# The $\rho(770,1450)\to\omega\pi$ contributions for three-body decays
$B\to\bar{D}^{(*)}\omega\pi$

Yu-Shan Ren1,3 Ai-Jun Ma2 Wen-Fei Wang4,1<EMAIL_ADDRESS>1Institute of
Theoretical Physics and State Key Laboratory of Quantum Optics and Quantum
Optics Devices, Shanxi University, Taiyuan, Shanxi 030006, China
2School of Mathematics and Physics, Nanjing Institute of Technology, Nanjing,
Jiangsu 211167, China
3School of Physics, University of Electronic Science and Technology of China,
Chengdu 610054, China
4Departament de Física Quàntica i Astrofísica and Institut de Ciències del
Cosmos (ICCUB), Facultat de Física, Universitat de Barcelona, Martí i Franquès
1, 08028, Barcelona, Spain

###### Abstract

The decays $B\to\bar{D}^{(*)}\omega\pi$ are very important for the
investigation of $\rho$ excitations and the test of factorization hypothesis
for $B$ meson decays. The $B^{+}\to\bar{D}^{(*)0}\omega\pi^{+}$ and $B^{0}\to
D^{(*)-}\omega\pi^{+}$ in particular have been measured by different
collaborations but without any predictions for their observables from
theoretical side. In this work, we study the contributions of
$\rho(770,1450)\to\omega\pi$ for the cascade decays
$B^{+}\to\bar{D}^{(*)0}\rho^{+}\to\bar{D}^{(*)0}\omega\pi^{+}$, $B^{0}\to
D^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$ and $B_{s}^{0}\to
D_{s}^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$. We introduce
$\rho(770,1450)\to\omega\pi$ subprocesses into the distribution amplitudes for
$\omega\pi$ system via the vector form factor $F_{\omega\pi}(s)$ and then
predict the branching fractions for the first time for concerned quasi-two-
body decays with $\rho(770,1450)\to\omega\pi$, as well as the corresponding
longitudinal polarization fractions $\Gamma_{L}/\Gamma$ for the cases with the
vector $\bar{D}^{*0}$ or $D_{(s)}^{*-}$ in their final states. The branching
fractions of these quasi-two-body decays are predicted at the order of
$10^{-3}$, which can be detected at the LHCb and Belle-II experiments. The
predictions for the decays
${B}^{0}\to{D}^{*-}\rho(770)^{+}\to{D}^{*-}\omega\pi^{+}$ and
${B}^{0}\to{D}^{*-}\rho(1450)^{+}\to{D}^{*-}\omega\pi^{+}$ agree well with the
measurements from Belle Collaboration. In order to avoid the pollution from
annihilation Feynman diagrams, we propose to take the decays $B_{s}^{0}\to
D_{s}^{*-}\rho(770,1450)^{+}$, which have only emission diagrams at quark
level, to test the factorization hypothesis for $B$ decays.

##  I Introduction

Three-body hadronic $B$ meson decay processes always provide us a rich field
to investigate various aspects of the strong and weak interactions. We may
rely on them to study dynamical models for the strong interaction, to analyse
hadron spectroscopy and explore the properties and substructures of resonant
states, to determine the fundamental parameters for quark mixing and to
understand the essence of $CP$ asymmetries. In recent years, experimental
efforts for these decay processes by employing Dalitz plot technique
prd94.1046  have revealed valuable information for the strong and weak
dynamics involved. But on the theoretical side, it is complicated to describe
the strong dynamics in these decays because of the rescattering processes
1512.09284 ; prd89.094013 ; epjc78-897 ; prd71.074016 , hadron-hadron
interactions and three-body effects npps199-341 ; prd84.094001  in the final
states. The resonance contributions in relevant decay channels, which are
associated with the scalar, vector and tensor intermediate states, could be
isolated from the total decay amplitudes and be studied within the quasi-two-
body framework plb763-29 ; 1605.03889 ; prd96.113003 .

Three-body decays $B\to\bar{D}^{(*)}\omega\pi$, with one open charm meson in
the final state of each channel, are relatively simple from the theoretical
point of view. The heavy $b$-quark weak decay in these processes receive
contributions only from tree-level $W$ exchange operators $O_{1}$ and $O_{2}$,
which can be described well by the effective Hamiltonian $\mathcal{H}_{\rm
eff}$ rmp68.1125  within the factorization method zpc34-103 . Among these
decays, $B^{+}\to\bar{D}^{(*)0}\omega\pi^{+}$ and
${B}^{0}\to{D}^{(*)-}\omega\pi^{+}$ were measured by CLEO Collaboration for
the first time twenty years ago prd64.092001 . The decay
${B}^{0}\to{D}^{*-}\omega\pi^{+}$ was studied later by BABAR and Belle
Collaborations with the updated total branching fractions
$\mathcal{B}=(2.88\pm 0.21({\rm stat.})\pm 0.31({\rm syst.}))\times 10^{-3}$
prd74.012001  and $\mathcal{B}=(2.31\pm 0.11({\rm stat.})\pm 0.14({\rm
syst.}))\times 10^{-3}$ prd92.012013 , respectively. The $\omega\pi$ pair in
the final states of $B\to\bar{D}^{(*)}\omega\pi$ decays is related to the
resonance $\rho(1450)$, the excitation of $\rho(770)$ PDG22 . In $B$ meson
decays, $\rho(1450)$ was actually observed for the first time in
$B\to\bar{D}^{(*)}\omega\pi$ decays by CLEO in prd64.092001 . In Ref.
prd92.012013 , the surprising large contribution for $\omega\pi$ from
$\rho(770)$ in ${B}^{0}\to{D}^{*-}\omega\pi^{+}$ decay was measured to be

$\displaystyle\mathcal{B}({B}^{0}\to{D}^{*-}\rho(770)^{+}\to{D}^{*-}\omega\pi^{+})=(1.48^{+0.37}_{-0.63})\times
10^{-3}$ (1)

as the branching fraction ($\mathcal{B}$), which is comparable to the
corresponding data prd92.012013 ; PDG22

$\displaystyle\mathcal{B}({B}^{0}\to{D}^{*-}\rho(1450)^{+}\to{D}^{*-}\omega\pi^{+})=(1.07^{+0.40}_{-0.34})\times
10^{-3}$ (2)

for the intermediate state $\rho(1450)$.

The natural decay mode of $\rho(770)\to\omega\pi$ is blocked as a result of
the resonance pole mass which is below the threshold of the $\omega\pi$ pair.
But the virtual contribution plb25-294 ; Dalitz62 ; prd94.072001 ; plb791-342
from the Breit-Wigner (BW) BW-model  tail for resonance $\rho(770)$ was found
playing a vital role in the production of $\omega\pi$ for the processes of
$e^{+}e^{-}\to\omega\pi^{0}$ plb174-453 ; plb486-29 ; plb562-173 ; plb669-223
; JETPL94-734 ; prd88.054013 ; prd94.112001 ; prd96.092009 ; plb813-136059 ;
2309.00280  and $\tau\to\omega\pi\nu_{\tau}$ plb185-223 ; prd61.072003 ;
rmp78.1043 ; prl103.041802 . The most precise determination for the mass and
width of the resonance $\rho(1450)$ comes actually from $e^{+}e^{-}$
annihilation and the related process of $\tau$ decay zpc62-455 . The mass of
$\rho(1450)$ is consistent with that for the $2S$ excitation of $\rho(770)$
prd55.4157 , but it has been suggested as a $2S$-hybrid mixture in Ref.
prd60.114011  because of its decay characters npb443-233 ; prd52.1706 ;
prd56.1584 . The study of $\rho(1450)$ in $B$ decays and the investigation of
its interference with its ground state would lead to a better understanding of
its properties prd92.012013 . Its contributions for the kaon pair have been
explored in Refs. prd101.111901 ; prd103.056021 ; prd103.016002 ; cpc46-053104
and in Refs. prd85.092016 ; prd93.052018 ; prd103.114028 ; prd104.116019  in
three-body $B$ and $D$ meson decays, respectively, in recent years.

In this paper, we shall concentrate on the cascade decays
$B^{+}\to\bar{D}^{(*)0}\rho^{+}\to\bar{D}^{(*)0}\omega\pi^{+}$, $B^{0}\to
D^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$ and $B_{s}^{0}\to
D_{s}^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$, where $\rho^{+}$ in this work
stands for the intermediate states $\rho(770)^{+}$ and $\rho(1450)^{+}$
decaying into $\omega\pi^{+}$. In the very recent study performed by SND
Collaboration for $e^{+}e^{-}\to\omega\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}\pi^{0}$
process in the energy range $1.05$-$2.00$ GeV, four isovector vector
resonances covering $\rho(770)$, $\rho(1450)$, $\rho(1700)$ and $\rho(2150)$
have been employed to parametrize the related form factor for the
$\rho\to\omega\pi$ transition 2309.00280 . But we noticed from the Born cross
section in Ref. 2309.00280  that the contributions for $\omega\pi$ from
$\rho(1700)$ and the so called $\rho(2150)$ state are not large and not
important when comparing with those from $\rho(770)$ and $\rho(1450)$. In
addition, the excited $\rho$ states around $2$ GeV are not well understood
plb813-136059 ; prd105.074035 . In this context we will leave the
contributions for $\omega\pi$ from $\rho(1700)$ and $\rho(2150)$ in the
concerned decays to future studies.

Figure 1: Schematic view of the cascade decays
$B^{+}\to\bar{D}^{(*)0}\rho^{+}\to\bar{D}^{(*)0}\omega\pi^{+}$, $B^{0}\to
D^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$ and $B_{s}^{0}\to
D_{s}^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$, here $\rho^{+}$ stands for the
intermediate states $\rho(770,1450)^{+}$ decaying into $\omega\pi^{+}$ in this
work.

The schematic diagram for the cascade decays
$B^{+}\to\bar{D}^{(*)0}\rho^{+}\to\bar{D}^{(*)0}\omega\pi^{+}$, $B^{0}\to
D^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$ and $B_{s}^{0}\to
D_{s}^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$ is shown in Fig. 1, with
$\rho^{+}\in\\{\rho(770)^{+},\rho(1450)^{+}\\}$. In the $B$ meson rest frame,
the initial state will decay into the intermediate resonance $\rho^{+}$ as
well as the bachelor state $\bar{D}^{(*)0}$ or $D_{(s)}^{(*)-}$, and then the
resonance decays into its daughters $\omega$ and $\pi^{+}$. The state $\omega$
can be independently reconstructed from its two channels
$\omega\to\pi^{+}\pi^{-}\pi^{0}$ and $\omega\to\pi^{0}\gamma$ plb562-173 ;
2309.00280 ; prd94.112001 ; prd96.092009 ; prd88.054013 ; JETPL94-734 . The
decay process $B^{0}\to D^{*-}\omega\pi^{+}$ has only been studied in Refs.
JHEP1109-129 ; JHEP2002-168  with the factorization hypothesis on the
theoretical side but without any observable predictions for its branching
fraction. In this work, we shall study these concerned cascade decays in the
perturbative QCD (PQCD) approach plb504-6 ; prd63.054008 ; prd63.074009 ;
ppnp51-85 ; prd70.054006 . The subprocesses $\rho(770,1450)\to\omega\pi$ in
these decays can not be calculated in PQCD approach; we will introduce them
into the distribution amplitudes for $\omega\pi$ system via the vector form
factor $F_{\omega\pi}$ which has been measured with related processes of
$e^{+}e^{-}$ annihilation and $\tau$ decay. In the first approximation in
isobar formalism pr135.B551 ; pr166.1731 ; prd11.3165 , one can neglect the
interaction between $\omega\pi$ system and the corresponding bachelor state in
relevant decay process, and then study the decays
$B\to\bar{D}^{(*)}\rho(770,1450)\to\bar{D}^{(*)}\omega\pi$ within the quasi-
two-body framework plb763-29 ; 1605.03889 ; prd96.113003 . The quasi-two-body
framework based on PQCD approach has been discussed in detail in plb763-29 ,
which has been followed in Refs. prd101.111901 ; jhep2003-162 ; prd96.036014 ;
prd95.056008 ; prd96.093011 ; npb923-54 ; epjc80-394 ; prd103.013005 ;
prd103.096016 ; prd103.016002 ; prd103.056021 ; cpc41-083105  for the quasi-
two-body $B$ meson decays in recent years. For relevant works on three-body
$B$ decays within the symmetries one is referred to Refs. plb564-90 ;
prd72.075013 ; prd72.094031 ; prd84.056002 ; plb727-136 ; plb726-337 ;
prd89.074043 ; plb728-579 ; prd91.014029 . Parallel analyses within QCD
factorization can be found in Refs. JHEP2011-103 ; jhep2006-073 ; plb622-207 ;
plb669-102 ; prd79.094005 ; prd72.094003 ; prd76.094006 ; prd88.114014 ;
prd102.053006 ; prd89.074025 ; prd94.094015 ; npb899-247 ; epjc75-536 ;
prd89.094007 ; prd87.076007 .

This paper is organized as follows. In Sec. II, we give a brief introduction
of the theoretical framework for the quasi-two-body decays
$B\to\bar{D}^{(*)}\rho(770,1450)\to\bar{D}^{(*)}\omega\pi$ within PQCD
approach. In Sec. III, we present our numerical results of the branching
fractions for $B^{+}\to\bar{D}^{(*)0}\rho^{+}\to\bar{D}^{(*)0}\omega\pi^{+}$,
$B^{0}\to D^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$ and $B_{s}^{0}\to
D_{s}^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$ along with some necessary
discussions. Summary of this work is given in Sec. IV. The wave functions
involved and factorization formulae for the related decay amplitudes are
collected in the Appendix.

