# The ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$, $B_{c}^{\ast}K$ decays
with perturbative QCD 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  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  Jinshu Huang  College of Physics
and Electronic Engineering, Nanyang Normal University, Nanyang 473061, China

###### Abstract

Besides the traditional strong and electromagnetic decay modes,
${\Upsilon}(nS)$ meson can also decay through the weak interactions within the
standard model of elementary particle. With anticipation of copious
${\Upsilon}(nS)$ data samples at the running LHC and coming SuperKEKB
experiments, the two-body nonleptonic bottom-changing ${\Upsilon}(nS)$ ${\to}$
$B_{c}^{\ast}{\pi}$, $B_{c}^{\ast}K$ decays ($n$ $=$ 1, 2, 3) are investigated
with perturbative QCD approach firstly. The absolute branching ratios for
${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$ and $B_{c}^{\ast}K$ decays are
estimated to reach up to about $10^{-10}$ and $10^{-11}$, respectively, which
might possibly be measured by the future experiments.

###### pacs:

13.25.Gv 12.39.St 14.40.Pq

##  I Introduction

The upsilon ${\Upsilon}(nS)$ meson is the spin-triplet $S$-wave state of
bottomonium (bound state consisting of bottom quark $b$ and anti-bottom quark
$\bar{b}$) with well-established quantum number of $I^{G}J^{PC}$ $=$
$0^{-}1^{--}$ pdg . The characteristic narrow decay widths of ${\Upsilon}(nS)$
mesons for $n$ $=$ 1, 2 and 3 provide insight into the study of strong
interactions. [see Table. 1, and note that for simplicity, ${\Upsilon}(nS)$
will denote ${\Upsilon}(1S)$, ${\Upsilon}(2S)$ and ${\Upsilon}(3S)$ mesons in
the following content if not specified definitely.] The mass of
${\Upsilon}(nS)$ meson is below the $B$ meson pair threshold. The
${\Upsilon}(nS)$ meson decays into bottomed hadrons through strong and
electromagnetic interactions are forbidden by the law of conservation of
flavor number. The bottom-changing ${\Upsilon}(nS)$ decays can occur only via
the weak interactions within the standard model, although with tiny incidence
probability. Both constituent quarks of upsilons can decay individually, which
provide an alternative system for investigating the weak decay of heavy-
flavored hadrons. In this paper, we will study the nonleptonic
${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}P$ ($P$ $=$ ${\pi}$ and $K$) weak decays
with perturbative QCD (pQCD) approach pqcd1 ; pqcd2 ; pqcd3 .

Table 1: Summary of mass, decay width, on(off)-peak luminosity and numbers of ${\Upsilon}(nS)$. | properties pdg  | luminosity ($fb^{-1}$) epjc74  | numbers ($10^{6}$) epjc74
---|---|---|---
meson | mass (MeV) | width (keV) | Belle | BaBar | Belle | BaBar
${\Upsilon}(1S)$ | $9460.30{\pm}0.26$ | $54.02{\pm}1.25$ | 5.7 (1.8) | …… | $102{\pm}2$ | ……
${\Upsilon}(2S)$ | $10023.26{\pm}0.31$ | $31.98{\pm}2.63$ | 24.9 (1.7) | 13.6 (1.4) | $158{\pm}4$ | $98.3{\pm}0.9$
${\Upsilon}(3S)$ | $10355.2{\pm}0.5$ | $20.32{\pm}1.85$ | 2.9 (0.2) | 28.0 (2.6) | $11{\pm}0.3$ | $121.3{\pm}1.2$

Experimentally, (1) over $10^{8}$ ${\Upsilon}(nS)$ data samples have been
accumulated at Belle and BaBar experiments epjc74 . More and more upsilon data
samples will be collected at the running hadron collider LHC and the
forthcoming $e^{+}e^{-}$ collider SuperKEKB111The SuperKEKB has started
commissioning test run (http://www.kek.jp/en/NewsRoom/Release).. There seems
to exist a realistic possibility to explore ${\Upsilon}(nS)$ weak decay at
future experiments. (2) Signals of the ${\Upsilon}(nS)$ ${\to}$
$B_{c}^{\ast}{\pi}$, $B_{c}^{\ast}K$ decays should be easily distinguished
with “charge tag” technique, due to the facts that the back-to-back final
states with different electric charges have definite momentum and energy in
the rest frame of ${\Upsilon}(nS)$ meson. (3) The $B_{c}^{\ast}$ meson has not
been observed experimentally by now. The $B_{c}^{\ast}$ meson production via
the strong interaction are suppressed due to the simultaneous presence of two
heavy quarks with different flavors and higher order in QCD coupling constant
${\alpha}_{s}$. The ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$,
$B_{c}^{\ast}K$ decays provide a novel pattern to study the $B_{c}^{\ast}$
meson production. The identification of a single explicitly flavored
$B_{c}^{\ast}$ meson could be used as an effective selection criterion to
detect upsilon weak decays. Moreover, the radiative decay of $B_{c}^{\ast}$
meson provide a useful extra signal and a powerful constraint222The
investigation on the radiative decay of $B_{c}^{\ast}$ meson can be found in,
for example, Ref. epjc73  with QCD sum rules.. Of course, any discernible
evidences of an anomalous production rate of single bottomed meson from
upsilon decays might be a hint of new physics.

Theoretically, many attractive QCD-inspired methods have been developed
recently to describe the exclusive nonleptonic decay of heavy-flavored mesons,
such as the pQCD approach pqcd1 ; pqcd2 ; pqcd3 , the QCD factorization
approach qcdf1 ; qcdf2 ; qcdf3 , soft and collinear effective theory scet1 ;
scet2 ; scet3 ; scet4 , and have been applied widely to vindicate measurements
on $B$ meson decays. The upsilon weak decay permits one to further constrain
parameters obtained from $B$ meson decay, and cross comparisons provide an
opportunity to test various phenomenological models. The upsilon weak decay
possess 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 ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}P$ decay belongs to the
favorable $b$ ${\to}$ $c$ transition, which should, in principle, have
relatively large branching ratio among upsilon weak decays. However, there is
still no theoretical study devoted to the ${\Upsilon}(nS)$ ${\to}$
$B_{c}^{\ast}P$ decay for the moment. In this paper, we will present a
phenomenological investigation on ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}P$
weak decay with the pQCD approach to supply a ready reference for the future
experiments.

