# Contributions of $K^{*}_{0}(1430)$ and $K^{*}_{0}(1950)$ in the three-body
decays $B\to K\pi h$

Wen-Fei Wang1,2<EMAIL_ADDRESS>Jian Chai1,2 Ai-Jun Ma3 1Institute of
Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China  2State
Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi
University, Taiyuan, Shanxi 030006, China  3Department of Mathematics and
Physics, Nanjing Institute of Technology, Nanjing, Jiangsu 211167, China

###### Abstract

We study the contributions of the resonant states $K_{0}^{*}(1430)$ and
$K_{0}^{*}(1950)$ in the three-body decays $B\to K\pi h$ (with $h=\pi,K$) in
the perturbative QCD approach. The crucial nonperturbative input $F_{K\pi}(s)$
in the distribution amplitudes of the $S$-wave $K\pi$ system is derived from
the matrix element of vacuum to $K\pi$ pair. The $CP$ averaged branching
fraction of the quasi-two-body decay process $B\to K^{*}_{0}(1950)h\to K\pi h$
is about one order smaller than that of the corresponding decay $B\to
K^{*}_{0}(1430)h\to K\pi h$. In view of the important contribution from the
$S$-wave $K\pi$ system for the $B\to K\pi h$ decays, it is not appropriate to
neglect the $K_{0}^{*}(1950)$ in the theoretical or experimental studies for
the relevant three-body $B$ meson decays. The predictions in this work for the
relevant decays are consistent with the existing experimental data.

###### pacs:

13.20.He, 13.25.Hw, 13.30.Eg

##  I Introduction

The charmless three-body hadronic $B$ meson decay processes provide us a field
to appraise different dynamical models of strong interaction, to investigate
hadronic final-state interactions and analyze hadron spectroscopy, to
determine the fundamental quark mixing parameters and understand $CP$
asymmetries. In order to extract the significative information from
experimental results and present the effective and accurate predictions for
the three-body $B$ decays, some methods have been adopted in abundant works,
such as the $U$-spin, isospin and flavor $SU(3)$ symmetries in plb564-90 ;
prd72-075013 ; prd72-094031 ; prd84-056002 ; plb727-136 ; plb726-337 ;
prd89-074043 ; plb728-579 ; ijmpa29-1450011 ; prd91-014029 , the QCD
factorization (QCDF) in plb622-207 ; prd74-114009 ; prd79-094005 ;
prd81-094033 ; appb42-2013 ; prd72-094003 ; prd76-094006 ; prd88-114014 ;
prd89-074025 ; prd89-094007 ; prd94-094015 ; epjc75-536 ; prd87-076007 ;
epjc78-845 ; npb899-247  and the perturbative QCD (PQCD) approach in
plb561-258 ; prd70-054006 ; prd89-074031 . The three-body decays $B\to K\pi
h$, with $h$ is the pion or kaon, have been studied by Belle prd71-092003 ;
prl96-251803 ; prd75-012006 ; prd79-072004 ; prd96-031101 ; prd99-031102 ;
prd100-011101 , BaBar prd72-072003 ; prl99-221801 ; prd78-012004 ;
prd78-052005 ; prd80-112001 ; prd82-031101 ; prd83-112010 ; prd96-072001  and
LHCb jhep1310-143 ; prl111-101801 ; prl112-011801 ; prd90-112004 ; plb765-307
; jhep1711-027 ; prl120-261801 ; jhep1906-114  Collaborations in recent years.
These decays especially the $B\to K\pi\pi$ were found to be a clean source for
the extraction of the Cabibbo-Kobayashi-Maskawa (CKM) CKM-C ; CKM-KM  angle
$\gamma$ prd74-051301 ; prd75-014002 ; prd76-073011 ; plb645-201 ;
prd84-034041 ; prd85-016010 ; plb728-206 ; jhep1504-154 . The relevant
processes also provide new possibilities for the measurements of the $CP$
violation in the $B$ decays prl96-251803 ; prl111-101801 ; prl112-011801 ;
prd90-112004 .

The total decay amplitude for the $B$ meson decays into three light mesons
$K,~{}\pi$ and $h$ as the final state can be described as the coherent sum of
the nonresonant and resonant contributions in the isobar formalism pr135-B551
; pr166-1731 ; prd11-3165 . The nonresonant contributions are spread all over
the phase space and play an important role in the corresponding decay
processes prd60-054029 ; prd66-054015 ; plb665-30 . The resonant contributions
from low energy scalar, vector and tensor resonances are known experimentally,
in most cases, to be the dominated proportion of the related decays and could
be studied in the quasi-two-body framework plb763-29 ; 1605-03889 ;
prd96-113003  when the rescattering effects 1512-09284  and three-body effects
npps199-341 ; prd84-094001  are neglected. For the three-body decays $B\to
K\pi h$, one has the resonant contributions from the $K\pi,~{}\pi h$ and $Kh$
pairs which are originated from different intermediate states and as well
containing the two-body final state interactions. And the $J^{P}=0^{+}$
component of the $K\pi$ spectrum, denoted as $(K\pi)^{*}_{0}$, is always found
very important for the relevant physical observables.

The kaon-pion scattering has been extensively studied in Refs epjc77-91 ;
prd93-074025 ; prd123-042002 ; npb932-29 ; prd91-054008 ; prd86-054508 ;
prd92-113002  in recent years. While the primary source of the information on
$I=1/2~{}S$-wave $K\pi$ system comes from the LASS experiment for the reaction
$K^{-}p\to K^{-}\pi^{+}n$ npb296-493 . The $K\pi$ $S$-wave amplitude has also
been studied in detail in the decays $D^{+}\to K^{-}\pi^{+}\pi^{+}$ by E791
prd73-032004 , FOCUS plb653-1 ; plb681-14  and CLEO prd78-052001 ,
$\eta_{c}\to K\bar{K}\pi$ by BaBar prd93-012005  and $\tau^{-}\to
K_{S}\pi^{-}\nu_{\tau}$ by Belle plb654-65  with the methods of Breit-Wigner
functions BW-model , K-matrix formalism pr70-15 ; npa189-417 ; ap4-404  or
model-independent partial-wave analysis. To describe the slowly increasing
phase as a function of the $K\pi$ invariant mass, the scalar $K\pi$ scattering
amplitude was written as the relativistic Breit-Wigner term BW-model  for the
resonance $K^{*}_{0}(1430)$ in the LASS parametrization together with an
effective range nonresonant component in npb296-493 , and the effective range
term has been applied a cutoff to the slowly varying part close to the charm
hadron mass at about $1.8$ GeV for the three-body $B$ decays in the
experimental studies prd78-012004 ; prd80-112001 ; prd83-112010 ; prd96-072001
. At about $1.95$ GeV one will find the presence of the resonance
$K^{*}_{0}(1950)$ in npb296-493  and also in the $\eta_{c}$ decays in
prd93-012005 ; prd89-112004 . This state was assigned as a radial excitation
of the $0^{+}$ member of the $L=1$ triplet in the LASS analysis npb296-493 .
The lowest-lying broad component of the $S$-wave $K\pi$ system is the
$K^{*}_{0}(700)$ PDG-2018 , also named as $\kappa$ or $K^{*}_{0}(800)$ in
literature prl89-121801 ; plb632-471 ; plb633-681 ; prd73-032004 ; epjc48-553
; prd78-052001 ; plb681-14 ; prd81-014002 ; plb698-183 , which has commonly
been placed together with the resonant states $\sigma$, $f_{0}(980)$ and
$a_{0}(980)$ into an $SU(3)$ flavor nonet, and they have been suspected to be
exotics prd15-267 ; prl48-659 ; prd41-2236 ; prl93-212002 ; pr389-61 ; pr409-1
; prd74-014028 .

In this work, we will focus on the contributions of the resonant state
$K^{*}_{0}(1430)$ in the $B\to K\pi h$ decay processes in the PQCD approach
based on the $k_{\rm T}$ factorization theorem plb504-6 ; prd63-054008 ;
prd63-074009 ; ppnp51-85 . The contributions of the resonant state
$K^{*}_{0}(1950)$ in the hadronic three-body $B$ meson decays involving
$S$-wave $K\pi$ pair have been ignored in the relevant theoretical studies and
be noticed only by LHCb Collaboration very recently in the works prd90-072003
; epjc78-1019 . We will systematically estimate, for the first time, the
contributions from the state $K^{*}_{0}(1950)$ for the $B\to K\pi h$ decays in
this work. As for the resonance $K^{*}_{0}(700)$, we shall leave to the future
studies in view of its ambiguous internal structure and the accompanying
complicated results for the three-body $B$ decays prd73-014017 , in addition,
the corresponding contributions have been covered up by the effective range
part of LASS line shape for the experimental results prd78-012004 ;
prd78-052005 ; prd80-112001 ; prd83-112010 ; prd96-072001 .

For the quasi-two-body decays $B\to K^{*}_{0}(1430,1950)h\to K\pi h$, the
subprocesses of the $B\to K\pi h$ decays, the intermediate state $K^{*}_{0}$,
as demonstrated in the Fig. 1, is generated in the hadronization of quark-
antiquark pair including one $s$ or $\bar{s}$-quark. The process $K^{*}_{0}\to
K\pi$, which can not be calculated in the PQCD approach, is always shrunken as
the decay constants in the twist-2 and twist-3 light-cone distribution
amplitudes of the scalar mesons prd73-014017 ; prd75-056001 ; epja49-78  in
the studies of the two-body $B$ meson decays involving the scalar mesons
$K^{*}_{0}(700)$ and $K^{*}_{0}(1430)$, see Ref. prd95-016011  and the
references therein for examples. While in the quasi-two-body framework based
on PQCD, one can easily introduce the nonperturbative subprocess $K^{*}_{0}\to
K\pi$ into a time-like form factor in the distribution amplitudes of the
$K\pi$ pair. The quasi-two-body framework based on PQCD has been discussed in
detail in plb763-29  and has been adopted in some studies on the quasi-two-
body $B$ meson decay processes recently prd100-014017 ; epjc79-37 ; epjc79-539
; plb791-342 ; plb788-468 ; prd96-093011 ; prd96-036014 ; epjc77-199 ;
prd95-056008 ; npb923-54 .

This work is organized as follows. In Sec. II, we give a brief introduction
for the theoretical framework. In Sec. III, we show the numerical results and
give some discussions. Conclusions are presented in Sec. IV. The factorization
formulas and functions for the related quasi-two-body decay amplitudes are
collected in the Appendix.

##  II Framework

Figure 1: Typical Feynman diagrams for the decay processes $B\to K_{0}^{*}h\to
K\pi h$, $h=$($\pi,K$). The symbol $\otimes$ is the weak vertex, $\times$
denotes possible attachments of hard gluons and the rectangle represents the
scalar resonances $K_{0}^{*}$.

In the rest frame of $B$ meson, we define its momentum $p_{B}$ and light
spectator quark momentum $k_{B}$ as

$\displaystyle p_{B}=\frac{m_{B}}{\sqrt{2}}(1,1,0_{\rm T}),\quad
k_{B}=\left(\frac{m_{B}}{\sqrt{2}}x_{B},0,k_{B{\rm T}}\right),$ (1)

in the light-cone coordinates, where $x_{B}$ is the momentum fraction and
$m_{B}$ is the mass. For the resonant states $K^{*}_{0}$ and the $K\pi$ pair
generated from it by the strong interaction as revealed in the Fig. 1, we
define their momentum $p=\frac{m_{B}}{\sqrt{2}}(\zeta,1,0)$. Its easy to
validate $\zeta=s/m_{B}^{2}$, where the invariant mass square
$s=p^{2}=m^{2}_{K\pi}$ for the $K\pi$ pair. The light spectator quark comes
from $B$ meson and goes into intermediate state in the hadronization of
$K^{*}_{0}$ as shown in Fig. 1 (a) has the momentum
$k=(0,\frac{m_{B}}{\sqrt{2}}z,k_{\rm T})$. For the bachelor final state $h$
and its spectator quark, their momenta $p_{3}$ and $k_{3}$ have the
definitions as

$\displaystyle p_{3}=\frac{m_{B}}{\sqrt{2}}(1-\zeta,0,0_{\rm T}),\quad
k_{3}=\left(\frac{m_{B}}{\sqrt{2}}(1-\zeta)x_{3},0,k_{3{\rm T}}\right).$ (2)

Where $x_{3}$ and $z$, which run from 0 to 1, are the corresponding momentum
fractions.

The matrix element from the vacuum to the $K^{+}\pi^{-}$ final state is given
by zpc29-637

$\displaystyle\langle
K^{+}(p_{1})\pi^{-}(p_{2})|\bar{d}\gamma_{\mu}(1-\gamma_{5})s|0\rangle=\left[(p_{1}-p_{2})_{\mu}-\frac{\Delta_{K\pi}}{p^{2}}p_{\mu}\right]F^{K\pi}_{+}(s)+\frac{\Delta_{K\pi}}{p^{2}}p_{\mu}F^{K\pi}_{0}(s),\;\;$
(3)

with the $p_{1}(p_{2})$ is the momentum for kaon(pion) in the $K\pi$ system,
$\Delta_{K\pi}=(m^{2}_{K}-m^{2}_{\pi})$ and $m_{K}(m_{\pi})$ is the mass of
$K(\pi)$ meson. The $F^{K\pi}_{+}(s)$ is the vector form factor which has been
discussed in detail in the Refs. plb654-65 ; epjc11-599 ; plb640-176 ;
plb664-78 ; epjc53-401 ; epjc59-821 ; jhep1009-031 ; jhep1409-042 ;
jhep1912-083 . The the scalar form factor $F^{K\pi}_{0}(s)$ is defined as
npb622-279 ; prd80-054007 ; plb730-336

$\displaystyle\langle
K\pi|\bar{q}s|0\rangle=C_{X}\frac{\Delta_{K\pi}}{m_{s}-m_{q}}F_{0}^{K\pi}(s)=B_{0}C_{X}F_{0}^{K\pi}(s)\;,$
(4)

where $q$ is the light quark $u$ or $d$, the isospin factor $C_{X}=1$ for
$X=\\{K^{+}\pi^{-},K^{0}\pi^{+}\\}$ and $C_{X}=1/\sqrt{2}$ for
$X=\\{K^{+}\pi^{0},K^{0}\pi^{0}\\}$. The constant $B_{0}$ equals to
$\Delta_{K\pi}/(m_{s}-m_{q})$. The form factor $F^{K\pi}_{0}(s)$ above is
suppose to be one when $s$ goes to zero. When the $K^{+}\pi^{-}$ pair
originated from the resonant state $K^{*}_{0}(1430)^{0}$, we have prd80-054007

$\displaystyle\langle K^{+}\pi^{-}|\bar{d}s|0\rangle\approx\langle
K^{+}\pi^{-}|K_{0}^{*0}\rangle\frac{1}{\mathcal{D}_{K^{*}_{0}}}\langle
K_{0}^{*0}|\bar{d}s|0\rangle=\Pi_{K_{0}^{*}K\pi}\langle
K_{0}^{*0}|\bar{d}s|0\rangle\;,$ (5)

and

$\displaystyle\Pi_{K_{0}^{*}K\pi}=\frac{g_{K_{0}^{*}K\pi}}{\mathcal{D}_{K^{*}_{0}}}\approx\frac{B_{0}}{\bar{f}_{K_{0}^{*}}m_{K_{0}^{*}}}F_{0}^{K\pi}(s)\;,$
(6)

with $\bar{f}_{K_{0}^{*}}=\frac{m_{K_{0}^{*}}}{m_{s}(\mu)-m_{d}(\mu)}\cdot
f_{K_{0}^{*}}$, the decay constants defined by $\langle
K_{0}^{*0}|\bar{d}s|0\rangle=m_{K_{0}^{*}}\bar{f}_{K_{0}^{*}}$ and $\langle
K_{0}^{*0}(p)|\bar{d}\gamma_{\mu}s|0\rangle=f_{K_{0}^{*}}p_{\mu}$ prd73-014017
, and the mass $m_{K_{0}^{*}}$ could be replaced by the invariant mass
$\sqrt{s}$ for the off-shell $K_{0}^{*}$. One can find different values of
$f_{K_{0}^{*}}$ for $K_{0}^{*}(1430)$ in prd78-114016 , we employ
$f_{K_{0}^{*}(1430)}m^{2}_{K_{0}^{*}(1430)}=0.0842\pm 0.0045$ GeV3 plb462-14
and $f_{K_{0}^{*}(1950)}m^{2}_{K_{0}^{*}(1950)}=0.0414$ GeV3 prd63-074017  in
this work. The Breit-Wigner formula for the denominator
${\mathcal{D}_{K^{*}_{0}}}=m^{2}_{K^{*}_{0}}-s-im_{K^{*}_{0}}\Gamma(s)$, with
the mass-dependent decay width
$\Gamma(s)=\Gamma_{0}\frac{q}{q_{0}}\frac{m_{K^{*}_{0}}}{\sqrt{s}}$ and
$\Gamma_{0}$ is the full width for resonant state $K_{0}^{*}$. In the rest
frame of the resonance $K_{0}^{*}$, its daughter kaon or pion has the
magnitude of the momentum as

$\displaystyle
q=\frac{1}{2}\sqrt{\left[s-(m_{K}+m_{\pi})^{2}\right]\left[s-(m_{K}-m_{\pi})^{2}\right]/s}\;.$
(7)

The $q_{0}$ in $\Gamma(s)$ is the value for $q$ at $s=m^{2}_{K_{0}^{*}}$. The
coupling constant $g_{K_{0}^{*}K\pi}=\langle K^{+}\pi^{-}|K_{0}^{*0}\rangle$,
one has prd88-114014

$\displaystyle g_{K_{0}^{*}K\pi}=\sqrt{\frac{8\pi
m^{2}_{K^{*}_{0}}\Gamma_{{K^{*}_{0}}\to K\pi}}{q_{0}}}\;,$ (8)

where the $\Gamma_{{K^{*}_{0}}\to K\pi}$ is the partial width for
${K^{*}_{0}}\to K\pi$.