##  II Framework

The relevant effective weak Hamiltonian $\mathcal{H}_{\rm eff}$ for the decays
$B\to\bar{D}^{(*)}\rho(770,1450)$ with subprocesses
$\rho(770,1450)\to\omega\pi$ via the $\bar{b}\to\bar{c}$ transition is written
as rmp68.1125

$\mathcal{H}_{\rm
eff}=\frac{G_{F}}{\sqrt{2}}V^{*}_{cb}V_{ud}\big{[}C_{1}(\mu)O^{c}_{1}(\mu)+C_{2}(\mu)O^{c}_{2}(\mu)\big{]},\;\;$
(3)

where $G_{F}=1.1663788(6)\times 10^{-5}$ GeV-2 PDG22  is the Fermi coupling
constant, $V_{cb}$ and $V_{ud}$ are the Cabibbo-Kobayashi-Maskawa (CKM) matrix
Cabibbo ; Kobayashi  elements. The Wilson coefficients $C_{1,2}(\mu)$ at scale
$\mu$ are always combined as $a_{1}=C_{1}+C_{2}/3$ and $a_{2}=C_{2}+C_{1}/3$.
The local four-quark operators ${O}^{c}_{1,2}$ are the products of two $V-A$
currents, and one has ${O^{c}_{1}}=(\bar{b}d)_{V-A}\,(\bar{u}c)_{V-A}$ and
${O^{c}_{2}}=(\bar{b}c)_{V-A}\,(\bar{u}d)_{V-A}$ rmp68.1125 .

In light cone coordinates the momentum $p_{B}$ is equal to
$\frac{m_{B}}{\sqrt{2}}(1,1,0_{\rm T})$ in the rest frame of $B$ meson, where
the mass $m_{B}$ stands for initial state $B^{+},B^{0}$ or $B^{0}_{s}$. In the
same coordinates, the resonances $\rho(770)$, its excited state $\rho(1450)$
and the $\omega\pi$ system generated from resonances by strong interaction
have the momentum $p=\frac{m_{B}}{\sqrt{2}}(\zeta,1-r^{2},0_{\rm T})$, with
the squared invariant mass $p^{2}=s$ for $\omega\pi$ system. For the bachelor
state $\bar{D}^{(*)}$ in the related processes, its momentum is defined as
$p_{3}=\frac{m_{B}}{\sqrt{2}}(1-\zeta,r^{2},0_{\rm T})$. The longitudinal
polarization vectors for the intermediate state and the $\bar{D}^{*}$ meson,
respectively, are

$\displaystyle\epsilon^{\rho}_{L}$ $\displaystyle=$
$\displaystyle\frac{m_{B}}{\sqrt{2s}}(-\zeta,1-r^{2},0_{\rm T}),$ (4)
$\displaystyle\epsilon^{D^{*}}_{L}$ $\displaystyle=$
$\displaystyle\frac{m_{B}}{\sqrt{2}m_{D}}(1-\zeta,-r^{2},0_{\rm T}),$ (5)

where the parameter $r$ will be satisfied by the relation
$p^{2}_{3}=m^{2}_{D}$, with the mass $m_{D}$ for the bachelor state
$\bar{D}^{(*)}$. The spectator quark comes out from initial state and goes
into the intermediate states in hadronization shown in Fig. 2 (a) has the
momenta $k_{B}=(\frac{m_{B}}{\sqrt{2}}x_{B},0,k_{B{\rm T}})$ and
$k=(0,\frac{m_{B}}{\sqrt{2}}x,k_{\rm T})$ in $B$ and $\rho$ states,
respectively, and the light quark in the $\bar{D}^{(*)}$ got the momentum
$k_{3}=(\frac{m_{B}}{\sqrt{2}}(1-\zeta)x_{3},0,k_{3{\rm T}})$. The $x_{B}$,
$x$ and $x_{3}$, which will run from zero to one in the calculations, are the
momentum fractions for the initial state $B$, the resonances $\rho(770,1450)$
and the bachelor final state $\bar{D}^{(*)}$, respectively.

Figure 2: Typical Feynman diagrams for the quasi-two-body decays
$B\to\bar{D}^{(*)}\rho\to\bar{D}^{(*)}\omega\pi$ at quark level, where
$q\in\\{u,d\\}$ and $q^{\prime}\in\\{u,d,s\\}$, the symbol $\otimes$ stands
for the weak interaction vertex.

In the PQCD approach, one has factorization formula of the decay amplitude
plb561-258 ; prd89.074031

$\displaystyle\mathcal{A}=\phi_{B}\otimes{\mathcal{H}}\otimes\phi^{P\text{-wave}}_{\omega\pi}\otimes\phi_{D^{(*)}}$
(6)

for the quasi-two-body decays $B\to\bar{D}^{(*)}\rho\to\bar{D}^{(*)}\omega\pi$
at leading order of the strong coupling $\alpha_{s}$ according to the Feynman
diagrams of Fig. 2. Here, the hard kernel ${\mathcal{H}}$ contains only one
hard gluon exchange, and the symbol $\otimes$ stands for the convolutions in
parton momenta. The expressions for the $B$ meson and bachelor final state
$\bar{D}^{(*)}$ distribution amplitudes $\phi_{B}$ and $\phi_{D^{(*)}}$ could
be found in the Appendix.

For the $P$-wave $\omega\pi$ system along with the subprocess
$\rho\to\omega\pi$, the distribution amplitudes hold the same structure of the
vector mesons and could be organized into prd76.074018 ; prd101.111901 ;
prd102.056017

$\displaystyle\phi^{P\text{-wave}}_{\omega\pi,L}(x,s)$ $\displaystyle=$
$\displaystyle\frac{-1}{\sqrt{2N_{c}}}\big{[}\sqrt{s}\,{\epsilon/}\\!_{L}\phi^{0}(x,s)+{\epsilon/}\\!_{L}{p/}\phi^{t}(x,s)+\sqrt{s}\phi^{s}(x,s)\big{]}\\!,$
(7) $\displaystyle\phi^{P\text{-wave}}_{\omega\pi,T}(x,s)$ $\displaystyle=$
$\displaystyle\frac{-1}{\sqrt{2N_{c}}}\big{[}\sqrt{s}\,{\epsilon/}\\!_{T}\phi^{v}(x,s)+{\epsilon/}\\!_{T}{p/}\phi^{T}(x,s)+\sqrt{s}\,i\epsilon_{\mu\nu\rho\sigma}\gamma_{5}\gamma^{\mu}\epsilon_{T}^{*\nu}n^{\rho}v^{\sigma}\phi^{a}(x)\big{]}\\!,$
(8)

with two dimensionless lightlike vectors $n=(1,0,{\bf 0}_{T})$ and
$v=(0,1,{\bf 0}_{T})$, and $N_{c}$ is the number of colors for QCD. We adopt
the convention $\epsilon^{0123}=1$ for the Levi-Civita tensor
$\epsilon^{\mu\nu\alpha\beta}$. The twist-$2$ distribution amplitude for a
longitudinally polarized $\rho$ state can be parametrized as prd76.074018

$\displaystyle\phi^{0}(x,s)$ $\displaystyle=$
$\displaystyle\frac{3f_{\omega\pi}(s)}{\sqrt{2N_{c}}}x(1-x)\big{[}1+a_{R}^{0}C^{3/2}_{2}(1-2x)\big{]}.~{}~{}$
(9)

where the Gegenbauer polynomial $C^{3/2}_{2}(t)=3/2(5t^{2}-1)$. The twist-$2$
transversely polarized distribution amplitude $\phi^{T}(x,s)$ has a similar
form as the longitudinally polarized one, we have prd76.074018

$\displaystyle\phi^{T}(x,s)$ $\displaystyle=$
$\displaystyle\frac{3f_{\omega\pi}^{T}(s)}{\sqrt{2N_{c}}}x(1-x)\big{[}1+a^{T}_{R}C^{3/2}_{2}(1-2x)\big{]}.\quad$
(10)

The form of the twist-3 distribution amplitudes are prd76.074018 ;
prd101.111901 ; prd102.056017

$\displaystyle\phi^{t}(x,s)$ $\displaystyle=$
$\displaystyle\frac{3f_{\omega\pi}^{T}(s)}{2\sqrt{2N_{c}}}(1-2x)^{2}\big{[}1+a_{R}^{t}C^{3/2}_{2}(1-2x)\big{]},$
(11) $\displaystyle\phi^{s}(x,s)$ $\displaystyle=$
$\displaystyle\frac{3f_{\omega\pi}^{T}(s)}{2\sqrt{2N_{c}}}(1-2x)\big{[}1+a_{R}^{s}(1-10x+10x^{2})\big{]},$
(12) $\displaystyle\phi^{v}(x,s)$ $\displaystyle=$
$\displaystyle\frac{3f_{\omega\pi}(s)}{8\sqrt{2N_{c}}}\left[1+(1-2x)^{2}\right]\\!,$
(13) $\displaystyle\phi^{a}(x,s)$ $\displaystyle=$
$\displaystyle\frac{3f_{\omega\pi}(s)}{4\sqrt{2N_{c}}}(1-2x).$ (14)

We adopt the same Gegenbauer moments for the $P$-wave $\omega\pi$ system in
this work as they are in Refs. prd101.111901 ; plb763-29 ; prd103.056021  for
the pion pair or kaon pair in view of the fact that these parameters are
employed to describe the formation of the intermediate states. And the value
of Gegenbauer moment $a^{T}_{R}$ in twist-$2$ transversely polarized
distribution amplitude $\phi^{T}(x,s)$ is set to the same as it of $a^{0}_{R}$
in this work. The form factor $F_{\omega\pi}^{T}$ for the twist-$3$
distribution amplitudes of $\phi^{P\text{-wave}}_{\omega\pi,L}(x,s)$ and the
twist-$2$ of $\phi^{P\text{-wave}}_{\omega\pi,T}(x,s)$ are deduced from the
relation $f_{\omega\pi}^{T}(s)\approx(f^{T}_{\rho}/f_{\rho})f_{\omega\pi}(s)$
plb763-29  with the result $f^{T}_{\rho}/f_{\rho}=0.687$ at the scale $\mu=2$
GeV prd78.114509 .

The factor $f_{\omega\pi}(s)$ in Eq. (9) is employed as the abbreviation of
the transition form factor for $\rho(770,1450)\to\omega\pi$ decays in the
concerned processes. The related effective Lagrangian is written as plb66-165
; prd30.594 ; prd46.1195

$\displaystyle\mathcal{L}_{\rho\omega\pi}=g_{\rho\omega\pi}\epsilon_{\mu\nu\alpha\beta}\partial^{\mu}\rho^{\nu}\partial^{\alpha}\omega^{\beta}\pi.$
(15)

With the help of this Lagrangian, we can define the form factor
$F_{\omega\pi}(s)=f_{\omega\pi}(s)m_{\rho}/f^{2}_{\rho}$ from the matrix
element epja38-331 ; prd92.014014 ; 2307.10357

$\displaystyle\langle\omega(p_{a},\lambda)\pi(p_{b})|j_{\mu}(0)|0\rangle=i\epsilon_{\mu\nu\alpha\beta}\varepsilon^{\nu}(p_{a},\lambda)p_{b}^{\alpha}p^{\beta}F_{\omega\pi}(s),$
(16)

where $j_{\mu}$ is the isovector part of the electromagnetic current,
$\lambda$ and $\varepsilon$ is the polarization and polarization vector for
$\omega$ meson, $p_{a}$ and $p_{b}$ are the momenta for $\omega$ and pion,
respectively, and $p=p_{a}+p_{b}$.

In the vector meson dominance model, the form factor $F_{\omega\pi}(s)$
defined from Eq. (16) is parametrized as plb486-29 ; prd88.054013 ;
prd94.112001 ; ppnp120-103884

$\displaystyle
F_{\omega\pi}(s)=\frac{g_{\rho\omega\pi}}{f_{\rho}}\sum\limits_{\rho_{i}}\frac{A_{i}e^{i\phi_{i}}m_{\rho_{i}}^{2}}{D_{\rho_{i}}(s)}\,,$
(17)

where the summation is over the isovector resonances
$\rho_{i}=\\{\rho(770),\rho(1450),\rho(1700),...\\}$ in $\rho$ family,
$A_{i}$, $\phi_{i}$ and $m_{\rho_{i}}$ are the weights, phases and masses for
these resonances, respectively, and one has $A=1$ and $\phi=0$ for
$\rho(770)$. Contributions from the excitations of $\omega$ meson can also be
include in Eq. (17), but their weights turn out to be negligibly small
jpg36-085008 . The parameter $f_{\rho}$ is the $\gamma^{*}\to\rho(770)$
coupling constant calculated from the decay width of $\rho(770)\to
e^{+}e^{-}$, the $g_{\rho\omega\pi}$ is the coupling constant for
$\rho(770)\to\omega\pi$ which can be calculated from the decay width of
$\omega\to\pi^{0}\gamma$ PDG22  or be estimated with the relation
$g_{\rho\omega\pi}\approx 3g_{\rho\pi\pi}^{2}/(8\pi^{2}F_{\pi})$ prc83.048201
, where $F_{\pi}=f_{\pi}/\sqrt{2}$ and $f_{\pi}$ is the decay constant for
pion. The denominator $D_{\rho_{i}}$ has a BW formula expression

$D_{\rho_{i}}(s)=m_{\rho_{i}}^{2}-s-i\sqrt{s}\,\Gamma_{\rho_{i}}(s).$ (18)

To describe the shape of the resonance $\rho(770)$, the energy-dependent width
is written as 2309.00280 ; prd88.054013

$\displaystyle\Gamma_{\rho(770)}(s)$ $\displaystyle=$
$\displaystyle\Gamma_{\rho(770)}\frac{m^{2}_{\rho(770)}}{s}\left(\frac{q_{\pi}(s)}{q_{\pi}(m^{2}_{\rho(770)})}\right)^{3}+\frac{g_{\rho\omega\pi}^{2}}{12\pi}q^{3}_{\omega}(s),$
(19)

where the first term corresponds to the decay $\rho(770)\to\pi\pi$, the second
to the decay of $\rho(770)\to\omega\pi$. And we have

$\displaystyle q_{\pi}(s)=\frac{1}{2}\sqrt{s-4m^{2}_{\pi}}\,,$ (20)
$\displaystyle
q_{\omega}(s)=\frac{1}{2\sqrt{s}}\sqrt{\left[s-(m_{\omega}+m_{\pi})^{2}\right]\left[s-(m_{\omega}-m_{\pi})^{2}\right]}\,.\qquad$
(21)

For the excited resonance $\rho(1450)$, the expression

$\displaystyle\Gamma_{\rho(1450)}(s)$ $\displaystyle=$
$\displaystyle\Gamma_{\rho(1450)}\Bigl{[}{\mathcal{B}}_{\rho(1450)\to\omega\pi}\Bigl{(}\frac{q_{\omega}(s)}{q_{\omega}(m^{2}_{\rho(1450)})}\Bigr{)}^{3}\qquad$
(22) $\displaystyle+$
$\displaystyle(1-{\mathcal{B}}_{\rho(1450)\to\omega\pi})\frac{m^{2}_{\rho(1450)}}{s}\Bigl{(}\frac{q_{\pi}(s)}{q_{\pi}(m^{2}_{\rho(1450)})}\Bigr{)}^{3}\Bigr{]}$

for the energy-dependent width is adopted in this work as it in Ref.
plb562-173 , where ${\mathcal{B}}_{\rho(1450)\to\omega\pi}$ is the branching
ratio of the $\rho(1450)\to\omega\pi$ decay, $\Gamma_{\rho(770)}$ and
$\Gamma_{\rho(1450)}$ are the full widths for $\rho(770)$ and $\rho(1450)$,
respectively.