This paper is organized as follows. Section II focus on theoretical framework
and decay amplitudes for ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$,
$B_{c}^{\ast}K$ weak decays. Section III is devoted to numerical results and
discussion. The last section is a summary.

##  II theoretical framework

###  II.1 The effective Hamiltonian

Theoretically, the ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$,
$B_{c}^{\ast}K$ weak decays are described by an effective bottom-changing
Hamiltonian based on operator product expansion 9512380 :

${\cal H}_{\rm eff}\ =\
\frac{G_{F}}{\sqrt{2}}\,\sum\limits_{q=d,s}V_{cb}V_{uq}^{\ast}\Big{\\{}C_{1}({\mu})\,O_{1}({\mu})+C_{2}({\mu})\,O_{2}({\mu})\Big{\\}}+{\rm
h.c.},$ (1)

where $G_{F}$ ${\simeq}$ $1.166{\times}10^{-5}\,{\rm GeV}^{-2}$ pdg  is the
Fermi coupling constant; the CKM factors $V_{cb}V_{ud}^{\ast}$ and
$V_{cb}V_{us}^{\ast}$ correspond to ${\Upsilon}(nS)$ ${\to}$
$B_{c}^{\ast}{\pi}$ and $B_{c}^{{\ast}}K$ decays, respectively; with the
Wolfenstein parameterization, the CKM factors are expanded as a power series
in a small Wolfenstein parameter ${\lambda}$ ${\sim}$ $0.2$ pdg :

$\displaystyle V_{cb}V_{ud}^{\ast}$ $\displaystyle=$ $\displaystyle
A{\lambda}^{2}-\frac{1}{2}A{\lambda}^{4}-\frac{1}{8}A{\lambda}^{6}+{\cal
O}({\lambda}^{7}),$ (2) $\displaystyle V_{cb}V_{us}^{\ast}$ $\displaystyle=$
$\displaystyle A{\lambda}^{3}+{\cal O}({\lambda}^{7}).$ (3)

The local tree operators $Q_{1,2}$ are defined as:

$\displaystyle O_{1}$ $\displaystyle=$
$\displaystyle[\bar{c}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\alpha}][\bar{q}_{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})u_{\beta}],$
(4) $\displaystyle O_{2}$ $\displaystyle=$
$\displaystyle[\bar{c}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\beta}][\bar{q}_{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})u_{\alpha}],$
(5)

where ${\alpha}$ and ${\beta}$ are color indices and the sum over repeated
indices is understood.

The scale ${\mu}$ factorizes physics contributions into short- and long-
distance dynamics. The Wilson coefficients $C_{i}(\mu)$ summarize the physics
contributions at scale higher than ${\mu}$, and are calculable with the
renormalization group improved perturbation theory. The hadronic matrix
elements (HME), where the local operators are inserted between initial and
final hadron states, embrace the physics contributions below scale of ${\mu}$.
To obtain decay amplitudes, the remaining work is to calculate HME properly by
separating from perturbative and nonperturbative contributions.

###  II.2 Hadronic matrix elements

Based on Lepage-Brodsky approach for exclusive processes prd22 , HME is
commonly expressed as a convolution integral of hard scattering subamplitudes
containing perturbative contributions with universal wave functions reflecting
nonperturbative contributions. In order to effectively regulate endpoint
singularities and provide a naturally dynamical cutoff on nonperturbative
contributions, transverse momentum of valence quarks is retained and the
Sudakov factor is introduced within the pQCD framework pqcd1 ; pqcd2 ; pqcd3 .
Phenomenologically, the pQCD’s decay amplitude could be divided into three
parts: the Wilson coefficients $C_{i}$ incorporating the hard contributions
above typical scale of $t$, process-dependent rescattering subamplitudes $T$
accounting for the heavy quark decay, and wave functions ${\Phi}$ of all
participating hadrons, which is expressed as

${\int}dk\,C_{i}(t)\,T(t,k)\,{\Phi}(k)e^{-S},$ (6)

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 ${\Upsilon}(nS)$ rest frame are
defined as follows.

$p_{\Upsilon}\,=\,p_{1}\,=\,\frac{m_{1}}{\sqrt{2}}(1,1,0),$ (7)
$p_{B_{c}^{\ast}}\,=\,p_{2}\,=\,(p_{2}^{+},p_{2}^{-},0),$ (8)
$p_{3}\,=\,(p_{3}^{-},p_{3}^{+},0),$ (9)
$p_{i}^{\pm}\,=\,(E_{i}\,{\pm}\,p)/\sqrt{2},$ (10)
$k_{i}\,=\,x_{i}\,p_{i}+(0,0,\vec{k}_{i{\perp}}),$ (11)
${\epsilon}_{1}^{\parallel}\,=\,\frac{p_{1}}{m_{1}}-\frac{m_{1}}{p_{1}{\cdot}n_{+}}n_{+},$
(12)
${\epsilon}_{2}^{\parallel}\,=\,\frac{p_{2}}{m_{2}}-\frac{m_{2}}{p_{2}{\cdot}n_{-}}n_{-},$
(13) ${\epsilon}_{1,2}^{\perp}\,=\,(0,0,\vec{1}),$ (14) $n_{+}=(1,0,0),$ (15)
$n_{-}=(0,1,0),$ (16) $s\,=\,2\,p_{2}{\cdot}p_{3},$ (17)
$t\,=\,2\,p_{1}{\cdot}p_{2}\,=\ 2\,m_{1}\,E_{2},$ (18)
$u\,=\,2\,p_{1}{\cdot}p_{3}\,=\ 2\,m_{1}\,E_{3},$ (19)
$p=\frac{\sqrt{[m_{1}^{2}-(m_{2}+m_{3})^{2}]\,[m_{1}^{2}-(m_{2}-m_{3})^{2}]}}{2\,m_{1}},$
(20)

where $x_{i}$ and $\vec{k}_{i{\perp}}$ are the longitudinal momentum fraction
and transverse momentum of valence quarks, respectively;
${\epsilon}_{i}^{\parallel}$ and ${\epsilon}_{i}^{\perp}$ are the longitudinal
and transverse polarization vectors, respectively, and satisfy relations
${\epsilon}_{i}^{2}$ $=$ $-1$ and ${\epsilon}_{i}{\cdot}p_{i}$ $=$ $0$; the
subscript $i$ on variables $p_{i}$, $E_{i}$, $m_{i}$, ${\epsilon}_{i}$
corresponds to participating hadrons, namely, $i$ $=$ $1$ for ${\Upsilon}(nS)$
meson, $i$ $=$ $2$ for the recoiled $B_{c}^{\ast}$ meson, $i$ $=$ $3$ for the
emitted pseudoscalar meson; $n_{+}$ and $n_{-}$ are positive and negative null
vectors, respectively; $s$, $t$ 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 , wave functions are defined as