The $S$-wave $K\pi$ system distribution amplitudes are collected into
plb730-336 ; prd73-014017 ; prd87-114001 ; prd91-094024

$\displaystyle\Phi_{K\pi}(z,s)=\frac{1}{\sqrt{2N_{c}}}\left[{p/}\phi(z,s)+\sqrt{s}\phi^{s}(z,s)+\sqrt{s}({v/}{n/}-1)\phi^{t}(z,s)\right],$
(9)

with the $v=(0,1,{0}_{\rm T})$ and $n=(1,0,{0}_{\rm T})$ being the
dimensionless vectors. The twist-2 light-cone distribution amplitude has the
form plb730-336 ; prd73-014017 ; prd87-114001

$\displaystyle\phi(z,s)=\frac{F_{K\pi}(s)}{2\sqrt{2N_{c}}}\left\\{6z(1-z)\left[a_{0}(\mu)+\sum^{\infty}_{m=1}a_{m}(\mu)C^{3/2}_{m}(2z-1)\right]\right\\}\;,$
(10)

with $C^{3/2}_{m}$ the Gegenbauer polynomials,
$a_{0}=(m_{s}(\mu)-m_{q}(\mu))/\sqrt{s}$ for ($K^{*-}_{0},\bar{K}^{*0}_{0}$)
and $a_{0}=(m_{q}(\mu)-m_{s}(\mu))/\sqrt{s}$ for ($K^{*+}_{0},K^{*0}_{0}$)
according to Ref. prd87-114001 . The $a_{m}$ are scale-dependent Gegenbauer
moments, with $a_{1}=-0.57\pm 0.13$ and $a_{3}=-0.42\pm 0.22$ at the scale
$\mu=1$ GeV for the resonance $K^{*}_{0}(1430)$, and the contributions from
the even terms could be neglected prd73-014017 . There is no available
Gegenbauer moments for the state $K^{*}_{0}(1950)$, we employ the scale-
dependent $a_{1}$ and $a_{3}$ of $K^{*}_{0}(1430)$ for the entire $S$-wave
$K\pi$ system in the numerical calculation. For the twist-3 light-cone
distribution amplitudes in this work, we take the asymptotic forms as

$\displaystyle\phi^{s}(z,s)=\frac{F_{K\pi}(s)}{2\sqrt{2N_{c}}}\;,\quad\quad\phi^{t}(z,s)=\frac{F_{K\pi}(s)}{2\sqrt{2N_{c}}}(1-2z)\;.$
(11)

The factor $F_{K\pi}(s)$ is related to scalar form factor $F^{K\pi}_{0}(s)$ by
$F_{K\pi}(s)=\frac{B_{0}}{m_{K_{0}^{*}}}F_{0}^{K\pi}(s)$.

The distribution amplitudes for $B$ meson and the bachelor final state $h$ in
this work are the same as those widely employed in the studies of the hadronic
$B$ meson decays in the PQCD approach, one can find their expressions and
parameters in the Appendix.

##  III Results and discussions

In the numerical calculation, we adopt the decay constants $f_{B}=0.189$ GeV,
$f_{B_{s}}=0.231$ GeV prd98-074512 , the mean lifetimes
$\tau_{B^{0}}=(1.520\pm 0.004)\times 10^{-12}$ s, $\tau_{B^{+}}=(1.638\pm
0.004)\times 10^{-12}$ s and $\tau_{B^{0}_{s}}=(1.509\pm 0.004)\times
10^{-12}$ s PDG-2018  for the $B^{0},B^{+}$ and $B^{0}_{s}$ mesons,
respectively. The masses and the decay constants for the relevant particles in
the numerical calculation in this work, the full widths for $K^{*}_{0}(1430)$
and $K^{*}_{0}(1950)$, and the Wolfenstein parameters of the CKM matrix are
presented in Table 1.

Table 1: Masses, decay constants, full widths of $K^{*}_{0}(1430)$ and
$K^{*}_{0}(1950)$ (in units of GeV) and Wolfenstein parameters PDG-2018 .
$m_{B^{0}}=5.280\quad\;\;m_{B^{\pm}}=5.279\quad\;m_{B^{0}_{s}}=5.367\quad\;m_{\pi^{\pm}}=0.140\quad\;m_{\pi^{0}}=0.135\;$
---
$m_{K^{\pm}}=0.494\quad\;m_{K^{0}}=0.498\quad\;\;f_{K}=0.156\quad\;\;\;f_{\pi}=0.130\;$
$m_{K^{*}_{0}(1430)}=1.425\pm 0.050\hskip
55.48277pt\Gamma_{K^{*}_{0}(1430)}=0.270\pm 0.080\;$
$m_{K^{*}_{0}(1950)}=1.945\pm 0.010\pm
0.020\quad\;\;\Gamma_{K^{*}_{0}(1950)}=0.201\pm 0.034\pm 0.079\;$
$\lambda=0.22453\pm 0.00044\;\;\;A=0.836\pm
0.015\;\;\;\bar{\rho}=0.122^{+0.018}_{-0.017}\;\;\;\bar{\eta}=0.355^{+0.012}_{-0.011}$

Utilizing the differential branching fraction Eq. (30) and the decay
amplitudes collected in Appendix A, we obtain the $CP$ averaged branching
fractions (${\mathcal{B}}$) and the direct $CP$ asymmetries
(${\mathcal{A}}_{CP}$) in Table 2 and Table 3 for the concerned quasi-two-body
decay processes involving the resonances $K_{0}^{*}(1430)$ and
$K_{0}^{*}(1950)$ as the intermediate states, respectively. The results for
those quasi-two-body decays with one daughter of the $K_{0}^{*}$ is the
neutral pion are omitted. One will get a half value of the ${\mathcal{B}}$ and
the same value of the ${\mathcal{A}}_{CP}$ of the corresponding result in
Tables 2, 3 for a decay with the subprocesses $K_{0}^{*}\to K\pi^{0}$
considering the isospin relation. For example, we have

$\displaystyle{\mathcal{B}}(B^{+}\to K_{0}^{*}(1430)^{+}\pi^{0}\to
K^{+}\pi^{0}\pi^{0})=\frac{1}{2}{\mathcal{B}}(B^{+}\to
K_{0}^{*}(1430)^{+}\pi^{0}\to K^{0}\pi^{+}\pi^{0}),$ (12)

while these two processes have the same direct $CP$ asymmetry.

Table 2: PQCD predictions of the $CP$ averaged branching fractions and the direct $CP$ asymmetries for the quasi-two-body $B\to K_{0}^{*}(1430)h\to K\pi h$ decays. Decay modes |  |  Quasi-two-body results
---|---|---
$B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$ | ${\mathcal{B}}(10^{-5})$ | $2.27\pm 0.59(\omega_{B})\pm 0.17(a_{3+1})\pm 0.34(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $-1.3\pm 0.2(\omega_{B})\pm 0.4(a_{3+1})\pm 0.2(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B^{+}\to K_{0}^{*}(1430)^{+}\pi^{0}\to K^{0}\pi^{+}\pi^{0}$ | ${\mathcal{B}}(10^{-6})$ | $7.86\pm 2.16(\omega_{B})\pm 0.55(a_{3+1})\pm 1.36(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $1.5\pm 0.4(\omega_{B})\pm 0.8(a_{3+1})\pm 0.4(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B^{+}\to K_{0}^{*}(1430)^{+}\bar{K}^{0}\to K^{0}\pi^{+}\bar{K}^{0}$ | ${\mathcal{B}}(10^{-7})$ | $2.33\pm 0.04(\omega_{B})\pm 1.29(a_{3+1})\pm 0.34(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $-18.4\pm 5.8(\omega_{B})\pm 2.7(a_{3+1})\pm 5.4(m^{K}_{0}{+}a^{K}_{2})$
$B^{+}\to\bar{K}_{0}^{*}(1430)^{0}K^{+}\to K^{-}\pi^{+}K^{+}$ | ${\mathcal{B}}(10^{-6})$ | $2.86\pm 0.54(\omega_{B})\pm 0.51(a_{3+1})\pm 0.42(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $17.9\pm 0.4(\omega_{B})\pm 8.0(a_{3+1})\pm 0.9(m^{K}_{0}{+}a^{K}_{2})$
$B^{0}\to K_{0}^{*}(1430)^{+}\pi^{-}\to K^{0}\pi^{+}\pi^{-}$ | ${\mathcal{B}}(10^{-5})$ | $2.07\pm 0.54(\omega_{B})\pm 0.14(a_{3+1})\pm 0.30(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $0.3\pm 0.5(\omega_{B})\pm 0.8(a_{3+1})\pm 0.1(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B^{0}\to K_{0}^{*}(1430)^{0}\pi^{0}\to K^{+}\pi^{-}\pi^{0}$ | ${\mathcal{B}}(10^{-5})$ | $1.39\pm 0.35(\omega_{B})\pm 0.11(a_{3+1})\pm 0.18(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $-1.8\pm 0.4(\omega_{B})\pm 0.2(a_{3+1})\pm 0.1(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B^{0}\to K_{0}^{*}(1430)^{+}K^{-}\to K^{0}\pi^{+}K^{-}$ | ${\mathcal{B}}(10^{-8})$ | $5.77\pm 2.38(\omega_{B})\pm 2.92(a_{3+1})\pm 0.62(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $4.9\pm 6.4(\omega_{B})\pm 3.7(a_{3+1})\pm 3.6(m^{K}_{0}{+}a^{K}_{2})$
$B^{0}\to K_{0}^{*}(1430)^{-}K^{+}\to\bar{K}^{0}\pi^{-}K^{+}$ | ${\mathcal{B}}(10^{-7})$ | $3.84\pm 1.48(\omega_{B})\pm 1.95(a_{3+1})\pm 0.09(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $-5.0\pm 2.6(\omega_{B})\pm 6.7(a_{3+1})\pm 3.0(m^{K}_{0}{+}a^{K}_{2})$
$B^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{0}\to K^{+}\pi^{-}\bar{K}^{0}$ | ${\mathcal{B}}(10^{-7})$ | $3.04\pm 0.15(\omega_{B})\pm 2.04(a_{3+1})\pm 0.36(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ |  -
$B^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0}\to K^{-}\pi^{+}K^{0}$ | ${\mathcal{B}}(10^{-6})$ | $2.89\pm 0.53(\omega_{B})\pm 0.65(a_{3+1})\pm 0.41(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ |  -
$B_{s}^{0}\to K_{0}^{*}(1430)^{-}\pi^{+}\to\bar{K}^{0}\pi^{-}\pi^{+}$ | ${\mathcal{B}}(10^{-5})$ | $3.77\pm 0.78(\omega_{B})\pm 0.51(a_{3+1})\pm 0.01(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $15.5\pm 1.6(\omega_{B})\pm 3.2(a_{3+1})\pm 1.0(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B_{s}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{0}\to K^{-}\pi^{+}\pi^{0}$ | ${\mathcal{B}}(10^{-7})$ | $5.03\pm 0.38(\omega_{B})\pm 1.52(a_{3+1})\pm 0.80(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $59.2\pm 3.2(\omega_{B})\pm 7.1(a_{3+1})\pm 2.5(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B_{s}^{0}\to K_{0}^{*}(1430)^{+}K^{-}\to K^{0}\pi^{+}K^{-}$ | ${\mathcal{B}}(10^{-5})$ | $1.44\pm 0.20(\omega_{B})\pm 0.26(a_{3+1})\pm 0.25(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $0.4\pm 0.3(\omega_{B})\pm 1.8(a_{3+1})\pm 1.2(m^{K}_{0}{+}a^{K}_{2})$
$B_{s}^{0}\to K_{0}^{*}(1430)^{-}K^{+}\to\bar{K}^{0}\pi^{-}K^{+}$ | ${\mathcal{B}}(10^{-5})$ | $1.74\pm 0.16(\omega_{B})\pm 0.84(a_{3+1})\pm 0.24(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $-51.1\pm 1.1(\omega_{B})\pm 6.7(a_{3+1})\pm 5.5(m^{K}_{0}{+}a^{K}_{2})$
$B_{s}^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{0}\to K^{+}\pi^{-}\bar{K}^{0}$ | ${\mathcal{B}}(10^{-5})$ | $1.47\pm 0.22(\omega_{B})\pm 0.24(a_{3+1})\pm 0.25(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ |  -
$B_{s}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0}\to K^{-}\pi^{+}K^{0}$ | ${\mathcal{B}}(10^{-5})$ | $1.19\pm 0.06(\omega_{B})\pm 0.71(a_{3+1})\pm 0.17(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ |  -

For the PQCD predictions in Tables 2, 3, the shape parameters
$\omega_{B}=0.40\pm 0.04$ or $\omega_{B_{s}}=0.50\pm 0.05$ in Eq. (25) for the
$B^{+,0}$ or $B^{0}_{s}$ contribute the first error. The second error for each
PQCD result comes from the Gegenbauer moments $a_{1}$ and $a_{3}$ in the Eq.
(10). The third one is induced by the chiral masses $m^{h}_{0}$ and the
Gegenbauer moment $a^{h}_{2}=0.25\pm 0.15$ of the bachelor final state pion or
kaon. The large uncertainties of the decay widths of the states
$K^{*}_{0}(1430)$ and $K^{*}_{0}(1950)$ in the Table 1 result in quite small
errors, which have been neglected, for these quasi-two-body predictions in the
Tables 2 and 3. The reason is that the variation effect of decay width
$\Gamma$ in the denominator ${\mathcal{D}}_{K^{*}_{0}}$ of Eq. (6) will be
mainly canceled out by the uncertainty of $\Gamma_{{K^{*}_{0}}\to K\pi}$
(equals to $\Gamma\cdot{{\mathcal{B}}({K^{*}_{0}}\to K\pi)}$) in the numerator
$g_{K_{0}^{*}K\pi}$. For instance, the corresponding errors for the decay
process $B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$ are
$0.04\times 10^{-5}$ and $0.2\%$ for its branching fraction and direct $CP$
asymmetry, respectively, while for $B^{+}\to K_{0}^{*}(1950)^{0}\pi^{+}\to
K^{+}\pi^{-}\pi^{+}$, the two errors are $0.01\times 10^{-6}$ and $0.1\%$.
There are other errors, which come from the uncertainties of the Wolfenstein
parameters of the CKM matrix, the parameters in the distribution amplitudes
for bachelor pion or kaon, the masses and the decay constants of the initial
and final states, etc. are small and have been neglected. One can find that
for those decay modes with the main contributions come from the annihilation
diagrams of Fig. 1, their branching fraction errors generated from the
variations of the $a_{1}$ and $a_{3}$ could be larger than the corresponding
errors from $\omega_{B}$ or $\omega_{B_{s}}$, because there is no shape
parameter for $B$ meson in the factorizable annihilation diagrams.

Figure 2: Differential branching fractions from threshold of $K\pi$ pair to
$3$ GeV for the $B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$
and $B^{+}\to K_{0}^{*}(1950)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$ decays.

In this work, the branching fraction of a quasi-two-body decay process
involving the resonant state $K_{0}^{*}(1950)$ is predicted to be roughly one
order smaller than the corresponding decay mode with the resonance
$K_{0}^{*}(1430)$. Or rather, the ratios between the $CP$ averaged branching
fractions for the decays in Table 3 and Table 2 with (without) the
factorizable emission diagrams of Fig. 1 (c) are about $12\%$-$15\%$
($6\%$-$9\%$). The difference mainly originated from the $(S-P)(S+P)$
amplitude the Eq. (65), which has the intermediate state invariant mass factor
$m_{B}\sqrt{\zeta}$ ($\equiv\sqrt{s}$). This factor makes the proportion
originated from Eq. (65) in the total branching ratio for a quasi-two-body
decay mode invloving $K_{0}^{*}(1950)$ larger than that of the corresponding
decay process including $K_{0}^{*}(1430)$ because of the larger pole mass of
the resonance $K_{0}^{*}(1950)$. Take the decays $B^{+}\to
K_{0}^{*}(1430,1950)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$ as the examples, when
we neglect the contribution from the $(S-P)(S+P)$ amplitude of the Eq. (65),
the ratio between two branching fractions of the decays $B^{+}\to
K_{0}^{*}(1950)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$ and $B^{+}\to
K_{0}^{*}(1430)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$ will drop to $8\%$ from
about $15\%$. From the lines of the differential branching fractions for
$B^{+}\to K_{0}^{*}(1950)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$ and $B^{+}\to
K_{0}^{*}(1430)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$ in Fig. 2, one can find
that the main portion of the branching fractions lies in the region around the
corresponding pole mass of the intermediate states.

