# Probing optical properties of metallic carbon nanotubes: beyond the standard
semi-empirical model for plasmonic predictions

Domitille Baux1, Patrick Hermet2, Stéphane Campidelli3, Jean-Louis
Bantignies1, Emmanuel Rousseau1,∗, Nicolas Izard1,∗ 1-Laboratoire Charles
Coulomb, UMR5521, CNRS-Université de Montpellier, 34095 Montpellier, France
2-ICGM, Univ. Montpellier, CNRS, ENSCM, Montpellier, France  3-Université
Paris-Saclay, CEA, CNRS, NIMBE, LICSEN, 91191, Gif-sur-Yvette, France
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Received )

###### Abstract

We investigate the intraband-transitions contribution to the optical
properties of metallic carbon nanotubes both experimentally and theoretically.
The experimental dielectric constant for metallic carbon nanotubes samples is
derived from reflectance measurements using among others Kramers-Kronig
relations. It is found in pretty good agreement with the predictions based on
ab initio calculations but departs from those based on an analytical model,
the surface conductivity model, widely used in the literature. We report new
theoretical predictions for the plasma frequency of metallic carbon nanotubes.
They are in notable agreement with experimental observations.

Plasmonics became in the last years an active field of research with the
promise of improvements in sensing[1], in particular for biological
applications[2]. The potential of highly integrated photonic devices[3], more
efficient solar cells,[4] and the enhancement of the radiative yield of
fluorophores[5] were also reported. In an effort of miniaturization, metallic
Single-Wall Carbon NanoTubes (m-SWCNT) could be envisioned as the smallest
metallic wire at the nanoscale with exceptional electrical properties. In
contrast to nanowires based on noble metals[6], individual m-SWCNT are defect-
free and present the highest-known DC-conductance[7]. Consequently, it would
be relevant to consider them as building blocks for plasmonic devices down to
the nanometric scale.

The knowledge of m-SWCNT complex dielectric constant is fundamental to
engineer practical nanotube-based optical devices. The experimental
measurements of the dielectric constant are mostly limited to unsorted samples
mixing semiconducting and metallic SWCNTs in the visible and the near-infrared
ranges[8, 9]. A wider range of frequencies can be found in ref.[10]. The
influence of the synthesis technique (HiPCO, arc-grown, CoMoCAT…) on the
optical properties of SWCNTs was also studied[11] but only reports the real
part of the refractive index, missing the imaginary part. The most complete
report to date[12] has measured the complex electrical-conductivity for
metallic and semiconducting SWCNT, but only for a narrow frequency range (0-40
THz, 0-0.16 eV) which does not allow to design m-SWCNT based optical antennas
in the visible or near-infrared range. m-SWCNT’s dielectric constant could
also be known from theoretical calculations. Most of the ab initio studies
focused on the optical properties of semiconducting SWCNT, being interested in
predicting their band-gap[13, 14, 15]. Some work concerns m-SWCNTs but the
contribution of the intraband transitions, namely the metallic behaviour of
m-SWCNTs, is missing[16]. In their work T. Movlarooy et al.[17] computes the
dielectric constants for chiralities (8,8) and (15,0) but does not give the
trend of variations of the optical properties depending on the diameter. An
analytical model, known as the surface conductivity model, has also been
developed[18] for predicting m-SWCNT’s optical properties. To be treatable at
the analytical level it relies on approximations that have not been validated
by a confrontation with experimental measurements. This paper aims at filling
these lacks by determining the complex dielectric constant over a wide
spectral range (0.05 to 5 eV) and comparing measurements with theoretical
predictions. We focus on the metallic behavior of m-SWCNT, highlighting the
Drude model contribution to the optical properties. We report new theoretical
predictions for the m-SWCNT plasma frequency, depending on the m-SWCNT
chirality, that are in agreement with the experimental measurements.

Figure 1: (a) Upper panel: AFM image of a typical m-SWCNT film. Lower panel:
mean thickness obtained by averaging on a 5 $\mu$m window. (b) Schematic of
the m-SWCNT film deposited on a thick CaF2 substrate. $d_{1}$ and $d_{2}$
denotes the thicknesses of the m-SWCNT film and substrate, respectively.
$\tilde{n}_{1}$ and $\tilde{n}_{2}$ are the complex refractive indices of the
m-SWCNT film and the CaF2 substrate. (c) Typical Raman spectra for excitation
at 633 nm (red curve) and 532 nm (green curve). The inset shows a mapping of
the G band at 1590 cm-1 over the m-SWCNT film (excitation at 633nm). (d)
Experimental reflectance of a m-SWCNT film on CaF2 substrate (red curve) and
of the reference CaF2 substrate (dot curve). The inset is a zoom on the
excitonic transitions.

HiPCO carbon nanotubes (Nano Integris) are sorted by column chromatography
following the procedure described by Tanaka et al. [19] to obtain enriched
m-SWCNT suspensions. They are filtered onto nitrocellulose membranes to form
thin films which are then transferred onto calcium fluoride $\text{CaF}_{2}$
substrates. Samples are further annealed under high vacuum ($10^{-6}$ Torr) at
250oC for 4 hours to remove solvent and impurities[11, 12]. The film’s
thickness, $d_{1}$, is determined by Atomic Force Microscopy (AFM) to be
$30\pm 5\text{~{}nm}$ (Fig.1(a)). Raman spectra (Fig.1(c)) performed at 532
and 633 nm feature a strong peak at $1590\text{~{}cm}^{-1}$, the so-called
G-band, typical of SWCNTs. The broad asymmetric Fano lineshape below the
G-band is a key characteristic of metallic carbon nanotubes[20], assessing the
sorting quality. The inset in Fig.1(c) displays a mapping of the intensity of
the G-band, underlying in conjunction with the AFM data the nanotube film
homogeneity in terms of thickness, density, and nanotube orientation. The
m-SWCNT diameter distribution is relatively broad but could be bracketed
between 0.7 and 1.2 nm (Fig S1 in supplementary material).

Broadband reflectance spectra are measured using an FTIR spectrometer (Bruker
IFS 66v/S) adapted with a combined transmission and specular reflection
apparatus from 0.02 eV up to 0.6 eV, and a UV-Vis-NIR spectrometer (Cary
Varian 5000) adapted with a VW absolute specular reflectance accessory from
0.6 eV up to 5 eV. Measurements are done at near-normal incidence in both
cases. Great care is taken to acquire the reflectance spectra which requires
an extremely fine alignment procedure to achieve a precision on the order of 1
%. Measurements are calibrated on materials whose optical properties are
known. Fig.1(d) displays the m-SWCNT film reflectance deposited on a CaF2
substrate (red curve), while the bare substrate reflectance is the dotted-gray
curve. The inset is a zoom on the 0.7 to 3 eV range displaying the carbon
nanotube excitonic transitions contribution[21, 11]. While the M11 transitions
(near 2 eV) contribute to the reflectance largely above the level due to the
substrate contribution, the S11 transitions (0.7-1.2 eV) contribute marginally
to this reflectance, underlining again sorting efficiency (see Fig. S2 in
supplementary material for additional comparison of the absorption spectra
before and after sorting).

The complex dielectric constant of the m-SWCNT layer is extracted from the
reflectance measurements using Kramers-Kronig relations[22, 23, 24] relating
the phase $\theta$ of a complex number $z=|z|e^{i\theta}$ to its modulus
$|z|$. The procedure requires the determination of the complex Fresnel
coefficient $r_{02}$ which describes light reflection by the m-SWCNT layer
deposited on a semi-infinite CaF2 substrate (Fig.1(b)). The reflectance
experiment does not measure $|r_{02}|^{2}$ directly. While thick compared to
any wavelength probes in this letter, the finite size of the substrate
($d_{2}=1$ mm) has to be taken into account, particularly in the spectral
range where CaF2 is transparent. Supplementary material S3 explains how to
relate the measured reflectance $R$ to the quantity $|r_{02}|^{2}$.

Once the modulus $|r_{02}|$ and the phase $\theta_{02}$ of $r_{02}$ are known,
the m-SWCNT refractive index could be extracted by solving:

$r_{02}[\tilde{n}_{1}(\omega)]=|r_{02}|e^{i\theta_{02}(\omega)}$ (1)

where $r_{02}[\tilde{n}_{1}(\omega)]$ is the analytical expression of the
Fresnel coefficient. The unknown to be found is the m-SWCNT complex
refractive-index, $\tilde{n}_{1}(\omega)$. The quantity on the right side of
Eq. (1) is derived from the experiment.

This experimental refractive index, and the corresponding dielectric constant,
found from this procedure are plotted in Fig.2, real and imaginary parts
respectively on the left and right side. The experimental confidence interval
is represented by a shaded area. This area includes the uncertainties on the
film thickness, the substrate refractive index, the reflectance spectral join
between the NIR / MIR range and the evaluation of $\theta$, the phase of the
Fresnel coefficient. The main source of errors comes from the uncertainty on
the film thickness. To check the accuracy of the experimentally determined
refractive index, the sample transmittance was measured in the 0.05 to 4 eV
range and compared with predictions based on this experimental refractive
index as shown in section S4 in supplementary materials. This measurement,
independent from the one used to extract the complex refractive index, shows a
good agreement with the predictions. It is a blind test to check the
consistency of the refractive index derived from the reflectance measurements.
The interband transitions above 1 eV are of particular interest. As displayed
in Fig.2(b), strong sharp peaks are observed corresponding to the m-SWCNT
absorption peaks in solution (Fig. S2). From an optical point of view, this
means the m-SWNT thin film behaves as a collection of individual nanotubes
without perceptible bundle effects. Therefore, it is of interest to compare
the experimental dielectric constant with calculations done on single
m-SWCNTs.

Figure 2: Experimental measurement of the (a) real part and (b) imaginary part
of the complex refractive index of the m-SWCNT layer (light-red shaded area,
including confidence interval). (c)(e) Real part and (d)(f) imaginary part of
the dielectric constant of the m-SWCNT layer, compared with predictions from
(c)(d) the analytical model (A.M.) and (e)(f) DFT calculations. Experimental
measurements in light-red shaded area, theoretical predictions for various
(n,m) chiralities in lines.

Particularly relevant for plasmonic applications is the cut-off energy
$E^{\star}$ below which a film made of m-SWCNT behaves like a metal. It is
defined as the energy for which the real part of the dielectric constant is
null and below which it is negative. This quantity is not clearly reported in
the literature but can be extracted from our experiments: we found
$E^{\star}=0.55\pm 0.03$ eV for the m-SWCNT thin film. In the energy range
below $E^{\star}$, the experimental dielectric constant shows an excellent
agreement with a Drude model[25] demonstrating that the m-SWCNT film indeed
behaves like a metal (Supplementary material S5).

We now aim at comparing the experimental optical properties of m-SWCNTs with
theoretical predictions. For this, SWCNT could be described as a rolled-up
graphene sheet with a diameter of $\sim$ 1 nm and a length between 500 nm and
1 $\mu$m. Due to this high aspect ratio they are anisotropic materials. Their
dielectric constant is a tensor with two different contributions: the axial
dielectric-constant $\varepsilon^{\parallel}$ along the nanotube axis and the
transverse dielectric constant $\varepsilon^{\bot}$ for the plane
perpendicular to the axis. Both intraband and interband transitions[26]
contribute to the dielectric constant. They could be expressed as a function
of the light angular frequency $\omega$ by:
$\varepsilon^{\parallel}(\omega)=\varepsilon^{\parallel}_{intra}(\omega)+\varepsilon^{\parallel}_{inter}(\omega)$
and
$\varepsilon^{\bot}(\omega)=\varepsilon^{\bot}_{intra}(\omega)+\varepsilon^{\bot}_{inter}(\omega)$.
Assuming orientational randomness (cf Fig.1(a)), the m-SWNT layer behaves as a
material with a uniform and isotropic dielectric constant
$\varepsilon(\omega)$ that reads:

$\displaystyle\varepsilon(\omega)=\frac{2}{3}\varepsilon^{\bot}(\omega)+\frac{1}{3}\varepsilon^{\parallel}(\omega)$
(2)

This quantity is extracted from reflectance measurements and plotted in
Fig.2(c)-(f). Two competing models are used to predict the complex dielectric
constant: an analytical model and an ab initio model based on Density
Functional Theory (DFT). The analytical model[18, 27] relies on the Boltzmann
kinetic equation. Based on tight-binding approximations[28], it shares many
similarities with the analytical model of graphene[29]. Importantly, this
model approximates m-SWCNT as infinitely thin wire, a pure 1D object, meaning
that the transverse dielectric constant $\varepsilon^{\bot}(\omega)$ is equal
to one. The axial component is computed with the help of the electrical
conductance $\Sigma^{\parallel}_{B}(\omega)$ along the SWCNT axis. It is
necessary to use a characteristic length-scale $t^{\star}$ to convert the
electrical conductance into electrical conductivity, a quantity related to the
dielectric constant. Following reference[30] the chosen characteristic length-
scale is the SWCNT diameter $t^{\star}=2R_{CNT}$ where $R_{CNT}$ is the
nanotube radius. Therefore, $\varepsilon^{\parallel}(\omega)$ is related to
the conductance derived from the Boltzmann formula
$\Sigma^{\parallel}_{K}(\omega)$ through the equation:

$\displaystyle\varepsilon^{\parallel}(\omega)=1+i\frac{\Sigma^{\parallel}_{B}(\omega)}{\varepsilon_{0}t^{\star}\omega}$
(3)

where $\varepsilon_{0}$ is the vacuum permittivity. The relaxation effects are
incorporated phenomenologically[31, p.154] through a damping rate
$\gamma=2\pi/\tau$, where $\tau=35\text{~{}fs}$ is the relaxation time[31,
27]. The numerical value is extracted from the fit with the Drude model
performed on the experimental results (See supplementary material S6). This
value is in agreement with the literature[27].