${\langle}0{|}b_{i}(z)\bar{b}_{j}(0){|}{\Upsilon}(p_{1},{\epsilon}_{1}^{{\parallel}}){\rangle}\,=\,\frac{f_{{\Upsilon}}}{4}{\int}dk_{1}\,e^{-ik_{1}{\cdot}z}\Big{\\{}\\!\\!\not{\epsilon}_{1}^{{\parallel}}\Big{[}m_{1}\,{\Phi}_{\Upsilon}^{v}(k_{1})-\\!\\!\not{p}_{1}\,{\Phi}_{\Upsilon}^{t}(k_{1})\Big{]}\Big{\\}}_{ji},$
(21)
${\langle}0{|}b_{i}(z)\bar{b}_{j}(0){|}{\Upsilon}(p_{1},{\epsilon}_{1}^{{\perp}}){\rangle}\,=\,\frac{f_{{\Upsilon}}}{4}{\int}dk_{1}\,e^{-ik_{1}{\cdot}z}\Big{\\{}\\!\\!\not{\epsilon}_{1}^{{\perp}}\Big{[}m_{1}\,{\Phi}_{\Upsilon}^{V}(k_{1})-\\!\\!\not{p}_{1}\,{\Phi}_{\Upsilon}^{T}(k_{1})\Big{]}\Big{\\}}_{ji},$
(22)
${\langle}B_{c}^{\ast}(p_{2},{\epsilon}_{2}^{{\parallel}}){|}\bar{c}_{i}(z)b_{j}(0){|}0{\rangle}\
=\
\frac{f_{B_{c}^{\ast}}}{4}{\int}_{0}^{1}dk_{3}\,e^{ik_{2}{\cdot}z}\Big{\\{}\\!\not{\epsilon}_{2}^{{\parallel}}\Big{[}m_{2}\,{\Phi}_{B_{c}^{\ast}}^{v}(k_{2})+\\!\not{p}_{2}\,{\Phi}_{B_{c}^{\ast}}^{t}(k_{2})\Big{]}\Big{\\}}_{ji},$
(23)
${\langle}B_{c}^{{\ast}}(p_{2},{\epsilon}_{2}^{{\perp}}){|}\bar{c}_{i}(z)b_{j}(0){|}0{\rangle}\
=\
\frac{f_{B_{c}^{\ast}}}{4}{\int}_{0}^{1}dk_{2}\,e^{ik_{2}{\cdot}z}\Big{\\{}\\!\not{\epsilon}_{2}^{{\perp}}\Big{[}m_{2}\,{\Phi}_{B_{c}^{\ast}}^{V}(k_{2})+\\!\not{p}_{2}\,{\Phi}_{B_{c}^{\ast}}^{T}(k_{2})\Big{]}\Big{\\}}_{ji},$
(24) $\displaystyle{\langle}P(p_{3}){|}u_{i}(0)\bar{q}_{j}(z){|}0{\rangle}$
(25) $\displaystyle=$
$\displaystyle\frac{i\,f_{P}}{4}{\int}dk_{3}\,e^{ik_{3}{\cdot}z}\Big{\\{}{\gamma}_{5}\Big{[}\\!\\!\not{p}_{3}\,{\Phi}_{P}^{a}(k_{3})+{\mu}_{P}{\Phi}_{P}^{p}(k_{3})+{\mu}_{P}(\\!\not{n}_{-}\\!\\!\not{n}_{+}\\!-\\!1)\,{\Phi}_{P}^{t}(k_{3})\Big{]}\Big{\\}}_{ji},$

where $f_{{\Upsilon}}$, $f_{B_{c}^{\ast}}$, $f_{P}$ are decay constants of
${\Upsilon}(nS)$, $B_{c}^{\ast}$, $P$ mesons, respectively.

Considering mass relations of $m_{{\Upsilon}(nS)}$ ${\simeq}$ $2m_{b}$ and
$m_{B_{c}^{\ast}}$ ${\simeq}$ $m_{b}$ $+$ $m_{c}$, it might assume that the
motion of heavy valence quarks in ${\Upsilon}(nS)$ and $B_{c}^{\ast}$ mesons
is nearly nonrelativistic. The wave functions of ${\Upsilon}(nS)$ and
$B_{c}^{\ast}$ mesons could be approximately described with nonrelativistic
quantum chromodynamics (NRQCD) prd46 ; prd51 ; rmp77  and time-independent
Schrödinger equation. For an isotropic harmonic oscillator potential, the
eigenfunctions of stationary state with quantum numbers $nL$ are written as
plb751

${\phi}_{1S}(\vec{k})\ {\sim}\ e^{-\vec{k}^{2}/2{\beta}^{2}},$ (26)
${\phi}_{2S}(\vec{k})\ {\sim}\
e^{-\vec{k}^{2}/2{\beta}^{2}}(2\vec{k}^{2}-3{\beta}^{2}),$ (27)
${\phi}_{3S}(\vec{k})\ {\sim}\
e^{-\vec{k}^{2}/2{\beta}^{2}}(4\vec{k}^{4}-20\vec{k}^{2}{\beta}^{2}+15{\beta}^{4}),$
(28)

where parameter ${\beta}$ determines the average transverse momentum, i.e.,
${\langle}nS{|}k^{2}_{\perp}{|}nS{\rangle}$ ${\sim}$ ${\beta}^{2}$. Employing
the substitution ansatz xiao ,

$\vec{k}^{2}\ {\to}\
\frac{1}{4}\sum\limits_{i}\frac{\vec{k}_{i\perp}^{2}+m_{q_{i}}^{2}}{x_{i}},$
(29)

where $x_{i}$ and $m_{q_{i}}$ are the longitudinal momentum fraction and mass
of valence quark, respectively, then integrating out $\vec{k}_{\perp}$ and
combining with their asymptotic forms, the distribution amplitudes (DAs) for
${\Upsilon}(nS)$ and $B_{c}^{\ast}$ mesons can be written as plb751 ,

