# Study of $B_{u,d,s}\to K_{0}^{*}(1430)P$ and $K_{0}^{*}(1430)V$ decays
within QCD factorization

Lili Chena, Mengfei Zhaoa, Yunyun Zhanga,b and Qin Changa
aInstitute of Particle and Nuclear Physics, Henan Normal University, Henan
453007, China
bNanyang Institute of Technology, Henan 473004, China

###### Abstract

We study the nonleptonic charmless $B_{u,d,s}\to K_{0}^{*}(1430)P$
$(P=K\,,\pi)$ and $K_{0}^{*}(1430)V$ $(V=K^{*}\,,\rho\,,\omega\,,\phi)$
decays. The amplitudes are calculated within the QCD factorization, and the
non-perturbative quantities are evaluated by using a covariant light-front
approach. The branching fractions and CP asymmetries of theses decay modes are
calculated, some decay modes are first predicted, and some useful relations
based on $SU(3)$ flavor symmetry are discussed. Comparing the theoretical
results with the current available experimental data, it is found that
$K_{0}^{*}(1430)$ can be described as the lowest-lying p-wave $(s,u/d)$ state
rather than the first excited one.

##  1 Introduction

The inner structure of scalar (S) meson has been studied for a long time, and
now it is still a hot topic in particle physics. There are several proposals
for their inner structure, e.g., $\bar{q}q$ state, tetra-quark states,
molecule states, glueball and hybrid states, while there is still no general
agreement on such issue. For the light scalar mesons, in the 2-quark scenario
(namely S1), it is suggested that the scalar mesons with mass below $1\,{\rm
GeV}$ (such as $a_{0}(980)$, $f_{0}(980)$, $\kappa/K_{0}^{*}(700)$, etc.) are
interpreted as the lowest lying $q\bar{q}$ states having a unit of orbital
angular momentum and form a $SU(3)$ nonet; while, the ones with mass above
$1\,{\rm GeV}$ (such as $a_{0}(1450)$, $f_{0}(1370)$, $K_{0}^{*}(1430)$, etc.)
are treated as the first excited $q\bar{q}$ states and are classified into
another $SU(3)$ nonet. On the contrary, in the tetra-quark scenario advocated
by Jaffe [1, 2], the former are predominately the $qq\bar{q}\bar{q}$ states
without introducing a unit of orbital angular momentum; accordingly, the later
are treated as the lowest lying $q\bar{q}$ p-wave states (namely S2). This
scenario is favored by some lattice calculations [3, 4] and mesonic
spectroscopy data [5]; it is also much more acceptable because the $0^{+}$
meson has a unit of orbital angular momentum and hence should have a higher
mass above $1\,{\rm GeV}$ in the $q\bar{q}$ model. Besides of studies of mass
spectra and decays of scalar mesons, the nonleptonic two-body $B$ meson decay
involving a scalar final state, $B\to SM$, provides another efficient way to
investigate the features and the possible inner structures of scalar mesons.

In recent years, although the identification of scalar mesons is difficult
experimentally, some experimental efforts have been devoted to measuring
$B_{u,d}\to SM$ decay modes. For instance, the light scalar $f_{0}(980)$ was
first observed in the $B\to f_{0}(980)K$ decay by the Belle [6] and BABAR [7]
collaborations in 2002 and 2004, respectively. Since then, more and more $B\to
SM$ decay modes have been observed by Belle [8, 9, 10, 11], BABAR [12, 13, 19,
15, 17, 16, 14, 20, 18] and LHCb [22, 21] collaborations. Motivated by the
rapid development of experiment, some theoretical studies on these decays are
made within some QCD inspired approaches, such as the generalized
factorization approach [23], QCD factorization (QCDF) [24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34], perturbative QCD approach (PQCD) [35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] and other methods [56,
57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 55]. Most of these previous works
mainly focus on the $B_{u,d}\to SM$ decay modes, but the $B_{s}\to SM$ decays
have not been fully studied theoretically because most of the $B_{s}\to SM$
decay modes have not been observed.

In 2019, the $B_{s}\to K_{0}^{*}(1430)^{+}K^{-}+c.c.$ and
$B_{s}\to\bar{K}_{0}^{*}(1430)^{0}K^{0}+c.c.$ decays are observed for the
first time by LHCb collaboration, each with significance over $10$ standard
deviations. The measured branching fractions are [21]

$\displaystyle{\cal B}(B_{s}^{0}\to K_{0}^{*}(1430)^{+}K^{-}+c.c.)=(31.3\pm
2.3\pm 0.7\pm 25.1\pm 3.3)\times 10^{-6},$ $\displaystyle{\cal
B}(B_{s}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0}+c.c.)=(33.0\pm 2.5\pm 0.9\pm
9.1\pm 3.5)\times 10^{-6},$ (1)

Unfortunately, there is no relevant theoretical prediction for these decays
before. It is expected that more $B_{s}\to SM$ decays will be observed by LHCb
and Belle-II collaborations in the near future. Therefore, we would like to
make a detailed study of $B_{s}\to K_{0}^{*}(1430)P$ and $K_{0}^{*}(1430)V$
($P$ denotes pseudoscalar meson, and $V$ denotes vector meson) decay modes
within the framework of QCDF in this paper. Besides, the $B_{u,d}\to
K_{0}^{*}(1430)P$ and $K_{0}^{*}(1430)V$ decays will also be investigated
111The $B_{u,d}\to SP$ and $SV$ decay modes have been investigated in detail
in Ref. [30] and Ref. [31], respectively. Some problems in these works are
corrected and the predictions are updated by the same authors in Ref. [33].
Besides the corrections made in Ref. [33], some essential improvements will be
made in this work.. In order to test whether $K_{0}^{*}(1430)$ is a lowest
lying $q\bar{q}$ state or a first excited state, our calculation and analysis
will be made within the just mentioned two scenarios (S1 and S2).

This paper is organized as follows. In section 2, we present the theoretical
framework and calculations for $B_{u,d,s}\to
K_{0}^{*}(1430)P,\,K_{0}^{*}(1430)V$ decays. In section 3, the values of
nonperturbative input parameters are calculated; after that, the numerical
results and discussions are presented. Finally, we give our summary in section
4. The definitions of decay constant, form factor and distribution amplitude
are given in appendix A, and the amplitudes of $B_{u,d,s}\to
K_{0}^{*}(1430)P,\,K_{0}^{*}(1430)V$ decays are summarized in appendix B.

##  2 Theoretical framework

In the Standard Model (SM), the effective weak Hamiltonian responsible for
$B_{u,d,s}\to M_{1}M_{2}$ decay induced by $b\to p$ transition is given as
[67]

$\displaystyle{\cal H}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\biggl{[}V_{ub}V_{up}^{*}\left(C_{1}O_{1}^{u}+C_{2}O_{2}^{u}\right)+V_{cb}V_{cp}^{*}\left(C_{1}O_{1}^{c}+C_{2}O_{2}^{c}\right)$
(2) $\displaystyle-
V_{tb}V_{tp}^{*}\,\big{(}\sum_{i=3}^{10}C_{i}O_{i}\big{.}\biggl{.}\biggl{.}\big{.}+C_{7\gamma}O_{7\gamma}+C_{8g}O_{8g}\big{)}\biggl{]}+{\rm
h.c.},$

where $V_{qb}V_{qp}^{\ast}$ ($q=u,c$ and $t$) are products of the Cabibbo-
Kobayashi-Maskawa (CKM) matrix elements, $C_{i}$ the Wilson coefficients, and
$O_{i}$ the relevant effective four-quark operators. To obtain the decay
amplitude, the main work is to calculate the hadronic matrix element, $\langle
M_{1}M_{2}|O_{i}|B\rangle$. In the QCDF, the hadronic matrix element of each
operator can be written as the convolution integrals of the scattering kernel
with the distribution amplitudes (DAs) of the participating mesons [68, 69,
70],

$\displaystyle\langle M_{1}M_{2}|Q_{i}|{B}\rangle=$
$\displaystyle\sum_{j}F_{j}^{B\to M_{1}}f_{M_{2}}\int\,dy\,{\cal
T}_{ij}^{I}(y)\,\varphi_{M_{2}}(y)+[M_{1}\leftrightarrow M_{2}]\,$
$\displaystyle+f_{B}f_{M_{1}}f_{M_{2}}\int dx\,dy\,dz\,{\cal
T}_{i}^{II}(x,y,z)\,\varphi_{M_{1}}(x)\,\varphi_{M_{2}}(y)\,\varphi_{B}(z)\,,$
(3)

where $x\,,y\,,z$ are the momentum fractions; $F_{j}^{B\to M_{1}}$ is an
appropriate form factor of $B\to M_{1}$ transition; $f_{B}$ and $f_{M}$ are
decay constants of $B$ and light mesons, respectively. The definitions of
decay constant, form factor and DAs are given in the appendix A. The kernels
${\cal T}_{ij}^{I}(y)$ and ${\cal T}_{i}^{II}(x,y,z)$ in Eq. (2) are hard-
scattering functions and are calculable in perturbation theory. The former
starts at tree level and contains the vertex, penguin corrections at next-to-
leading order in $\alpha_{s}$; the later contains the order $\alpha_{s}$
contributions caused by hard spectator-scattering and annihilation topologies.

Applying the factorization formula, one can obtain the amplitudes of
$B_{u,d,s}\to K_{0}^{*}(1430)P$ and $K_{0}^{*}(1430)V$ decay modes, which are
collected in the appendix B. In each amplitude, the quantity $A_{M1M2}$ is the
factorized matrix element and can be written as

$\displaystyle A_{M1M2}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}\begin{cases}-(m_{B}^{2}-m_{M_{1}}^{2})U_{0}^{B\to
M_{1}}(m_{M_{2}}^{2})f_{M_{2}}&\text{if}~{}M_{1}M_{2}=SP\,,\\\
(m_{B}^{2}-m_{M_{1}}^{2})F_{0}^{B\to
M_{1}}(m_{M_{2}}^{2})f_{M_{2}}&\text{if}~{}M_{1}M_{2}=PS\,,\\\
2m_{M_{2}}\epsilon^{*}\cdot p_{B}U_{1}^{B\to
M_{1}}(m_{M_{2}}^{2})f_{M_{2}}&\text{if}~{}M_{1}M_{2}=SV\,,\\\
-2m_{M_{1}}\epsilon^{*}\cdot p_{B}A_{0}^{B\to
M_{1}}(m_{M_{2}}^{2})f_{M_{2}}&\text{if}~{}M_{1}M_{2}=VS\,.\end{cases}$ (4)

Here, it has been assumed that the final-state meson $M_{1}$ carries away the
spectator quark from $B$ meson, and the other one is $M_{2}$. The coefficients
of flavor operators $\alpha_{i}^{p}$ appeared in the amplitudes are expressed
in terms of the effective coefficients $a_{i}^{p}$ as

$\displaystyle\alpha_{1}(M_{1}M_{2})$
$\displaystyle=a_{1}(M_{1}M_{2}),\qquad\alpha_{2}(M_{1}M_{2})=a_{2}(M_{1}M_{2}),$
(5) $\displaystyle\alpha_{3}^{p}(M_{1}M_{2})$
$\displaystyle=\begin{cases}a_{3}^{p}(M_{1}M_{2})+a_{5}^{p}(M_{1}M_{2})&\text{if}~{}M_{1}M_{2}=PS,SV,VS,\\\
a_{3}^{p}(M_{1}M_{2})-a_{5}^{p}(M_{1}M_{2})&\text{if}~{}M_{1}M_{2}=SP,\end{cases}$
(6) $\displaystyle\alpha_{4}^{p}(M_{1}M_{2})$
$\displaystyle=\begin{cases}a_{4}^{p}(M_{1}M_{2})-r_{\chi}^{M_{2}}a_{6}^{p}(M_{1}M_{2})&\text{if}~{}M_{1}M_{2}=PS,SP,SV,\\\
a_{4}^{p}(M_{1}M_{2})+r_{\chi}^{M_{2}}a_{6}^{p}(M_{1}M_{2})&\text{if}~{}M_{1}M_{2}=VS,\end{cases}$
(7) $\displaystyle\alpha_{3,EW}^{p}(M_{1}M_{2})$
$\displaystyle=\begin{cases}a_{9}^{p}(M_{1}M_{2})+a_{7}^{p}(M_{1}M_{2})&\text{if}~{}M_{1}M_{2}=PS,SV,VS,\\\
a_{9}^{p}(M_{1}M_{2})-a_{7}^{p}(M_{1}M_{2})&\text{if}~{}M_{1}M_{2}=SP,\end{cases}$
(8) $\displaystyle\alpha_{4,EW}^{p}(M_{1}M_{2})$
$\displaystyle=\begin{cases}a_{10}^{p}(M_{1}M_{2})-r_{\chi}^{M_{2}}a_{8}^{p}(M_{1}M_{2})&\text{if}~{}M_{1}M_{2}=PS,SP,SV,\\\
a_{10}^{p}(M_{1}M_{2})+r_{\chi}^{M_{2}}a_{8}^{p}(M_{1}M_{2})&\text{if}~{}M_{1}M_{2}=VS\,,\end{cases}$
(9)

where the ratio $r_{\chi}$ is defined as

$\displaystyle r_{\chi}^{P}(\mu)=$
$\displaystyle\frac{2m_{P}^{2}}{m_{b}(\mu)(m_{1}(\mu)+m_{2}(\mu))},\qquad
r_{\chi}^{V}(\mu)=\frac{2m_{V}}{m_{b}(\mu)}\frac{f_{V}^{\perp}(\mu)}{f_{V}},$
(10) $\displaystyle r_{\chi}^{S}(\mu)=$
$\displaystyle\frac{2m_{S}}{m_{b}(\mu)}\frac{\bar{f}_{S}(\mu)}{f_{S}}=\frac{2m_{S}^{2}}{m_{b}(\mu)(m_{1}(\mu)-m_{2}(\mu))}\,.$
(11)

In Eqs. (5-9), the general form of effective coefficient $a_{i}^{p}$ at next-
to-leading order in $\alpha_{s}$ is

$\displaystyle a_{i}^{p}(M_{1}M_{2})=\left(C_{i}+\frac{C_{i\pm
1}}{N_{c}}\right)N_{i}(M_{2})+\frac{C_{i\pm
1}}{N_{c}}\frac{C_{F}\alpha_{s}}{4\pi}\left[V_{i}(M_{2})+\frac{4\pi^{2}}{N_{c}}H_{i}(M_{1}M_{2})\right]+P_{i}^{p}(M_{2}),$
(12)

where $i=1,...,10$, the upper (lower) sign applies when $i$ is odd (even),
$C_{F}=(N_{c}^{2}-1)/(2N_{c})$ with $N_{c}=3$, and $N_{i}(M_{2})=1$ except for
$N_{6,8}(V)=0$. The terms proportional to $N_{i}(M_{2})$ are the leading order
contributions, and are the same as the results obtained by naive factorization
approach; the $\alpha_{s}$ corrections are encoded in the quantities
$V_{i}(M_{2})$, $H_{i}(M_{1}M_{2})$ and $P_{i}^{p}(M_{2})$, which are obtained
by calculating vertex, hard-spectator and penguin diagrams, and can be written
as the convolution integrals of the hard-scattering kernels with meson light-
cone DAs. The convolution integrals for these quantities can be evaluated by
using expansions of the DAs in terms of Gegenbauer polynomials. In this work,
we include the first four terms in the Gegenbauer expansion for the twist-2
DAs due to the nontrivial contribution related to the odd Gegenbauer moments
of scalar meson.

