29 , and the nonfactorizable
corrections to HTME are considered 1605.01629 ; 1605.01631  based on the
collinear-based and QCD-improved factorization (QCDF) approach qcdf1 ; qcdf2 ;
qcdf3 ; plb.509.263 ; prd64.014036 . In this paper, we will study the
nonleptonic $B_{q}^{\ast}$ decay into the pseudoscalar charmed-meson pair
$\overline{D}D$ with the perturbative QCD factorization (pQCD) approach pqcd1
; pqcd2 ; pqcd3 , just to provide a ready reference for the future
experimental research. In addition, as is well known, the production ratio for
the $B_{q}^{\ast}$ meson is comparable with that for the $B_{q}$ meson (see
Table 3), the $B_{q}^{\ast}$ and $B_{q}$ mesons have nearly equal mass. Hence,
the study of the $B_{q}^{\ast}$ ${\to}$ $\overline{D}D$ decays will
undoubtedly be helpful to the experimental background analysis on the $B_{q}$
${\to}$ $\overline{D}D$ decays.

This paper is organized as follows. The theoretical framework and amplitudes
for $B_{q}^{\ast}$ ${\to}$ $\overline{D}D$ decays with pQCD approach are given
in section II. Section III is devoted to numerical results and discussion. The
final section is a summary.

##  II theoretical framework

###  II.1 The effective Hamiltonian

The effective Hamiltonian describing the $B_{q}^{\ast}$ ${\to}$
$\overline{D}D$ weak decay is written as 9512380

${\cal H}_{\rm
eff}\,=\,\frac{G_{F}}{\sqrt{2}}\sum\limits_{q=d,s}\Big{\\{}\sum\limits_{p=u,c}V_{pb}^{\ast}V_{pq}\sum\limits_{i=1}^{2}C_{i}({\mu})\,Q_{i}({\mu})-V_{tb}^{\ast}V_{tq}\,\sum\limits_{k=3}^{10}C_{k}({\mu})\,Q_{k}({\mu})\Big{\\}}+{\rm
h.c.},$ (1)

where the Fermi coupling constant $G_{F}$ ${\simeq}$
$1.166{\times}10^{-5}\,{\rm GeV}^{-2}$ pdg ; $V_{pb}^{\ast}V_{pq}$ and
$V_{tb}^{\ast}V_{tq}$ are the CKM factors; The scale ${\mu}$ factorizes the
physical contributions into two parts: the Wilson coefficients $C_{i}$ and the
local four-fermion operators $Q_{i}$. The operators are defined as follows.

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle(\bar{b}_{\alpha}\,p_{\alpha})_{V-A}\,(\bar{p}_{\beta}\,q_{\beta})_{V-A},\qquad\qquad\quad\
\ Q_{2}\ =\
(\bar{b}_{\alpha}\,p_{\beta})_{V-A}\,(\bar{p}_{\beta}\,q_{\alpha})_{V-A},$ (2)
$\displaystyle Q_{3}$ $\displaystyle=$
$\displaystyle\sum\limits_{q^{\prime}}\,(\bar{b}_{\alpha}\,q_{\alpha})_{V-A}\,(\bar{q}^{\prime}_{\beta}\,q^{\prime}_{\beta})_{V-A},\qquad\qquad
Q_{4}\ =\
\sum\limits_{q^{\prime}}\,(\bar{b}_{\alpha}\,q_{\beta})_{V-A}\,(\bar{q}^{\prime}_{\beta}\,q^{\prime}_{\alpha})_{V-A},$
(3) $\displaystyle Q_{5}$ $\displaystyle=$
$\displaystyle\sum\limits_{q^{\prime}}\,(\bar{b}_{\alpha}\,q_{\alpha})_{V-A}\,(\bar{q}^{\prime}_{\beta}\,q^{\prime}_{\beta})_{V+A},\qquad\qquad
Q_{6}\ =\
\sum\limits_{q^{\prime}}\,(\bar{b}_{\alpha}\,q_{\beta})_{V-A}\,(\bar{q}^{\prime}_{\beta}\,q^{\prime}_{\alpha})_{V+A},$
(4) $\displaystyle Q_{7}$ $\displaystyle=$
$\displaystyle\sum\limits_{q^{\prime}}\,\frac{3}{2}\,Q_{q^{\prime}}\,(\bar{b}_{\alpha}\,q_{\alpha})_{V-A}\,(\bar{q}^{\prime}_{\beta}\,q^{\prime}_{\beta})_{V+A},\quad\
Q_{8}\ =\
\sum\limits_{q^{\prime}}\,\frac{3}{2}\,Q_{q^{\prime}}\,(\bar{b}_{\alpha}\,q_{\beta})_{V-A}\,(\bar{q}^{\prime}_{\beta}\,q^{\prime}_{\alpha})_{V+A},$
(5) $\displaystyle Q_{9}$ $\displaystyle=$
$\displaystyle\sum\limits_{q^{\prime}}\,\frac{3}{2}\,Q_{q^{\prime}}\,(\bar{b}_{\alpha}\,q_{\alpha})_{V-A}\,(\bar{q}^{\prime}_{\beta}\,q^{\prime}_{\beta})_{V-A},\quad
Q_{10}\ =\
\sum\limits_{q^{\prime}}\,\frac{3}{2}\,Q_{q^{\prime}}\,(\bar{b}_{\alpha}\,q_{\beta})_{V-A}\,(\bar{q}^{\prime}_{\beta}\,q^{\prime}_{\alpha})_{V-A},$
(6)

where $Q_{1,2}$ are tree operators arising from the $W$-boson exchange;
$Q_{3,{\cdots},6}$ and $Q_{7,{\cdots},10}$ are called the QCD and electroweak
penguin operators, respectively; $(\bar{q}_{1}\,q_{2})_{V{\pm}A}$ ${\equiv}$
$\bar{q}_{1}\,{\gamma}_{\mu}(1{\pm}{\gamma}_{5})\,q_{2}$; ${\alpha}$ and
${\beta}$ are color indices; $q^{\prime}$ denotes all the active quarks at the
scale of ${\cal O}(m_{b})$, i.e., $q^{\prime}$ $=$ $u$, $d$, $c$, $s$, $b$;
and $Q_{q^{\prime}}$ is the electric charge of quark $q^{\prime}$ in the unit
of ${|}e{|}$.

The Wilson coefficients $C_{i}(\mu)$, which summarize the physical
contributions above the scale of ${\mu}$, have been properly calculated at the
next-to-leading order with the renormalization group equation assisted
perturbation theory 9512380 . Due to the presence of long-distance QCD effects
and the entanglement of nonperturbative and perturbative shares, the main
obstacle to evaluate the $B_{q}^{(\ast)}$ weak decays is the treatment of
physical contributions below the scale of ${\mu}$ which are included in the
HTME of local operators.

###  II.2 Hadronic matrix elements

Some phenomenological models have recently been developed to improve the
sketchy treatment with naive factorization scheme npb133 ; zpc34 . These
models are generally based on the Lepage-Brodsky approach prd22  and some
power counting rules in parameters of ${\alpha}_{s}$ and ${\Lambda}_{\rm
QCD}/m_{Q}$ (where ${\alpha}_{s}$ is the strong coupling, ${\Lambda}_{\rm
QCD}$ is the QCD characteristic scale, and $m_{Q}$ is the mass of a heavy
quark), and express the HTME as a convolution integral of universal wave
functions and hard scattering subamplitudes, such as the QCDF approach qcdf1 ;
qcdf2 ; qcdf3 , pQCD approach pqcd1 ; pqcd2 ; pqcd3 , the soft and collinear
effective theory scet1 ; scet2 ; scet3 ; scet4 , and so on, which have been
extensively employed in the interpretation of the $B$ weak decays. To wipe out
the endpoint singularities appearing in the collinear approximation qcdf1 ;
qcdf2 ; qcdf3 , it is suggested by the pQCD approach pqcd1 ; pqcd2 ; pqcd3
that the transverse momentum $k_{T}$ of valence quarks should be retaken, and
a Sudakov factor should be introduced for each wave function to further
suppress the soft contributions and make the hard scattering more
perturbative. Finally, a decay amplitude is written as a multidimensional
integral of many parts pqcd2 ; pqcd3 , including the Wilson coefficients
$C_{i}$, the heavy quark decay subamplitudes ${\cal H}$, and the universal
wave functions ${\Phi}$,

${\cal A}\,{\sim}\,\sum_{i}{\int}{\prod_{j}}dk_{j}\,C_{i}(t)\,{\cal
H}_{i}(t,k_{j})\,{\Phi}_{j}(k_{j})\,e^{-S_{j}},$ (7)

where $t$ is a typical scale; $k_{j}$ is the momentum of a valence quark;
$e^{-S_{j}}$ is a Sudakov factor.

###  II.3 Kinematic variables

The light-cone variables in the rest frame of the $B^{\ast}$ meson are defined
as follows.

$p_{B^{\ast}}\,=\,p_{1}\,=\,\frac{m_{1}}{\sqrt{2}}(1,1,0),$ (8)
$p_{\overline{D}}\,=\,p_{2}\,=\,(p_{2}^{+},p_{2}^{-},0),$ (9)
$p_{D}\,=\,p_{3}\,=\,(p_{3}^{-},p_{3}^{+},0),$ (10)
$k_{i}\,=\,x_{i}\,p_{i}+(0,0,\vec{k}_{i{T}}),$ (11)
$p_{i}^{\pm}\,=\,(E_{i}\,{\pm}\,p)/\sqrt{2},$ (12)
${\epsilon}_{B^{\ast}}^{{\parallel}}\,=\,\frac{1}{\sqrt{2}}(-1,1,0),$ (13)
$s\,=\,2\,p_{2}{\cdot}p_{3},$ (14) $t\,=\,2\,p_{1}{\cdot}p_{2}\,=\
2\,m_{1}\,E_{2},$ (15) $u\,=\,2\,p_{1}{\cdot}p_{3}\,=\ 2\,m_{1}\,E_{3},$ (16)
$s\,t+s\,u-t\,u-4\,m_{1}^{2}\,p^{2}\ =\ 0,$ (17)

where the subscripts $i$ $=$ $1$, $2$, $3$ of variables (energy $E_{i}$,
momentum $p_{i}$ and mass $m_{i}$) correspond to $B^{\ast}$, $\overline{D}$
and $D$ mesons, respectively; $k_{i}$ is the momentum of the valence quark
with the longitudinal momentum fraction $x_{i}$ and the transverse momentum
$\vec{k}_{iT}$; ${\epsilon}_{B^{\ast}}^{{\parallel}}$ is the longitudinal
polarization vector; $p$ is the momentum of the final states; $s$, $t$ and $u$
are the Lorentz invariant parameters. The notation is displayed in Fig.2.