Table 3: PQCD predictions of the $CP$ averaged branching fractions and the direct $CP$ asymmetries for the quasi-two-body $B\to K_{0}^{*}(1950)h\to K\pi h$ decays. Decay modes |  |  Quasi-two-body results
---|---|---
$B^{+}\to K_{0}^{*}(1950)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$ | ${\mathcal{B}}(10^{-6})$ | $3.36\pm 0.86(\omega_{B})\pm 0.24(a_{3+1})\pm 0.51(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $1.5\pm 0.3(\omega_{B})\pm 0.2(a_{3+1})\pm 0.3(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B^{+}\to K_{0}^{*}(1950)^{+}\pi^{0}\to K^{0}\pi^{+}\pi^{0}$ | ${\mathcal{B}}(10^{-6})$ | $1.19\pm 0.32(\omega_{B})\pm 0.08(a_{3+1})\pm 0.21(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $3.5\pm 0.1(\omega_{B})\pm 0.4(a_{3+1})\pm 0.2(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B^{+}\to K_{0}^{*}(1950)^{+}\bar{K}^{0}\to K^{0}\pi^{+}\bar{K}^{0}$ | ${\mathcal{B}}(10^{-8})$ | $1.86\pm 0.04(\omega_{B})\pm 0.60(a_{3+1})\pm 0.38(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $-9.2\pm 5.3(\omega_{B})\pm 4.0(a_{3+1})\pm 2.8(m^{K}_{0}{+}a^{K}_{2})$
$B^{+}\to\bar{K}_{0}^{*}(1950)^{0}K^{+}\to K^{-}\pi^{+}K^{+}$ | ${\mathcal{B}}(10^{-7})$ | $3.59\pm 0.66(\omega_{B})\pm 0.54(a_{3+1})\pm 0.54(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $19.2\pm 0.1(\omega_{B})\pm 7.4(a_{3+1})\pm 1.4(m^{K}_{0}{+}a^{K}_{2})$
$B^{0}\to K_{0}^{*}(1950)^{+}\pi^{-}\to K^{0}\pi^{+}\pi^{-}$ | ${\mathcal{B}}(10^{-6})$ | $2.99\pm 0.77(\omega_{B})\pm 0.20(a_{3+1})\pm 0.45(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $1.9\pm 0.5(\omega_{B})\pm 0.5(a_{3+1})\pm 0.1(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B^{0}\to K_{0}^{*}(1950)^{0}\pi^{0}\to K^{+}\pi^{-}\pi^{0}$ | ${\mathcal{B}}(10^{-6})$ | $2.01\pm 0.50(\omega_{B})\pm 0.15(a_{3+1})\pm 0.26(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $0.4\pm 0.6(\omega_{B})\pm 0.3(a_{3+1})\pm 0.3(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B^{0}\to K_{0}^{*}(1950)^{+}K^{-}\to K^{0}\pi^{+}K^{-}$ | ${\mathcal{B}}(10^{-9})$ | $5.14\pm 1.90(\omega_{B})\pm 1.66(a_{3+1})\pm 0.29(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $-2.8\pm 10(\omega_{B})\pm 10.6(a_{3+1})\pm 3.3(m^{K}_{0}{+}a^{K}_{2})$
$B^{0}\to K_{0}^{*}(1950)^{-}K^{+}\to\bar{K}^{0}\pi^{-}K^{+}$ | ${\mathcal{B}}(10^{-8})$ | $2.36\pm 0.95(\omega_{B})\pm 1.10(a_{3+1})\pm 0.06(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $-1.0\pm 2.4(\omega_{B})\pm 8.5(a_{3+1})\pm 1.3(m^{K}_{0}{+}a^{K}_{2})$
$B^{0}\to K_{0}^{*}(1950)^{0}\bar{K}^{0}\to K^{+}\pi^{-}\bar{K}^{0}$ | ${\mathcal{B}}(10^{-8})$ | $2.22\pm 0.08(\omega_{B})\pm 1.05(a_{3+1})\pm 0.35(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ |  -
$B^{0}\to\bar{K}_{0}^{*}(1950)^{0}K^{0}\to K^{-}\pi^{+}K^{0}$ | ${\mathcal{B}}(10^{-7})$ | $3.36\pm 0.64(\omega_{B})\pm 0.58(a_{3+1})\pm 0.48(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ |  -
$B_{s}^{0}\to K_{0}^{*}(1950)^{-}\pi^{+}\to\bar{K}^{0}\pi^{-}\pi^{+}$ | ${\mathcal{B}}(10^{-6})$ | $3.35\pm 0.59(\omega_{B})\pm 0.37(a_{3+1})\pm 0.01(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $12.9\pm 7.0(\omega_{B})\pm 3.1(a_{3+1})\pm 0.8(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B_{s}^{0}\to\bar{K}_{0}^{*}(1950)^{0}\pi^{0}\to K^{-}\pi^{+}\pi^{0}$ | ${\mathcal{B}}(10^{-8})$ | $3.74\pm 0.35(\omega_{B})\pm 1.01(a_{3+1})\pm 0.48(m^{\pi}_{0}{+}a^{\pi}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $57.1\pm 4.0(\omega_{B})\pm 8.1(a_{3+1})\pm 5.5(m^{\pi}_{0}{+}a^{\pi}_{2})$
$B_{s}^{0}\to K_{0}^{*}(1950)^{+}K^{-}\to K^{0}\pi^{+}K^{-}$ | ${\mathcal{B}}(10^{-6})$ | $2.03\pm 0.31(\omega_{B})\pm 0.19(a_{3+1})\pm 0.32(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $0.6\pm 0.2(\omega_{B})\pm 1.0(a_{3+1})\pm 0.9(m^{K}_{0}{+}a^{K}_{2})$
$B_{s}^{0}\to K_{0}^{*}(1950)^{-}K^{+}\to\bar{K}^{0}\pi^{-}K^{+}$ | ${\mathcal{B}}(10^{-6})~{}$ | $1.26\pm 0.15(\omega_{B})\pm 0.54(a_{3+1})\pm 0.20(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $-45.1\pm 1.3(\omega_{B})\pm 4.6(a_{3+1})\pm 5.7(m^{K}_{0}{+}a^{K}_{2})$
$B_{s}^{0}\to K_{0}^{*}(1950)^{0}\bar{K}^{0}\to K^{+}\pi^{-}\bar{K}^{0}$ | ${\mathcal{B}}(10^{-6})$ | $2.13\pm 0.33(\omega_{B})\pm 0.19(a_{3+1})\pm 0.33(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ |  -
$B_{s}^{0}\to\bar{K}_{0}^{*}(1950)^{0}K^{0}\to K^{-}\pi^{+}K^{0}$ | ${\mathcal{B}}(10^{-7})$ | $7.65\pm 0.54(\omega_{B})\pm 4.56(a_{3+1})\pm 1.48(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ |  -

We must stress that the ratios between the corresponding branching fractions
in Table 3 and Table 2, and also the branching fractions in Table 3 for the
quasi-two-body decays involving $K_{0}^{*}(1950)$ are squared dependent on the
result $f_{K_{0}^{*}(1950)}m^{2}_{K_{0}^{*}(1950)}=0.0414$ GeV3 prd63-074017 .
If the value $0.0414$ becomes two times larger, the ratios and the branching
fractions in Table 3 will become four times larger than their current values.
In Ref. epjc78-1019 , there are two branching fractions measured by LHCb to be

$\displaystyle{\mathcal{B}}(B^{0}\to\eta_{c}K^{*}_{0}(1950)^{0}\to\eta_{c}K^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle(2.18\pm 1.04\pm 0.04^{+0.80}_{-1.43}\pm
0.25)\times 10^{-5},$ (13)
$\displaystyle{\mathcal{B}}(B^{0}\to\eta_{c}K^{*}_{0}(1430)^{0}\to\eta_{c}K^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle(14.50\pm 2.10\pm 0.28^{+2.01}_{-1.60}\pm
1.67)\times 10^{-5}.$ (14)

The two central values above give us the ratio about $0.15$ between these two
branching factions, but there is no diagrams like Fig. 1 (c) for
$B^{0}\to\eta_{c}K^{*0}_{0}$ decays. Because of the large errors for
$B^{0}\to\eta_{c}K^{*}_{0}(1950)^{0}$, we can not extract the decay constant
$f_{K^{*}_{0}(1950)}$ from this measurement. While from the data of the fit
fractions for $\eta_{c}\to K^{0}_{S}K^{\pm}\pi^{\mp}$ in prd93-012005  and
$\eta_{c}\to K^{+}K^{-}\pi^{0}$ in prd89-112004  both from BaBar, one can
expect a larger value than $0.0414$ GeV3 for the
$f_{K_{0}^{*}(1950)}m^{2}_{K_{0}^{*}(1950)}$.

Table 4: Comparison of the extracted predictions with the experimental measurements for the relevant two-body branching fractions (in units of $10^{-6}$). The first error for the theoretical results is added in quadrature from the errors in Table 2, the second error comes from the uncertainty of ${\mathcal{B}}(K^{*}_{0}(1430)\to K\pi)=0.93\pm 0.10$ PDG-2018 . Two-body decays |  This work | Data | Ref.
---|---|---|---
$B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}$ | $36.6\pm 11.3\pm 3.9$ | $34.6\pm 3.3\pm 4.2^{+1.9}_{-1.8}$ | BaBar prd96-072001
|  | $32.0\pm 1.2\pm 2.7^{+9.1}_{-1.4}\pm 5.2$ | BaBar prd78-012004
|  | $51.6\pm 1.7\pm 6.8^{+1.8}_{-3.1}$ | Belle prl96-251803
$B^{+}\to K_{0}^{*}(1430)^{+}\pi^{0}$ | $12.7\pm 4.2\pm 1.4$ | $11.9\pm 1.7\pm 1.0^{+0.0}_{-1.3}$ | BaBar prd96-072001
$B^{0}\to K_{0}^{*}(1430)^{+}\pi^{-}$ | $33.4\pm 10.2\pm 3.6$ | $29.9^{+2.3}_{-1.7}\pm 1.6\pm 0.6\pm 3.2$ | BaBar prd80-112001
|  | $49.7\pm 3.8\pm 6.7^{+1.2}_{-4.8}$ | Belle prd75-012006

The two-body branching fractions for $B\to K^{*}_{0}h$ can be extracted from
the quasi-two-body predictions of this work with the relation

$\displaystyle\Gamma(B\to K^{*}_{0}h\to K\pi h)=\Gamma(B\to
K^{*}_{0}h)\times{\mathcal{B}}(K^{*}_{0}\to K\pi).$ (15)

In Ref. prd67-054021 , a parameter $\eta$ was defined to measure the violation
of the factorization relation the Eq. (15) in the $D$ meson decays. For the
$B\to K_{0}^{*}(1430)h$ and $B\to K_{0}^{*}(1430)h\to K\pi h$ decays, we have

$\displaystyle\eta$ $\displaystyle=$ $\displaystyle\frac{\Gamma(B\to
K_{0}^{*}(1430)h\to K\pi h)}{\Gamma(B\to
K_{0}^{*}(1430)h)\times{\mathcal{B}}(K_{0}^{*}\to K\pi)}$ (16)
$\displaystyle\approx$ $\displaystyle\frac{m^{2}_{K_{0}^{*}(1430)}}{4\pi
m_{B}}\frac{\Gamma_{K_{0}^{*}(1430)}}{\hat{q}_{h}q_{0}}\int^{(m_{B}-m_{h})^{2}}_{(m_{K}+m_{\pi})^{2}}\frac{ds}{s}\frac{\lambda^{1/2}(m^{2}_{B},s,m^{2}_{h})\lambda^{1/2}(s,m^{2}_{K},m^{2}_{\pi})}{(s-m^{2}_{K_{0}^{*}(1430)})^{2}+(m_{K_{0}^{*}(1430)}\Gamma_{K^{*}_{0}}(s))^{2}},\;\;$

where $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$, the $\hat{q}_{h}$ is the
expression of Eq. (31) in the rest frame of $B$ meson and fixed at
$s=m^{2}_{K_{0}^{*}(1430)}$. With Eq. (16), we have $\eta=0.90$ for the decays
$B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}$, which means the violation of the
factorization relation is not large when neglecting the effect of the
invariant mass $s$ in the decay amplitudes of the quasi-two-body decays. In
order to check this conclusion, we calculate the decay $B^{+}\to
K_{0}^{*}(1430)^{0}\pi^{+}$ in the two-body framework of the PQCD approach,
and we have ${\mathcal{B}}(B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+})=35.2\times
10^{-6}$, which is about $96.2\%$ of the result in Table 4 extracted with Eq
(15), and ${\mathcal{A}}_{CP}(B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+})=-1.0\%$ is
consistent with the $-1.3\%$ in Table 2.

The comparison of the PQCD branching fractions with the experimental
measurements for the two-body decays $B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}$,
$B^{+}\to K_{0}^{*}(1430)^{+}\pi^{0}$ and $B^{0}\to
K_{0}^{*}(1430)^{+}\pi^{-}$ are shown in the Table 4, with the first error
added in quadrature from the errors in Table 2 and the second error comes from
the uncertainty of ${\mathcal{B}}(K^{*}_{0}(1430)\to K\pi)=0.93\pm 0.10$
PDG-2018  for these theoretical results. The branching fraction and direct
$CP$ asymmetry for $B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}$ in Review of Particle
Physics PDG-2018  averaged from the results in prd96-072001 ; prd78-012004 ;
prl96-251803  are $39^{+6}_{-5}\times 10^{-6}$ and $0.061\pm 0.032$,
respectively, which are consistent with the predictions $(36.6\pm 11.3\pm
3.9)\times 10^{-6}$ in Table 4 and $(-1.3\pm 0.5)\%$ in Table 2. Because of
the large uncertainty of the ${\mathcal{A}}_{CP}=0.26^{+0.18}_{-0.14}$ for
$B^{+}\to K_{0}^{*}(1430)^{+}\pi^{0}$ in PDG-2018 , we can not evaluate the
significance of the prediction $(1.5\pm 1.0)\%$, but our branching fraction
agrees very well with BaBar’s result in prd96-072001  for this decay mode. For
the decay $B^{0}\to K_{0}^{*}(1430)^{+}\pi^{-}$, one has two results as listed
in Table 4 from BaBar and Belle Collaborations, its average ${\mathcal{B}}$ is
presented to be $(33\pm 7)\times 10^{-6}$ in Review of Particle Physics
PDG-2018 , this value agrees well with the PQCD prediction $(33.4\pm 10.2\pm
3.6)\times 10^{-6}$. There is an upper limit of $2.2\times 10^{-6}$ for the
decay $B^{+}\to\bar{K}^{*}_{0}(1430)^{0}K^{+}$, which is below our
expectation. Our predictions in this work will be tested by future
experiments. In the very recent work, LHCb Collaboration presented the
branching fractions for the combined decays $B^{0}_{s}\to$
$($$\overline{K}$$)$ ${}^{0}\pi^{\pm}K^{\mp}$ as jhep1906-114

$\displaystyle{\mathcal{B}}(B^{0}_{s}\to
K^{*}_{0}(1430)^{\pm}K^{\mp}\to\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip
0.28453pt^{0}\pi^{\pm}K^{\mp})\\!=\\!(19.4\pm 1.4\pm 0.4\pm 15.6\pm 2.0\pm
0.3)\times 10^{-6},\quad\,$ (17)
$\displaystyle{\mathcal{B}}(B^{0}_{s}\to\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip
0.28453pt^{*}_{0}(1430)^{0}\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip
0.28453pt^{0}\to
K^{\mp}\pi^{\pm}\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip
0.28453pt^{0})\\!=\\!(20.5\pm 1.6\pm 0.6\pm 5.7\pm 2.2\pm 0.3)\times
10^{-6},\quad\;$ (18)

which are in agreement with the PQCD predictions in Table 5.

Table 5: PQCD predictions of the $CP$ averaged branching fractions and the direct $CP$ asymmetries for the combined decays $B^{0}_{s}\to K\pi K$, with the resonances $K^{*}_{0}(1430)$ and $K^{*}_{0}(1950)$ as the intermediate states. Decay modes |  |  Quasi-two-body results
---|---|---
$B^{0}_{s}\to K^{*}_{0}(1430)^{\pm}K^{\mp}\to\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip 0.28453pt^{0}\pi^{\pm}K^{\mp}$ | ${\mathcal{B}}(10^{-5})$ | $1.97\pm 0.45(\omega_{B})\pm 0.10(a_{3+1})\pm 0.43(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $-7.7\pm 1.5(\omega_{B})\pm 1.3(a_{3+1})\pm 4.3(m^{K}_{0}{+}a^{K}_{2})$
$B^{0}_{s}\to\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip 0.28453pt^{*}_{0}(1430)^{0}\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip 0.28453pt^{0}\to K^{\mp}\pi^{\pm}\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip 0.28453pt^{0}$ | ${\mathcal{B}}(10^{-5})$ | $1.50\pm 0.36(\omega_{B})\pm 0.09(a_{3+1})\pm 0.40(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ |  -
$B^{0}_{s}\to K^{*}_{0}(1950)^{\pm}K^{\mp}\to\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip 0.28453pt^{0}\pi^{\pm}K^{\mp}$ | ${\mathcal{B}}(10^{-6})$ | $3.20\pm 0.69(\omega_{B})\pm 0.21(a_{3+1})\pm 0.62(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ | $3.1\pm 0.2(\omega_{B})\pm 2.9(a_{3+1})\pm 1.6(m^{K}_{0}{+}a^{K}_{2})$
$B^{0}_{s}\to\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip 0.28453pt^{*}_{0}(1950)^{0}\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip 0.28453pt^{0}\to K^{\mp}\pi^{\pm}\text{{\raisebox{8.5pt}[0.0pt][0.0pt]{$($}}$\overline{K}${\raisebox{8.5pt}[0.0pt][0.0pt]{$)$}}}\hskip 0.28453pt^{0}$ | ${\mathcal{B}}(10^{-6})$ | $2.67\pm 0.59(\omega_{B})\pm 0.25(a_{3+1})\pm 0.61(m^{K}_{0}{+}a^{K}_{2})$
| ${\mathcal{A}}_{CP}(\%)$ |  -

On the experimental side, the LASS parametrization npb296-493 ; prd72-072003

$\displaystyle R(s)=\frac{\sqrt{s}}{q{\rm
cot}\delta_{B}-iq}+e^{2i\delta_{B}}\frac{m_{0}\Gamma_{0}\frac{m_{0}}{q_{0}}}{m^{2}_{0}-s-im_{0}\Gamma_{0}\frac{q}{m}\frac{m_{0}}{q_{0}}}\;,$
(19)

are employed in most cases to describe the $S$-wave $K\pi$ system, where
$m_{0}$ and $\Gamma_{0}$ are now the pole mass and full width for
$K^{*}_{0}(1430)$, and ${\rm cot}\delta_{B}=\frac{1}{aq}+\frac{1}{2}rq$ with
the parameters $a=2.07\pm 0.10$ GeV-1 and $r=3.32\pm 0.34$ GeV-1 prd72-072003
. The relativistic Breit-Wigner term of Eq. (19) is different from Eq. (6).
Before the $F_{K\pi}(s)$ in Eqs. (10)-(11) be replaced by the LASS expression,
a coefficient is needed for $R(s)$. We have the replacement

$\displaystyle
F_{K\pi}(s)\to\hat{R}(s)=\frac{q_{0}}{m^{2}_{0}\Gamma_{0}}g_{K_{0}^{*}(1430)K\pi}\bar{f}_{K^{*}_{0}(1430)}R(s)$
(20)

on the theoretical side. With $\hat{R}(s)$ in the concerned quasi-two-body
decay amplitudes, one could in principle have the predictions for the decays
$B\to(K\pi)^{*}_{0}h$, including the results as same as the values in the
Table 2 for the resonance $K^{*}_{0}(1430)$ and the contributions from the
nonresonant effective range term. But we argue that, considering the
nonresonant term of a three-body decay amplitude should not be included in the
resonance distribution amplitudes the Eq. (9), it’s improper for the effective
range term of the Eq. (19) to be studied in the quasi-two-body framework with
the same expressions of the decay amplitudes in Appendix A.

