# The next-to-leading order corrections to $B\to\rho$ transition in the
$k_{T}$ factorization

Jun Hua1<EMAIL_ADDRESS>Ya-Lan Zhang2<EMAIL_ADDRESS>Zhen-Jun Xiao1,3
<EMAIL_ADDRESS>1\. Department of Physics and Institute of Theoretical
Physics, Nanjing Normal University, Nanjing, Jiangsu 210023, People’s Republic
of China,  2\. Department of Faculty of Mathematics and Physics, Huaiyin
Institute of Technology, Huaian, Jiangsu 223001, People’s Republic of China,
3\. Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex
Systems, Nanjing Normal University, Nanjing 210023, People’s Republic of China

###### Abstract

In this paper, we investigate the factorization hypothesis step by step for
the exclusive process $B\to\rho$ at next-to-leading order (NLO) with the
collinear factorization approach, and then we extend our results to the
$k_{T}$ factorization frame. We show that the soft divergence from the
specific NLO diagrams will cancel each other at the quark level, while the
remaining collinear divergence can be absorbed into the NLO wave functions.
The full NLO amplitudes can be factorized into two parts: the NLO $B$ and
$\rho$ meson wave functions containing the collinear divergence and the
leading order (LO) finite hard kernels. We give the general form of the
nonlocal hadron matrix for the NLO $B$ and $\rho$ meson wave functions and all
results of factorization for different twists’ combinations.

###### pacs:

11.80.Fv, 12.38.Bx, 12.38.Cy, 12.39.St

##  I Introduction

As the foundation of perturbative QCD formalism, the factorization theorem
plb87-359 ; prd55-272 ; prp112-173  claims that hard part of the QCD
interactions is finite and can be calculated perturbatively, meanwhile the
non-perturbative part can be factorized into the universal wave functions
defined in an infinite momentum frame. The perturbative QCD (PQCD) approachkt-
fact  based on the $k_{T}$ factorization theorem is proposed to eliminate
endpoint divergence by picking up the transversal momentum of the propagators
together with the Sudakov resummation techniquePQCD-fact1 ; prove  .

In recent years, several exclusive processes $\pi\gamma^{*}\to\gamma(\pi)$,
$\rho\gamma^{*}\to\pi$ and $B\to\gamma(\pi)l\bar{\nu}$ have been investigated
by many authors for example in Refs. PQCD-fact1 ; prove ; Zhang-Hua ; PQCD-
NLO-Li ; PQCD-NLO-Cheng  at the leading order (LO) and next-to-leading order
(NLO). The calculations of the space like and time like form factors PQCD-NLO-
timelike  will be a great help for high precision studies for B meson decays.

In this paper, taking the $\rho$ meson as one example for those light vector
mesons, we consider the $B\to\rho$ transition process, give the proof of
collinear factorization at the NLO level for leading twist parts. Fierz
identity and eikonal approximation will be taken into account to factorize the
fermion flow and momentum flow respectively. The power counting for various
gamma matrices we discussed in prd95-076005  are also considered in order to
give the right color factors with triple gluon vertex diagrams. By summing up
the contributions from all sub-diagrams, we can prove that the soft divergence
will cancel each other between quark diagrams, and the remaining collinear
divergence can also be absorbed into the NLO meson wave functions. The
convolutions of the NLO wave function and hard kernel have two forms: the one
has additional gluon emitted from external quarks flowing into hard kernel;
another has no additional gluon flowing into hard kernel. We will finally
write down the hadronic matrix elements for $B$ and $\rho$ meson wave
functions with collinear parts in $k_{T}$ factorization.

This paper is organized as follows. In section ii@, we show the dynamical
analysis and the leading order amplitudes for $B\to\rho$ transition. In
section iii@, we present the proof of collinear factorization for $B\to\rho$
transition at the NLO step by step, and give the hadronic matrix elements for
wave functions in $k_{T}$ factorization. In section iv@, finally, a brief
summary and some discussions will be given.

##  II Leading order hard kernel

Figure 1: The leading-order quark diagrams for the $B\to\rho$ transition form
factors with the symbol $\otimes$ representing the weak vertex.

In this section, we calculate the leading order $O(\alpha_{s})$ hard kernel
for $B\to\rho$ transition, and the topological diagrams of leading-order
process are displayed in Fig. 1. The definition of the kinematics in light
cone coordinate are of the following form:

$\displaystyle p_{1\mu}$ $\displaystyle=$
$\displaystyle\frac{m_{B}}{\sqrt{2}}(1,1,\mathbf{0_{T}}),\qquad
k_{1\mu}=\frac{m_{B}}{\sqrt{2}}(x_{1},0,\mathbf{0_{T}});$
$\displaystyle\epsilon_{1\mu}(L)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(1,0,\mathbf{0_{T}}),~{}~{}~{}~{}~{}\quad\epsilon_{1\mu}(T)=(0,0,\mathbf{1_{T}});$
(1) $\displaystyle p_{2\mu}$ $\displaystyle=$ $\displaystyle\frac{\eta
m_{B}}{\sqrt{2}}(0,1,\mathbf{0_{T}}),~{}~{}~{}~{}~{}\quad k_{2\mu}=\frac{\eta
m_{B}}{\sqrt{2}}(0,x_{2},\mathbf{0_{T}});$ $\displaystyle\epsilon_{2\mu}(L)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}\gamma_{\rho}}(-\gamma_{\rho}^{2},1,\mathbf{0_{T}}),~{}~{}~{}~{}~{}\epsilon_{2\mu}(T)=(0,0,\mathbf{1_{T}}),$
(2)

where the $m_{B}$ is the mass of $B$-meson, the momentum $P_{1}$ is chosen as
${m_{B}}/\sqrt{2}(1,1,\mathbf{0_{T}})$, both $p^{+}$ and $p^{-}$ parts count
due to the large scale of $m_{B}$. The energy fraction $\eta$ of the $\rho$
meson in large recoil region is of order $\mathbf{1}$, and the polarization
vector with the definition $\gamma_{\rho}=m_{\rho}/Q$ is defined by condition
$\epsilon^{2}_{i}(L/T)=-1$. The momentum transfer squared
$Q^{2}=-(p_{1}-p_{2})^{2}\quad(Q^{2}>0)$ is taken to represent evolution
behaviors of form factors.

The leading twist (twist-2) wave functions of B and $\rho$ mesons are chosen
in the same form as those in Refs. WfB ; WFsrho :

$\displaystyle\Phi_{B}(x_{1},p_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2N_{c}}}(p/_{1}+M_{B})\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})+\frac{1}{\sqrt{2}}\left(n/_{-}-k^{+}_{1}\gamma^{\nu}_{\perp}\frac{\partial}{\partial\mathbf{k}^{\nu}_{1T}}\right)\phi^{-}_{B}(x_{1})\right],$
$\displaystyle\Phi_{\rho}(p_{2},\epsilon_{2L})$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2N_{c}}}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right],$
$\displaystyle\Phi_{\rho}(p_{2},\epsilon_{2T})$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2N_{c}}}\left[M_{\rho}\epsilon/_{2T}\phi^{v}_{\rho}(x_{2})\right],$
(3)

where $N_{c}=3$ is the number of colors, $n_{+}=(1,0,0)$ and $n_{-}=(0,1,0)$
are the unit vector in the light cone coordinate. The wave functions of the
vector $\rho$ meson contain both the longitudinal and transversal component
$\Phi_{\rho}(p_{2},\epsilon_{2L})$ and $\Phi_{\rho}(p_{2},\epsilon_{2T})$. To
insure the gauge invariance and the accuracy of calculation, both
contributions from $\phi^{+}_{B}$ and $\phi^{-}_{B}$ will be taken into
account in this work.

Because of the symmetry relations, the contributions from Fig. 1(c,d) can be
simply derived from those for Fig. 1(a,b) by exchange of momentum fraction for
up (down) quarks. Then the LO transition amplitude of Fig. 1(a) and (b) can be
simplified to the following forms:

$\displaystyle H^{(0)}_{aL}$ $\displaystyle=$
$\displaystyle-\frac{ieg^{2}C_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})]\gamma_{\alpha}(p/_{2}-k/_{1})\gamma_{\mu}(p/_{1}+m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})+\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{(p_{2}-k_{1})^{2}(k_{1}-k_{2})^{2}}\bigg{\\}}$
(4) $\displaystyle=$ $\displaystyle-
ieg^{2}C_{F}m_{B}\mathbf{Tr}\left[\frac{\epsilon/_{2L}k/_{1}\gamma_{\mu}\gamma_{5}\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})}{(p_{2}-k_{1})^{2}(k_{1}-k_{2})^{2}}\right],$
$\displaystyle H^{(0)}_{bL}$ $\displaystyle=$ $\displaystyle-
ieg^{2}C_{F}m_{B}\mathbf{Tr}\left[\frac{M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\gamma_{\mu}k/_{2}\frac{n/+}{\sqrt{2}}\phi^{+}_{B}(x_{1})\gamma_{5}+M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\gamma_{\mu}m_{b}\phi^{-}_{B}(x_{1})\gamma_{5}}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}}\right],$
(5) $\displaystyle H^{(0)}_{bT}$ $\displaystyle=$ $\displaystyle-
ieg^{2}C_{F}m_{B}\mathbf{Tr}\left[\frac{\Big{[}\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\Big{]}\gamma_{\mu}p/_{1}\Big{[}\phi^{+}_{B}(x_{1})+\phi^{-}_{B}(x_{1})\Big{]}\gamma_{5}-\Big{[}\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\Big{]}\gamma_{\mu}m_{b}\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\gamma_{5}}{\Big{[}(p_{1}-k_{2})^{2}-m^{2}_{b}\Big{]}(k_{1}-k_{2})^{2}}\right],$
(6)

where $C_{F}=4/3$ is the color factor. Since $H^{(0)}_{aT}=0$ for Fig. 1(a),
we do not show $H^{(0)}_{aT}$ explicitly in Eq. (4). For $H^{(0)}_{aL}$, only
the $k/_{1}$ in quark propagator and $m_{B}$ component in B wave function have
contributions, which also requires $\gamma^{\alpha}$ and $\gamma_{\mu}$ should
be $\gamma_{\perp}$ and $\gamma^{+}$. $H^{(0)}_{bL}$ contains two components
corresponding to the two parts of the $B$ wave function $\phi^{+}_{B}$ and
$\phi^{-}_{B}$ respectively, which leads to different restrictions on the
vertex with $\gamma_{\alpha}$ and $\gamma_{\mu}$. Similarly, $H^{(0)}_{bT}$
contains three components, two for $\phi^{+}_{B}$ and one for $\phi^{-}_{B}$.
Every part of the combinations requires unique choice for $P_{1}$ momentum
$(P^{-}_{1}/P^{+}_{1})$ and gamma matrix $\gamma_{\mu}$ and $\gamma_{\alpha}$.
We should consider each part independently.