For the differential branching fraction ($\mathcal{B}$), one has the formula
PDG22

$\displaystyle\frac{d{\mathcal{B}}}{ds}=\tau_{B}\frac{s\,|\mathbf{p}_{\pi}|^{3}|\mathbf{p}_{D}|^{3}}{24\pi^{3}m^{7}_{B}}{|{\mathcal{A}}|^{2}}$
(23)

for the quasi-two-body decays
$B\to\bar{D}^{(*)}\rho\to\bar{D}^{(*)}\omega\pi$, where $\mathcal{\tau}_{B}$
is the mean lifetime for $B$ meson, $s$ is the squared invariant mass for
$\omega\pi$ system. One should note that the phase space factor in Eq. (23) is
different from that for the decays with subprocesses of $\rho\to\pi\pi$ and
$\rho\to K\bar{K}$ as a result of the definition of $F_{\omega\pi}(s)$ in Eq.
(16); the relations

$\displaystyle\sum_{\lambda=0,\pm}\varepsilon^{\mu}{(p,\lambda)}\varepsilon^{\nu}{(p,\lambda)}=-g^{\mu\nu}+\frac{p^{\mu}p^{\nu}}{p^{2}},\qquad$
(24)
$\displaystyle\sum_{\lambda=0,\pm}|\epsilon_{\mu\nu\alpha\beta}p_{3}^{\mu}\varepsilon^{\nu}(p_{\omega},\lambda)p_{\pi}^{\alpha}p^{\beta}|^{2}=s\,|\mathbf{p}_{\pi}|^{2}|\mathbf{p}_{D}|^{2}(1-\cos^{2}{\theta})$
(25)

are employed for the derivation of Eq. (23), where $\theta$ is the angle
between the $\omega$ and bachelor state $\bar{D}^{(*)}$ three-momenta. In the
rest frame of the intermediate states, the magnitude of the momenta are
written as

$\displaystyle|\mathbf{p}_{\pi}|$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\left[s-(m_{\pi}+m_{\omega})^{2}\right]\left[s-(m_{\pi}-m_{\omega})^{2}\right]}}{2\sqrt{s}},\;\;$
(26) $\displaystyle|\mathbf{p}_{D}|$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\left[m^{2}_{B}-(s+m_{D})^{2}\right]\left[m^{2}_{B}-(s-m_{D})^{2}\right]}}{2\sqrt{s}},\quad$
(27)

for pion and the bachelor meson $\bar{D}^{(*)}$, where $m_{\pi}$, $m_{\omega}$
and $m_{D}$ are the masses for the pion, $\omega$ and the bachelor meson,
respectively. The Lorentz invariant decay amplitudes according to Fig. 2 for
the concerned decays are given in the Appendix.

##  III Results and Discussions

In the numerical calculation, we employ the decay constants $f_{\rho}=0.216\pm
0.003$ GeV jhep1608-098  for $\rho(770)$ and
$f_{\rho(1450)}=0.185^{+0.030}_{-0.035}$ GeV plb763-29  resulting from the
data zpc62-455  for $\rho(1450)$, the mean lives $\tau_{B^{\pm}}=1.638\times
10^{-12}$ s, $\tau_{B^{0}}=1.519\times 10^{-12}$ s and
$\tau_{B_{s}^{0}}=1.520\times 10^{-12}$ s for the initial states $B^{\pm}$,
$B^{0}$ and $B_{s}^{0}$ PDG22 , respectively. The masses for particles in
relevant decay processes, the decay constants for $B_{(s)}$, $D_{(s)}$ and
$D^{*}_{(s)}$ mesons, the full widths for resonances $\rho(770)$ and
$\rho(1450)$ (in units of GeV), and the Wolfenstein parameters for CKM matrix
elements are presented in Table 1.

Table 1: Masses, decay constants and full widths (in units of GeV) for relevant states as well as the Wolfenstein parameters for CKM matrix elements from Review of Particle Physics PDG22 , the $f_{D^{*}}$ and $f_{D^{*}_{s}}$ are cited from prd96.034524 . $m_{B^{\pm}}=5.279$ | $m_{B^{0}}=5.280$
---|---
$m_{B^{0}_{s}}=5.367$ | $m_{D^{\pm}}=1.870$
$m_{D^{0}}=1.865$ | $m_{D^{\pm}_{s}}=1.968$
$m_{D^{*\pm}}=2.010$ | $m_{D^{*0}}=2.007$
$m_{D^{*\pm}_{s}}=2.112$ | $m_{\pi^{\pm}}=0.140$
$m_{\omega}=0.783$ | $f_{B^{\pm,0}}=0.190$
$f_{B_{s}^{0}}=0.230$ | $f_{\pi^{+}}=0.130$
$f_{D^{\pm,0}}=0.2120$ | $f_{D_{s}^{\pm}}=0.2499$
$f_{D^{*\pm,0}}=0.2235$ | $f_{D_{s}^{*\pm}}=0.2688$
$m_{\rho(770)}=0.775$ | $m_{\rho(1450)}=1.465\pm 0.025$
$\Gamma_{\rho(770)}=0.1491$ | $\Gamma_{\rho(1450)}=0.400\pm 0.060$
$A=0.826^{+0.018}_{-0.015}$ | $\lambda=0.22500\pm 0.00067$
Table 2: PQCD results for the quasi-two-body decays $B^{+}\to\bar{D}^{(*)0}[\rho(770)^{+}\to]\pi^{+}\pi^{0}$, $B^{0}\to D^{(*)-}[\rho(770)^{+}\to]\pi^{+}\pi^{0}$ and $B_{s}^{0}\to D_{s}^{(*)-}[\rho(770)^{+}\to]\pi^{+}\pi^{0}$, along with their corresponding two-body data from Review of Particle Physics PDG22 . Decay modes |  Units | PQCD |  Data PDG22
---|---|---|---
$B^{+}\to\bar{D}^{0}[\rho(770)^{+}\to]\pi\pi^{+}$ | $\%$ |  $1.21^{+0.66+0.10+0.05}_{-0.42-0.12-0.06}$ | $1.34\pm 0.18$
$B^{0}\to D^{-}[\rho(770)^{+}\to]\pi\pi^{+}$ | $10^{-3}$ |  $7.63^{+4.82+0.97+0.34}_{-2.81-0.73-0.21}$ | $7.6\pm 1.2$
$B_{s}^{0}\to D_{s}^{-}[\rho(770)^{+}\to]\pi\pi^{+}$ | $10^{-3}$ |  $7.36^{+4.45+0.54+0.03}_{-2.64-0.60-0.09}$ | $6.8\pm 1.4$
$B^{+}\to\bar{D}^{*0}[\rho(770)^{+}\to]\pi\pi^{+}$ | $10^{-3}$ |  $9.03^{+4.30+0.73+0.51}_{-2.76-0.64-0.46}$ | $9.8\pm 1.7$
$B^{0}\to D^{*-}[\rho(770)^{+}\to]\pi\pi^{+}$ | $10^{-3}$ |  $8.15^{+4.24+0.64+0.03}_{-2.72-0.62-0.07}$ | $6.8\pm 0.9$
$B_{s}^{0}\to D_{s}^{*-}[\rho(770)^{+}\to]\pi\pi^{+}$ | $10^{-3}$ |  $7.12^{+3.56+0.56+0.01}_{-2.27-0.56-0.03}$ | $9.5\pm 2.0$

The crucial input $g_{\rho\omega\pi}$ for the form factor $F_{\omega\pi}(s)$
in Eq. (17) has been fitted to be $15.9\pm 0.4$ GeV-1 and $16.5\pm 0.2$ GeV-1
in prd94.112001 , respectively, by SND Collaboration recently with different
models for the form factor. This input can also be calculated from the decay
width of $\omega\to\pi^{0}\gamma$ prd96.054033 ; prd56.4084 ; with the
relation $g_{\rho\omega\pi}\approx 3g_{\rho\pi\pi}^{2}/(8\pi^{2}F_{\pi})$
prc83.048201 , it’s easy to get its value $14.8$ GeV-1. In the numerical
calculation of this work, we adopt $g_{\rho\omega\pi}=16.0\pm 2.0$ GeV-1 by
taking into account the corresponding values in Refs. prd94.112001 ;
prd86.057301 ; prd61.072003 ; JETPL94-734 ; prd55.249 ; prd77.113011 ;
prd86.037302 ; prd88.054013  for it. The weight $A_{1}$ in Eq. (17) for the
subprocess $\rho(1450)\to\omega\pi$ moves a lot in the literature, it has been
measured to be $0.584\pm 0.003$ and $0.164\pm 0.003$ in 2309.00280 , $0.175\pm
0.016$, $0.137\pm 0.006$ and $0.251\pm 0.006$ in prd94.112001 , $0.26\pm 0.01$
and $0.11\pm 0.01$ in prd88.054013  with different models for
$F_{\omega\pi}(s)$ in recent years. In view of the expression for
$F_{\omega\pi}(s)$ in Eq. (17), we have a constraint

$\displaystyle
A_{1}=\frac{g_{\rho(1450)\omega\pi}f_{\rho(1450)}m_{\rho(770)}}{g_{\rho(770)\omega\pi}f_{\rho(770)}m_{\rho(1450)}}$
(28)

for its value. With the relation

$\displaystyle f_{\rho(1450)}g_{\rho(1450)\omega\pi}=\sqrt{12\pi
f^{2}_{\rho(1450)}\mathcal{B}(\rho(1450)\to\omega\pi)\Gamma_{\rho(1450)}/p_{c}^{3}}\;,$
(29)

where $p_{c}=q_{\omega}(m^{2}_{\rho(1450)})$, and the measured result
$f^{2}_{\rho(1450)}\mathcal{B}(\rho(1450)\to\omega\pi)=0.011\pm 0.003$ GeV2
prd64.092001 , one has $A_{1}=0.171\pm 0.036$, where the error comes from the
uncertainties of mass and full width for $\rho(1450)$ in Table 1, the coupling
$16.0\pm 2.0$ GeV-1 and the measured result $0.011\pm 0.003$ GeV2 in
prd64.092001 . The value for $A_{1}$ from Eq. (28) is close to the results
$0.164\pm 0.003$ in 2309.00280  and $0.175\pm 0.016$ in prd94.112001 .

Table 3: PQCD predictions of the branching fractions (in units of $10^{-3}$) for the quasi-two-body decays $B_{(s)}\to\bar{D}_{(s)}\rho^{+}\to\bar{D}_{(s)}\omega\pi^{+}$, where $\rho^{+}$ means the resonance $\rho(770)^{+}$ or $\rho(1450)^{+}$. Decay modes |  $\mathcal{B}$ (in $10^{-3}$)
---|---
$B^{+}\to\bar{D}^{0}[\rho(770)^{+}\to]\omega\pi^{+}$ |  $1.42^{+0.81+0.15+0.11+0.10}_{-0.47-0.13-0.09-0.10}$
$B^{+}\to\bar{D}^{0}[\rho(1450)^{+}\to]\omega\pi^{+}$ |  $0.96^{+0.54+0.09+0.08+0.40}_{-0.32-0.09-0.08-0.40}$
$B^{0}\to D^{-}[\rho(770)^{+}\to]\omega\pi^{+}$ |  $0.80^{+0.58+0.12+0.06+0.07}_{-0.30-0.09-0.02-0.07}$
$B^{0}\to D^{-}[\rho(1450)^{+}\to]\omega\pi^{+}$ |  $0.52^{+0.35+0.06+0.03+0.22}_{-0.21-0.06-0.03-0.22}$
$B_{s}^{0}\to D_{s}^{-}[\rho(770)^{+}\to]\omega\pi^{+}$ |  $0.88^{+0.56+0.07+0.00+0.06}_{-0.32-0.07-0.01-0.06}$
$B_{s}^{0}\to D_{s}^{-}[\rho(1450)^{+}\to]\omega\pi^{+}$ |  $0.59^{+0.38+0.05+0.00+0.25}_{-0.22-0.04-0.00-0.25}$

When the subprocess $\rho(770)^{+}\to\omega\pi^{+}$ shrink into meson
$\rho(770)^{+}$, the six quasi-two-body decays of
$B\to\bar{D}^{(*)}\rho(770)^{+}\to\bar{D}^{(*)}\omega\pi^{+}$ will turned into
the two-body decay channels $B\to\bar{D}^{(*)}\rho(770)^{+}$. These six two-
body decays with $\rho(770)^{+}$ have been measured, one finds their branching
fractions in Table 2. In view of
$\mathcal{B}({\rho(770)^{+}\to\pi^{+}\pi^{0}})\approx 100\%$ PDG22 , the PQCD
results in Table 2 for the decays with subprocess
$\rho(770)^{+}\to\pi^{+}\pi^{0}$ could be seen as a way to test the framework
and inputs of this work. Obviously, these PQCD results in Table 2 agree with
the data quite well.