${\phi}_{{\Upsilon}(1S)}^{v,T}(x)=A\,x\bar{x}\,{\exp}\Big{\\{}-\frac{m_{b}^{2}}{8\,{\beta}_{1}^{2}\,x\,\bar{x}}\Big{\\}},$
(30)
${\phi}_{{\Upsilon}(1S)}^{t}(x)=B\,t^{2}\,{\exp}\Big{\\{}-\frac{m_{b}^{2}}{8\,{\beta}_{1}^{2}\,x\,\bar{x}}\Big{\\}},$
(31)
${\phi}_{{\Upsilon}(1S)}^{V}(x)=C\,(1+t^{2})\,{\exp}\Big{\\{}-\frac{m_{b}^{2}}{8\,{\beta}_{1}^{2}\,x\,\bar{x}}\Big{\\}},$
(32)
${\phi}_{{\Upsilon}(2S)}^{v,t,V,T}(x)=D\,{\phi}_{{\Upsilon}(1S)}^{v,t,V,T}(x)\,\Big{\\{}1+\frac{m_{b}^{2}}{2\,{\beta}_{1}^{2}\,x\,\bar{x}}\Big{\\}},$
(33)
${\phi}_{{\Upsilon}(3S)}^{v,t,V,T}(x)=E\,{\phi}_{{\Upsilon}(1S)}^{v,t,V,T}(x)\,\Big{\\{}\Big{(}1-\frac{m_{b}^{2}}{2\,{\beta}_{1}^{2}\,x\,\bar{x}}\Big{)}^{2}+6\Big{\\}},$
(34)
${\phi}_{B_{c}^{\ast}}^{v,T}(x)=F\,x\bar{x}\,{\exp}\Big{\\{}-\frac{\bar{x}\,m_{c}^{2}+x\,m_{b}^{2}}{8\,{\beta}_{2}^{2}\,x\,\bar{x}}\Big{\\}},$
(35)
${\phi}_{B_{c}^{\ast}}^{t}(x)=G\,t^{2}\,{\exp}\Big{\\{}-\frac{\bar{x}\,m_{c}^{2}+x\,m_{b}^{2}}{8\,{\beta}_{2}^{2}\,x\,\bar{x}}\Big{\\}},$
(36)
${\phi}_{B_{c}^{\ast}}^{V}(x)=H\,(1-t^{2})\,{\exp}\Big{\\{}-\frac{\bar{x}\,m_{c}^{2}+x\,m_{b}^{2}}{8\,{\beta}_{2}^{2}\,x\,\bar{x}}\Big{\\}},$
(37)

where $\bar{x}$ $=$ $1$ $-$ $x$; $t$ $=$ $x$ $-$ $\bar{x}$. According to NRQCD
power counting rules prd46 , ${\beta}_{i}$ ${\simeq}$
${\xi}_{i}\,{\alpha}_{s}({\xi}_{i})$ with ${\xi}_{i}$ $=$ $m_{i}/2$ and QCD
coupling constant ${\alpha}_{s}$. The exponential function represents
$k_{\perp}$ distribution. Parameters of $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$
are normalization coefficients satisfying with the conditions

${\int}_{0}^{1}dx\,{\phi}_{{\Upsilon}(nS)}^{i}(x)={\int}_{0}^{1}dx\,{\phi}^{i}_{B_{c}^{\ast}}(x)=1\quad\text{for}\
\ i=v,t,V,T.$ (38)

Figure 1: The normalized distribution amplitudes for ${\Upsilon}(nS)$ and
$B_{c}^{\ast}$ mesons.

The shape lines of normalized DAs for ${\Upsilon}(nS)$ and $B_{c}^{\ast}$
mesons are showed in Fig. 1. It is clearly seen that (1) DAs for
${\Upsilon}(nS)$ and $B_{c}^{\ast}$ mesons fall quickly down to zero at
endpoint $x$, $\bar{x}$ ${\to}$ $0$ due to suppression from exponential
functions; (2) DAs for ${\Upsilon}(nS)$ meson are symmetric under the
interchange of momentum fractions $x$ ${\leftrightarrow}$ $\bar{x}$, and DAs
for $B_{c}^{\ast}$ meson are basically consistent with the feature that
valence quarks share momentum fractions according to their masses.

Our study shows that only the leading twist (twist-2) DAs of the emitted light
pseudoscalar meson $P$ is involved in decay amplitudes (see Appendix A). The
twist-2 DAs has the expansion jhep0605 :

${\phi}_{P}^{a}(x)=6\,x\,\bar{x}\,\sum\limits_{i=0}a_{i}\,C_{i}^{3/2}(t),$
(39)

and are normalized as

${\int}_{0}^{1}{\phi}_{P}^{a}(x)\,dx=1,$ (40)

where $C_{i}^{3/2}(t)$ are Gegenbauer polynomials,

$C_{0}^{3/2}(t)=1,\quad C_{1}^{3/2}(t)=3\,t,\quad
C_{2}^{3/2}(t)=\frac{3}{2}\,(5\,t^{2}-1),\quad\cdots$ (41)

and each term corresponds to a nonperturbative Gegenbauer moment $a_{i}$; note
that $a_{0}$ $=$ $1$ due to the normalization condition Eq.(40); the
$G$-parity invariance of the pion DAs requires Gegenbauer moment $a_{i}$ $=$
$0$ for $i$ $=$ $1$, $3$, $5$ ${\cdots}$.

###  II.5 Decay amplitudes

The Feynman diagrams for ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$ weak
decay are shown in Fig. 2. There are two types. One is factorizable emission
topology where gluon attaches to quarks in the same meson, and the other is
nonfactorizable emission topology where gluon connects to quarks between
different mesons.

Figure 2: Feynman diagrams for ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$
decay with the pQCD approach, including factorizable emission diagrams (a,b)
and nonfactorizable emission diagrams (c,d).