###  II.4 Wave functions

Wave functions are the basic input parameters with the pQCD approach. Although
wave functions contain soft and nonperturbative contributions, they are
universal, i.e., process independent. Wave functions and/or distribution
amplitudes (DAs) determined by nonperturbative methods or extracted from data,
can be employed here to make predictions.

Following the notations in Refs. npb529 ; prd65 ; plb752sun ; ijmpa31yang ,
HTME of the diquark operators is defined as

${\langle}0{|}\bar{b}_{i}(0)q_{j}(z){|}B^{\ast}(p,{\epsilon}^{\parallel}){\rangle}\,=\,\frac{f_{B^{\ast}}}{4}{\int}d^{4}k\,e^{-ik{\cdot}z}\Big{\\{}\\!\\!\not{\\!\epsilon}^{\parallel}\,\Big{[}m_{B^{\ast}}\,{\Phi}_{B^{\ast}}^{v}(k)-\\!\\!\not{p}\,{\Phi}_{B^{\ast}}^{t}(k)\Big{]}\Big{\\}}_{ji},$
(18)
${\langle}\overline{D}(p){|}c_{i}(0)\bar{q}_{j}(z){|}0{\rangle}\,=\,\frac{i\,f_{D}}{4}{\int}d^{4}k\,e^{+ik{\cdot}z}\,\Big{\\{}{\gamma}_{5}\,\Big{[}\\!\\!\not{p}\,{\Phi}_{\bar{D}}^{a}(k)+m_{\bar{D}}\,{\Phi}_{\bar{D}}^{p}(k)\Big{]}\Big{\\}}_{ji},$
(19)

where $f_{B^{\ast}}$ and $f_{D}$ are decay constants; the wave functions
${\Phi}_{B^{\ast}}^{v}$ and ${\Phi}_{\bar{D}}^{a}$ are twist-2; and
${\Phi}_{B^{\ast}}^{t}$ and ${\Phi}_{\bar{D}}^{p}$ are twist-3. Due to the
kinematic relation ${\epsilon}_{B^{\ast}}^{\perp}{\cdot}p_{i}$ $=$ $0$, the
transversely polarized $B^{\ast}$ meson contributes nothing to the amplitudes
for the $B^{\ast}$ ${\to}$ $\overline{D}D$ decays. The expressions for DAs of
the $B^{\ast}$ and $D$ mesons are plb752sun ; ijmpa31yang

${\phi}_{B^{\ast}}^{v}(x)=A\,x\,\bar{x}\,{\exp}\Big{\\{}-\frac{\bar{x}\,m_{q}^{2}+x\,m_{b}^{2}}{8\,{\omega}_{1}^{2}\,x\,\bar{x}}\Big{\\}},$
(20)
${\phi}_{B^{\ast}}^{t}(x)=B\,(\bar{x}-x)^{2}\,{\exp}\Big{\\{}-\frac{\bar{x}\,m_{q}^{2}+x\,m_{b}^{2}}{8\,{\omega}_{1}^{2}\,x\,\bar{x}}\Big{\\}},$
(21)
${\phi}_{\bar{D}}^{a}(x)=C\,x\,\bar{x}\,{\exp}\Big{\\{}-\frac{\bar{x}\,m_{q}^{2}+x\,m_{c}^{2}}{8\,{\omega}_{2}^{2}\,x\,\bar{x}}\Big{\\}},$
(22)
${\phi}_{\bar{D}}^{p}(x)=D\,{\exp}\Big{\\{}-\frac{\bar{x}\,m_{q}^{2}+x\,m_{c}^{2}}{8\,{\omega}_{2}^{2}\,x\,\bar{x}}\Big{\\}},$
(23)

where $x$ and $\bar{x}$ (${\equiv}$ $1$ $-$ $x$) are the parton momentum
fractions; ${\omega}_{i}$ determines the average transverse momentum of
partons and ${\omega}_{i}$ ${\simeq}$ $m_{i}\,{\alpha}_{s}(m_{i})$; parameters
$A$, $B$, $C$, $D$ are the normalization coefficients satisfying the
conditions

${\int}_{0}^{1}dx\,{\phi}_{B^{\ast}}^{v,t}(x)=1,$ (24)
${\int}_{0}^{1}dx\,{\phi}_{\bar{D}}^{a,p}(x)=1.$ (25)

In fact, there are many phenomenological models of DAs for the charmed meson,
for example, some of them have been listed by Eq.(30) in Ref.prd78.014018 .
One of the favorable models from the experimental data within the pQCD
framework has the expression prd78.014018

${\phi}_{D}(x)=6\,x\bar{x}\,\big{\\{}1+C_{D}(1-2x)\big{\\}},$ (26)

where parameter $C_{D}$ $=$ $0.5$ for the $D_{u,d}$ meson, and $C_{D}$ $=$
$0.4$ for the $D_{s}$ meson. In the actual calculation prd78.014018 ;
prd81.034006 ; jpg31.273 ; jpg37 , there is no distinction between twist-2 and
twist-3 DAs.

The shape lines of the normalized DAs ${\phi}_{B^{\ast}}^{v,t}(x)$ and
${\phi}_{D}^{a,p}(x)$ are illustrated in Fig.1. It is clearly seen from Fig.1
that (1) the shape lines of DAs in Eqs.(20)-(23) have a broad peak in the
small $x$ regions, which is generally consistent with an ansatz in which a
light quark carries fewer parton momentum fractions than a heavy quark in a
heavy-light system. (2) Due to the suppression from exponential functions, the
DAs of Eqs.(20)-(23) converge quickly to zero at endpoint $x$, $\bar{x}$
${\to}$ $0$, which supplies the soft contributions with an effective cutoff.
(3) The flavor symmetry breaking effects, and especially the distinction
between different the twist DAs, are highlighted, compared with the nearly
symmetric distribution Eq.(26).

Figure 1: The distributions of DAs ${\phi}_{B^{\ast}}^{v,t}(x)$ [Eqs.(20,21)],
${\phi}_{D}^{a,p}(x)$ [Eqs.(22,23)], and ${\phi}_{D}(x)$ [Eq.(26)]. Figure 2:
Feynman diagrams for $B^{{\ast}0}$ ${\to}$ $D^{+}D^{-}$ decay, where (a,b) are
the factorizable emission topologies, (c,d) are the nonfactorizable emission
topologies, (e,f,i,j) are the factorizable annihilation topologies, and
(g,h,k,l) are the nonfactorizable annihilation topologies.

###  II.5 Decay amplitudes

The Feynman diagrams for the $B^{{\ast}0}$ ${\to}$ $D^{+}D^{-}$ decay with the
pQCD approach are shown in Fig.2, including the factorizable emission
topologies (a,b) where one gluon links with the initial and the recoiled
states, the nonfactorizable emission topologies (c,d) where one gluon is
exchanged between the spectator quark and the emitted states, the factorizable
annihilation topologies (e,f,i,j) where one gluon is conjoined with the final
states, and the nonfactorizable annihilation topologies (g,h,k,l) where one
gluon is transmitted between the initial and the final states.

Generally, the amplitudes for the factorizable emission topologies in
Fig.2(a,b) can be written as the $D$ meson decay constant and the space-like
$B^{\ast}$ ${\to}$ $D$ transition form factor, and the amplitudes for the
factorizable annihilation topologies in Fig.2(e,f,i,j) can be written as the
$B^{\ast}$ meson decay constant and the time-like transition form factor
between two charmed mesons. The amplitudes for the nonfactorizable topologies
have more complicated structures, and can be written as the convolution
integral of all participating meson wave functions. Compared with the
contributions of the emission topologies in Fig.2(a-d), the contributions of
the annihilation topologies in Fig.2(e-l) are assumed to be power suppressed,
as stated in Ref.qcdf2 . In addition, different topologies have different
scales. The gluons of the emission topologies in Fig.2(a-d) are time-like,
while the gluons of the annihilation topologies in Fig.2(e-l) are space-like.
The gluon virtuality of creating a pair of heavy charm quarks from the vacuum
for the annihilation topologies in Fig.2(i-l), $k_{g}^{2}$ ${\geq}$
$(2m_{c})^{2}$, should be much larger than that of producing a pair of light
quarks for the annihilation topologies in Fig.2(e-h). Thus, it is not hard to
figure out that the contributions of the annihilation topologies in Fig.2(i-l)
might be very small relative to the others, because of the nature of the
asymptotic freedom of the QCD at the unltrahigh energy.

After a straightforward calculation, the amplitudes for the $B_{q}^{\ast}$
${\to}$ $\overline{D}D$ decays are expressed as below.