Fig.2(c) and (d) compare the measurements with predictions based on the
analytical model shown as solid lines. We focus the comparison on the range
[0.2,4] eV but our conclusions are valid for the whole energy range. Indeed at
low energy, the absolute value of the dielectric constant is large shrinking
the curves variations if the whole energy range [0.05,4] eV is plotted. The
experimental sample is described by a distribution of metallic nanotube
species: armchair ($n$,$n$), zigzag ($n$,0) and chiral ($n$,$m$), in the
diameter range [0.7,1.2] nm determined from the Radial Breathing Modes study
(See supplementary material S1). To compare, we chose the (9,9) and (9,0)
nanotubes, with a respective diameter of 1.24 nm and 0.71 nm, which are the
extrema that bound the diameter distribution. The chirality (6,6) whose
diameter of 0.81 nm is close to the center of the distribution, is chosen as a
representative species. This comparison does not include any chiral species
since the analytical solution for the interband transitions is not given
explicitly for this type of m-SWCNT[27, 18]. Results show that the comparison
is only qualitative. It under-estimates the real-part of the dielectric
constant for energies larger than 0.3 eV while it over-estimates the
imaginary-part of the dielectric constant in the energy range between 0.3 and
0.6 eV. This analytical model does not predict accurately the dielectric
constant of a film of m-SWCNT.

The optical properties of m-SWCNT can also be computed from a first-principles
approach based on the Density Functional Theory (DFT). (See supplementary
material S6) This method relaxes some assumptions inherent to the analytical
model. Indeed, the electron dynamics takes into account both the
$\sigma$-electrons and the $\pi$-electrons and is not limited to the first
three adjacent atoms in the hexagonal structure. Then, SWCNT are no more
assumed to be infinitely thin perpendicular to its axis (1D-object), so that
$\varepsilon^{\bot}(\omega)\neq 1$. Finally, DFT takes into account curvature
effects, which lacks in the analytical model.

In this paper, we compute both intraband and interband transitions to the
complex dielectric constant for a large range of chiral indexes. Concerning
interband transitions, both $\varepsilon^{\bot}(\omega)$ and
$\varepsilon^{\parallel}(\omega)$ are characterized by a series of absorption
lines in the high-energy range. It is worth highlighting that
$\varepsilon^{\bot}(\omega)$ differs from unity. Armchairs are the only
species to be truly metallic. Zigzag and chiral m-SWCNT are semi-metallic,
with a low energy gap arising from curvature[11, 32]. This gap arises at very
low energy ($e<0.05$ eV) below our measurement range, but the tail of their
contribution could contribute to the dielectric constant in our measurement
range.

The intraband transition contributes to the dielectric constant as a Drude-
like term[25] with a damping factor $\gamma=2\pi/\tau$ ($\tau=35$ fs for all
nanotube species) to take into account electronic relaxation.

The comparison between the experimental dielectric constant and DFT
calculations is shown in Fig.2(e) and (f). The chiral nanotube (12,3) of
diameter 1.09 nm was also included. The experimental measurements fall in the
range bounded by the two species (9,0) and (9,9). It follows quite closely the
predictions for the chirality (6,6) that is close to the center of the
diameter distribution. Overall, a very good agreement is achieved between the
DFT calculations and the experimental values.

Figure 3: (a) Cut-off energy as a function of the m-SWCNT diameter. Filled
circles are predictions from the DFT calculations. They are delimited by the
blue-shaded area. Filled squares are predictions from the analytical model.
They are delimited by the red-shaded area. Armchair (m,m), zigzag (m,0) and
chiral (2m+n=3q) species are specified by the color blue, red or black,
respectively. (b) Plasma frequency as a function of the m-SWCNT diameters
predicted from the DFT calculations (filled circles) and the analytical model
(filled squares). Results from DFT are fitted with a linear curve and results
from the analytical model follow a hyperbola.

The cut-off energy $E^{\star}$ allows a quantitative assessment of the
theoretical predictions compared to the experimental reality. Fig.3(a)
displays $E^{\star}$ as a function of m-SWCNT diameter. Experimental results
are represented as a black shaded area, with a confidence interval of $\pm$
0.03 eV. DFT calculations for various chiralities inside the experimental
diameter distribution are represented by solid circles, while the value
predicted by the analytical model are represented by solid squares.
Experimental values are clearly within the interval of DFT calculations and
clearly outside the values predicted by the analytical model, its lowest value
being 0.17 eV too high.

To understand more deeply the discrepancies between those two models, the
Drude model can be considered. Based on eq. (2), the cut-off energy
$E^{\star}$ reads (see supplementary material S7):

$\displaystyle
E^{\star}=\hbar\sqrt{\frac{\omega_{p}^{2}}{\varepsilon_{b}^{\parallel}+2\varepsilon_{b}^{\bot}}-\gamma^{2}}$
(4)

with the plasma frequency $\omega_{p}$, the loss rate $\gamma$ and the
background dielectric constant $\varepsilon_{b}^{\parallel}$ and
$\varepsilon_{b}^{\bot}$, respectively parallel and perpendicular to the
nanotube axis. Except for the loss rate that is considered to be the same in
the two models, all other quantities differ. The quantity
$\varepsilon_{b}^{\parallel}$ differs marginally in the two theories by
roughly 10-20 $\%$. This difference can not explain the large discrepancy
between the experimental value and the theory. As a consequence, we focus the
analysis on the two other quantities $\omega_{p}$ and
$\varepsilon_{b}^{\bot}$.

The plasma energy $\hbar\omega_{p}$ is plotted in Fig.3(b) for a wide
distribution of nanotube diameters, wider than experimentally available. DFT
calculations are represented by solid circles while the analytical model
values are displayed as solid squares. They strikingly differ for small
diameter nanotubes and seem to converge toward the largest diameters. The
nanotube diameter dependence differs. For the analytical model,
$\hbar\omega_{p}$ varies as the inverse of the nanotube diameter[27, 18],
while it seems rather to decrease linearly for the DFT results . This is
independent of the nanotube type: armchair, zigzag and chiral SWCNT align on
the same line. This is a new result leading to qualitative and quantitative
differences between the two theories. The second parameter that differs in the
two model’s predictions is $\varepsilon_{b}^{\bot}$. The analytical model
assumes $\varepsilon_{b}^{\bot}=1$ whereas DFT predicts that m-SWCNTs are also
polarizable objects in the transverse directions despite their small diameter
compared to the light wavelength. For energy smaller than 0.5 eV, the DFT
calculations predict a value ranging between 2 and 3, significantly different
from 1.

Last but not least, DFT calculations take into account the curvature-induced
low-energy gap existing in non-armchair SWCNT while the analytical model does
not. This effect is not captured by the Drude model. As observed in Fig.3, it
leads to higher $E^{\star}$ for zigzag and chiral compared to armchair
nanotubes, independently of the diameter effect. Consequently, it could not
explain the discrepancy between the measurement and the predictions of the
analytical model.

We reported the measurement of the complex dielectric constant of metallic
carbon nanotubes films over a wide range of energies ( 0.05 to 4 eV) and the
comparison with predictions from two theoretical models. This allowed us to
investigate intraband transitions in m-SWCNT and in particular to extract the
fundamental quantities that are the cut-off frequency $E^{\star}=0.55\pm 0.03$
eV, below which a m-SWCNT film behaves like a metal and plasma frequencies for
a large range of chiralities. We highlight a remarkable agreement with DFT
calculations and exclude prediction based on the analytical model. A similar
conclusion was reached for graphene[29]. The assumptions performed in the
analytical model[18] are too crude to describe correctly the optical
properties of m-SWCNT. Our methodology strengthens the need for ab initio
calculations to predict accurately the electronic and optical properties of
SWCNT. Based on this enhanced knowledge of m-SWCNT optical properties, we
envision the use of metallic carbon nanotubes as fundamental building blocks
for nanometric scale plasmonic devices in the near infrared range, in
particular telecom relevant optical windows.

ER and NI contributed equally to this work. They initiated and lead this
research. The samples were sorted by DB under the supervision of NI with
insights from SC. The reflectance measurements were performed by DB under the
supervision of NI and JLB. The Kramers-Kronig analysis were performed by DB
and ER. The DFT calculations were performed by PH. ER and NI wrote a draft of
the paper. All the authors discussed the results, agreed with the conclusions,
and contributed to the final version of the letter. Authors thanks M. Ramonda
from CTM Montpellier for his help with AFM measurements. Reflectance
measurements were performed on the technological platform IRRAMAN of the
University of Montpellier.

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# Study of the ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$ decay with pQCD
approach

Junfeng Sun  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  Qingxia Li
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  Jinshu Huang  College of Physics
and Electronic Engineering, Nanyang Normal University, Nanyang 473061, China
Qin Chang  Institute of Particle and Nuclear Physics, Henan Normal University,
Xinxiang 453007, China

###### Abstract

The ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$ weak decay is studied with
the perturbative QCD approach firstly. It is found that (1) main contributions
to branching ratio come from the longitudinal and parallel helicity
amplitudes, (2) branching ratio, longitudinal and parallel polarization
fractions are sensitive to the wave functions of the ${\Upsilon}(1S)$ meson,
(3) branching ratio for the ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$ decay
can reach up to $10^{-9}$, which might be promisingly measured by the future
experiments.

###### pacs:

13.25.Gv 12.39.St 14.40.Pq

##  I Introduction

The ${\Upsilon}(1S)$ meson is the ground spin-triplet $S$-wave state of
bottomonium (bound state consists of both the bottom quark $b$ and the anti-
bottom quark $\bar{b}$) with the quantum number of $I^{G}J^{PC}$ $=$
$0^{-}1^{--}$ pdg . The ${\Upsilon}(1S)$ meson lies below the kinematic open-
bottom threshold. It is generally believed that the ${\Upsilon}(1S)$ meson
decays mainly through the annihilation of the $b\bar{b}$ pairs into three
gluons, one photon, two gluons plus one photon, with branching ratios pdg  of
some $(81.7{\pm}0.7)\%$, $(3+R){\cal B}r_{{\ell}{\ell}}$, $(2.2{\pm}0.6)\%$,
respectively, where $R$ ${\simeq}$ $10/3$ is the ratio of the hadron
production rate to the lepton ${\mu}^{+}{\mu}^{-}$ pair production rate at the
energy scale of $m_{{\Upsilon}(1S)}$, and ${\cal B}r_{{\ell}{\ell}}$ is the
branching ratio for the pure leptonic ${\Upsilon}(1S)$ ${\to}$
${\ell}^{+}{\ell}^{-}$ decay. One of the prominent features is its narrow
width, ${\Gamma}_{{\Upsilon}(1S)}$ $=$ $54.02{\pm}1.25$ keV pdg , while the
${\Upsilon}(1S)$ meson is about ten times heavier than the nucleon,
$m_{{\Upsilon}(1S)}$ $=$ $9460.30{\pm}0.26$ MeV pdg . This fact can be
explained by the following argument. The hadronic decay ${\Upsilon}(1S)$
${\to}$ $ggg$ is suppressed by the phenomenological Okubo-Zweig-Iizuka (OZI)
rule o ; z ; i . The branching ratio ${\cal B}r_{{\ell}{\ell}}$ is
proportional (a) to the square of the electric charge of the bottom quark,
$e_{b}^{2}$ where $e_{b}$ $=$ $-1/3$ in the unit of ${|}e{|}$; (b) to the
square of the electromagnetic coupling constant, ${\alpha}^{2}$ where
${\alpha}$ ${\approx}$ $1/128$; and (c) to the energy dependence of the photon
propagator, $1/m_{{\Upsilon}(1S)}^{2}$ annu1993 .