Table 4: Same as in Table 3 but with the different bachelor mesons $\bar{D}^{*0}$ and $D_{(s)}^{*-}$; the results in column $\Gamma_{L}/\Gamma$ are the predictions for the corresponding longitudinal polarization fraction. Decay modes |  $\mathcal{B}$ (in $10^{-3}$) | $\Gamma_{L}/\Gamma$
---|---|---
$B^{+}\to\bar{D}^{*0}[\rho(770)^{+}\to]\omega\pi^{+}$ |  $1.21^{+0.63+0.09+0.05+0.07}_{-0.39-0.09-0.03-0.07}$ |  $0.74^{+0.02}_{-0.02}$
$B^{+}\to\bar{D}^{*0}[\rho(1450)^{+}\to]\omega\pi^{+}$ |  $0.87^{+0.47+0.07+0.03+0.37}_{-0.29-0.07-0.02-0.37}$ |  $0.67^{+0.02}_{-0.02}$
$B^{0}\to D^{*-}[\rho(770)^{+}\to]\omega\pi^{+}$ |  $1.20^{+0.67+0.09+0.02+0.07}_{-0.40-0.08-0.01-0.07}$ |  $0.68^{+0.02}_{-0.02}$
$B^{0}\to D^{*-}[\rho(1450)^{+}\to]\omega\pi^{+}$ |  $0.89^{+0.50+0.06+0.02+0.38}_{-0.31-0.06-0.02-0.38}$ |  $0.63^{+0.01}_{-0.01}$
$B_{s}^{0}\to D_{s}^{*-}[\rho(770)^{+}\to]\pi\pi^{+}$ |  $1.03^{+0.58+0.08+0.00+0.05}_{-0.35-0.08-0.00-0.05}$ |  $0.65^{+0.01}_{-0.01}$
$B_{s}^{0}\to D_{s}^{*-}[\rho(1450)^{+}\to]\pi\pi^{+}$ |  $0.77^{+0.43+0.06+0.00+0.32}_{-0.26-0.06-0.00-0.32}$ |  $0.59^{+0.01}_{-0.01}$

Utilizing differential branching fractions the Eq. (23) and the decay
amplitudes collected in Appendix, we obtain the branching fractions in Tables
3, 4 for the concerned quasi-two-body decays with $\rho(770)^{+}$ and
$(1450)^{+}$ decaying into $\omega\pi^{+}$. For these PQCD branching fractions
in Tables 2-4, their first error comes from the uncertainties of the shape
parameter $\omega_{B}=0.40\pm 0.04$ or $\omega_{B_{s}}=0.50\pm 0.05$ for the
$B^{\pm,0}$ or $B^{0}_{s}$ meson; the Gegenbauer moments $C_{D}=0.6\pm 0.15$
or $C_{D^{*}}=0.5\pm 0.10$ for $D_{(s)}$ or $D^{*}_{(s)}$ mesons contribute
the second error; the third one is induced by the Gegenbauer moments
$a_{R}^{0}=0.25\pm 0.10$, $a_{R}^{t}=-0.60\pm 0.20$ and $a_{R}^{s}=0.75\pm
0.25$ plb763-29  for the intermediate states; the fourth one for the decay
results with $\rho\to\omega\pi$ comes from the uncertainties of the coupling
$g_{\rho\omega\pi}$ or $A_{1}$ in Eq. (17). There are other errors for the
PQCD predictions in this work, which come from the uncertainties of the masses
and the decay constants of the initial and final states, from the
uncertainties of the Wolfenstein parameters, etc., are small and have been
neglected.

Table 5: Experimental data for the relevant three-body branching fractions from Review of Particle Physics PDG22 . Decay mode |  ${\mathcal{B}}$ PDG22
---|---
$B^{+}\to\bar{D}^{0}\omega\pi^{+}$ |  $(4.1\pm 0.9)\times 10^{-3}$
$B^{+}\to\bar{D}^{*0}\omega\pi^{+}$ |  $(4.5\pm 1.2)\times 10^{-3}$
$B^{0}\to{D}^{-}\omega\pi^{+}$ |  $(2.8\pm 0.6)\times 10^{-3}$
$B^{0}\to{D}^{*-}\omega\pi^{+}$ |  $(2.8\pm 0.6)\times 10^{-3}$

The four decay channels $B^{+}\to\bar{D}^{(*)0}\omega\pi^{+}$ and
${B}^{0}\to{D}^{(*)-}\omega\pi^{+}$ have been observed by CLEO Collaboration
in Ref. prd64.092001 , the updated studies for the decay
${B}^{0}\to{D}^{*-}\omega\pi^{+}$ were presented later by BABAR and Belle
Collaborations in Refs. prd74.012001 ; prd92.012013 . In these measurements,
the $\omega\pi^{+}$ system in the final states showed a preference for the
$1^{-}$ resonances. The relevant data from Review of Particle Physics PDG22
are found in Table 5. In addition to the total branching fraction for
${B}^{0}\to{D}^{*-}\omega\pi^{+}$ decay, one finds the fitted branching
fractions

$\displaystyle\mathcal{B}$ $\displaystyle=$ $\displaystyle(1.48\pm
0.27^{+0.15+0.21}_{-0.09-0.56})\times 10^{-3}$ (30) $\displaystyle\mathcal{B}$
$\displaystyle=$
$\displaystyle(1.07^{+0.15+0.06+0.40}_{-0.31-0.13-0.02})\times 10^{-3}$ (31)

in Ref. prd92.012013  for the quasi-two-body decays
${B}^{0}\to{D}^{*-}\rho(770)^{+}\to{D}^{*-}\omega\pi^{+}$ and
${B}^{0}\to{D}^{*-}\rho(1450)^{+}\to{D}^{*-}\omega\pi^{+}$, respectively,
where the first error is statistical, the second is systematic and the third
is the model error. One can find that the predictions

$\displaystyle\mathcal{B}(B^{0}\to D^{*-}[\rho(770)^{+}\to]\omega\pi^{+})$
$\displaystyle=$
$\displaystyle(1.20^{+0.67+0.09+0.02+0.07}_{-0.40-0.08-0.01-0.07})\times
10^{-3},$ (32) $\displaystyle\mathcal{B}(B^{0}\to
D^{*-}[\rho(1450)^{+}\to]\omega\pi^{+})$ $\displaystyle=$
$\displaystyle(0.89^{+0.50+0.06+0.02+0.38}_{-0.31-0.06-0.02-0.38})\times
10^{-3}$ (33)

in Table 4 for the corresponding two quasi-two-body decays are in agreement
with these two branching fractions presented by Belle Collaboration in
prd92.012013 . In consideration of the fitted branching fractions for
${B}^{0}\to{D}^{*-}\rho(770)^{+}\to{D}^{*-}\omega\pi^{+}$ and
${B}^{0}\to{D}^{*-}\rho(1450)^{+}\to{D}^{*-}\omega\pi^{+}$ in prd92.012013
and the data in Table 5 for the three-body decay
$B^{0}\to{D}^{*-}\omega\pi^{+}$, one finds that the contributions from
subprocesses $\rho(770)^{+}\to\omega\pi^{+}$ and
$\rho(1450)^{+}\to\omega\pi^{+}$ are dominant for this three-body process.

By examining the fraction of the longitudinal polarization $\Gamma_{L}/\Gamma$
at a fixed value of the momentum transfer, the decays
$B^{0}\to{D}^{*-}\rho(770,1450)^{+}\to{D}^{*-}\omega\pi^{+}$ can be employed
to test the factorization hypothesis for $B$ meson decays plb89-105 ;
prd67.112002 . The measurement of the fraction of longitudinal polarization in
Ref. prd67.112002  for the decays $B^{0}\to{D}^{*-}\rho(770)^{+}$ and
$B^{+}\to\bar{D}^{*0}\rho(770)^{+}$ confirmed the validity of the
factorization assumption at relatively low region of the momentum transfer. In
Ref. plb507-142 , the authors proposed that if the $\omega\pi^{+}$ system in
the ${B}\to{D}^{*}\omega\pi^{+}$ decays is composed of two or more particles
not dominated by a single narrow resonance, factorization can be tested in
different kinematic regions. In Table 4, we list PQCD predictions for the
corresponding longitudinal polarization fractions $\Gamma_{L}/\Gamma$ for the
relevant decays. The errors, which are added in quadrature, for these
longitudinal polarization fractions are quite small from the uncertainties of
$\omega_{B_{(s)}}$, $C_{D^{(*)}}$, Gegenbauer moments for resonances, coupling
$g_{\rho\omega\pi}$ or the weight parameter $A_{1}$. The explanation is that
the increase or decrease for the relevant numerical results from the
uncertainties of these parameters will result in nearly identical change of
the weight for the numerator and denominator of the corresponding
$\Gamma_{L}/\Gamma$ predictions. In Ref. prd64.092001 , the longitudinal
polarization fraction for $B^{0}\to{D}^{*-}\omega\pi^{+}$ was measured to be
$\Gamma_{L}/\Gamma=0.63\pm 0.09$; for the same decay channel in mass region of
$1.1$-$1.9$ GeV for $\omega\pi^{+}$, the result $\Gamma_{L}/\Gamma=0.654\pm
0.042(\rm stat.)\pm 0.016(\rm syst.)$ was provided by BABAR in Ref.
prd74.012001 . These two measurements agree well with the corresponding
predictions in Table 4.

Figure 3: The invariant mass $\sqrt{s}$ dependent $\Gamma_{L}/\Gamma$ for
$B_{s}^{0}\to D_{s}^{*-}\rho(770)^{+}$ decay, with the subprocess
$\rho(770)^{+}\to\omega\pi^{+}$.

When employing $B^{0}\to{D}^{*-}\rho(770,1450)^{+}\to{D}^{*-}\omega\pi^{+}$ to
test of the factorization hypothesis, we should keep in mind that there are
contributions from the annihilation Feynman diagrams as shown in Fig. 2-(c)
for these two decay processes. By comparing the data
$\mathcal{B}=(3.2^{+1.5}_{-1.3})\times 10^{-5}$ for the pure annihilation
decay $B^{0}\to D_{s}^{*-}K^{*+}$ PDG22  with the results in Table 2 for
$B^{0}\to{D}^{*-}\rho(770)^{+}$, one can roughly take the annihilation diagram
contributions to be around a few percent at the decay amplitude level. In
order to avoid the pollution from annihilation Feynman diagrams, we propose to
take the decays $B_{s}^{0}\to D_{s}^{*-}\rho(770,1450)^{+}$ with
$\rho(770,1450)^{+}$ decay into $\pi^{+}\pi^{0}$ or $\omega\pi^{+}$ to test of
the factorization hypothesis, in view of these decay channels have only
emission diagrams the Fig. 2-(b) at quark level. We plot the invariant mass
$\sqrt{s}$ dependent $\Gamma_{L}/\Gamma$ in Fig. 3 for the decay $B_{s}^{0}\to
D_{s}^{*-}\rho(770)^{+}$ with the subprocess $\rho(770)^{+}\to\omega\pi^{+}$.
One finds that the $\Gamma_{L}/\Gamma$ for $B_{s}^{0}\to
D_{s}^{*-}\rho(770)^{+}$ is going down as the increase of the invariant mass
$\sqrt{s}$ for $\omega\pi^{+}$ system. Since the subprocesses
$\rho(770)^{+}\to\pi^{+}\pi^{0}$ and $\rho(770)^{+}\to\omega\pi^{+}$ are
described by the electromagnetic form factors $F_{\pi}$ and $F_{\omega\pi}$,
respectively, in the quasi-two-body decay amplitudes, they are independence
from the weak interaction in the related decay processes and wouldn’t disturb
the measurement results of $\Gamma_{L}/\Gamma$ for relevant channels.

Figure 4: The differential branching fraction for the quasi-two-body decay
$B^{0}\to{D}^{*-}\rho^{+}\to{D}^{*-}\omega\pi^{+}$, with
$\rho^{+}\in\\{\rho(770)^{+},\rho(1450)^{+},\rho(770)^{+}\&\rho(1450)^{+}\\}$.