The integral forms of $V_{i}(M_{2})$ and $P_{i}^{p}(M_{2})$ for $B\to SP$ and
$SV$ modes are the same as the ones for $B\to PP$ and $PV$ modes, which have
been obtained in Ref. [70]. After integrating out the momentum fraction, we
obtain

$\displaystyle
V_{i}(M_{2})=(12\ln\frac{m_{b}}{\mu}-\frac{37}{2}-3i\pi)a_{0}^{M_{2}}+(\frac{11}{2}-3i\pi)a_{1}^{M_{2}}-\frac{21}{20}a_{2}^{M_{2}}+(\frac{79}{36}-\frac{2i\pi}{3})a_{3}^{M_{2}}$
(13)

for $i=1-4,9,10$, and

$\displaystyle V_{i}({M_{2}})=$
$\displaystyle(-12\ln\frac{m_{b}}{\mu}+\frac{13}{2}+3i\pi)a_{0}^{M_{2}}+(\frac{11}{2}-3i\pi)a_{1}^{M_{2}}+\frac{21}{20}a_{2}^{M_{2}}+(\frac{79}{36}-\frac{2i\pi}{3})a_{3}^{M_{2}}$
(14)

for $i=5,7$ when $M_{2}=P\,,S\,,V$; while, for $i=6$ and $8$, we have

$\displaystyle
V_{i}(M)=\begin{cases}-6&\text{if}~{}M_{2}=P~{}\text{and}~{}S\,,\\\ 9-6\pi
i&\text{if}~{}M_{2}=V\,.\end{cases}$ (15)

It is noted that our result given by Eq. (14) is different from the one given
in Ref. [30] but is consistent with the one given by the same authors in Ref.
[31]. For the penguin functions $G_{M_{2}}(s)$ appeared in $P_{i}^{p}(M_{2})$
(one may refer to Ref. [70] for detail), we have the following analytic
results,

$\displaystyle G_{M_{2}}(0)=$
$\displaystyle\left(\frac{5}{3}+\frac{2i\pi}{3}\right)a_{0}^{{M_{2}}}+\frac{a_{1}^{{M_{2}}}}{2}+\frac{a_{2}^{{M_{2}}}}{5}+\frac{a_{3}^{{M_{2}}}}{9},$
(16) $\displaystyle G_{M_{2}}(s_{c})=$
$\displaystyle(\frac{5}{3}-\frac{2}{3}\ln
s_{c})a_{0}^{{M_{2}}}+\frac{a_{1}^{{M_{2}}}}{2}+\frac{a_{2}^{{M_{2}}}}{5}+\frac{a_{3}^{{M_{2}}}}{9}+\frac{4}{3}\left(8a_{0}^{{M_{2}}}+9a_{1}^{{M_{2}}}+9a_{2}^{{M_{2}}}+9a_{3}^{{M_{2}}}\right)s_{c}$
$\displaystyle+2(8a_{0}^{{M_{2}}}+63a_{1}^{{M_{2}}}+214a_{2}^{{M_{2}}}+\frac{1505a_{3}^{{M_{2}}}}{3})s_{c}^{2}-24(9a_{1}^{{M_{2}}}+80a_{2}^{{M_{2}}}+\frac{9490a_{3}^{{M_{2}}}}{27})s_{c}^{3}$
$\displaystyle+(2880a_{2}^{{M_{2}}}+\frac{91000a_{3}^{{M_{2}}}}{3})s_{c}^{4}-39200a_{3}^{{M_{2}}}s_{c}^{5}-\frac{2}{3}\sqrt{1-4s_{c}}\,\bigg{[}(1+2s_{c})a_{0}^{{M_{2}}}$
$\displaystyle+6(4a_{0}^{{M_{2}}}+27a_{1}^{{M_{2}}}+78a_{2}^{{M_{2}}}+160a_{3}^{{M_{2}}})s_{c}^{2}-36(9a_{1}^{{M_{2}}}+70a_{2}^{{M_{2}}}+\frac{2440a_{3}^{{M_{2}}}}{9})s_{c}^{3}$
$\displaystyle+(4320a_{2}^{{M_{2}}}+40600a_{3}^{{M_{2}}})s_{c}^{4}-58800a_{3}^{{M_{2}}}s_{c}^{5}\bigg{]}\left(2\,\mbox{arctanh}\sqrt{1-4s_{c}}-i\pi\right)+12s_{c}^{2}\,\bigg{[}a_{0}^{{M_{2}}}$
$\displaystyle+3a_{1}^{{M_{2}}}+6a_{2}^{{M_{2}}}+10a_{3}^{{M_{2}}}-\frac{4}{3}\left(a_{0}^{{M_{2}}}+9a_{1}^{{M_{2}}}+36a_{2}^{{M_{2}}}+100a_{3}^{{M_{2}}}\right)s_{c}+18(a_{1}^{{M_{2}}}+10a_{2}^{{M_{2}}}+50a_{3}^{{M_{2}}})s_{c}^{2}$
$\displaystyle-(240a_{2}^{{M_{2}}}+2800a_{3}^{{M_{2}}})s_{c}^{3}+\frac{9800a_{3}^{{M_{2}}}}{3}s_{c}^{4}\bigg{]}\left(2\,\mbox{arctanh}\sqrt{1-4s_{c}}-i\pi\right)^{2}\,,$
(17) $\displaystyle G_{M_{2}}(1)=$
$\displaystyle\left(\frac{85}{3}-6\sqrt{3}\,\pi+\frac{4\pi^{2}}{9}\right)a_{0}^{{M_{2}}}-\left(\frac{155}{2}-36\sqrt{3}\,\pi+12\pi^{2}\right)a_{1}^{{M_{2}}}$
$\displaystyle+\left(\frac{7001}{5}-504\sqrt{3}\,\pi+136\pi^{2}\right)a_{2}^{{M_{2}}}-\left(\frac{146581}{9}-6000\sqrt{3}\,\pi+\frac{14920}{9}\pi^{2}\right)a_{3}^{{M_{2}}}\,.$
(18)

where, $s_{c}=(m_{c}/m_{b})^{2}$; while, the functions $\hat{G}_{M_{2}}(s)$
read

$\displaystyle\hat{G}_{M_{2}}(s_{c})=$
$\displaystyle\frac{16}{9}(1-3s_{c})-\frac{2}{3}\text{ln}s_{c}-\frac{2}{3}(1-4s_{c})^{3/2}(2\text{arctanh}\sqrt{1-4s_{c}}-i\pi)\,,$
$\displaystyle\hat{G}_{M_{2}}(0)=$
$\displaystyle\frac{16}{9}+\frac{2\pi}{3}i\,,\quad\hat{G}_{M_{2}}(1)=\frac{2\pi}{\sqrt{3}}-\frac{32}{9}\,,$
(19)

when $M_{2}=P\,,S$, and

$\displaystyle\hat{G}_{M_{2}}(s_{c})=$ $\displaystyle
1-36s_{c}+12s_{c}\sqrt{1-4s_{c}}(2\text{arctanh}\sqrt{1-4s_{c}}-i\pi)-12s_{c}^{2}(2\text{arctanh}\sqrt{1-4s_{c}}-i\pi)^{2}\,,$
$\displaystyle\hat{G}_{M_{2}}(0)=$ $\displaystyle
1\,,\quad\hat{G}_{M_{2}}(1)=-35+4\sqrt{3}\pi+\frac{4\pi^{2}}{3}\,,$ (20)

when $M_{2}=V$.

The hard-spectator corrections can be written as

$\displaystyle
H_{i}(M_{1}M_{2})=\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\int_{0}^{1}\frac{d\xi}{\xi}\Phi_{B}(\xi)\int_{0}^{1}dx\int_{0}^{1}dy\left[\frac{\Phi_{M_{2}}(x)\Phi_{M_{1}}(y)}{\bar{x}\bar{y}}\pm\gamma_{\chi}^{M_{1}}\frac{\Phi_{M_{2}}(x)\phi_{M_{1}}(y)}{x\bar{y}}\right]$
(21)

for $i=1-4,9,10$,

$\displaystyle H_{i}(M_{1}M_{2})=$
$\displaystyle-\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\int_{0}^{1}\frac{d\xi}{\xi}\Phi_{B}(\xi)\int_{0}^{1}dx\int_{0}^{1}dy\left[\frac{\Phi_{M_{2}}(x)\Phi_{M_{1}}(y)}{x\bar{y}}\pm\gamma_{\chi}^{M_{1}}\frac{\Phi_{M_{2}}(x)\phi_{M_{1}}(y)}{\bar{x}\bar{y}}\right]$
(22)

for $i=5,7$, and $H_{i}(M_{1}M_{2})=0$ for $i=6,8$, where $\bar{x}=1-x$,
$\bar{y}=1-y$. The upper (lower) sign should be applied when $M_{1}=V\,,P$
($M_{1}=S$). The quantity $B_{M_{1}M_{2}}$ is defined as

$\displaystyle B_{M1M2}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}\begin{cases}f_{B_{q}}f_{M_{1}}f_{M_{2}}&\text{if}~{}M_{1}M_{2}=PS,SP,\\\
-f_{B_{q}}f_{M_{1}}f_{M_{2}}&\text{if}~{}M_{1}M_{2}=VS,SV.\end{cases}$ (23)

Integrating out the momentum fraction, we can finally obtain

$\displaystyle
H_{i}=\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\frac{m_{B}}{\lambda_{B}}\cdot$
$\displaystyle\Big{[}3(a_{0}^{M_{1}}+a_{1}^{M_{1}}+a_{2}^{M_{1}}+a_{3}^{M_{1}})\cdot
3(a_{0}^{M_{2}}+a_{1}^{M_{2}}+a_{2}^{M_{2}}+a_{3}^{M_{2}})$ $\displaystyle
H_{i}=\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\frac{m_{B}}{\lambda_{B}}\cdot$
$\displaystyle~{}~{}\pm
3\gamma_{\chi}^{M_{1}}(a_{0}^{M_{2}}-a_{1}^{M_{2}}+a_{2}^{M_{2}}-a_{3}^{M_{2}})X_{H}\Big{]},$
$M_{1}=P~{}(S)$, (24) $\displaystyle
H_{i}=\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\frac{m_{B}}{\lambda_{B}}\cdot$
$\displaystyle\Big{[}3(a_{0}^{M_{1}}+a_{1}^{M_{1}}+a_{2}^{M_{1}}+a_{3}^{M_{1}})\cdot
3(a_{0}^{M_{2}}+a_{1}^{M_{2}}+a_{2}^{M_{2}}+a_{3}^{M_{2}})$ $\displaystyle
H_{i}=\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\frac{m_{B}}{\lambda_{B}}\cdot$
$\displaystyle~{}~{}+3\gamma_{\chi}^{M_{1}}(a_{0}^{M_{2}}-a_{1}^{M_{2}}+a_{2}^{M_{2}}-a_{3}^{M_{2}})(3X_{H}-6)\Big{]},$
$M_{1}=V$, (25)

for $i=1-4,9,10$, and

$\displaystyle
H_{i}=-\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\frac{m_{B}}{\lambda_{B}}\cdot$
$\displaystyle\Big{[}3(a_{0}^{M_{1}}+a_{1}^{M_{1}}+a_{2}^{M_{1}}+a_{3}^{M_{1}})\cdot
3(a_{0}^{M_{2}}-a_{1}^{M_{2}}+a_{2}^{M_{2}}-a_{3}^{M_{2}})$ $\displaystyle
H_{i}=-\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\frac{m_{B}}{\lambda_{B}}\cdot$
$\displaystyle~{}~{}\pm
3\gamma_{\chi}^{M_{1}}(a_{0}^{M_{2}}+a_{1}^{M_{2}}+a_{2}^{M_{2}}+a_{3}^{M_{2}})X_{H}\Big{]},$
$M_{1}=P~{}(S)$, (26) $\displaystyle
H_{i}=-\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\frac{m_{B}}{\lambda_{B}}\cdot$
$\displaystyle\Big{[}3(a_{0}^{M_{1}}+a_{1}^{M_{1}}+a_{2}^{M_{1}}+a_{3}^{M_{1}})\cdot
3(a_{0}^{M_{2}}-a_{1}^{M_{2}}+a_{2}^{M_{2}}-a_{3}^{M_{2}})$ $\displaystyle
H_{i}=-\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\frac{m_{B}}{\lambda_{B}}\cdot$
$\displaystyle~{}~{}+3\gamma_{\chi}^{M_{1}}(a_{0}^{M_{2}}+a_{1}^{M_{2}}+a_{2}^{M_{2}}+a_{3}^{M_{2}})(3X_{H}-6)\Big{]},$
$M_{1}=V$, (27)

for $i=5$ and $7$, where, the upper (lower) sign in Eqs. (24) and (26) is
applied when $M_{1}=P$ ($S$). As has been discussed in many previous works
[70, 33, 56, 71, 72, 73, 74, 75, 76], hard-spectator corrections suffer from
the end-point divergence, which is usually parameterized by the end-point
parameter

$\displaystyle
X_{H}=\ln\left(\frac{m_{B}}{\Lambda_{h}}\right)(1+\rho_{H}e^{i\phi_{H}})\,.$
(28)

The parameters $\rho_{H}$ and $\phi_{H}$ reflect the strength and possible
strong phase of the end-point contributions, respectively.

The amplitudes of $B_{u,d,s}\to K_{0}^{*}(1430)P$ and $K_{0}^{*}(1430)V$
decays collected in the appendix B also receives the contributions of weak
annihilation, which are involved in the effective coefficients $\beta_{i}^{p}$
defined as

$\displaystyle\beta_{i}^{p}(M_{1}M_{2})\equiv\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}b_{i}^{p}\,,$
(29)

where,222 In Refs. [30, 31], the superscript of the last terms in $b_{4}^{p}$
and $b_{3,EW}^{p}$ (Eq. (3.12)) should be corrected, i.e., $A_{2}^{f}\to
A_{2}^{i}$ and $A_{3}^{i}\to A_{3}^{f}$.