Besides the above-mentioned strong, electromagnetic and radiative decay
mechanisms, the ${\Upsilon}(1S)$ meson can also decay via the weak interaction
within the standard model. As it is well known, over $10^{8}$ ${\Upsilon}(1S)$
data samples have been accumulated at Belle epjc74 . More and more upsilon
data samples are hopefully expected at the running LHC and the forthcoming
SuperKEKB. Although branching ratio for the ${\Upsilon}(1S)$ weak decay is
tiny, about $2/{\tau}_{B}{\Gamma}_{{\Upsilon}(1S)}$ ${\sim}$ ${\cal
O}(10^{-8})$ where ${\tau}_{B}$ is the lifetime of the $B_{u,d}$ meson, there
seems to exist a realistic possibility to search for the ${\Upsilon}(1S)$ weak
decay at future experiments. In this paper, we will study the ${\Upsilon}(1S)$
${\to}$ $B_{c}D_{s}^{\ast}$ weak decay with the perturbative QCD (pQCD)
approach pqcd1 ; pqcd2 ; pqcd3 .

Experimentally, branching ratios for the two-body leptonic ${\Upsilon}(1S)$
decays, some OZI-suppressed hadronic ${\Upsilon}(1S)$ decays and radiative
decays have been measured, but there is still no measurement reprot on the
magnetic dipole transition decay ${\Upsilon}(1S)$ ${\to}$ ${\gamma}{\eta}_{b}$
and weak decays for the moment pdg . The ${\Upsilon}(1S)$ weak decay posses a
unique structure due to the Cabibbo-Kobayashi-Maskawa (CKM) matrix properties
which predicts the channels with one $B_{c}^{(\ast)}$ meson are dominant. The
signals for the ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$ weak decay
should, in principle, be easily distinguished from possibly intricate
background, due to the facts that the back-to-back final states with opposite
electric charges have definite momentum and energy in the rest frame of the
${\Upsilon}(1S)$ meson. The identification of a single flavored either $B_{c}$
or $D_{s}^{\ast}$ meson could be used as an effective selection criterion.
Moreover, the radiative decay of the $D_{s}^{\ast}$ meson can provide a useful
extra signal and a powerful constraint. Of course, any evidences of an
abnormally large branching ratio for the ${\Upsilon}(1S)$ weak decay might be
a hint of new physics.

Theoretically, in recent years, many attractive methods have been fully
developed, such as the pQCD approach pqcd1 ; pqcd2 ; pqcd3 , the QCD
factorization (QCDF) approach qcdf1 ; qcdf2 ; qcdf3 , soft and collinear
effective theory scet1 ; scet2 ; scet3 ; scet4 , and widely applied to
accommodate measurements on the $B$ meson weak decays. The ${\Upsilon}(1S)$
weak decays permit one to cross check parameters obtained from the $B$ meson
decay, and to test various phenomenological models. The ${\Upsilon}(1S)$ weak
decays into final states containing one $B_{c}$ meson are favorable processes
due to the CKM factor $V_{cb}$, which also provide an additional occasion to
scrutinize the underlying structure of doubly-heavy hadrons, and to improve
our understanding on the short- and long-distance contributions in heavy quark
weak decay. The semileptonic decays ${\Upsilon}(1S)$ ${\to}$
$B_{c}{\ell}^{-}\bar{\nu}_{\ell}$ (${\ell}$ $=$ $e$, ${\mu}$, ${\tau}$) have
been studied based on the Bauer-Stech-Wirbel model rohit . The two-body
nonleptonic decays ${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ ($M$ $=$ ${\pi}$,
$K^{(\ast)}$, ${\rho}$) have been investigated recently by employing the
factorization scheme, such as the naive factorization approximation rohit ;
ijmpa1999 , the QCD-improved QCDF formulation adv-sun ; jpg-sun  and the pQCD
approach prd92 ; plb751 . The ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$
weak decay is favored by color and the CKM factor ${|}V_{cb}V_{cs}^{\ast}{|}$,
so it should, in principle, have relatively large branching ratio among the
${\Upsilon}(1S)$ weak decays. However, there is still no theoretical study
devoted to the ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$ weak decay now. In
this paper, we will investigate the ${\Upsilon}(1S)$ ${\to}$
$B_{c}D_{s}^{\ast}$ decay with the pQCD approach to offer a ready reference
for the future experiments.

This paper is organized as follows. The theoretical framework and the
amplitudes for the ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$ decay are
presented in section II. The numerical results and discussion are given in
section III. The last section is a summary.

##  II theoretical framework

###  II.1 The effective Hamiltonian

The effective Hamiltonian responsible for the ${\Upsilon}(1S)$ ${\to}$
$B_{c}D_{s}^{\ast}$ weak decay is 9512380

${\cal H}_{\rm
eff}\,=\,\frac{G_{F}}{\sqrt{2}}\,\Big{\\{}V_{cb}V_{cs}^{\ast}\sum\limits_{i=1}^{2}C_{i}({\mu})\,Q_{i}({\mu})-V_{tb}V_{ts}^{\ast}\sum\limits_{j=3}^{10}C_{j}({\mu})\,Q_{j}({\mu})\Big{\\}}+{\rm
H.c.},$ (1)

where the Fermi coupling constant $G_{F}$ ${\simeq}$
$1.166{\times}10^{-5}\,{\rm GeV}^{-2}$ pdg ; with the Wolfenstein
parameterization, the CKM factors are written as pdg ,

$\displaystyle V_{cb}V_{cs}^{\ast}$ $\displaystyle=$
$\displaystyle+A{\lambda}^{2}-\frac{1}{2}A{\lambda}^{4}-\frac{1}{8}A{\lambda}^{6}(1+4A^{2})+{\cal
O}({\lambda}^{7}),$ (2) $\displaystyle V_{tb}V_{ts}^{\ast}$ $\displaystyle=$
$\displaystyle-V_{cb}V_{cs}^{\ast}-A{\lambda}^{4}({\rho}-i{\eta})+{\cal
O}({\lambda}^{7}).$ (3)

The local tree operators $Q_{1,2}$ and penguin operators $Q_{3,{\cdots},10}$
are defined below.

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle[\bar{c}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\alpha}][\bar{s}_{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})c_{\beta}],$
(4) $\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle[\bar{c}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\beta}][\bar{s}_{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})c_{\alpha}],$
(5) $\displaystyle Q_{3}$ $\displaystyle=$
$\displaystyle\sum\limits_{q}[\bar{s}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\alpha}][\bar{q}_{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})q_{\beta}],$
(6) $\displaystyle Q_{4}$ $\displaystyle=$
$\displaystyle\sum\limits_{q}[\bar{s}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\beta}][\bar{q}_{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})q_{\alpha}],$
(7) $\displaystyle Q_{5}$ $\displaystyle=$
$\displaystyle\sum\limits_{q}[\bar{s}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\alpha}][\bar{q}_{\beta}{\gamma}^{\mu}(1+{\gamma}_{5})q_{\beta}],$
(8) $\displaystyle Q_{6}$ $\displaystyle=$
$\displaystyle\sum\limits_{q}[\bar{s}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\beta}][\bar{q}_{\beta}{\gamma}^{\mu}(1+{\gamma}_{5})q_{\alpha}],$
(9) $\displaystyle Q_{7}$ $\displaystyle=$
$\displaystyle\sum\limits_{q}\frac{3}{2}e_{q}\,[\bar{s}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\alpha}][\bar{q}_{\beta}{\gamma}^{\mu}(1+{\gamma}_{5})q_{\beta}],$
(10) $\displaystyle Q_{8}$ $\displaystyle=$
$\displaystyle\sum\limits_{q}\frac{3}{2}e_{q}\,[\bar{s}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\beta}][\bar{q}_{\beta}{\gamma}^{\mu}(1+{\gamma}_{5})q_{\alpha}],$
(11) $\displaystyle Q_{9}$ $\displaystyle=$
$\displaystyle\sum\limits_{q}\frac{3}{2}e_{q}\,[\bar{s}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\alpha}][\bar{q}_{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})q_{\beta}],$
(12) $\displaystyle Q_{10}$ $\displaystyle=$
$\displaystyle\sum\limits_{q}\frac{3}{2}e_{q}\,[\bar{s}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\beta}][\bar{q}_{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})q_{\alpha}],$
(13)

where ${\alpha}$ and ${\beta}$ are color indices; $q$ denotes all the active
quarks at the scale of ${\mu}$ ${\sim}$ ${\cal O}(m_{b})$, i.e., $q$ $=$ $u$,
$d$, $s$, $c$, $b$; and $e_{q}$ is the electric charge of the $q$ quark in the
unit of ${|}e{|}$.

The Wilson coefficients $C_{i}(\mu)$ summarize the physical contributions
above the scale of ${\mu}$, and could be reliably calculated with the
renormalization group improved perturbation theory. The physical contributions
below the scale of ${\mu}$ are included in the hadronic matrix elements (HME)
where the local operators sandwiched between initial and final hadron states.
To obtain the decay amplitudes, the remaining work is to calculate HME
properly.

###  II.2 Hadronic matrix elements

Phenomenologically, combining the $k_{T}$ factorization theorem npb366  with
the collinear factorization hypothesis, and using the Lepage-Brodsky approach
for exclusive processes prd22 , HME can be written as the convolution of
universal wave functions reflecting the nonperturbative contributions with
hard scattering subamplitudes containing the perturbative contributions within
the pQCD framework, where the transverse momentum of valence quarks is
retained and the Sudakov factor is introduced, in order to regulate the
endpoint singularities and provide a naturally dynamical cutoff on the
nonperturbative contributions pqcd1 ; pqcd2 ; pqcd3 . Generally, the decay
amplitude can be separated into three parts: the Wilson coefficients $C_{i}$
incorporating the hard contributions above the typical scale of $t$, the
process-dependent scattering amplitudes $T$ accounting for the heavy quark
decay, and the universal wave functions ${\Phi}$ including the soft and long-
distance contributions, i.e.,

${\int}dk\,C_{i}(t)\,T(t,k)\,{\Phi}(k)e^{-S},$ (14)

where $k$ is the momentum of valence quarks, and $e^{-S}$ is the Sudakov
factor.

###  II.3 Kinematic variables

The light cone kinematic variables in the ${\Upsilon}(1S)$ rest frame are
defined as follows.

$p_{{\Upsilon}(1S)}\,=\,p_{1}\,=\,\frac{m_{1}}{\sqrt{2}}(1,1,0),$ (15)
$p_{B_{c}}\,=\,p_{2}\,=\,(p_{2}^{+},p_{2}^{-},0),$ (16)
$p_{D_{s}^{\ast}}\,=\,p_{3}\,=\,(p_{3}^{-},p_{3}^{+},0),$ (17)
$k_{i}\,=\,x_{i}\,p_{i}+(0,0,\vec{k}_{iT}),$ (18)
${\epsilon}_{i}^{\parallel}\,=\,\frac{p_{i}}{m_{i}}-\frac{m_{i}}{p_{i}{\cdot}n_{+}}n_{+},$
(19) ${\epsilon}_{i}^{\perp}\,=\,(0,0,\vec{1}),$ (20) $n_{+}=(1,0,0),$ (21)
$p_{i}^{\pm}\,=\,(E_{i}\,{\pm}\,p)/\sqrt{2},$ (22)
$s\,=\,2\,p_{2}{\cdot}p_{3},$ (23) $t\,=\,2\,p_{1}{\cdot}p_{2}\,=\
2\,m_{1}\,E_{2},$ (24) $u\,=\,2\,p_{1}{\cdot}p_{3}\,=\ 2\,m_{1}\,E_{3},$ (25)
$p=\frac{\sqrt{[m_{1}^{2}-(m_{2}+m_{3})^{2}]\,[m_{1}^{2}-(m_{2}-m_{3})^{2}]}}{2\,m_{1}},$
(26)

where $x_{i}$ and $\vec{k}_{iT}$ are the longitudinal momentum fraction and
transverse momentum of the valence quark, respectively;
${\epsilon}_{i}^{\parallel}$ and ${\epsilon}_{i}^{\perp}$ are the longitudinal
and transverse polarization vectors, respectively, satisfying with the
relations ${\epsilon}_{i}^{2}$ $=$ $-1$ and ${\epsilon}_{i}{\cdot}p_{i}$ $=$
$0$; the subscript $i$ $=$ $1$, $2$, $3$ on variables $p_{i}$, $E_{i}$,
$m_{i}$, and ${\epsilon}_{i}^{{\parallel},{\perp}}$ correspond to the
${\Upsilon}(1S)$, $B_{c}$, $D_{s}^{\ast}$ mesons, respectively; $n_{+}$ is the
null vector; $s$, $t$ and $u$ are the Lorentz-invariant variables; $p$ is the
common momentum of final states. The notation of momentum is displayed in
Fig.2(a).