The two-body decays $B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}$, $B^{+}\to
K_{0}^{*}(1430)^{+}\pi^{0}$, $B^{0}\to K_{0}^{*}(1430)^{+}\pi^{-}$ and
$B^{0}\to K_{0}^{*}(1430)^{0}\pi^{0}$ have been studied in Ref. prd73-014017
and updated in prd87-114001  in the QCDF with $K_{0}^{*}(1430)$ being the
first excited states of $K_{0}^{*}(700)$ (scenario $1$) or the lowest lying
scalar state (scenario $2$), and in the scenario $2$ $K_{0}^{*}(700)$ is
treated as a four-quark state. In view of the discussions for $K_{0}^{*}(700)$
in prd15-267 ; prl48-659 ; prd41-2236 ; prl93-212002 ; pr389-61 ; pr409-1 ;
prd74-014028 , we will consider only the results for the $K_{0}^{*}(1430)$ in
the scenario $2$ in this work. The branching fractions in prd73-014017 ;
prd87-114001  for the four decays invloving the $K_{0}^{*}(1430)$ are all
smaller when comparing with the measurements and our results but with quite
large errors as shown in Table 6. The difference between our predictions and
the concerned results in prd73-014017 ; prd87-114001  may be partly due to the
dynamical enhancement of penguin contributions in PQCD approach as discussed
in detail in plb504-6 ; prd63-054008 ; prd74-094020 , and also due to the
small values for the decay constant $f_{K_{0}^{*}(1430)}$ and the $B\to
K_{0}^{*}(1430)$ and $B\to\pi$ transition form factors $F^{BK_{0}^{*}}_{0,1}$
and $F^{B\pi}_{0,1}$ in prd73-014017 ; prd87-114001 . The value
$f_{K_{0}^{*}(1430)}=34$ MeV prd73-014017  will make our branching ratios
involving $K_{0}^{*}(1430)$ $1.5$ times smaller than the results in Table 6.
With the parameters in this work, the form factors $F^{B\pi}_{0,1}(0)=0.26$
and $F^{BK_{0}^{*}(1430)}_{0,1}(0)=0.42$ could be induced in the PQCD
approach. The value $0.26$ is close to $0.25$ for $F^{B\pi}_{0,1}(0)$ in
prd73-014017 ; prd87-114001 , but the $0.42$ for $F^{BK_{0}^{*}}_{0,1}(0)$ is
two times larger than the value $0.21$ in Refs. prd73-014017 ; prd87-114001 .
In the PQCD approach, the two-body decays $B\to K^{*}_{0}(1430)\pi$ were
studied in epjc50-877 , with the branching fractions larger than the
corresponding results of this work except the decay $B^{0}\to
K_{0}^{*}(1430)^{0}\pi^{0}$ which is $18.4^{+4.4+1.5+4.0}_{-3.9-1.4-2.9}\times
10^{-6}$ in epjc50-877  as listed in Table 6. The result
$28.8^{+6.8+1.9+3.2}_{-6.1-1.9-3.5}\times 10^{-6}$ in epjc50-877  is about
double of our prediction and BaBar’s measurement prd96-072001  for the decay
$B^{+}\to K_{0}^{*}(1430)^{+}\pi^{0}$. The difference between the results in
epjc50-877  and our predictions could be be explained as the different input
parameters. The decays $B\to K^{*}_{0}(1430)K$ have been studied in the QCDF
in prd91-074022 . One can find the comparison of relevant branching fractions
in Table 7. The
${\mathcal{A}}_{CP}=-22.51^{+4.90+5.63+19.61}_{-7.57-9.36-22.86}\%$ for the
decay $B^{+}\to K_{0}^{*}(1430)^{+}\bar{K}^{0}$ in prd91-074022  is consistent
with the result $(-18.4\pm 5.8\pm 2.7\pm 5.4)\%$ in Table 2, while the
${\mathcal{A}}_{CP}=-2.60^{+1.61+0.59+3.52}_{-1.76-0.59-5.47}\%$ for
$B^{+}\to\bar{K}_{0}^{*}(1430)^{0}K^{+}$ in prd91-074022  is smaller than the
PQCD prediction $(17.9\pm 0.4\pm 8.0\pm 0.9)\%$ in this work and with an
opposite sign.

Table 6: Comparison of the extracted predictions with the results in literature for the relevant two-body branching fractions (in units of $10^{-6}$). The sources of the errors of our results are the same as in Table 4. Two-body decays |  This work | Theory | Ref.
---|---|---|---
$B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}$ | $36.6\pm 11.3\pm 3.9$ | $11.0^{+10.3+7.5+49.9}_{-6.0-3.5-10.1}$ | prd73-014017
|  | $12.9^{+4.6+4.1+38.5}_{-3.7-3.4-9.1}$ | prd87-114001
|  | $47.6^{+11.3+3.7+6.9}_{-10.1-3.6-5.1}$ | epjc50-877
$B^{+}\to K_{0}^{*}(1430)^{+}\pi^{0}$ | $12.7\pm 4.2\pm 1.4$ | $5.3^{+4.7+1.6+22.3}_{-2.8-1.7-4.7}$ | prd73-014017
|  | $7.4^{+2.4+2.1+20.1}_{-1.9-1.8-5.0}$ | prd87-114001
|  | $28.8^{+6.8+1.9+3.2}_{-6.1-1.9-3.5}$ | epjc50-877
$B^{0}\to K_{0}^{*}(1430)^{+}\pi^{-}$ | $33.4\pm 10.2\pm 3.6$ | $11.3^{+9.4+3.7+45.8}_{-5.8-3.7-9.9}$ | prd73-014017
|  | $13.8^{+4.5+4.1+38.3}_{-3.6-3.5-9.5}$ | prd87-114001
|  | $43.0^{+10.2+3.1+7.0}_{-9.1-2.9-5.2}$ | epjc50-877
$B^{0}\to K_{0}^{*}(1430)^{0}\pi^{0}$ | $22.4\pm 6.6\pm 2.4$ | $6.4^{+5.4+2.2+26.1}_{-3.3-2.1-5.7}$ | prd73-014017
|  | $5.6^{+2.6+2.4+18.8}_{-1.3-1.2-3.9}$ | prd87-114001
|  | $18.4^{+4.4+1.5+4.0}_{-3.9-1.4-2.9}$ | epjc50-877
Table 7: Comparison of the extracted predictions with the QCDF results in prd91-074022  for the relevant two-body branching fractions (in units of $10^{-7}$). The sources of the errors of our results are the same as in Table 4. Two-body decays |  This work | QCDF prd91-074022
---|---|---
$B^{+}\to K_{0}^{*}(1430)^{+}\bar{K}^{0}$ | $3.76\pm 2.16\pm 0.40$ | $1.14^{+0.54+1.40+1.17}_{-0.38-0.56-0.92}$
$B^{+}\to\bar{K}_{0}^{*}(1430)^{0}K^{+}$ | $39.9\pm 13.8\pm 4.3$ | $33.70^{+10.33+5.52+3.37}_{-8.47-4.82-3.94}$
$B^{0}\to K_{0}^{*}(1430)^{+}K^{-}$ | $0.93\pm 0.61\pm 0.10$ | $1.07^{+0.72+0.03+2.27}_{-0.47-0.04-0.97}$
$B^{0}\to K_{0}^{*}(1430)^{-}K^{+}$ | $6.19\pm 3.95\pm 0.67$ | $0.58^{+0.45+0.02+0.14}_{-0.29-0.03-0.05}$
$B^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{0}$ | $4.90\pm 3.34\pm 0.53$ | $2.39^{+1.20+1.95+2.67}_{-0.85-0.90-2.00}$
$B^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0}$ | $46.1\pm 15.0\pm 5.0$ | $40.47^{+13.36+6.09+6.06}_{-10.77-5.38-6.16}$

With $m_{K\pi}$ in the region $(0.64\\!\sim\\!1.76)$ GeV, the branching ratios
of the decay processes $B^{-}\to[\bar{K}^{*}_{0}(1430)^{0}\to
K^{-}\pi^{+}]\pi^{-}$ and
$\bar{B}^{0}\to[K^{*}_{0}(1430)^{-}\to\bar{K}^{0}\pi^{-}]\pi^{+}$ were
calculated in QCDF in Ref. prd79-094005  with the predictions $(11.6\pm
0.6)\times 10^{-6}$ and $(11.1\pm 0.5)\times 10^{-6}$, respectively. These two
decays have also been studied in Ref. prd81-094033  in the $m_{K\pi}$ region
$(1.0\sim 1.76)$ GeV and the branching ratios are $(12.11\pm 0.32)\times
10^{-6}$ and $(11.05\pm 0.25)\times 10^{-6}$, respectively. In the PQCD
approach we have $(16.6\pm 5.3)\times 10^{-6}$ and $(15.2\pm 4.7)\times
10^{-6}$ in the region $m_{K\pi}\\!\in\\!(0.64\sim 1.76)$ GeV, $(16.4\pm
5.1)\times 10^{-6}$ and $(15.0\pm 4.6)\times 10^{-6}$ in the region
$m_{K\pi}\\!\in\\!(1.0\sim 1.76)$ GeV for the branching ratios of the decays
$B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$ and $B^{0}\to
K_{0}^{*}(1430)^{+}\pi^{-}\to K^{0}\pi^{+}\pi^{-}$, respectively, which are
consistent with the results in Refs. prd79-094005 ; prd81-094033  within
errors. The three-body decays $B\to K\pi h$ have been discussed in detail in
Refs. prd88-114014 ; prd94-094015  in QCDF. The comparison of PQCD predictions
in this work with the related results in prd88-114014 ; prd94-094015  are
listed in Table 8. From Table 4 and Table 8, one can find that the PQCD
predictions are totally larger than the QCDF results prd88-114014 ;
prd94-094015  but closer to the available data.

Table 8: Comparison of the PQCD predictions with the theoretical results for the relevant quasi-two-body branching fractions (in units of $10^{-6}$). The errors of this work have been added in quadrature. Decay modes | This work | Theory |  Ref.
---|---|---|---
$B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}\to K^{+}\pi^{-}\pi^{+}$ | $22.7\pm 7.0$ | $11.3^{+0.0+3.3+0.1}_{-0.0-2.8-0.1}$ |  prd88-114014
|  | $11.5^{+0.0+3.3+0.0}_{-0.0-2.8-0.0}$ |  prd94-094015
$B^{+}\to K_{0}^{*}(1430)^{+}\pi^{0}\to K^{0}\pi^{+}\pi^{0}$ | $7.86\pm 2.61$ | $5.4^{+0.0+1.6+0.1}_{-0.0-1.4-0.1}$ |  prd88-114014
|  | $5.6^{+0.0+1.6+0.0}_{-0.0-1.4-0.0}$ |  prd94-094015
$B^{+}\to\bar{K}_{0}^{*}(1430)^{0}K^{+}\to K^{-}\pi^{+}K^{+}$ | $2.86\pm 0.85$ | $1.0^{+0.0+0.2+0.0}_{-0.0-0.2-0.0}$ |  prd88-114014
|  | $1.0^{+0.0+0.2+0.0}_{-0.0-0.2-0.0}$ |  prd94-094015
$B^{0}\to K_{0}^{*}(1430)^{+}\pi^{-}\to K^{0}\pi^{+}\pi^{-}$ | $20.7\pm 6.3$ | $10.3^{+0.0+2.9+0.0}_{-0.0-2.5-0.0}$ |  prd88-114014
|  | $10.6^{+0.0+3.0+0.0}_{-0.0-2.6-0.0}$ |  prd94-094015
$B^{0}\to K_{0}^{*}(1430)^{0}\pi^{0}\to K^{+}\pi^{-}\pi^{0}$ | $13.9\pm 4.1$ | $4.1^{+0.0+1.4+0.0}_{-0.0-1.2-0.0}$ |  prd88-114014
|  | $4.2^{+0.0+1.4+0.0}_{-0.0-1.2-0.0}$ |  prd94-094015

Figure 3: Differential direct $CP$ asymmetry for the decay $B_{s}^{0}\to
K_{0}^{*}(1430)^{-}K^{+}\to\bar{K}^{0}\pi^{-}K^{+}$.

There is no direct $CP$ asymmetries for $B^{0}_{(s)}\to K^{*0}_{0}\bar{K}^{0}$
and $B^{0}_{(s)}\to\bar{K}^{*0}_{0}K^{0}$ in Tables 2, 3, because these decays
have contributions only from the penguin operators in their decay amplitudes.
For the decays $B^{0}\to K_{0}^{*}(1430)^{+}\pi^{-}\to K^{0}\pi^{+}\pi^{-}$
and $B^{0}\to K_{0}^{*}(1430)^{0}\pi^{0}\to K^{+}\pi^{-}\pi^{0}$ via the $b\to
sq\bar{q}$ transition at quark level, the very small proportion of the total
branching ratio from the current-current operators led to the small direct
$CP$ asymmetries for these two decays as shown in Table 2. The same pattern
will appear again for the decays $B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}\to
K^{+}\pi^{-}\pi^{+}$ and $B^{+}\to K_{0}^{*}(1430)^{+}\pi^{0}\to
K^{0}\pi^{+}\pi^{0}$, and also for the corresponding decays with the
$K_{0}^{*}(1430)$ be replaced by the $K_{0}^{*}(1950)$ as the intermediate,
but not for the decays $B_{s}^{0}\to
K_{0}^{*}(1430)^{-}\pi^{+}\to\bar{K}^{0}\pi^{-}\pi^{+}$ and
$B_{s}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{0}\to K^{-}\pi^{+}\pi^{0}$ via the
$b\to dq\bar{q}$ transition. The interference between the weak and the strong
phases of the decay amplitudes from current-current and penguin operators
results in the large direct $CP$ asymmetries for the $B_{s}^{0}\to
K_{0}^{*}(1430)^{-}\pi^{+}\to\bar{K}^{0}\pi^{-}\pi^{+}$ and
$B_{s}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{0}\to K^{-}\pi^{+}\pi^{0}$ decays.
As an example, we display the differential distribution curve of the
${\mathcal{A}}_{CP}$ in $m_{K\pi}$ for the decay process $B_{s}^{0}\to
K_{0}^{*}(1430)^{-}K^{+}\to\bar{K}^{0}\pi^{-}K^{+}$ in Fig. 3.

For the decays $B^{+}\to K_{0}^{*}(1430)^{0}\pi^{+}$ and $B^{0}\to
K_{0}^{*}(1430)^{+}\pi^{-}$, with the isospin limit, one has the ratio
epjc50-877

$\displaystyle R=\frac{\tau_{B^{0}}}{\tau_{B^{+}}}\frac{{\mathcal{B}}(B^{+}\to
K_{0}^{*}(1430)^{0}\pi^{+})}{{\mathcal{B}}(B^{0}\to
K_{0}^{*}(1430)^{+}\pi^{-})}\approx 1\,.$ (21)

With the predictions in Table 6, we have the ratio $R=1.017\pm 0.003$ in this
work. The small error for $R$ is because the cancellation between the errors
of two branching ratios, which means the increase or the decrease of the
parameters that caused the errors will result in nearly identical change of
the weight for the numerator and denominator of $R$. For the decays $B^{+}\to
K_{0}^{*}(1430)^{+}\pi^{0}$ and $B^{0}\to K_{0}^{*}(1430)^{0}\pi^{0}$, the
diagrams of Fig. 1 (a), (c), (d) will contribute to the branching fractions,
the decay amplitudes from Fig. 1 (a) are same for both $B^{+}\to
K_{0}^{*}(1430)^{+}\pi^{0}$ and $B^{0}\to K_{0}^{*}(1430)^{0}\pi^{0}$, but the
decay amplitudes from Fig. 1 (c), (d) have the opposite sign considering the
difference for $\bar{u}u$ and $\bar{d}d$ to form a neutral pion. It is not
strange for the ratio between the branching fractions of $B^{+}\to
K_{0}^{*}(1430)^{+}\pi^{0}$ and $B^{0}\to K_{0}^{*}(1430)^{0}\pi^{0}$ away
from unity.