$\displaystyle b_{1}$ $\displaystyle=\frac{C_{F}}{N_{c}^{2}}C_{1}A_{1}^{i},$
$\displaystyle b_{2}$ $\displaystyle=\frac{C_{F}}{N_{c}^{2}}C_{2}A_{1}^{i},$
$\displaystyle b_{3}^{p}$
$\displaystyle=\frac{C_{F}}{N_{c}^{2}}[C_{3}A_{1}^{i}+C_{5}(A_{3}^{i}+A_{3}^{f})+N_{c}C_{6}A_{3}^{f}],$
$\displaystyle b_{4}^{p}$
$\displaystyle=\frac{C_{F}}{N_{c}^{2}}[C_{4}A_{1}^{i}+C_{6}A_{2}^{i}],$
$\displaystyle b_{3,EW}^{p}$
$\displaystyle=\frac{C_{F}}{N_{c}^{2}}[C_{9}A_{1}^{i}+C_{7}(A_{3}^{i}+A_{3}^{f})+N_{c}C_{8}A_{3}^{f}],$
$\displaystyle b_{4,EW}^{p}$
$\displaystyle=\frac{C_{F}}{N_{c}^{2}}[C_{10}A_{1}^{i}+C_{8}A_{2}^{i}].$ (30)

The subscripts $n=1\,,2\,,3$ of $A_{n}^{i,f}$ correspond to the possible Dirac
structures $(V-A)(V-A)$, $(V-A)(V+A)$ and $(S-P)(S+P)$, respectively; and the
supscripts $i$ and $f$ refer to gluon emission from the initial- and final-
state quarks, respectively. The explicit expressions of $A_{n}^{i,f}$ for
$B\to PS$ and $SP$ decays can be written as

$\displaystyle A_{1}^{i}$
$\displaystyle=\pi\alpha_{s}\int_{0}^{1}dxdy\left\\{\Phi_{M_{2}}(x)\Phi_{M_{1}}(y)\left[\frac{1}{y(1-x\bar{y})}+\frac{1}{\bar{x}^{2}y}\right]\mp\gamma_{\chi}^{M_{1}}\gamma_{\chi}^{M_{2}}\phi_{M_{2}}(x)\phi_{M_{1}}(y)\frac{2}{\bar{x}y}\right\\}\,,$
(31) $\displaystyle A_{1}^{f}$ $\displaystyle=0\,;$ (32) $\displaystyle
A_{2}^{i}$
$\displaystyle=\pi\alpha_{s}\int_{0}^{1}dxdy\left\\{-\Phi_{M_{2}}(x)\Phi_{M_{1}}(y)\left[\frac{1}{\bar{x}(1-x\bar{y})}+\frac{1}{\bar{x}y^{2}}\right]\pm\gamma_{\chi}^{M_{1}}\gamma_{\chi}^{M_{2}}\phi_{M_{2}}(x)\phi_{M_{1}}(y)\frac{2}{\bar{x}y}\right\\}\,,$
(33) $\displaystyle A_{2}^{f}$ $\displaystyle=0,$ (34) $\displaystyle
A_{3}^{i}$
$\displaystyle=\pi\alpha_{s}\int_{0}^{1}dxdy\left\\{\pm\gamma_{\chi}^{M_{1}}\Phi_{M_{2}}(x)\phi_{M_{1}}(y)\frac{2\bar{y}}{\bar{x}y(1-x\bar{y})}+\gamma_{\chi}^{M_{2}}\phi_{M_{2}}(x)\Phi_{M_{1}}(y)\frac{2x}{\bar{x}y(1-x\bar{y})}\right\\}\,,$
(35) $\displaystyle A_{3}^{f}$
$\displaystyle=\pi\alpha_{s}\int_{0}^{1}dxdy\left\\{\pm\gamma_{\chi}^{M_{1}}\Phi_{M_{2}}(x)\phi_{M_{1}}(y)\frac{2(1+\bar{x})}{\bar{x}^{2}y}-\gamma_{\chi}^{M_{2}}\phi_{M_{2}}(x)\Phi_{M_{1}}(y)\frac{2(1+y)}{\bar{x}y^{2}}\right\\}\,,$
(36)

where, the upper and lower signs are applied to $B\to PS$ and $SP$ decays,
respectively. The $A_{n}^{i,f}$ for $B\to VS$ decay can be obtained from the
$A_{n}^{i,f}$ for $B\to PS$ decay by changing the sign of second term in
$A_{1,3}^{i}$ and $A_{3}^{f}$, and the first term in $A_{2}^{i}$; while, the
$A_{n}^{i,f}$ for $B\to SV$ decay is the same as $A_{n}^{i,f}$ for $B\to SP$
decay except for the overall sign of $A_{2}^{i}$. After integrating out the
momentum fractions, we can finally obtain

$\displaystyle A_{1}^{i}(PS)$ $\displaystyle\approx
2\pi\alpha_{s}\bigg{\\{}9\bigg{[}a_{0}^{{M_{2}}}(X_{A}-4+\frac{\pi^{2}}{3})+a_{1}^{{M_{2}}}(3X_{A}+4-\pi^{2})+a_{2}^{{M_{2}}}(6X_{A}-\frac{107}{3}+2\pi^{2})$
$\displaystyle+a_{3}^{{M_{2}}}(10X_{A}+\frac{23}{18}-\frac{10}{3}\pi^{2})\bigg{]}-\gamma_{\chi}^{P}\gamma_{\chi}^{S}X_{A}^{2}\bigg{\\}},$
(37) $\displaystyle A_{2}^{i}(PS)$ $\displaystyle\approx
2\pi\alpha_{s}\bigg{\\{}-9\bigg{[}a_{0}^{{M_{2}}}(X_{A}-4+\frac{\pi^{2}}{3})+a_{1}^{{M_{2}}}(X_{A}+29-3\pi^{2})+a_{2}^{{M_{2}}}(X_{A}-119+12\pi^{2})$
$\displaystyle+a_{3}^{{M_{2}}}(X_{A}+\frac{2956}{9}-\frac{100}{3}\pi^{2})\bigg{]}+\gamma_{\chi}^{P}\gamma_{\chi}^{S}X_{A}^{2}\bigg{\\}},$
(38) $\displaystyle A_{3}^{i}(PS)$ $\displaystyle\approx
6\pi\alpha_{s}\bigg{\\{}\gamma_{\chi}^{P}\bigg{[}a_{0}^{{M_{2}}}(X_{A}^{2}-2X_{A}+\frac{\pi^{2}}{3})+3a_{1}^{{M_{2}}}(X_{A}^{2}-4X_{A}+4+\frac{\pi^{2}}{3})$
$\displaystyle+6a_{2}^{{M_{2}}}(X_{A}^{2}-\frac{16}{3}X_{A}+\frac{15}{2}+\frac{\pi^{2}}{3})+10a_{3}^{{M_{2}}}(X_{A}^{2}-\frac{19}{3}X_{A}+\frac{191}{18}+\frac{\pi^{2}}{3})\bigg{]}$
$\displaystyle+\gamma_{\chi}^{S}(X_{A}^{2}-2X_{A}+\frac{\pi^{2}}{3})\bigg{\\}},$
(39) $\displaystyle A_{3}^{f}(PS)$ $\displaystyle\approx
6\pi\alpha_{s}X_{A}\bigg{\\{}\gamma_{\chi}^{P}\bigg{[}a_{0}^{{M_{2}}}(2X_{A}-1)+a_{1}^{{M_{2}}}(6X_{A}-11)+a_{2}^{{M_{2}}}(12X_{A}-31)$
$\displaystyle+a_{3}^{{M_{2}}}(20X_{A}-\frac{187}{3})\bigg{]}-\gamma_{\chi}^{S}(2X_{A}-1)\bigg{\\}}$
(40)

for $M_{1}M_{2}=PS$, and

$\displaystyle A_{1}^{i}(VS)$ $\displaystyle\approx
6\pi\alpha_{s}\bigg{\\{}3\bigg{[}a_{0}^{{M_{2}}}(X_{A}-4+\frac{\pi^{2}}{3})+a_{1}^{{M_{2}}}(3X_{A}+4-\pi^{2})+a_{2}^{{M_{2}}}(6X_{A}-\frac{107}{3}+2\pi^{2})$
$\displaystyle+a_{3}^{{M_{2}}}(10X_{A}+\frac{23}{18}-\frac{10}{3}\pi^{2})\bigg{]}-\gamma_{\chi}^{V}\gamma_{\chi}^{S}X_{A}(X_{A}-2)\bigg{\\}},$
(41) $\displaystyle A_{2}^{i}(VS)$ $\displaystyle\approx
6\pi\alpha_{s}\bigg{\\{}3\bigg{[}a_{0}^{{M_{2}}}(X_{A}-4+\frac{\pi^{2}}{3})+a_{1}^{{M_{2}}}(X_{A}+29-3\pi^{2})+a_{2}^{{M_{2}}}(X_{A}-119+12\pi^{2})$
$\displaystyle+a_{3}^{{M_{2}}}(X_{A}+\frac{2956}{9}-\frac{100}{3}\pi^{2})\bigg{]}-\gamma_{\chi}^{V}\gamma_{\chi}^{S}X_{A}(X_{A}-2)\bigg{\\}},$
(42) $\displaystyle A_{3}^{i}(VS)$ $\displaystyle\approx
6\pi\alpha_{s}\bigg{\\{}-\gamma_{\chi}^{V}\bigg{[}3a_{0}^{{M_{2}}}(X_{A}^{2}-2X_{A}+4-\frac{\pi^{2}}{3})+9a_{1}^{{M_{2}}}(X_{A}^{2}-4X_{A}-4+\pi^{2})$
$\displaystyle+3a_{2}^{{M_{2}}}(6X_{A}^{2}-32X_{A}+79-2\pi^{2})+10a_{3}^{{M_{2}}}(3X_{A}^{2}-19X_{A}+\frac{61}{6}+3\pi^{2})\bigg{]}$
$\displaystyle-\gamma_{\chi}^{S}(X_{A}^{2}-2X_{A}+\frac{\pi^{2}}{3})\bigg{\\}},$
(43) $\displaystyle A_{3}^{f}(VS)$ $\displaystyle\approx
6\pi\alpha_{s}\bigg{\\{}-3\gamma_{\chi}^{V}(X_{A}-2)\bigg{[}a_{0}^{{M_{2}}}(2X_{A}-1)+a_{1}^{{M_{2}}}(6X_{A}-11)+a_{2}^{{M_{2}}}(12X_{A}-31)$
$\displaystyle+a_{3}^{{M_{2}}}(20X_{A}-\frac{187}{3})\bigg{]}+\gamma_{\chi}^{S}X_{A}(2X_{A}-1)\bigg{\\}}$
(44)

for $M_{1}M_{2}=VS$, where, $X_{A}$ is the endpoint parameter and is defined
in the same manner as $X_{H}$ given by Eq. (28). The results for the cases of
$M_{1}M_{2}=SP$ and $M_{1}M_{2}=SV$ can be obtained via the relation

$\displaystyle A_{1}^{i}(SP)$
$\displaystyle=A_{2}^{i}(PS),~{}A_{2}^{i}(SP)=A_{1}^{i}(PS),~{}A_{3}^{i}(SP)=-A_{3}^{i}(PS),~{}A_{3}^{f}(SP)=A_{3}^{f}(PS),$
(45) $\displaystyle A_{1}^{i}(SV)$
$\displaystyle=-A_{2}^{i}(VS),~{}A_{2}^{i}(SV)=-A_{1}^{i}(VS),~{}A_{3}^{i}(SV)=A_{3}^{i}(VS),~{}A_{3}^{f}(SV)=-A_{3}^{f}(VS)\,,$
(46)

but with different sign of $a_{0}^{{M_{2}}}$ and $a_{2}^{{M_{2}}}$. In above
evaluation, the asymptotic DAs of $P$ and $V$ mesons are used for simplicity;
while, for the DA of $S$ meson, the first four terms in Gegenbauer expansion
are included because of the dominance of $a_{1}^{S}$. In the previous works
[30, 31, 32, 33], the contributions related to $a_{0}^{S}$ and $a_{2}^{S}$ are
neglected, which is reasonable for the cases of scalar $(u,d)$ state and
quarkonium because $a_{0,2}^{S}\simeq 0$. However, for the case of
$K^{*}_{0}(1430)$, the contributions related to non-zero $a_{0,2}^{S}$ are
possibly nontrivial compared with the ones related to $a_{1,3}^{S}$, and hence
are considered in this work.

##  3 Numerical Results and Discussions

Before presenting our numerical results, we would like to clarify the input
parameters used in the evaluations. For the CKM matrix elements, we adopt the
Wolfenstein parameterization and choose the four parameters as [77]

$\displaystyle
A=0.790^{+0.017}_{-0.012}\,,\quad\lambda=0.22650^{+0.00048}_{-0.00048}\,,\quad\bar{\rho}=0.141^{+0.016}_{-0.017},\quad\bar{\eta}=0.357^{+0.011}_{-0.011}.$
(47)

As for the quark masses, we take [77]

$\displaystyle m_{s}(\mu)/m_{q}(\mu)=27.3^{+0.7}_{-1.3}\,,\quad m_{s}(2{\rm
GeV})=93^{+11}_{-5}\,{\rm MeV}\,,\quad m_{b}(m_{b})=4.18^{+0.03}_{-0.02}\,{\rm
GeV}\,,$ (48) $\displaystyle m_{c}=1.67\pm 0.07\,{\rm GeV}\,,\quad
m_{b}=4.78\pm 0.06\,{\rm GeV}\,,\quad m_{t}=172.76\pm 0.30\,{\rm GeV}\,,$ (49)

where $m_{q}\equiv(m_{u}+m_{d})/2$. For the well-determined Fermi coupling
constant, masses of mesons and lifetimes of $B$ mesons, we take their default
values given by PDG [77].

The nonperturbative inputs used in this work include decay constant,
Gegenbauer moment and form factor. Unfortunately, for the $B_{u,d,s}\to
K_{0}^{*}(1430)P$ and $K_{0}^{*}(1430)V$ decays concerned in this work, some
of these nonperturbative inputs are not known. The standard light-front (SLF)
approach [78, 79, 80, 81] provides a conceptually simple and
phenomenologically feasible framework for calculating the non-perturbative
quantities of hadrons. However, it is powerless for determining the zero-mode
contributions by itself and the Lorentz covariance is lost. In order to cover
the shortages of SLF approach, a manifestly covariant light-front (CLF)
approach is exploited [82, 83, 56] with the help of the manifestly covariant
Bethe-Salpeter (BS) approach, and has been applied to study the $B_{u,d}\to
SP$ and $SV$ decays [30, 31, 32, 33]. Unfortunately, this traditional CLF
approach has some self-consistence problems, and the covariance in fact can
not be strictly guaranteed due to the residual spurious $\omega$-dependent
contribution [56, 84, 85]. In order to resolve these problems, a self-
consistent scheme is presented in Ref. [84] by improving the correspondence
between CLF and BS calculation, and has been tested in, for instance, Refs.
[87, 93, 89, 86, 90, 85, 92, 88, 91]. Most of the results based on such
improved self-consistent CLF approach for the decay constant and form factors
generally agree with the experimental data and the predictions obtained by
using Lattice QCD (LQCD) and light cone sum rules (LCSR) (some examples can be
found in Refs. [86, 90, 88, 89, 87]) , while the self-consistent CLF results
for some DAs of light $P$-mesons ( for instance, the twist-3 DAs of $\pi$ and
$K$ mesons [93, 94]) are different from the QCD sum rule (QCD SR) results.

In the previous works for $B_{u,d}\to SP$ and $SV$ decays [30, 31, 32, 33],
the decay constant and Gegenbauer moments are evaluated by using the QCD SR,
while and the form factor is evaluated by using the traditional CLF approach.
In this work, all of the nonperturbative parameters will be calculated by
using the self-consistent CLF approach for consistence. The theoretical
framework for the decay constant of $S$, $P$ and $V$ mesons and the form
factors of $P\to(S\,,P\,,V)$ transitions within the CLF approach has been
given in, for instance, Refs. [56, 86], and the self-consistent correspondence
relation between the BS and the LF approaches has been discussed in detail in
Refs. [84, 87, 93, 89, 86, 90, 85, 92]. Using the Gaussian-type wavefunctions
333Our sign convention for $\psi_{2p}(x,{\bf k}_{\bot})$ is different from the
one in Ref. [95], and ensures the positive decay constant of $S(2P)$ and form
factor of $P\to S(2P)$ transition. It should be noted that the different
conventions do not affect the final results of observables.

$\displaystyle\psi_{1s}(x,{\bf k}_{\bot})$ $\displaystyle=$
$\displaystyle\frac{4\pi^{\frac{3}{4}}}{\beta^{\frac{3}{2}}}\sqrt{\frac{\partial
k_{z}}{\partial
x}}\exp\left[-\frac{\overrightarrow{k}^{2}}{2\beta^{2}}\right]\,,$ (50)
$\displaystyle\psi_{1p}(x,{\bf k}_{\bot})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{\beta}\psi_{1s}(x,{\bf k}_{\bot})\,,$ (51)
$\displaystyle\psi_{2p}(x,{\bf k}_{\bot})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{5}{2}}\left(\frac{2\overrightarrow{k}^{2}}{5\beta^{2}}-1\right)\psi_{1p}(x,{\bf
k}_{\bot})\,,$ (52)

and the values of input parameters collected in Ref. [85], we obtain (in units
of MeV)

$\displaystyle f_{K_{0}^{*}(1430)}=18\pm
5\qquad\text{S1},\qquad\qquad\,\,\,f_{K_{0}^{*}(1430)}=43\pm
10\qquad\text{S2},$ (53) $\displaystyle f_{B_{u,d}}=186\pm 7,\qquad\quad
f_{B_{s}}=224\pm 9,\qquad\quad f_{\pi}=131\pm 7,\qquad\quad\,f_{K}=156\pm 5,$
(54) $\displaystyle f_{K^{*}}=205\pm 8,\qquad\quad\,\,f_{\rho}=210\pm
4,\qquad\quad\,\,~{}f_{\omega}=210\pm 4,\qquad\quad\,f_{\phi}=228\pm 5,$ (55)
$\displaystyle f_{K^{*}}^{\perp}=173\pm 6,\qquad\quad\,f_{\rho}^{\perp}=167\pm
4,\qquad\quad\,~{}f_{\omega}^{\perp}=167\pm 4,\qquad\quad
f_{\phi}^{\perp}=198\pm 4,$ (56)

where, as has been mentioned in the introduction, S1 and S2 are the two
assumptions for $K_{0}^{*}(1430)$ meson that

  * •

S1: $K_{0}^{*}(1430)$ is the first excited p-wave two-quark state;

  * •

S2: $K_{0}^{*}(1430)$ is the lowest-lying p-wave two-quark state.