###  II.4 Wave functions

With the notation in jhep0605 ; jhep0703 , the definitions of the diquark
operator HME are

${\langle}0{|}b_{i}(z)\bar{b}_{j}(0){|}{\Upsilon}(p_{1},{\epsilon}_{1}^{{\parallel}}){\rangle}\,=\,\frac{f_{{\Upsilon}(1S)}}{4}{\int}dk_{1}\,e^{-ik_{1}{\cdot}z}\Big{\\{}\\!\\!\not{\epsilon}_{1}^{{\parallel}}\Big{[}m_{1}\,{\phi}_{\Upsilon}^{v}(k_{1})-\\!\\!\not{p}_{1}\,{\phi}_{\Upsilon}^{t}(k_{1})\Big{]}\Big{\\}}_{ji},$
(27)
${\langle}0{|}b_{i}(z)\bar{b}_{j}(0){|}{\Upsilon}(p_{1},{\epsilon}_{1}^{{\perp}}){\rangle}\,=\,\frac{f_{{\Upsilon}(1S)}}{4}{\int}dk_{1}\,e^{-ik_{1}{\cdot}z}\Big{\\{}\\!\\!\not{\epsilon}_{1}^{{\perp}}\Big{[}m_{1}\,{\phi}_{\Upsilon}^{V}(k_{1})-\\!\\!\not{p}_{1}\,{\phi}_{\Upsilon}^{T}(k_{1})\Big{]}\Big{\\}}_{ji},$
(28)
${\langle}B_{c}(p_{2}){|}\bar{c}_{i}(z)b_{j}(0){|}0{\rangle}\,=\,\frac{if_{B_{c}}}{4}{\int}dk_{2}\,e^{ik_{2}{\cdot}z}\,\Big{\\{}{\gamma}_{5}\Big{[}\\!\\!\not{p}_{2}+m_{2}\Big{]}{\phi}_{B_{c}}(k_{2})\Big{\\}}_{ji},$
(29)
${\langle}D_{s}^{{\ast}}(p_{3},{\epsilon}_{3}^{{\parallel}}){|}c_{i}(0)\bar{s}_{j}(z){|}0{\rangle}\
=\
\frac{f_{D_{s}^{\ast}}}{4}{\int}_{0}^{1}dk_{3}\,e^{ik_{3}{\cdot}z}\Big{\\{}\\!\not{\epsilon}_{3}^{{\parallel}}\Big{[}m_{3}\,{\phi}_{D_{s}^{\ast}}^{v}(k_{3})+\\!\not{p}_{3}\,{\phi}_{D_{s}^{\ast}}^{t}(k_{3})\Big{]}\Big{\\}}_{ji},$
(30)
${\langle}D_{s}^{{\ast}}(p_{3},{\epsilon}_{3}^{{\perp}}){|}c_{i}(0)\bar{s}_{j}(z){|}0{\rangle}\
=\
\frac{f_{D_{s}^{\ast}}}{4}{\int}_{0}^{1}dk_{3}\,e^{ik_{3}{\cdot}z}\Big{\\{}\\!\not{\epsilon}_{3}^{{\perp}}\Big{[}m_{3}\,{\phi}_{D_{s}^{\ast}}^{V}(k_{3})+\\!\not{p}_{3}\,{\phi}_{D_{s}^{\ast}}^{T}(k_{3})\Big{]}\Big{\\}}_{ji},$
(31)

where $f_{{\Upsilon}(1S)}$, $f_{B_{c}}$, $f_{D_{s}^{\ast}}$ are decay
constants.

Because of the mass relations, $m_{{\Upsilon}(1S)}$ ${\simeq}$ $2m_{b}$,
$m_{B_{c}}$ ${\simeq}$ $m_{b}$ $+$ $m_{c}$, and $m_{D_{s}^{\ast}}$ ${\simeq}$
$m_{c}$ $+$ $m_{s}$ (see Table 1), it might assume that the motion of the
valence quarks in all participating mesons is nearly nonrelativistic. The wave
functions of the ${\Upsilon}(1S)$, $B_{c}$, $D_{s}^{\ast}$ mesons could be
approximately described with the nonrelativistic quantum chromodynamics prd46
; prd51 ; rmp77  and Schrödinger equation. Combining the wave functions of a
nonrelativistic isotropic harmonic oscillator potential with their asymptotic
forms jhep0605 ; jhep0703 , we obtain prd92 ,

${\phi}_{\Upsilon}^{v}(x)={\phi}_{\Upsilon}^{T}(x)=A\,x\bar{x}\,{\exp}\Big{\\{}-\frac{m_{b}^{2}}{8\,{\beta}_{1}^{2}\,x\,\bar{x}}\Big{\\}},$
(32)
${\phi}_{\Upsilon}^{t}(x)=B\,t^{2}\,{\exp}\Big{\\{}-\frac{m_{b}^{2}}{8\,{\beta}_{1}^{2}\,x\,\bar{x}}\Big{\\}},$
(33)
${\phi}_{\Upsilon}^{V}(x)=C\,(1+t^{2})\,{\exp}\Big{\\{}-\frac{m_{b}^{2}}{8\,{\beta}_{1}^{2}\,x\,\bar{x}}\Big{\\}},$
(34)
${\phi}_{B_{c}}(x)=D\,x\bar{x}\,{\exp}\Big{\\{}-\frac{\bar{x}\,m_{c}^{2}+x\,m_{b}^{2}}{8\,{\beta}_{2}^{2}\,x\,\bar{x}}\Big{\\}},$
(35)
${\phi}_{D_{s}^{\ast}}^{v}(x)={\phi}_{D_{s}^{\ast}}^{T}(x)=E\,x\bar{x}\,{\exp}\Big{\\{}-\frac{\bar{x}\,m_{s}^{2}+x\,m_{c}^{2}}{8\,{\beta}_{3}^{2}\,x\,\bar{x}}\Big{\\}},$
(36)
${\phi}_{D_{s}^{\ast}}^{t}(x)=F\,t^{2}\,{\exp}\Big{\\{}-\frac{\bar{x}\,m_{s}^{2}+x\,m_{c}^{2}}{8\,{\beta}_{3}^{2}\,x\,\bar{x}}\Big{\\}},$
(37)
${\phi}_{D_{s}^{\ast}}^{V}(x)=G\,(1+t^{2})\,{\exp}\Big{\\{}-\frac{\bar{x}\,m_{s}^{2}+x\,m_{c}^{2}}{8\,{\beta}_{3}^{2}\,x\,\bar{x}}\Big{\\}},$
(38)

where $\bar{x}$ $=$ $1$ $-$ $x$; $t$ $=$ $x$ $-$ $\bar{x}$; ${\beta}_{i}$ $=$
${\xi}_{i}\,{\alpha}_{s}({\xi}_{i})$ with ${\xi}_{i}$ $=$ $m_{i}/2$ based on
the NRQCD power counting rules prd46 ; ${\alpha}_{s}$ is the QCD coupling
constant; the exponential function represents the $k_{T}$ distribution;
parameters $A$, $B$, $C$, $D$, $E$, $F$, $G$ are the normalization
coefficients satisfying the conditions

${\int}_{0}^{1}dx\,{\phi}_{B_{c}}(x)=1,$ (39)
${\int}_{0}^{1}dx\,{\phi}_{\Upsilon}^{i}(x)=1,\quad\text{for}\ \ i=v,t,V,T,$
(40) ${\int}_{0}^{1}dx\,{\phi}_{D_{s}^{\ast}}^{i}(x)=1,\quad\text{for}\ \
i=v,t,V,T.$ (41)

Figure 1: The distribution amplitudes for the ${\Upsilon}(1S)$, $B_{c}$,
$D_{s}^{\ast}$ mesons in (a), (b), (c), respectively.

The shape lines of the normalized distribution amplitudes for the
${\Upsilon}(1S)$, $B_{c}$, $D_{s}^{\ast}$ mesons are showed in Fig.1. It is
clearly seen that (1) distribution amplitudes for the ${\Upsilon}(1S)$,
$B_{c}$, $D_{s}^{\ast}$ mesons shrink rapidly to zero at the endpoint $x$
${\to}$ $0$, $1$ due to the suppression from the exponential functions, (2)
although the nonrelativistic model of wave functions is crude, distribution
amplitudes Eq.(32)-Eq.(38) can reflect, at least to some extent, the feature
that the valence quarks share momentum fractions according to their masses.

###  II.5 Decay amplitudes

The Feynman diagrams for the ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$ weak
decay are shown in Fig.2. There are two types. One is the emission topology,
and the other is annihilation topology. Each type is further subdivided into
factorizable diagram where gluon attaches to quarks in the same meson, and
nonfactorizable diagrams where gluon connects to quarks between different
mesons.

Figure 2: Feynman diagrams for the ${\Upsilon}(1S)$ ${\to}$
$B_{c}D_{s}^{\ast}$ decay with the pQCD approach, where (a,b) are the
factorizable emission diagrams, (c,d) are the nonfactorizable emission
diagrams, (e,f) are the nonfactorizable annihilation diagrams, and (g,h) are
the factorizable annihilation diagrams.

The amplitude for the ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$ weak decay
is defined as below prd66 ,

${\cal A}({\Upsilon}(1S){\to}B_{c}D_{s}^{\ast})\ =\ {\cal
A}_{L}({\epsilon}_{1}^{{\parallel}},{\epsilon}_{3}^{{\parallel}})+{\cal
A}_{N}({\epsilon}_{1}^{{\perp}},{\epsilon}_{3}^{{\perp}})+i\,{\cal
A}_{T}\,{\varepsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,{\epsilon}_{1}^{{\mu}}\,{\epsilon}_{3}^{{\nu}}\,p_{1}^{\alpha}\,p_{3}^{\beta},$
(42)

which is conventionally written as the helicity amplitudes prd66 ,

${\cal A}_{0}\ =\ -C_{\cal A}\,{\cal
A}_{L}({\epsilon}_{1}^{{\parallel}},{\epsilon}_{3}^{{\parallel}}),$ (43)
${\cal A}_{\parallel}\ =\ \sqrt{2}\,C_{\cal A}{\cal
A}_{N}({\epsilon}_{1}^{{\perp}},{\epsilon}_{3}^{{\perp}}),$ (44) ${\cal
A}_{\perp}\ =\ \sqrt{2}\,C_{\cal A}\,m_{1}\,p\,{\cal A}_{T},$ (45) $C_{\cal
A}\ =\
i\frac{G_{F}}{\sqrt{2}}\,\frac{C_{F}}{N_{c}}\,{\pi}\,f_{{\Upsilon}(1S)}\,f_{B_{c}}\,f_{D_{s}^{\ast}},$
(46)