In Fig. 4, we show the differential branching fraction for the quasi-two-body
decays $B^{0}\to{D}^{*-}\rho^{+}\to{D}^{*-}\omega\pi^{+}$, with
$\rho^{+}\in\\{\rho(770)^{+},\rho(1450)^{+},\rho(770)^{+}\&\rho(1450)^{+}\\}$.
The phase difference $\phi_{1}$ between $\rho(770)$ and $\rho(1450)$ for Eq.
(17) will generate different shapes for the curves of differential branching
fractions and branching fractions of the processes with
$\rho(770\&1450)^{+}\to\omega\pi^{+}$. In Refs. plb486-29 ; plb562-173 ;
JETPL94-734 ; prd61.072003 , a phase difference of $\phi_{1}=\pi$ were adopted
or fitted for $F_{\omega\pi}(s)$ between $\rho(770)$ and $\rho(1450)$; the
measurements in prd88.054013 ; prd94.112001 ; prd92.012013  showing the
results very close to $\pi$ for this phase $\phi_{1}$. In this context, we
adopt $\phi_{1}=\pi$ in this work. With this choice of the phase we find the
shapes of these curves in Fig. 4 are consistent with the Fig. 12 in Ref.
prd92.012013  for the same quasi-two-body decay channels measured by Belle
Collaboration. From Fig. 4, one finds that the curve for
$B^{0}\to{D}^{*-}[\rho(770)^{+}\&\rho(1450)^{+}\to]\omega\pi^{+}$ is seriously
affected by the interference between $\rho(770)^{+}$ and $\rho(1450)^{+}$.
Since the phase difference of $\phi_{1}=\pi$, there is essentially a minus
sign between $\rho(770)$ and $\rho(1450)$ components of Eq. (17) the form
factor $F_{\omega\pi}(s)$. Take into consideration of the denominator
$D_{\rho_{i}}$ for BW formula, in the invariant mass region of $\omega\pi^{+}$
system well below the mass of $\rho(1450)$, the form factor $F_{\omega\pi}(s)$
will be instructive for the branching fraction of
$B^{0}\to{D}^{*-}[\rho(770)^{+}\&\rho(1450)^{+}\to]\omega\pi^{+}$, but when
invariant mass is much larger than the mass for $\rho(1450)$, the real parts
of denominator $D_{\rho_{i}}$ for $\rho(770)^{+}$ and $\rho(1450)^{+}$ will
have the same sign, the $F_{\omega\pi}(s)$ will be destructive even if we take
the influence of the full width of $\rho(1450)$ into account. The branching
fractions with the subprocess $\rho(770\&1450)^{+}\to\omega\pi^{+}$ could also
verify our analysis above. For example, we have the prediction

$\displaystyle\mathcal{B}=(1.03^{+0.58+0.07+0.01+0.15}_{-0.35-0.08-0.01-0.15})\times
10^{-3}$ (34)

for $B^{0}\to D^{*-}[\rho(770\&1450)^{+}\to]\omega\pi^{+}$ decay, it is much
smaller than the sum of two branching fractions from the subprocesses
$\rho(770)^{+}\to\omega\pi^{+}$ and $\rho(1450)^{+}\to\omega\pi^{+}$ given in
Table 4.

Because the threshold for $\omega\pi^{+}$ is larger than the mass of
$\rho(770)^{+}$, we don’t see a typical BW shape for the curve with the
subprocess $\rho(770)^{+}\to\omega\pi^{+}$ in Fig. 4, the bump of the curve is
attributed to kinematic characteristics in the corresponding decay process
rather than the properties of the resonant state $\rho(770)^{+}$ involved. The
resonance $\rho(770)^{+}$ as a virtual bound state Dalitz62 ; plb25-294  in
the process $\rho(770)^{+}\to\omega\pi^{+}$ can not completely present its
properties in the concerned processes because of the phase space of the
relevant decay processes. But the quantum number of the involved resonance
could be fixed from its decay daughters the $\omega\pi^{+}$ system. The exact
resonant source for $\omega\pi^{+}$ makes the cascade decay like
$B^{0}\to{D}^{*-}\rho(770)^{+}\to{D}^{*-}\omega\pi^{+}$ to be a quasi-two-body
process, although the invariant mass region for the $\omega\pi^{+}$ system is
excluded from the region around pole mass of $\rho(770)$. The resonance
$\rho(1450)^{+}$ with the mass larger than the threshold of $\omega\pi^{+}$
contribute a normal BW shape for the curve of the differential branching
fraction for the decay
$B^{0}\to{D}^{*-}\rho(1450)^{+}\to{D}^{*-}\omega\pi^{+}$ in Fig. 4. But in the
decay $D^{+}_{s}\to\omega\pi^{+}\eta$ which has been measured by BESIII
prd107.052010  recently, since the initial decaying state does not have enough
energy to make $\rho(1450)$ demonstrate its intact properties, it will provide
only the virtual contribution for $\omega\pi^{+}$ system in this three-body
$D^{+}_{s}$ decay process, we shall leave the detailed discussion of it to
future study.

##  IV Summary

In this work we studied the subprocesses $\rho(770,1450)^{+}\to\omega\pi^{+}$
contributions in the cascade decays
$B^{+}\to\bar{D}^{(*)0}\rho^{+}\to\bar{D}^{(*)0}\omega\pi^{+}$, $B^{0}\to
D^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$ and $B_{s}^{0}\to
D_{s}^{(*)-}\rho^{+}\to D^{(*)-}\omega\pi^{+}$ within the PQCD approach. These
decays are important for the investigation of the properties for $\rho$
excitations and are very valuable for the test of the factorization hypothesis
for $B$ meson decays. The decays $B^{+}\to\bar{D}^{(*)0}\omega\pi^{+}$ and
$B^{0}\to D^{(*)-}\omega\pi^{+}$ have been measured by different
collaborations but without any predictions for their observables on
theoretical side.

With one open charm meson in the final state of each decay channel, the decay
amplitudes of these processes were described well by effective Hamiltonian
$\mathcal{H}_{\rm eff}$ with the tree-level $W$ exchange operators $O_{1}$ and
$O_{2}$ in the quasi-two-body framework. The subprocesses
$\rho(770,1450)^{+}\to\omega\pi^{+}$, which are related to the processes
$e^{+}e^{-}\to\omega\pi^{0}$ and $\tau\to\omega\pi\nu_{\tau}$ and can not be
calculated in PQCD, were introduced into the distribution amplitudes for
$\omega\pi$ system in this work via the vector form factor $F_{\omega\pi}(s)$
which has measured by different collaborations recently.

With the parameters $g_{\rho\omega\pi}=16.0\pm 2.0$ GeV-1 and $A_{1}=0.171\pm
0.036$ for form factor $F_{\omega\pi}(s)$, we predicted the branching
fractions for the first time on theoretical side for $12$ quasi-two-body
decays with $\rho(770,1450)^{+}\to\omega\pi^{+}$, as well as the corresponding
longitudinal polarization fractions $\Gamma_{L}/\Gamma$ for the cases with the
vector $\bar{D}^{(*)0}$ or $D_{(s)}^{(*)-}$ in their final states. The
branching fractions of these quasi-two-body decays are at the order of
$10^{-3}$, which may be detected at the LHCb and Belle-II experiments. Our
results
$\mathcal{B}=(1.20^{+0.67+0.09+0.02+0.07}_{-0.40-0.08-0.01-0.07})\times
10^{-3}$ and
$\mathcal{B}=(0.89^{+0.50+0.06+0.02+0.38}_{-0.31-0.06-0.02-0.38})\times
10^{-3}$ for the decays
${B}^{0}\to{D}^{*-}\rho(770)^{+}\to{D}^{*-}\omega\pi^{+}$ and
${B}^{0}\to{D}^{*-}\rho(1450)^{+}\to{D}^{*-}\omega\pi^{+}$ agree with the
measurements $\mathcal{B}=(1.48\pm 0.27^{+0.15+0.21}_{-0.09-0.56})\times
10^{-3}$ and $\mathcal{B}=(1.07^{+0.15+0.06+0.40}_{-0.31-0.13-0.02})\times
10^{-3}$, respectively, from Belle Collaboration.

The decay $B^{0}\to{D}^{*-}\omega\pi^{+}$ has been employed in literature to
test the factorization hypothesis for $B$ meson decays by examining the
fraction of the longitudinal polarization $\Gamma_{L}/\Gamma$ at a fixed value
of the momentum transfer. But we should care about contributions from the
annihilation Feynman diagrams for this decay process. In order to avoid the
pollution from annihilation Feynman diagrams, we propose to take the decays
$B_{s}^{0}\to D_{s}^{*-}\rho(770,1450)^{+}$ with $\rho(770,1450)^{+}$ decay
into $\pi^{+}\pi^{0}$ or $\omega\pi^{+}$ to test the factorization hypothesis
for $B$ decays. These decay channels have only emission diagrams with
$B_{s}\to D_{s}^{*-}$ transition at quark level, and the subprocesses which
can be described with the corresponding electromagnetic form factors would not
disturb the measurement results for $\Gamma_{L}/\Gamma$.

The resonance $\rho(770)^{+}$ in the concerned quasi-two-decays of this work
decaying to $\omega\pi^{+}$ system in the final states can not completely
present its properties and contribute only the virtual contribution for the
total branching fraction for corresponding three-body decay channels, because
of the threshold for $\omega\pi^{+}$ and phase space limitation. But the
quantum number of the involved resonance could be fixed from its decay
daughters the $\omega\pi^{+}$ system. We want to stress here that the virtual
contributions from specific known intermediate states are different from the
nonresonant contributions demarcated in the experimental studies.

###### Acknowledgements.

W. F. Wang thanks Prof. Àngels Ramos for helpful discussions and for carefully
reading the manuscript. This work was supported in part by the National
Natural Science Foundation of China under Grants No. 12205148 and No.
11947011, the Fund for Shanxi “1331 Project” Key Subjects Construction, the
Natural Science Foundation of Jiangsu Province under Grant No. BK20191010, and
the Qing Lan Project of Jiangsu Province.

##  Appendix A Decay amplitudes for
$B\to\bar{D}^{(*)}\rho\to\bar{D}^{(*)}\omega\pi$ decays

The $B$ meson light-cone matrix element in the decay amplitudes of
$B\to\bar{D}^{(*)}\rho\to\bar{D}^{(*)}\omega\pi$ decays can be decomposed as
prd55.272 ; npb592-3 ; plb523-111

$\displaystyle\Phi_{B}=\frac{i}{\sqrt{2N_{c}}}(p{/}_{B}+m_{B})\gamma_{5}\phi_{B}(k_{B}),$
(35)

where the distribution amplitude $\phi_{B}$ is of the form

$\displaystyle\phi_{B}(x_{B},b_{B})$ $\displaystyle=$ $\displaystyle
N_{B}x_{B}^{2}(1-x_{B})^{2}\mathrm{exp}\left[-\frac{(x_{B}m_{B})^{2}}{2\omega_{B}^{2}}-\frac{1}{2}(\omega_{B}b_{B})^{2}\right]\\!,$
(36)

with two shape parameters $\omega_{B}=0.40\pm 0.04$ GeV for $B^{\pm,0}$ and
$\omega_{B_{s}}=0.50\pm 0.05$ for $B^{0}_{s}$, respectively, the $N_{B}$ is a
normalization factor.

The wave functions for $\bar{D}^{*0}$ and $D^{*-}_{(s)}$ have been discussed
in detail in Ref. prd78.014018 . Up to twist-3 accuracy, their two-particle
light-cone distribution amplitudes are defined as

$\displaystyle\langle
D_{(s)}(p)|q_{\alpha}(z)\bar{c}_{\beta}(0)|0\rangle=\frac{i}{\sqrt{2N_{C}}}\int_{0}^{1}dxe^{ixp\cdot
z}\left[\gamma_{5}\left(p/+m\right)\phi_{D_{(s)}}(x,b)\right]_{\alpha\beta}\\!,\quad$
(37) $\displaystyle\langle
D_{(s)}^{*}(p)|q_{\alpha}(z)\bar{c}_{\beta}(0)|0\rangle=-\frac{1}{\sqrt{2N_{C}}}\int_{0}^{1}dxe^{ixp\cdot
z}\big{[}\epsilon/_{L}(p/+m)\phi^{L}_{D^{*}_{(s)}}(x,b)$
$\displaystyle\phantom{\langle
D_{(s)}^{*}(p)|q_{\alpha}(z)\bar{c}_{\beta}(0)|0\rangle}+\epsilon/_{T}(p/+m)\phi^{T}_{D^{*}_{(s)}}(x,b)\big{]}_{\alpha\beta},$
(38)

with the normalization conditions

$\displaystyle\int_{0}^{1}dx\phi_{D}(x)$ $\displaystyle=$
$\displaystyle\frac{f_{D}}{2\sqrt{2N_{c}}},$ (39)
$\displaystyle\int_{0}^{1}dx\phi_{D^{*}}^{L}(x)$ $\displaystyle=$
$\displaystyle\frac{f_{D^{*}}}{2\sqrt{2N_{c}}},$ (40)
$\displaystyle\int_{0}^{1}dx\phi_{D^{*}}^{T}(x)$ $\displaystyle=$
$\displaystyle\frac{f_{D^{*}}^{T}}{2\sqrt{2N_{c}}}.$ (41)

The distribution amplitude for the $\bar{D}^{0}$ and $D^{-}_{(s)}$ mesons is
prd78.014018 ; prd90.094018

$\displaystyle\phi_{D_{(s)}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2N_{C}}}f_{D_{(s)}}6x(1-x)\left[1+C_{D_{(s)}}(1-2x)\right]\exp\left[-\frac{\omega_{D_{(s)}}^{2}b^{2}}{2}\right]\\!,$
(42)

In this work, we also adopt the same distribution amplitude for both the open
charm vector mesons and open charm pseudoscalar mesons because of their small
mass difference prd78.014018 , but with different Gegenbauer moments
$C_{D}=0.6\pm 0.15$ and $C_{D^{*}}=0.5\pm 0.10$ for $D_{(s)}$ and
$D^{*}_{(s)}$, respectively, considering the different decay constants for
them as they in this work and in Ref. prd78.014018 .