A relation for the direct $CP$ asymmetries of the two-body decays $B^{+}\to
K^{+}\pi^{0},\,B^{+}\to K^{0}\pi^{+},\,B^{0}\to K^{+}\pi^{-}$ and $B^{0}\to
K^{0}\pi^{0}$ was suggested in Ref. plb627-82  as

$\displaystyle{\mathcal{A}}_{CP}(B^{+}$ $\displaystyle\to$ $\displaystyle
K^{+}\pi^{0})\frac{2{\mathcal{B}}(B^{+}\to
K^{+}\pi^{0})}{{\mathcal{B}}(B^{0}\to
K^{+}\pi^{-})}\frac{\tau_{B^{0}}}{\tau_{B^{+}}}+{\mathcal{A}}_{CP}(B^{0}\to
K^{0}\pi^{0})\frac{2{\mathcal{B}}(B^{0}\to
K^{0}\pi^{0})}{{\mathcal{B}}(B^{0}\to K^{+}\pi^{-})}$ (22) $\displaystyle=$
$\displaystyle{\mathcal{A}}_{CP}(B^{0}\to
K^{+}\pi^{-})+{\mathcal{A}}_{CP}(B^{+}\to
K^{0}\pi^{+})\frac{{\mathcal{B}}(B^{+}\to
K^{0}\pi^{+})}{{\mathcal{B}}(B^{0}\to
K^{+}\pi^{-})}\frac{\tau_{B^{0}}}{\tau_{B^{+}}}.\;$

Considering the same transitions at quark level, one could extend the Eq. (22)
to the $B\to K^{*}_{0}(1430)\pi$ decays with the replacement $K\to
K^{*}_{0}(1430)$. This relation is satisfied within errors with the
${\mathcal{A}}_{CP}(B^{0}\to K^{*}_{0}(1430)^{+}\pi^{-})=(0.3\pm 0.9)\%$,
${\mathcal{A}}_{CP}(B^{+}\to K^{*}_{0}(1430)^{0}\pi^{+})=(-1.3\pm 0.5)\%$,
${\mathcal{A}}_{CP}(B^{+}\to K^{*}_{0}(1430)^{+}\pi^{0})=(1.5\pm 1.0)\%$ and
${\mathcal{A}}_{CP}(B^{0}\to K^{*}_{0}(1430)^{0}\pi^{0})=(-1.8\pm 0.5)\%$, and
relevant branching fractions in Table 2. One can find that the relation Eq.
(22) will also hold for $B\to K^{*}_{0}(1950)\pi$ decays with the values in
Table 3.

##  IV Conclusion

In this work, we studied the contributions from the resonant state
$K_{0}^{*}(1430)$ and, for the first time, from the resonance
$K_{0}^{*}(1950)$ in the three-body decays $B\to K\pi h$ in the PQCD approach.
The crucial nonperturbative input factor $F_{K\pi}(s)$ in the distribution
amplitudes of the $S$-wave $K\pi$ system was derived from the matrix element
of the vacuum to $K\pi$ final state and was related to the scalar time-like
form factor $F_{0}^{K\pi}(s)$ by the relation
$F_{K\pi}(s)={B_{0}}/{m_{K_{0}^{*}}}F_{0}^{K\pi}(s)$. This relation also means
that the LASS parametrization for the $(K\pi)^{*}_{0}$ system which frequently
appeared in the experimental works cannot be adopted directly for the $K\pi$
system distribution amplitudes in the PQCD approach.

With $f_{K_{0}^{*}(1430)}m^{2}_{K_{0}^{*}(1430)}=0.0842\pm 0.0045$ GeV3 and
$f_{K_{0}^{*}(1950)}m^{2}_{K_{0}^{*}(1950)}=0.0414$ GeV3, the branching
fractions and the direct $CP$ asymmetries for the concerned quasi-two-body
decays $B\to K^{*}_{0}(1430,1950)h\to K\pi h$ were calculated. An important
conclusion is that the $CP$ averaged branching fraction of a quasi-two-body
process with $K_{0}^{*}(1950)$ as the intermediate state is about one order
smaller than the corresponding decay mode involving the resonance
$K_{0}^{*}(1430)$. In view of the important contribution from the $S$-wave
$K\pi$ system for the $B\to K\pi h$ decays, it is not appropriate to neglect
the $K_{0}^{*}(1950)$ in the theoretical or experimental studies for the
relevant three-body $B$ meson decays. We compared our predictions with the
related results in literature and found the predictions in this work for the
relevant decays agree well with the existing experimental results from BaBar,
Belle and LHCb Collaborations.

###### Acknowledgements.

This work was supported in part by the National Natural Science Foundation of
China under Grants No. 11547038, No. 11575110 and No. 11947011; by the Natural
Science Foundation of Jiangsu Province under Grant No. BK20191010.

##  Appendix A Decay amplitudes

The Lorentz invariant decay amplitude ${\mathcal{A}}$ for the quasi-two-body
decay $B\to K^{*}_{0}h\to K\pi h$ in the PQCD approach, according to Fig. 1,
is given by plb561-258 ; plb763-29

$\displaystyle{\mathcal{A}}=\Phi_{B}\otimes
H\otimes\Phi_{h}\otimes\Phi_{K\pi}\;.$ (23)

The symbol $\otimes$ here means convolutions in parton momenta, the hard
kernel $H$ contains one hard gluon exchange at the leading order in strong
coupling $\alpha_{s}$ as in the two-body formalism. The distribution
amplitudes $\Phi_{B},\Phi_{h}$ and $\Phi_{K\pi}$ absorb the nonperturbative
dynamics in the relevant decay processes.

The $B$ meson light-cone matrix element can be decomposed as npb592-3 ;
epjc28-515 ; prd76-074018

$\displaystyle\Phi_{B}=\frac{i}{\sqrt{2N_{c}}}(p{/}_{B}+m_{B})\gamma_{5}\phi_{B}(k_{B}),$
(24)

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

$\displaystyle\phi_{B}(x_{B},b_{B})=N_{B}x_{B}^{2}(1-x_{B})^{2}\mathrm{exp}\left[-\frac{(x_{B}m_{B})^{2}}{2\omega_{B}^{2}}-\frac{1}{2}(\omega_{B}b_{B})^{2}\right],$
(25)

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

The light-cone wave functions for pion and kaon are written as jhep9809-005 ;
jhep9901-010 ; prd71-014015 ; jhep0605-004

$\displaystyle\Phi_{h}=\frac{i}{\sqrt{2N_{c}}}\gamma_{5}\left[p{/}_{3}\phi^{A}(x_{3})+m^{h}_{0}\phi^{P}(x_{3})+m^{h}_{0}(n/v/-1)\phi^{T}(x_{3})\right].$
(26)

The distribution amplitudes of $\phi^{A}(x_{3}),\phi^{P}(x_{3})$ and
$\phi^{T}(x_{3})$ are

$\displaystyle\phi^{A}(x_{3})$ $\displaystyle=$
$\displaystyle\frac{f_{h}}{2\sqrt{2N_{c}}}6x_{3}(1-x_{3})\left[1+a_{1}^{h}C_{1}^{3/2}(t)+a_{2}^{h}C_{2}^{3/2}(t)+a_{4}^{h}C_{4}^{3/2}(t)\right],$
(27) $\displaystyle\phi^{P}(x_{3})$ $\displaystyle=$
$\displaystyle\frac{f_{h}}{2\sqrt{2N_{c}}}\left[1+(30\eta_{3}-\frac{5}{2}\rho^{2}_{h})C_{2}^{1/2}(t)-3\big{[}\eta_{3}\omega_{3}+\frac{9}{20}\rho^{2}_{h}(1+6a_{2}^{h})\big{]}C_{4}^{1/2}(t)\right],$
(28) $\displaystyle\phi^{T}(x_{3})$ $\displaystyle=$
$\displaystyle\frac{f_{h}}{2\sqrt{2N_{c}}}(-t)\left[1+6\left(5\eta_{3}-\frac{1}{2}\eta_{3}\omega_{3}-\frac{7}{20}\rho^{2}_{h}-\frac{3}{5}\rho^{2}_{h}a_{2}^{h}\right)(1-10x_{3}+10x_{3}^{2})\right],\quad\;$
(29)

with $t=2x_{3}-1$, $C^{1/2}_{2,4}(t)$ and $C^{3/2}_{1,2,4}(t)$ are Gegenbauer
polynomials. The chiral masses $m^{h}_{0}$ for pion and kaon are
$m_{0}^{\pi}=(1.4\pm 0.1)$ GeV and $m_{0}^{K}=(1.6\pm 0.1)$ GeV as they in
Ref. prd86-114025 . The Gegenbauer moments
$a_{1}^{\pi}=0,a_{1}^{K}=0.06,a_{2}^{h}=0.25,a_{4}^{h}=-0.015$ and the
parameters $\rho_{h}=m_{h}/m_{0}^{h},\eta_{3}=0.015,\omega_{3}=-3$ are adopted
in the numerical calculation.

For the the differential branching fraction, we have PDG-2018

$\displaystyle\frac{d\mathcal{B}}{d\zeta}=\tau_{B}\frac{q_{h}q}{64\pi^{3}m_{B}}\overline{|{\cal
A}|^{2}},$ (30)

The magnitude momentum for the bachelor $h$ is

$\displaystyle
q_{h}=\frac{1}{2}\sqrt{\big{[}\left(m^{2}_{B}-m_{h}^{2}\right)^{2}-2\left(m^{2}_{B}+m_{h}^{2}\right)s+s^{2}\big{]}/s}$
(31)

in the center-of-mass frame of the $K^{*}_{0}$, where $m_{h}$ is the mass of
the bachelor state. The direct $CP$ asymmetry ${\mathcal{A}}_{CP}$ is defined
as

$\displaystyle{\mathcal{A}}_{CP}=\frac{{\mathcal{B}}(\bar{B}\to\bar{f})-{\mathcal{B}}(B\to
f)}{{\mathcal{B}}(\bar{B}\to\bar{f})+{\mathcal{B}}(B\to f)}$ (32)

For errors of the $\mathcal{B}$ and ${\mathcal{A}}_{CP}$ induced by the
parameter ${\mathcal{P}}\pm\Delta{\mathcal{P}}$ in this work, we employ the
formulas

$\displaystyle\Delta\mathcal{B}=\left|\frac{\partial\mathcal{B}}{\partial{\mathcal{P}}}\right|\Delta{\mathcal{P}},\qquad\Delta{\mathcal{A}}_{CP}=\left|\frac{\partial\mathcal{A}_{CP}}{\partial{\mathcal{P}}}\right|\Delta{\mathcal{P}}=\frac{2(\mathcal{B}\Delta\overline{\mathcal{B}}-\overline{\mathcal{B}}\Delta\mathcal{B})}{(\overline{\mathcal{B}}+\mathcal{B})^{2}}.$
(33)

With the subprocesses $K^{*+}_{0}\to\\{K^{0}\pi^{+},\sqrt{2}K^{+}\pi^{0}\\}$,
$K^{*0}_{0}\to\\{K^{+}\pi^{-},\sqrt{2}K^{0}\pi^{0}\\}$,
$K^{*-}_{0}\to\\{\bar{K}^{0}\pi^{-},\sqrt{2}K^{-}\pi^{0}\\}$ and
$\bar{K}^{*0}_{0}\to\\{K^{-}\pi^{+},\sqrt{2}\bar{K}^{0}\pi^{0}\\}$, and the
$K^{*}_{0}$ is $K^{*}_{0}(1430)$ or $K^{*}_{0}(1950)$, the concerned quasi-
two-body decay amplitudes are given as follows:

$\displaystyle{\cal A}\left(B^{+}\to K^{*0}_{0}\pi^{+}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{ub}^{*}V_{us}[a_{1}F^{LL}_{Ah}+C_{1}M^{LL}_{Ah}]-V_{tb}^{*}V_{ts}[(a_{4}-\frac{a_{10}}{2})F^{LL}_{Th}+(a_{6}-\frac{a_{8}}{2})F^{SP}_{Th}$
(34) $\displaystyle+$
$\displaystyle(C_{3}-\frac{C_{9}}{2})M^{LL}_{Th}+(C_{5}-\frac{C_{7}}{2})M^{LR}_{Th}+(a_{4}+a_{10})F^{LL}_{Ah}+(a_{6}+a_{8})F^{SP}_{Ah}$
$\displaystyle+$
$\displaystyle(C_{3}+C_{9})M^{LL}_{Ah}+(C_{5}+C_{7})M^{LR}_{Ah}]\big{\\}},$
$\displaystyle{\cal A}\left(B^{+}\to K^{*+}_{0}\pi^{0}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{2}\big{\\{}V_{ub}^{*}V_{us}[a_{2}F^{LL}_{TK^{*}_{0}}+C_{2}M^{LL}_{TK^{*}_{0}}+a_{1}(F^{LL}_{Th}+F^{LL}_{Ah})$
(35) $\displaystyle+$ $\displaystyle
C_{1}(M^{LL}_{Th}+M^{LL}_{Ah})]-V_{tb}^{*}V_{ts}[(\frac{3}{2}(a_{9}-a_{7})F^{LL}_{TK^{*}_{0}}+\frac{3C_{10}}{2}M^{LL}_{TK^{*}_{0}}$
$\displaystyle+$
$\displaystyle\frac{3C_{8}}{2}M^{SP}_{TK^{*}_{0}}+(a_{4}+a_{10})(F^{LL}_{Th}+F^{LL}_{Ah})+(a_{6}+a_{8})(F^{SP}_{Th}+F^{SP}_{Ah})$
$\displaystyle+$
$\displaystyle(C_{3}+C_{9})(M^{LL}_{Th}+M^{LL}_{Ah})+(C_{5}+C_{7})(M^{LR}_{Th}+M^{LR}_{Ah})]\big{\\}},$
$\displaystyle{\cal A}\left(B^{+}\to K^{*+}_{0}\bar{K}^{0}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{ub}^{*}V_{ud}[a_{1}F^{LL}_{AK^{*}_{0}}+C_{1}M^{LL}_{AK^{*}_{0}}]-V_{tb}^{*}V_{td}[(a_{4}-\frac{a_{10}}{2})F^{LL}_{TK^{*}_{0}}$
(36) $\displaystyle+$
$\displaystyle(a_{6}-\frac{a_{8}}{2})F^{SP}_{TK^{*}_{0}}+(C_{3}-\frac{C_{9}}{2})M^{LL}_{TK^{*}_{0}}+(C_{5}-\frac{C_{7}}{2})M^{LR}_{TK^{*}_{0}}$
$\displaystyle+$
$\displaystyle(a_{4}+a_{10})F^{LL}_{AK^{*}_{0}}+(a_{6}+a_{8})F^{SP}_{AK^{*}_{0}}+(C_{3}+C_{9})M^{LL}_{AK^{*}_{0}}$
$\displaystyle+$ $\displaystyle(C_{5}+C_{7})M^{LR}_{AK^{*}_{0}}]\big{\\}},$
$\displaystyle{\cal A}\left(B^{+}\to\bar{K}^{*0}_{0}K^{+}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{ub}^{*}V_{ud}[a_{1}F^{LL}_{Ah}+C_{1}M^{LL}_{Ah}]-V_{tb}^{*}V_{td}[(a_{4}-\frac{a_{10}}{2})F^{LL}_{Th}$
(37) $\displaystyle+$
$\displaystyle(a_{6}-\frac{a_{8}}{2})F^{SP}_{Th}+(C_{3}-\frac{C_{9}}{2})M^{LL}_{Th}+(C_{5}-\frac{C_{7}}{2})M^{LR}_{Th}$
$\displaystyle+$
$\displaystyle(a_{4}+a_{10})F^{LL}_{Ah}+(a_{6}+a_{8})F^{SP}_{Ah}+(C_{3}+C_{9})M^{LL}_{Ah}$
$\displaystyle+$ $\displaystyle(C_{5}+C_{7})M^{LR}_{Ah}]\big{\\}},$
$\displaystyle{\cal A}\left(B^{0}\to K^{*+}_{0}\pi^{-}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{ub}^{*}V_{us}[a_{1}F^{LL}_{Th}+C_{1}M^{LL}_{Th}]-V_{tb}^{*}V_{ts}[(a_{4}+a_{10})F^{LL}_{Th}$
(38) $\displaystyle+$
$\displaystyle(a_{6}+a_{8})F^{SP}_{Th}+(C_{3}+C_{9})M^{LL}_{Th}+(C_{5}+C_{7})M^{LR}_{Th}$
$\displaystyle+$
$\displaystyle(a_{4}-\frac{a_{10}}{2})F^{LL}_{Ah}+(a_{6}-\frac{a_{8}}{2})F^{SP}_{Ah}+(C_{3}-\frac{C_{9}}{2})M^{LL}_{Ah}$
$\displaystyle+$ $\displaystyle(C_{5}-\frac{C_{7}}{2})M^{LR}_{Ah}]\big{\\}},$
$\displaystyle{\cal A}\left(B^{0}\to K^{*0}_{0}\pi^{0}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{2}\big{\\{}V_{ub}^{*}V_{us}[a_{2}F^{LL}_{TK^{*}_{0}}+C_{2}M^{LL}_{TK^{*}_{0}}]-V_{tb}^{*}V_{ts}[(\frac{3}{2}(a_{9}-a_{7})F^{LL}_{TK^{*}_{0}}$
(39) $\displaystyle+$
$\displaystyle\frac{3C_{10}}{2}M^{LL}_{TK^{*}_{0}}+\frac{3C_{8}}{2}M^{SP}_{TK^{*}_{0}}-(a_{4}-\frac{a_{10}}{2})(F^{LL}_{Th}+F^{LL}_{Ah})$
$\displaystyle-$
$\displaystyle(a_{6}-\frac{a_{8}}{2})(F^{SP}_{Th}+F^{SP}_{Ah})-(C_{3}-\frac{C_{9}}{2})(M^{LL}_{Th}+M^{LL}_{Ah})$
$\displaystyle-$
$\displaystyle(C_{5}-\frac{C_{7}}{2})(M^{LR}_{Th}+M^{LR}_{Ah})]\big{\\}},$
$\displaystyle{\cal A}\left(B^{0}\to K^{*+}_{0}K^{-}\right)$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{ub}^{*}V_{ud}[a_{2}F^{LL}_{AK^{*}_{0}}+C_{2}M^{LL}_{AK^{*}_{0}}]-V_{tb}^{*}V_{td}[(a_{3}+a_{9}-a_{5}-a_{7})F^{LL}_{AK^{*}_{0}}$
(40) $\displaystyle+$
$\displaystyle(C_{4}+C_{10})M^{LL}_{AK^{*}_{0}}+(C_{6}+C_{8})M^{SP}_{AK^{*}_{0}}+(a_{3}-\frac{a_{9}}{2}-a_{5}+\frac{a_{7}}{2})F^{LL}_{Ah}$
$\displaystyle+$
$\displaystyle(C_{4}-\frac{C_{10}}{2})M^{LL}_{Ah}+(C_{6}-\frac{C_{8}}{2})M^{SP}_{Ah}]\big{\\}},$
$\displaystyle{\cal A}\left(B^{0}\to K^{*-}_{0}K^{+}\right)$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{ub}^{*}V_{ud}[a_{2}F^{LL}_{Ah}+C_{2}M^{LL}_{Ah}]-V_{tb}^{*}V_{td}[(a_{3}+a_{9}-a_{5}-a_{7})F^{LL}_{Ah}$
(41) $\displaystyle+$
$\displaystyle(C_{4}+C_{10})M^{LL}_{Ah}+(C_{6}+C_{8})M^{SP}_{Ah}+(a_{3}-\frac{a_{9}}{2}-a_{5}+\frac{a_{7}}{2})F^{LL}_{AK^{*}_{0}}$
$\displaystyle+$
$\displaystyle(C_{4}-\frac{C_{10}}{2})M^{LL}_{AK^{*}_{0}}+(C_{6}-\frac{C_{8}}{2})M^{SP}_{AK^{*}_{0}}]\big{\\}},$
$\displaystyle{\cal A}\left(B^{0}\to K^{*0}_{0}\bar{K}^{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{tb}^{*}V_{td}[(a_{4}-\frac{a_{10}}{2})F^{LL}_{TK^{*}_{0}}+(a_{6}-\frac{a_{8}}{2})(F^{SP}_{TK^{*}_{0}}+F^{SP}_{AK^{*}_{0}})$
(42) $\displaystyle+$
$\displaystyle(C_{3}-\frac{C_{9}}{2})M^{LL}_{TK^{*}_{0}}+(C_{5}-\frac{C_{7}}{2})(M^{LR}_{TK^{*}_{0}}+M^{LR}_{AK^{*}_{0}})+(\frac{4}{3}(C_{3}+C_{4}$
$\displaystyle-$
$\displaystyle\frac{C_{9}+C_{10}}{2})-a_{5}+\frac{a_{7}}{2})F^{LL}_{AK^{*}_{0}}+(C_{3}+C_{4}-\frac{C_{9}+C_{10}}{2})M^{LL}_{AK^{*}_{0}}$
$\displaystyle+$
$\displaystyle(C_{6}-\frac{C_{8}}{2})(M^{SP}_{AK^{*}_{0}}+M^{SP}_{Ah})+(a_{3}-\frac{a_{9}}{2}-a_{5}+\frac{a_{7}}{2})F^{LL}_{Ah}$
$\displaystyle+$ $\displaystyle(C_{4}-\frac{C_{10}}{2})M^{LL}_{Ah}]\big{\\}},$
$\displaystyle{\cal A}\left(B^{0}\to\bar{K}^{*0}_{0}K^{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{tb}^{*}V_{td}[(a_{4}-\frac{a_{10}}{2})F^{LL}_{Th}+(a_{6}-\frac{a_{8}}{2})(F^{SP}_{Th}+F^{SP}_{Ah})$
(43) $\displaystyle+$
$\displaystyle(C_{3}-\frac{C_{9}}{2})M^{LL}_{Th}+(C_{5}-\frac{C_{7}}{2})(M^{LR}_{Th}+M^{LR}_{Ah})+(\frac{4}{3}(C_{3}+C_{4}$
$\displaystyle-$
$\displaystyle\frac{C_{9}+C_{10}}{2})-a_{5}+\frac{a_{7}}{2})F^{LL}_{Ah}+(C_{3}+C_{4}-\frac{C_{9}+C_{10}}{2})M^{LL}_{Ah}$
$\displaystyle+$
$\displaystyle(C_{6}-\frac{C_{8}}{2})(M^{SP}_{Ah}+M^{SP}_{AK^{*}_{0}})+(a_{3}-\frac{a_{9}}{2}-a_{5}+\frac{a_{7}}{2})F^{LL}_{AK^{*}_{0}}$
$\displaystyle+$
$\displaystyle(C_{4}-\frac{C_{10}}{2})M^{LL}_{AK^{*}_{0}}]\big{\\}},$
$\displaystyle{\cal A}\left(B_{s}^{0}\to K^{*-}_{0}\pi^{+}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{ub}^{*}V_{ud}[a_{1}F^{LL}_{TK^{*}_{0}}+C_{1}M^{LL}_{TK^{*}_{0}}]-V_{tb}^{*}V_{td}[(a_{4}+a_{10})F^{LL}_{TK^{*}_{0}}$
(44) $\displaystyle+$
$\displaystyle(a_{6}+a_{8})F^{SP}_{TK^{*}_{0}}+(C_{3}+C_{9})M^{LL}_{TK^{*}_{0}}+(C_{5}+C_{7})M^{LR}_{TK^{*}_{0}}$
$\displaystyle+$
$\displaystyle(a_{4}-\frac{a_{10}}{2})F^{LL}_{AK^{*}_{0}}+(a_{6}-\frac{a_{8}}{2})F^{SP}_{AK^{*}_{0}}+(C_{3}-\frac{C_{9}}{2})M^{LL}_{AK^{*}_{0}}$
$\displaystyle+$
$\displaystyle(C_{5}-\frac{C_{7}}{2})M^{LR}_{AK^{*}_{0}}]\big{\\}},$
$\displaystyle{\cal A}\left(B_{s}^{0}\to\bar{K}^{*0}_{0}\pi^{0}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{2}\big{\\{}V_{ub}^{*}V_{ud}[a_{2}F^{LL}_{TK^{*}_{0}}+C_{2}M^{LL}_{TK^{*}_{0}}]-V_{tb}^{*}V_{td}[(-a_{4}-\frac{3a_{7}}{2}$
(45) $\displaystyle+$
$\displaystyle\frac{5C_{9}}{3}+C_{10})F^{LL}_{TK^{*}_{0}}-(a_{6}-\frac{a_{8}}{2})F^{SP}_{TK^{*}_{0}}+(-C_{3}+\frac{3a_{10}}{2})M^{LL}_{TK^{*}_{0}}$
$\displaystyle-$
$\displaystyle(C_{5}-\frac{C_{7}}{2})M^{LR}_{TK^{*}_{0}}+\frac{3C_{8}}{2}M^{SP}_{TK^{*}_{0}}-(a_{4}-\frac{a_{10}}{2})F^{LL}_{AK^{*}_{0}}-(a_{6}$
$\displaystyle-$
$\displaystyle\frac{a_{8}}{2})F^{SP}_{AK^{*}_{0}}-(C_{3}-\frac{C_{9}}{2})M^{LL}_{AK^{*}_{0}}-(C_{5}-\frac{C_{7}}{2})M^{LR}_{AK^{*}_{0}}]\big{\\}},$
$\displaystyle{\cal A}\left(B_{s}^{0}\to K^{*+}_{0}K^{-}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{ub}^{*}V_{us}[a_{1}F^{LL}_{Th}+C_{1}M^{LL}_{Th}+a_{2}F^{LL}_{AK^{*}_{0}}+C_{2}M^{LL}_{AK^{*}_{0}}]$
(46) $\displaystyle-$ $\displaystyle
V_{tb}^{*}V_{ts}[(a_{4}+a_{10})F^{LL}_{Th}+(a_{6}+a_{8})F^{SP}_{Th}+(C_{3}+C_{9})M^{LL}_{Th}$
$\displaystyle+$
$\displaystyle(C_{5}+C_{7})M^{LR}_{Th}+(\frac{4}{3}(C_{3}+C_{4}-\frac{C_{9}+C_{10}}{2})-a_{5}+\frac{a_{7}}{2})F^{LL}_{Ah}$
$\displaystyle+$
$\displaystyle(a_{6}-\frac{a_{8}}{2})F^{SP}_{Ah}+(C_{3}+C_{4}-\frac{C_{9}+C_{10}}{2})M^{LL}_{Ah}+(C_{5}-\frac{C_{7}}{2})M^{LR}_{Ah}$
$\displaystyle+$
$\displaystyle(C_{6}-\frac{C_{8}}{2})M^{SP}_{Ah}+(a_{3}+a_{9}-a_{5}-a_{7})F^{LL}_{AK^{*}_{0}}+(C_{4}+C_{10})M^{LL}_{AK^{*}_{0}}$
$\displaystyle+$ $\displaystyle(C_{6}+C_{8})M^{SP}_{AK^{*}_{0}}]\big{\\}},$
$\displaystyle{\cal A}\left(B_{s}^{0}\to K^{*-}_{0}K^{+}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{ub}^{*}V_{us}[a_{1}F^{LL}_{TK^{*}_{0}}+C_{1}M^{LL}_{TK^{*}_{0}}+a_{2}F^{LL}_{Ah}+C_{2}M^{LL}_{Ah}]$
(47) $\displaystyle-$ $\displaystyle
V_{tb}^{*}V_{ts}[(a_{4}+a_{10})F^{LL}_{TK^{*}_{0}}+(a_{6}+a_{8})F^{SP}_{TK^{*}_{0}}+(C_{3}+C_{9})M^{LL}_{TK^{*}_{0}}$
$\displaystyle+$
$\displaystyle(C_{5}+C_{7})M^{LR}_{TK^{*}_{0}}+(\frac{4}{3}(C_{3}+C_{4}-\frac{C_{9}+C_{10}}{2})-a_{5}+\frac{a_{7}}{2})F^{LL}_{AK^{*}_{0}}$
$\displaystyle+$
$\displaystyle(a_{6}-\frac{a_{8}}{2})F^{SP}_{AK^{*}_{0}}+(C_{3}+C_{4}-\frac{C_{9}+C_{10}}{2})M^{LL}_{AK^{*}_{0}}+(C_{5}-\frac{C_{7}}{2})M^{LR}_{AK^{*}_{0}}$
$\displaystyle+$
$\displaystyle(C_{6}-\frac{C_{8}}{2})M^{SP}_{AK^{*}_{0}}+(a_{3}+a_{9}-a_{5}-a_{7})F^{LL}_{Ah}+(C_{4}+C_{10})M^{LL}_{Ah}$
$\displaystyle+$ $\displaystyle(C_{6}+C_{8})M^{SP}_{Ah}]\big{\\}},$
$\displaystyle{\cal A}\left(B_{s}^{0}\to K^{*0}_{0}\bar{K}^{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{tb}^{*}V_{ts}[(a_{4}-\frac{a_{10}}{2})F^{LL}_{Th}+(a_{6}-\frac{a_{8}}{2})(F^{SP}_{Th}+F^{SP}_{Ah})$
(48) $\displaystyle+$
$\displaystyle(C_{3}-\frac{C_{9}}{2})M^{LL}_{Th}+(C_{5}-\frac{C_{7}}{2})(M^{LR}_{Th}+M^{LR}_{Ah})+(\frac{4}{3}(C_{3}+C_{4}$
$\displaystyle-$
$\displaystyle\frac{C_{9}+C_{10}}{2})-a_{5}+\frac{a_{7}}{2})F^{LL}_{Ah}+(C_{3}+C_{4}-\frac{C_{9}+C_{10}}{2})M^{LL}_{Ah}$
$\displaystyle+$
$\displaystyle(C_{6}-\frac{C_{8}}{2})(M^{SP}_{Ah}+M^{SP}_{AK^{*}_{0}})+(a_{3}-\frac{a_{9}}{2}-a_{5}+\frac{a_{7}}{2})F^{LL}_{AK^{*}_{0}}$
$\displaystyle+$
$\displaystyle(C_{4}-\frac{C_{10}}{2})M^{LL}_{AK^{*}_{0}}]\big{\\}},$
$\displaystyle{\cal A}\left(B_{s}^{0}\to\bar{K}^{*0}_{0}K^{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}\big{\\{}V_{tb}^{*}V_{ts}[(a_{4}-\frac{a_{10}}{2})F^{LL}_{TK^{*}_{0}}+(a_{6}-\frac{a_{8}}{2})(F^{SP}_{TK^{*}_{0}}+F^{SP}_{AK^{*}_{0}})$
(49) $\displaystyle+$
$\displaystyle(C_{3}-\frac{C_{9}}{2})M^{LL}_{TK^{*}_{0}}+(C_{5}-\frac{C_{7}}{2})(M^{LR}_{TK^{*}_{0}}+M^{LR}_{AK^{*}_{0}})+(\frac{4}{3}(C_{3}+C_{4}$
$\displaystyle-$
$\displaystyle\frac{C_{9}+C_{10}}{2})-a_{5}+\frac{a_{7}}{2})F^{LL}_{AK^{*}_{0}}+(C_{3}+C_{4}-\frac{C_{9}+C_{10}}{2})M^{LL}_{AK^{*}_{0}}$
$\displaystyle+$
$\displaystyle(C_{6}-\frac{C_{8}}{2})(M^{SP}_{AK^{*}_{0}}+M^{SP}_{Ah})+(a_{3}-\frac{a_{9}}{2}-a_{5}+\frac{a_{7}}{2})F^{LL}_{Ah}$
$\displaystyle+$ $\displaystyle(C_{4}-\frac{C_{10}}{2})M^{LL}_{Ah}]\big{\\}},$

in which $G_{F}$ is the Fermi coupling constant, $V$’s are the CKM matrix
elements. The combinations $a_{i}$ of the Wilson coefficients are defined as

$\displaystyle
a_{1}=C_{2}+\frac{C_{1}}{3},\;a_{2}=C_{1}+\frac{C_{2}}{3},\;a_{3}=C_{3}+\frac{C_{4}}{3},\;a_{4}=C_{4}+\frac{C_{3}}{3},\;\;\,a_{5}\,=C_{5}+\frac{C_{6}}{3},\;$
(50) $\displaystyle
a_{6}=C_{6}+\frac{C_{5}}{3},\;a_{7}=C_{7}+\frac{C_{8}}{3},\;a_{8}=C_{8}+\frac{C_{7}}{3},\;a_{9}=C_{9}+\frac{C_{10}}{3},\;a_{10}=C_{10}+\frac{C_{9}}{3}.$
(51)

It should be understood that the Wilson coefficients $C$ and the amplitudes
$F$ and $M$ for the factorizable and nonfactorizable contributions,
respectively, appear in convolutions in momentum fractions and impact
parameters $b$.

The general amplitudes for the decays $B\to K^{*}_{0}h\to K\pi h$ in the decay
amplitudes Eq. (34)-Eq. (49) are given according to the Fig. 1, the typical
Feynman diagrams in the PQCD approach. In the following expressions, we will
employ $LL$ and $LR$ to denote the contributions from $(V-A)(V-A)$ and
$(V-A)(V+A)$ operators, respectively. For the contribution from $(S-P)(S+P)$
operators which come from the Fierz transformation of the $(V-A)(V+A)$
operators, we will use $SP$ to denote it. The emission diagrams are depicted
in Fig. 1 (a) and (c) with $B\to K^{*}_{0}$ and $B\to h$ transitions, and
described as the subscripts $TK^{*}_{0}$ and $Th$ in their amplitudes,
respectively. The factorizable and nonfactorizable diagrams have been merged
in Fig. 1, which could be distinguished easily from the attachments of the
hard gluons. Those diagrams with two attachments of the hard gluon passed the
weak vertex are nonfactorizable diagrams, we name their expressions with $M$,
while the others are factorizable, and we name their expressions with $F$.
There are two similar merged annihilation diagrams, the Fig. 1 (b) and (d),
with the subscripts $AK^{*}_{0}$ and $Ah$ in their amplitudes, respectively,
which demonstrate the $W$ annihilation and $W$-exchange, space-like penguin
and time-like penguin annihilation-type diagrams.