With the pQCD master formula Eq.(6), the amplitude for ${\Upsilon}(nS)$
${\to}$ $B_{c}^{\ast}P$ decay can be expressed as prd66 ,

${\cal A}({\Upsilon}(nS){\to}B_{c}^{\ast}P)\ =\ {\cal
A}_{L}({\epsilon}_{1}^{{\parallel}},{\epsilon}_{2}^{{\parallel}})+{\cal
A}_{N}({\epsilon}_{1}^{{\perp}},{\epsilon}_{2}^{{\perp}})+i\,{\cal
A}_{T}\,{\varepsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,{\epsilon}_{1}^{{\mu}}\,{\epsilon}_{2}^{{\nu}}\,p_{1}^{\alpha}\,p_{2}^{\beta},$
(42)

which is conventionally written as the helicity amplitudes prd66 ,

${\cal A}_{0}\ =\ -C_{\cal A}\,\sum\limits_{j}{\cal
A}_{L}^{j}({\epsilon}_{1}^{{\parallel}},{\epsilon}_{2}^{{\parallel}}),$ (43)
${\cal A}_{\parallel}\ =\ \sqrt{2}\,C_{\cal A}\,\sum\limits_{j}{\cal
A}_{N}^{j}({\epsilon}_{1}^{{\perp}},{\epsilon}_{2}^{{\perp}}),$ (44) ${\cal
A}_{\perp}\ =\ \sqrt{2}\,C_{\cal A}\,m_{1}\,p\,\sum\limits_{j}{\cal
A}_{T}^{j},$ (45) $C_{\cal A}\ =\
i\,V_{cb}V_{uq}^{\ast}\,\frac{G_{F}}{\sqrt{2}}\,\frac{C_{F}}{N_{c}}\,{\pi}\,f_{{\Upsilon}}\,f_{B_{c}^{\ast}}\,f_{P},$
(46)

where $C_{F}$ $=$ $4/3$ and the color number $N_{c}$ $=$ $3$; the subscript
$i$ on $A_{i}^{j}$ corresponds to three different helicity amplitudes, i.e.,
$i$ $=$ $L$, $N$, $T$; the superscript $j$ on $A_{i}^{j}$ denotes to indices
of Fig. 2. The explicit expressions of building blocks ${\cal A}_{i}^{j}$ are
collected in Appendix A.

##  III Numerical results and discussion

In the center-of-mass of ${\Upsilon}(nS)$ meson, branching ratio ${\cal B}r$
for ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}P$ decay are defined as

${\cal B}r\ =\
\frac{1}{12{\pi}}\,\frac{p}{m_{{\Upsilon}}^{2}{\Gamma}_{{\Upsilon}}}\,\Big{\\{}{|}{\cal
A}_{0}{|}^{2}+{|}{\cal A}_{\parallel}{|}^{2}+{|}{\cal
A}_{\perp}{|}^{2}\Big{\\}}.$ (47)

The input parameters are listed in Table 1 and 2. If not specified explicitly,
we will take their central values as the default inputs. Our numerical results
are collected in Table. 3, where the first uncertainty comes from scale
$(1{\pm}0.1)t_{i}$ and the expression of $t_{i}$ is given in Eq.(67) and
Eq.(68); the second uncertainty is from mass $m_{b}$ and $m_{c}$; the third
uncertainty is from hadronic parameters including decay constants and
Gegenbauer moments; the fourth uncertainty is from CKM parameters. The
followings are some comments.

Table 2: The numerical values of input parameters. The Wolfenstein parameters
---
$A$ $=$ $0.814^{+0.023}_{-0.024}$ pdg , |  ${\lambda}$ $=$ $0.22537{\pm}0.00061$ pdg ,
Mass, decay constant and Gegenbauer moments
$m_{b}$ $=$ $4.78{\pm}0.06$ GeV pdg , |  $f_{\pi}$ $=$ $130.41{\pm}0.20$ MeV pdg ,
$m_{c}$ $=$ $1.67{\pm}0.07$ GeV pdg , |  $f_{K}$ $=$ $156.2{\pm}0.7$ MeV pdg ,
$m_{B_{c}^{\ast}}$ $=$ $6332{\pm}9$ MeV mbc , |  $f_{B_{c}^{\ast}}$ $=$ $422{\pm}13$ MeV fbc 333The decay constant $f_{B_{c}^{\ast}}$ cannot be extracted from the experimental data because of no measurement on $B_{c}^{\ast}$ weak decay at the present time. Theoretically, the value of $f_{B_{c}^{\ast}}$ has been estimated, for example, in Ref. epja49  with the QCD sum rules. From Table. 3 of Ref. epja49 , one can see that the value of $f_{B_{c}^{\ast}}$ are model-dependent. In our calculation, we will take the latest value given by the lattice QCD approach fbc  just to offer an order of magnitude estimation on branching ratio for ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}P$ decays.,
$a_{1}^{K}$ (1GeV) $=$ $-0.06{\pm}0.03$ jhep0605 , |  $f_{{\Upsilon}(1S)}$ $=$ $676.4{\pm}10.7$ MeV plb751 ,
$a_{2}^{K}$ (1GeV) $=$ $0.25{\pm}0.15$ jhep0605 , |  $f_{{\Upsilon}(2S)}$ $=$ $473.0{\pm}23.7$ MeV plb751 ,
$a_{2}^{\pi}$ (1GeV) $=$ $0.25{\pm}0.15$ jhep0605 , |  $f_{{\Upsilon}(3S)}$ $=$ $409.5{\pm}29.4$ MeV plb751 .
Table 3: Branching ratio for ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}P$ decays. modes |  ${\Upsilon}(1S)$ ${\to}$ $B_{c}^{\ast}{\pi}$ |  ${\Upsilon}(2S)$ ${\to}$ $B_{c}^{\ast}{\pi}$ |  ${\Upsilon}(3S)$ ${\to}$ $B_{c}^{\ast}{\pi}$
---|---|---|---
$10^{10}{\times}{\cal B}r$ | $4.35^{+0.29+0.19+0.44+0.17}_{-0.24-0.41-0.31-0.30}$ | $2.28^{+0.13+0.26+0.40+0.09}_{-0.03-0.35-0.16-0.15}$ | $2.14^{+0.12+0.09+0.48+0.07}_{-0.12-0.41-0.15-0.15}$
modes |  ${\Upsilon}(1S)$ ${\to}$ $B_{c}^{\ast}K$ |  ${\Upsilon}(2S)$ ${\to}$ $B_{c}^{\ast}K$ |  ${\Upsilon}(3S)$ ${\to}$ $B_{c}^{\ast}K$
$10^{11}{\times}{\cal B}r$ | $3.45^{+0.23+0.13+0.38+0.13}_{-0.21-0.35-0.27-0.25}$ | $1.91^{+0.11+0.07+0.36+0.07}_{-0.09-0.31-0.15-0.14}$ | $1.65^{+0.09+0.08+0.40+0.05}_{-0.21-0.33-0.13-0.12}$