It is found that our result for $f_{K_{0}^{*}(1430)}$ in S2 is in agreement
with the results $f_{K_{0}^{*}}=34\pm 7\,\rm MeV$ [96] and $42\pm 8\,\rm MeV$
[97] obtained within QCD SR and $f_{K_{0}^{*}}=42\pm 2\,\rm MeV$ [98] obtained
within the finite-energy sum rules. The heavy lepton $\tau$ decay process
$\tau\to K^{*}_{0}(1430)\nu_{\tau}$ is very suitable for testing the value of
$f_{K_{0}^{*}}$ because there is only one hadron evolved in this decay, the
interaction in the leptonic vertex can be calculated with high precision, and
its branching fraction is dependent on $f_{K_{0}^{*}}$ directly. Using our
prediction $f_{K_{0}^{*}}=43\,{\rm MeV}$ and the formulae given in Ref. [97],
we obtain ${\cal B}(\tau\to K^{*}_{0}(1430)\nu)\simeq 0.81\times 10^{-4}$
(S2), which is allowed by current data ${\cal B}(\tau\to
K^{*}_{0}(1430)\nu)^{\rm exp.}<5\times 10^{-4}$ [77].

Table 1: Form factors of $B\to K_{0}^{*}(1430),K,K^{*},\pi,\rho,\omega$ and $B_{s}\to K_{0}^{*}(1430),K,K^{*},\phi$ transitions in S1 (lower entry) and S2 (upper entry) by using the self-consistent CLF approach. F | $\quad F(0)$ | a | b |  F | $\,\,\,F(0)$ | a | b
---|---|---|---|---|---|---|---
$U_{1}^{B\to K_{0}^{*}}$ | $0.29\pm 0.02$ | 1.27 | 0.33 | $U_{0}^{B\to K_{0}^{*}}$ | $0.29\pm 0.02$ | 0.16 | 0.11
| $0.18\pm 0.01$ | 1.03 | 0.15 |  | $0.18\pm 0.01$ | $-0.23$ | 0.29
$U_{1}^{B_{s}\to K_{0}^{*}}$ | $0.28\pm 0.02$ | 1.58 | 0.84 | $U_{0}^{B_{s}\to K_{0}^{*}}$ | $0.28\pm 0.02$ | 0.55 | 0.20
| $0.23\pm 0.02$ | 0.92 | 0.29 |  | $0.23\pm 0.02$ | $-0.23$ | 0.36
$F_{0}^{B\to\pi}$ | $0.27\pm 0.03$ | 0.65 | 0.03 | $F_{0}^{B\to K}$ | $0.34\pm 0.03$ | 0.66 | 0.07
$F_{0}^{B_{s}\to K}$ | $0.23\pm 0.03$ | 0.95 | 0.28 | $A_{0}^{B\to\omega}$ | $0.29\pm 0.03$ | 1.54 | 0.74
$A_{0}^{B\to\rho}$ | $0.29\pm 0.03$ | 1.54 | 0.74 | $A_{0}^{B\to K^{*}}$ | $0.34\pm 0.04$ | 1.52 | 0.65
$A_{0}^{B_{s}\to K^{*}}$ | $0.21\pm 0.04$ | 1.94 | 1.62 | $A_{0}^{B_{s}\to\phi}$ | $0.28\pm 0.04$ | 1.80 | 1.29
Table 2: Form factors of $B\to K,K^{*},\pi,\rho,\omega$ and $B_{s}\to K,K^{*},\phi$ transitions at $q^{2}=0\,{\rm GeV}^{2}$ obtained in this and some previous works . | This work | LCSR | SCET | LFQM | pQCD | CCQM
---|---|---|---|---|---|---
|  | [99, 100] | [101] | [102] | [103, 104] | [105]
$F_{0}^{B\to\pi}$ | $0.27\pm 0.03$ | $0.258\pm 0.030$ | 0.247 | $0.25$ | $0.26^{+0.05}_{-0.05}$ | $0.283\pm 0.019$
$F_{0}^{B\to K}$ | $0.34\pm 0.03$ | $0.331\pm 0.040$ | 0.297 | $0.34$ | $0.31^{+0.05}_{-0.05}$ | —
$F_{0}^{B_{s}\to K}$ | $0.23\pm 0.03$ | — | 0.290 | $0.23$ | $0.26^{+0.05}_{-0.05}$ | $0.247\pm 0.015$
$A_{0}^{B\to\rho}$ | $0.29\pm 0.03$ | $0.303\pm 0.028$ | 0.260 | $0.32$ | $0.25^{+0.07}_{-0.06}$ | $0.266\pm 0.013$
$A_{0}^{B\to\omega}$ | $0.29\pm 0.03$ | $0.281\pm 0.030$ | 0.240 | $0.28$ | $0.23^{+0.06}_{-0.05}$ | $0.236\pm 0.011$
$A_{0}^{B\to K^{*}}$ | $0.34\pm 0.04$ | $0.374\pm 0.034$ | 0.283 | $0.38$ | $0.31^{+0.09}_{-0.07}$ | —
$A_{0}^{B_{s}\to K^{*}}$ | $0.21\pm 0.04$ | $0.363\pm 0.034$ | 0.279 | $0.25$ | $0.24^{+0.06}_{-0.05}$ | $0.225\pm 0.090$
$A_{0}^{B_{s}\to\phi}$ | $0.28\pm 0.04$ | $0.474\pm 0.037$ | 0.279 | $0.31$ | $0.31^{+0.09}_{-0.07}$ | —

Our results for the form factors of $B\to
K_{0}^{*}(1430),K,K^{*},\pi,\rho,\omega$ and $B_{s}\to
K_{0}^{*}(1430),K,K^{*},\phi$ transitions are collected in Table 1. Our
results $U_{0,1}^{B\to K_{0}^{*}}=$$0.29\pm 0.02$(S2) and $0.18\pm 0.01$(S1)
are a little larger and smaller than the traditional CLF results
$U_{0,1}^{B\to K_{0}^{*}}=$$0.26$(S2) and $0.21$(S1) [30]. The results for
$B_{s}\to K_{0}^{*}$ transition are first obtained in this work. The previous
results for the $B\to K,K^{*},\pi,\rho,\omega$ and $B_{s}\to K,K^{*},\phi$
transitions at $q^{2}=0\,{\rm GeV}^{2}$ obtained via other approaches are
collected in Table 2 for comparison. It can be found that these results based
on different approaches are generally consistent with each other except for
the large $A_{0}^{B_{s}\to(K^{*},\phi)}=(0.363\pm 0.034\,,0.474\pm 0.037)$
predicted by LCSR [99, 100] (relatively small results
$A_{0}^{B_{s}\to(K^{*},\phi)}=(0.314\pm 0.048\,,0.389\pm 0.045)$ are obtained
in Ref. [110]). From above discussion and Table 2, we can roughly conclude
that the uncertainties of form factors associated with $B$ and $B_{s}$ decays
caused by different approaches are less than about $20\%$ and $35\%$,
respectively.

Table 3: Gegenbauer moments of pseudoscalar and vector mesons at $\mu=1\rm GeV$. |  | $a_{1}$ | $a_{2}$ | $a_{3}$
---|---|---|---|---
$\pi$ | this work | $0$ | $0.10\pm 0.04$ | $0$
| LQCD[106] | $0$ | $0.116\pm 0.020$ | $0$
| QCD SR[107] | $0$ | $0.25\pm 0.15$ | $0$
| linear(HO)[108] | $0$ | $0.12(0.05)$ | $0$
$\bar{K}$ | this work | $0.09\pm 0.01$ | $0.02\pm 0.02$ | $0.05\pm 0.01$
| LQCD[106] | $0.053^{+0.031}_{-0.033}$ | $0.106\pm 0.016$ | —
| QCD SR[107] | $0.06\pm 0.03$ | $0.25\pm 0.15$ | —
| linear(HO)[108] | $0.09(0.13)$ | $0.03(-0.03)$ | $0.06(0.04)$
$\rho$ | this work | $0$ | $-0.01\pm 0.02$ | $0$
| QCD SR[100] | $0$ | $0.09^{+0.10}_{-0.07}$ | $0$
| QCD SR[109] | $0$ |  $0.15\pm 0.07$ | $0$
| linear(HO)[108] | $0$ |  $0.02(-0.02)$ | $0$
$\bar{K}^{*}$ | this work | $0.13\pm 0.04$ | $-0.07\pm 0.00$ | $0.02\pm 0.00$
| QCD SR[100] | $0.10\pm 0.07$ | $0.07^{+0.09}_{-0.07}$ | —
| QCD SR[109] | $0.03\pm 0.02$ | $0.11\pm 0.09$ | —
| linear(HO)[108] | $0.11(0.14)$ | $-0.03(-0.07)$ | $0.03(0.02)$
$\phi$ | this work | $0$ | $-0.13\pm 0.03$ | $0$
| QCD SR[100] | $0$ | $0.06^{+0.09}_{-0.07}$ | $0$
| QCD SR[109] | $0$ | $0.18\pm 0.08$ | $0$
Table 4: Gegenbauer moments of ${K}_{0}^{*}(1430)$ at $\mu=1\rm GeV$ in S1 (lower entry) and S2 (upper entry). | $a_{1}$ | $a_{2}$ | $a_{3}$
---|---|---|---
this work | $-1.65\pm 0.38$ | $-0.38\pm 0.02$ | $-0.06\pm 0.01$
| $-1.93\pm 0.69$ | $0.75\pm 0.13$ | $-1.13\pm 0.72$
QCD SR [30] | $-7.21\pm 1.64$ | — | $-5.31\pm 2.70$
| $7.33\pm 0.89$ | — | $-15.17\pm 1.01$

Figure 1: The DA of ${K}_{0}^{*}(1430)$ predicted in this work and QCD SR
[30]. The dashed and solid lines correspond to the truncations up to $n=1$ and
$n=3$, respectively, according to Eq. (81).

The DAs of mesons have been studied within the LF approaches in, for instance,
Refs. [84, 94, 93]. The Gegenbauer moments can be extracted from DAs via

$\displaystyle a_{n}^{M}=\frac{2(2n+3)}{3(n+1)(n+2)}\int_{0}^{1}{\rm
d}x\,C_{n}^{3/2}(x-\bar{x})\Phi(x)\,,$ (57)

where, $C_{n}^{3/2}$ are Gegenbauer polynomials. Our numerical results for the
Gegenbauer moments at $\mu=1\rm GeV$ are collected in Tables 3 and 4. The
results obtained by using LQCD [106], QCD SR [107, 109, 100, 30] and
traditional LFQM with linear confining and harmonic oscillator (HO) potentials
[108] are also collected in these tables for comparison. For the pseudoscalar
and vector mesons, our results are generally agree with the results based on
the other approaches within errors except for the signs of
$a_{2}^{\rho,K^{*},\phi}$, which is caused by the fact that the DAs predicted
by LF approaches are usually different from the QCD SR predictions especially
at $x\to 0$ or $1$ (some examples can be found in, for instance, Refs. [93,
108, 111]). For the ${K}_{0}^{*}(1430)$ meson, it is found from Table 4 that
our results are much smaller than the ones given by QCD SR [30]. In order to
clearly show their difference, we plot the DA of ${K}_{0}^{*}(1430)$ in Fig.
2. For the case of $n=1$ (dashed lines), our results in S1 and S2 are similar,
while the QCD SR results in S1 and S2 are totally different with each other
due to the different signs of $a_{1}^{{K}_{0}^{*}}$ (QCD SR, $-7.21$ vs.
$7.33$). In addition, the $a_{3}^{{K}_{0}^{*}}$ correction to
$\Phi_{{K}_{0}^{*}}(x)$ in our evaluation is not as significant as the one
predicted by QCD SR. More theoretical and experimental efforts are needed for
a clear picture of $\Phi_{{K}_{0}^{*}}(x)$. 444The $\gamma\gamma\to
K^{*+}_{0}(1430)K^{*-}_{0}(1430)\to 2K2\pi$ process may provide a way to test
the DA of $K^{*}_{0}(1430)$, which is similar to the cases of
$\gamma\gamma\to\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ processes for testing the DAs
of $\pi$ and $K$ mesons [112, 113]. Such process is in the scope of Belle(-II)
in principle [114], but the specific measurement has not been made.

Figure 2: The dependences of measured ${\cal B}\left(B_{u,d,s}\to
K_{0}^{*}(1430)(\pi\,,\rho\,,\omega\,,\phi)\right)$ on $\phi_{A}$ with
$(\rho_{A},\rho_{H})={(1,0)}$ and $\phi_{H}$ with
$(\rho_{A},\rho_{H})={(0,1)}$ are shown by the dash and dash-dotted lines,
respectively. The solid lines are plotted with the simplification that
$X_{H}(\rho_{H},\phi_{H})=X_{A}(\rho_{A},\phi_{A})$ with $\rho_{A}=1$. The
shaded region is the experimental result with $1\sigma$ error bar. See text
for further discussion.

Figure 3: The dependences of measured $A_{CP}\left(B_{u,d}\to
K_{0}^{*}(1430)(\pi\,,\rho\,,\omega\,,\phi)\right)$ on $\phi_{A}$ with
$\rho_{A}=1$ in S1 and S2. The shaded region is the experimental result with
$1\sigma$ error bar. See text for further discussion.

Using the inputs given above, we then present our numerical results and
discussion. The QCDF approach itself cannot give the exact values of the end-
point parameters $X_{H}(\rho_{H},\phi_{H})$ in the hard-spectator scattering
corrections and $X_{A}(\rho_{A},\phi_{A})$ in the annihilation amplitudes,
which can only be determined by fitting to the experimental data. The current
data indicate that $\rho_{A(H)}$ is at the level of 1, but $\phi_{A(H)}$ is
generally different for kinds of decay types. For instance,
$\phi_{A}=-55^{\circ}$, $-22^{\circ}$ and $-70^{\circ}$ with $\rho_{A}=1$ (the
so-called “Scenario S4” in Ref. [70]) are favored by the $B\to PP$, $PV$ and
$VP$ decays, respectively [70, 71, 72]. In order to clearly show the effects
of end-point contributions, we plot the dependences of the measured ${\cal
B}(B_{u,d}\to K_{0}^{*}(1430)(\pi\,,\rho\,,\omega\,,\phi))$ on the end-point
parameters in Fig. 2, in addition, the experimental data are also shown in
such figure for comparison. The dash and dash-dotted lines show the
dependences on $\phi_{A}$ with $(\rho_{A},\rho_{H})={(1,0)}$ and $\phi_{H}$
with $(\rho_{A},\rho_{H})={(0,1)}$, respectively. Comparing the dash with the
dash-dotted lines, it can be found that the measured decay modes are very
sensitive to $X_{A}(\rho_{A},\phi_{A})$, but the effect of
$X_{H}(\rho_{H},\phi_{H})$ is trivial. It can be easily understood because
these decay modes are penguin-dominated, and the hard-spectator scattering
corrections may be significant only when the decays are dominated by the
color-suppressed tree contribution [70]. Therefore, we assume that the end-
point parameters are universal in the annihilation and hard-spectator
scattering corrections, i.e.,
$X_{H}(\rho_{H},\phi_{H})=X_{A}(\rho_{A},\phi_{A})$, in our following
discussions for simplification. 555In the decay modes considered in this work,
$\bar{B}_{s}\to K_{0}^{*0}(\pi^{0},\rho^{0},\omega)$ decays are color-
suppressed tree dominated, thus the simplification
$X_{H}(\rho_{H},\phi_{H})=X_{A}(\rho_{A},\phi_{A})$ may affect our following
predictions for these decays.  Under this assumption, the dependences of the
measured branching fractions and direct CP asymmetries of $B_{u,d}\to
K_{0}^{*}(1430)(\pi\,,\rho\,,\omega\,,\phi)$ decays on the end-point
parameters are shown in Figs. 2 and 3 (solid lines), respectively.