With the effective weak Hamiltonian $\mathcal{H}_{\rm eff}$ in Eq. (3), the
total decay amplitudes for the concerned quasi-two-body decays are then
written as

$\displaystyle\mathcal{A}(B^{+}\to\bar{D}^{0}[\rho^{+}\to]\omega\pi^{+})=\frac{G_{F}}{\sqrt{2}}V^{*}_{cb}V_{ud}\big{[}a_{1}F_{T\rho}^{LL}+C_{2}M_{T\rho}^{LL}+a_{2}F_{TD}^{LL}+C_{1}M_{TD}^{LL}\big{]},\quad$
(43)
$\displaystyle\mathcal{A}(B^{0}\to{D^{-}}[\rho^{+}\to]\omega\pi^{+})=\frac{G_{F}}{\sqrt{2}}V^{*}_{cb}V_{ud}\big{[}a_{2}F_{TD}^{LL}+C_{1}M_{TD}^{LL}+a_{1}F_{a\rho}^{LL}+C_{2}M_{a\rho}^{LL}\big{]},\quad$
(44) $\displaystyle\mathcal{A}(B_{s}^{0}\to
D_{s}^{-}[\rho^{+}\to]\omega\pi^{+})=\frac{G_{F}}{\sqrt{2}}V^{*}_{cb}V_{ud}\big{[}a_{2}F_{TD}^{LL}+C_{1}M_{TD}^{LL}\big{]},$
(45)

by combining various of contributions from the related Feynman diagrams in
Fig. 2. Where $\rho^{+}$ stands for the $\rho(770)^{+}$ or $\rho(1450)^{+}$ in
the relevant decays. The other three decay amplitudes for the corresponding
$B^{+}$, $B^{0}$ and $B_{s}^{0}$ decays with $\bar{D}^{*0}$, $D^{*-}$ and
$D_{s}^{*-}$, respectively, can be obtained from Eq. (43)-(45) with the
replacements of $D_{(s)}$ meson wave function by the $D^{*}_{(s)}$ wave
function. As has been done in two-body decays of $B$ to two vector meson as
the final state, the decay amplitudes for
$B\to\bar{D}^{*}\rho^{+}\to\bar{D}^{*}\omega\pi^{+}$ in this work can be
decomposed as
$\mathcal{A}^{(\lambda)}=M^{(\lambda)}\cdot{\langle\omega\pi|\rho_{i}\rangle}/{D_{\rho_{i}}(s)}$
with prd76.074018

$\displaystyle M^{(\lambda)}$ $\displaystyle=$
$\displaystyle\epsilon_{\bar{D}^{*}\mu}^{*}(\lambda)\epsilon_{\rho\nu}^{*}(\lambda)\left[a\,g^{\mu\nu}+{b\over
m_{D}\sqrt{s}}P_{B}^{\mu}P_{B}^{\nu}+i{c\over
m_{D}\sqrt{s}}\epsilon^{\mu\nu\alpha\beta}P_{\alpha}P_{3\beta}\right]\\!,$
(46) $\displaystyle\equiv$ $\displaystyle
M_{L}+M_{N}\epsilon^{*}_{\bar{D}^{*}}(\lambda=T)\cdot\epsilon^{*}_{\rho}(\lambda=T)+i\frac{M_{T}}{m_{B}^{2}}\epsilon^{\alpha\beta\gamma\rho}\epsilon^{*}_{\rho\alpha}(\lambda)\epsilon^{*}_{D^{*}\beta}(\lambda)P_{\gamma}P_{3\rho}.$

According to the polarized decay amplitudes, one has
$|A|^{2}=|A_{L}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2}$, and
$\Gamma_{L}/\Gamma=|A_{L}|^{2}/(|A_{L}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2})$,
the amplitudes $A_{L},A_{\parallel}$ and $A_{\perp}$ are related to the
$M_{L},M_{N}$ and $M_{T}$, respectively. For the detailed discussion, one is
referred to Refs. prd76.074018 ; prd45.193 ; prd46.2969 ; zpc55-497 ;
prd61.074031 .

With the subprocesses $\rho^{+}\to\omega\pi^{+}$, where $\rho$ is $\rho(770)$
or $\rho(1450)$, the specific expressions in PQCD approach for the Lorentz
invariant decay amplitudes of these general amplitudes $F$’s and $M$’s for
$B\to\bar{D}^{(*)}\rho\to\bar{D}^{(*)}\omega\pi$ decays are given as follows:

The amplitudes from Fig. 2-(a) for the decays with a pseudoscalar
$\bar{D}^{0}$ or $D^{-}_{(s)}$ meson in the final state are written as

$\displaystyle F_{T\rho}^{LL}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{D}\int dx_{B}dx\int
b_{B}db_{B}bdb\phi_{B}\big{\\{}\big{[}[r^{2}-\bar{\zeta}(x(r^{2}-1)^{2}+1)]\phi^{0}-\sqrt{\zeta}[(r^{2}+\bar{\zeta}$
(47)
$\displaystyle+2\bar{\zeta}x(r^{2}-1))\phi^{s}-(r^{2}-1)\bar{\zeta}(2x(r^{2}-1)+1)-r^{2})\phi^{t}\big{]}E_{e}(t_{a})h_{a}(x_{B},x,b,b_{B})S_{t}(x)$
$\displaystyle+\big{[}(r^{2}-1)[\zeta\bar{\zeta}-r^{2}(\zeta-
x_{B})]\phi^{0}-2\sqrt{\zeta}[\bar{\zeta}-r^{2}(x_{B}-2\zeta+1)]\phi^{s}\big{]}$
$\displaystyle\times
E_{e}(t_{b})h_{b}(x_{B},x,b_{B},b)S_{t}(|x_{B}-\zeta|)\big{\\}},$
$\displaystyle M_{T\rho}^{LL}$ $\displaystyle=$ $\displaystyle
16\sqrt{\frac{2}{3}}\pi C_{F}m_{B}^{4}\int dx_{B}dxdx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}\phi_{D}\big{\\{}\big{[}-[(\bar{\zeta}+r^{2})((r^{2}-1)(x_{3}\bar{\zeta}+x_{B})$
(48) $\displaystyle+r^{2}(\zeta
x-1)-\zeta(x+1)+1)+rr_{c}(r^{2}-\bar{\zeta})]\phi^{0}-\sqrt{\zeta}[(r^{2}(\bar{\zeta}(x_{3}+x-2)+x_{B})-x\bar{\zeta}$
$\displaystyle+4rr_{c})\phi^{s}+(r^{2}-1)(r^{2}(\bar{\zeta}(x-x_{3})-x_{B})-x\bar{\zeta})\phi^{t}]\big{]}E_{n}(t_{c})h_{c}(x_{B},x,x_{3},b_{B},b_{3})$
$\displaystyle+\big{[}x(r^{2}-1)[(r^{2}-\bar{\zeta})\phi^{0}+\sqrt{\zeta}\bar{\zeta}(\phi^{s}-(r^{2}-1)\phi^{t})]-(x_{3}\bar{\zeta}-x_{B})[(r^{2}-\bar{\zeta})\phi^{0}$
$\displaystyle+\sqrt{\zeta}r^{2}((r^{2}-1)\phi^{t}+\phi^{s})]\big{]}E_{n}(t_{d})h_{d}(x_{B},x,x_{3},b_{B},b_{3})\big{\\}},$

with the symbol $\bar{\zeta}=1-\zeta$, the mass ratios $r=m_{D^{(*)}}/m_{B}$
and $r_{c}=m_{c}/m_{B}$. The amplitudes from Fig. 2-(b) are written as

$\displaystyle F_{TD}^{LL}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{\rho}\int dx_{B}dx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}\phi_{D}\big{\\{}\big{[}(r+1)[r^{2}-\bar{\zeta}-x_{3}\bar{\zeta}(r-1)(2r-\bar{\zeta})]\big{]}$
(49) $\displaystyle\times
E_{e}(t_{m})h_{m}(x_{B},x_{3},b_{3},b_{B})S_{t}(x_{3})+\big{[}(r^{2}-\bar{\zeta})[2r(r_{c}+1)-r^{2}\bar{\zeta}-r_{c}]-\zeta
x_{B}(2r-\bar{\zeta})\big{]}$ $\displaystyle\times
E_{e}(t_{n})h_{n}(x_{B},x_{3},b_{B},b_{3})S_{t}(x_{B})\big{\\}},$
$\displaystyle M_{TD}^{LL}$ $\displaystyle=$ $\displaystyle
16\sqrt{\frac{2}{3}}\pi C_{F}m_{B}^{4}\int dx_{B}dxdx_{3}\int
b_{B}db_{B}bdb\phi_{B}\phi_{D}\phi^{0}\big{\\{}\big{[}x_{B}[\bar{\zeta}^{2}-\bar{\zeta}r^{2}+\zeta
r]+\bar{\zeta}x_{3}r(\zeta r$ (50)
$\displaystyle+(r+1)(r-1)^{2})-\zeta(r-1)^{2}(r+1)[(r+2)x-2(r+1)]+\zeta^{2}[x-r^{2}(x-2)-1]$
$\displaystyle+(x-1)(r^{2}-1)^{2}\big{]}E_{n}(t_{o})h_{o}(x_{B},x,x_{3},b_{B},b)+\big{[}(r-1)(\bar{\zeta}+r)[x_{B}+(r^{2}-1)x]$
$\displaystyle+\bar{\zeta}x_{3}[(r-1)^{2}(r+1)-\zeta]\big{]}E_{n}(t_{p})h_{p}(x_{B},x,x_{3},b_{B},b)\big{\\}}.$

The amplitudes from Fig. 2-(c) the annihilation diagrams are written as

$\displaystyle F_{A\rho}^{LL}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{B}\int dx_{3}dx\int
bdbb_{3}db_{3}\phi_{D}\big{\\{}\big{[}((2rr_{c}-1)(r^{2}-\bar{\zeta})-(r^{2}-1)^{2}x\bar{\zeta})\phi^{0}+\sqrt{\zeta}$
(51)
$\displaystyle\times[(r^{2}-1)(r_{c}(r^{2}-\bar{\zeta})-2r(r^{2}-1)x)\phi^{t}+(r_{c}(r^{2}-\zeta+1)+2r(x-xr^{2}-2))\phi^{s}]\big{]}$
$\displaystyle\times
E_{a}(t_{e})h_{e}(x,x_{3},b,b_{3})S_{t}(x)+\big{[}(r^{2}-1)[x_{3}\bar{\zeta}^{2}-\zeta(r^{2}-\bar{\zeta})]\phi^{0}+2\sqrt{\zeta}r(x_{3}\bar{\zeta}+\zeta$
$\displaystyle-r^{2}+1)\phi^{s}\big{]}E_{a}(t_{f})h_{f}(x,x_{3},b_{3},b)S_{t}(|\bar{\zeta}x_{3}+\zeta|)\big{\\}},$
$\displaystyle M_{A\rho}^{LL}$ $\displaystyle=$
$\displaystyle-16\sqrt{\frac{2}{3}}\pi C_{F}m_{B}^{4}\int dx_{B}dxdx_{3}\int
b_{B}db_{B}bdb\phi_{B}\phi_{D}\big{\\{}\big{[}(r^{2}-1)[r^{2}(x_{B}+x_{3}-1)+x_{B}$
(52)
$\displaystyle+x_{3}]\phi^{0}+\zeta[r^{4}x-(r^{2}-1)x_{B}+\zeta((r^{2}-1)x_{3}-xr^{2}+x+1)-(r^{4}+r^{2}-2)x_{3}$
$\displaystyle-x-1]\phi^{0}+\zeta^{3/2}r(1-x_{3})[(r^{2}-1)\phi^{t}+\phi^{s}]+\sqrt{\zeta}r[\phi^{s}(x_{B}+r^{2}(x-1)+x_{3}-x+3)$
$\displaystyle+(r^{2}-1)(x_{B}-xr^{2}+r^{2}+x_{3}+x-1)\phi^{t}]\big{]}E_{n}(t_{g})h_{g}(x_{B},x,x_{3},b,b_{B})+\big{[}(r^{2}-\bar{\zeta})$
$\displaystyle\times[r^{2}(x_{B}-x_{3}-x+1)+\zeta(r^{2}(x_{3}+x-2)-x+1)+x-1]\phi^{0}+\sqrt{\zeta}r[(x_{B}-x_{3}\bar{\zeta}$
$\displaystyle-\zeta+(r^{2}-1)(1-x))\phi^{s}+(1-r^{2})(x_{B}-x_{3}\bar{\zeta}-\zeta+(r^{2}-1)(x-1))\phi^{t}]\big{]}$
$\displaystyle\times E_{n}(t_{h})h_{h}(x_{B},x,x_{3},b,b_{B})\big{\\}}.$

Where the $T\rho$, $TD$ and $A\rho$ in the subscript of above expressions
stand for $B\to\rho$, $B\to D$ transitions and the annihilation Feynman
diagrams, respectively. The $F$’s stand for those factorizable diagrams and
$M$’s for the nonfactorizable diagrams in Fig. 2.