(1) Branching ratio for ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$ decay is
about ${\cal O}(10^{-10})$ with pQCD approach, which is well within the
measurement potential of LHC and SuperKEKB. For example, experimental studies
have showed that production cross sections for ${\Upsilon}(nS)$ meson in p-p
and p-Pb collisions are a few ${\mu}b$ at the LHCb epjc74lhcb ; jhep1407  and
ALICE prd87 ; plb740  detectors. Consequently, there will be more than
$10^{12}$ ${\Upsilon}(nS)$ data samples per $ab^{-1}$ data collected by the
LHCb and ALICE, corresponding to a few hundreds of ${\Upsilon}(nS)$ ${\to}$
$B_{c}^{\ast}{\pi}$ events. Branching ratio for ${\Upsilon}(nS)$ ${\to}$
$B_{c}^{\ast}K$ decay, ${\cal O}(10^{-11})$, is generally less than that for
${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$ decay by one order of magnitude
due to the CKM suppression, ${|}V_{us}^{\ast}/V_{ud}^{\ast}{|}^{2}$ ${\sim}$
${\lambda}^{2}$.

Figure 3: The contributions to branching ratios for ${\Upsilon}(1S)$ ${\to}$
$B_{c}^{\ast}{\pi}$ decay (a), ${\Upsilon}(2S)$ ${\to}$ $B_{c}^{\ast}{\pi}$
decay (b) and ${\Upsilon}(3S)$ ${\to}$ $B_{c}^{\ast}{\pi}$ decay (c) from
different region of ${\alpha}_{s}/{\pi}$ (horizontal axises), where the
numbers over histogram denote the percentage of the corresponding
contributions.

(2) As it is well known, due to the large mass of $B_{c}^{\ast}$, the momentum
transition in the ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}P$ 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 for ${\Upsilon}(nS)$
${\to}$ $B_{c}^{\ast}{\pi}$ decay from different ${\alpha}_{s}/{\pi}$ region
are showed in Fig. 3. It can be clearly seen that more than 93% (97%)
contributions come from the ${\alpha}_{s}/{\pi}$ ${\leq}$ $0.2$ (0.3) region,
implying that the ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$ decay is
computable with the pQCD approach. As the discussion in pqcd1 ; pqcd2 ; pqcd3
, there are many factors for this, for example, the choice of the typical
scale, retaining the quark transverse moment and introducing the Sudakov
factor to suppress the nonperturbative contributions, which deserve much
attention and further investigation.

(3) Because of the relations among masses $m_{{\Upsilon}(3S)}$ $>$
$m_{{\Upsilon}(2S)}$ $>$ $m_{{\Upsilon}(1S)}$ resulting in the fact that phase
space increases with the radial quantum number $n$, in addition, the relations
among decay widths ${\Gamma}_{{\Upsilon}(3S)}$ $<$ ${\Gamma}_{{\Upsilon}(2S)}$
$<$ ${\Gamma}_{{\Upsilon}(1S)}$, in principle, there should be relations among
branching ratios ${\cal B}r({\Upsilon}(3S){\to}B_{c}^{\ast}P)$ $>$ ${\cal
B}r({\Upsilon}(2S){\to}B_{c}^{\ast}P)$ $>$ ${\cal
B}r({\Upsilon}(1S){\to}B_{c}^{\ast}P)$ for the same pseudoscalar meson $P$.
But the numerical results in Table. 3 are beyond such expectation. Why? The
reason is that the factor of $p/m_{{\Upsilon}(nS)}^{2}$ in Eq.(47) has almost
the same value for $n$ ${\leq}$ $3$, so branching ratio is proportional to
factor $f_{{\Upsilon}(nS)}^{2}/{\Gamma}_{{\Upsilon}(nS)}$ with the maximal
value $f_{{\Upsilon}(1S)}^{2}/{\Gamma}_{{\Upsilon}(1S)}$ for $n$ ${\leq}$ $3$.
Besides, contributions from ${\alpha}_{s}/{\pi}$ ${\in}$ $[0.2,0.3]$ regions
decrease with $n$ (see Fig. 3), which enhance the decay amplitudes.

(4) Besides the uncertainties listed in Table 3, 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 ${\Upsilon}(nS)$ decay via the weak interaction, as a complementary to
strong and electromagnetic decay mechanism, is allowable within the standard
model. Based on the potential prospects of ${\Upsilon}(nS)$ physics at high-
luminosity collider experiment, ${\Upsilon}(nS)$ decay into
$B_{c}^{\ast}{\pi}$ and $B_{c}^{\ast}K$ final states is investigated with the
pQCD approach firstly. It is found that (1) the dominant contributions come
from perturbative regions ${\alpha}_{s}/{\pi}$ ${\leq}$ $0.3$, which might
imply that the pQCD calculation is practicable and workable; (2) there is a
promiseful possibility of searching for ${\Upsilon}(nS)$ ${\to}$
$B_{c}^{\ast}{\pi}$ ($B_{c}^{\ast}K$) decay with branching ratio about
$10^{-10}$ ($10^{-11}$) at the future experiments.

## Acknowledgments

The work is supported by the National Natural Science Foundation of China
(Grant Nos. 11547014, 11475055, U1332103 and 11275057).

##  Appendix A Building blocks for ${\Upsilon}$ ${\to}$ $B_{c}^{\ast}P$ decays

The building blocks ${\cal A}_{i}^{j}$, where the superscript $j$ corresponds
to indices of Fig. 2 and the subscript $i$ relates with different helicity
amplitudes, are expressed as follows.