  * •

From Figs. 2 and 3, it can be found that the scenarios S1 and S2 give
significantly different results for the branching fractions, but the direct CP
asymmetries are similar with each other. As shown by Fig. 2, the results based
on S2 could agree well with the experimental data at the level of $1\sigma$
when a proper value of $\phi_{A}$ is taken; however, the results for most of
decay modes based on S1 are much smaller than data for any value of
$\phi_{A}$. This implies that the measured $B_{u,d}\to K_{0}^{*}(1430)P$ and
$K_{0}^{*}(1430)V$ decays favor $K_{0}^{*}(1430)$ as the lowest-lying
$(s,u/d)$ state rather than it as the first excited one. In addition, this
conclusion would not change even if the $20\%$ uncertainties of form factors
caused by different approaches are considered. Therefore, our following
discussion is mainly based on S2, unless otherwise stated.

  * •

In S2, it can be found from Figs. 2 that
$\phi_{A}\sim[-50^{\circ},-100^{\circ}]$ and $[50^{\circ},100^{\circ}]$ is
allowed by the branching fractions, while a negative $\phi_{A}$ is favored by
the CP asymmetries as shown by Fig. 3. In our following numerical calculation,
we take

$\displaystyle(\rho_{A},\phi_{A})=(1,-55^{\circ})\quad\text{and}\quad(1,-64^{\circ})$
(58)

for $B\to SP$ and $SV$ decay modes, respectively.

Table 5: Branching fractions (in units of ${10}^{-6}$) of $B_{u,d,s}\to K_{0}^{*}(1430)P\,(P=\pi,K)$ decays.  Decay modes | S2 | S1 | QCDF(S2) | pQCD | data
---|---|---|---|---|---
$B^{-}\to K_{0}^{*}(1430)^{-}K^{0}$ | $0.08^{+0.00+0.01+0.03}_{-0.00-0.02-0.03}$ | $0.06^{+0.00+0.01+0.04}_{-0.00-0.01-0.03}$ |  $0.11^{+0.05}_{-0.04}$ [29] |  $0.38\pm 0.22$ [116] |
$B^{-}\to K_{0}^{*}(1430)^{0}K^{-}$ | $3.19^{+0.18+0.36+1.73}_{-0.14-0.62-1.39}$ | $0.54^{+0.03+0.06+0.35}_{-0.02-0.11-0.27}$ |  $3.37^{+1.03}_{-0.85}$ [29] |  $3.99\pm 1.38$ [116] | $0.38\pm 0.13$
$B^{-}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{-}$ | $33.37^{+1.48+4.28+18.52}_{-1.04-7.29-14.95}$ | $5.85^{+0.26+0.75+3.88}_{-0.18-1.27-2.99}$ |  $12.9^{+4.6}_{-3.7}$ [33] |  $47.6^{+11.3}_{-10.1}$ [117] | $39^{+6}_{-5}$
|  |  |  |  $36.6\pm 11.3$ [116] |
$B^{-}\to K_{0}^{*}(1430)^{-}\pi^{0}$ | $18.42^{+0.82+2.18+9.52}_{-0.58-3.73-7.80}$ | $3.41^{+0.15+0.39+2.03}_{-0.11-0.67-1.61}$ |  $7.4^{+2.4}_{-1.9}$ [33] |  $28.8^{+6.8}_{-6.1}$ [117] | ${11.9^{+2.0}_{-2.3}}$
|  |  |  |  $12.7\pm 4.2$ [116] |
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0}$ | $0.08^{+0.00+0.01+0.04}_{-0.00-0.02-0.03}$ | $0.06^{+0.00+0.01+0.04}_{-0.00-0.01-0.03}$ |  $0.24^{+0.12}_{-0.09}$ [29] |  $0.49\pm 0.33$ [116] |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{0}$ | $2.97^{+0.16+0.33+1.61}_{-0.13-0.57-1.29}$ | $0.50^{+0.03+0.06+0.33}_{-0.02-0.10-0.25}$ |  $4.05^{+1.34}_{-1.08}$ [29] |  $4.61\pm 1.50$ [116] |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{-}K^{+}$ | $0.08^{+0.01+0.02+0.04}_{-0.01-0.02-0.03}$ | $0.02^{+0.00+0.00+0.01}_{-0.00-0.00-0.01}$ |  $0.11^{+0.07}_{-0.05}$ [29] |  $0.09\pm 0.06$ [116] |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{+}K^{-}$ | $0.09^{+0.01+0.02+0.05}_{-0.01-0.03-0.04}$ | $0.05^{+0.00+0.01+0.04}_{-0.00-0.01-0.03}$ |  $0.06^{+0.05}_{-0.03}$ [29] |  $0.62\pm 0.40$ [116] |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{-}\pi^{+}$ | $30.55^{+1.36+3.85+16.97}_{-0.97-6.56-13.70}$ | $5.31^{+0.24+0.66+3.52}_{-0.17-1.13-2.72}$ |  $13.8^{+4.5}_{-3.6}$ [33] |  $43.0^{+10.2}_{-9.1}$ [117] | $33.5^{+3.9}_{-3.8}$
|  |  |  |  $33.4\pm 10.2$ [116] |
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{0}$ | $13.78^{+0.61+1.89+8.24}_{-0.44-3.20-6.54}$ | $2.26^{+0.10+0.32+1.68}_{-0.07-0.54-1.25}$ |  $5.6^{+2.6}_{-1.3}$ [33] |  $18.4^{+4.4}_{-3.9}$ [117] |
|  |  |  |  $22.4\pm 6.6$ [116] |
$\bar{B}_{s}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0}$ | $29.70^{+1.31+3.74+16.51}_{-0.93-6.37-13.32}$ | $5.40^{+0.24+0.66+3.59}_{-0.17-1.13-2.76}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{0}$ | $3.08^{+0.14+0.44+1.51}_{-0.10-0.72-1.17}$ | $1.22^{+0.05+0.22+0.78}_{-0.04-0.35-0.59}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{-}K^{+}$ | $29.79^{+1.33+3.70+16.56}_{-0.95-6.30-13.36}$ | $5.37^{+0.24+0.65+3.57}_{-0.17-1.11-2.75}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{+}K^{-}$ | $3.94^{+0.18+0.47+1.57}_{-0.13-0.79-1.21}$ | $1.82^{+0.09+0.24+0.81}_{-0.07-0.40-0.63}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{+}\pi^{-}$ | $6.41^{+0.49+0.04+1.22}_{-0.44-0.02-1.14}$ | $4.51^{+0.35+0.01+0.98}_{-0.31-0.01-0.91}$ |  |  $37.0^{+14.0}_{-10.0}$ [51] |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\pi^{0}$ | $0.70^{+0.05+0.07+0.31}_{-0.04-0.12-0.26}$ | $0.21^{+0.01+0.02+0.09}_{-0.01-0.03-0.07}$ |  |  $0.41^{+0.10}_{-0.07}$ [51] |
Table 6: Branching fractions (in units of ${10}^{-6}$) of $B_{u,d,s}\to K_{0}^{*}(1430)V\,(V=K^{*}\,,\rho\,,\omega\,,\phi)$ decays.  Decay modes | S2 | S1 | QCDF(S2) | pQCD(S2) | data
---|---|---|---|---|---
$B^{-}\to K_{0}^{*}(1430)^{-}K^{*0}$ | $0.24^{+0.01+0.01+0.05}_{-0.01-0.01-0.04}$ | $0.07^{+0.00+0.00+0.01}_{-0.00-0.00-0.01}$ |  $0.01^{+0.01}_{-0.01}$ [29] |  $1.3^{+0.5}_{-0.3}$ [45] |
$B^{-}\to K_{0}^{*}(1430)^{0}K^{*-}$ | $2.48^{+0.14+0.30+1.38}_{-0.11-0.52-1.12}$ | $0.43^{+0.02+0.05+0.29}_{-0.02-0.09-0.22}$ |  $2.17^{+0.55}_{-0.47}$ [29] |  $1.5^{+0.5}_{-0.3}$ [45] |
$B^{-}\to\bar{K}_{0}^{*}(1430)^{0}\rho^{-}$ | $51.81^{+2.29+5.64+28.39}_{-1.62-9.71-22.86}$ | $8.66^{+0.38+0.97+5.71}_{-0.27-1.66-4.40}$ |  $39.0^{+34.5}_{-35.8}$ [33] |  $12.1^{+2.8}_{-0.0}$ [52] |
$B^{-}\to K_{0}^{*}(1430)^{-}\rho^{0}$ | $20.99^{+0.93+2.55+12.72}_{-0.66-4.35-10.02}$ | $3.15^{+0.14+0.41+2.44}_{-0.10-0.70-1.80}$ |  $14.8^{+3.7}_{-3.2}$ [33] |  $8.4^{+2.3}_{-0.0}$ [52] |
$B^{-}\to K_{0}^{*}(1430)^{-}\omega$ | $25.08^{+1.11+2.82+13.98}_{-0.78-4.84-11.20}$ | $4.11^{+0.18+0.48+2.79}_{-0.13-0.82-2.13}$ |  $21.5^{+5.8}_{-4.9}$ [33] |  $7.4^{+2.1}_{-1.5}$ [52] | $24.0\pm 5.1$
$B^{-}\to K_{0}^{*}(1430)^{-}\phi$ | $4.97^{+0.22+0.21+1.08}_{-0.16-0.37-0.96}$ | $1.29^{+0.06+0.08+0.35}_{-0.04-0.11-0.22}$ |  $3.80^{+0.7}_{-0.6}$ [33] |  $25.6^{+6.2}_{-5.4}$ [118] | $7.0\pm 1.6$
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{*0}$ | $0.22^{+0.01+0.01+0.05}_{-0.01-0.01-0.04}$ | $0.06^{+0.00+0.00+0.01}_{-0.00-0.00-0.01}$ |  |  | $<3.3$
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{*0}$ | $2.30^{+0.13+0.28+1.28}_{-0.10-0.48-1.03}$ | $0.39^{+0.02+0.05+0.26}_{-0.02-0.08-0.20}$ |  $2.01^{+0.54}_{-0.45}$ [29] |  |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{-}K^{*+}$ | $0.03^{+0.00+0.00+0.02}_{-0.00-0.00-0.01}$ | $0.01^{+0.00+0.00+0.01}_{-0.00-0.00-0.01}$ |  $0.18^{+0.13}_{-0.08}$ [29] |  |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{+}K^{*-}$ | $0.04^{+0.00+0.00+0.03}_{-0.00-0.00-0.02}$ | $0.05^{+0.00+0.00+0.04}_{-0.00-0.00-0.03}$ |  $0.01^{+0.01}_{-0.01}$ [29] |  |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{-}\rho^{+}$ | $45.92^{+2.03+5.05+25.14}_{-1.43-8.69-20.24}$ | $7.67^{+0.34+0.86+5.05}_{-0.24-1.48-3.89}$ |  $36.3^{+8.5}_{-7.4}$ [33] |  $10.5^{+2.7}_{-0.0}$ [52] | $28.0\pm 11.0$
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\rho^{0}$ | $27.95^{+1.24+2.78+13.95}_{-0.87-4.82-11.45}$ | $5.10^{+0.23+0.50+2.91}_{-0.16-0.86-2.32}$ |  $23.4^{+5.1}_{-4.5}$ [33] |  $4.8^{+1.1}_{-0.0}$ [52] | $27.0\pm 5.0$
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\omega$ | $24.71^{+1.10+2.74+13.76}_{-0.78-4.71-11.03}$ | $4.06^{+0.18+0.46+2.75}_{-0.13-0.80-2.10}$ |  $21.9^{+5.9}_{-5.0}$ [33] |  $9.3^{+2.1}_{-2.0}$ [52] | $16.0\pm 3.4$
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\phi$ | $4.80^{+0.21+0.21+1.46}_{-0.15-0.38-1.18}$ | $1.25^{+0.06+0.08+0.42}_{-0.04-0.11-0.29}$ |  $3.7^{+0.8}_{-0.6}$ [33] |  $23.6^{+5.6}_{-5.0}$ [118] | $4.2\pm 0.5$
$\bar{B}_{s}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{*0}$ | $29.87^{+1.32+3.28+17.95}_{-0.93-5.64-14.51}$ | $5.08^{+0.22+0.56+3.57}_{-0.16-0.97-2.77}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{*0}$ | $5.11^{+0.23+0.29+1.43}_{-0.16-0.50-1.17}$ | $2.15^{+0.09+0.07+0.50}_{-0.07-0.12-0.42}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{-}K^{*+}$ | $28.39^{+1.26+3.15+17.04}_{-0.89-5.42-13.77}$ | $4.83^{+0.21+0.54+3.39}_{-0.15-0.93-2.63}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{+}K^{*-}$ | $5.42^{+0.28+0.32+1.63}_{-0.21-0.56-1.34}$ | $2.17^{+0.13+0.08+0.51}_{-0.11-0.13-0.42}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{+}\rho^{-}$ | $17.77^{+1.36+0.12+2.74}_{-1.22-0.07-2.56}$ | $12.57^{+0.97+0.05+2.30}_{-0.86-0.03-2.12}$ |  |  $108^{+25}_{-23}$ [53] |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\rho^{0}$ | $1.28^{+0.10+0.11+0.56}_{-0.09-0.20-0.46}$ | $0.35^{+0.03+0.02+0.16}_{-0.02-0.04-0.13}$ |  |  $0.96^{+0.22}_{-0.20}$ [53] |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\omega$ | $1.45^{+0.11+0.12+0.57}_{-0.10-0.20-0.48}$ | $0.44^{+0.03+0.03+0.17}_{-0.03-0.05-0.14}$ |  |  $0.86^{+0.21}_{-0.18}$ [53] |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\phi$ | $1.90^{+0.11+0.22+1.03}_{-0.08-0.38-0.84}$ | $0.35^{+0.02+0.04+0.21}_{-0.01-0.07-0.17}$ |  |  $0.95^{+0.25}_{-0.17}$ [53] |
Table 7: The direct CP asymmetries (in units of $\%$) of $B_{u,d,s}\to K_{0}^{*}(1430)P\,(P=\pi,K)$decays. Decay modes | S2 | S1 | QCDF(S2) | pQCD(S2) | data
---|---|---|---|---|---
$B^{-}\to K_{0}^{*}(1430)^{-}K^{0}$ | $30.32^{+1.08+0.73+1.23}_{-1.11-0.42-1.43}$ | $18.58^{+0.71+1.79+2.67}_{-0.72-0.79-1.94}$ |  $-22.51^{+4.90}_{-7.57}$ [29] |  |
$B^{-}\to K_{0}^{*}(1430)^{0}K^{-}$ | $-16.56^{+0.64+0.28+0.88}_{-0.63-0.56-0.84}$ | $-12.25^{+0.48+1.42+2.23}_{-0.47-1.47-2.25}$ |  $-2.60^{+1.61}_{-1.76}$ [29] |  | $10\pm 17$
$B^{-}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{-}$ | $0.65^{+0.02+0.02+0.05}_{-0.02-0.01-0.06}$ | $0.57^{+0.02+0.07+0.13}_{-0.02-0.07-0.13}$ |  $1.3^{+0.1}_{-0.1}$ [33] |  | $6.1\pm 3.2$
$B^{-}\to K_{0}^{*}(1430)^{-}\pi^{0}$ | $4.71^{+0.15+0.30+0.43}_{-0.15-0.14-0.34}$ | $5.20^{+0.16+0.37+0.64}_{-0.16-0.18-0.46}$ |  $3.0^{+0.4}_{-0.4}$ [33] |  $3.5$ [117] | $26^{+18}_{-14}$
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0}$ | $0.68^{+0.03+3.08+1.34}_{-0.03-1.48-2.62}$ | $0.08^{+0.00+2.05+0.81}_{-0.00-0.97-1.81}$ |  |  |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{0}$ | $-20.75^{+0.79+0.13+0.61}_{-0.77-0.28-0.59}$ | $-19.68^{+0.74+1.27+1.38}_{-0.72-1.35-1.29}$ |  |  |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{-}K^{+}$ | $-1.38^{+0.05+0.16+0.28}_{-0.05-0.38-0.28}$ | $-9.01^{+0.32+0.80+4.82}_{-0.32-1.77-4.43}$ |  |  |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{+}K^{-}$ | $1.46^{+0.06+0.33+0.14}_{-0.06-0.15-0.16}$ | $3.56^{+0.14+0.25+0.79}_{-0.14-0.13-1.46}$ |  |  |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{-}\pi^{+}$ | $2.25^{+0.07+0.21+0.17}_{-0.07-0.09-0.16}$ | $2.59^{+0.08+0.31+0.50}_{-0.08-0.22-0.53}$ |  $0.21^{+0.06}_{-0.06}$ [33] |  | $9\pm 8$
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{0}$ | $-2.04^{+0.06+0.07+0.38}_{-0.06-0.16-0.54}$ | $-2.42^{+0.08+0.23+0.85}_{-0.08-0.31-1.11}$ |  $-1.9^{+0.4}_{-0.5}$ [33] |  | $-15\pm 11$
$\bar{B}_{s}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0}$ | $0.82^{+0.03+0.01+0.04}_{-0.03-0.00-0.05}$ | $0.77^{+0.02+0.06+0.09}_{-0.02-0.06-0.10}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{0}$ | $0.03^{+0.00+0.04+0.06}_{-0.00-0.07-0.04}$ | $0.17^{+0.01+0.06+0.15}_{-0.01-0.14-0.10}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{-}K^{+}$ | $1.25^{+0.04+0.41+0.14}_{-0.04-0.19-0.14}$ | $1.78^{+0.06+0.50+0.77}_{-0.06-0.28-0.86}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{+}K^{-}$ | $-63.79^{+1.59+3.12+7.34}_{-1.56-6.56-4.83}$ | $-61.91^{+1.80+3.82+8.67}_{-1.83-8.20-2.54}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{+}\pi^{-}$ | $25.23^{+0.92+1.51+6.69}_{-0.91-2.83-6.57}$ | $15.14^{+0.55+0.92+6.42}_{-0.55-1.72-6.37}$ |  |  $21.0^{+3.4}_{-3.1}$ [51] |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\pi^{0}$ | $-58.32^{+1.82+3.32+3.83}_{-1.80-1.62-4.91}$ | $-71.72^{+2.14+4.17+10.77}_{-2.16-1.97-10.12}$ |  |  $95.5^{+1.2}_{-8.7}$ [51] |
Table 8: The direct CP asymmetries (in units of $\%$) of $B_{u,d,s}\to K_{0}^{*}(1430)V\,(V=K^{*}\,,\rho\,,\omega\,,\phi)$ decays. Decay modes | S2 | S1 | QCDF(S2) | pQCD(S2) | data
---|---|---|---|---|---
$B^{-}\to K_{0}^{*}(1430)^{-}K^{*0}$ | $-13.24^{+0.46+0.39+2.34}_{-0.45-0.77-2.36}$ | $-11.37^{+0.39+1.24+6.04}_{-0.38-1.46-5.30}$ |  $-31.02^{+4.67}_{-6.56}$ [29] |  $-34.9^{+5.0}_{-4.5}$ [45] |
$B^{-}\to K_{0}^{*}(1430)^{0}K^{*-}$ | $-19.99^{+0.75+0.19+0.66}_{-0.73-0.10-0.66}$ | $-26.27^{+0.94+1.19+3.06}_{-0.92-1.28-3.02}$ |  $0.64^{+2.54}_{-2.64}$ [29] |  $-67.9^{+4.9}_{-5.2}$ [45] |
$B^{-}\to\bar{K}_{0}^{*}(1430)^{0}\rho^{-}$ | $1.02^{+0.03+0.01+0.03}_{-0.03-0.00-0.03}$ | $1.13^{+0.04+0.06+0.17}_{-0.04-0.06-0.16}$ |  $0.32^{+0.34}_{-0.29}$ [33] |  $-7.1^{+0.0}_{-0.0}$ [52] |
$B^{-}\to K_{0}^{*}(1430)^{-}\rho^{0}$ | $-4.64^{+0.14+0.18+0.65}_{-0.14-0.39-0.93}$ | $-6.10^{+0.19+0.30+1.09}_{-0.19-0.64-1.88}$ |  $1.6^{+0.6}_{-0.6}$ [33] |  $6.3^{+0.0}_{-0.1}$ [52] |
$B^{-}\to K_{0}^{*}(1430)^{-}\omega$ | $-4.25^{+0.13+0.14+0.53}_{-0.13-0.32-0.73}$ | $-5.34^{+0.17+0.23+0.83}_{-0.17-0.47-1.32}$ |  $0.55^{+0.35}_{-0.34}$ [33] |  $6.2^{+0.0}_{-0.0}$ [52] | $-10\pm 9$
$B^{-}\to K_{0}^{*}(1430)^{-}\phi$ | $0.68^{+0.02+0.05+0.17}_{-0.02-0.02-0.16}$ | $0.42^{+0.01+0.07+0.46}_{-0.01-0.06-0.47}$ |  $0.64^{+0.02}_{-0.03}$ [33] |  $1.9$ [118] | $4\pm 15$
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{*0}$ | $-3.95^{+0.14+0.50+2.61}_{-0.14-1.01-3.21}$ | $-4.49^{+0.16+1.78+5.25}_{-0.16-2.04-5.59}$ |  |  |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{*0}$ | $-15.78^{+0.60+0.38+0.64}_{-0.59-0.17-0.61}$ | $-15.61^{+0.59+1.40+1.07}_{-0.58-1.36-1.05}$ |  |  |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{-}K^{*+}$ | $-5.10^{+0.19+0.19+0.35}_{-0.20-0.38-0.39}$ | $-1.37^{+0.06+0.64+1.74}_{-0.06-0.30-2.18}$ |  |  $-83.9\pm 0.7$ [45] |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{+}K^{*-}$ | $-4.27^{+0.16+0.14+0.43}_{-0.16-0.29-0.52}$ | $-3.64^{+0.14+0.02+0.61}_{-0.15-0.03-0.68}$ |  |  $38.5^{+1.1}_{-0.8}$ [45] |
$\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{-}\rho^{+}$ | $-0.15^{+0.00+0.05+0.15}_{-0.00-0.11-0.16}$ | $-0.30^{+0.01+0.08+0.27}_{-0.01-0.15-0.30}$ |  $1.1^{+0.0}_{-0.0}$ [33] |  $-4.8^{+0.3}_{-0.0}$ [52] |
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\rho^{0}$ | $4.85^{+0.15+0.13+0.53}_{-0.15-0.06-0.39}$ | $5.64^{+0.18+0.21+0.87}_{-0.18-0.10-0.59}$ |  $0.54^{+0.45}_{-0.46}$ [33] |  $-24.2^{+0.2}_{-0.0}$ [52] |
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\omega$ | $-3.30^{+0.10+0.09+0.44}_{-0.10-0.19-0.63}$ | $-4.24^{+0.13+0.23+0.73}_{-0.13-0.35-1.18}$ |  $0.03^{+0.37}_{-0.35}$ [33] |  $10.0^{+0.1}_{-0.0}$ [52] | $-7\pm 9$
$\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\phi$ | $0.15^{+0.00+0.06+0.20}_{-0.00-0.03-0.14}$ | $0.03^{+0.00+0.10+0.40}_{-0.00-0.09-0.30}$ |  $0.43^{+0.04}_{-0.04}$ [33] |  | $12.4\pm 8.1$
$\bar{B}_{s}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{*0}$ | $0.68^{+0.02+0.00+0.04}_{-0.02-0.01-0.06}$ | $0.67^{+0.02+0.05+0.06}_{-0.02-0.05-0.07}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{*0}$ | $0.20^{+0.01+0.05+0.17}_{-0.01-0.02-0.13}$ | $0.54^{+0.02+0.10+0.20}_{-0.02-0.09-0.20}$ |  |  |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{-}K^{*+}$ | $0.41^{+0.01+0.09+0.17}_{-0.01-0.19-0.21}$ | $0.15^{+0.00+0.13+0.45}_{-0.00-0.28-0.54}$ |  |  $1.7^{+0.3}_{-0.4}$ [45] |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{+}K^{*-}$ | $-46.13^{+1.54+0.32+2.53}_{-1.49-0.31-1.11}$ | $-36.91^{+1.47+3.09+12.56}_{-1.44-2.53-10.14}$ |  |  $-63.6^{+10.7}_{-8.4}$ [45] |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{+}\rho^{-}$ | $15.11^{+0.63+0.87+4.48}_{-0.61-1.61-4.37}$ | $6.85^{+0.29+0.67+2.92}_{-0.28-0.94-2.86}$ |  |  $12.6^{+0.0}_{-0.0}$ [53] |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\rho^{0}$ | $-7.83^{+0.25+2.14+8.79}_{-0.25-1.02-6.17}$ | $8.98^{+0.29+3.37+15.96}_{-0.28-3.47-11.94}$ |  |  $84.5^{+0.1}_{-0.1}$ [53] |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\omega$ | $-5.01^{+0.22+1.50+8.30}_{-0.24-3.11-11.42}$ | $-26.89^{+1.14+2.24+13.59}_{-1.24-3.71-17.23}$ |  |  $-86.7^{+0.1}_{-0.1}$ [53] |
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\phi$ | $-13.11^{+0.50+0.44+0.93}_{-0.49-0.20-0.80}$ | $-11.48^{+0.43+1.27+1.49}_{-0.42-1.14-1.30}$ |  |  |