##  III Factorization Of $B\to\rho$ At Next-To-Leading Order

In this section, we will give the proof of collinear factorization for the
exclusive process $B\to\rho$ at NLO level, considering both momentum space and
color space factorization but neglecting the transverse momentum first. At the
end of this section, we will pick up again the transverse momentum and discuss
how to deal with the infrared contribution carrying by $k_{T}$-related part.

The fermion flow in momentum space can be factorized by inserting the Fierz
identity

$\displaystyle I_{ij}I_{lk}$ $\displaystyle=$
$\displaystyle\frac{1}{4}I_{ik}I_{lj}+\frac{1}{4}(\gamma_{5})_{ik}(\gamma_{5})_{lj}+\frac{1}{4}(\gamma^{\alpha})_{ik}(\gamma^{\alpha})_{lj}$
(7)
$\displaystyle+\frac{1}{4}(\gamma_{5}\gamma^{\alpha})_{ik}(\gamma_{\alpha}\gamma_{5})_{lj}+\frac{1}{8}(\sigma^{\alpha\beta})_{ik}(\sigma_{\alpha\beta})_{lj},$

into the proper place of the matrix elements, here
$\sigma^{\alpha\beta}=i[\gamma^{\alpha},\gamma^{\beta}]/2$. The different
terms on the right hand side correspond to the contributions from different
twists of $B(\rho)$ meson. The eikonal approximation is taken before inserting
the Fierz identity to reformulate the singularity propagator into a simplified
form and reduce the gamma matrix for convenience meanwhile. All the possible
diagrams for a gluon radiation should be considered and finally be resumed to
collect the color factors and maintain the gauge invariance.

###  III.1 Collinear factorization for the NLO(${{\cal O}}(\alpha^{2}_{s})$)
corrections to $H^{(0)}_{aL}$

Figure 2: The quark diagrams for NLO corrections to Fig. 1(a) with additional
gluon emitted from the quark(antiquark) lines of initial B meson

We will only show the factorization of the NLO corrections for Fig. 1(a,b),
for the reason that Fig. 1(c,d) can be derived from Fig. 1(a,b) by symmetry.
Taking into account the different twists’ combinations, there exist six
different structures from the decay amplitudes as listed Eqs. (4),(5). Each
structure has two terms corresponding to the additional gluon lines emitted
from the initial $B$ meson or the finial $\rho$ meson.

The topological diagrams for NLO corrections to Fig. 1(a) with additional
gluon emitted from initial B meson are showed in Fig. 2. Generally, there
exists two kinds of infrared divergence: (a) the soft divergence when the
momentum of additional gluon is small at
$l=(l^{+},l^{-},l_{\perp})\sim(\lambda,\lambda,\lambda)$ for
$\lambda\sim\Lambda_{QCD}$; and (b) the collinear divergence when the
additional gluon is parallel with one longitudinal direction
$l=(l^{+},l^{-},l_{\perp})\sim(\lambda^{2}/Q^{2},Q,\lambda^{2})$jhep0802-002 ;
npb844-199 . The third kind infrared divergence appeared in the Glauber region
of NLO spectator amplitudes of B meson two-body nonleptonic decaysprd90-074018
; prd91-114019  does not exist here. We rewrite the Eq. (4) explicitly in the
following form:

$\displaystyle H^{(0)}_{aL}$ $\displaystyle=$
$\displaystyle-\frac{ieg^{2}C_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})]\gamma_{\alpha}(-k/_{1})\gamma_{\mu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1})^{2}(k_{1}-k_{2})^{2}}\bigg{\\}},$
(8)

in which we can see only $k/_{1}$ in internal quark propagator and $m_{B}$
part in $B$ meson wave function can contribute in this channel, and
$\gamma_{\mu}$ and $\gamma^{\alpha}$ should be $\gamma^{+}$ and
$\gamma_{\perp}$.

We can straightly write down the NLO amplitudes of Fig. 2(a,b,c) since no
radiated gluon lines attached to internal line, and therefore no momentum will
flow into the structure of LO hard kernel. So factorization can be derived
simply by inserting the Fierz identity, neatly cutting on the external quark
(antiquark) line.

$\displaystyle G^{(1)}_{aL,2a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{eg^{4}C^{2}_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\alpha}(-k/_{1})\gamma_{\mu}(p/_{1}-k/_{1}+m_{b})\gamma^{\nu}(p/_{1}-k/_{1}-l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1})^{2}(k_{1}-k_{2})^{2}[(p_{1}-k_{1})^{2}-m^{2}_{b}][(p_{1}-k_{1}-l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(9) $\displaystyle=$ $\displaystyle\frac{1}{2}\phi^{(1)}_{B,a}\otimes
G^{(0)}_{aL}(x_{1},x_{2}),$ $\displaystyle G^{(1)}_{aL,2b}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}C^{2}_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\alpha}(-k/_{1}+l/)\gamma_{\mu}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1}+l)^{2}(k_{1}-k_{2}-l)^{2}(k_{1}-l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(10) $\displaystyle=$ $\displaystyle\phi^{(1)}_{B,b}\otimes
G^{(0)}_{aL}(\xi_{1},x_{2}),$ $\displaystyle G^{(1)}_{aL,2c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{eg^{4}C^{2}_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma_{\nu}(k/_{1})\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\alpha}(-k/_{1})\gamma_{\mu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1})^{2}(k_{1}-k_{2})^{2}(k_{1}-l)^{2}(k_{1})l^{2}}\bigg{\\}}$
(11) $\displaystyle=$ $\displaystyle\frac{1}{2}\phi^{(1)}_{B,c}\otimes
G^{(0)}_{aL}(x_{1},x_{2}),$

with

$\displaystyle\phi^{(1)}_{B,a}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{4}\frac{\gamma_{5}\gamma^{\rho}\gamma_{\rho}\gamma_{5}(p/_{1}-k/_{1}+m_{b})\gamma^{\nu}(p/_{1}-k/_{1}-l/+m_{b})\gamma_{\nu}}{[(p_{1}-k_{1})^{2}-m^{2}_{b}][(p_{1}-k_{1}-l)^{2}-m^{2}_{b}]l^{2}},$
(12) $\displaystyle\phi^{(1)}_{B,b}$ $\displaystyle=$
$\displaystyle\frac{ig^{2}C_{F}}{4}\frac{\gamma_{5}\gamma^{\rho}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\nu}\gamma_{5}\gamma_{\rho}\gamma_{\nu}(k/_{1}-l/)}{(k_{1}-l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}},$
(13) $\displaystyle\phi^{(1)}_{B,c}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{4}\frac{\gamma_{5}\gamma_{\rho}\gamma^{\nu}(k/_{1}-l/)\gamma_{\nu}(k/_{1})\gamma^{\rho}\gamma_{5}}{(k_{1}-l)^{2}(k_{1})l^{2}}.$
(14)

The factor $1/2$ appeared in Eqs. (9,11) comes from the symmetry of Fig. 2(a)
and 2(c). From the QCD diagrams and dynamics, no independent soft divergences
exist in reducible diagrams, then the collinear divergences can be absorbed
into NLO wave functions concisely. These infrared singularities from
amplitudes of Figs. 2(a,b,c) are absorbed into NLO wave functions
$\phi^{(1)}_{B,i}(i=a,b,c)$ and can be re-expressed as in Eqs. (12-14). For
the reason that the infrared divergence in reducible diagrams without mixing
can be simply matched one by one between QCD diagrams and effective
diagramsPQCD-NLO-Li , these results are independent out of these irreducible
diagrams. So, we just pay more attention to irreducible diagrams.