$\displaystyle{\cal A}_{L}^{a}$ $\displaystyle=$
$\displaystyle{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}\,H_{ab}({\alpha}_{e},{\beta}_{a},b_{1},b_{2})\,E_{ab}(t_{a})\,{\phi}_{\Upsilon}^{v}(x_{1})\,{\alpha}_{s}(t_{a})$
(48) $\displaystyle
a_{1}(t_{a})\,\Big{\\{}{\phi}_{B_{c}^{\ast}}^{v}(x_{2})\,\Big{[}m_{1}^{2}\,s-(4\,m_{1}^{2}\,p^{2}+m_{2}^{2}\,u)\,\bar{x}_{2}\Big{]}+{\phi}_{B_{c}^{\ast}}^{t}(x_{2})\,m_{2}\,m_{b}\,u\Big{\\}},$
$\displaystyle{\cal A}_{N}^{a}$ $\displaystyle=$
$\displaystyle{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}\,H_{ab}({\alpha}_{e},{\beta}_{a},b_{1},b_{2})\,E_{ab}(t_{a})\,{\phi}_{\Upsilon}^{V}(x_{1})$
(49)
$\displaystyle{\alpha}_{s}(t_{a})\,a_{1}(t_{a})\,m_{1}\,\Big{\\{}{\phi}_{B_{c}^{\ast}}^{V}(x_{2})\,m_{2}\,(u-s\,\bar{x}_{2})+{\phi}_{B_{c}^{\ast}}^{T}(x_{2})\,m_{b}\,s\Big{\\}},$
$\displaystyle{\cal A}_{T}^{a}$ $\displaystyle=$
$\displaystyle-2\,m_{1}\,{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}\,H_{ab}({\alpha}_{e},{\beta}_{a},b_{1},b_{2})\,E_{ab}(t_{a})$
(50)
$\displaystyle{\phi}_{\Upsilon}^{V}(x_{1})\,{\alpha}_{s}(t_{a})\,a_{1}(t_{a})\,\Big{\\{}{\phi}_{B_{c}^{\ast}}^{T}(x_{2})\,m_{b}+{\phi}_{B_{c}^{\ast}}^{V}(x_{2})\,m_{2}\,x_{2}\Big{\\}},$
$\displaystyle{\cal A}_{L}^{b}$ $\displaystyle=$
$\displaystyle{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}\,H_{ab}({\alpha}_{e},{\beta}_{b},b_{2},b_{1})\,E_{ab}(t_{b})\,{\phi}_{B_{c}^{\ast}}^{v}(x_{2})$
(51)
$\displaystyle{\alpha}_{s}(t_{b})\,a_{1}(t_{b})\,\Big{\\{}{\phi}_{\Upsilon}^{v}(x_{1})\,\Big{[}m_{2}^{2}\,u-m_{1}^{2}\,(s-4\,p^{2})\,\bar{x}_{1}\Big{]}+{\phi}_{\Upsilon}^{t}(x_{1})\,m_{1}\,m_{c}\,s\Big{\\}},$
$\displaystyle{\cal A}_{N}^{b}$ $\displaystyle=$
$\displaystyle{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}\,H_{ab}({\alpha}_{e},{\beta}_{b},b_{2},b_{1})\,E_{ab}(t_{b})\,{\phi}_{B_{c}^{\ast}}^{V}(x_{2})$
(52)
$\displaystyle{\alpha}_{s}(t_{b})\,a_{1}(t_{b})\,m_{2}\,\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,m_{1}\,(s-u\,\bar{x}_{1})+{\phi}_{\Upsilon}^{T}(x_{1})\,m_{c}\,u\Big{\\}},$
$\displaystyle{\cal A}_{T}^{b}$ $\displaystyle=$
$\displaystyle-2\,m_{2}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{1}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}\,H_{ab}({\alpha}_{e},{\beta}_{b},b_{2},b_{1})\,E_{ab}(t_{b})$
(53)
$\displaystyle{\phi}_{B_{c}^{\ast}}^{V}(x_{2})\,{\alpha}_{s}(t_{b})\,a_{1}(t_{b})\,\Big{\\{}{\phi}_{\Upsilon}^{V}(x_{1})\,m_{1}\,x_{1}+{\phi}_{\Upsilon}^{T}(x_{1})\,m_{c}\Big{\\}},$
$\displaystyle{\cal A}_{L}^{c}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,H_{cd}({\alpha}_{e},{\beta}_{c},b_{2},b_{3})$
(54) $\displaystyle
E_{cd}(t_{c})\,{\phi}_{P}^{a}(x_{3})\,{\alpha}_{s}(t_{c})\,C_{2}(t_{c})\,\Big{\\{}{\phi}_{\Upsilon}^{v}(x_{1})\,{\phi}_{B_{c}^{\ast}}^{v}(x_{2})\,4\,m_{1}^{2}\,p^{2}\,(x_{1}-\bar{x}_{3})$
$\displaystyle+{\phi}_{\Upsilon}^{t}(x_{1})\,{\phi}_{B_{c}^{\ast}}^{t}(x_{2})\,m_{1}\,m_{2}\,(u\,x_{1}-s\,x_{2}-2\,m_{3}^{2}\,\bar{x}_{3})\Big{\\}}{\delta}(b_{1}-b_{2}),$
$\displaystyle{\cal A}_{N}^{c}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,H_{cd}({\alpha}_{e},{\beta}_{c},b_{2},b_{3})\,E_{cd}(t_{c})\,C_{2}(t_{c})$
(55)
$\displaystyle{\alpha}_{s}(t_{c})\,{\delta}(b_{1}-b_{2})\,{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{B_{c}^{\ast}}^{T}(x_{2})\,{\phi}_{P}^{a}(x_{3})\,\Big{\\{}m_{1}^{2}\,s\,(x_{1}-\bar{x}_{3})+m_{2}^{2}\,u\,(\bar{x}_{3}-x_{2})\Big{\\}},$
$\displaystyle{\cal A}_{T}^{c}$ $\displaystyle=$
$\displaystyle\frac{2}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,H_{cd}({\alpha}_{e},{\beta}_{c},b_{2},b_{3})\,E_{cd}(t_{c})\,C_{2}(t_{c})$
(56)
$\displaystyle{\alpha}_{s}(t_{c})\,{\delta}(b_{1}-b_{2})\,{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{B_{c}^{\ast}}^{T}(x_{2})\,{\phi}_{P}^{a}(x_{3})\,\Big{\\{}m_{1}^{2}\,(\bar{x}_{3}-x_{1})+m_{2}^{2}\,(x_{2}-\bar{x}_{3})\Big{\\}},$
$\displaystyle{\cal A}_{L}^{d}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,H_{cd}({\alpha}_{e},{\beta}_{d},b_{2},b_{3})$
(57) $\displaystyle
E_{cd}(t_{d})\,{\phi}_{P}^{a}(x_{3})\,{\alpha}_{s}(t_{d})\,C_{2}(t_{d})\,\Big{\\{}{\phi}_{\Upsilon}^{v}(x_{1})\,{\phi}_{B_{c}^{\ast}}^{v}(x_{2})\,4\,m_{1}^{2}\,p^{2}\,(x_{3}-x_{2})$
$\displaystyle+{\phi}_{\Upsilon}^{t}(x_{1})\,{\phi}_{B_{c}^{\ast}}^{t}(x_{2})\,m_{1}\,m_{2}\,(s\,x_{2}+2\,m_{3}^{2}\,x_{3}-u\,x_{1})\Big{\\}}\,{\delta}(b_{1}-b_{2}),$
$\displaystyle{\cal A}_{N}^{d}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,H_{cd}({\alpha}_{e},{\beta}_{d},b_{2},b_{3})\,E_{cd}(t_{d})\,C_{2}(t_{d})$
(58)
$\displaystyle{\alpha}_{s}(t_{d})\,{\delta}(b_{1}-b_{2})\,{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{B_{c}^{\ast}}^{T}(x_{2})\,{\phi}_{P}^{a}(x_{3})\,\Big{\\{}m_{1}^{2}\,s\,(x_{3}-x_{1})+m_{2}^{2}\,u\,(x_{2}-x_{3})\Big{\\}},$
$\displaystyle{\cal A}_{T}^{d}$ $\displaystyle=$
$\displaystyle\frac{2}{N_{c}}{\int}_{0}^{1}dx_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}{\int}_{0}^{\infty}db_{1}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^{\infty}b_{3}db_{3}\,H_{cd}({\alpha}_{e},{\beta}_{d},b_{2},b_{3})\,E_{cd}(t_{d})\,C_{2}(t_{d})$
(59)
$\displaystyle{\alpha}_{s}(t_{d})\,{\delta}(b_{1}-b_{2})\,{\phi}_{\Upsilon}^{T}(x_{1})\,{\phi}_{B_{c}^{\ast}}^{T}(x_{2})\,{\phi}_{P}^{a}(x_{3})\,\Big{\\{}m_{1}^{2}\,(x_{1}-x_{3})-m_{2}^{2}\,(x_{2}-x_{3})\Big{\\}},$