$\displaystyle{\cal A}(B^{{\ast}+}{\to}\overline{D}^{0}D_{q}^{+})$ (27)
$\displaystyle=$ $\displaystyle{\cal
F}\,\Big{\\{}V_{cb}^{\ast}\,V_{cq}\,\Big{[}a_{1}\,{\cal
A}_{a+b}^{LL}+C_{2}\,{\cal
A}_{c+d}^{LL}\Big{]}+V_{ub}^{\ast}\,V_{uq}\,\Big{[}a_{1}\,{\cal
A}_{i+j}^{LL}+C_{2}\,{\cal A}_{k+l}^{LL}\Big{]}$ $\displaystyle\quad-
V_{tb}^{\ast}\,V_{tq}\,\Big{[}(a_{4}+a_{10})\,{\cal
A}_{a+b+i+j}^{LL}+(a_{6}+a_{8})\,{\cal A}_{a+b}^{SP}$
$\displaystyle\qquad\qquad+\ (C_{3}+C_{9})\,{\cal
A}_{c+d+k+l}^{LL}+(C_{5}+C_{7})\,{\cal A}_{c+d+k+l}^{SP}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}(B_{q}^{{\ast}0}{\to}\overline{D}^{0}D^{0})$ (28)
$\displaystyle=$ $\displaystyle{\cal
F}\,\Big{\\{}V_{cb}^{\ast}\,V_{cq}\,\Big{[}a_{2}\,{\cal
A}_{e+f}^{LL}+C_{1}\,{\cal
A}_{g+h}^{LL}\Big{]}+V_{ub}^{\ast}\,V_{uq}\,\Big{[}a_{2}\,{\cal
A}_{i+j}^{LL}+C_{1}\,{\cal A}_{k+l}^{LL}\Big{]}$ $\displaystyle\quad-
V_{tb}^{\ast}\,V_{tq}\,\Big{[}(a_{3}+a_{9})\,{\cal
A}_{e+f+i+j}^{LL}+(a_{5}+a_{7})\,{\cal A}_{e+f+i+j}^{LR}$
$\displaystyle\qquad\ \ \quad+\ (C_{4}+C_{10})\,{\cal
A}_{g+h+k+l}^{LL}+(C_{6}+C_{8})\,{\cal A}_{g+h+k+l}^{LR}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}(B_{q}^{{\ast}0}{\to}D_{q}^{-}D_{q}^{+})$ (29)
$\displaystyle=$ $\displaystyle{\cal
F}\,\Big{\\{}V_{cb}^{\ast}\,V_{cq}\,\Big{[}a_{1}\,{\cal
A}_{a+b}^{LL}+C_{2}\,{\cal A}_{c+d}^{LL}+a_{2}\,{\cal
A}_{e+f}^{LL}+C_{1}\,{\cal A}_{g+h}^{LL}\Big{]}$ $\displaystyle\quad-
V_{tb}^{\ast}\,V_{tq}\,\Big{[}(a_{4}+a_{10})\,{\cal
A}_{a+b}^{LL}+(a_{6}+a_{8})\,{\cal A}_{a+b}^{SP}+(a_{3}+a_{9})\,{\cal
A}_{e+f}^{LL}$ $\displaystyle\quad+\ (C_{3}+C_{9})\,{\cal
A}_{c+d}^{LL}+(C_{5}+C_{7})\,{\cal A}_{c+d}^{SP}+(C_{4}+C_{10})\,{\cal
A}_{g+h}^{LL}$ $\displaystyle\quad+\ (a_{5}-\frac{1}{2}\,a_{7})\,{\cal
A}_{i+j}^{LR}+(C_{5}-\frac{1}{2}\,C_{7})\,{\cal
A}_{k+l}^{SP}+(C_{6}-\frac{1}{2}\,C_{8})\,{\cal A}_{k+l}^{LR}$
$\displaystyle\quad+\ (a_{5}+a_{7})\,{\cal
A}_{e+f}^{LR}+(a_{3}+a_{4}-\frac{1}{2}\,a_{9}-\frac{1}{2}\,a_{10})\,{\cal
A}_{i+j}^{LL}$ $\displaystyle\quad+\ (C_{6}+C_{8})\,{\cal
A}_{g+h}^{LR}+(C_{3}+C_{4}-\frac{1}{2}\,C_{9}-\frac{1}{2}\,C_{10})\,{\cal
A}_{k+l}^{LL}\Big{]}\Big{\\}},$ $\displaystyle{\cal
A}(B_{d}^{{\ast}0}{\to}D^{-}D_{s}^{+})$ (30) $\displaystyle=$
$\displaystyle{\cal F}\,\Big{\\{}V_{cb}^{\ast}\,V_{cs}\,\Big{[}a_{1}\,{\cal
A}_{a+b}^{LL}+C_{2}\,{\cal
A}_{c+d}^{LL}\Big{]}-V_{tb}^{\ast}\,V_{ts}\,\Big{[}(a_{4}+a_{10})\,{\cal
A}_{a+b}^{LL}$ $\displaystyle\quad+\ (a_{4}-\frac{1}{2}\,a_{10})\,{\cal
A}_{i+j}^{LL}+(C_{3}-\frac{1}{2}\,C_{9})\,{\cal
A}_{k+l}^{LL}+(C_{5}-\frac{1}{2}\,C_{7})\,{\cal A}_{k+l}^{SP}$
$\displaystyle\quad+\ (a_{6}+a_{8})\,{\cal A}_{a+b}^{SP}+(C_{3}+C_{9})\,{\cal
A}_{c+d}^{LL}+(C_{5}+C_{7})\,{\cal A}_{c+d}^{SP}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}(B_{s}^{{\ast}0}{\to}D_{s}^{-}D^{+})$ (31)
$\displaystyle=$ $\displaystyle{\cal
F}\,\Big{\\{}V_{cb}^{\ast}\,V_{cd}\,\Big{[}a_{1}\,{\cal
A}_{a+b}^{LL}+C_{2}\,{\cal
A}_{c+d}^{LL}\Big{]}-V_{tb}^{\ast}\,V_{td}\,\Big{[}(a_{4}+a_{10})\,{\cal
A}_{a+b}^{LL}$ $\displaystyle\quad+\ (a_{4}-\frac{1}{2}\,a_{10})\,{\cal
A}_{i+j}^{LL}+(C_{3}-\frac{1}{2}\,C_{9})\,{\cal
A}_{k+l}^{LL}+(C_{5}-\frac{1}{2}\,C_{7})\,{\cal A}_{k+l}^{SP}$
$\displaystyle\quad+\ (a_{6}+a_{8})\,{\cal A}_{a+b}^{SP}+(C_{3}+C_{9})\,{\cal
A}_{c+d}^{LL}+(C_{5}+C_{7})\,{\cal A}_{c+d}^{SP}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}(B_{d}^{{\ast}0}{\to}D_{s}^{-}D_{s}^{+})$ (32)
$\displaystyle=$ $\displaystyle{\cal
F}\,\Big{\\{}V_{cb}^{\ast}\,V_{cd}\,\Big{[}a_{2}\,{\cal
A}_{e+f}^{LL}+C_{1}\,{\cal
A}_{g+h}^{LL}\Big{]}-V_{tb}^{\ast}\,V_{td}\,\Big{[}(a_{3}+a_{9})\,{\cal
A}_{e+f}^{LL}$ $\displaystyle\quad+\ (a_{5}+a_{7})\,{\cal
A}_{e+f}^{LR}+(C_{4}+C_{10})\,{\cal A}_{g+h}^{LL}+(C_{6}+C_{8})\,{\cal
A}_{g+h}^{LR}$ $\displaystyle\quad+\ (a_{3}-\frac{1}{2}\,a_{9})\,{\cal
A}_{i+j}^{LL}+(C_{4}-\frac{1}{2}\,C_{10})\,{\cal A}_{k+l}^{LL}$
$\displaystyle\quad+\ (a_{5}-\frac{1}{2}\,a_{7})\,{\cal
A}_{i+j}^{LR}+(C_{6}-\frac{1}{2}\,C_{8})\,{\cal
A}_{k+l}^{LR}\Big{]}\Big{\\}},$ $\displaystyle{\cal
A}(B_{s}^{{\ast}0}{\to}D^{-}D^{+})$ (33) $\displaystyle=$ $\displaystyle{\cal
F}\,\Big{\\{}V_{cb}^{\ast}\,V_{cs}\,\Big{[}a_{2}\,{\cal
A}_{e+f}^{LL}+C_{1}\,{\cal
A}_{g+h}^{LL}\Big{]}-V_{tb}^{\ast}\,V_{ts}\,\Big{[}(a_{3}+a_{9})\,{\cal
A}_{e+f}^{LL}$ $\displaystyle\quad+\ (a_{5}+a_{7})\,{\cal
A}_{e+f}^{LR}+(C_{4}+C_{10})\,{\cal A}_{g+h}^{LL}+(C_{6}+C_{8})\,{\cal
A}_{g+h}^{LR}$ $\displaystyle\quad+\ (a_{3}-\frac{1}{2}\,a_{9})\,{\cal
A}_{i+j}^{LL}+(C_{4}-\frac{1}{2}\,C_{10})\,{\cal A}_{k+l}^{LL}$
$\displaystyle\quad+\ (a_{5}-\frac{1}{2}\,a_{7})\,{\cal
A}_{i+j}^{LR}+(C_{6}-\frac{1}{2}\,C_{8})\,{\cal
A}_{k+l}^{LR}\Big{]}\Big{\\}},$ ${\cal
F}\,=\,\sqrt{2}\,G_{F}\,\frac{{\pi}\,C_{F}}{N_{c}}\,f_{B_{q}^{\ast}}\,f_{\bar{D}}\,f_{D}\,m_{1}\,({\epsilon}_{B_{q}^{\ast}}{\cdot}p_{\bar{D}}),$
(34)

where $C_{i}$ is the Wilson coefficient; the parameter $a_{i}$ is defined as

$a_{i}=\left\\{\begin{array}[]{l}C_{i}+C_{i+1}/N_{c}\quad\text{for odd
$i$};\\\ C_{i}+C_{i-1}/N_{c}\quad\text{for even $i$},\end{array}\right.$ (35)

and ${\cal A}_{m_{1}+m_{2}+{\cdots}}^{n}$ is an abbreviation for ${\cal
A}_{m_{1}}^{n}$ $+$ ${\cal A}_{m_{2}}^{n}$ $+$ ${\cdots}$, where the subscript
$m_{i}$ corresponds to one of indices of Fig.2; the superscript $n$ refers to
three possible Dirac structures ${\Gamma}_{1}{\otimes}{\Gamma}_{2}$ of the
operators
$(\bar{q}_{1}q_{2})_{{\Gamma}_{1}}(\bar{q}_{3}q_{4})_{{\Gamma}_{2}}$, namely
$n$ $=$ $LL$ for $(V-A){\otimes}(V-A)$, $n$ $=$ $LR$ for
$(V-A){\otimes}(V+A)$, and $n$ $=$ $SP$ for $-2(S-P){\otimes}(S+P)$. The
explicit expressions of the building blocks ${\cal A}_{m_{i}}^{n}$ are
collected in Appendix.

##  III Numerical results and discussion

In the rest frame of the $B_{q}^{\ast}$ meson, the branching ratio is defined
as

${\cal B}r\ =\
\frac{1}{24{\pi}}\,\frac{p}{m_{B^{\ast}}^{2}{\Gamma}_{B_{q}^{\ast}}}\,{|}{\cal
A}(B^{\ast}{\to}\overline{D}D){|}^{2},$ (36)

where ${\Gamma}_{B_{q}^{\ast}}$ is the full decay width of the $B_{q}^{\ast}$
meson.

Table 1: The numerical values of the input parameters. CKM parameter111The
relations between the CKM parameters (${\rho}$, ${\eta}$) and ($\bar{\rho}$,
$\bar{\eta}$) are pdg : $({\rho}+i{\eta})$ $=$
$\displaystyle\frac{\sqrt{1-A^{2}{\lambda}^{4}}(\bar{\rho}+i\bar{\eta})}{\sqrt{1-{\lambda}^{2}}[1-A^{2}{\lambda}^{4}(\bar{\rho}+i\bar{\eta})]}$.
pdg
---
${\lambda}$ $=$ $0.22506{\pm}0.00050$, $A$ $=$ $0.811{\pm}0.026$, $\bar{\rho}$
$=$ $0.124^{+0.019}_{-0.018}$, $\bar{\eta}$ $=$ $0.356{\pm}0.011$;
mass and decay constant
$m_{B_{s}^{\ast}}$ $=$ $5415.4^{+1.8}_{-1.5}$ MeV pdg , |  $f_{B_{s}^{\ast}}$ $=$ $213{\pm}7$ MeV prd91.114509 , |  ${\Lambda}^{(5)}_{\rm QCD}$ $=$ $210{\pm}14$ MeV pdg ,
$m_{B_{u,d}^{\ast}}$ $=$ $5324.65{\pm}0.25$ MeV pdg , |  $f_{B_{u,d}^{\ast}}$ $=$ $175{\pm}6$ MeV prd91.114509 , |  ${\Lambda}^{(4)}_{\rm QCD}$ $=$ $292{\pm}16$ MeV pdg ,
$m_{D_{s}}$ $=$ $1968.27{\pm}0.10$ MeV pdg , |  $f_{D_{s}}$ $=$ $249.0{\pm}1.2$ MeV pdg , |  $m_{b}$ $=$ $4.78{\pm}0.06$ GeV pdg ,
$m_{D_{d}}$ $=$ $1869.58{\pm}0.09$ MeV pdg , |  $f_{D_{u,d}}$ $=$ $211.9{\pm}1.1$ MeV pdg , |  $m_{c}$ $=$ $1.67{\pm}0.07$ GeV pdg ,
$m_{D_{u}}$ $=$ $1864.83{\pm}0.05$ MeV pdg , |  $m_{u,d}$ ${\simeq}$ $0.31$ GeV uds , |  $m_{s}$ ${\simeq}$ $0.51$ GeV uds .