For Figs. 2(d,e) we have

$\displaystyle G^{(1)}_{aL,2d}$ $\displaystyle=$
$\displaystyle\frac{ieg^{4}Tr[T^{a}T^{b}T^{c}]f_{abc}}{2N_{c}}$
$\displaystyle\times\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma^{\beta}(p/_{2}-k/_{1}+l/)\gamma_{\mu}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\gamma}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}(p_{2}-k_{1}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
$\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{B,d}\otimes\left[G^{(0)}_{aL}(\xi_{1},x_{1},x_{2})-G^{(0)}_{aL}(\xi_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{aL,2e}$ $\displaystyle=$
$\displaystyle\frac{ieg^{4}Tr[T^{a}T^{b}T^{c}]f_{abc}}{2N_{c}}$ (15)
$\displaystyle\times\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{1}-l/)\gamma^{\beta}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma^{\gamma}(p/_{2}-k/_{1})\gamma_{\mu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}(p_{2}-k_{1})^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
$\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{B,e}\otimes\left[G^{(0)}_{aL}(x_{1},x_{2})-G^{(0)}_{aL}(x_{1},\xi_{1},x_{2})\right],$

with

$\displaystyle\phi^{(1)}_{B,d}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{4}\frac{\gamma^{+}\gamma_{5}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\rho}\gamma_{5}\gamma^{-}}{[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\frac{\nu_{\rho}}{\nu\cdot
l},$ $\displaystyle\phi^{(1)}_{B,e}$ $\displaystyle=$
$\displaystyle\frac{ig^{2}C_{F}}{4}\frac{\gamma_{5}\gamma^{-}\gamma^{\rho}(k/_{1}-l/)\gamma^{+}\gamma_{5}}{(k_{1}-l)^{2}l^{2}}\frac{\nu_{\rho}}{\nu\cdot
l}.$ (16)

For Fig. 2(d) where the triple gluon vertex appeared, as we examined in Ref.
prd95-076005 , only the terms in $G^{(1)}_{aL,2d}$ proportional to
$g_{\alpha\beta}$ in
$F_{\alpha\beta\gamma}=g_{\alpha\beta}(2k_{2}-2k_{1}+l)_{\gamma}+g_{\beta\gamma}(k_{1}-k_{2}-2l)_{\alpha}+g_{\gamma\alpha}(k_{1}-k_{2}+l)_{\beta}$
give the right NLO corrections to the LO hard kernel. The other terms are
power suppressed by power counting of matrix elements and also unphysical in
topological diagrams. For Fig. 2(e), similarly, only the terms proportional to
$g_{\beta\gamma}$ in
$F_{\alpha\beta\gamma}=g_{\alpha\beta}(k_{1}-k_{2}-2l)_{\gamma}+g_{\beta\gamma}(2k_{2}-2k_{1}+l)_{\alpha}+g_{\gamma\alpha}(k_{1}-k_{2}+l)_{\beta}$
contribute effectively. Then the NLO $B$ meson wave function containing
infrared divergence with $``l"$ emitted from quark (antiquark) line can be
represented as in Eq. (16). Because of the large scale of $m_{b}$, the
$\phi^{(1)}_{B,d}$ contains the $m_{b}$ term in propagator when $``l"$ emits
from the $b$ quark .

The factorization of amplitudes for remaining diagrams Fig. 2(f-k) can be
expressed as follows:

$\displaystyle G^{(1)}_{aL,2f}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}Tr[T^{a}T^{a}T^{c}T^{c}]}{2N_{c}}$ (17)
$\displaystyle\times\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\alpha}(p/_{2}-k/_{1})\gamma^{\nu}(p/_{2}-k/_{1}+l/)\gamma_{\mu}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1})^{2}(k_{1}-k_{2})^{2}(p_{2}-k_{1}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
$\displaystyle\sim 0,$ $\displaystyle G^{(1)}_{aL,2g}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}Tr[T^{a}T^{a}T^{c}T^{c}]}{2N_{c}}$ (18)
$\displaystyle\times\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\alpha}(p/_{2}-k/_{1}+l/)\gamma_{\nu}(p/_{2}-k/_{1})\gamma_{\mu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1})^{2}(k_{1}-k_{2}-l)^{2}(p_{2}-k_{1}+l)^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
$\displaystyle\sim 0,$ $\displaystyle G^{(1)}_{aL,2h}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}Tr[T^{a}T^{c}T^{a}T^{c}]}{2N_{c}}$ (19)
$\displaystyle\times\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}+l/)\gamma_{\alpha}(p/_{2}-k/_{1}+l/)\gamma_{\mu}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1}+l)^{2}(k_{1}-k_{2})^{2}(p_{2}-k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
$\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{B,d}\otimes
G^{(0)}_{aL}(\xi_{1},x_{1},x_{2}),$ $\displaystyle G^{(1)}_{aL,2i}$
$\displaystyle=$ $\displaystyle\frac{eg^{4}Tr[T^{a}T^{c}T^{a}T^{c}]}{2N_{c}}$
(20)
$\displaystyle\times\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\alpha}(p/_{2}-k/_{1}+l/)\gamma_{\mu}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1}+l)^{2}(k_{1}-k_{2}-l)^{2}(k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
$\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{B,d}\otimes
G^{(0)}_{aL}(\xi_{1},x_{2}),$ $\displaystyle G^{(1)}_{aL,2j}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}Tr[T^{a}T^{c}T^{a}T^{c}]}{2N_{c}}$ (21)
$\displaystyle\times\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}(k/_{2}-l/)\gamma_{\nu}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\alpha}(p/_{2}-k/_{1})\gamma_{\mu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1}+l)^{2}(k_{1}-k_{2})^{2}(k_{1}-l)^{2}(k_{2}-l)^{2}l^{2}}\bigg{\\}}$
$\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{B,e}\otimes
G^{(0)}_{aL}(x_{1},x_{2}),$ $\displaystyle G^{(1)}_{aL,2k}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}Tr[T^{a}T^{c}T^{a}T^{c}]}{2N_{c}}$ (22)
$\displaystyle\times\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\alpha}(p/_{2}-k/_{1})\gamma_{\mu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1})^{2}(k_{1}-k_{2}-l)^{2}(p_{2}-k_{2}-l)^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
$\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{B,e}\otimes
G^{(0)}_{aL}(x_{1},\xi_{1},x_{2}),$

where the infrared divergence in Figs. 2(f,g) are suppressed by kinematics,
and we briefly set them to zero for we concentrate on infrared divergence only
in factorization theorem here. All the diagrams Figs. 2(d-g) generate no soft
divergence, for the reason that soft divergence only occurs when the
additional gluon bridge the initial and final quarks as Figs. 2(h-k). Then
these diagrams Figs. 2(h-k) will generate the soft divergence as well as the
collinear divergence. But fortunately, we know that these soft divergence will
cancel between Fig. 2(h) and Fig. 2(i), Fig. 2(j) and Fig. 2(k) by a simple
deformation of propagators. This conclusion is also supported by analytic
calculations for quark diagrams as shown in Refs. PQCD-NLO-Cheng ; PQCD-NLO-Li
; Zhang-Hua .

We will sum up all the irreducible diagrams to correct the color factors and
ensure the gauge invariance. All the diagrams Figs. 2(d,f,h,i) with the
additional gluon emitted from the up-quark lines of $B$ meson give the result:

$\displaystyle
G^{(1)}_{up,aL}(x_{1},x_{2})=\phi^{(1)}_{B,d}(x_{1},\xi_{1})\otimes\left[G^{(0)}_{aL}(\xi_{1},x_{1},x_{2})-G^{(0)}_{aL}(\xi_{1},x_{2})\right].$
(23)

The diagrams Figs. 2(e,g,j,k) with the additional gluon emitted from the down-
quark lines of $B$ meson give the result:

$\displaystyle
G^{(1)}_{down,aL}(x_{1},x_{2})=\phi^{(1)}_{B,e}(x_{1},\xi_{1})\otimes\left[G^{(0)}_{aL}(x_{1},x_{2})-G^{(0)}_{aL}(x_{1},\xi_{1},x_{2})\right].$
(24)

We take the similar representation as our previous work prove , the function
$G^{(0)}_{aL}(\xi_{1},x_{1},x_{2})$, $G^{(0)}_{aL}(x_{1},\xi_{1},x_{2})$ and
$G^{(0)}_{aL}(x_{1},\xi_{1},x_{2})$ in Eqs. (23,24) represent the different
forms of LO hard kernel with gluon momentum. The different location of
$\xi_{1}$ in vertex represents tiny difference of types as gluon momentum
flowing into hard kernel.

$\displaystyle G^{(0)}_{aL}(\xi_{1},x_{1},x_{2})$ $\displaystyle=$
$\displaystyle-\frac{ieg^{2}C_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})]\gamma_{\alpha}(p/_{2}-k/_{1}+l/)\gamma_{\mu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1}+l)^{2}(k_{1}-k_{2})^{2}}\bigg{\\}},$
(25) $\displaystyle G^{(0)}_{aL}(x_{1},\xi_{1},x_{2})$ $\displaystyle=$
$\displaystyle-\frac{ieg^{2}C_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})]\gamma_{\alpha}(p/_{2}-k/_{1})\gamma_{\mu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1})^{2}(k_{1}-k_{2}-l)^{2}}\bigg{\\}},$
(26) $\displaystyle G^{(0)}_{aL}(\xi_{1},x_{2})$ $\displaystyle=$
$\displaystyle-\frac{ieg^{2}C_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})]\gamma_{\alpha}(p/_{2}-k/_{1}+l/)\gamma_{\mu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{(p_{2}-k_{1}+l)^{2}(k_{1}-k_{2}-l)^{2}}\bigg{\\}}.$
(27)

Figure 3: The effective diagrams for the NLO initial $B$ meson.

After the cancellation of soft divergence, the remaining collinear divergence
will be absorbed into those effective wave functions of the mesons involved.
The effective diagrams for the NLO initial $B$ meson are shown as Fig. 3,
where the Figs. 3(a,c) represent diagrams with no gluon momentum flows into LO
hard kernel, relatively, Figs. 3(b,d) represent diagrams with the gluon
momentum flowing into LO hard kernel. The nonlocal hadronic matrix element for
the B meson wave function can be written as:

$\displaystyle\Phi^{(1)}_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{2N_{c}P^{+}_{1}}\int\frac{dy^{-}}{2\pi}e^{-ixp^{+}_{1}y^{-}}<0|\overline{q}(y^{-})\Gamma(-ig_{s})\int^{y^{-}}_{0}dzvA(z\nu)q(0)|h_{\nu}\bar{q}(p_{1}))>,$
(28)

where $A(z\nu)$ and $h_{\nu}$ represent the gauge field and the effective
heavy-quark field respectively, and $\Gamma$ represents gamma matrix decided
by specific part of twists, which can be chosen as $\gamma_{5}\gamma^{-}/2$
here in this channel.