The longitudinal polarization amplitudes from Fig. 2-(a) for the decays with a
vector $\bar{D}^{*0}$ or $D^{*-}_{(s)}$ meson in the final state are written
as

$\displaystyle F_{T\rho,L}^{LL}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{D^{*}}\int dx_{B}dx\int
b_{B}db_{B}bdb\phi_{B}\big{\\{}\big{[}[\bar{\zeta}+\bar{\zeta}x(r^{2}-1)^{2}+(2\zeta-1)r^{2}]\phi^{0}$
(53)
$\displaystyle+\sqrt{\zeta}[(1-r^{2})(2\bar{\zeta}x(r^{2}-1)+\bar{\zeta}+r^{2})\phi^{t}+(2\bar{\zeta}x(r^{2}-1)+\bar{\zeta}-r^{2})\phi^{s}]\big{]}$
$\displaystyle\times
E_{e}(t_{a})h_{a}(x_{B},x,b,b_{B})S_{t}(x)+\big{[}(r^{2}-1)[r^{2}x_{B}+\zeta^{2}-\zeta(r^{2}+1)]\phi^{0}$
$\displaystyle-2\sqrt{\zeta}[r^{2}(1-x_{B})-\bar{\zeta}]\phi^{s}\big{]}E_{e}(t_{b})h_{b}(x_{B},x,b_{B},b)S_{t}(|x_{B}-\zeta|)\big{\\}},$
$\displaystyle M_{T\rho,L}^{LL}$ $\displaystyle=$ $\displaystyle
16\sqrt{\frac{2}{3}}\pi C_{F}m_{B}^{4}\int dx_{B}dxdx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}\phi_{D^{*}}\big{\\{}\big{[}[rr_{c}(1-\bar{\zeta}r^{2}-\zeta^{2})-(r^{2}-\bar{\zeta})$
(54) $\displaystyle\times(\bar{\zeta}x_{3}(r^{2}-1)+x_{B}(r^{2}-1)+(\zeta
x-1)r^{2}-\zeta(x+1)+1)]\phi^{0}-\sqrt{\zeta}[(r^{2}(x_{3}\bar{\zeta}-\bar{\zeta}x$
$\displaystyle+x_{B})-\zeta
x+x)\phi^{s}+(r^{2}-1)(\bar{\zeta}x(1-r^{2})-r^{2}((x_{3}-2)\bar{\zeta}+x_{B}))\phi^{t}]\big{]}$
$\displaystyle\times
E_{n}(t_{c})h_{c}(x_{B},x,x_{3},b_{B},b_{3})+\big{[}x_{B}[(\bar{\zeta}+(2\zeta-1)r^{2})\phi^{0}+\sqrt{\zeta}r^{2}((r^{2}-1)\phi^{t}+\phi^{s})]$
$\displaystyle-\bar{\zeta}x_{3}[(\bar{\zeta}+(2\zeta-1)r^{2})\phi^{0}+\sqrt{\zeta}r^{2}((r^{2}-1)\phi^{t}+\phi^{s})]+x(r^{2}-1)[(\bar{\zeta}+(2\zeta-1)r^{2})\phi^{0}$
$\displaystyle-\sqrt{\zeta}\bar{\zeta}(\phi^{s}-(r^{2}-1)\phi^{t})]\big{]}E_{n}(t_{d})h_{d}(x_{B},x,x_{3},b_{B},b_{3})\big{\\}}.$

The longitudinal polarization amplitudes from Fig. 2-(b) are

$\displaystyle F_{TD^{*},L}^{LL}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{\rho}\int dx_{B}dx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}\phi_{D^{*}}\big{\\{}\big{[}\bar{\zeta}+(2r-1)(r^{2}-1)x_{3}\bar{\zeta}^{2}+r$
(55)
$\displaystyle\times[\zeta(r^{2}+2r-\zeta)-r^{2}-r+1]\big{]}E_{e}(t_{m})h_{m}(x_{B},x_{3},b_{3},b_{B})S_{t}(x_{3})+\big{[}r^{2}[r_{c}(2\zeta-1)$
$\displaystyle-\zeta^{2}+1]-\bar{\zeta}(\zeta
x_{B}-r_{c}+r^{4})\big{]}E_{e}(t_{n})h_{n}(x_{B},x_{3},b_{B},b_{3})S_{t}(x_{B})\big{\\}},$
$\displaystyle M_{TD^{*},L}^{LL}$ $\displaystyle=$
$\displaystyle-16\sqrt{\frac{2}{3}}\pi C_{F}m_{B}^{4}\int dx_{B}dxdx_{3}\int
b_{B}db_{B}bdb\phi_{B}\phi_{D^{*}}\phi^{0}\big{\\{}\big{[}\bar{\zeta}x_{B}(1-r)(\bar{\zeta}+r)-\bar{\zeta}x_{3}r$
$\displaystyle\times(r^{3}-\bar{\zeta}(r^{2}+r-1))-\zeta^{2}(2r^{3}-x(r+1)(r-1)^{2}-2r+1)+(x-1)(r^{2}-1)^{2}$
$\displaystyle-\zeta(r+1)(r-1)^{2}(rx+2x-2)\big{]}E_{n}(t_{o})h_{o}(x_{B},x,x_{3},b_{B},b)+\big{[}\bar{\zeta}x_{3}[r^{2}(r\bar{\zeta}+2\zeta-1)$
$\displaystyle+\bar{\zeta}-r\bar{\zeta}]-(x_{B}+(r^{2}-1)x)[\bar{\zeta}-\bar{\zeta}\zeta
r+(2\zeta-1)r^{2}]\big{]}E_{n}(t_{p})h_{p}(x_{B},x,x_{3},b_{B},b)\big{\\}}.$

The longitudinal polarization amplitudes from Fig. 2-(c) are

$\displaystyle F_{A\rho,L}^{LL}$ $\displaystyle=$ $\displaystyle-8\pi
C_{F}m^{4}_{B}f_{B}\int dx_{3}dx\int
bdbb_{3}db_{3}\phi_{D^{*}}\big{\\{}\big{[}\sqrt{\zeta}r_{c}[(r^{4}-\zeta
r^{2}-\bar{\zeta})\phi^{t}+(r^{2}-\bar{\zeta})\phi^{s}]$ (57)
$\displaystyle+[\bar{\zeta}(1-x(r^{2}-1)^{2})+r^{2}(2\zeta-1)]\phi^{0}\big{]}E_{a}(t_{e})h_{e}(x,x_{3},b,b_{3})S_{t}(x)+\big{[}2r\sqrt{\zeta}\bar{\zeta}$
$\displaystyle\times((x_{3}-1)\bar{\zeta}+r^{2})\phi^{s}+(r^{2}-1)[\zeta(r^{2}+\bar{\zeta}(1-x_{3})-x_{3})+x_{3}]\phi^{0}\big{]}$
$\displaystyle\times
E_{a}(t_{f})h_{f}(x,x_{3},b_{3},b)S_{t}(|\bar{\zeta}x_{3}+\zeta|)\big{\\}},$
$\displaystyle M_{A\rho,L}^{LL}$ $\displaystyle=$ $\displaystyle
16\sqrt{\frac{2}{3}}\pi C_{F}m_{B}^{4}\int dx_{B}dxdx_{3}\int
b_{B}db_{B}bdb\phi_{B}\phi_{D^{*}}\big{\\{}\big{[}-(x_{3}\bar{\zeta}+x_{B})[(r^{2}-1)(r^{2}-\bar{\zeta})\phi^{0}$
(58)
$\displaystyle-\sqrt{\zeta}\bar{\zeta}r(r^{2}-1)\phi^{t}-\sqrt{\zeta}\bar{\zeta}r\phi^{s}]-\zeta\bar{\zeta}(x+1)\phi^{0}+\sqrt{\zeta}\bar{\zeta}r^{5}(x-1)\phi^{t}+\sqrt{\zeta}\bar{\zeta}r^{3}((\zeta-2x)\phi^{t}$
$\displaystyle-(x-1)\phi^{s})+\sqrt{\zeta}\bar{\zeta}r((x-\bar{\zeta})\phi^{s}+(x+\bar{\zeta})\phi^{t})-r^{4}(\zeta
x-1)\phi^{0}-r^{2}(\zeta x(\zeta-2)+1)\phi^{0}\big{]}$ $\displaystyle\times
E_{n}(t_{g})h_{g}(x_{B},x,x_{3},b,b_{B})-\big{[}(r^{2}-\bar{\zeta})(r^{2}(x_{B}-x_{3}\bar{\zeta})+r^{2}(\bar{\zeta}x-1)-\bar{\zeta}(x-1))\phi^{0}$
$\displaystyle-\sqrt{\zeta}\bar{\zeta}r[((x_{3}-1)\bar{\zeta}+(1-x)r^{2}+x-x_{B})\phi^{s}-(r^{2}-1)(x_{3}\bar{\zeta}+\zeta-(1-x)r^{2}-x$
$\displaystyle-
x_{B}+1)\phi^{t}]\big{]}E_{n}(t_{h})h_{h}(x_{B},x,x_{3},b,b_{B})\big{\\}}.$

The normal and transverse polarization amplitudes from Fig. 2 for the decays
with a vector $\bar{D}^{*0}$ or $D^{*-}_{(s)}$ are written as

$\displaystyle F_{T\rho,T}^{LL}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{D^{*}}r\int dx_{B}dx\int
b_{B}db_{B}bdb\phi_{B}\big{\\{}\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[\sqrt{\zeta}(x(r^{2}-1)(\phi^{a}-\phi^{v})+2\phi^{v})$
(59)
$\displaystyle+\zeta(2x(r^{2}-1)+1)\phi^{T}+(1-r^{2})\phi^{T}]-i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}[\sqrt{\zeta}((x(r^{2}-1)-2)\phi^{a}$
$\displaystyle-x(r^{2}-1)\phi^{v})+\zeta(2x(r^{2}-1)+1)\phi^{T}+(r^{2}-1)\phi^{T}]\big{]}E_{e}(t_{a})h_{a}(x_{B},x,b,b_{B})S_{t}(x)$
$\displaystyle+\sqrt{\zeta}\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[(\zeta-
x_{B}-r^{2}+1)\phi^{v}+(\bar{\zeta}+x_{B}-r^{2})\phi^{a}]+i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}[(\zeta-
x_{B}-r^{2}$ $\displaystyle+1)\phi^{a}-(\zeta-
x_{B}+r^{2}-1)\phi^{v}]\big{]}E_{e}(t_{b})h_{b}(x_{B},x,b_{B},b)S_{t}(|x_{B}-\zeta|)\big{\\}},$
$\displaystyle M_{T\rho,T}^{LL}$ $\displaystyle=$ $\displaystyle
16\sqrt{\frac{2}{3}}\pi C_{F}m_{B}^{4}\int dx_{B}dxdx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}\phi_{D^{*}}\big{\\{}\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[\zeta^{3/2}r_{c}(\phi^{a}-\phi^{v})$
(60)
$\displaystyle+\sqrt{\zeta}r_{c}((r^{2}-1)\phi^{a}+(r^{2}+1)\phi^{v})+r(r^{2}-1)(x_{B}+x_{3}-1)\phi^{T}-\zeta
r((r^{2}-1)$
$\displaystyle\times(x_{3}+x)-2r^{2}+1)\phi^{T}]-i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}[\zeta^{3/2}r_{c}(\phi^{a}-\phi^{v})-\sqrt{\zeta}r_{c}((r^{2}+1)\phi^{a}$
$\displaystyle+(r^{2}-1)\phi^{v})-r(r^{2}-1)(x_{B}+x_{3}-1)\phi^{T}+\zeta
r((x_{3}-x)(r^{2}-1)+1)\phi^{T}]\big{]}$ $\displaystyle\times
E_{n}(t_{c})h_{c}(x_{B},x,x_{3},b_{B},b_{3})+r\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[2\sqrt{\zeta}(x_{B}+x(r^{2}-1)-x_{3}\bar{\zeta})\phi^{v}$
$\displaystyle+(r^{2}-1)(x_{B}-x\zeta-
x_{3}\bar{\zeta})\phi^{T}]+i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}[2\sqrt{\zeta}(x_{B}+x(r^{2}-1)-x_{3}\bar{\zeta})\phi^{a}$
$\displaystyle+(r^{2}-1)(x_{B}+x\zeta-
x_{3}\bar{\zeta})\phi^{T}]\big{]}E_{n}(t_{d})h_{d}(x_{B},x,x_{3},b_{B},b_{3})\big{\\}},$
$\displaystyle F_{TD^{*},T}^{LL}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{\rho}\sqrt{\zeta}\int dx_{B}dx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}\phi_{D^{*}}\big{\\{}\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[x(r^{2}-1)(2\bar{\zeta}-r)$
(61)
$\displaystyle+\bar{\zeta}+r^{2}+2r]-i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}[x(r^{2}-1)(r-2\bar{\zeta})-\bar{\zeta}+r^{2}]\big{]}E_{e}(t_{m})h_{m}(x_{B},x_{3},b_{3},b_{B})S_{t}(x_{3})$
$\displaystyle+r\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[\zeta-
x_{B}+2r_{c}-r^{2}+1]-i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}[\bar{\zeta}+x_{B}-r^{2}]\big{]}$
$\displaystyle\times
E_{e}(t_{n})h_{n}(x_{B},x_{3},b_{B},b_{3})S_{t}(x_{B})\big{\\}},$
$\displaystyle M_{TD^{*},T}^{LL}$ $\displaystyle=$ $\displaystyle
16\sqrt{\frac{2}{3}}\pi C_{F}m_{B}^{4}\sqrt{\zeta}\int dx_{B}dxdx_{3}\int
b_{B}db_{B}bdb\phi_{B}\phi_{D^{*}}\big{\\{}\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[r^{2}(\bar{\zeta}((2-x)\phi^{v}$
(62)
$\displaystyle-x\phi^{a})+(\bar{\zeta}x_{3}+x_{B})(\phi^{a}-\phi^{v}))+\bar{\zeta}x(\phi^{a}+\phi^{v})]-i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}[r^{2}((\bar{\zeta}(x-x_{3})$
$\displaystyle-
x_{B})\phi^{v}+(\bar{\zeta}(x_{3}+x-2)+x_{B})\phi^{a})-\bar{\zeta}x(\phi^{a}+\phi^{v})]\big{]}E_{n}(t_{o})h_{o}(x_{B},x,x_{3},b_{B},b)$
$\displaystyle+\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[(r^{2}(x_{B}-x_{3}\bar{\zeta})-x\bar{\zeta}(r^{2}-1))\phi^{a}+(x(r^{2}-1)(2r-\bar{\zeta})-(r-2)$
$\displaystyle\times
r(x_{B}-x_{3}\bar{\zeta}))\phi^{v}]+i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}[(x(r^{2}-1)(2r-\bar{\zeta})-r(r-2)(x_{B}-x_{3}\bar{\zeta}))\phi^{a}$
$\displaystyle+(r^{2}(x_{B}-x_{3}\bar{\zeta})-\bar{\zeta}x(r^{2}-1))\phi^{v}]\big{]}E_{n}(t_{p})h_{p}(x_{B},x,x_{3},b_{B},b)\big{\\}},$
$\displaystyle F_{A\rho,T}^{LL,L}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{B}r\int dx_{3}dx\int
bdbb_{3}db_{3}\phi_{D^{*}}\big{\\{}\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[\sqrt{\zeta}(x(r^{2}-1)(\phi^{a}-\phi^{v})-2\phi^{v})$
(63) $\displaystyle-
r_{c}(r^{2}-\zeta-1)\phi^{T}]+i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}[\sqrt{\zeta}(x(r^{2}-1)\phi^{v}-(x(r^{2}-1)+2)\phi^{a})+(r^{2}-\bar{\zeta})$
$\displaystyle\times
r_{c}\phi^{T}]\big{]}E_{a}(t_{e})h_{e}(x,x_{3},b,b_{3})S_{t}(x)+\sqrt{\zeta}\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[(\bar{\zeta}x_{3}+\zeta-r^{2}+1)\phi^{v}+(\bar{\zeta}x_{3}+\zeta$
$\displaystyle+r^{2}-1)\phi^{a}]+i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}[(\bar{\zeta}x_{3}+\zeta+r^{2}-1)\phi^{v}+(\bar{\zeta}x_{3}+\zeta-r^{2}+1)\phi^{a}]\big{]}$
$\displaystyle\times
E_{a}(t_{f})h_{f}(x,x_{3},b_{3},b)S_{t}(|\bar{\zeta}x_{3}+\zeta|)\big{\\}},$
$\displaystyle M_{A\rho,T}^{LL}$ $\displaystyle=$ $\displaystyle
16\sqrt{\frac{2}{3}}\pi C_{F}m_{B}^{4}\int dx_{B}dxdx_{3}\int
b_{B}db_{B}bdb\phi_{B}\phi_{D^{*}}\big{\\{}\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[(r^{2}x_{B}(r^{2}-1)$
(64)
$\displaystyle+\bar{\zeta}r^{2}((r^{2}-1)(x_{3}-1)+\zeta)-\bar{\zeta}\zeta
x(r^{2}-1))\phi^{T}-2\sqrt{\zeta}r\phi^{v}]+i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}[(r^{2}(\bar{\zeta}r^{2}$
$\displaystyle-\bar{\zeta}^{2}-(\bar{\zeta}x_{3}+x_{B})(r^{2}-1))-\bar{\zeta}\zeta
x(r^{2}-1))\phi^{T}-2\sqrt{\zeta}r\phi^{a}]\big{]}E_{n}(t_{g})h_{g}(x_{B},x,x_{3},b,b_{B})$
$\displaystyle+(r^{2}-1)\big{[}\epsilon_{T}^{D^{*}}\cdot\epsilon_{T}^{\rho}[r^{2}(x_{B}-x_{3})+\zeta(r^{2}(x_{3}-1)+x-1)+\zeta^{2}(1-x)]-i\epsilon^{nv\epsilon_{T}^{D^{*}}\epsilon_{T}^{\rho}}$
$\displaystyle\times[r^{2}(x_{B}-x_{3})+\zeta(r^{2}(x_{3}-1)-\bar{\zeta}(x-1))]\big{]}\phi^{T}E_{n}(t_{h})h_{h}(x_{B},x,x_{3},b,b_{B})\big{\\}}.$