and the polarization amplitude ${\cal A}_{j}$ is written as

$\displaystyle{\cal A}_{j}$ $\displaystyle=$ $\displaystyle
V_{cb}V_{cs}^{\ast}\,\Big{\\{}\Big{(}{\cal A}_{a,j}^{LL}+{\cal
A}_{b,j}^{LL}\Big{)}a_{1}+\Big{(}{\cal A}_{c,j}^{LL}+{\cal
A}_{d,j}^{LL}\Big{)}C_{2}\Big{\\}}$ (47) $\displaystyle-$ $\displaystyle
V_{tb}V_{tq}^{\ast}\,\Big{\\{}\Big{(}{\cal A}_{a,j}^{LL}+{\cal
A}_{b,j}^{LL}\Big{)}\,(a_{4}+a_{10})+\Big{(}{\cal A}_{c,j}^{LL}+{\cal
A}_{d,j}^{LL}\Big{)}\,(C_{3}+C_{9})$ $\displaystyle\quad+\Big{(}{\cal
A}_{e,j}^{LL}+{\cal
A}_{f,j}^{LL}\Big{)}\,(C_{3}+C_{4}-\frac{1}{2}C_{9}-\frac{1}{2}C_{10})$
$\displaystyle\quad+\Big{(}{\cal A}_{g,j}^{LL}+{\cal
A}_{h,j}^{LL}\Big{)}\,(a_{3}+a_{4}-\frac{1}{2}a_{9}-\frac{1}{2}a_{10})$
$\displaystyle\quad+\Big{(}{\cal A}_{e,j}^{LR}+{\cal
A}_{f,j}^{LR}\Big{)}\,(C_{6}-\frac{1}{2}C_{8})+\Big{(}{\cal
A}_{g,j}^{LR}+{\cal A}_{h,j}^{LR}\Big{)}\,(a_{5}-\frac{1}{2}a_{7})$
$\displaystyle\quad+\Big{(}{\cal A}_{c,j}^{SP}+{\cal
A}_{d,j}^{SP}\Big{)}\,(C_{5}+C_{7})+\Big{(}{\cal A}_{e,j}^{SP}+{\cal
A}_{f,j}^{SP}\Big{)}\,(C_{5}-\frac{1}{2}C_{7})\Big{\\}},$

where $C_{F}$ $=$ $4/3$ and the color number $N_{c}$ $=$ $3$; for the building
blocks ${\cal A}_{i,j}^{k}$, the first subscript $i$ corresponds to the
indices of Fig.2; the second subscript $j$ $=$ $L$, $N$, $T$ denotes to three
different helicity amplitudes; the superscript $k$ refers to three possible
Dirac structures ${\Gamma}_{1}{\otimes}{\Gamma}_{2}$ of the four-quark
operator $(\bar{q}_{1}{\Gamma}_{1}q_{2})(\bar{q}_{1}{\Gamma}_{2}q_{2})$,
namely $k$ $=$ $LL$ for $(V-A){\otimes}(V-A)$, $k$ $=$ $LR$ for
$(V-A){\otimes}(V+A)$, and $k$ $=$ $SP$ for $-2(S-P){\otimes}(S+P)$; The
explicit expressions of building blocks ${\cal A}_{i,j}^{k}$ are collected in
Appendix A. The parameter $a_{i}$ is defined as follows.

$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.$ (48)

##  III Numerical results and discussion

In the rest frame of the ${\Upsilon}(1S)$ meson, branching ratio (${\cal
B}r$), polarization fractions ($f_{0,{\parallel},{\perp}}$) and relative
phases (${\phi}_{{\parallel},{\perp}}$) between helicity amplitudes (${\cal
A}_{0,{\parallel},{\perp}}$) for the ${\Upsilon}(1S)$ ${\to}$
$B_{c}D_{s}^{\ast}$ weak decay are defined as

${\cal B}r\ =\
\frac{1}{12{\pi}}\,\frac{p}{m_{{\Upsilon}(1S)}^{2}{\Gamma}_{{\Upsilon}(1S)}}\,\Big{\\{}{|}{\cal
A}_{0}{|}^{2}+{|}{\cal A}_{\parallel}{|}^{2}+{|}{\cal
A}_{\perp}{|}^{2}\Big{\\}},$ (49) $f_{0,{\parallel},{\perp}}\ =\
\frac{{|}{\cal A}_{0,{\parallel},{\perp}}{|}^{2}}{{|}{\cal
A}_{0}{|}^{2}+{|}{\cal A}_{\parallel}{|}^{2}+{|}{\cal A}_{\perp}{|}^{2}},$
(50) ${\phi}_{{\parallel},{\perp}}\ =\ {\arg}({\cal
A}_{{\parallel},{\perp}}/{\cal A}_{0}).$ (51)

Table 1: The numerical values of input parameters. The Wolfenstein
parameters111The relation between parameters (${\rho}$, ${\eta}$) and
($\bar{\rho}$, $\bar{\eta}$) is pdg : $({\rho}+i{\eta})$ $=$
$\displaystyle\frac{\sqrt{1-A^{2}{\lambda}^{4}}(\bar{\rho}+i\bar{\eta})}{\sqrt{1-{\lambda}^{2}}[1-A^{2}{\lambda}^{4}(\bar{\rho}+i\bar{\eta})]}$.
---
$A$ $=$ $0.814^{+0.023}_{-0.024}$ pdg , |  ${\lambda}$ $=$ $0.22537{\pm}0.00061$ pdg ,
$\bar{\rho}$ $=$ $0.117{\pm}0.021$ pdg , |  $\bar{\eta}$ $=$ $0.353{\pm}0.013$ pdg ,
Mass, width and decay constant
$m_{b}$ $=$ $4.78{\pm}0.06$ GeV pdg , |  $m_{c}$ $=$ $1.67{\pm}0.07$ GeV pdg ,
$m_{s}$ ${\simeq}$ $510$ MeV book , |  ${\Gamma}_{{\Upsilon}(1S)}$ $=$ $54.02{\pm}1.25$ keV pdg ,
$m_{{\Upsilon}(1S)}$ $=$ $9460.30{\pm}0.26$ MeV pdg  |  $f_{{\Upsilon}(1S)}$ $=$ $676.4{\pm}10.7$ MeV prd92
$m_{B_{c}}$ $=$ $6275.6{\pm}1.1$ MeV pdg , |  $f_{B_{c}}$ $=$ $427{\pm}6$ MeV fbc ,
$m_{D_{s}^{\ast}}$ $=$ $2112.1{\pm}0.4$ MeV pdg , |  $f_{D_{s}^{\ast}}$ $=$ $274{\pm}6$ MeV fds .

The input parameters are listed in Table 1. If not specified explicitly, we
will take their central values as the default inputs. Our numerical results
are collected in Table. 2, where the first uncertainty comes from the CKM
parameters, the second uncertainty is from the choice of the typical scale
$(1{\pm}0.1)t_{i}$ and $t_{i}$ is given in Eqs.(108-111); the third
uncertainty is from the variation of mass $m_{b}$ and $m_{c}$. The following
are some comments.

Table 2: Branching ratio, polarization fractions, and relative phases for different cases, where we use the wave functions of Eqs.(32-34) and Eqs.(36-38) for the ${\Upsilon}(1S)$ and $D_{s}^{\ast}$ meson, respectively in case A; we use the same wave functions for both the transversal and longitudinal polarization ${\Upsilon}(1S)$ meson in case B, i.e., ${\phi}_{\Upsilon}^{v,t,V,T}$ = Eq.(32); case C for ${\phi}_{D_{s}^{\ast}}^{v,t,V,T}$ = Eq.(36); case D for ${\phi}_{\Upsilon}^{v,t,V,T}$ = Eq.(32) and ${\phi}_{D_{s}^{\ast}}^{v,t,V,T}$ = Eq.(36). | case A | case B | case C | case D
---|---|---|---|---
$10^{9}{\times}{\cal B}r$ | $1.68^{+0.12+0.24+0.10}_{-0.11-0.11-0.22}$ | $2.10^{+0.14+0.29+0.09}_{-0.14-0.14-0.25}$ | $1.63^{+0.11+0.18+0.10}_{-0.11-0.08-0.20}$ | $2.04^{+0.14+0.22+0.08}_{-0.13-0.10-0.24}$
$10^{2}{\times}f_{0}$ | $35.6^{+0.0+0.2+1.3}_{-0.0-0.1-1.4}$ | $47.8^{+0.0+0.1+0.2}_{-0.0-0.1-0.1}$ | $35.8^{+0.0+0.1+1.3}_{-0.0-0.0-1.4}$ | $48.0^{+0.0+0.1+0.2}_{-0.0-0.0-0.1}$
$10^{2}{\times}f_{\parallel}$ | $56.5^{+0.0+0.1+0.9}_{-0.0-0.2-0.9}$ | $46.0^{+0.0+0.0+0.1}_{-0.0-0.1-0.5}$ | $56.3^{+0.0+0.0+0.9}_{-0.0-0.1-0.9}$ | $45.8^{+0.0+0.0+0.1}_{-0.0-0.1-0.5}$
$10^{2}{\times}f_{\perp}$ | $7.9^{+0.0+0.0+0.5}_{-0.0-0.1-0.4}$ | $6.2^{+0.0+0.0+0.2}_{-0.0-0.0-0.2}$ | $7.8^{+0.0+0.0+0.5}_{-0.0-0.0-0.4}$ | $6.2^{+0.0+0.0+0.2}_{-0.0-0.0-0.2}$
${\phi}_{\parallel}$ |  ${\simeq}$ $1.9^{\circ}$ |  ${\simeq}$ $0.4^{\circ}$ |  ${\simeq}$ $2.1^{\circ}$ |  ${\simeq}$ $0.4^{\circ}$
${\phi}_{\perp}$ |  ${\simeq}$ $-174.0^{\circ}$ |  ${\simeq}$ $-175.6^{\circ}$ |  ${\simeq}$ $-173.7^{\circ}$ |  ${\simeq}$ $-175.2^{\circ}$

(1) Branching ratio for the ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$ decay
can reach up to ${\cal O}(10^{-9})$ with the pQCD approach, which might be
promisingly measurable at the running LHC and forthcoming SuperKEKB. For
example, the ${\Upsilon}(1S)$ production cross section in p-Pb collision is
about a few ${\mu}b$ at the LHCb jhep1407  and ALICE plb740  detectors. So,
more than $10^{12}$ ${\Upsilon}(1S)$ data samples could be in principle
available per $ab^{-1}$ data collected by the LHCb and ALICE detectors,
corresponding to a few thousands of the ${\Upsilon}(1S)$ ${\to}$
$B_{c}D_{s}^{\ast}$ events.

(2) The contributions to branching ratio mainly come from the longitudinal and
parallel polarization helicity amplitudes, $f_{0}$ $+$ $f_{{\parallel}}$
${\gtrsim}$ 90%, while the perpendicular polarization fraction $f_{{\perp}}$
is generally less than 10%. From Table.2, it is seen that the polarization
fractions are not sensitive to the input parameters. However, we find that
branching ratio, polarization fractions $f_{0}$ and $f_{{\parallel}}$, are
sensitive to the ${\Upsilon}(1S)$ wave functions (see Table.2). This might
imply that the polarization measurement on the ${\Upsilon}(1S)$ ${\to}$
$B_{c}D_{s}^{\ast}$ decay would provide some information on the wave functions
and thus the interquark binding forces responsible for the ${\Upsilon}(1S)$
meson.

(3) The relative phase ${\phi}_{{\parallel}}$ is very small. This is
consistent with prediction of the QCD factorization approach qcdf1 ; qcdf2 ,
where the strong phase arising from nonfactorizable contributions is
suppressed by color and ${\alpha}_{s}$ for the $a_{1}$-dominated processes.
The relative phases, if they could be determined experimentally, will improve
our understanding on nonfactorizable contributions, factorization mechanism,
and the strong dynamics at different energy scales.

Figure 3: The contributions to the branching ratio from different region of
${\alpha}_{s}/{\pi}$ (horizontal axises), where the numbers over histogram
denote the percentage of the corresponding contributions.

(4) As it is well known, due to the large mass of final states, the momentum
transition in the ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$ decay may be
not large enough. One might naturally wonder whether the pQCD approach is
applicable and whether the perturbative calculation is reliable. Therefore, it
is necessary to check what percentage of the contributions comes from the
perturbative region. The contributions to branching ratio from different
region of ${\alpha}_{s}/{\pi}$ are showed in Fig.3. It can be clearly seen
that more than 90% contributions to branching ratio come from the
${\alpha}_{s}/{\pi}$ ${\leq}$ $0.3$ region, implying that the calculation with
the pQCD approach is reliable. As the discussion in pqcd1 ; pqcd2 ; pqcd3 ,
there are many factors for this, for example, the choice of the typical scale
in Eqs.(108-111), retaining the quark transverse moment and introducing the
Sudakov factor to suppress the nonperturbative contributions, which deserve
much attention and further investigation, but beyond the scope of this paper.

(5) Besides the uncertainties listed in Table 2, the decay constants,
$f_{{\Upsilon}(1S)}$, $f_{B_{c}}$, and $f_{D_{s}^{\ast}}$, can bring about 6%
uncertainties to branching ratios. Other factors, such as the models of wave
functions, contributions of higher order corrections to HME, relativistic
effects, and so on, deserve the dedicated study. Our results just provide an
order of magnitude estimation.