Using the values of end-point parameters given above, we then present our
numerical results and discussions. The branching fractions and the direct $CP$
asymmetries of $B_{u,d,s}\to K_{0}^{*}(1430)P\,(P=\pi,K)$ and
$K_{0}^{*}(1430)V\,(V=K^{*}\,,\rho\,,\omega\,,\phi)$ decays are given in
tables 5, 6, 7, 8, 9 and 10. For each result, the first error is caused by the
uncertainties of CKM parameters, the second error comes from the variation of
quark masses, and the third error from the decay constants, form factors and
Gegenbauer moments. In each table, the experimental data given by PDG [77] or
HFAG [115] and the theoretical results obtained in previous works are also
listed for comparison.

  * •

From tables 5 and 6, it can be found that the branching fractions obtained
based on S2 are generally much larger than the ones based on S1 because
relatively larger decay constant of $K_{0}^{*}(1430)$ and form factors of
$B\to K_{0}^{*}(1430)$ transition are predicated in S2. In addition, most of
predictions based on S2 (S1) are favored (disfavored) by experimental data,
the only exception is $B^{-}\to K_{0}^{*}(1430)^{0}K^{-}$ decay. Our result
${\cal B}(B^{-}\to
K_{0}^{*}(1430)^{0}K^{-})=(3.19^{+0.18+0.36+1.73}_{-0.14-0.62-1.39})\times
10^{-6}$ in S2 is consistent with the results obtained in the previous works,
$(3.37^{+1.03}_{-0.85})\times 10^{-6}$ (QCDF) [29] and $(3.99\pm 1.38)\times
10^{-6}$ (pQCD) [116], however all of these theoretical predictions are an
order of magnitude larger than experimental data $(0.38\pm 0.13)\times
10^{-6}$. The reason will be analyzed in the next item.

  * •

$B^{-}\to K_{0}^{*}(1430)^{0}K^{-}$ and $K_{0}^{*}(1430)^{-}K^{0}$ decays are
penguin dominated. After neglecting the power suppressed contribution, their
simplified amplitudes can be written as

$\displaystyle{\cal A}(B^{-}\to
K_{0}^{*}(1430)^{0}K^{-})\sim(a_{4}^{p}-\gamma_{\chi}^{K_{0}^{*0}}a_{6}^{p})\,A_{K^{-}K_{0}^{*}(1430)^{0}}\,,$
(59) $\displaystyle{\cal A}(B^{-}\to
K_{0}^{*}(1430)^{-}K^{0})\sim(a_{4}^{p}-\gamma_{\chi}^{K^{0}}a_{6}^{p})\,A_{K_{0}^{*}(1430)^{0}K^{-}}\,.$
(60)

A significant feature of scalar meson is that its chiral factor is
proportional to $M_{S}^{2}/(m_{1}(\mu)-m_{2}(\mu))$, which results in that
$\gamma_{\chi}^{K_{0}^{*0}}$ is much larger than $\gamma_{\chi}^{K^{0}}$.
Numerically, we obtain
$\gamma_{\chi}^{K_{0}^{*0}}:\gamma_{\chi}^{K^{0}}\approx 12.6:1.4$ at $\mu\sim
m_{b}$. As a result, it is expected that ${\cal B}(B^{-}\to
K_{0}^{*}(1430)^{0}K^{-})\gg{\cal B}(B^{-}\to K_{0}^{*}(1430)^{-}K^{0})$.
Moreover, for all of the penguin dominated $B\to SP$ and $SV$ decays, the
decay modes with $M_{2}=S$ ($M_{2}$ is the emitted meson) generally have
relatively larger branching fractions than the ones with $M_{2}=(P,V)$, which
can also be found from the numerical results listed in tables 5 and 6.

In addition, the large $\gamma_{\chi}^{K_{0}^{*0}}$ also results in the large
theoretical predictions for ${\cal B}(B^{-}\to K_{0}^{*}(1430)^{0}K^{-})$
compared with data, which has been mentioned in the last item. It should be
noted that the significance of data, $(0.38\pm 0.13)\times 10^{-6}$, is
smaller than $3\sigma$, thus more precise measurement on ${\cal B}(B^{-}\to
K_{0}^{*}(1430)^{0}K^{-})$ is required for confirming or refuting such
possible anomaly.

  * •

The $\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{-}K^{(*)+}$ and
$K_{0}^{*}(1430)^{+}K^{(*)-}$ decays are pure annihilation processes, and thus
are very suitable for probing the effects of annihilation corrections.
However, these decays have very small branching fractions $\sim 10^{-7}$
because their amplitudes are power suppressed, and therefore are not easy to
be precisely measured in the near future.

  * •

The $\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{+}\pi^{-}/\rho^{-}$ and
$\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{0}\pi^{0}/\rho^{0}(\omega)$ decays are
tree-dominated, while the former are color-allowed and the later are color-
suppressed. As a result, the branching fractions of former are an order of
magnitude larger than the later.