The involved evolution factors $E_{e}(t)$, $E_{a}(t)$ and $E_{n}(t)$ are given
by

$\displaystyle E_{e}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{\rho}(t)],$ $\displaystyle
E_{a}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{D^{(*)}}(t)-S_{\rho}(t)],$ $\displaystyle
E_{n}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{\rho}(t)-S_{D^{(*)}}(t)],$ (65)

in which the Sudakov exponents are defined as

$\displaystyle S_{B}$ $\displaystyle=$ $\displaystyle
S(x_{B}\frac{m_{B}}{\sqrt{2}},b_{B})+\frac{5}{3}\int^{t}_{1/b_{B}}\frac{d\bar{\mu}}{\bar{\mu}}\gamma_{q}(\alpha_{s}(\bar{\mu})),$
(66) $\displaystyle S_{\rho}$ $\displaystyle=$ $\displaystyle
S(x(1-r^{2})\frac{m_{B}}{\sqrt{2}},b)+S((1-x)(1-r^{2})\frac{m_{B}}{\sqrt{2}},b)+2\int^{t}_{1/b}\frac{d\bar{\mu}}{\bar{\mu}}\gamma_{q}(\alpha_{s}(\bar{\mu})),$
(67) $\displaystyle S_{D^{(*)}}$ $\displaystyle=$ $\displaystyle
S(x_{3}(1-\zeta)\frac{m_{B}}{\sqrt{2}},b_{3})+2\int^{t}_{1/b_{3}}\frac{d\bar{\mu}}{\bar{\mu}}\gamma_{q}(\alpha_{s}(\bar{\mu})),$
(68)

with the quark anomalous dimension $\gamma_{q}=-\alpha_{s}/\pi$. The explicit
form for the function $s(Q,b)$ is prd76.074018

$\displaystyle s(Q,b)$ $\displaystyle=$
$\displaystyle\frac{A^{(1)}}{2\beta_{1}}\hat{q}\ln\left(\frac{\hat{q}}{\hat{b}}\right)-\frac{A^{(1)}}{2\beta_{1}}\left(\hat{q}-\hat{b}\right)$
(69) $\displaystyle+$
$\displaystyle\frac{A^{(2)}}{4\beta_{1}^{2}}\left(\frac{\hat{q}}{\hat{b}}-1\right)-\left[\frac{A^{(2)}}{4\beta_{1}^{2}}-\frac{A^{(1)}}{4\beta_{1}}\ln\left(\frac{e^{2\gamma_{E}-1}}{2}\right)\right]\ln\left(\frac{\hat{q}}{\hat{b}}\right)$
$\displaystyle+$
$\displaystyle\frac{A^{(1)}\beta_{2}}{4\beta_{1}^{3}}\hat{q}\left[\frac{\ln(2\hat{q})+1}{\hat{q}}-\frac{\ln(2\hat{b})+1}{\hat{b}}\right]+\frac{A^{(1)}\beta_{2}}{8\beta_{1}^{3}}\left[\ln^{2}(2\hat{q})-\ln^{2}(2\hat{b})\right],$

with the variables

$\displaystyle\hat{q}\ \ \equiv\
\mbox{ln}[Q/(\sqrt{2}\Lambda)],\qquad\qquad\hat{b}\ \ \equiv\
\mbox{ln}[1/(b\Lambda)],$ (70)

and the coefficients $A^{(i)}$ and $\beta_{i}$ are

$\displaystyle\beta_{1}$ $\displaystyle=$
$\displaystyle\frac{33-2n_{f}}{12},\qquad\quad\beta_{2}\ =\
\frac{153-19n_{f}}{24},$ $\displaystyle A^{(1)}$ $\displaystyle=$
$\displaystyle\frac{4}{3},\qquad\qquad\qquad A^{(2)}=\
\frac{67}{9}-\frac{\pi^{2}}{3}-\frac{10}{27}n_{f}+\frac{8}{3}\beta_{1}\mbox{ln}(\frac{1}{2}e^{\gamma_{E}}),$
(71)

where $n_{f}$ is the number of the quark flavors and $\gamma_{E}$ is the Euler
constant.

The hard scale, denoted as $t_{i}$, are determined by selecting the maximum
value of the virtuality associated with the internal momentum transition in
the hard amplitudes, the specific expressions for the hard scales are given by
:

$\displaystyle t_{a}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|a_{1}|},m_{B}\sqrt{|a_{2}|},1/b,1/b_{B}\\},$
$\displaystyle t_{b}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|b_{1}|},m_{B}\sqrt{|b_{2}|},1/b_{B},1/b\\},$
$\displaystyle t_{c}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|c_{1}|},m_{B}\sqrt{|c_{2}|},1/b_{B},1/b_{3}\\},$
$\displaystyle t_{d}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|d_{1}|},m_{B}\sqrt{|d_{2}|},1/b_{B},1/b_{3}\\},$
$\displaystyle t_{e}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|e_{1}|},m_{B}\sqrt{|e_{2}|},1/b,1/b_{3}\\},$
$\displaystyle t_{f}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|f_{1}|},m_{B}\sqrt{|f_{2}|},1/b_{3},1/b\\};$
$\displaystyle t_{g}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|g_{1}|},m_{B}\sqrt{|g_{2}|},1/b,1/b_{B}\\},$
$\displaystyle t_{h}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|h_{1}|},m_{B}\sqrt{|h_{2}|},1/b,1/b_{B}\\};$
$\displaystyle t_{m}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|m_{1}|},m_{B}\sqrt{|m_{2}|},1/b_{3},1/b_{B}\\},$
$\displaystyle t_{n}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|n_{1}|},m_{B}\sqrt{|n_{2}|},1/b_{3},1/b_{B}\\},$
$\displaystyle t_{o}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|o_{1}|},m_{B}\sqrt{|o_{2}|},1/b_{B},1/b\\},$
$\displaystyle t_{p}$ $\displaystyle=$
$\displaystyle\max\\{m_{B}\sqrt{|p_{1}|},m_{B}\sqrt{|p_{2}|},1/b_{B},1/b\\}.$
(72)

with the factors

$\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle(1-r^{2})x,\qquad\qquad\quad\quad\quad a_{2}=(1-r^{2})x_{B}x,$
$\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle(1-r^{2})(x_{B}-\zeta),\qquad\qquad b_{2}=a_{2},$ $\displaystyle
c_{1}$ $\displaystyle=$ $\displaystyle a_{2},\qquad\qquad\qquad\qquad\qquad
c_{2}=r_{c}^{2}-[(1-x)r^{2}+x][(1-\zeta)(1-x_{3})-x_{B}],$ $\displaystyle
d_{1}$ $\displaystyle=$ $\displaystyle a_{2},\qquad\qquad\qquad\qquad\qquad
d_{2}=(1-r^{2})x[x_{B}-(1-\zeta)x_{3}],$ $\displaystyle e_{1}$
$\displaystyle=$ $\displaystyle r_{c}^{2}-[1-x(1-r^{2})],\quad\quad\ \ \
e_{2}=(1-r^{2})(1-x)[(\zeta-1)x_{3}-\zeta],$ $\displaystyle f_{1}$
$\displaystyle=$
$\displaystyle(1-r^{2})[(\zeta-1)x_{3}-\zeta],\quad\;f_{2}=e_{2},$
$\displaystyle g_{1}$ $\displaystyle=$ $\displaystyle
e_{2},\qquad\qquad\qquad\qquad\qquad
g_{2}=1-[(1-x)r^{2}+x][(1-\zeta)(1-x_{3})-x_{B}],$ $\displaystyle h_{1}$
$\displaystyle=$ $\displaystyle e_{2},\qquad\qquad\qquad\qquad\qquad
h_{2}=(1-r^{2})(1-x)[(\zeta-1)x_{3}-\zeta+x_{B}],$ $\displaystyle m_{1}$
$\displaystyle=$ $\displaystyle(1-\zeta)x_{3},\qquad\qquad\quad\quad\quad
m_{2}=(1-\zeta)x_{3}x_{B};$ $\displaystyle n_{1}$ $\displaystyle=$
$\displaystyle r_{c}^{2}-(r^{2}-x_{B})(1-\zeta),\quad\quad n_{2}=m_{2},$
$\displaystyle o_{1}$ $\displaystyle=$ $\displaystyle
m_{2},\qquad\qquad\qquad\qquad\qquad
o_{2}=[(\zeta-1)x_{3}-\zeta][(1-x)(1-r^{2})-x_{B}],$ $\displaystyle p_{1}$
$\displaystyle=$ $\displaystyle m_{2},\qquad\qquad\qquad\qquad\qquad
p_{2}=(1-\zeta)x_{3}[x_{B}-(1-r^{2})x].$ (73)

The threshold resummation factor $S_{t}(x)$ is of the form prd65.014007 :

$\displaystyle
S_{t}(x)=\frac{2^{1+2c}\Gamma(3/2+c)}{\sqrt{\pi}\Gamma(1+c)}[x(1-x)]^{c},$
(74)

with the parameter $c$ adopted to be $0.3$.

The expressions of the hard functions $h_{i}$ with
$i\in\\{a,b,c,d,e,f,g,h,m,n,o,p\\}$ are obtained through the Fourier transform
of the hard kernel:

$\displaystyle h_{i}(x_{1},x_{2},x_{3},b_{1},b_{2})$ $\displaystyle=$
$\displaystyle h_{1}(\beta,b_{2})\times h_{2}(\alpha,b_{1},b_{2}),$
$\displaystyle h_{1}(\beta,b_{2})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}K_{0}(\sqrt{\beta}b_{2}),&\quad\quad\beta>0\\\
K_{0}(i\sqrt{-\beta}b_{2}),&\quad\quad\beta<0\end{array}\right.$ (77)
$\displaystyle h_{2}(\alpha,b_{1},b_{2})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\theta(b_{2}-b_{1})I_{0}(\sqrt{\alpha}b_{1})K_{0}(\sqrt{\alpha}b_{2})+(b_{1}\leftrightarrow
b_{2}),&\quad\alpha>0\\\
\theta(b_{2}-b_{1})I_{0}(\sqrt{-\alpha}b_{1})K_{0}(i\sqrt{-\alpha}b_{2})+(b_{1}\leftrightarrow
b_{2}),&\quad\alpha<0\end{array}\right.$ (80)

where $K_{0}$, $I_{0}$ are modified Bessel function with
$K_{0}(ix)=\frac{\pi}{2}(-N_{0}(x)+iJ_{0}(x))$ and $J_{0}$ is the Bessel
function, $\alpha$ and $\beta$ are the factors $i_{1},i_{2}$.

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