Figure 4: The quark diagrams for NLO corrections to Fig. 1(b) with additional
gluon emitted from the quark(antiquark) lines of initial B meson.

Now we consider the NLO corrections to one of the structures of Fig. 1(b). The
first part of the LO amplitude $H^{(0)}_{bL}$ as given in Eq. 5 can be written
in the form of

$\displaystyle G^{(0)}_{bL,1}$ $\displaystyle=$
$\displaystyle\frac{-ieg^{2}C_{F}m_{B}}{2}\mathbf{Tr}\left[\frac{\gamma^{\alpha}M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\gamma_{\mu}k/_{2}\gamma_{\alpha}\gamma_{5}\frac{n/+}{\sqrt{2}}\phi^{+}_{B}(x_{1})}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}}\right].$
(29)

We retain the $\gamma^{\alpha}$ matrix which should be $\gamma_{\perp}$ in Eq.
(29) for convenience. The factorization about NLO corrections to this part of
the LO hard kernel gives the results as follows. The NLO $B$ meson wave
function containing infrared divergence is of the form as defined in Eq. (33).
Each sub-diagram of this channel contributes without dynamical forbidden.
While, the change of weak vertex’s location leads to different color structure
for Figs. 4(f,g) comparing with those in Fig. 2.

$\displaystyle G^{(1)}_{bL1,2d}$ $\displaystyle=$ $\displaystyle
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}k/_{2}\gamma^{\beta}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\gamma}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}][(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(30) $\displaystyle=$
$\displaystyle-\frac{9}{8}\phi^{(1)}_{B,d}\otimes\left[G^{(0)}_{bL,1}(x_{1},x_{2})-G^{(0)}_{bL,1}(x_{1},\xi_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bL1,2e}$ $\displaystyle=$ $\displaystyle-
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{1}-l/)\gamma^{\beta}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}k/_{2}\gamma^{\gamma}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(31) $\displaystyle=$
$\displaystyle-\frac{9}{8}\phi^{(1)}_{B,e}\otimes\left[G^{(0)}_{bL,1}(x_{1},x_{2})-G^{(0)}_{bL,1}(x_{1},\xi_{1},x_{2})\right],$

with

$\displaystyle\phi^{(1)}_{B,d}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{4}\frac{\gamma^{+}\gamma_{5}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\rho}\gamma_{5}\gamma^{-}}{[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\frac{\nu_{\rho}}{\nu\cdot
l},$ (32) $\displaystyle\phi^{(1)}_{B,e}$ $\displaystyle=$
$\displaystyle\frac{ig^{2}C_{F}}{4}\frac{\gamma_{5}\gamma^{-}\gamma^{\rho}(k/_{1}-l/)\gamma^{+}\gamma_{5}}{(k_{1}-l)^{2}l^{2}}\frac{\nu_{\rho}}{\nu\cdot
l}.$ (33)

and

$\displaystyle G^{(1)}_{bL1,2f}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}k/_{2}\gamma^{\nu}(p/_{1}-k/_{2}+l/+m_{b})\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}][(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(34) $\displaystyle=$
$\displaystyle\frac{1}{8}\phi^{(1)}_{B,d}\otimes\left[G^{(0)}_{bL,1}(x_{1},x_{2})-G^{(0)}_{bL,1}(\xi_{1},x_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bL1,2g}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}k/_{2}\gamma_{\nu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\alpha}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(35) $\displaystyle=$
$\displaystyle-\frac{1}{8}\phi^{(1)}_{B,e}\otimes\left[G^{(0)}_{bL,1}(x_{1},\xi_{1},x_{2})-G^{(0)}_{bL,1}(\xi_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bL1,2h}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}+l/)\gamma_{\mu}(k/_{2}-l/)\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(p_{2}-k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(36) $\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{B,d}\otimes
G^{(0)}_{bL,1}(\xi_{1},x_{1},x_{2}),$ $\displaystyle G^{(1)}_{bL1,2i}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}k/_{2}\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}+l)^{2}(k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(37) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{B,d}\otimes
G^{(0)}_{bL,1}(x_{1},\xi_{1},x_{2}),$ $\displaystyle G^{(1)}_{bL1,2j}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}(k/_{2}-l/)\gamma_{\nu}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}k/_{2}\gamma_{\alpha}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(k_{1}-l)^{2}(k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(38) $\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{B,e}\otimes
G^{(0)}_{bL,1}(x_{1},x_{2}),$ $\displaystyle G^{(1)}_{bL1,2k}$
$\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(k/_{2}+l/)\gamma_{\alpha}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}(p_{2}-k_{2}-l)^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(39) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{B,e}\otimes
G^{(0)}_{bL,1}(\xi_{1},x_{2}).$

We also sum up all the contributions from the diagrams with additional gluon
emitted from up-quark (down-quark) lines of initial $B$ meson of this channel
and the results are of the following form:

$\displaystyle
G^{(1)}_{up,bL1}(x_{1},x_{2})=-\phi^{(1)}_{B,d}(x_{1},\xi_{1})\otimes\left[G^{(0)}_{aL}(x_{1},x_{2})-G^{(0)}_{aL}(x_{1},\xi_{1},x_{2})\right],$
$\displaystyle
G^{(1)}_{down,bL1}(x_{1},x_{2})=-\phi^{(1)}_{B,e}(x_{1},\xi_{1})\otimes\left[G^{(0)}_{aL}(x_{1},x_{2})-G^{(0)}_{aL}(x_{1},\xi_{1},x_{2})\right].$
(40)

The minus sign here is caused by the $-k/_{2}$ in gluon propagator at LO.
Although the Figs. 4(f,g) now give the different color structures, the
summation of all the sub-diagrams still give the right color factor which we
can also be obtained from the effective diagrams directly. As we did in
$G^{1}_{aL}$ or other terms, after the cancellation of soft divergence, the
remaining collinear divergence can be factorized into NLO $B$ meson wave
function without any other theoretical problems.

###  III.2 $k_{T}$ factorization of NLO corrections to $B\to\rho$ transition

Figure 5: The graph for the deviation of the integral (Wilson line) by b in
light cone coordinate for the two-parton meson wave function.

We pick up the transversal momentum $``k_{T}"$ in $k_{T}$ factorization frame
to remove the endpoint singularity emerged in small $x$ region. The Sudakov
factor $e^{-S(t)}$ is introduced ppnp51-85  in order to suppress the collinear
divergence by resummation of the large logarithmic terms which $k_{T}$ is
concerned.

As we proposed in $k_{T}$ factorization frame, the $k_{T}$ works only in small
$x$ region, which indicates that we have the hierarchy $k^{2}_{iT}\ll
k_{1}\cdot k_{2}$. Comparing with the expressions as shown in Eqs. (4) and
(6), we can also keep the same forms of LO hard kernel in $k_{T}$
factorization frame. After the inclusion of the NLO corrections, the only
difference we should pay attention to is the transverse momentum
$\mathbf{l_{T}}$ in propagators which can be understood as a small momentum
shift.

The ratio term $\nu_{\rho}/{\nu\cdot l}$ in Eq. (16) represents the Feynman
rule related to Wilson line. Considering the $\mathbf{l_{T}}$, we rewrite the
hadronic matrix element in Eq. (28) containing Wilson line through Fourier
transformation for the gauge field form from $A(z\nu)$ to $\widetilde{A}(l)$:

$\displaystyle\int^{\infty}_{0}dz\;v\cdot A(zv)$ $\displaystyle\to$
$\displaystyle\int^{\infty}_{0}dz\int dl~{}e^{iz(v\cdot
l+i\epsilon)}~{}v\cdot\widetilde{A}(l)$ (41) $\displaystyle\to$ $\displaystyle
i\int dl~{}\frac{v_{\rho}}{v\cdot l}\widetilde{A}^{\rho}(l),$
$\displaystyle\int^{y^{-}}_{\infty}dz\;v\cdot A(zv)$ $\displaystyle\to$
$\displaystyle\int^{y^{-}}_{0}dz\int dl~{}e^{[iz(v\cdot
l+i\epsilon)-i\mathbf{l}_{T}\cdot\mathbf{b}]}~{}v\cdot\widetilde{A}(l)$ (42)
$\displaystyle\to$ $\displaystyle-i\int dl~{}\frac{v_{\rho}}{v\cdot
l}~{}e^{[il^{+}y^{-}-i\mathbf{l}_{T}\cdot\mathbf{b}]}~{}\widetilde{A}^{\rho}(l),$

The factor $e^{il^{+}y^{-}}$ will generate a delta function
$\delta(\xi_{1}-x_{1}+\frac{l^{+}}{p_{1}})$ which describes a momentum shift
when the $l$ flowing into hard kernel. The other term $e^{i\mathbf{l}_{T}}$
represents a small transversal part flowing into the hard kernel. The NLO $B$
and $\rho$ meson wave functions can be finally written as a general form:

$\displaystyle\Phi^{(1)}_{B}(x_{1},\xi_{1};\mathbf{b_{1}})$ $\displaystyle=$
$\displaystyle\frac{1}{2N_{c}P^{+}_{1}}\int\frac{dy^{-}}{2\pi}\frac{d\mathbf{b_{1}}}{(2\pi)^{2}}e^{-ixp^{+}_{1}y^{-}+i\mathbf{k_{1T}}\cdot\mathbf{b_{1}}}$
(43)
$\displaystyle\cdot<0|\overline{q}(y^{-})\Gamma\cdot(-ig_{s})\int^{y}_{0}dz\;n\cdot
A(zn)q(0)|h_{\nu}B(p_{1})>,$
$\displaystyle\Phi^{(1)}_{\rho}(x_{2},\xi_{2};\mathbf{b_{2}})=$
$\displaystyle\frac{1}{2N_{c}P^{-}_{2}}\int\frac{dy^{-}}{2\pi}\frac{d\mathbf{b_{2}}}{(2\pi)^{2}}e^{-ixp^{+}_{2}y^{-}+i\mathbf{k_{2T}}\cdot\mathbf{b_{2}}}$
(44)
$\displaystyle\cdot<0|\overline{q}(y^{-})\Gamma^{\prime}\cdot(-ig_{s})\int^{y}_{0}dz\;n\cdot
A(zn)q(0)|\rho(p_{1})>.$

where $\Gamma$ is the gamma matrix depending on the twist of the $B$ and
$\rho$ wave functions.