With the ratio $r_{0}=m^{h}_{0}/m_{B}$, the amplitudes from Fig. 1 (a) are
written as

$\displaystyle F_{TK_{0}^{*}}^{LL}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{K(\pi)}(\zeta-1)\int dx_{B}dz\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})$ (52) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}\sqrt{\zeta}(2z-1)(\phi^{s}+\phi^{t})-(z+1)\phi\big{]}E_{a12}(t_{a1})h_{a1}(x_{B},z,b_{B},b)$
$\displaystyle+$
$\displaystyle\left(\zeta\phi-2\sqrt{\zeta}\phi^{s}\right)E_{a12}(t_{a2})h_{a2}(x_{B},z,b_{B},b)\bigg{\\}},$
$\displaystyle F_{TK_{0}^{*}}^{LR}$ $\displaystyle=$ $\displaystyle-
F_{TK_{0}^{*}}^{LL},$ (53) $\displaystyle F_{TK_{0}^{*}}^{SP}$
$\displaystyle=$ $\displaystyle 16\pi C_{F}m^{4}_{B}r_{0}f_{K(\pi)}\int
dx_{B}dz\int b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})$ (54) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}\phi[\zeta(2z-1)-1]+\sqrt{\zeta}[z\phi^{t}-(z+2)\phi^{s}]\big{]}E_{a12}(t_{a1})h_{a1}(x_{B},z,b_{B},b)$
$\displaystyle+$ $\displaystyle\left[\phi(2\zeta-
x_{B})-2\sqrt{\zeta}\phi^{s}(\zeta-
x_{B}+1)\right]E_{a12}(t_{a2})h_{a2}(x_{B},z,b_{B},b)\bigg{\\}},$
$\displaystyle M_{TK_{0}^{*}}^{LL}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}(\zeta-1)\int dx_{B}dzdx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})\phi^{A}$ (55)
$\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}[\zeta(1-x_{3}-z)+x_{B}+x_{3}-1]\phi+\sqrt{\zeta}z(\phi^{s}-\phi^{t})\big{]}E_{a34}(t_{a3})h_{a3}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle+$
$\displaystyle\big{[}[x_{3}(1-\zeta)-x_{B}]\phi+z[\phi-\sqrt{\zeta}(\phi^{s}+\phi^{t})]\big{]}E_{a34}(t_{a4})h_{a4}(x_{B},z,x_{3},b_{B},b_{3})\bigg{\\}},$
$\displaystyle M_{TK_{0}^{*}}^{LR}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}r_{0}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})$ (56) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}[\zeta(1-x_{3})+x_{B}+x_{3}-1][\phi+\sqrt{\zeta}(\phi^{s}-\phi^{t})](\phi^{P}+\phi^{T})+\sqrt{\zeta}z(\sqrt{\zeta}\phi+\phi^{s}+\phi^{t})$
$\displaystyle\times$
$\displaystyle(\phi^{T}-\phi^{P})\big{]}E_{a34}(t_{a3})h_{a3}(x_{B},z,x_{3},b_{B},b_{3})+\big{[}[(1-\zeta)x_{3}-x_{B}][\phi+\sqrt{\zeta}(\phi^{s}-\phi^{t})]$
$\displaystyle\times$
$\displaystyle(\phi^{P}-\phi^{T})+\sqrt{\zeta}z(\sqrt{\zeta}\phi+\phi^{s}+\phi^{t})(\phi^{P}+\phi^{T})\big{]}E_{a34}(t_{a4})h_{a4}(x_{B},z,x_{3},b_{B},b_{3})\bigg{\\}},\quad$
$\displaystyle M_{TK_{0}^{*}}^{SP}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}(\zeta-1)\int dx_{B}dzdx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})\phi^{A}$ (57)
$\displaystyle\times$ $\displaystyle\bigg{\\{}\big{[}[(x_{3}-1)\zeta-
x_{B}+z-x_{3}+1]\phi-\sqrt{\zeta}z(\phi^{s}+\phi^{t})\big{]}E_{a34}(t_{a3})h_{a3}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle+$
$\displaystyle\big{[}[x_{B}+x_{3}(\zeta-1)]\phi-z\sqrt{\zeta}(\sqrt{\zeta}\phi-\phi^{s}+\phi^{t})\big{]}E_{a34}(t_{a4})h_{a4}(x_{B},z,x_{3},b_{B},b_{3})\bigg{\\}},$

with the color factor $C_{F}=4/3$. The amplitudes from Fig. 1 (b) are written
as

$\displaystyle F_{AK_{0}^{*}}^{LL}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{B}\int dzdx_{3}\int bdbb_{3}db_{3}$ (58)
$\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}(1-\zeta)(z-1)\phi\phi^{A}+2\sqrt{\zeta}r_{0}[(2-z)\phi^{s}+z\phi^{t}]\phi^{P}\big{]}E_{b12}(t_{b1})h_{b1}(z,x_{3},b,b_{3})$
$\displaystyle+$
$\displaystyle\big{[}(1-\zeta)[x_{3}(1-\zeta)+\zeta]\phi\phi^{A}+2\sqrt{\zeta}r_{0}\phi^{s}[\big{(}\zeta(x_{3}-1)-x_{3}\big{)}(\phi^{P}+\phi^{T})-(\phi^{P}-\phi^{T})]\big{]}$
$\displaystyle\times$ $\displaystyle
E_{b12}(t_{b2})h_{b2}(z,x_{3},b,b_{3})\bigg{\\}},$ $\displaystyle
F_{AK_{0}^{*}}^{LR}$ $\displaystyle=$ $\displaystyle-F_{AK_{0}^{*}}^{LL},$
(59) $\displaystyle F_{AK_{0}^{*}}^{SP}$ $\displaystyle=$ $\displaystyle 16\pi
C_{F}m^{4}_{B}f_{B}\int dzdx_{3}\int bdbb_{3}db_{3}$ (60)
$\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}(\zeta-1)\sqrt{\zeta}(z-1)(\phi^{s}+\phi^{t})\phi^{A}+2r_{0}\phi\phi^{P}[\zeta(z-1)-1]\big{]}E_{b12}(t_{b1})h_{b1}(z,x_{3},b,b_{3})$
$\displaystyle+$
$\displaystyle\big{[}2\sqrt{\zeta}(1-\zeta)\phi^{s}\phi^{A}+x_{3}r_{0}\phi(\zeta-1)(\phi^{P}-\phi^{T})-2\zeta
r_{0}\phi\phi^{P}\big{]}E_{b12}(t_{b2})$ $\displaystyle\times$ $\displaystyle
h_{b2}(z,x_{3},b,b_{3})\bigg{\\}},$ $\displaystyle M_{AK_{0}^{*}}^{LL}$
$\displaystyle=$ $\displaystyle 32\pi C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int
dx_{B}dzdx_{3}\int b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})$ (61)
$\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}[{\zeta}^{2}(1-z-x_{3})+\zeta(x_{B}+2x_{3}+z-1)-(x_{B}+x_{3})]\phi\phi^{A}$
$\displaystyle+$
$\displaystyle\sqrt{\zeta}r_{0}[(x_{B}+(1-\zeta)(x_{3}-1))(\phi^{s}-\phi^{t})(\phi^{P}+\phi^{T})+z(\phi^{s}+\phi^{t})(\phi^{T}-\phi^{P})+4\phi^{s}\phi^{P}]\big{]}$
$\displaystyle\times$ $\displaystyle
E_{b34}(t_{b3})h_{b3}(x_{B},z,x_{3},b_{B},b)+\big{[}({\zeta}^{2}-1)(z-1)\phi\phi^{A}+\sqrt{\zeta}r_{0}[(\zeta(x_{3}-1)-x_{3}+x_{B})$
$\displaystyle\times$
$\displaystyle(\phi^{s}+\phi^{t})(\phi^{P}-\phi^{T})+(z-1)(\phi^{s}-\phi^{t})(\phi^{P}+\phi^{T})]\big{]}E_{b34}(t_{b4})h_{b4}(x_{B},z,x_{3},b_{B},b)\bigg{\\}},$
$\displaystyle M_{AK_{0}^{*}}^{LR}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})$ (62) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}(\zeta-1)(z+1)\sqrt{\zeta}(\phi^{s}-\phi^{t})\phi^{A}+r_{0}\phi[(x_{3}(1-\zeta)+x_{B}-2)(\phi^{P}+\phi^{T})+\zeta
z(\phi^{T}-\phi^{P})$ $\displaystyle+$ $\displaystyle
2\zeta\phi^{T}]\big{]}E_{b34}(t_{b3})h_{b3}(x_{B},z,x_{3},b_{B},b)+\big{[}(1-z)\sqrt{\zeta}(\zeta-1)(\phi^{s}-\phi^{t})\phi^{A}+r_{0}\phi[(\zeta
x_{3}$ $\displaystyle-$ $\displaystyle(x_{3}-x_{B}))(\phi^{P}+\phi^{T})+\zeta
z(\phi^{P}-\phi^{T})-2\zeta\phi^{P}]\big{]}E_{b34}(t_{b4})h_{b4}(x_{B},z,x_{3},b_{B},b)\bigg{\\}},$
$\displaystyle M_{AK_{0}^{*}}^{SP}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})$ (63) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}(z\zeta+z-1)(\zeta-1)\phi\phi^{A}+\sqrt{\zeta}r_{0}[(\zeta(1-x_{3})+x_{B}+x_{3}-1)(\phi^{s}+\phi^{t})(\phi^{T}-\phi^{P})$
$\displaystyle+$ $\displaystyle
z(\phi^{s}-\phi^{t})(\phi^{P}+\phi^{T})-4\phi^{s}\phi^{P}]\big{]}E_{b34}(t_{b3})h_{b3}(x_{B},z,x_{3},b_{B},b)$
$\displaystyle+$
$\displaystyle\big{[}(1-\zeta)[\zeta(z-2)+x_{3}(\zeta-1)+x_{B}]\phi\phi^{A}+r_{0}\sqrt{\zeta}[(1-z)(\phi^{s}+\phi^{t})(\phi^{P}-\phi^{T})$
$\displaystyle+$ $\displaystyle((1-\zeta)x_{3}+\zeta-
x_{B})(\phi^{s}-\phi^{t})(\phi^{P}+\phi^{T})]\big{]}E_{b34}(t_{b4})h_{b4}(x_{B},z,x_{3},b_{B},b)\bigg{\\}},$

The amplitudes from Fig. 1 (c) are

$\displaystyle F_{Th}^{LL}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}F_{K\pi}(s)/\mu_{s}\int dx_{B}dx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})$ (64) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}(1-\zeta)((\zeta-1)x_{3}-1)\phi^{A}-r_{0}[\big{(}2x_{3}(\zeta-1)+\zeta+1\big{)}\phi^{P}+(\zeta-1)(2x_{3}-1)\phi^{T}]\big{]}$
$\displaystyle\times$ $\displaystyle
E_{c12}(t_{c1})h_{c1}(x_{B},x_{3},b_{B},b_{3})+\left[\zeta(\zeta-1)x_{B}\phi^{A}+2r_{0}(\zeta
x_{B}+\zeta-1)\phi^{P}\right]E_{c12}(t_{c2})$ $\displaystyle\times$
$\displaystyle h_{c2}(x_{B},x_{3},b_{B},b_{3})\bigg{\\}},$ $\displaystyle
F_{Th}^{SP}$ $\displaystyle=$ $\displaystyle 16\pi
C_{F}m^{4}_{B}\sqrt{\zeta}F_{K\pi}(s)\int dx_{B}dx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})$ (65) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}(\zeta-1)\phi^{A}+r_{0}[x_{3}(\zeta-1)(\phi^{P}-\phi^{T})-2\phi^{P}]\big{]}E_{c12}(t_{c1})h_{c1}(x_{B},x_{3},b_{B},b_{3})$
$\displaystyle+$
$\displaystyle\left[(\zeta-1)x_{B}\phi^{A}+2r_{0}(\zeta+x_{B}-1)\phi^{P}\right]E_{c12}(t_{c2})h_{c2}(x_{B},x_{3},b_{B},b_{3})\bigg{\\}},$
$\displaystyle M_{Th}^{LL}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})\phi$ (66) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}({\zeta}^{2}-1)(x_{B}+z-1)\phi^{A}+r_{0}[\zeta(x_{B}+z)(\phi^{P}+\phi^{T})+x_{3}(\zeta-1)(\phi^{P}-\phi^{T})$
$\displaystyle-$ $\displaystyle
2\zeta\phi^{P}]\big{]}E_{c34}(t_{c3})h_{c3}(x_{B},z,x_{3},b_{B},b)+\big{[}(1-\zeta)x_{3}[(\zeta-1)\phi^{A}+r_{0}(\phi^{P}+\phi^{T})]$
$\displaystyle-$ $\displaystyle(x_{B}-z)[(\zeta-1)\phi^{A}+\zeta
r_{0}(\phi^{P}-\phi^{T})]\big{]}E_{c34}(t_{c4})h_{c4}(x_{B},z,x_{3},b_{B},b)\bigg{\\}},$
$\displaystyle M_{Th}^{LR}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}\sqrt{\zeta}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})$ (67) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}(\zeta-1)(x_{B}+z-1)(\phi^{s}+\phi^{t})\phi^{A}+r_{0}[(\zeta(1-x_{3})+x_{3})(\phi^{s}-\phi^{t})(\phi^{P}+\phi^{T})$
$\displaystyle+$
$\displaystyle(x_{B}+z-1)(\phi^{s}+\phi^{t})(\phi^{T}-\phi^{P})]\big{]}E_{c34}(t_{c3})h_{c3}(x_{B},z,x_{3},b_{B},b)$
$\displaystyle+$
$\displaystyle\big{[}(z-x_{B})(\phi^{s}-\phi^{t})[(\zeta-1)\phi^{A}+r_{0}(\phi^{T}-\phi^{P})]+(\zeta-1)r_{0}x_{3}(\phi^{s}+\phi^{t})$
$\displaystyle\times$
$\displaystyle(\phi^{P}+\phi^{T})]\big{]}E_{c34}(t_{c4})h_{c4}(x_{B},z,x_{3},b_{B},b)\bigg{\\}},$

The amplitudes from Fig. 1 (d) are

$\displaystyle F_{Ah}^{LL}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{B}\int dzdx_{3}\int bdbb_{3}db_{3}$ (68)
$\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}[(\zeta-1)[(\zeta-1)x_{3}+1]\phi\phi^{A}-2\sqrt{\zeta}r_{0}\phi^{s}[(\zeta-1)x_{3}(\phi^{P}-\phi^{T})+2\phi^{P}]\big{]}E_{d12}(t_{d1})$
$\displaystyle\times$ $\displaystyle
h_{d1}(z,x_{3},b,b_{3})+\big{[}z[2\sqrt{\zeta}r_{0}(\phi^{s}+\phi^{t})\phi^{P}+(1-\zeta)\phi\phi^{A}]-2(\zeta-1)\sqrt{\zeta}r_{0}(\phi^{s}-\phi^{t})$
$\displaystyle\times$
$\displaystyle\phi^{P}\big{]}E_{d12}(t_{d2})h_{d2}(z,x_{3},b,b_{3})\bigg{\\}},$
$\displaystyle F_{Ah}^{LR}$ $\displaystyle=$ $\displaystyle-F_{Ah}^{LL},$ (69)
$\displaystyle F_{Ah}^{SP}$ $\displaystyle=$ $\displaystyle 16\pi
C_{F}m^{4}_{B}f_{B}\int dzdx_{3}\int bdbb_{3}db_{3}$ (70)
$\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}2(1-\zeta)\sqrt{\zeta}\phi^{s}\phi^{A}+r_{0}\phi[((\zeta-1)x_{3}+1)(\phi^{P}+\phi^{T})+\zeta(\phi^{P}-\phi^{T})]\big{]}E_{d12}(t_{d1})$
$\displaystyle\times$ $\displaystyle
h_{d1}(z,x_{3},b,b_{3})+\left[(1-\zeta)\sqrt{\zeta}z(\phi^{s}-\phi^{t})\phi^{A}+2r_{0}(\zeta(z-1)+1)\phi\phi^{P}\right]E_{d12}(t_{d2})$
$\displaystyle\times$ $\displaystyle h_{d2}(z,x_{3},b,b_{3})\bigg{\\}},$
$\displaystyle M_{Ah}^{LL}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})$ (71) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}[(x_{B}+z-1){\zeta}^{2}+\zeta-(x_{B}+z)]\phi\phi^{A}+\sqrt{\zeta}r_{0}[(x_{3}-\zeta(x_{3}-1))(\phi^{s}-\phi^{t})(\phi^{P}$
$\displaystyle+$
$\displaystyle\phi^{T})+(x_{B}+z-1)(\phi^{s}+\phi^{t})(\phi^{T}-\phi^{P})-4\phi^{s}\phi^{P}]\big{]}E_{d34}(t_{d3})h_{d3}(x_{B},z,x_{3},b_{B},b)$
$\displaystyle+$
$\displaystyle\big{[}(1-\zeta)[\zeta(x_{3}-x_{B}+z-1)-x_{3}+1]\phi\phi^{A}-\sqrt{\zeta}r_{0}[(x_{B}-z)(\phi^{s}-\phi^{t})(\phi^{P}+\phi^{T})$
$\displaystyle+$
$\displaystyle(1-x_{3})(\zeta-1)(\phi^{s}+\phi^{t})(\phi^{P}-\phi^{T})]\big{]}E_{d34}(t_{d4})h_{d4}(x_{B},z,x_{3},b_{B},b)\bigg{\\}},$
$\displaystyle M_{Ah}^{LR}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})$ (72) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}(\zeta-1)\sqrt{\zeta}(x_{B}+z-2)(\phi^{s}+\phi^{t})\phi^{A}+r_{0}\phi[\zeta(x_{B}+z-1)(\phi^{P}+\phi^{T})$
$\displaystyle+$ $\displaystyle(\zeta
x_{3}-x_{3}-1)(\phi^{P}-\phi^{T})-2\zeta\phi^{P}]\big{]}E_{d34}(t_{d3})h_{d3}(x_{B},z,x_{3},b_{B},b)$
$\displaystyle+$
$\displaystyle\big{[}\sqrt{\zeta}(\zeta-1)(x_{B}-z)(\phi^{s}+\phi^{t})\phi^{A}+r_{0}\phi[\zeta(x_{B}-z)(\phi^{P}+\phi^{T})$
$\displaystyle+$
$\displaystyle(1-x_{3})(\zeta-1)(\phi^{P}-\phi^{T})]\big{]}E_{d34}(t_{d4})h_{d4}(x_{B},z,x_{3},b_{B},b)\bigg{\\}},$
$\displaystyle M_{Ah}^{SP}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})$ (73) $\displaystyle\times$
$\displaystyle\bigg{\\{}\big{[}(1-\zeta)[\zeta(x_{B}+z+x_{3}-2)-x_{3}+1]\phi\phi^{A}+\sqrt{\zeta}r_{0}[(x_{B}+z-1)(\phi^{s}-\phi^{t})(\phi^{P}$
$\displaystyle+$ $\displaystyle\phi^{T})+(x_{3}\zeta-\zeta-
x_{3})(\phi^{s}+\phi^{t})(\phi^{P}-\phi^{T})+4\phi^{s}\phi^{P}]\big{]}E_{d34}(t_{d3})h_{d3}(x_{B},z,x_{3},b_{B},b)$
$\displaystyle+$
$\displaystyle\big{[}(1-{\zeta}^{2})(x_{B}-z)\phi\phi^{A}+r_{0}\sqrt{\zeta}[(1-x_{3})(\zeta-1)(\phi^{s}-\phi^{t})(\phi^{P}+\phi^{T})$
$\displaystyle+$
$\displaystyle(x_{B}-z)(\phi^{s}+\phi^{t})(\phi^{P}-\phi^{T})]\big{]}E_{d34}(t_{d4})h_{d4}(x_{B},z,x_{3},b_{B},b)\bigg{\\}},$

##  Appendix B PQCD functions

In this section, we group the functions which appear in the factorization
formulas of this work.