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 $k_{i{\perp}}$; and ${\alpha}_{s}(t)$ is the QCD
coupling at the scale of $t$; $a_{1}$ $=$ $C_{1}$ $+$ $C_{2}/N_{c}$.

The function $H_{i}$ are defined as follows plb751 .

$\displaystyle H_{ab}({\alpha}_{e},{\beta},b_{i},b_{j})$ $\displaystyle=$
$\displaystyle
K_{0}(\sqrt{-{\alpha}}b_{i})\Big{\\{}{\theta}(b_{i}-b_{j})K_{0}(\sqrt{-{\beta}}b_{i})I_{0}(\sqrt{-{\beta}}b_{j})+(b_{i}{\leftrightarrow}b_{j})\Big{\\}},$
(60) $\displaystyle H_{cd}({\alpha}_{e},{\beta},b_{2},b_{3})$ $\displaystyle=$
$\displaystyle\Big{\\{}{\theta}(-{\beta})K_{0}(\sqrt{-{\beta}}b_{3})+\frac{{\pi}}{2}{\theta}({\beta})\Big{[}iJ_{0}(\sqrt{{\beta}}b_{3})-Y_{0}(\sqrt{{\beta}}b_{3})\Big{]}\Big{\\}}$
(61) $\displaystyle{\times}$
$\displaystyle\Big{\\{}{\theta}(b_{2}-b_{3})K_{0}(\sqrt{-{\alpha}}b_{2})I_{0}(\sqrt{-{\alpha}}b_{3})+(b_{2}{\leftrightarrow}b_{3})\Big{\\}}$

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 plb751 .

$\displaystyle{\alpha}$ $\displaystyle=$
$\displaystyle\bar{x}_{1}^{2}\,m_{1}^{2}+\bar{x}_{2}^{2}\,m_{2}^{2}-\bar{x}_{1}\,\bar{x}_{2}\,t,$
(62) $\displaystyle{\beta}_{a}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{b}^{2}+\bar{x}_{2}^{2}\,m_{2}^{2}-\bar{x}_{2}\,t,$ (63)
$\displaystyle{\beta}_{b}$ $\displaystyle=$ $\displaystyle
m_{2}^{2}-m_{c}^{2}+\bar{x}_{1}^{2}\,m_{1}^{2}-\bar{x}_{1}\,t,$ (64)
$\displaystyle{\beta}_{c}$ $\displaystyle=$ $\displaystyle
x_{1}^{2}\,m_{1}^{2}+x_{2}^{2}\,m_{2}^{2}+\bar{x}_{3}^{2}\,m_{3}^{2}$ (65)
$\displaystyle-$ $\displaystyle
x_{1}\,x_{2}\,t-x_{1}\,\bar{x}_{3}\,u+x_{2}\,\bar{x}_{3}\,s,$
$\displaystyle{\beta}_{d}$ $\displaystyle=$ $\displaystyle
x_{1}^{2}\,m_{1}^{2}+x_{2}^{2}\,m_{2}^{2}+x_{3}^{2}\,m_{3}^{2}$ (66)
$\displaystyle-$ $\displaystyle
x_{1}\,x_{2}\,t-x_{1}\,x_{3}\,u+x_{2}\,x_{3}\,s.$

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.

$\displaystyle t_{a(b)}$ $\displaystyle=$
$\displaystyle{\max}(\sqrt{-{\alpha}},\sqrt{-{\beta}_{a(b)}},1/b_{1},1/b_{2}),$
(67) $\displaystyle t_{c(d)}$ $\displaystyle=$
$\displaystyle{\max}(\sqrt{-{\alpha}},\sqrt{{|}{\beta}_{c(d)}{|}},1/b_{2},1/b_{3}),$
(68) $\displaystyle E_{ab}(t)$ $\displaystyle=$
$\displaystyle{\exp}\\{-S_{{\Upsilon}}(t)-S_{B_{c}^{\ast}}(t)\\},$ (69)
$\displaystyle E_{cd}(t)$ $\displaystyle=$
$\displaystyle{\exp}\\{-S_{{\Upsilon}}(t)-S_{B_{c}^{\ast}}(t)-S_{P}(t)\\},$
(70) $S_{{\Upsilon}}(t)\ =\
s(x_{1},p_{1}^{+},1/b_{1})+2{\int}_{1/b_{1}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q},$
(71) $S_{B_{c}^{\ast}}(t)\ =\
s(x_{2},p_{2}^{+},1/b_{2})+2{\int}_{1/b_{2}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q},$
(72) $S_{{\pi},K}(t)\ =\
s(x_{3},p_{3}^{+},1/b_{3})+s(\bar{x}_{3},p_{3}^{+},1/b_{3})+2{\int}_{1/b_{3}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q},$
(73)

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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