  * •

The $SU(3)$ flavor symmetry indicates some useful relations between the decays
considered in this work. Taking $B_{u,d,s}\to SP$ decays as examples, after
applying $SU(3)$ flavor symmetry on the spectator quark, one may expect that

$\displaystyle{\cal A}(B^{-}\to K_{0}^{*}(1430)^{-}K^{0})\approx$
$\displaystyle{\cal A}(\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0})\,,$
(61) $\displaystyle{\cal A}(B^{-}\to K_{0}^{*}(1430)^{0}K^{-})\approx$
$\displaystyle{\cal A}(\bar{B}_{d}^{0}\to K_{0}^{*}(1430)^{0}\bar{K}^{0})\,,$
(62) $\displaystyle{\cal A}(B^{-}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{-})\approx$
$\displaystyle{\cal
A}(\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{0})\approx{\cal
A}(\bar{B}_{s}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0})\,,$ (63)
$\displaystyle{\cal A}(B^{-}\to K_{0}^{*}(1430)^{-}\pi^{0})\approx$
$\displaystyle{\cal A}(B_{d}^{0}\to K_{0}^{*}(1430)^{-}\pi^{+})\approx{\cal
A}(\bar{B}_{s}^{0}\to K_{0}^{*}(1430)^{-}K^{+})\,,$ (64)

which further imply that

$\displaystyle R_{1}$ $\displaystyle\equiv\frac{\Gamma_{B^{-}\to
K_{0}^{*}(1430)^{-}K^{0}}}{\Gamma_{\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0}}}\approx
1\,,\qquad R_{2}\equiv\frac{\Gamma_{B^{-}\to
K_{0}^{*}(1430)^{0}K^{-}}}{\Gamma_{\bar{B}_{d}^{0}\to
K_{0}^{*}(1430)^{0}\bar{K}^{0})}}\approx 1\,,$ (65) $\displaystyle R_{3}$
$\displaystyle\equiv\frac{\Gamma_{B^{-}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{-}}}{2\Gamma_{\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{0}}}\approx
1\,,\qquad
R_{3}^{\prime}\equiv\frac{\Gamma_{B^{-}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{-}}}{\Gamma_{\bar{B}_{s}^{0}\to\bar{K}_{0}^{*}(1430)^{0}K^{0}}}\approx
1\,,$ (66) $\displaystyle R_{4}$ $\displaystyle\equiv\frac{2\Gamma_{B^{-}\to
K_{0}^{*}(1430)^{-}\pi^{0}}}{\Gamma_{B_{d}^{0}\to
K_{0}^{*}(1430)^{-}\pi^{+}}}\approx 1\,,\qquad
R_{4}^{\prime}\equiv\frac{2\Gamma_{B^{-}\to
K_{0}^{*}(1430)^{-}\pi^{0}}}{\Gamma_{\bar{B}_{s}^{0}\to
K_{0}^{*}(1430)^{-}K^{+}}}\approx 1\,.$ (67)

Besides, the $SU(3)$ flavor symmetry also expects that

$\displaystyle R_{5}\equiv\frac{\Gamma_{\bar{B}_{d}^{0}\to
K_{0}^{*}(1430)^{-}\pi^{+}}}{2\Gamma_{\bar{B}_{d}^{0}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{0}}}\approx
1\,,\qquad
R_{6}\equiv\frac{\Gamma_{B^{-}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{-}}}{2\Gamma_{B^{-}\to
K_{0}^{*}(1430)^{-}\pi^{0}}}\approx 1\,.$ (68)

It is found in our calculation (S2) that

$\displaystyle R_{1}$ $\displaystyle=0.93\,,\quad R_{2}=1.00\,,\quad
R_{3}=1.12\,,\quad R_{3}^{\prime}=1.10\,,\quad R_{4}=0.99\,,\quad
R_{4}^{\prime}=1.08\,,$ (69) $\displaystyle R_{5}$ $\displaystyle=1.11\,,\quad
R_{6}=0.90\,,$ (70)

which generally agree with the expectations of $SU(3)$ flavor symmetry. The
flavor symmetry breaking effect is mainly ascribed to the remaining suppressed
contributions. Taking $R_{6}$ as an example, the amplitude of $B^{-}\to
K_{0}^{*}(1430)^{-}\pi^{0}$ decay receives additional CKM-suppressed
contributions proportional to $\delta_{pu}\alpha_{1}$, as well as CKM- and
color-suppressed $\delta_{pu}\alpha_{2}$, compared with with
$B^{-}\to\bar{K}_{0}^{*}(1430)^{0}\pi^{-}$ decay.

  * •

The ${\cal O}(\alpha_{s}^{2})$ corrections to the amplitudes of non-leptonic
two-body $B$ decays have been evaluated in recent years [119, 120, 121, 122,
123, 124, 125, 126, 127, 128, 129]. In Ref. [119], it is found that the next-
to-next-to-leading order (NNLO) vertex correction to the color-suppressed
amplitude $\alpha_{2}$ is sizable, but when combined with the ${\cal
O}(\alpha_{s}^{2})$ correction to spectator scattering, the overall NNLO
corrections to the color-allowed and -suppressed tree amplitudes are small due
to the large cancellation. Therefore, the ${\cal O}(\alpha_{s}^{2})$
corrections to the tree-dominated $\bar{B}_{s}^{0}\to
K_{0}^{*}(1430)^{+}\pi^{-}/\rho^{-}$ and $\bar{B}_{s}^{0}\to
K_{0}^{*}(1430)^{0}\pi^{0}/\rho^{0}(\omega)$ decays would not be significant.
The NNLO correction to the penguin amplitude has also been studied in Refs.
[120, 121]. It is found that the NNLO contributions from current-current and
penguin operators are sizable, but there is a strong cancellation between
them, which results in a much reduced overall NNLO corrections to the penguin
amplitude $a_{4}^{u,c}$. As a consequence the full NNLO result for
$a_{4}^{u,c}$ is very close to the NLO result [121] (an example,
$a_{4}^{u,c}(\pi\bar{K})$, is shown by Fig. 8 in Ref. [121] ). Therefore,
based on these previous works on the ${\cal O}(\alpha_{s}^{2})$ correction, we
can expect that the ${\cal O}(\alpha_{s}^{2})$ corrections do not affect the
main findings of this work.

Table 9: The branching fractions (in units of ${10}^{-6}$) of ${\bar{B}}^{0}_{d,s}\to\bar{K}_{0}^{*}(1430){K}^{(*)}+c.c.$ decays in S2. Decay modes | this work | pQCD[45] | data
---|---|---|---
$\bar{B}_{d}^{0}\to K_{0}^{*0}\bar{K}^{0}\,\,+c.c.$ | $3.05^{+0.17+0.35+1.61}_{-0.13-0.59-1.29}$ |  |
${\bar{B}}^{0}_{d}\to K_{0}^{*+}K^{-}+c.c.$ | $0.17^{+0.01+0.03+0.07}_{-0.01-0.05-0.05}$ |  |
${\bar{B}}^{0}_{s}\to K_{0}^{*0}\bar{K}^{0}\,\,+c.c.$ | $32.78^{+1.45+4.17+16.59}_{-1.02-7.08-13.39}$ |  | $33.0\pm 10.1$
${\bar{B}}^{0}_{s}\to K_{0}^{*+}K^{-}+c.c.$ | $33.73^{+1.51+4.17+16.65}_{-1.07-7.09-13.43}$ |  | $31.3\pm 25.4$
$\bar{B}_{d}^{0}\to K_{0}^{*0}\bar{K}^{*0}\,\,+c.c.$ | $2.52^{+0.14+0.29+1.28}_{-0.11-0.49-1.03}$ | $0.59^{+0.1}_{-0.1}$ |
${\bar{B}}^{0}_{d}\to K_{0}^{*+}K^{*-}+c.c.$ | $0.07^{+0.01+0.01+0.03}_{-0.00-0.01-0.02}$ | $1.1^{+0.1}_{-0.1}$ |
${\bar{B}}^{0}_{s}\to K_{0}^{*0}\bar{K}^{*0}\,\,+c.c.$ | $34.98^{+1.55+3.56+18.02}_{-1.09-6.14-14.57}$ | $13.0^{+2.0}_{-2.0}$ |
${\bar{B}}^{0}_{s}\to K_{0}^{*+}K^{*-}+c.c.$ | $33.81^{+1.51+3.47+17.13}_{-1.07-5.98-13.85}$ | $15.0^{+4.0}_{-3.0}$ |
Table 10: The $CP$ asymmetry parameters (in units of ${10}^{-2}$) of ${\bar{B}}^{0}_{d,s}\to\bar{K}_{0}^{*}(1430){K}^{(*)}+c.c.$ decays in S2. | $A_{CP}$ | C | ${\Delta}C$ | S | ${\Delta}S$
---|---|---|---|---|---
${\bar{B}}^{0}_{d}\to\bar{K}_{0}^{*0}{K}^{0}+c.c.$ | $-20.24^{+0.77}_{-0.75}$ | $10.04^{+0.37}_{-0.38}$ | $-10.72^{+0.41}_{-0.40}$ | $2.15^{+0.08}_{-0.08}$ | $8.70^{+0.36}_{-0.36}$
${\bar{B}}^{0}_{d}\to K_{0}^{*-}K^{+}+c.c.$ | $1.42^{+0.06}_{-0.05}$ | $-0.04^{+0.00}_{-0.00}$ | $1.42^{+0.05}_{-0.05}$ | $4.76^{+8.13}_{-7.64}$ | $6.35^{+0.22}_{-0.25}$
$\bar{B}_{s}^{0}\to\bar{K}_{0}^{*0}{K}^{0}+c.c.$ | $-0.74^{+0.02}_{-0.02}$ | $-0.42^{+0.01}_{-0.01}$ | $-0.40^{+0.01}_{-0.01}$ | $0.03^{+0.00}_{-0.00}$ | $0.31^{+0.01}_{-0.01}$
${\bar{B}}^{0}_{s}\to K_{0}^{*-}K^{+}+c.c.$ | $-8.56^{+0.26}_{-0.26}$ | $31.27^{+0.76}_{-0.78}$ | $-32.52^{+0.82}_{-0.80}$ | $-11.39^{+0.60}_{-0.56}$ | $3.49^{+0.50}_{-0.54}$
${\bar{B}}^{0}_{d}\to\bar{K}_{0}^{*0}{K}^{*0}+c.c.$ | $-14.03^{+0.53}_{-0.52}$ | $9.87^{+0.36}_{-0.37}$ | $-5.91^{+0.23}_{-0.22}$ | $-14.20^{+0.54}_{-0.53}$ | $-6.58^{+0.24}_{-0.23}$
${\bar{B}}^{0}_{d}\to K_{0}^{*-}K^{*+}+c.c.$ | $-0.52^{+0.02}_{-0.02}$ | $4.69^{+0.18}_{-0.18}$ | $0.42^{+0.02}_{-0.02}$ | $5.61^{+8.16}_{-7.67}$ | $-0.75^{+0.03}_{-0.03}$
$\bar{B}_{s}^{0}\to\bar{K}_{0}^{*0}{K}^{*0}+c.c.$ | $-0.55^{+0.02}_{-0.02}$ | $-0.44^{+0.01}_{-0.01}$ | $-0.24^{+0.01}_{-0.01}$ | $0.70^{+0.02}_{-0.02}$ | $-0.23^{+0.01}_{-0.01}$
${\bar{B}}^{0}_{s}\to K_{0}^{*-}K^{*+}+c.c.$ | $-7.74^{+0.23}_{-0.23}$ | $22.86^{+0.74}_{-0.77}$ | $-23.27^{+0.77}_{-0.75}$ | $30.72^{+0.69}_{-0.71}$ | $-23.55^{+0.49}_{-0.47}$

Different from the other decay modes, ${\bar{B}}^{0}_{s}$ or
${\bar{B}}^{0}_{d}$ can decay into $\bar{K}_{0}^{*}(1430){K}^{(*)}$ and its CP
conjugate state simultaneously. The sum of the
$\bar{K}_{0}^{*}(1430){K}^{(*)}$ and $K_{0}^{*}(1430)\bar{K}^{(*)}$ decay
rates is measured more accurately than the individual rates. Our prediction
for ${\bar{B}}^{0}_{d,s}\to\bar{K}_{0}^{*}(1430){K}^{(*)}+c.c.$ decays are
given in table 9, the LHCb data [21] given by Eq. (1) and the pQCD predictions
[45] are also listed for comparision. In addition, the CP violations of
${\bar{B}}^{0}_{d,s}\to\bar{K}_{0}^{*}(1430){K}^{(*)}+c.c.$ decays are much
more complicated because $\bar{K}_{0}^{*}(1430){K}^{(*)}+c.c.$ are not CP
eigenstates. The system of these decays define five CP asymmetry parameters,
$C$, ${\Delta}C$, $S$, ${\Delta}S$ and $A_{CP}$, where $S$ is referred to as
mixing-induced CP asymmetry, $C$ is the direct CP asymmetry, ${\Delta}C$ and
${\Delta}S$ are CP conserving quantities, and $A_{CP}$ is the time-integrated
charge asymmetry. Our results for these observables are first given in table
10.

The ${\bar{B}}^{0}_{d}\to K_{0}^{*-}K^{(*)+}+c.c.$ decays are caused by the
pure annihilation transition, and therefore have very small branching
fractions $\sim{\cal O}(10^{-7})$. The
${\bar{B}}^{0}_{d}\to\bar{K}_{0}^{*0}{K}^{(*)0}+c.c.$ decays are dominated by
penguin contributions, but are CKM-suppressed. The penguin-dominated
${\bar{B}}^{0}_{s}\to K_{0}^{*-}K^{(*)+}+c.c.$ and
$\bar{K}_{0}^{*0}{K}^{(*)0}+c.c.$ decays have relatively large branching
fractions, and our results for the $SP$ modes are in good agreement with the
data obtained by the LHCb collaboration with significance over $10$ standard
deviations [21]. However, some systematic variations do impact strongly on the
need to include tensor resonances in the fit model, and thus the model
uncertainties of experimental data are very large [21]. It is expected that
the future refined measurements for the
${\bar{B}}^{0}_{d,s}\to\bar{K}_{0}^{*}(1430){K}^{(*)}+c.c.$ decays can provide
serious test on the theoretical predictions.

##  4 Summary

In this paper, the nonleptonic charmless $B_{u,d,s}\to K_{0}^{*}(1430)P$
$(P=K\,,\pi)$ and $K_{0}^{*}(1430)V$ $(V=K^{*}\,,\rho\,,\omega\,,\phi)$ decays
are studied. The amplitudes are calculated by using the QCD factorization
approach, and the non-perturbative quantities (form factor, decay constant and
distribution amplitudes) are evaluated by using a covariant light-front
approach. The branching fractions and CP asymmetries of theses decay modes are
calculated, our main theoretical results are collected in tables 5-10. Some
decay modes are first predicted in this work. In order to test the underlying
structure of $K_{0}^{*}(1430)$ meson, our calculation are made based on two
different scenarios that $K_{0}^{*}(1430)$ is the first excited (scenario S1)
and the lowest-lying (scenario S2) p-wave two-quark state. Comparing our
results with data, we find that the scenarios S1 and S2 lead to significantly
different results for branching factions, and the results based on scenario S2
agree well with the experimental data, which implies that $K_{0}^{*}(1430)$ as
the lowest-lying (scenario S2) p-wave $(s,u/d)$ state is favored by data.
However, the theoretical results for ${\cal B}(B^{-}\to
K_{0}^{*}(1430)^{0}K^{-})$ are much larger than data due to the large chiral
factor $\gamma_{\chi}^{K_{0}^{*0}}$. The future refined measurement on ${\cal
B}(B^{-}\to K_{0}^{*}(1430)^{0}K^{-})$ is required for confirming or refuting
such possible anomaly. Besides, some useful relations based on $SU(3)$ flavor
symmetry are studied.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
(Grant No. 11875122), Excellent Youth Foundation of Henan Province (Grant No.
212300410010), The Youth Talent Support Program of Henan Province (Grant No.
ZYQR201912178) and the Program for Innovative Research Team in University of
Henan Province (Grant No. 19IRTSTHN018).

## Appendix A: Decay constant, form factor and DAs

The decay constant, form factor and DAs are essential nonperturbative inputs
for the amplitudes of $B\to SP$ and $SV$. In this sections, we would like to
clarify our convention for the definitions of these nonperturbative
quantities.