With $\bar{\zeta}=(1-\zeta)$, $\bar{x}_{3}=(1-x_{3})$ and $\bar{z}=(1-z)$, the
involved hard scales are chosen as

$\displaystyle t_{a1}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{z},\;1/b_{B},\;1/b\right\\},\quad
t_{a2}=\max\left\\{m_{B}\sqrt{|x_{B}-\zeta|},\;1/b_{B},\;1/b\right\\},$ (74)
$\displaystyle t_{a3}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{x_{B}z},\;m_{B}\sqrt{z|\bar{\zeta}\bar{x}_{3}-x_{B}|},\;1/b_{B},\;1/b_{3}\right\\},$
(75) $\displaystyle t_{a4}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{x_{B}z},\;m_{B}\sqrt{z|x_{B}-x_{3}\bar{\zeta}|},\;1/b_{B},\;1/b_{3}\right\\},$
(76) $\displaystyle t_{b1}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{1-z},\;1/b,\;1/b_{3}\right\\},\quad
t_{b2}=\max\left\\{m_{B}\sqrt{\zeta+x_{3}\bar{\zeta}},\;1/b,\;1/b_{3}\right\\},$
(77) $\displaystyle t_{b3}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{\bar{z}(\zeta+x_{3}\bar{\zeta})},\;m_{B}\sqrt{1-z(\bar{x}_{3}\bar{\zeta}-x_{B})},\;1/b_{B},\;1/b\right\\},$
(78) $\displaystyle t_{b4}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{\bar{z}(\zeta+x_{3}\bar{\zeta})},\;m_{B}\sqrt{\bar{z}|x_{B}-\zeta-
x_{3}\bar{\zeta}|},\;1/b_{B},\;1/b\right\\},$ (79) $\displaystyle t_{c1}$
$\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{x_{3}\bar{\zeta}},\;1/b_{B},\;1/b_{3}\right\\},\quad
t_{c2}=\max\left\\{m_{B}\sqrt{x_{B}\bar{\zeta}},\;1/b_{B},\;1/b_{3}\right\\},$
(80) $\displaystyle t_{c3}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{x_{B}x_{3}\bar{\zeta}},\;m_{B}\sqrt{|1-x_{B}-z|[x_{3}\bar{\zeta}+\zeta]},\;1/b_{B},\;1/b\right\\},$
(81) $\displaystyle t_{c4}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{x_{B}x_{3}\bar{\zeta}},\;m_{B}\sqrt{|x_{B}-z|x_{3}\bar{\zeta}},\;1/b_{B},\;1/b\right\\},$
(82) $\displaystyle t_{d1}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{1-x_{3}\bar{\zeta}},\;1/b,\;1/b_{3}\right\\},\quad
t_{d2}=\max\left\\{m_{B}\sqrt{z\bar{\zeta}},\;1/b,\;1/b_{3}\right\\},$ (83)
$\displaystyle t_{d3}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{\bar{x}_{3}z\bar{\zeta}},\;m_{B}\sqrt{1-(x_{3}\bar{\zeta}+\zeta)(1-x_{B}-z)},\;1/b_{B},\;1/b\right\\},$
(84) $\displaystyle t_{d4}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{\bar{x}_{3}z\bar{\zeta}},\;m_{B}\sqrt{|x_{B}-z|\bar{x}_{3}\bar{\zeta}},\;1/b_{B},\;1/b\right\\}.$
(85)

The hard functions are written as

$\displaystyle h_{a1}(x_{B},z,b_{B},b)$ $\displaystyle=$ $\displaystyle
K_{0}(m_{B}\sqrt{x_{B}z}b_{B})\big{[}\theta(b_{B}-b)K_{0}(m_{B}\sqrt{z}b_{B})I_{0}(m_{B}\sqrt{z}b)$
(86) $\displaystyle+$ $\displaystyle(b\leftrightarrow b_{B})\big{]}S_{t}(z),$
$\displaystyle h_{a2}(x_{B},z,b_{B},b)$ $\displaystyle=$ $\displaystyle
K_{0}(m_{B}\sqrt{x_{B}z}b)S_{t}(x_{B})$ (91) $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{i\pi}{2}\big{[}\theta(b-b_{B})H_{0}^{(1)}(m_{B}\sqrt{\zeta-
x_{B}}b)J_{0}(m_{B}\sqrt{\zeta-x_{B}}b_{B})\\\
\;\quad\quad\;+(b\leftrightarrow
b_{B})\big{]},\quad\quad\quad\;x_{B}<\zeta,\\\
\big{[}\theta(b-b_{B})K_{0}(m_{B}\sqrt{x_{B}-\zeta}b)I_{0}(m_{B}\sqrt{x_{B}-\zeta}b_{B})\\\
\;\quad\quad\;+(b\leftrightarrow
b_{B})\big{]},\quad\quad\quad\;x_{B}\geq\zeta,\\\ \end{array}\right.$
$\displaystyle h_{a3}(x_{B},z,x_{3},b_{B},b_{3})$ $\displaystyle=$
$\displaystyle\big{[}\theta(b_{B}-b_{3})K_{0}(m_{B}\sqrt{x_{B}z}b_{B})I_{0}(m_{B}\sqrt{x_{B}z}b_{3})+(b_{B}\leftrightarrow
b_{3})\big{]}$ (94) $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{z[\bar{\zeta}\bar{x}_{3}-x_{B}]}b_{3}),\quad\quad\;\;\bar{\zeta}\bar{x}_{3}>x_{B},\\\
K_{0}(m_{B}\sqrt{z[x_{B}-\bar{\zeta}\bar{x}_{3}]}b_{3}),\quad\quad\quad\quad\bar{\zeta}\bar{x}_{3}\leq
x_{B},\end{array}\right.$ $\displaystyle h_{a4}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle=$
$\displaystyle\big{[}\theta(b_{B}-b_{3})K_{0}(m_{B}\sqrt{x_{B}z}b_{B})I_{0}(m_{B}\sqrt{x_{B}z}b_{3})+(b_{B}\leftrightarrow
b_{3})\big{]}$ (97) $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{z[x_{3}\bar{\zeta}-x_{B}]}b_{3}),~{}\quad\quad
x_{3}\bar{\zeta}>x_{B},\\\
K_{0}(m_{B}\sqrt{z[x_{B}-x_{3}\bar{\zeta}]}b_{3}),~{}~{}~{}\quad\quad\quad
x_{3}\bar{\zeta}\leq x_{B},\end{array}\right.$ $\displaystyle
h_{b1}(z,x_{3},b,b_{3})$ $\displaystyle=$
$\displaystyle\left(\frac{i\pi}{2}\right)^{2}H_{0}^{(1)}(m_{B}\sqrt{\bar{z}(\zeta+x_{3}\bar{\zeta})}b_{3})S_{t}(z)$
(98) $\displaystyle\times$
$\displaystyle\big{[}\theta(b-b_{3})H_{0}^{(1)}(m_{B}\sqrt{1-z}b)J_{0}(m_{B}\sqrt{1-z}b_{3})+(b\leftrightarrow
b_{3})\big{]},$ $\displaystyle h_{b2}(z,x_{3},b,b_{3})$ $\displaystyle=$
$\displaystyle\left(\frac{i\pi}{2}\right)^{2}H_{0}^{(1)}(m_{B}\sqrt{\bar{z}(\zeta+x_{3}\bar{\zeta})}b)S_{t}(x_{3})\big{[}\theta(b-b_{3})$
(99) $\displaystyle\times$ $\displaystyle
H_{0}^{(1)}(m_{B}\sqrt{\zeta+x_{3}\bar{\zeta}}b)J_{0}(m_{B}\sqrt{\zeta+x_{3}\bar{\zeta}}b_{3})+(b\leftrightarrow
b_{3})\big{]},$ $\displaystyle h_{b3}(x_{B},z,x_{3},b_{B},b)$ $\displaystyle=$
$\displaystyle\frac{i\pi}{2}K_{0}(m_{B}\sqrt{1-z(\bar{x}_{3}\bar{\zeta}-x_{B})}b_{B})\big{[}\theta(b_{B}-b)$
(100) $\displaystyle\times$ $\displaystyle
H_{0}^{(1)}(m_{B}\sqrt{\bar{z}(\zeta+x_{3}\bar{\zeta})}b_{B})J_{0}(m_{B}\sqrt{\bar{z}(\zeta+x_{3}\bar{\zeta})}b)+(b_{B}\leftrightarrow
b)\big{]},\;\;$ $\displaystyle h_{b4}(x_{B},z,x_{3},b_{B},b)$ $\displaystyle=$
$\displaystyle\frac{i\pi}{2}\big{[}\theta(b_{B}-b)H_{0}^{(1)}(m_{B}\sqrt{\bar{z}(\zeta+x_{3}\bar{\zeta})}b_{B})$
(103) $\displaystyle\times$ $\displaystyle
J_{0}(m_{B}\sqrt{\bar{z}(\zeta+x_{3}\bar{\zeta})}b)+(b_{B}\leftrightarrow
b)\big{]}$ $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{\bar{z}(\zeta+x_{3}\bar{\zeta}-x_{B})}b_{B}),~{}\quad
x_{B}<\zeta+x_{3}\bar{\zeta},\\\ K_{0}(m_{B}\sqrt{\bar{z}(x_{B}-\zeta-
x_{3}\bar{\zeta})}b_{B}),~{}~{}~{}\quad\quad
x_{B}\geq\zeta+x_{3}\bar{\zeta},\end{array}\right.$ $\displaystyle
h_{c1}(x_{B},x_{3},b_{B},b_{3})$ $\displaystyle=$ $\displaystyle
K_{0}(m_{B}\sqrt{x_{B}x_{3}\bar{\zeta}}b_{B})\big{[}\theta(b_{B}-b_{3})K_{0}(m_{B}\sqrt{x_{3}\bar{\zeta}}b_{B})$
(104) $\displaystyle\times$ $\displaystyle
I_{0}(m_{B}\sqrt{x_{3}\bar{\zeta}}b_{3})+(b_{3}\leftrightarrow
b_{B})\big{]}S_{t}(x_{3}),$ $\displaystyle h_{c2}(x_{B},x_{3},b_{B},b_{3})$
$\displaystyle=$ $\displaystyle h_{c1}(x_{3},x_{B},b_{3},b_{B}),$ (105)
$\displaystyle h_{c3}(x_{B},z,x_{3},b_{B},b)$ $\displaystyle=$
$\displaystyle\big{[}\theta(b_{B}-b)K_{0}(m_{B}\sqrt{x_{B}x_{3}\bar{\zeta}}b_{B})I_{0}(m_{B}\sqrt{x_{B}x_{3}\bar{\zeta}}b)+(b_{B}\leftrightarrow
b)\big{]}$ (108) $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{(1-x_{B}-z)[x_{3}\bar{\zeta}+\zeta]}b),\;\quad
x_{B}+z<1,\\\
K_{0}(m_{B}\sqrt{(x_{B}+z-1)[x_{3}\bar{\zeta}+\zeta]}b),~{}~{}\quad\;\quad
x_{B}+z\geq 1,\end{array}\right.$ $\displaystyle
h_{c4}(x_{B},z,x_{3},b_{B},b)$ $\displaystyle=$
$\displaystyle\big{[}\theta(b_{B}-b)K_{0}(m_{B}\sqrt{x_{B}x_{3}\bar{\zeta}}b_{B})I_{0}(m_{B}\sqrt{x_{B}x_{3}\bar{\zeta}}b)+(b_{B}\leftrightarrow
b)\big{]}$ (111) $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{x_{3}(z-x_{B})\bar{\zeta}}b),~{}\quad\quad
x_{B}<z,\\\
K_{0}(m_{B}\sqrt{x_{3}(x_{B}-z)\bar{\zeta}}b),~{}~{}~{}\quad\quad\quad
x_{B}\geq z,\end{array}\right.$ $\displaystyle h_{d1}(z,x_{3},b,b_{3})$
$\displaystyle=$
$\displaystyle\left(\frac{i\pi}{2}\right)^{2}H_{0}^{(1)}(m_{B}\sqrt{\bar{x}_{3}z\bar{\zeta}}b)S_{t}(x_{3})\big{[}\theta(b-b_{3})$
(112) $\displaystyle\times$ $\displaystyle
H_{0}^{(1)}(m_{B}\sqrt{1-x_{3}\bar{\zeta}}b)J_{0}(m_{B}\sqrt{1-x_{3}\bar{\zeta}}b_{3})+(b\leftrightarrow
b_{3})\big{]},$ $\displaystyle h_{d2}(z,x_{3},b,b_{3})$ $\displaystyle=$
$\displaystyle\left(\frac{i\pi}{2}\right)^{2}H_{0}^{(1)}(m_{B}\sqrt{\bar{x}_{3}z\bar{\zeta}}b_{3})S_{t}(z)$
(113) $\displaystyle\times$
$\displaystyle\big{[}\theta(b-b_{3})H_{0}^{(1)}(m_{B}\sqrt{z\bar{\zeta}}b)J_{0}(m_{B}\sqrt{z\bar{\zeta}}b_{3})+(b\leftrightarrow
b_{3})\big{]},$ $\displaystyle h_{d3}(x_{B},z,x_{3},b_{B},b)$ $\displaystyle=$
$\displaystyle\frac{i\pi}{2}K_{0}(m_{B}\sqrt{1-x_{3}(1-x_{B}-z)\bar{\zeta}+(x_{B}+z-1)\zeta}b_{B})$
(114) $\displaystyle\times$
$\displaystyle\big{[}\theta(b_{B}-b)H_{0}^{(1)}(m_{B}\sqrt{\bar{x}_{3}z\bar{\zeta}}b_{B})J_{0}(m_{B}\sqrt{\bar{x}_{3}z\bar{\zeta}}b)+(b_{B}\leftrightarrow
b)\big{]},$ $\displaystyle h_{d4}(x_{B},z,x_{3},b_{B},b)$ $\displaystyle=$
$\displaystyle\frac{i\pi}{2}\big{[}\theta(b_{B}-b)H_{0}^{(1)}(m_{B}\sqrt{\bar{x}_{3}z\bar{\zeta}}b_{B})J_{0}(m_{B}\sqrt{\bar{x}_{3}z\bar{\zeta}}b)+(b_{B}\leftrightarrow
b)\big{]}$ (117) $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{\bar{x}_{3}(z-x_{B})\bar{\zeta}}b_{B}),~{}\quad\quad
x_{B}<z,\\\
K_{0}(m_{B}\sqrt{\bar{x}_{3}(x_{B}-z)\bar{\zeta}}b_{B}),~{}~{}~{}\quad\quad\quad
x_{B}\geq z,\end{array}\right.$

where $H_{0}^{(1)}(\chi)=J_{0}(\chi)+iY_{0}(\chi)$. The factor $S_{t}(\chi)$
with the expression prd65-014007

$\displaystyle
S_{t}(\chi)=\frac{2^{1+2c}\Gamma(3/2+c)}{\sqrt{\pi}\Gamma(1+c)}[\chi(1-\chi)]^{c},$
(118)

resums the threshold logarithms ln${}^{2}\chi$ appearing in the hard kernels
to all orders, and the parameter $c$ has its expression as
$c=0.04Q^{2}-0.51Q+1.87$ with $Q^{2}$ the invariant mass square of the final
state $f$ in the $B\to f$ transition prd91-094024 ; prd80-074024 .

The evolution factors in the factorization expressions are given by

$\displaystyle E_{a12}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{K_{0}^{*}}(t)],$ (119)
$\displaystyle E_{a34}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{K_{0}^{*}}(t)-S_{h}]|_{b=b_{B}},$
(120) $\displaystyle E_{b12}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{K_{0}^{*}}-S_{h}(t)],$ (121)
$\displaystyle E_{b34}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{K_{0}^{*}}(t)-S_{h}]|_{b_{3}=b},$
(122) $\displaystyle E_{c12}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{K_{0}^{*}}(t)],$ (123)
$\displaystyle E_{c34}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{K_{0}^{*}}(t)-S_{h}]|_{b_{3}=b_{B}},$
(124) $\displaystyle E_{d12}(t)$ $\displaystyle=$ $\displaystyle E_{b12}(t),$
(125) $\displaystyle E_{d34}(t)$ $\displaystyle=$ $\displaystyle E_{b34}(t),$
(126)

in which the Sudakov exponents are defined as

$\displaystyle S_{B}$ $\displaystyle=$ $\displaystyle
s\left(x_{B}\frac{m_{B}}{\sqrt{2}},b_{B}\right)+\frac{5}{3}\int^{t}_{1/b_{B}}\frac{d\bar{\mu}}{\bar{\mu}}\gamma_{q}(\alpha_{s}(\bar{\mu})),$
(127) $\displaystyle S_{K_{0}^{*}}$ $\displaystyle=$ $\displaystyle
s\left(z\frac{m_{B}}{\sqrt{2}},b\right)+s\left((1-z)\frac{m_{B}}{\sqrt{2}},b\right)+2\int^{t}_{1/b}\frac{d\bar{\mu}}{\bar{\mu}}\gamma_{q}(\alpha_{s}(\bar{\mu})),$
(128) $\displaystyle S_{h}$ $\displaystyle=$ $\displaystyle
s\left(x_{3}\frac{m_{B}}{\sqrt{2}},b_{3}\right)+s\left((1-x_{3})\frac{m_{B}}{\sqrt{2}},b_{3}\right)+2\int^{t}_{1/b_{3}}\frac{d\bar{\mu}}{\bar{\mu}}\gamma_{q}(\alpha_{s}(\bar{\mu})),$
(129)

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

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

with the variables are

$\displaystyle\hat{q}\equiv\mbox{ln}[Q/(\sqrt{2}\Lambda)],\quad\hat{b}\equiv\mbox{ln}[1/(b\Lambda)],$
(131)

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

$\displaystyle\beta_{1}=\frac{33-2n_{f}}{12},\quad\beta_{2}=\frac{153-19n_{f}}{24},\quad
A^{(1)}=\frac{4}{3},$ $\displaystyle
A^{(2)}=\frac{67}{9}-\frac{\pi^{2}}{3}-\frac{10}{27}n_{f}+\frac{8}{3}\beta_{1}\mbox{ln}(\frac{1}{2}e^{\gamma_{E}}),$
(132)

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

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