##  IV Summary

The ${\Upsilon}(1S)$ weak decay is allowable within the standard model. With
anticipation of the potential prospects of the ${\Upsilon}(1S)$ physics at
high-luminosity dedicated heavy-flavor factories, the ${\Upsilon}(1S)$ ${\to}$
$B_{c}D_{s}^{\ast}$ weak decay is studied with the pQCD approach firstly. It
is found that (1) the longitudinal plus parallel polarization fractions are
main shares, but sensitive to the ${\Upsilon}(1S)$ wave functions; (2)
branching ratio for the ${\Upsilon}(1S)$ ${\to}$ $B_{c}D_{s}^{\ast}$ weak
decay can reach up to ${\cal O}(10^{-9})$, which might be measurable at the
future experiments.

## Acknowledgments

We thank Professor Dongsheng Du (IHEP@CAS) and Professor Yadong Yang (CCNU)
for helpful discussion. The work is supported by the National Natural Science
Foundation of China (Grant Nos. 11547014, 11475055, U1332103 and 11275057).

##  Appendix A The building blocks of decay amplitudes

For the sake of simplicity, we decompose the decay amplitude Eq.(47) into some
building blocks ${\cal A}_{i,j}^{k}$, where the subscript $i$ corresponds to
the indices of Fig.2; the subscript $j$ relates with different helicity
amplitudes; the superscript $k$ refers to one of the three possible Dirac
structures ${\Gamma}_{1}{\otimes}{\Gamma}_{2}$ of the four-quark operator
$(\bar{q}_{1}{\Gamma}_{1}q_{2})(\bar{q}_{1}{\Gamma}_{2}q_{2})$, namely $k$ $=$
$LL$ for $(V-A){\otimes}(V-A)$, $k$ $=$ $LR$ for $(V-A){\otimes}(V+A)$, and
$k$ $=$ $SP$ for $-2(S-P){\otimes}(S+P)$. The explicit expressions of ${\cal
A}_{i,j}^{k}$ are written as follows.