The numerical values of some input parameters are listed in Table 1, where if
it is not specified explicitly, their central values will be fixed as the
default inputs. Besides, the full decay width of the $B_{q}^{\ast}$ meson,
${\Gamma}_{B_{q}^{\ast}}$, is also an essential parameter. Unfortunately, an
experimental measurement on ${\Gamma}_{B_{q}^{\ast}}$ is unavailable now,
because the soft photon from the $B_{q}^{\ast}$ ${\to}$ $B_{q}{\gamma}$
process is usually beyond the detection capability of electromagnetic
calorimeters sitting at existing high energy colliders. It is well known that
the electromagnetic radiation process $B_{q}^{\ast}$ ${\to}$ $B_{q}{\gamma}$
dominates the decay of the $B_{q}^{\ast}$ meson. So, for the time being, the
full decay width will be approximated by the radiative partial width, i.e.,
${\Gamma}_{B_{q}^{\ast}}$ ${\simeq}$
${\Gamma}(B_{q}^{\ast}{\to}B_{q}{\gamma})$. At present, the information on
${\Gamma}(B_{q}^{\ast}{\to}B_{q}{\gamma})$ comes mainly from theoretical
estimation. Theoretically, the partial decay width of the M1 transition (spin-
flip) process has the expression jhep1404.177 ; epja52.90

${\Gamma}(B_{q}^{\ast}{\to}B_{q}{\gamma})\ =\
\frac{4}{3}\,{\alpha}\,k_{\gamma}^{3}\,{\mu}^{2}_{h},$ (37)

where ${\alpha}$ is the fine structure constant; $k_{\gamma}$ $=$
$(m_{B_{q}^{\ast}}^{2}-m_{B_{q}}^{2})/2m_{B_{q}^{\ast}}$ is the photon
momentum in the rest frame of the $B_{q}^{\ast}$ meson; ${\mu}_{h}$ is the M1
moment of the $B_{q}^{\ast}$ meson. There are plenty of theoretical
predictions on ${\Gamma}(B_{q}^{\ast}{\to}B_{q}{\gamma})$, for example, the
numbers in Table 7 in Ref.jhep1404.177  and Tables 3 and 4 in Ref.epja52.90 ,
but these estimation suffer from large uncertainties due to our insufficient
understanding on the M1 moments of mesons. In principle, the M1 moment of a
meson should be a combination of the M1 moments of the constituent quark and
antiquark. For a heavy-light meson, the M1 moment of a heavy quark might be
negligible relative to the M1 moment of a light quark, because it is widely
assumed that the mass of a heavy quark is usually much larger than the mass of
a light quark, and that the M1 moment is inversely proportional to the mass of
a charged particle. With the M1 moment relations among light $u$, $d$, $s$
quarks, ${|}{\mu}_{u}{|}$ $>$ ${|}{\mu}_{d}{|}$ $>$ ${|}{\mu}_{s}{|}$ uds ,
one could expect to have ${\Gamma}(B_{u}^{\ast}{\to}B_{u}{\gamma})$ $>$
${\Gamma}(B_{d}^{\ast}{\to}B_{d}{\gamma})$ $>$
${\Gamma}(B_{s}^{\ast}{\to}B_{s}{\gamma})$, and so ${\Gamma}_{B_{u}^{\ast}}$
$>$ ${\Gamma}_{B_{d}^{\ast}}$ $>$ ${\Gamma}_{B_{s}^{\ast}}$. Of course, more
details about the width ${\Gamma}_{B_{q}^{\ast}}$ is beyond the scope of this
paper. In our calculation, in order to give a quantitative estimation of the
branching ratios for the $B^{\ast}$ ${\to}$ $\overline{D}D$ decays, we will
fix

${\Gamma}_{B_{u}^{\ast}}\ {\sim}\ {\Gamma}(B_{u}^{\ast}{\to}B_{u}{\gamma})\
{\sim}\ 450\,\text{eV},$ (38) ${\Gamma}_{B_{d}^{\ast}}\ {\sim}\
{\Gamma}(B_{d}^{\ast}{\to}B_{d}{\gamma})\ {\sim}\ 150\,\text{eV},$ (39)
${\Gamma}_{B_{s}^{\ast}}\ {\sim}\ {\Gamma}(B_{s}^{\ast}{\to}B_{s}{\gamma})\
{\sim}\ 100\,\text{eV},$ (40)

which is basically consistent with the recent results in Refs.jhep1404.177 ;
epja52.90 .

In order to investigate the effects from different DA models, we explore three
scenarios,

  * •

Scenario I: ${\phi}_{B^{\ast}}^{v}$ $=$ Eq.(20), ${\phi}_{B^{\ast}}^{t}$ $=$
Eq.(21), ${\phi}_{\bar{D},D}^{a}$ $=$ Eq.(22) and ${\phi}_{\bar{D},D}^{p}$ $=$
Eq.(23).

  * •

Scenario II: ${\phi}_{B^{\ast}}^{v}$ $=$ ${\phi}_{B^{\ast}}^{t}$ $=$ Eq.(20),
and ${\phi}_{\bar{D},D}^{a}$ $=$ ${\phi}_{\bar{D},D}^{p}$ $=$ Eq.(22).

  * •

Scenario III: ${\phi}_{B^{\ast}}^{v}$ $=$ ${\phi}_{B^{\ast}}^{t}$ $=$ Eq.(20),
${\phi}_{\bar{D},D}^{a}$ $=$ ${\phi}_{\bar{D},D}^{p}$ $=$ Eq.(26).

Our numerical results are presented in Table 2, where the uncertainties come
from the typical scale $(1{\pm}0.1)t_{i}$, mass $m_{c}$ and $m_{b}$, and the
CKM parameters, respectively. The following are some comments.

Table 2: The branching ratios for the $B^{\ast}$ ${\to}$ $\overline{D}D$ decays, where the theoretical uncertainties come from scale $(1{\pm}0.1)t_{i}$, mass $m_{c}$ and $m_{b}$, and the CKM parameters, respectively; the numbers in columns correspond to different DA scenarios. | class | I | II | III
---|---|---|---|---
${\cal B}r(B_{u}^{{\ast}+}{\to}\overline{D}_{u}^{0}D_{d}^{+}){\times}10^{11}$ | B | $7.65^{+1.81+0.04+0.62}_{-0.68-0.78-0.59}$ | $2.21^{+0.40+0.17+0.18}_{-0.17-0.21-0.17}$ | $1.24^{+0.20+0.15+0.10}_{-0.09-0.16-0.09}$
${\cal B}r(B_{u}^{{\ast}+}{\to}\overline{D}_{u}^{0}D_{s}^{+}){\times}10^{9}$ | A | $1.89^{+0.45+0.01+0.14}_{-0.17-0.19-0.13}$ | $0.57^{+0.11+0.04+0.04}_{-0.05-0.05-0.04}$ | $0.31^{+0.05+0.04+0.02}_{-0.02-0.04-0.02}$
${\cal B}r(B_{d}^{{\ast}0}{\to}D_{d}^{-}D_{s}^{+}){\times}10^{9}$ | A | $5.68^{+1.36+0.01+0.42}_{-0.51-0.54-0.40}$ | $1.72^{+0.33+0.13+0.13}_{-0.14-0.17-0.12}$ | $0.94^{+0.16+0.12+0.07}_{-0.07-0.12-0.07}$
${\cal B}r(B_{s}^{{\ast}0}{\to}D_{s}^{-}D_{d}^{+}){\times}10^{10}$ | B | $6.34^{+1.40+0.07+0.51}_{-0.54-0.45-0.49}$ | $2.09^{+0.38+0.12+0.17}_{-0.16-0.18-0.16}$ | $1.15^{+0.19+0.12+0.09}_{-0.08-0.13-0.09}$
${\cal B}r(B_{d}^{{\ast}0}{\to}D_{d}^{-}D_{d}^{+}){\times}10^{10}$ | B | $2.27^{+0.55+0.00+0.18}_{-0.21-0.21-0.17}$ | $0.68^{+0.12+0.05+0.05}_{-0.05-0.06-0.05}$ | $0.39^{+0.06+0.05+0.03}_{-0.03-0.05-0.03}$
${\cal B}r(B_{s}^{{\ast}0}{\to}D_{s}^{-}D_{s}^{+}){\times}10^{8}$ | A | $1.51^{+0.34+0.03+0.11}_{-0.13-0.12-0.11}$ | $0.53^{+0.10+0.03+0.04}_{-0.04-0.04-0.04}$ | $0.30^{+0.05+0.03+0.02}_{-0.02-0.03-0.02}$
${\cal B}r(B_{d}^{{\ast}0}{\to}\overline{D}_{u}^{0}D_{u}^{0}){\times}10^{14}$ | D | $1.53^{+0.73+1.45+0.19}_{-0.61-0.21-0.17}$ | $0.43^{+0.14+0.53+0.07}_{-0.12-0.34-0.06}$ | $4.10^{+0.12+0.26+0.48}_{-0.02-0.25-0.44}$
${\cal B}r(B_{d}^{{\ast}0}{\to}D_{s}^{-}D_{s}^{+}){\times}10^{13}$ | D | $1.11^{+0.16+0.22+0.09}_{-0.19-0.20-0.09}$ | $0.99^{+0.06+0.17+0.08}_{-0.06-0.18-0.08}$ | $0.66^{+0.03+0.04+0.05}_{-0.02-0.04-0.05}$
${\cal B}r(B_{s}^{{\ast}0}{\to}\overline{D}_{u}^{0}D_{u}^{0}){\times}10^{13}$ | C | $7.57^{+3.07+6.57+0.58}_{-2.52-1.66-0.55}$ | $1.97^{+0.67+2.71+0.15}_{-0.58-1.68-0.15}$ | $10.35^{+0.64+1.00+0.79}_{-0.39-1.01-0.76}$
${\cal B}r(B_{s}^{{\ast}0}{\to}D_{d}^{-}D_{d}^{+}){\times}10^{13}$ | C | $7.62^{+3.07+6.51+0.57}_{-2.53-1.61-0.55}$ | $1.99^{+0.67+2.64+0.15}_{-0.58-1.67-0.14}$ | $10.41^{+0.60+0.98+0.77}_{-0.37-1.00-0.74}$

(1) Generally, the $B^{\ast}$ ${\to}$ $\overline{D}D$ decay modes may be
divided into four classes. The tree contributions of classes A, B, C, D are
proportional to the factors of $V_{cb}^{\ast}\,V_{cs}\,a_{1}$ ${\sim}$
$A{\lambda}^{2}\,a_{1}$, $V_{cb}^{\ast}\,V_{cd}\,a_{1}$ ${\sim}$
$A{\lambda}^{3}\,a_{1}$, $V_{cb}^{\ast}\,V_{cs}\,a_{2}$ ${\sim}$
$A{\lambda}^{2}\,a_{2}$, $V_{cb}^{\ast}\,V_{cd}\,a_{2}$ ${\sim}$
$A{\lambda}^{3}\,a_{2}$, respectively. Classes C and D are pure annihilation
processes. There is a clear hierarchy of branching ratios, i.e., ${\cal
B}r(\text{class A})$ ${\gtrsim}$ $10^{-9}$, ${\cal B}r(\text{class B})$
${\gtrsim}$ $10^{-11}$, ${\cal B}r(\text{class C})$ ${\gtrsim}$ $10^{-13}$,
${\cal B}r(\text{class D})$ ${\lesssim}$ $10^{-13}$.