The whole process of the integration and modification in Eqs. (41,42) can be
understood by graphs in Fig. 5. Both the Wilson lines in Fig. 5(a,b) parallel
to the light cone coordinate will generate light cone singularity. To avoid
this singularity, we can slightly rotate the direction of Wilson line as in
Fig. 5(c), then such a singularity can be regularized by $n^{2}(n^{2}\neq 0)$.
The choice of number $n^{2}$ is scheme dependent, which has been demonstrated
to be small li1302  and we can simply set $n^{2}=1$ for convenience in
calculation. While this rotation and the choose of $n^{2}$ will generate a
pinched singularity during the integration for Wilson lines. To solve this
problem, the transverse-momentum-dependent(TMD) wave function with soft
substraction factor with a square root is introduced to the un-subtracted wave
function ijmpcs4-85 , and a more elegant TMD wave function involves two pieces
of non-light-like Wilson links is proposed in Ref.jhep1506-013 . We pay more
attention to the factorization of the collinear divergence in this paper, more
discussion on these TMD wave function will be given in future work.

##  IV SUMMARY

In this paper, we investigate the $B\to\rho$ transition process and give the
proof of factorization at NLO level in collinear factorization approach.
Because of the different structures in the initial $B$ meson and the final
$\rho$ meson state, three amplitudes with six parts of twists’ forms for LO
diagrams as shown in Fig. 1(a,b) should be considered. Also, both the
corrections to initial state and finial state should be considered. The
calculation of all these six channels will be much complex, which suggests us
an estimate for proportion for every channel in LO will be necessary before
numerical calculations done at the NLO level.

We have verified that with the right power counting for triple gluon vertex,
the amplitudes of the NLO corrections to the LO hard kernel with an additional
gluon emitted from the initial $B$ meson or the final $\rho$ meson can be
separated into a convolution of the NLO wave function with LO hard kernel. The
LO hard kernel here can be distinguished as two forms, with gluon momentum
flowing into the hard kernel, or with no gluon momentum flowing into the hard
kernel. We then extend these results to $k_{T}$ factorization frame and the
conclusion can still be stable by a momentum shift. The form of nonlocal two
quarks hadron matrices of NLO $B(\rho)$ meson are given in section<EMAIL_ADDRESS>This
work will also play an essential role for future numerical calculations of
$B\to\rho$ transition form factors.

## ACKNOWLEDEMENT

This work is supported by the National Natural Science Foundation of China
under Grants No. 11775117 and 11235005, and also by the Practice Innovation
Program of Jiangsu Province under Grant No. KYCX18-1184.

##  Appendix A Factorization of the NLO amplitudes

In this Appendix, we present the explicit expressions for NLO corrections to
the remaining LO amplitude $H^{0}_{bL}$ and $H^{0}_{bT}$ . The function
$G^{(0)}_{bL2}$ corresponds to the second part of $H^{0}_{bL}$ as given in Eq.
(5). The function $G^{(0)}_{bT1}$, $G^{(0)}_{bT2}$ and $G^{(0)}_{bT3}$ come
from the three components of $H^{0}_{bT}$ as defined in Eq. (6): two for
$\phi_{B}^{+}$ and one for $\phi_{B}^{-}$ component. For the sake of
clearness, we write these four functions in their original non-reduced form:

$\displaystyle G^{(0)}_{bL2}$ $\displaystyle=$
$\displaystyle-\frac{ieg^{2}C_{F}}{2}\mathbf{Tr}\left\\{\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}(m_{b})\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}}\right\\},$
(45) $\displaystyle G^{(0)}_{bT1}$ $\displaystyle=$
$\displaystyle-\frac{ieg^{2}C_{F}}{2}\mathbf{Tr}\left\\{[\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}(p/_{1})\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{\Big{[}(p_{1}-k_{2})^{2}-m^{2}_{b}\Big{]}(k_{1}-k_{2})^{2}}\right\\},$
(46) $\displaystyle G^{(0)}_{bT2}$ $\displaystyle=$
$\displaystyle-\frac{ieg^{2}C_{F}}{2}\mathbf{Tr}\left\\{\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}(p/_{1})\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{\Big{[}(p_{1}-k_{2})^{2}-m^{2}_{b}\Big{]}(k_{1}-k_{2})^{2}}\right\\},$
(47) $\displaystyle G^{(0)}_{bT3}$ $\displaystyle=$
$\displaystyle-\frac{ieg^{2}C_{F}}{2}\mathbf{Tr}\left\\{\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}(m_{b})\gamma_{\alpha}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{\Big{[}(p_{1}-k_{2})^{2}-m^{2}_{b}\Big{]}(k_{1}-k_{2})^{2}}\right\\}.$
(48)

The NLO corrections with additional gluon emitted from the final $\rho$ meson
are also supplied. In Fig. 6 and 7 , we show the Feynman diagrams for NLO
corrections to Fig. 1(a) and Fig. 1(b) respectively, where the second blue
gluon are emitted from the quark and antiquark lines of the finial $\rho$
meson. Notice that the $G^{(0)}_{aL}$ and $G^{(0)}_{bL1}$ with the gluon
emitted from the final $\rho$ meson have the similar structure with the ones
for $\rho\to\rho$ decays as defined in Refs. prove , so we here do not present
them explicitly.

Figure 6:  The Feynman diagrams for NLO corrections to Fig. 1(a) with the
additional gluon emitted from the quark (antiquark) lines of the finial $\rho$
meson.  Figure 7: The Feynman diagrams for NLO corrections to Fig. 1(b) with
the additional gluon emitted from the quark (antiquark) lines of the finial
$\rho$ meson.

###  A.1 The NLO amplitudes for $G^{(0)}_{bL2}$

The amplitudes for the NLO contributions from the irreducible diagrams to
$G^{(0)}_{bL2}$ are the following:

$\displaystyle G^{(1)}_{bL2,2d}$ $\displaystyle=$ $\displaystyle-
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma^{\beta}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}][(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(49) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{B,d}\otimes\left[G^{(0)}_{bL,2}(x_{1},x_{2})-G^{(0)}_{bL,2}(x_{1},\xi_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bL2,2e}$ $\displaystyle=$ $\displaystyle-
eg^{4}\times\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{1}-l/)\gamma^{\beta}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(50) $\displaystyle\sim 0.$

The contribution of $G^{(1)}_{bL2,2e}$ is suppressed by dynamics and we simply
set it $0$ here. For other contributions we have:

$\displaystyle G^{(1)}_{bL2,2f}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma^{\nu}(p/_{1}-k/_{2}+l/+m_{b})\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}][(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}},$
(51) $\displaystyle\sim 0,$ $\displaystyle G^{(1)}_{bL2,2g}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\nu}(p/_{1}-k/_{2}+l/+m_{b})\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}](k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(52) $\displaystyle\sim 0,$ $\displaystyle G^{(1)}_{bL2,2h}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}+l/)\gamma_{\mu}(m_{b}+l/)\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(p_{2}-k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(53) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{B,d}\otimes
G^{(0)}_{bL,1}(x_{1},x_{2}),$ $\displaystyle G^{(1)}_{bL2,2i}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}+l)^{2}(k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(54) $\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{B,d}\otimes
G^{(0)}_{bL,1}(x_{1},\xi_{1},x_{2}),$ $\displaystyle G^{(1)}_{bL2,2j}$
$\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}(k/_{2}-l/)\gamma_{\nu}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(k_{1}-l)^{2}(k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(55) $\displaystyle\sim 0,$ $\displaystyle G^{(1)}_{bL2,2k}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(m_{b}-l/)\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}(p_{2}-k_{2}-l)^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(56) $\displaystyle\sim 0,$

with

$\displaystyle\phi^{(1)}_{B,d}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{8}\frac{\gamma^{-}\gamma_{5}\gamma^{+}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\rho}\gamma^{-}\gamma_{5}\gamma^{+}}{[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\frac{\nu_{\rho}}{\nu\cdot
l}.$ (57)

The summation over above contributions gives the result:

$\displaystyle
G^{(1)}_{up,bL2}(x_{1},x_{2})=\phi^{(1)}_{B,d}(x_{1},\xi_{1})\otimes\left[G^{(0)}_{aL}(x_{1},x_{2})-G^{(0)}_{aL}(x_{1},\xi_{1},x_{2})\right].$
(58)

###  A.2 The NLO amplitudes for $G^{(0)}_{bT1}$

The amplitudes for the NLO contributions from the irreducible diagrams to
$G^{(0)}_{bT1}$ are the following:

$\displaystyle G^{(1)}_{bT1,2d}$ $\displaystyle=$ $\displaystyle-
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma^{\beta}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}][(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(59) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{B,d}\otimes\left[G^{(0)}_{bT,1}(x_{1},x_{2})-G^{(0)}_{bT,1}(x_{1},\xi_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bT1,2e}$ $\displaystyle=$ $\displaystyle
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{1}-l/)\gamma^{\beta}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(60) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{B,e}\otimes\left[G^{(0)}_{bT,1}(x_{1},x_{2})-G^{(0)}_{bT,1}(x_{1},\xi_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bT1,2f}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma^{\nu}(p/_{1}-k/_{2}+l/+m_{b})\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}][(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(61) $\displaystyle=$
$\displaystyle-\frac{1}{8}\phi^{(1)}_{B,d}\otimes\left[G^{(0)}_{bT,1}(x_{1},x_{2})-G^{(0)}_{bT,1}(\xi_{1},x_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bT1,2g}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\nu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(62) $\displaystyle\sim 0,$ $\displaystyle G^{(1)}_{bT1,2h}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}+l/)\gamma_{\mu}(p/_{1}+l/)\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(p_{2}-k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(63) $\displaystyle\sim 0,$ $\displaystyle G^{(1)}_{bT1,2i}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}+l)^{2}(k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(64) $\displaystyle\sim 0,$ $\displaystyle G^{(1)}_{bT1,2j}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}(k/_{2}-l/)\gamma_{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(k_{1}-l)^{2}(k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(65) $\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{B,e}\otimes
G^{(0)}_{bT,1}(x_{1},x_{2}),$ $\displaystyle G^{(1)}_{bT1,2k}$
$\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(p/_{1}-l/)\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}(p_{2}-k_{2}-l)^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(66) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{B,e}\otimes
G^{(0)}_{bT,1}(\xi_{1},x_{2}),$

with the wave functions $\phi^{(1)}_{B,d}$ and $\phi^{(1)}_{B,e}$ are of the
following form:

$\displaystyle\phi^{(1)}_{B,d}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{8}\frac{\gamma^{+}\gamma_{5}\gamma^{-}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\rho}\gamma^{+}\gamma_{5}\gamma^{-}}{[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\frac{\nu_{\rho}}{\nu\cdot
l},$ (67) $\displaystyle\phi^{(1)}_{B,e}$ $\displaystyle=$
$\displaystyle\frac{ig^{2}C_{F}}{8}\frac{\gamma^{+}\gamma_{5}\gamma^{-}\gamma^{\rho}(k/_{1}-l/)\gamma^{+}\gamma_{5}\gamma^{-}}{(k_{1}-l)^{2}l^{2}}\frac{\nu_{\rho}}{\nu\cdot
l}.$ (68)

The direct summation of above contributions gives the following result:

$\displaystyle G^{(1)}_{up,bT3}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\phi^{(1)}_{B,d}(x_{1},\xi_{1})\otimes\left[G^{(0)}_{bT}(x_{1},x_{2})-G^{(0)}_{bT}(x_{1},\xi_{1},x_{2})\right]$
(69)
$\displaystyle+\frac{1}{8}\phi^{(1)}_{B,d}(x_{1},\xi_{1})\otimes\left[G^{(0)}_{bT}(\xi_{1},x_{1},x_{2})-G^{(0)}_{bT}(x_{1},\xi_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{down,aL}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\phi^{(1)}_{B,e}(x_{1},\xi_{1})\otimes\left[G^{(0)}_{aL}(x_{1},x_{2})-G^{(0)}_{aL}(x_{1},\xi_{1},x_{2})\right],$
(70)

where the $1/8$ term seems abnormal here. Fortunately, as we demonstrated in
Eqs. (25-27) , the difference between $G^{(0)}_{bT}(\xi_{1},x_{1},x_{2})$ and
$G^{(0)}_{bT}(x_{1},\xi_{1},x_{2})$ is the form of the propagator in
denominator. The flow $l$ here can be recognized as a momentum shift for
$x^{\prime}_{1}=x_{1}-l^{+}/p_{1}$ and with the symmetry by choice of $P_{2}$
and $k_{2}$, the infrared structure between the two terms will be canceled
each other.

###  A.3 The NLO amplitudes for $G^{(0)}_{bT2}$

The amplitudes for the NLO contributions from the irreducible diagrams to
$G^{(0)}_{bT2}$ are the following:

$\displaystyle G^{(1)}_{bT2,2d}$ $\displaystyle=$ $\displaystyle-
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma^{\beta}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}][(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(71) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{B,d}\otimes\left[G^{(0)}_{bT,2}(x_{1},x_{2})-G^{(0)}_{bT,2}(x_{1},\xi_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bT2,2e}$ $\displaystyle=$ $\displaystyle
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{1}-l/)\gamma^{\beta}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(72) $\displaystyle\sim 0,$ $\displaystyle G^{(1)}_{bT2,2f}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma^{\nu}(p/_{1}-k/_{2}+l/+m_{b})\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}][(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(73) $\displaystyle=$
$\displaystyle-\frac{1}{8}\phi^{(1)}_{B,d}\otimes\left[G^{(0)}_{bT,2}(x_{1},x_{2})-G^{(0)}_{bT,2}(\xi_{1},x_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bT2,2g}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\nu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(74) $\displaystyle\sim 0,$ $\displaystyle G^{(1)}_{bT2,2h}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}+l/)\gamma_{\mu}(p/_{1}+l/)\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(p_{2}-k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(75) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{B,d}\otimes
G^{(0)}_{bT,2}(\xi_{1},x_{1},x_{2}),$ $\displaystyle G^{(1)}_{bT2,2i}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}+l)^{2}(k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(76) $\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{B,d}\otimes
G^{(0)}_{bT,2}(x_{1},\xi_{1},x_{2}),$ $\displaystyle G^{(1)}_{bT2,2j}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}(k/_{2}-l/)\gamma_{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(k_{1}-l)^{2}(k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(77) $\displaystyle\sim 0,$ $\displaystyle G^{(1)}_{bT2,2k}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(p/_{1}-l/)\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}(p_{2}-k_{2}-l)^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(78) $\displaystyle\sim 0,$

with the wave function $\phi^{(1)}_{B,d}$ as the form of

$\displaystyle\phi^{(1)}_{B,d}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{8}\frac{\gamma^{-}\gamma_{5}\gamma^{+}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\rho}\gamma^{-}\gamma_{5}\gamma^{+}}{[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\frac{\nu_{\rho}}{\nu\cdot
l}.$ (79)

The summation of the contributions from these irreducible sub-diagrams gives
the following result:

$\displaystyle
G^{(1)}_{up,bT2}(x_{1},x_{2})=\phi^{(1)}_{B,d}(x_{1},\xi_{1})\otimes\left[G^{(0)}_{bT}(x_{1},x_{2})-G^{(0)}_{bT}(x_{1},\xi_{1},x_{2})\right].$
(80)

###  A.4 The NLO amplitudes for $G^{(0)}_{bT3}$

The amplitudes for the NLO contributions from the irreducible diagrams to
$G^{(0)}_{bT3}$ are the following:

$\displaystyle G^{(1)}_{bT3,2d}$ $\displaystyle=$ $\displaystyle-
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma^{\beta}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\gamma}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}][(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(81) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{B,d}\otimes\left[G^{(0)}_{bT,3}(x_{1},x_{2})-G^{(0)}_{bT,3}(x_{1},\xi_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bT3,2e}$ $\displaystyle=$ $\displaystyle
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{1}-l/)\gamma^{\beta}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma^{\gamma}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(82) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{B,e}\otimes\left[G^{(0)}_{bT,3}(x_{1},x_{2})-G^{(0)}_{bT,3}(x_{1},\xi_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bT3,2f}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma^{\nu}(p/_{1}-k/_{2}+l/+m_{b})\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}][(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(83) $\displaystyle=$
$\displaystyle-\frac{1}{8}\phi^{(1)}_{B,d}\otimes\left[G^{(0)}_{bT,3}(x_{1},x_{2})-G^{(0)}_{bT,3}(\xi_{1},x_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bT3,2g}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\nu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\alpha}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(84) $\displaystyle=$
$\displaystyle-\frac{1}{8}\phi^{(1)}_{B,e}\otimes\left[G^{(0)}_{bT,3}(x_{1},\xi_{1},x_{2})-G^{(0)}_{bT,3}(\xi_{1},x_{2})\right],$
$\displaystyle G^{(1)}_{bT3,2h}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}+l/)\gamma_{\mu}(m_{b}+l/)\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(p_{2}-k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(85) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{B,d}\otimes
G^{(0)}_{bT,3}(\xi_{1},x_{1},x_{2}),$ $\displaystyle G^{(1)}_{bT3,2i}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}+l)^{2}(k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(86) $\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{B,d}\otimes
G^{(0)}_{bT,3}(x_{1},\xi_{1},x_{2}),$ $\displaystyle G^{(1)}_{bT3,2j}$
$\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}(k/_{2}-l/)\gamma_{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\alpha}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(k_{1}-l)^{2}(k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(87) $\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{B,e}\otimes
G^{(0)}_{bT,3}(x_{1},x_{2}),$ $\displaystyle G^{(1)}_{bT3,2k}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(m_{b}-l/)\gamma_{\alpha}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}(p_{2}-k_{2}-l)^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(88) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{B,e}\otimes
G^{(0)}_{bT,3}(\xi_{1},x_{2}),$

with the wave functions $\phi^{(1)}_{B,d}$ and $\phi^{(1)}_{B,e}$ in the form
of

$\displaystyle\phi^{(1)}_{B,d}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{4}\frac{\gamma^{+}\gamma_{5}(p/_{1}-k/_{1}+l/+m_{b})\gamma^{\rho}\gamma_{5}\gamma^{-}}{[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\frac{\nu_{\rho}}{\nu\cdot
l},$ $\displaystyle\phi^{(1)}_{B,e}$ $\displaystyle=$
$\displaystyle\frac{ig^{2}C_{F}}{4}\frac{\gamma_{5}\gamma^{-}\gamma^{\rho}(k/_{1}-l/)\gamma^{+}\gamma_{5}}{(k_{1}-l)^{2}l^{2}}\frac{\nu_{\rho}}{\nu\cdot
l}.$ (89)

The summation of above terms from the irreducible sub-diagrams gives the
following result:

$\displaystyle
G^{(1)}_{up,bT3}(x_{1},x_{2})=\phi^{(1)}_{B,d}(x_{1},\xi_{1})\otimes\left[G^{(0)}_{bT}(x_{1},x_{2})-G^{(0)}_{bT}(x_{1},\xi_{1},x_{2})\right],$
(90) $\displaystyle
G^{(1)}_{down,aL}(x_{1},x_{2})=\phi^{(1)}_{B,e}(x_{1},\xi_{1})\otimes\left[G^{(0)}_{aL}(x_{1},x_{2})-G^{(0)}_{aL}(x_{1},\xi_{1},x_{2})\right].$
(91)

###  A.5 The NLO amplitudes for $G^{(0^{\prime})}_{bL2}$

The amplitudes for the NLO contributions from the irreducible diagrams to
$G^{(0^{\prime})}_{bL2}$ with the additional gluon emitted from the final
$\rho$ meson are the following:

$\displaystyle G^{(1^{\prime})}_{bL2,2d}$ $\displaystyle=$ $\displaystyle-
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma^{\beta}(p/_{2}-k/_{2}-l/)\gamma_{\mu}m_{b}\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](p_{2}-k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(92) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{\rho,d}\otimes\left[G^{(0)}_{bL,2}(x_{1},\xi_{2},x_{2})-G^{(0)}_{bL,2}(x_{1},\xi_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bL2,2e}$ $\displaystyle=$ $\displaystyle
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\beta}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(93) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{\rho,e}\otimes\left[-G^{(0)}_{bL,2}(x_{1},x_{2})+G^{(0)}_{bL,2}(x_{1},x_{2},\xi_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bL2,2f}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}C^{2}_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\nu}m_{b}\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](p_{2}-k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(94) $\displaystyle\sim 0,$ $\displaystyle G^{(1^{\prime})}_{bL2,2g}$
$\displaystyle=$
$\displaystyle\frac{eg^{4}C^{2}_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\nu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(95) $\displaystyle=$
$\displaystyle\phi^{(1)}_{\rho,e}\otimes\left[-G^{(0)}_{bL,2}(x_{1},x_{2},\xi_{2})+G^{(0)}_{bL,2}(x_{1},\xi_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bL2,2h}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(m_{b}-l/)\gamma_{\alpha}(p/_{1}-k/_{1}-l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(p_{2}-k_{2}-l)^{2}[(p_{1}-k_{1}-l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(96) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{\rho,d}\otimes
G^{(0)}_{bL,2}(x_{1},\xi_{2},x_{2}),$ $\displaystyle
G^{(1^{\prime})}_{bL2,2i}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(m_{b}-l/)\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}(p_{2}-k_{2}-l)^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(97) $\displaystyle\sim 0,$ $\displaystyle G^{(1^{\prime})}_{bL2,2j}$
$\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}+l/)\gamma^{\alpha}(k/_{2}+l/)\gamma_{\nu}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(k_{1}+l)^{2}(k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(98) $\displaystyle\sim 0,$ $\displaystyle G^{(1^{\prime})}_{bL2,2k}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[M_{\rho}\epsilon/_{2L}\phi_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(99) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{\rho,e}\otimes
G^{(0)}_{bL,2}(x_{1},x_{2},\xi_{2}),$

with the wave functions $\phi^{(1)}_{\rho,d}$ and $\phi^{(1)}_{\rho,e}$ in the
form of

$\displaystyle\phi^{(1)}_{\rho,d}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{4}\frac{\gamma^{+}\gamma^{\rho}(p/_{2}-k/_{2}-l/)\gamma^{-}}{(p_{2}-k_{2}-l)^{2}l^{2}}\frac{n_{\rho}}{n\cdot
l},$ $\displaystyle\phi^{(1)}_{\rho,e}$ $\displaystyle=$
$\displaystyle\frac{ig^{2}C_{F}}{4}\frac{\gamma^{-}(k/_{2}+l/)\gamma^{\rho}\gamma^{+}}{(k_{2}+l)^{2}l^{2}}\frac{n_{\rho}}{n\cdot
l}.$ (100)

Analogous to the ones in Eq. (69), the $1/8$ terms also appear in this
channel. After the summation of the contributions from these irreducible sub-
diagrams, one finds the following result:

$\displaystyle
G^{(1^{\prime})}_{up,bL2}(x_{1},x_{2})=\phi^{(1)}_{\rho,d}\otimes\left[G^{(0)}_{bL,2}(x_{1},\xi_{2},x_{2})-G^{(0)}_{bL,2}(x_{1},\xi_{2})\right],$
(101) $\displaystyle
G^{(1^{\prime})}_{down,bL2}(x_{1},x_{2})=\phi^{(1)}_{\rho,e}\otimes\left[-G^{(0)}_{bL,2}(x_{1},x_{2})+G^{(0)}_{bL,2}(x_{1},x_{2},\xi_{2})\right].$
(102)

###  A.6 The NLO amplitudes for $G^{(0^{\prime})}_{bT1}$

The amplitudes for the NLO contributions from the irreducible diagrams to
$G^{(0^{\prime})}_{bT1}$ with the additional gluon emitted from the final
$\rho$ meson are the following:

$\displaystyle G^{(1^{\prime})}_{bT1,2d}$ $\displaystyle=$ $\displaystyle-
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\beta}(p/_{2}-k/_{2}-l/)\gamma_{\mu}p/_{1}\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](p_{2}-k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(103) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{\rho,d}\otimes\left[G^{(0)}_{bT,1}(x_{1},\xi_{2},x_{2})-G^{(0)}_{bT,1}(x_{1},\xi_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bT1,2e}$ $\displaystyle=$ $\displaystyle
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\beta}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(104) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{\rho,e}\otimes\left[-G^{(0)}_{bT,1}(x_{1},x_{2})+G^{(0)}_{bT,1}(x_{1},x_{2},\xi_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bT1,2f}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}C^{2}_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\nu}p/_{1}\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](p_{2}-k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(105) $\displaystyle=$
$\displaystyle\phi^{(1)}_{\rho,d}\otimes\left[G^{(0)}_{bT,1}(x_{1},x_{2})-G^{(0)}_{bT,1}(x_{1},\xi_{2},x_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bT1,2g}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}C^{2}_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\nu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(106) $\displaystyle=$
$\displaystyle\phi^{(1)}_{\rho,e}\otimes\left[-G^{(0)}_{bT,1}(x_{1},x_{2},\xi_{2})+G^{(0)}_{bT,1}(x_{1},\xi_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bT1,2h}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}+l/)\gamma_{\mu}(p/_{1}+l/)\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(p_{2}-k_{2}+l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(107) $\displaystyle\sim 0,$ $\displaystyle G^{(1^{\prime})}_{bT1,2i}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(p/_{1}-l/)\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}(p_{2}-k_{2}-l)^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(108) $\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{\rho,d}\otimes
G^{(0)}_{bT,1}(x_{1},\xi_{2}),$ $\displaystyle G^{(1^{\prime})}_{bT1,2j}$
$\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}+l/)\gamma^{\alpha}(k/_{2}+l/)\gamma_{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(k_{1}+l)^{2}(k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(109) $\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{\rho,e}\otimes
G^{(0)}_{bT,2}(x_{1},x_{2}),$ $\displaystyle G^{(1^{\prime})}_{bT1,2k}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(110) $\displaystyle\sim 0,$

with the wave functions $\phi^{(1)}_{\rho,d}$ and $\phi^{(1)}_{\rho,e}$ in the
form of

$\displaystyle\phi^{(1)}_{\rho,d}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{4}\frac{\gamma_{\perp}\gamma^{+}\gamma^{\rho}(p/_{2}-k/_{2}-l/)\gamma^{-}\gamma_{\perp}}{(p_{2}-k_{2}-l)^{2}l^{2}}\frac{n_{\rho}}{n\cdot
l}$ $\displaystyle\phi^{(1)}_{\rho,e}$ $\displaystyle=$
$\displaystyle\frac{ig^{2}C_{F}}{4}\frac{\gamma^{-}\gamma_{\perp}(k/_{2}+l/)\gamma^{\rho}\gamma_{\perp}\gamma^{+}}{(k_{2}+l)^{2}l^{2}}\frac{n_{\rho}}{n\cdot
l}.$ (111)

The summation of the contributions from above irreducible sub-diagrams gives
the following result:

$\displaystyle
G^{(1^{\prime})}_{up,bT1}(x_{1},x_{2})=\phi^{(1)}_{\rho,d}\otimes\left[G^{(0)}_{bT,1}(x_{1},\xi_{2},x_{2})-G^{(0)}_{bT,1}(x_{1},\xi_{2})\right],$
(112) $\displaystyle
G^{(1^{\prime})}_{down,bT1}(x_{1},x_{2})=\phi^{(1)}_{\rho,e}\otimes\left[-G^{(0)}_{bT,1}(x_{1},x_{2})+G^{(0)}_{bT,1}(x_{1},x_{2},\xi_{2})\right].$
(113)

###  A.7 The NLO amplitudes for $G^{(0^{\prime})}_{bT2}$

The NLO amplitudes for $G^{(0^{\prime})}_{bT2}$ with the additional gluon
emitted from the final state $\rho$ meson are the following:

$\displaystyle G^{(1^{\prime})}_{bT2,2d}$ $\displaystyle=$ $\displaystyle-
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\beta}(p/_{2}-k/_{2}-l/)\gamma_{\mu}p/_{1}\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](p_{2}-k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(114) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{\rho,d}\otimes\left[G^{(0)}_{bT,2}(x_{1},\xi_{2},x_{2})-G^{(0)}_{bT,2}(x_{1},\xi_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bT2,2e}$ $\displaystyle=$ $\displaystyle
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\beta}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma^{\gamma}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(115) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{\rho,e}\otimes\left[-G^{(0)}_{bT,2}(x_{1},x_{2})+G^{(0)}_{bT,2}(x_{1},x_{2},\xi_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bT2,2f}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}C^{2}_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\nu}p/_{1}\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](p_{2}-k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(116) $\displaystyle=$
$\displaystyle\phi^{(1)}_{\rho,d}\otimes\left[G^{(0)}_{bT,2}(x_{1},x_{2})-G^{(0)}_{bT,2}(x_{1},\xi_{2},x_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bT2,2g}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}C^{2}_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\nu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(117) $\displaystyle=$
$\displaystyle\phi^{(1)}_{\rho,e}\otimes\left[-G^{(0)}_{bT,2}(x_{1},x_{2},\xi_{2})+G^{(0)}_{bT,2}(x_{1},\xi_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bT2,2h}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(p/_{1}+l/)\gamma_{\alpha}(p/_{1}-k/_{1}-l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2}+l)^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(p_{2}-k_{2}+l)^{2}[(p_{1}-k_{1}-l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(118) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{\rho,d}\otimes
G^{(0)}_{bT,2}(x_{1},\xi_{2},x_{2}),$ $\displaystyle
G^{(1^{\prime})}_{bT2,2i}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(p/_{1}-l/)\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}(p_{2}-k_{2}-l)^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(119) $\displaystyle\sim 0,$ $\displaystyle G^{(1^{\prime})}_{bT2,2j}$
$\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}+l/)\gamma^{\alpha}(k/_{2}+l/)\gamma_{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\alpha}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(k_{1}+l)^{2}(k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(120) $\displaystyle\sim 0,$ $\displaystyle G^{(1^{\prime})}_{bT2,2k}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}p/_{1}\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(p/_{1})\gamma_{5}\left[\frac{n/_{-}}{\sqrt{2}}\phi^{-}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(121) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{\rho,e}\otimes
G^{(0)}_{bT,2}(x_{1},x_{2},\xi_{2}),$

with the wave functions $\phi^{(1)}_{\rho,d}$ and $\phi^{(1)}_{\rho,e}$ in the
form of

$\displaystyle\phi^{(1)}_{\rho,d}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{4}\frac{\gamma_{\perp}\gamma^{+}\gamma^{\rho}(p/_{2}-k/_{2}-l/)\gamma^{-}\gamma_{\perp}}{(p_{2}-k_{2}-l)^{2}l^{2}}\frac{n_{\rho}}{n\cdot
l},$ $\displaystyle\phi^{(1)}_{\rho,e}$ $\displaystyle=$
$\displaystyle\frac{ig^{2}C_{F}}{4}\frac{\gamma^{-}\gamma_{\perp}(k/_{2}+l/)\gamma^{\rho}\gamma_{\perp}\gamma^{+}}{(k_{2}+l)^{2}l^{2}}\frac{n_{\rho}}{n\cdot
l}.$ (122)

The summation over the contributions from above irreducible sub-diagrams gives
the following result:

$\displaystyle
G^{(1^{\prime})}_{up,bT2}(x_{1},x_{2})=\phi^{(1)}_{\rho,d}\otimes\left[G^{(0)}_{bT,2}(x_{1},\xi_{2},x_{2})-G^{(0)}_{bT,2}(x_{1},\xi_{2})\right],$
(123) $\displaystyle
G^{(1^{\prime})}_{down,bT1}(x_{1},x_{2})=\phi^{(1)}_{\rho,e}\otimes\left[-G^{(0)}_{bT,2}(x_{1},x_{2})+G^{(0)}_{bT,2}(x_{1},x_{2},\xi_{2})\right].$
(124)

###  A.8 The NLO amplitudes for $G^{(0^{\prime})}_{bT3}$

The NLO amplitudes for $G^{(0^{\prime})}_{bT3}$ with the additional gluon
emitted from the final $\rho$ meson are the following:

$\displaystyle G^{(1^{\prime})}_{bT3,2d}$ $\displaystyle=$ $\displaystyle-
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\beta}(p/_{2}-k/_{2}-l/)\gamma_{\mu}m_{b}\gamma^{\gamma}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](p_{2}-k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(125) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{\rho,d}\otimes\left[G^{(0)}_{bT,3}(x_{1},\xi_{2},x_{2})-G^{(0)}_{bT,3}(x_{1},\xi_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bT3,2e}$ $\displaystyle=$ $\displaystyle
eg^{4}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\beta}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma^{\gamma}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]F_{\alpha\beta\gamma}}{(k_{1}-k_{2})^{2}(k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(126) $\displaystyle=$
$\displaystyle\frac{9}{8}\phi^{(1)}_{\rho,e}\otimes\left[-G^{(0)}_{bT,3}(x_{1},x_{2})+G^{(0)}_{bT,3}(x_{1},x_{2},\xi_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bT3,2f}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}C^{2}_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\nu}m_{b}\gamma_{\alpha}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](p_{2}-k_{2}-l)^{2}l^{2}}\bigg{\\}}$
(127) $\displaystyle=$
$\displaystyle\phi^{(1)}_{\rho,d}\otimes\left[G^{(0)}_{bT,3}(x_{1},x_{2})-G^{(0)}_{bT,3}(x_{1},\xi_{2},x_{2})\right],$
$\displaystyle G^{(1^{\prime})}_{bT3,2g}$ $\displaystyle=$
$\displaystyle\frac{eg^{4}C^{2}_{F}}{2}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\nu}(p/_{1}-k/_{2}-l/+m_{b})\gamma_{\alpha}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(128) $\displaystyle\sim 0,$ $\displaystyle G^{(1^{\prime})}_{bT3,2h}$
$\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma^{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(m_{b}-l/)\gamma_{\alpha}(p/_{1}-k/_{1}-l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(p_{2}-k_{2}-l)^{2}[(p_{1}-k_{1}-l)^{2}-m^{2}_{b}]l^{2}}\bigg{\\}}$
(129) $\displaystyle=$ $\displaystyle-\phi^{(1)}_{\rho,d}\otimes
G^{(0)}_{bT,3}(x_{1},\xi_{2},x_{2}),$ $\displaystyle
G^{(1^{\prime})}_{bT3,2i}$ $\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}-l/)\gamma^{\alpha}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\nu}(p/_{2}-k/_{2}-l/)\gamma_{\mu}(m_{b}-l/)\gamma_{\alpha}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2}-l)^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}(p_{2}-k_{2}-l)^{2}(k_{1}-l)^{2}l^{2}}\bigg{\\}}$
(130) $\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{\rho,d}\otimes
G^{(0)}_{bT,3}(x_{1},\xi_{2}),$ $\displaystyle G^{(1^{\prime})}_{bT3,2j}$
$\displaystyle=$
$\displaystyle\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\nu}(k/_{1}+l/)\gamma^{\alpha}(k/_{2}+l/)\gamma_{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\alpha}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2})^{2}(k_{1}+l)^{2}(k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(131) $\displaystyle=$ $\displaystyle\frac{1}{8}\phi^{(1)}_{\rho,e}\otimes
G^{(0)}_{bT,3}(x_{1},x_{2}),$ $\displaystyle G^{(1^{\prime})}_{bT3,2k}$
$\displaystyle=$
$\displaystyle-\frac{eg^{4}}{9}\mathbf{Tr}\bigg{\\{}\frac{\gamma^{\alpha}(k/_{2}+l/)\gamma^{\nu}\left[\epsilon/_{2T}p/_{2}\phi^{T}_{\rho}(x_{2})\right]\gamma_{\mu}m_{b}\gamma_{\alpha}(p/_{1}-k/_{1}+l/+m_{b})\gamma_{\nu}(m_{B})\gamma_{5}\left[\frac{n/_{+}}{\sqrt{2}}\phi^{+}_{B}(x_{1})\right]}{[(p_{1}-k_{2})^{2}-m^{2}_{b}](k_{1}-k_{2}-l)^{2}[(p_{1}-k_{1}+l)^{2}-m^{2}_{b}](k_{2}+l)^{2}l^{2}}\bigg{\\}}$
(132) $\displaystyle=$ $\displaystyle-\frac{1}{8}\phi^{(1)}_{\rho,e}\otimes
G^{(0)}_{bT,3}(x_{1},x_{2},\xi_{2}),$

with the wave functions $\phi^{(1)}_{\rho,d}$ and $\phi^{(1)}_{\rho,e}$ in the
form of

$\displaystyle\phi^{(1)}_{\rho,d}$ $\displaystyle=$
$\displaystyle\frac{-ig^{2}C_{F}}{4}\frac{\gamma_{\perp}\gamma^{+}\gamma^{\rho}(p/_{2}-k/_{2}-l/)\gamma^{-}\gamma_{\perp}}{(p_{2}-k_{2}-l)^{2}l^{2}}\frac{n_{\rho}}{n\cdot
l},$ $\displaystyle\phi^{(1)}_{\rho,e}$ $\displaystyle=$
$\displaystyle\frac{ig^{2}C_{F}}{4}\frac{\gamma^{-}\gamma_{\perp}(k/_{2}+l/)\gamma^{\rho}\gamma_{\perp}\gamma^{+}}{(k_{2}+l)^{2}l^{2}}\frac{n_{\rho}}{n\cdot
l}.$ (133)

The summation of the contributions from above irreducible sub-diagrams gives
the following result:

$\displaystyle
G^{(1^{\prime})}_{up,bT3}(x_{1},x_{2})=\phi^{(1)}_{\rho,d}\otimes\left[G^{(0)}_{bT,3}(x_{1},\xi_{2},x_{2})-G^{(0)}_{bT,3}(x_{1},\xi_{2})\right],$
(134) $\displaystyle
G^{(1^{\prime})}_{down,bT3}(x_{1},x_{2})=\phi^{(1)}_{\rho,e}\otimes\left[-G^{(0)}_{bT,3}(x_{1},x_{2})+G^{(0)}_{bT,3}(x_{1},x_{2},\xi_{2})\right].$
(135)

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