$\displaystyle{\cal A}_{a,L}^{LL}$ $\displaystyle=$
$\displaystyle{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}\,{\phi}_{\Upsilon}^{v}(x_{1})\,$
(52)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{a}(t_{a})\,H_{ab}({\alpha}_{e},{\beta}_{a},b_{1},b_{2})\,{\alpha}_{s}(t_{a})$
$\displaystyle\Big{\\{}m_{1}^{2}\,s+m_{2}\,m_{b}\,u-(4\,m_{1}^{2}\,p^{2}+m_{2}^{2}\,u)\,\bar{x}_{2}\Big{\\}},$
$\displaystyle{\cal A}_{a,N}^{LL}$ $\displaystyle=$ $\displaystyle
m_{1}\,m_{3}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}$
(53)
$\displaystyle{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{B_{c}}(x_{2})\,E_{a}(t_{a})\,H_{ab}({\alpha}_{e},{\beta}_{a},b_{1},b_{2})$
$\displaystyle{\alpha}_{s}(t_{a})\,\Big{\\{}2\,m_{2}^{2}\,\bar{x}_{2}-2\,m_{2}\,m_{b}-t\Big{\\}},$
$\displaystyle{\cal A}_{a,T}^{LL}$ $\displaystyle=$ $\displaystyle
2\,m_{1}\,m_{3}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}$
(54)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{B_{c}}(x_{2})\,E_{a}(t_{a})\,H_{ab}({\alpha}_{e},{\beta}_{a},b_{1},b_{2})\,{\alpha}_{s}(t_{a}),$
$\displaystyle{\cal A}_{b,L}^{LL}$ $\displaystyle=$
$\displaystyle{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}$
(55)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{b}(t_{b})\,H_{ab}({\alpha}_{e},{\beta}_{b},b_{2},b_{1})\,{\alpha}_{s}(t_{b})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{v}(x_{1})\,\Big{[}m_{1}^{2}\,(s-4\,p^{2})\,\bar{x}_{1}+2\,m_{2}\,m_{c}\,u-m_{2}^{2}\,u\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{t}(x_{1})\,m_{1}\,\Big{[}s\,(2\,m_{2}-m_{c})-2\,m_{2}\,u\,\bar{x}_{1}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{b,N}^{LL}$ $\displaystyle=$ $\displaystyle
m_{3}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}$
(56)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{b}(t_{b})\,H_{ab}({\alpha}_{e},{\beta}_{b},b_{2},b_{1})\,{\alpha}_{s}(t_{b})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,m_{1}\,\Big{[}2\,m_{2}^{2}-4\,m_{2}\,m_{c}-t\,\bar{x}_{1}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,\Big{[}t\,(m_{c}-2\,m_{2})+4\,m_{1}^{2}\,m_{2}\,\bar{x}_{1}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{b,T}^{LL}$ $\displaystyle=$
$\displaystyle-2\,m_{3}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}$
(57)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{b}(t_{b})\,H_{ab}({\alpha}_{e},{\beta}_{b},b_{2},b_{1})\,{\alpha}_{s}(t_{b})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,m_{1}\,\bar{x}_{1}+{\phi}_{\Upsilon}^{T}(x_{1})\,(m_{c}-2\,m_{2})\Big{\\}},$
$\displaystyle{\cal A}_{c,L}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(58)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,E_{c}(t_{c})\,H_{cd}({\alpha}_{e},{\beta}_{c},b_{2},b_{3})\,{\alpha}_{s}(t_{c})\,$
$\displaystyle{\delta}(b_{1}-b_{2})\,\Big{\\{}{\phi}_{\Upsilon}^{v}(x_{1})\,u\,\Big{[}t\,x_{1}-2\,m_{2}^{2}\,x_{2}-s\,\bar{x}_{3}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{t}(x_{1})\,m_{1}\,m_{2}\,\Big{[}s\,x_{2}+2\,m_{3}^{2}\,\bar{x}_{3}-u\,x_{1}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{c,N}^{LL}$ $\displaystyle=$
$\displaystyle\frac{m_{3}}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(59)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,E_{c}(t_{c})\,H_{cd}({\alpha}_{e},{\beta}_{c},b_{2},b_{3})\,{\alpha}_{s}(t_{c})\,$
$\displaystyle{\delta}(b_{1}-b_{2})\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,2\,m_{1}\,\Big{[}2\,m_{2}^{2}\,x_{2}+s\,\bar{x}_{3}-t\,x_{1}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,m_{2}\,\Big{[}2\,m_{1}^{2}\,x_{1}-t\,x_{2}-u\,\bar{x}_{3}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{c,T}^{LL}$ $\displaystyle=$
$\displaystyle\frac{2\,m_{2}\,m_{3}}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}$
(60)
$\displaystyle{\int}_{0}^{\infty}b_{3}db_{3}\,{\delta}(b_{1}-b_{2})\,{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{B_{c}}(x_{2})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})$
$\displaystyle
E_{c}(t_{c})\,H_{cd}({\alpha}_{e},{\beta}_{c},b_{2},b_{3})\,{\alpha}_{s}(t_{c})\,(x_{2}-\bar{x}_{3}),$
$\displaystyle{\cal A}_{c,L}^{SP}$ $\displaystyle=$
$\displaystyle\frac{m_{3}}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(61)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,E_{c}(t_{c})\,H_{cd}({\alpha}_{e},{\beta}_{c},b_{2},b_{3})\,{\alpha}_{s}(t_{c})\,$
$\displaystyle{\delta}(b_{1}-b_{2})\,\Big{\\{}{\phi}_{\Upsilon}^{v}(x_{1})\,m_{2}\,\Big{[}2\,m_{1}^{2}\,x_{1}-t\,x_{2}-u\,\bar{x}_{3}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{t}(x_{1})\,m_{1}\,\Big{[}2\,m_{2}^{2}\,x_{2}+s\,\bar{x}_{3}-t\,x_{1}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{c,N}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(62)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,E_{c}(t_{c})\,H_{cd}({\alpha}_{e},{\beta}_{c},b_{2},b_{3})\,{\alpha}_{s}(t_{c})\,$
$\displaystyle{\delta}(b_{1}-b_{2})\,\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,m_{1}\,m_{2}\,\Big{[}s\,x_{2}+2\,m_{3}^{2}\,\bar{x}_{3}-u\,x_{1}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,\Big{[}m_{1}^{2}\,s\,x_{1}+(m_{2}^{2}\,u-s\,t)\,x_{2}-m_{3}^{2}\,t\,\bar{x}_{3}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{c,T}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(63)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,E_{c}(t_{c})\,H_{cd}({\alpha}_{e},{\beta}_{c},b_{2},b_{3})\,{\alpha}_{s}(t_{c})\,$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{T}(x_{1})\,\Big{[}(s+t)\,x_{2}+2\,m_{3}^{2}\,\bar{x}_{3}-(t+u)\,x_{1}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{V}(x_{1})\,2\,m_{1}\,m_{2}\,(x_{1}-x_{2})\Big{\\}}\,{\delta}(b_{1}-b_{2}),$
$\displaystyle{\cal A}_{d,L}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(64)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{d}(t_{d})\,H_{cd}({\alpha}_{e},{\beta}_{d},b_{2},b_{3})\,{\alpha}_{s}(t_{d})\,{\delta}(b_{1}-b_{2})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{t}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,m_{1}\,m_{2}\,\Big{[}s\,x_{2}+2\,m_{3}^{2}\,x_{3}-u\,x_{1}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{v}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,4\,m_{1}^{2}\,p^{2}\,(x_{3}-x_{2})$
$\displaystyle-{\phi}_{\Upsilon}^{v}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,m_{3}\,m_{c}\,t\Big{\\}},$
$\displaystyle{\cal A}_{d,N}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(65)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{d}(t_{d})\,H_{cd}({\alpha}_{e},{\beta}_{d},b_{2},b_{3})\,{\alpha}_{s}(t_{d})\,{\delta}(b_{1}-b_{2})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{2}\,m_{3}\,\Big{[}2\,m_{1}^{2}\,x_{1}-t\,x_{2}\,-u\,x_{3}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,m_{1}\,m_{c}\,s\Big{\\}},$
$\displaystyle{\cal A}_{d,T}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(66)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{d}(t_{d})\,H_{cd}({\alpha}_{e},{\beta}_{d},b_{2},b_{3})\,{\alpha}_{s}(t_{d})\,{\delta}(b_{1}-b_{2})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,2\,m_{2}\,m_{3}\,(x_{2}-x_{3})$
$\displaystyle-{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,2\,m_{1}\,m_{c}\Big{\\}},$
$\displaystyle{\cal A}_{d,L}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(67)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{d}(t_{d})\,H_{cd}({\alpha}_{e},{\beta}_{d},b_{2},b_{3})\,{\alpha}_{s}(t_{d})\,{\delta}(b_{1}-b_{2})$
$\displaystyle\Big{\\{}{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,m_{c}\Big{[}{\phi}_{\Upsilon}^{t}(x_{1})\,m_{1}\,s-{\phi}_{\Upsilon}^{v}(x_{1})\,m_{2}\,u\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{v}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,m_{2}\,m_{3}\Big{[}2\,m_{1}^{2}\,x_{1}-t\,x_{2}-u\,x_{3}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{t}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,m_{1}\,m_{3}\Big{[}2\,m_{2}^{2}\,x_{2}+s\,x_{3}-t\,x_{1}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{d,N}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(68)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{d}(t_{d})\,H_{cd}({\alpha}_{e},{\beta}_{d},b_{2},b_{3})\,{\alpha}_{s}(t_{d})\,{\delta}(b_{1}-b_{2})$
$\displaystyle\Big{\\{}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{3}\,m_{c}\Big{[}{\phi}_{\Upsilon}^{V}(x_{1})\,2\,m_{1}\,m_{2}-{\phi}_{\Upsilon}^{T}(x_{1})\,t\Big{]}$
$\displaystyle+{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\Big{[}{\phi}_{\Upsilon}^{V}(x_{1})\,m_{1}\,m_{2}\,(s\,x_{2}+2\,m_{3}^{2}\,x_{3}-u\,x_{1})$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,\\{m_{1}^{2}\,s\,x_{1}+(m_{2}^{2}\,u-s\,t)\,x_{2}-m_{3}^{2}\,t\,x_{3}\\}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{d,T}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(69)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{d}(t_{d})\,H_{cd}({\alpha}_{e},{\beta}_{d},b_{2},b_{3})\,{\alpha}_{s}(t_{d})\,{\delta}(b_{1}-b_{2})$
$\displaystyle\Big{\\{}{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\Big{[}{\phi}_{\Upsilon}^{V}(x_{1})\,2\,m_{1}\,m_{2}\,(x_{1}-x_{2})$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,\\{2\,m_{3}^{2}\,x_{3}+(s+t)\,x_{2}-(u+t)\,x_{1}\\}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,2\,m_{3}\,m_{c}\Big{\\}},$
$\displaystyle{\cal A}_{e,L}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(70)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{e}(t_{e})\,H_{ef}({\alpha}_{a},{\beta}_{e},b_{1},b_{2})\,{\alpha}_{s}(t_{e})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{v}(x_{1})\,\Big{[}{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,u\,\Big{(}t\,x_{1}-2\,m_{2}^{2}\,x_{2}-s\,\bar{x}_{3}\Big{)}$
$\displaystyle+{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,m_{2}\,m_{3}\,\Big{(}2\,m_{1}^{2}\,x_{1}-t\,x_{2}-u\,\bar{x}_{3}\Big{)}\Big{]}$
$\displaystyle-{\phi}_{\Upsilon}^{t}(x_{1})\,m_{1}\,m_{b}\,\Big{[}{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,s+{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,4\,m_{2}\,m_{3}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{e,N}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(71)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{e}(t_{e})\,H_{ef}({\alpha}_{a},{\beta}_{e},b_{1},b_{2})\,{\alpha}_{s}(t_{e})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,\Big{[}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,2\,m_{1}\,m_{3}\,\Big{(}2\,m_{2}^{2}\,x_{2}+s\,\bar{x}_{3}-t\,x_{1}\Big{)}$
$\displaystyle+{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,m_{1}\,m_{2}\,\Big{(}s\,x_{2}+2\,m_{3}^{2}\,\bar{x}_{3}-u\,x_{1}\Big{)}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,m_{b}\,\Big{[}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{3}\,t+{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,2\,m_{2}\,u\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{e,T}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(72)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{e}(t_{e})\,H_{ef}({\alpha}_{a},{\beta}_{e},b_{1},b_{2})\,{\alpha}_{s}(t_{e})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,2\,m_{1}\,m_{2}\,(x_{1}-x_{2})$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,2\,m_{b}\,\Big{[}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{3}-{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,2\,m_{2}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{e,L}^{LR}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(73)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{e}(t_{e})\,H_{ef}({\alpha}_{a},{\beta}_{e},b_{1},b_{2})\,{\alpha}_{s}(t_{e})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,\Big{[}{\phi}_{\Upsilon}^{v}(x_{1})\,4\,m_{1}^{2}\,p^{2}\,(x_{2}-x_{1})+{\phi}_{\Upsilon}^{t}(x_{1})\,m_{1}\,m_{b}\,s\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{v}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,m_{2}\,m_{3}\,\Big{[}t\,x_{2}+u\,\bar{x}_{3}-2\,m_{1}^{2}\,x_{1}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{e,N}^{LR}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(74)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{e}(t_{e})\,H_{ef}({\alpha}_{a},{\beta}_{e},b_{1},b_{2})\,{\alpha}_{s}(t_{e})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,m_{1}\,m_{2}\,\Big{[}u\,x_{1}-s\,x_{2}-2\,m_{3}^{2}\,\bar{x}_{3}\Big{]}$
$\displaystyle-{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{3}\,m_{b}\,t\Big{\\}},$
$\displaystyle{\cal A}_{e,T}^{LR}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(75)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{e}(t_{e})\,H_{ef}({\alpha}_{a},{\beta}_{e},b_{1},b_{2})\,{\alpha}_{s}(t_{e})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,2\,m_{1}\,m_{2}\,(x_{2}-x_{1})$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,2\,m_{3}\,m_{b}\Big{\\}},$
$\displaystyle{\cal A}_{e,L}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(76)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{e}(t_{e})\,H_{ef}({\alpha}_{a},{\beta}_{e},b_{1},b_{2})\,{\alpha}_{s}(t_{e})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{t}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,m_{1}\,m_{2}\,(u\,x_{1}-s\,x_{2}-2\,m_{3}^{2}\,\bar{x}_{3})$
$\displaystyle+{\phi}_{\Upsilon}^{t}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,m_{1}\,m_{3}\,(t\,x_{1}-2\,m_{2}^{2}\,x_{2}-s\,\bar{x}_{3})$
$\displaystyle-{\phi}_{\Upsilon}^{v}(x_{1})\,m_{b}\,\Big{[}{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,m_{2}\,u+{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,m_{3}\,t\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{e,N}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(77)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{e}(t_{e})\,H_{ef}({\alpha}_{a},{\beta}_{e},b_{1},b_{2})\,{\alpha}_{s}(t_{e})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,m_{1}\,m_{b}\,\Big{[}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,2\,m_{2}\,m_{3}+{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,s\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,\Big{[}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{2}\,m_{3}\,(t\,x_{2}+u\,\bar{x}_{3}-2\,m_{1}^{2}\,x_{1})$
$\displaystyle+{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,\\{(s\,t-m_{2}^{2}\,u)\,x_{2}+m_{3}^{2}\,t\,\bar{x}_{3}-m_{1}^{2}\,s\,x_{1}\\}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{e,T}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(78)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{e}(t_{e})\,H_{ef}({\alpha}_{a},{\beta}_{e},b_{1},b_{2})\,{\alpha}_{s}(t_{e})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{T}(x_{1})\,\Big{[}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,2\,m_{2}\,m_{3}\,(x_{2}-\bar{x}_{3})$
$\displaystyle+{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\\{(s+t)\,x_{2}+(u-s)\,\bar{x}_{3}-2\,m_{1}^{2}\,x_{1}\\}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,2\,m_{1}\,m_{b}\Big{\\}},$
$\displaystyle{\cal A}_{f,L}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(79)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{f}(t_{f})\,H_{ef}({\alpha}_{a},{\beta}_{f},b_{1},b_{2})\,{\alpha}_{s}(t_{f})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{v}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,m_{2}\,m_{3}\,\Big{[}2\,m_{1}^{2}\,\bar{x}_{1}-t\,x_{2}-u\,\bar{x}_{3}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{v}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,4\,m_{1}^{2}\,p^{2}\,(\bar{x}_{1}-x_{2})$
$\displaystyle-{\phi}_{\Upsilon}^{t}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,m_{1}\,m_{b}\,s\Big{\\}},$
$\displaystyle{\cal A}_{f,N}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(80)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{f}(t_{f})\,H_{ef}({\alpha}_{a},{\beta}_{f},b_{1},b_{2})\,{\alpha}_{s}(t_{f})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,m_{1}\,m_{2}\,(s\,x_{2}+2\,m_{3}^{2}\,\bar{x}_{3}-u\,\bar{x}_{1})$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{3}\,m_{b}\,t\Big{\\}},$
$\displaystyle{\cal A}_{f,T}^{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(81)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{f}(t_{f})\,H_{ef}({\alpha}_{a},{\beta}_{f},b_{1},b_{2})\,{\alpha}_{s}(t_{f})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,2\,m_{1}\,m_{2}\,(\bar{x}_{1}-x_{2})$
$\displaystyle-{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,2\,m_{3}\,m_{b}\Big{\\}},$
$\displaystyle{\cal A}_{f,L}^{LR}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(82)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{f}(t_{f})\,H_{ef}({\alpha}_{a},{\beta}_{f},b_{1},b_{2})\,{\alpha}_{s}(t_{f})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{v}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,u\,\Big{[}2\,m_{2}^{2}\,x_{2}+s\,\bar{x}_{3}-t\,\bar{x}_{1}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{v}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,m_{2}\,m_{3}\,\Big{[}t\,x_{2}+u\,\bar{x}_{3}-2\,m_{1}^{2}\,\bar{x}_{1}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{t}(x_{1})\,m_{1}\,m_{b}\,\Big{[}{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,s+{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,4\,m_{2}\,m_{3}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{f,N}^{LR}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(83)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{f}(t_{f})\,H_{ef}({\alpha}_{a},{\beta}_{f},b_{1},b_{2})\,{\alpha}_{s}(t_{f})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,2\,m_{1}\,m_{3}\,\Big{[}t\,\bar{x}_{1}-2\,m_{2}^{2}\,x_{2}-s\,\bar{x}_{3}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,m_{1}\,m_{2}\,\Big{[}u\,\bar{x}_{1}-s\,x_{2}-2\,m_{3}^{2}\,\bar{x}_{3}\Big{]}$
$\displaystyle-{\phi}_{\Upsilon}^{T}(x_{1})\,m_{b}\,\Big{[}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{3}\,t+{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,2\,m_{2}\,u\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{f,T}^{LR}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(84)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{f}(t_{f})\,H_{ef}({\alpha}_{a},{\beta}_{f},b_{1},b_{2})\,{\alpha}_{s}(t_{f})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,2\,m_{1}\,m_{2}\,(x_{2}-\bar{x}_{1})$
$\displaystyle+{\phi}_{\Upsilon}^{T}(x_{1})\,2\,m_{b}\,\Big{[}{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,2\,m_{2}-{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{3}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{f,L}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(85)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{f}(t_{f})\,H_{ef}({\alpha}_{a},{\beta}_{f},b_{1},b_{2})\,{\alpha}_{s}(t_{f})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{t}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,m_{1}\,m_{3}\,\Big{[}t\,\bar{x}_{1}-2\,m_{2}^{2}\,x_{2}-s\,\bar{x}_{3}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{t}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,m_{1}\,m_{2}\,\Big{[}u\,\bar{x}_{1}-s\,x_{2}-2\,m_{3}^{2}\,\bar{x}_{3}\Big{]}$
$\displaystyle-{\phi}_{\Upsilon}^{v}(x_{1})\,m_{b}\,\Big{[}{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,m_{2}\,u+{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,m_{3}\,t\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{f,N}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(86)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{f}(t_{f})\,H_{ef}({\alpha}_{a},{\beta}_{f},b_{1},b_{2})\,{\alpha}_{s}(t_{f})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{T}(x_{1})\,\Big{[}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{2}\,m_{3}\,(t\,x_{2}+u\,\bar{x}_{3}-2\,m_{1}^{2}\,\bar{x}_{1})$
$\displaystyle+{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,\\{(s\,t-m_{2}^{2}\,u)\,x_{2}+m_{3}^{2}\,t\,\bar{x}_{3}-m_{1}^{2}\,s\,\bar{x}_{1}\\}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{V}(x_{1})\,m_{1}\,m_{b}\,\Big{[}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,2\,m_{2}\,m_{3}+{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,s\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{f,T}^{SP}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}db_{3}$
(87)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{f}(t_{f})\,H_{ef}({\alpha}_{a},{\beta}_{f},b_{1},b_{2})\,{\alpha}_{s}(t_{f})\,{\delta}(b_{2}-b_{3})$
$\displaystyle\Big{\\{}{\phi}_{\Upsilon}^{T}(x_{1})\,\Big{[}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,2\,m_{2}\,m_{3}\,(x_{2}-\bar{x}_{3})$
$\displaystyle+{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,\\{(s+t)\,x_{2}+(u-s)\,\bar{x}_{3}-2\,m_{1}^{2}\,\bar{x}_{1}\\}\Big{]}$
$\displaystyle+{\phi}_{\Upsilon}^{V}(x_{1})\,{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,2\,m_{1}\,m_{b}\Big{\\}},$
$\displaystyle{\cal A}_{g,L}^{LL}$ $\displaystyle=$ $\displaystyle{\cal
A}_{g,L}^{LR}\ =\
{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(88)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,E_{g}(t_{g})\,H_{gh}({\alpha}_{a},{\beta}_{g},b_{2},b_{3})$
$\displaystyle{\alpha}_{s}(t_{g})\,\Big{\\{}m_{3}^{2}\,t+(4\,m_{1}^{2}\,p^{2}+m_{2}^{2}\,u)\,x_{2}\Big{\\}},$
$\displaystyle{\cal A}_{g,N}^{LL}$ $\displaystyle=$ $\displaystyle{\cal
A}_{g,N}^{LR}\ =\
-{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(89)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,E_{g}(t_{g})\,H_{gh}({\alpha}_{a},{\beta}_{g},b_{2},b_{3})$
$\displaystyle{\alpha}_{s}(t_{g})\,m_{1}\,m_{3}\,\Big{\\{}s+2\,m_{2}^{2}\,x_{2}\Big{\\}},$
$\displaystyle{\cal A}_{g,T}^{LL}$ $\displaystyle=$ $\displaystyle{\cal
A}_{g,T}^{LR}\ =\
{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(90)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,E_{g}(t_{g})\,H_{gh}({\alpha}_{a},{\beta}_{g},b_{2},b_{3})$
$\displaystyle{\alpha}_{s}(t_{g})\,2\,m_{1}\,m_{3},$ $\displaystyle{\cal
A}_{h,L}^{LL}$ $\displaystyle=$ $\displaystyle{\cal A}_{h,L}^{LR}\ =\
{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(91)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{h}(t_{h})\,H_{gh}({\alpha}_{a},{\beta}_{h},b_{3},b_{2})\,{\alpha}_{s}(t_{h})$
$\displaystyle\Big{\\{}{\phi}_{D_{s}^{\ast}}^{t}(x_{3})\,\Big{[}2\,m_{2}\,m_{3}(t+u\,\bar{x}_{3})-m_{3}\,m_{b}\,t\Big{]}$
$\displaystyle+{\phi}_{D_{s}^{\ast}}^{v}(x_{3})\,\Big{[}(4\,m_{1}^{2}\,p^{2}+m_{3}^{2}\,t)\,\bar{x}_{3}+m_{2}^{2}\,u$
$\displaystyle\qquad\qquad-2\,m_{2}\,m_{b}\,u\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{h,N}^{LL}$ $\displaystyle=$ $\displaystyle{\cal
A}_{h,N}^{LR}\ =\
{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(92)
$\displaystyle{\phi}_{B_{c}}(x_{2})\,E_{h}(t_{h})\,H_{gh}({\alpha}_{a},{\beta}_{h},b_{3},b_{2})\,{\alpha}_{s}(t_{h})$
$\displaystyle\Big{\\{}{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{1}\,m_{3}\,\Big{[}4\,m_{2}\,m_{b}-2\,m_{2}^{2}-s\,\bar{x}_{3}\Big{]}$
$\displaystyle+{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,m_{1}\,\Big{[}m_{b}\,s-2\,m_{2}\,s-4\,m_{2}\,m_{3}^{2}\,\bar{x}_{3}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{h,T}^{LL}$ $\displaystyle=$ $\displaystyle{\cal
A}_{h,T}^{LR}\ =\
{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}$
(93) $\displaystyle
2\,m_{1}\,{\phi}_{B_{c}}(x_{2})\,E_{h}(t_{h})\,H_{gh}({\alpha}_{a},{\beta}_{h},b_{3},b_{2})\,{\alpha}_{s}(t_{h})$
$\displaystyle\Big{\\{}{\phi}_{D_{s}^{\ast}}^{T}(x_{3})\,(2\,m_{2}-m_{b})-{\phi}_{D_{s}^{\ast}}^{V}(x_{3})\,m_{3}\,\bar{x}_{3}\Big{\\}},$