In addition, for each class, the magnitude relations between the decay
constants $f_{B_{s}^{{\ast}}}$ $>$ $f_{B_{u,d}^{{\ast}}}$ and $f_{D_{s}}$ $>$
$f_{D_{u,d}}$, and relations among the decay widths ${\Gamma}_{B_{s}^{\ast}}$
$<$ ${\Gamma}_{B_{d}^{\ast}}$ $<$ ${\Gamma}_{B_{u}^{\ast}}$, result in the
size relations among the branching ratios, i.e.,

${\cal B}r(B_{s}^{{\ast}0}{\to}D_{s}^{-}D_{s}^{+})>{\cal
B}r(B_{d}^{{\ast}0}{\to}D_{d}^{-}D_{s}^{+})>{\cal
B}r(B_{u}^{{\ast}+}{\to}\overline{D}_{u}^{0}D_{s}^{+}),$ (41) ${\cal
B}r(B_{s}^{{\ast}0}{\to}D_{s}^{-}D_{d}^{+})>{\cal
B}r(B_{d}^{{\ast}0}{\to}D_{d}^{-}D_{d}^{+})>{\cal
B}r(B_{u}^{{\ast}+}{\to}\overline{D}_{u}^{0}D_{d}^{+}),$ (42) ${\cal
B}r(B_{d}^{{\ast}0}{\to}D_{s}^{-}D_{s}^{+})>{\cal
B}r(B_{d}^{{\ast}0}{\to}\overline{D}_{u}^{0}D_{u}^{0}).$ (43)

(2) Due to the isospin symmetry, there are some approximate relations among
the branching ratios, for example,

$\frac{{\cal B}r(B_{d}^{{\ast}0}{\to}D_{d}^{-}D_{s}^{+})}{{\cal
B}r(B_{u}^{{\ast}+}{\to}\overline{D}_{u}^{0}D_{s}^{+})}\ {\approx}\
\frac{{\Gamma}_{B_{u}^{\ast}}}{{\Gamma}_{B_{d}^{\ast}}}\ {\approx}\ 3,$ (44)
$\frac{{\cal B}r(B_{d}^{{\ast}0}{\to}D_{d}^{-}D_{d}^{+})}{{\cal
B}r(B_{u}^{{\ast}+}{\to}\overline{D}_{u}^{0}D_{d}^{+})}\ {\approx}\
\frac{{\Gamma}_{B_{u}^{\ast}}}{{\Gamma}_{B_{d}^{\ast}}}\ {\approx}\ 3,$ (45)
$\frac{{\cal B}r(B_{s}^{{\ast}0}{\to}\overline{D}_{u}^{0}D_{u}^{0})}{{\cal
B}r(B_{s}^{{\ast}0}{\to}D_{d}^{-}D_{d}^{+})}\ {\approx}\ 1.$ (46)

In addition, there are some other approximate relations, for example,

$\frac{{\cal B}r(B_{u}^{{\ast}+}{\to}\overline{D}_{u}^{0}D_{s}^{+})}{{\cal
B}r(B_{u}^{{\ast}+}{\to}\overline{D}_{u}^{0}D_{d}^{+})}\ {\approx}\
\frac{f_{D_{s}}^{2}}{f_{D_{d}}^{2}}\,\frac{{|}V_{cs}{|}^{2}}{{|}V_{cd}{|}^{2}},$
(47) $\frac{{\cal B}r(B_{d}^{{\ast}0}{\to}D_{d}^{-}D_{s}^{+})}{{\cal
B}r(B_{d}^{{\ast}0}{\to}D_{d}^{-}D_{d}^{+})}\ {\approx}\
\frac{f_{D_{s}}^{2}}{f_{D_{d}}^{2}}\,\frac{{|}V_{cs}{|}^{2}}{{|}V_{cd}{|}^{2}},$
(48) $\frac{{\cal B}r(B_{s}^{{\ast}0}{\to}D_{s}^{-}D_{s}^{+})}{{\cal
B}r(B_{s}^{{\ast}0}{\to}D_{s}^{-}D_{d}^{+})}\ {\approx}\
\frac{f_{D_{s}}^{2}}{f_{D_{d}}^{2}}\,\frac{{|}V_{cs}{|}^{2}}{{|}V_{cd}{|}^{2}}.$
(49)

(3) Our study shows that (i) both the emission topologies and the annihilation
topologies contribute to the decay channels of Classes A and B. Furthermore,
the contributions from the emission topologies are dominant over those from
the annihilation topologies. (ii) For the pure annihilation decay channels of
Classes C and D, the factorizable contributions are color-suppressed, so the
nonfactorizable contributions are the main ones. In addition, the
interferences among different topologies are important. (iii) Compared with
the contributions of the tree operators, the contributions of the penguin
operators are small because of the suppression from the small Wilson
coefficients. The contributions of the topologies in Fig.2(i-l) are much less
than those of the topologies in Fig.2(e-h). (iv) For the decay channels with
the final states $\overline{D}_{q}D_{q}$, the contribution of the factorizable
annihilation topology in Fig.2(e) [Fig.2(i)] is the same magnitude as that in
Fig.2(f) [Fig.2(j)] due to the flavor symmetry. (v) The interferences between
factorizable topologies in Fig.2(e) and Fig.2(f) [Fig.2(i) and Fig.2(j)] are
constructive, while the interferences between nonfactorizable topologies in
Fig.2(g) and Fig.2(h) [Fig.2(k) and Fig.2(l)] are destructive.

Table 3: The fractions (in the unit of %) of the different $b$-hadron species. channels | $(b\bar{s})(\bar{b}s)$ | $B_{s}^{{\ast}0}\overline{B}_{s}^{{\ast}0}$ | $B_{s}^{{\ast}0}\overline{B}_{s}^{0}+{\rm c.c.}$ | $~{}~{}(b\bar{u})(\bar{b}u)\atop+(b\bar{d})(\bar{b}d)$ | $B^{\ast}\overline{B}^{\ast}$ | $B^{\ast}\overline{B}+{\rm c.c.}$ | $B^{\ast}\overline{B}{\pi}+{\rm c.c.}$ | $B^{\ast}\overline{B}^{\ast}{\pi}$
---|---|---|---|---|---|---|---|---
fractions | ${\cal B}_{s}$ | ${\cal B}_{B_{s}^{{\ast}0}\overline{B}_{s}^{{\ast}0}}$ | ${\cal B}_{B_{s}^{{\ast}0}\overline{B}_{s}^{0}}$ |  ${\cal B}_{u}$ $+$ ${\cal B}_{d}$ | ${\cal B}_{B^{\ast}\overline{B}^{\ast}}$ |  |  |
${\Upsilon}(5S)$ epjc74.3026  | $19.5^{+3.0}_{-2.3}$ | $17.6{\pm}2.8$ | $1.4{\pm}0.7$ | $73.7{\pm}6.0$ | $37.5{\pm}3.7$ | $13.7{\pm}1.7$ | $7.3{\pm}2.4$ | $1.0{\pm}1.5$
Tevatron 1412.7515  | $10.0{\pm}1.0$ |  |  | $70.0{\pm}4.0$ |  |  |  |

(4) The $B_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{-}D_{s}^{+}$, $B_{d}^{{\ast}0}$
${\to}$ $D_{d}^{-}D_{s}^{+}$, $B_{u}^{{\ast}+}$ ${\to}$
$\overline{D}_{u}^{0}D_{s}^{+}$ decays, belonging to class A, have relatively
large branching ratios, ${\cal B}r(\text{class A})$ ${\gtrsim}$ $10^{-9}$. The
numbers of the $B_{q}^{\ast}$ mesons in a data sample can be estimated by

$N(B_{q}^{\ast})\ =\ {\cal L}_{\rm
int}\,{\times}\,{\sigma}_{b\bar{b}}\,{\times}\,{\cal
B}_{q}\,{\times}\,\frac{{\cal B}_{B_{q}^{\ast}}}{{\cal B}_{q}},$ (50) ${\cal
B}_{B_{q}^{\ast}}\ =\ 2\,{\times}\,{\cal
B}_{B_{q}^{\ast}\overline{B}_{q}^{\ast}}+2\,{\times}\,{\cal
B}_{B_{q}^{\ast}\overline{B}_{q}^{\ast}{\pi}}+{\cal
B}_{B_{q}^{\ast}\overline{B}_{q}+{\rm c.c.}}+{\cal
B}_{B_{q}^{\ast}\overline{B}_{q}{\pi}+{\rm c.c}}+{\cdots},$ (51)