where $\bar{x}_{i}$ $=$ $1$ $-$ $x_{i}$; variable $x_{i}$ is the longitudinal
momentum fraction of the valence quark; $b_{i}$ is the conjugate variable of
the transverse momentum $k_{iT}$; and ${\alpha}_{s}(t)$ is the QCD coupling at
the scale of $t$.

The function $H_{i}$ are defined as follows plb752 .

$\displaystyle H_{ab}({\alpha}_{e},{\beta},b_{i},b_{j})$ $\displaystyle=$
$\displaystyle
K_{0}(\sqrt{-{\alpha}_{e}}b_{i})\Big{\\{}{\theta}(b_{i}-b_{j})K_{0}(\sqrt{-{\beta}}b_{i})I_{0}(\sqrt{-{\beta}}b_{j})+(b_{i}{\leftrightarrow}b_{j})\Big{\\}},$
(94) $\displaystyle H_{cd}({\alpha}_{e},{\beta},b_{2},b_{3})$ $\displaystyle=$
$\displaystyle\Big{\\{}{\theta}(-{\beta})K_{0}(\sqrt{-{\beta}}b_{3})+\frac{{\pi}}{2}{\theta}({\beta})\Big{[}iJ_{0}(\sqrt{{\beta}}b_{3})-Y_{0}(\sqrt{{\beta}}b_{3})\Big{]}\Big{\\}}$
(95) $\displaystyle{\times}$
$\displaystyle\Big{\\{}{\theta}(b_{2}-b_{3})K_{0}(\sqrt{-{\alpha}_{e}}b_{2})I_{0}(\sqrt{-{\alpha}_{e}}b_{3})+(b_{2}{\leftrightarrow}b_{3})\Big{\\}},$
$\displaystyle H_{ef}({\alpha}_{a},{\beta},b_{1},b_{2})$ $\displaystyle=$
$\displaystyle\Big{\\{}{\theta}(-{\beta})K_{0}(\sqrt{-{\beta}}b_{1})+\frac{{\pi}}{2}{\theta}({\beta})\Big{[}iJ_{0}(\sqrt{{\beta}}b_{1})-Y_{0}(\sqrt{{\beta}}b_{1})\Big{]}\Big{\\}}$
(96)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\times}\frac{{\pi}}{2}\Big{\\{}{\theta}(b_{1}-b_{2})\Big{[}iJ_{0}(\sqrt{{\alpha}_{a}}b_{1})-Y_{0}(\sqrt{{\alpha}_{a}}b_{1})\Big{]}J_{0}(\sqrt{{\alpha}_{a}}b_{2})+(b_{1}{\leftrightarrow}b_{2})\Big{\\}},$
$\displaystyle H_{hg}({\alpha}_{a},{\beta},b_{i},b_{j})$ $\displaystyle=$
$\displaystyle\frac{{\pi}^{2}}{4}\Big{\\{}iJ_{0}(\sqrt{{\alpha}_{a}}b_{j})-Y_{0}(\sqrt{{\alpha}_{a}}b_{j})\Big{\\}}$
(97) $\displaystyle{\times}$
$\displaystyle\Big{\\{}{\theta}(b_{i}-b_{j})\Big{[}iJ_{0}(\sqrt{{\beta}}b_{i})-Y_{0}(\sqrt{{\beta}}b_{i})\Big{]}J_{0}(\sqrt{{\beta}}b_{j})+(b_{i}{\leftrightarrow}b_{j})\Big{\\}},$

where $J_{0}$ and $Y_{0}$ ($I_{0}$ and $K_{0}$) are the (modified) Bessel
function of the first and second kind, respectively; ${\alpha}_{e}$
(${\alpha}_{a}$) is the gluon virtuality of the emission (annihilation)
topological diagrams; the subscript of the quark virtuality ${\beta}_{i}$
corresponds to the indices of Fig.2. The definition of the particle virtuality
is listed as follows plb752 .

$\displaystyle{\alpha}_{e}$ $\displaystyle=$
$\displaystyle\bar{x}_{1}^{2}\,m_{1}^{2}+\bar{x}_{2}^{2}\,m_{2}^{2}-\bar{x}_{1}\,\bar{x}_{2}\,t,$
(98) $\displaystyle{\alpha}_{a}$ $\displaystyle=$ $\displaystyle
x_{2}^{2}\,m_{2}^{2}+\bar{x}_{3}^{2}\,m_{3}^{2}+x_{2}\,\bar{x}_{3}\,s,$ (99)
$\displaystyle{\beta}_{a}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{b}^{2}+\bar{x}_{2}^{2}\,m_{2}^{2}-\bar{x}_{2}\,t,$ (100)
$\displaystyle{\beta}_{b}$ $\displaystyle=$ $\displaystyle
m_{2}^{2}-m_{c}^{2}+\bar{x}_{1}^{2}\,m_{1}^{2}-\bar{x}_{1}\,t,$ (101)
$\displaystyle{\beta}_{c}$ $\displaystyle=$ $\displaystyle
x_{1}^{2}\,m_{1}^{2}+x_{2}^{2}\,m_{2}^{2}+\bar{x}_{3}^{2}\,m_{3}^{2}$ (102)
$\displaystyle-$ $\displaystyle
x_{1}\,x_{2}\,t-x_{1}\,\bar{x}_{3}\,u+x_{2}\,\bar{x}_{3}\,s,$
$\displaystyle{\beta}_{d}$ $\displaystyle=$ $\displaystyle
x_{1}^{2}\,m_{1}^{2}+x_{2}^{2}\,m_{2}^{2}+x_{3}^{2}\,m_{3}^{2}-m_{c}^{2}$
(103) $\displaystyle-$ $\displaystyle
x_{1}\,x_{2}\,t-x_{1}\,x_{3}\,u+x_{2}\,x_{3}\,s,$ $\displaystyle{\beta}_{e}$
$\displaystyle=$ $\displaystyle
x_{1}^{2}\,m_{1}^{2}+x_{2}^{2}\,m_{2}^{2}+\bar{x}_{3}^{2}\,m_{3}^{2}-m_{b}^{2}$
(104) $\displaystyle-$ $\displaystyle
x_{1}\,x_{2}\,t-x_{1}\,\bar{x}_{3}\,u+x_{2}\,\bar{x}_{3}\,s,$
$\displaystyle{\beta}_{f}$ $\displaystyle=$
$\displaystyle\bar{x}_{1}^{2}\,m_{1}^{2}+x_{2}^{2}\,m_{2}^{2}+\bar{x}_{3}^{2}\,m_{3}^{2}-m_{b}^{2}$
(105) $\displaystyle-$
$\displaystyle\bar{x}_{1}\,x_{2}\,t-\bar{x}_{1}\,\bar{x}_{3}\,u+x_{2}\,\bar{x}_{3}\,s,$
$\displaystyle{\beta}_{g}$ $\displaystyle=$ $\displaystyle
x_{2}^{2}\,m_{2}^{2}+m_{3}^{2}+x_{2}\,s,$ (106) $\displaystyle{\beta}_{h}$
$\displaystyle=$
$\displaystyle\bar{x}_{3}^{2}\,m_{3}^{2}+m_{2}^{2}+\bar{x}_{3}\,s-m_{b}^{2}.$
(107)

The typical scale $t_{i}$ and the Sudakov factor $E_{i}$ are defined as
follows, where the subscript $i$ corresponds to the indices of Fig.2.

$\displaystyle t_{a(b)}$ $\displaystyle=$
$\displaystyle{\max}(\sqrt{-{\alpha}_{e}},\sqrt{-{\beta}_{a(b)}},1/b_{1},1/b_{2}),$
(108) $\displaystyle t_{c(d)}$ $\displaystyle=$
$\displaystyle{\max}(\sqrt{-{\alpha}_{e}},\sqrt{{|}{\beta}_{c(d)}{|}},1/b_{2},1/b_{3}),$
(109) $\displaystyle t_{e(f)}$ $\displaystyle=$
$\displaystyle{\max}(\sqrt{{\alpha}_{a}},\sqrt{{|}{\beta}_{e(f)}{|}},1/b_{1},1/b_{2}),$
(110) $\displaystyle t_{g(h)}$ $\displaystyle=$
$\displaystyle{\max}(\sqrt{{\alpha}_{a}},\sqrt{{\beta}_{g(h)}},1/b_{2},1/b_{3}),$
(111)
$E_{i}(t)=\left\\{\begin{array}[]{lll}{\exp}\\{-S_{{\Upsilon}(1S)}(t)-S_{B_{c}}(t)\\},&{}&i=a,b\\\
{\exp}\\{-S_{{\Upsilon}(1S)}(t)-S_{B_{c}}(t)-S_{D_{s}^{\ast}}(t)\\},&&i=c,d,e,f\\\
{\exp}\\{-S_{B_{c}}(t)-S_{D_{s}^{\ast}}(t)\\},&&i=g,h\end{array}\right.,$
(112) $\displaystyle S_{{\Upsilon}(1S)}(t)$ $\displaystyle=$ $\displaystyle
s(x_{1},p_{1}^{+},1/b_{1})+2{\int}_{1/b_{1}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q},$
(113) $\displaystyle S_{B_{c}}(t)$ $\displaystyle=$ $\displaystyle
s(x_{2},p_{2}^{+},1/b_{2})+2{\int}_{1/b_{2}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q},$
(114) $\displaystyle S_{D_{s}^{\ast}}(t)$ $\displaystyle=$ $\displaystyle
s(x_{3},p_{3}^{+},1/b_{3})+2{\int}_{1/b_{3}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q},$
(115)

where ${\gamma}_{q}$ $=$ $-{\alpha}_{s}/{\pi}$ is the quark anomalous
dimension; the explicit expression of $s(x,Q,1/b)$ can be found in the
appendix of Ref.pqcd1 .

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