where ${\cal L}_{\rm int}$ is the integrated luminosity, ${\sigma}_{b\bar{b}}$
denotes the $b\bar{b}$ pair production cross section, ${\cal B}_{q}$ refers to
the fragmentation fraction of $(b\bar{q})(\bar{b}q)$ events, and ${\cal
B}_{B_{q}^{\ast}\overline{B}_{q}^{\ast}}$, ${\cal
B}_{B_{q}^{\ast}\overline{B}_{q}^{\ast}X}$, …. represent the production
fractions of specific modes (see Table 3). With a large production cross
section of the process $e^{+}e^{-}$ ${\to}$ $b\bar{b}$ at the ${\Upsilon}(5S)$
peak ${\sigma}_{b\bar{b}}$ $=$ $(0.340{\pm}0.016)\,{\rm nb}$ epjc74.3026  and
a high luminosity $8{\times}10^{35}\,{\rm cm}^{-2}\,{\rm s}^{-1}$ at the
forthcoming SuperKEKB 1002.5012 , it is expected that some $3.3{\times}10^{9}$
$B_{u,d}^{\ast}$ and $1.2{\times}10^{9}$ $B_{s}^{\ast}$ mesons could be
available per $10\,{\rm ab}^{-1}$ ${\Upsilon}(5S)$ dataset, corresponding to a
few events of the $B_{d}^{{\ast}0}$ ${\to}$ $D_{d}^{-}D_{s}^{+}$ and
$B_{u}^{{\ast}+}$ ${\to}$ $\overline{D}_{u}^{0}D_{s}^{+}$ decays and dozens of
the $B_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{-}D_{s}^{+}$ decay. At high energy
hadron colliders, for example, with a visible $b$-hadron cross section at the
LHCb ${\sigma}_{b\bar{b}}$ ${\sim}$ $100\,{\rm{\mu}b}$ pdg ; crp16.435 , a
similar ratio ${\cal B}_{q}$ at Tevatron and a similar ratio ${\cal
B}_{B_{q}^{\ast}}/{\cal B}_{q}$ at ${\Upsilon}(5S)$, some $9.2{\times}10^{13}$
$B_{u,d}^{\ast}$ and $1.9{\times}10^{13}$ $B_{s}^{\ast}$ events per ${\rm
ab}^{-1}$ dataset should be available at the LHCb, corresponding to more than
$10^{5}$ (class A) $B_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{-}D_{s}^{+}$,
$B_{d}^{{\ast}0}$ ${\to}$ $D_{d}^{-}D_{s}^{+}$, and $B_{u}^{{\ast}+}$ ${\to}$
$\overline{D}_{u}^{0}D_{s}^{+}$ decay events and over $10^{4}$ (class B)
$B_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{-}D_{d}^{+}$, $B_{d}^{{\ast}0}$ ${\to}$
$D_{d}^{-}D_{d}^{+}$ $B_{u}^{{\ast}+}$ ${\to}$ $\overline{D}_{u}^{0}D_{d}^{+}$
decay events, which are measurable at the future LHCb experiments. However,
the search for pure annihilation processes (classes C and D) at LHC and
SuperKEKB should still be very challenging.

(5) Compared with the branching ratios ${\gtrsim}$ $10^{-5}$ for the $B$
${\to}$ $\overline{D}D$ decays pdg ; prd81.034006 , the branching ratios for
the $B^{\ast}$ ${\to}$ $\overline{D}D$ decays are smaller by at least three
orders of magnitude. This fact might imply that the background from the
$B^{\ast}$ ${\to}$ $\overline{D}D$ decays could be safely neglected when one
analyzes the $B$ ${\to}$ $\overline{D}D$ decays, but not vice versa, i.e., one
of main pollution backgrounds for the $B^{\ast}$ ${\to}$ $\overline{D}D$
decays would come from the $B$ ${\to}$ $\overline{D}D$ decays, even if the
invariant mass of the $\overline{D}D$ meson pair could be used to distinguish
the $B^{\ast}$ meson from the $B$ meson experimentally.

(6) For the $B_{q}^{\ast}$ decays of classes A and B, our estimation of the
branching ratios agrees well with that based on the naive factorization
approach 1605.01630 . One of the important reasons is that these processes are
all $a_{1}$-dominated (color favored), and in general, insensitive to
nonfactorizable corrections to the HTME. Of course, one fact is clear that
there are many theoretical uncertainties, especially, regarding the
discrepancy among different DA scenarios, which results from our uncertain
knowledge of the long-distance QCD effects and the underlying dynamics of low
energy hadron interactions. Moreover, as aforementioned, there are large
uncertainties of the decay width ${\Gamma}_{B_{q}^{\ast}}$. With a different
value of ${\Gamma}_{B_{q}^{\ast}}$, the branching ratios in Table 2 should be
multiplied by the factors of ${450\,{\rm eV}}/{{\Gamma}_{B_{u}^{\ast}}}$,
${150\,{\rm eV}}/{{\Gamma}_{B_{d}^{\ast}}}$, ${100\,{\rm
eV}}/{{\Gamma}_{B_{s}^{\ast}}}$ for the $B_{u}^{\ast}$, $B_{d}^{\ast}$,
$B_{s}^{\ast}$ decays, respectively. In addition, many other factors, such as
the final state interactions, models for the $B^{\ast}$ and $D$ meson wave
functions222In principle, one can do a global fit on the $B^{\ast}$ and $D$
meson wave functions with experimental measurements in the future, analogous
to that with the ${\chi}^{2}$ method in Ref.prd78.014018 . The fitting will be
a very time-consuming work, because the amplitudes for the $B^{\ast}$ ${\to}$
$\overline{D}D$ decays are expressed as the multidimensional integral with the
pQCD approach. In addition, there is no measurement report on the $B^{\ast}$
${\to}$ $\overline{D}D$ decays at the moment., higher order corrections to the
HTME, and so on, are not carefully scrutinized here, but deserve much
dedicated study. Our estimation may be just an order of magnitude.

##  IV Summary

With the running LHC and the forthcoming SuperKEKB, a large amount of
$B^{\ast}$ data should be in stock soon, which will make it seemingly possible
to explore the $B^{\ast}$ weak decays experimentally. A theoretical study is
necessary in order to offer a timely reference, and is helpful in clearing up
some of puzzles surrounding heavy meson weak decays. In this paper, we
investigated the $B^{\ast}$ ${\to}$ $\overline{D}D$ decays with the
phenomenological pQCD approach. It is found that the $B_{s}^{{\ast}0}$ ${\to}$
$D_{s}^{-}D_{s}^{+}$, $B_{d}^{{\ast}0}$ ${\to}$ $D_{d}^{-}D_{s}^{+}$, and
$B_{u}^{{\ast}+}$ ${\to}$ $\overline{D}_{u}^{0}D_{s}^{+}$ decays have
branching ratios ${\gtrsim}$ $10^{-9}$, and will be promisingly accessible at
the future high luminosity experiments, with help of a sophisticated
experimental analytical technique to effectively suppress or exclude the
background from the $B$ ${\to}$ $\overline{D}D$ decays.

## Acknowledgments

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

##  Appendix A Amplitude building blocks for $B^{\ast}$ ${\to}$
$\overline{D}D$ decays

$\displaystyle{\cal A}_{a}^{LL}$ $\displaystyle=$
$\displaystyle{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}\,{\alpha}_{s}(t_{a})\,H_{ef}({\alpha}_{e},{\beta}_{a},b_{1},b_{2})$
(52)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\times}\,E_{ef}(t_{a})\,{\phi}_{B^{\ast}}^{v}(x_{1})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,(m_{1}^{2}\,\bar{x}_{2}+m_{3}^{2}\,x_{2})+{\phi}_{\bar{D}}^{p}(x_{2})\,m_{2}\,m_{b}\Big{\\}},$
$\displaystyle{\cal A}_{a}^{SP}$ $\displaystyle=$
$\displaystyle-2\,m_{3}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}\,H_{ef}({\alpha}_{e},{\beta}_{a},b_{1},b_{2})$
(53) $\displaystyle{\times}$
$\displaystyle{\alpha}_{s}(t_{a})\,E_{ef}(t_{a})\,{\phi}_{B^{\ast}}^{v}(x_{1})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,m_{b}+{\phi}_{\bar{D}}^{p}(x_{2})\,m_{2}\,x_{2}\Big{\\}},$
$\displaystyle{\cal A}_{b}^{LL}$ $\displaystyle=$
$\displaystyle{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}\,{\alpha}_{s}(t_{b})\,H_{ef}({\alpha}_{e},{\beta}_{a},b_{2},b_{1})$
(54) $\displaystyle{\times}$ $\displaystyle
E_{ef}(t_{b})\,\Big{\\{}{\phi}_{B^{\ast}}^{t}(x_{1})\,\Big{[}{\phi}_{\bar{D}}^{p}(x_{2})\,2\,m_{1}\,m_{2}\,\bar{x}_{1}-{\phi}_{\bar{D}}^{a}(x_{2})\,m_{1}\,m_{c}\Big{]}$
$\displaystyle+$
$\displaystyle{\phi}_{B^{\ast}}^{v}(x_{1})\,\Big{[}{\phi}_{\bar{D}}^{p}(x_{2})\,2\,m_{2}\,m_{c}-{\phi}_{\bar{D}}^{a}(x_{2})\,(m_{2}^{2}\,\bar{x}_{1}+m_{3}^{2}\,x_{1})\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{b}^{SP}$ $\displaystyle=$ $\displaystyle
2\,m_{3}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}\,{\alpha}_{s}(t_{b})\,H_{ef}({\alpha}_{e},{\beta}_{a},b_{2},b_{1})$
(55)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\times}\,E_{ef}(t_{b})\,\Big{\\{}{\phi}_{B^{\ast}}^{v}(x_{1})\,\Big{[}{\phi}_{\bar{D}}^{a}(x_{2})\,m_{c}-{\phi}_{\bar{D}}^{p}(x_{2})\,2\,m_{2}\Big{]}+{\phi}_{B^{\ast}}^{t}(x_{1})\,{\phi}_{\bar{D}}^{a}(x_{2})\,m_{1}\,x_{1}\Big{\\}},$
$\displaystyle{\cal A}_{c}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,{\alpha}_{s}(t_{c})\,E_{n}(t_{c})$
(56) $\displaystyle{\times}$ $\displaystyle
H_{en}({\alpha}_{e},{\beta}_{c},b_{3},b_{2},b_{1})\,\Big{\\{}{\phi}_{B^{\ast}}^{t}(x_{1})\,{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,m_{1}\,m_{2}\,(x_{1}-x_{2})$
$\displaystyle+{\phi}_{B^{\ast}}^{v}(x_{1})\,{\phi}_{\bar{D}}^{a}(x_{2})\,\Big{[}{\phi}_{D}^{a}(x_{3})\,s\,(x_{2}-\bar{x}_{3})-{\phi}_{D}^{p}(x_{3})\,m_{3}\,m_{c}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{c}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,H_{en}({\alpha}_{e},{\beta}_{c},b_{3},b_{2},b_{1})$
(57) $\displaystyle{\times}$ $\displaystyle
E_{n}(t_{c})\,{\alpha}_{s}(t_{c})\,\Big{\\{}{\phi}_{B^{\ast}}^{v}(x_{1})\,{\phi}_{\bar{D}}^{p}(x_{2})\,m_{2}\,\Big{[}{\phi}_{D}^{p}(x_{3})\,m_{3}\,(x_{2}-\bar{x}_{3})-{\phi}_{D}^{a}(x_{3})\,m_{c}\Big{]}$
$\displaystyle+{\phi}_{B^{\ast}}^{t}(x_{1})\,{\phi}_{\bar{D}}^{a}(x_{2})\,m_{1}\,\Big{[}{\phi}_{D}^{a}(x_{3})\,m_{c}+{\phi}_{D}^{p}(x_{3})\,m_{3}\,(\bar{x}_{3}-x_{1})\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{d}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,{\alpha}_{s}(t_{d})\,E_{n}(t_{d})$
(58) $\displaystyle{\times}$ $\displaystyle
H_{en}({\alpha}_{e},{\beta}_{d},b_{3},b_{2},b_{1})\,{\phi}_{D}^{a}(x_{3})\,\Big{\\{}{\phi}_{B^{\ast}}^{t}(x_{1})\,{\phi}_{\bar{D}}^{p}(x_{2})\,m_{1}\,m_{2}\,(x_{1}-x_{2})$
$\displaystyle+{\phi}_{B^{\ast}}^{v}(x_{1})\,{\phi}_{\bar{D}}^{a}(x_{2})\,(2\,m_{2}^{2}\,x_{2}+s\,x_{3}-t\,x_{1})\Big{\\}},$
$\displaystyle{\cal A}_{d}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,{\alpha}_{s}(t_{d})\,E_{n}(t_{d})$
(59) $\displaystyle{\times}$ $\displaystyle
H_{en}({\alpha}_{e},{\beta}_{d},b_{3},b_{2},b_{1})\,{\phi}_{D}^{p}(x_{3})\,\Big{\\{}{\phi}_{B^{\ast}}^{v}(x_{1})\,{\phi}_{\bar{D}}^{p}(x_{2})\,m_{2}\,m_{3}\,(x_{3}-x_{2})$
$\displaystyle+{\phi}_{B^{\ast}}^{t}(x_{1})\,{\phi}_{\bar{D}}^{a}(x_{2})\,m_{1}\,m_{3}\,(x_{1}-x_{3})\Big{\\}},$
$\displaystyle{\cal A}_{e}^{LL}$ $\displaystyle=$ $\displaystyle{\cal
A}_{e}^{LR}\ =\
{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,H_{af}({\alpha}_{q},{\beta}_{e},b_{2},b_{3})$
(60) $\displaystyle{\times}$
$\displaystyle{\alpha}_{s}(t_{e})\,E_{af}(t_{e})\,\Big{\\{}{\phi}_{D}^{p}(x_{3})\,2\,m_{3}\,\Big{[}{\phi}_{\bar{D}}^{a}(x_{2})\,m_{c}+{\phi}_{\bar{D}}^{p}(x_{2})\,m_{2}\,\bar{x}_{2}\Big{]}$
$\displaystyle-{\phi}_{D}^{a}(x_{3})\,\Big{[}{\phi}_{\bar{D}}^{a}(x_{2})\,(m_{1}^{2}\,x_{2}+m_{3}^{2}\bar{x}_{2})+{\phi}_{\bar{D}}^{p}(x_{2})\,m_{2}\,m_{c}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{f}^{LL}$ $\displaystyle=$ $\displaystyle{\cal
A}_{f}^{LR}\ =\
{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,H_{af}({\alpha}_{q},{\beta}_{f},b_{3},b_{2})$
(61) $\displaystyle{\times}$
$\displaystyle{\alpha}_{s}(t_{f})\,E_{af}(t_{f})\,\Big{\\{}{\phi}_{\bar{D}}^{p}(x_{2})\,2\,m_{2}\,\Big{[}{\phi}_{D}^{a}(x_{3})\,m_{c}+{\phi}_{D}^{p}(x_{3})\,m_{3}\,\bar{x}_{3}\Big{]}$
$\displaystyle-{\phi}_{\bar{D}}^{a}(x_{2})\,\Big{[}{\phi}_{D}^{a}(x_{3})\,(m_{1}^{2}\,x_{3}+m_{2}^{2}\bar{x}_{3})+{\phi}_{D}^{p}(x_{3})\,m_{3}\,m_{c}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{g}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}\,H_{an}({\alpha}_{q},{\beta}_{g},b_{1},b_{2},b_{3})$
(62) $\displaystyle{\times}$ $\displaystyle
E_{n}(t_{g})\,{\alpha}_{s}(t_{g})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,\Big{[}{\phi}_{B^{\ast}}^{v}(x_{1})\,(s\,x_{2}+2\,m_{3}^{2}\,x_{3}-u\,\bar{x}_{1})$
$\displaystyle+{\phi}_{B^{\ast}}^{t}(x_{1})\,m_{1}\,m_{b}\Big{]}+{\phi}_{B^{\ast}}^{v}(x_{1})\,{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,m_{2}\,m_{3}\,(x_{2}-x_{3})\Big{\\}},$
$\displaystyle{\cal A}_{g}^{LR}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}\,H_{an}({\alpha}_{q},{\beta}_{g},b_{1},b_{2},b_{3})$
(63) $\displaystyle{\times}$ $\displaystyle
E_{n}(t_{g})\,{\alpha}_{s}(t_{g})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,\Big{[}{\phi}_{B^{\ast}}^{v}(x_{1})\,(t\,\bar{x}_{1}-2\,m_{2}^{2}\,x_{2}-s\,x_{3})$
$\displaystyle-{\phi}_{B^{\ast}}^{t}(x_{1})\,m_{1}\,m_{b}\Big{]}+{\phi}_{B^{\ast}}^{v}(x_{1})\,{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,m_{2}\,m_{3}\,(x_{2}-x_{3})\Big{\\}},$
$\displaystyle{\cal A}_{h}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}\,E_{n}(t_{h})$
(64) $\displaystyle{\times}$
$\displaystyle{\alpha}_{s}(t_{h})\,{\phi}_{B^{\ast}}^{v}(x_{1})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,(2\,m_{2}^{2}\,x_{2}+s\,x_{3}-t\,x_{1})$
$\displaystyle+$
$\displaystyle{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,m_{2}\,m_{3}\,(x_{3}-x_{2})\Big{\\}}\,H_{an}({\alpha}_{q},{\beta}_{h},b_{1},b_{2},b_{3}),$
$\displaystyle{\cal A}_{h}^{LR}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}\,E_{n}(t_{h})$
(65) $\displaystyle{\times}$
$\displaystyle{\alpha}_{s}(t_{h})\,{\phi}_{B^{\ast}}^{v}(x_{1})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,(u\,x_{1}-s\,x_{2}-2\,m_{3}^{2}\,x_{3})$
$\displaystyle+$
$\displaystyle{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,m_{2}\,m_{3}\,(x_{3}-x_{2})\Big{\\}}\,H_{an}({\alpha}_{q},{\beta}_{h},b_{1},b_{2},b_{3}),$
$\displaystyle{\cal A}_{i}^{LL}$ $\displaystyle=$ $\displaystyle{\cal
A}_{i}^{LR}\ =\
{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,{\alpha}_{s}(t_{i})\,H_{af}({\alpha}_{c},{\beta}_{i},b_{2},b_{3})$
(66)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\times}\,E_{af}(t_{i})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,(m_{1}^{2}\,\bar{x}_{2}+m_{3}^{2}\,x_{2})-{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,2\,m_{2}\,m_{3}\,x_{2}\Big{\\}},$
$\displaystyle{\cal A}_{j}^{LL}$ $\displaystyle=$ $\displaystyle{\cal
A}_{j}^{LR}\ =\
{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,{\alpha}_{s}(t_{j})\,H_{af}({\alpha}_{c},{\beta}_{j},b_{3},b_{2})$
(67)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\times}\,E_{af}(t_{j})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,(m_{1}^{2}\,\bar{x}_{3}+m_{2}^{2}\,x_{3})-{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,2\,m_{2}\,m_{3}\,x_{3}\Big{\\}},$
$\displaystyle{\cal A}_{k}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}\,H_{an}({\alpha}_{c},{\beta}_{k},b_{1},b_{2},b_{3})$
(68) $\displaystyle{\times}$ $\displaystyle
E_{n}(t_{k})\,{\alpha}_{s}(t_{k})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,\Big{[}{\phi}_{B^{\ast}}^{v}(x_{1})\,(2\,m_{2}^{2}\,x_{2}+s\,x_{3}-t\,x_{1})$
$\displaystyle-{\phi}_{B^{\ast}}^{t}(x_{1})\,m_{1}\,m_{b}\Big{]}+{\phi}_{B^{\ast}}^{v}(x_{1})\,{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,m_{2}\,m_{3}\,(x_{3}-x_{2})\Big{\\}},$
$\displaystyle{\cal A}_{k}^{LR}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}\,H_{an}({\alpha}_{c},{\beta}_{k},b_{1},b_{2},b_{3})$
(69) $\displaystyle{\times}$ $\displaystyle
E_{n}(t_{k})\,{\alpha}_{s}(t_{k})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,\Big{[}{\phi}_{B^{\ast}}^{v}(x_{1})\,(u\,x_{1}-s\,x_{2}-2\,m_{3}^{2}\,x_{3})$
$\displaystyle+{\phi}_{B^{\ast}}^{t}(x_{1})\,m_{1}\,m_{b}\Big{]}+{\phi}_{B^{\ast}}^{v}(x_{1})\,{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,m_{2}\,m_{3}\,(x_{3}-x_{2})\Big{\\}},$
$\displaystyle{\cal A}_{k}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}\,H_{an}({\alpha}_{c},{\beta}_{k},b_{1},b_{2},b_{3})$
(70) $\displaystyle{\times}$ $\displaystyle
E_{n}(t_{k})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,m_{3}\,\Big{[}{\phi}_{B^{\ast}}^{v}(x_{1})\,m_{b}+{\phi}_{B^{\ast}}^{t}(x_{1})\,m_{1}\,(x_{1}-x_{3})\Big{]}$
$\displaystyle+{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,m_{2}\,\Big{[}{\phi}_{B^{\ast}}^{v}(x_{1})\,m_{b}+{\phi}_{B^{\ast}}^{t}(x_{1})\,m_{1}\,(x_{1}-x_{2})\Big{]}\Big{\\}}\,{\alpha}_{s}(t_{k}),$
$\displaystyle{\cal A}_{l}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}\,E_{n}(t_{l})$
(71) $\displaystyle{\times}$
$\displaystyle{\alpha}_{s}(t_{l})\,{\phi}_{B^{\ast}}^{v}(x_{1})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,(u\,x_{1}-s\,\bar{x}_{2}-2\,m_{3}^{2}\,\bar{x}_{3})$
$\displaystyle+$
$\displaystyle{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,m_{2}\,m_{3}\,(x_{2}-x_{3})\Big{\\}}\,H_{an}({\alpha}_{c},{\beta}_{l},b_{1},b_{2},b_{3}),$
$\displaystyle{\cal A}_{l}^{LR}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}\,E_{n}(t_{l})$
(72) $\displaystyle{\times}$
$\displaystyle{\alpha}_{s}(t_{l})\,{\phi}_{B^{\ast}}^{v}(x_{1})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,(2\,m_{2}^{2}\,\bar{x}_{2}+s\,\bar{x}_{3}-t\,x_{1})$
$\displaystyle+$
$\displaystyle{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,m_{2}\,m_{3}\,(x_{2}-x_{3})\Big{\\}}\,H_{an}({\alpha}_{c},{\beta}_{l},b_{1},b_{2},b_{3}),$
$\displaystyle{\cal A}_{l}^{SP}$ $\displaystyle=$
$\displaystyle\frac{m_{1}}{N_{c}}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(73) $\displaystyle{\times}$ $\displaystyle
E_{n}(t_{l})\,{\alpha}_{s}(t_{l})\,{\phi}_{B^{\ast}}^{t}(x_{1})\,\Big{\\{}{\phi}_{\bar{D}}^{a}(x_{2})\,{\phi}_{D}^{p}(x_{3})\,m_{3}\,(\bar{x}_{3}-x_{1})$
$\displaystyle+$
$\displaystyle{\phi}_{\bar{D}}^{p}(x_{2})\,{\phi}_{D}^{a}(x_{3})\,m_{2}\,(\bar{x}_{2}-x_{1})\Big{\\}}\,H_{an}({\alpha}_{c},{\beta}_{l},b_{1},b_{2},b_{3}),$

where the subscript $i$ of ${\cal A}_{i}^{j}$ corresponds to the indices of
Fig.2; the superscript $j$ refers to three possible Dirac structures
${\Gamma}_{1}{\otimes}{\Gamma}_{2}$ of the operators
$(\bar{q}_{1}q_{2})_{{\Gamma}_{1}}(\bar{q}_{3}q_{4})_{{\Gamma}_{2}}$, namely
$j$ $=$ $LL$ for $(V-A){\otimes}(V-A)$, $j$ $=$ $LR$ for
$(V-A){\otimes}(V+A)$, and $j$ $=$ $SP$ for $-2(S-P){\otimes}(S+P)$.

The function $H_{i}$ and the Sudakov factor $E_{i}$ are defined as

$H_{ef}({\alpha}_{e},{\beta},b_{m},b_{n})\,=\,K_{0}(b_{m}\sqrt{-{\alpha}_{e}})\,\Big{\\{}{\theta}(b_{m}-b_{n})K_{0}(b_{m}\sqrt{-{\beta}})\,I_{0}(b_{n}\sqrt{-{\beta}})+(b_{m}{\leftrightarrow}b_{n})\Big{\\}},$
(74) $\displaystyle H_{en}({\alpha}_{e},{\beta},b_{3},b_{2},b_{1})$
$\displaystyle=$
$\displaystyle\Big{\\{}{\theta}(-{\beta})\,K_{0}(b_{3}\sqrt{-{\beta}})+\frac{{\pi}}{2}\,{\theta}({\beta})\,\Big{[}i\,J_{0}(b_{3}\sqrt{{\beta}})-Y_{0}(b_{3}\sqrt{{\beta}})\Big{]}\Big{\\}}$
(75)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\times}\,\Big{\\{}{\theta}(b_{2}-b_{3})\,K_{0}(b_{2}\sqrt{-{\alpha}_{e}})\,I_{0}(b_{3}\sqrt{-{\alpha}_{e}})+(b_{2}{\leftrightarrow}b_{3})\Big{\\}}\,{\delta}(b_{1}-b_{2}),$
$\displaystyle H_{af}({\alpha},{\beta},b_{m},b_{n})$ $\displaystyle=$
$\displaystyle\Big{\\{}{\theta}(b_{m}-b_{n})\Big{[}i\,J_{0}(b_{m}\sqrt{{\beta}})-Y_{0}(b_{m}\sqrt{{\beta}})\Big{]}J_{0}(b_{n}\sqrt{{\beta}})$
(76)
$\displaystyle+\,(b_{m}{\leftrightarrow}b_{n})\Big{\\}}\,\Big{\\{}i\,J_{0}(b_{n}\sqrt{{\alpha}})-Y_{0}(b_{n}\sqrt{{\alpha}})\Big{\\}}\,\frac{{\pi}^{2}}{4},$
$\displaystyle H_{an}({\alpha},{\beta},b_{1},b_{2},b_{3})$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}\,\Big{\\{}{\theta}(-{\beta})\,K_{0}(b_{1}\sqrt{-{\beta}})+\frac{\pi}{2}{\theta}({\beta})\,\Big{[}i\,J_{0}(b_{1}\sqrt{{\beta}})-Y_{0}(b_{1}\sqrt{{\beta}})\Big{]}\Big{\\}}$
(77)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\times}\,\Big{\\{}{\theta}(b_{1}-b_{2})\Big{[}i\,J_{0}(b_{1}\sqrt{{\alpha}})-Y_{0}(b_{1}\sqrt{{\alpha}})\Big{]}J_{0}(b_{2}\sqrt{{\alpha}})+(b_{1}{\leftrightarrow}b_{2})\Big{\\}}\,{\delta}(b_{2}-b_{3}),$
$E_{ef}(t)\ =\ {\exp}\\{-S_{B^{\ast}}(t)-S_{\bar{D}}(t)\\},$ (78) $E_{af}(t)\
=\ {\exp}\\{-S_{\bar{D}}(t)-S_{D}(t)\\},$ (79) $E_{n}(t)\ =\
{\exp}\\{-S_{B^{\ast}}(t)-S_{\bar{D}}(t)-S_{D}(t)\\},$ (80)
$S_{B^{\ast}}(t)\,=\,s(x_{1},b_{1},p_{1}^{+})+2{\int}_{1/b_{1}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q},$
(81)
$S_{\bar{D}}(t)\,=\,s(x_{2},b_{2},p_{2}^{+})+s(\bar{x}_{2},b_{2},p_{2}^{+})+2{\int}_{1/b_{2}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q},$
(82)
$S_{D}(t)\,=\,s(x_{3},b_{3},p_{3}^{+})+s(\bar{x}_{3},b_{3},p_{3}^{+})+2{\int}_{1/b_{3}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q},$
(83)

where the subscript $i$ $=$ $ef$, $en$, $af$, $an$ corresponds to the
factorizable emission topologies, the nonfactorizable emission topologies, the
factorizable annihilation topologies, and the nonfactorizable annihilation
topologies, respectively; $I_{0}$, $J_{0}$, $K_{0}$ and $Y_{0}$ are Bessel
functions; ${\gamma}_{q}$ $=$ $-{\alpha}_{s}/{\pi}$ is the quark anomalous
dimension; the expression of $s(x,b,Q)$ can be found in of Ref.pqcd1 ;
${\alpha}$ and ${\beta}$ are the virtualities of gluon and quarks. the
subscript of ${\beta}_{i}$ corresponds to the indices of Fig.2. The
definitions of the particle virtuality and typical scale $t_{i}$ are given as
follows.

${\alpha}_{e}\ =\ x_{1}^{2}\,m_{1}^{2}+x_{2}^{2}\,m_{2}^{2}-x_{1}\,x_{2}\,t,$
(84) ${\alpha}_{q}\ =\
x_{2}^{2}\,m_{2}^{2}+x_{3}^{2}\,m_{3}^{2}+x_{2}\,x_{3}\,s,$ (85)
${\alpha}_{c}\ =\
\bar{x}_{2}^{2}\,m_{2}^{2}+\bar{x}_{3}^{2}\,m_{3}^{2}+\bar{x}_{2}\,\bar{x}_{3}\,s,$
(86) ${\beta}_{a}\ =\ m_{1}^{2}+x_{2}^{2}\,m_{2}^{2}-x_{2}\,t-m_{b}^{2},$ (87)
${\beta}_{b}\ =\ m_{2}^{2}+x_{1}^{2}\,m_{1}^{2}-x_{1}\,t-m_{c}^{2},$ (88)
${\beta}_{c}\ =\
{\alpha}_{e}+\bar{x}_{3}^{2}\,m_{3}^{2}-x_{1}\,\bar{x}_{3}\,u+x_{2}\,\bar{x}_{3}\,s-m_{c}^{2},$
(89) ${\beta}_{d}\ =\
{\alpha}_{e}+x_{3}^{2}\,m_{3}^{2}-x_{1}\,x_{3}\,u+x_{2}\,x_{3}\,s,$ (90)
${\beta}_{e}\ =\ m_{3}^{2}+x_{2}^{2}\,m_{2}^{2}+x_{2}\,s-m_{c}^{2}\ >\ 0,$
(91) ${\beta}_{f}\ =\ m_{2}^{2}+x_{3}^{2}\,m_{3}^{2}+x_{3}\,s-m_{c}^{2}\ >\
0,$ (92) ${\beta}_{g}\ =\
{\alpha}_{q}+\bar{x}_{1}^{2}\,m_{1}^{2}-\bar{x}_{1}\,x_{2}\,t-\bar{x}_{1}\,x_{3}\,u-m_{b}^{2},$
(93) ${\beta}_{h}\ =\
{\alpha}_{q}+x_{1}^{2}\,m_{1}^{2}-x_{1}\,x_{2}\,t-x_{1}\,x_{3}\,u,$ (94)
${\beta}_{i}\ =\ m_{3}^{2}+\bar{x}_{2}^{2}\,m_{2}^{2}+\bar{x}_{2}\,s\ >\ 0,$
(95) ${\beta}_{j}\ =\ m_{2}^{2}+\bar{x}_{3}^{2}\,m_{3}^{2}+\bar{x}_{3}\,s\ >\
0,$ (96) ${\beta}_{k}\ =\
{\alpha}_{c}+\bar{x}_{1}^{2}\,m_{1}^{2}-\bar{x}_{1}\,\bar{x}_{2}\,t-\bar{x}_{1}\,\bar{x}_{3}\,u-m_{b}^{2},$
(97) ${\beta}_{l}\ =\
{\alpha}_{c}+x_{1}^{2}\,m_{1}^{2}-x_{1}\,\bar{x}_{2}\,t-x_{1}\,\bar{x}_{3}\,u,$
(98) $t_{a,b}\ =\
{\max}\\{\sqrt{-{\alpha}_{e}},\sqrt{{|}{\beta}_{a,b}{|}},1/b_{1},1/b_{2}\\},$
(99) $t_{c,d}\ =\
{\max}\\{\sqrt{-{\alpha}_{e}},\sqrt{{|}{\beta}_{c,d}{|}},1/b_{2},1/b_{3}\\},$
(100) $t_{e,f}\ =\
{\max}\\{\sqrt{{\alpha}_{q}},\sqrt{{|}{\beta}_{e,f}{|}},1/b_{2},1/b_{3}\\},$
(101) $t_{g,h}\ =\
{\max}\\{\sqrt{{\alpha}_{q}},\sqrt{{|}{\beta}_{g,h}{|}},1/b_{1},1/b_{2}\\},$
(102) $t_{i,j}\ =\
{\max}\\{\sqrt{{\alpha}_{c}},\sqrt{{|}{\beta}_{i,j}{|}},1/b_{2},1/b_{3}\\},$
(103) $t_{k,l}\ =\
{\max}\\{\sqrt{{\alpha}_{c}},\sqrt{{|}{\beta}_{k,l}{|}},1/b_{1},1/b_{2}\\}.$
(104)

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