### A.1 Decay constant

The decay constants of pseudoscalar (P), vector (V) and scalar (S) mesons are
defined as

$\displaystyle\langle P(p)|\bar{q}_{1}\gamma^{\mu}\gamma^{5}q_{2}|0\rangle$
$\displaystyle=-if_{P}p_{\mu}\,,\quad\langle
V(p,\epsilon)|\bar{q}_{1}\gamma^{\mu}q_{2}|0\rangle=-if_{V}m_{V}\epsilon^{*\mu}\,,$
$\displaystyle\langle S(p)|\bar{q}_{1}\gamma^{\mu}q_{2}|0\rangle$
$\displaystyle=f_{S}p_{\mu}\,,\qquad\langle
S(p)|\bar{q}_{1}q_{2}|0\rangle=m_{S}\bar{f}_{S}(\mu)\,.$ (71)

The scale-dependent scalar decay constant $\bar{f}_{S}(\mu)$ and the vector
decay constant $f_{S}$ are related by the equation of motion, and thus have
the following relation,

$\displaystyle\bar{f}_{S}=\frac{\mu_{S}}{m_{S}}f_{S}\equiv\bar{\mu}_{S}f_{S}\,,\qquad\text{with}\qquad\mu_{S}=\frac{m_{S}^{2}}{m_{1}(\mu)-m_{2}(\mu)}\,,$
(72)

where $m_{1}(\mu)$ and $m_{2}(\mu)$ are running masses of current quarks.

### A.2 Form factor

The form factors of $B\to S\,,P\,,V$ transitions concerned in this paper can
be defined as

$\displaystyle{\langle}S(p^{\prime\prime})|{\bar{q}}\gamma_{\mu}\gamma_{5}b|B(p^{\prime}){\rangle}=$
$\displaystyle-i\bigg{[}\bigg{(}P_{\mu}-\frac{m_{B}^{2}-m_{S}^{2}}{q^{2}}q_{\mu}\bigg{)}U_{1}(q^{2})+\frac{m_{B}^{2}-m_{S}^{2}}{q^{2}}q_{\mu}U_{0}(q^{2})\bigg{]}\,,$
(73)
$\displaystyle{\langle}P(p^{\prime\prime})|{\bar{q}}\gamma_{\mu}b|B(p^{\prime}){\rangle}=$
$\displaystyle\bigg{(}P_{\mu}-\frac{m_{B}^{2}-m_{P}^{2}}{q^{2}}q_{\mu}\bigg{)}F_{1}(q^{2})+\frac{m_{B}^{2}-m_{P}^{2}}{q^{2}}q_{\mu}F_{0}(q^{2})\,,$
(74)
$\displaystyle{\langle}V(\epsilon,p^{\prime\prime})|{\bar{q}}\gamma_{\mu}b|B(p^{\prime}){\rangle}=$
$\displaystyle-i\frac{V(q^{2})}{m_{B}+m_{V}}\varepsilon_{\mu\nu\alpha\beta}\epsilon^{*\nu}P^{\alpha}q^{\beta}\,,$
(75)
$\displaystyle{\langle}V(\epsilon,p^{\prime\prime})|{\bar{q}}\gamma_{\mu}\gamma_{5}b|B(p^{\prime}){\rangle}=$
$\displaystyle 2m_{V}\frac{\epsilon^{*}\cdot
P}{q^{2}}q_{\mu}A_{0}(q^{2})+(m_{B}+m_{V})\bigg{(}\epsilon_{\mu}^{*}-\frac{\epsilon^{*}\cdot
P}{q^{2}}q_{\mu}\bigg{)}A_{1}(q^{2})$ $\displaystyle-\frac{\epsilon^{*}\cdot
P}{m_{B}+m_{V}}\bigg{(}P_{\mu}-\frac{m_{B}^{2}-m_{V}^{2}}{q^{2}}q_{\mu}\bigg{)}A_{2}(q^{2})\,,$
(76)

where
$P_{\mu}=(p^{\prime}+p^{\prime\prime}),\,q_{\mu}=(p^{\prime}-p^{\prime\prime})$
and $\varepsilon_{0123}=-1$. The momentum dependence of form factor can be
parameterized as

$\displaystyle
F(q^{2})=\frac{F(0)}{1-a\,(q^{2}/m^{2}_{B})+b\,(q^{2}/m^{2}_{B})^{2}}\,.$ (77)

The values of parameters $a$, $b$ and $F(0)$ for the transition concerned in
this work are summarized in Table 1.

### A.2 Distribution amplitude

For the light-cone distribution amplitudes of pseudoscalar and vector mesons,
we take the same definition and convention used in the Ref. [70]. For the
scalar meson, the twist-2 and -3 light-cone distribution amplitudes are given
by

$\displaystyle\langle
S(p)|{\bar{q}_{2}}(z_{2})\gamma_{\mu}q_{1}(z_{1})|0\rangle$
$\displaystyle=f_{S}p_{\mu}\int_{0}^{1}dxe^{i(xp\cdot z_{2}+\bar{x}p\cdot
z_{1})}\Phi_{S}(x),$ (78) $\displaystyle\langle
S(p)|{\bar{q}_{2}}(z_{2})q_{1}(z_{1})|0\rangle$
$\displaystyle=f_{S}\mu_{S}\int_{0}^{1}dxe^{i(xp\cdot z_{2}+\bar{x}p\cdot
z_{1})}\phi_{S}(x)\,,$ (79) $\displaystyle\langle
S(p)|{\bar{q}_{2}}(z_{2})\sigma_{\mu\nu}q_{1}(z_{1})|0\rangle$
$\displaystyle=-f_{S}\mu_{S}(p_{\mu}z_{\nu}-p_{\nu}z_{\mu})\int_{0}^{1}dxe^{i(xp\cdot
z_{2}+\bar{x}p\cdot z_{1})}\frac{\phi_{S}^{\sigma}(x)}{6}\,.$ (80)

The leading-twist DAs are conventionally expended in Gegenbauer polynomials,

$\displaystyle\Phi_{M}(x,\mu)=$ $\displaystyle
6x(1-x)\left[1+\sum^{\infty}_{n=1}a_{n}^{M}(\mu)C_{n}^{3/2}(x-\bar{x})\right],$
(81)

where $M=P\,,S\,,V$. Equivalently, for the scalar meson, one can also use

$\displaystyle\Phi_{S}(x)=$ $\displaystyle
6x(1-x)\bar{\mu}_{S}(\mu)\left[b_{0}^{S}+\sum^{\infty}_{n=1}b_{n}^{S}(\mu)C_{n}^{3/2}(x-\bar{x})\right],$
(82)

where, $a_{n}^{S}(\mu)={\bar{\mu}_{S}}(\mu)b_{n}^{S}(\mu)$ and
$\bar{\mu}_{S}(\mu)$ can be conveniently absorbed by $f_{S}$ via Eq. (72).

When three-particle contributions are neglected, the twist-3 two-particle DAs
of P and S mesons are determined completely by the equation of motion, which
then require

$\displaystyle\phi_{P,S}=1\,,\qquad\phi_{P,S}^{\sigma}=6x\bar{x}\,;$ (83)

For the vector meson, the twist-3 DAs can be expressed in terms of the
leading-twist DA $\Phi_{\bot}(x)$ of transversely polarized state, are usually
expressed by the function, $\phi_{V}$, defined as

$\displaystyle\phi_{V}(x,\mu)\equiv$
$\displaystyle\int_{0}^{x}dv\frac{\Phi_{\perp}(v)}{\bar{v}}-\int_{x}^{1}dv\frac{\Phi_{\perp}(v)}{v}=3\sum_{n=0}^{\infty}a_{n,\perp}^{V}(\mu)P_{n+1}(2x-1),$
(84)

where $a_{0,\perp}^{V}=1$ and $P_{n}(x)$ are the Legendre polynomials.

## Appendix B: Decay amplitudes

The amplitudes of $B_{u,d,s}\to K_{0}^{*}(1430)P$ $(P=K\,,\pi)$ decays are
written as

$\displaystyle{\cal A}_{B^{-}\to K_{0}^{*-}K^{0}}=$ $\displaystyle
A_{K_{0}^{*}K}[\delta_{pu}\beta_{2}+\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}]\,,$
(85) $\displaystyle{\cal A}_{B^{-}\to K_{0}^{*0}K^{-}}=$ $\displaystyle
A_{KK_{0}^{*}}[\delta_{pu}\beta_{2}+\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}]\,,$
(86) $\displaystyle{\cal A}_{B^{-}\to\bar{K}_{0}^{*0}\pi^{-}}=$ $\displaystyle
A_{\pi
K_{0}^{*}}[\delta_{pu}\beta_{2}+\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}]\,,$
(87) $\displaystyle{\cal A}_{B^{-}\to K_{0}^{*-}\pi^{0}}=$
$\displaystyle\frac{1}{\sqrt{2}}A_{\pi
K_{0}^{*}}[\delta_{pu}(\alpha_{1}+\beta_{2})+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}]$
$\displaystyle+\frac{1}{\sqrt{2}}A_{K_{0}^{*}\pi}[\delta_{pu}\alpha_{2}+\frac{3}{2}\alpha_{3,EW}^{p}]\,,$
(88) $\displaystyle{\cal A}_{\bar{B}^{0}\to\bar{K}_{0}^{*0}K^{0}}=$
$\displaystyle
A_{K_{0}^{*}K}[\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{KK_{0}^{*}}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$ (89)
$\displaystyle{\cal A}_{\bar{B}^{0}\to K_{0}^{*0}\bar{K}^{0}}=$ $\displaystyle
A_{KK_{0}^{*}}[\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{K_{0}^{*}K}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$ (90)
$\displaystyle{\cal A}_{\bar{B}^{0}\to K_{0}^{*-}K^{+}}=$ $\displaystyle
A_{K_{0}^{*}K}[\delta_{pu}\beta_{1}+\beta_{4}^{p}+\beta_{4,EW}^{p}]+B_{KK_{0}^{*}}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$
(91) $\displaystyle{\cal A}_{\bar{B}^{0}\to K_{0}^{*+}K^{-}}=$ $\displaystyle
A_{KK_{0}^{*}}[\delta_{pu}\beta_{1}+\beta_{4}^{p}+\beta_{4,EW}^{p}]+B_{K_{0}^{*}K}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$
(92) $\displaystyle{\cal A}_{\bar{B}^{0}\to K_{0}^{*-}\pi^{+}}=$
$\displaystyle A_{\pi
K_{0}^{*}}[\delta_{pu}\alpha_{1}+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}-\frac{1}{2}\beta_{3,EW}^{p}]\,,$
(93) $\displaystyle{\cal A}_{\bar{B}^{0}\to\bar{K}_{0}^{*0}\pi^{0}}=$
$\displaystyle\frac{1}{\sqrt{2}}A_{\pi
K_{0}^{*}}[-\alpha_{4}^{p}+\frac{1}{2}\alpha_{4,EW}^{p}-\beta_{3}^{p}+\frac{1}{2}\beta_{3,EW}^{p}]$
$\displaystyle+\frac{1}{\sqrt{2}}A_{K_{0}^{*}\pi}[\delta_{pu}\alpha_{2}+\frac{3}{2}\alpha_{3,EW}^{p}]\,,$
(94) $\displaystyle{\cal A}_{\bar{B}_{s}^{0}\to\bar{K}_{0}^{*0}K^{0}}=$
$\displaystyle
A_{KK_{0}^{*}}[\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{K_{0}^{*}K}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$ (95)
$\displaystyle{\cal A}_{\bar{B}_{s}^{0}\to K_{0}^{*0}\bar{K}^{0}}=$
$\displaystyle
A_{K_{0}^{*}K}[\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{KK_{0}^{*}}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$ (96)
$\displaystyle{\cal A}_{\bar{B}_{s}^{0}\to K_{0}^{*-}K^{+}}=$ $\displaystyle
A_{KK_{0}^{*}}[\delta_{pu}\alpha_{1}+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{K_{0}^{*}K}[\delta_{pu}b_{1}+b_{4}^{p}+b_{4,EW}^{p}]\,,$
(97) $\displaystyle{\cal A}_{\bar{B}_{s}^{0}\to K_{0}^{*+}K^{-}}=$
$\displaystyle
A_{K_{0}^{*}K}[\delta_{pu}\alpha_{1}+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{KK_{0}^{*}}[\delta_{pu}b_{1}+b_{4}^{p}+b_{4,EW}^{p}]\,,$
(98) $\displaystyle{\cal A}_{\bar{B}_{s}^{0}\to K_{0}^{*+}\pi^{-}}=$
$\displaystyle
A_{K_{0}^{*}\pi}[\delta_{pu}\alpha_{1}+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}-\frac{1}{2}\beta_{3,EW}^{p}]\,,$
(99) $\displaystyle{\cal A}_{\bar{B}_{s}^{0}\to K_{0}^{*0}\pi^{0}}=$
$\displaystyle\frac{1}{\sqrt{2}}A_{K_{0}^{*}\pi}[\delta_{pu}\alpha_{2}-\alpha_{4}^{p}+\frac{3}{2}\alpha_{3,EW}^{p}+\frac{1}{2}\alpha_{4,EW}^{p}-\beta_{3}^{p}+\frac{1}{2}\beta_{3,EW}^{p}]\,.$
(100)

The amplitudes of $B_{u,d,s}\to K_{0}^{*}(1430)V$
$(V=\rho\,,K^{*}\,,\omega\,,\phi)$ decays are written as

$\displaystyle{\cal A}_{B^{-}\to K_{0}^{*-}K^{*0}}=$ $\displaystyle
A_{K_{0}^{*}K^{*}}[\delta_{pu}\beta_{2}+\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}]\,,$
(101) $\displaystyle{\cal A}_{B^{-}\to K_{0}^{*0}K^{*-}}=$ $\displaystyle
A_{K^{*}K_{0}^{*}}[\delta_{pu}\beta_{2}+\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}]\,,$
(102) $\displaystyle{\cal A}_{B^{-}\to\bar{K}_{0}^{*0}\rho^{-}}=$
$\displaystyle A_{\rho
K_{0}^{*}}[\delta_{pu}\beta_{2}+\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}]\,,$
(103) $\displaystyle{\cal A}_{B^{-}\to K_{0}^{*-}\rho^{0}}=$
$\displaystyle\frac{1}{\sqrt{2}}A_{\rho
K_{0}^{*}}[\delta_{pu}(\alpha_{1}+\beta_{2})+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}]$
$\displaystyle+\frac{1}{\sqrt{2}}A_{K_{0}^{*}\rho}[\delta_{pu}\alpha_{2}+\frac{3}{2}\alpha_{3,EW}^{p}]\,,$
(104) $\displaystyle{\cal A}_{B^{-}\to K_{0}^{*-}\omega}=$
$\displaystyle\frac{1}{\sqrt{2}}A_{\omega
K_{0}^{*}}[\delta_{pu}(\alpha_{1}+\beta_{2})+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}]$
$\displaystyle+\frac{1}{\sqrt{2}}A_{K_{0}^{*}\omega}[\delta_{pu}\alpha_{2}+2\alpha_{3}^{p}+\frac{1}{2}\alpha_{3,EW}^{p}]\,,$
(105) $\displaystyle{\cal A}_{B^{-}\to K_{0}^{*-}\phi}=$ $\displaystyle
A_{K_{0}^{*}\phi}[\delta_{pu}\beta_{2}+\alpha_{3}^{p}+\alpha_{4}^{p}-\frac{1}{2}\alpha_{3,EW}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}]\,,$
(106) $\displaystyle{\cal A}_{\bar{B}^{0}\to\bar{K}_{0}^{*0}K^{*0}}=$
$\displaystyle
A_{K_{0}^{*}K^{*}}[\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{K^{*}K_{0}^{*}}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$ (107)
$\displaystyle{\cal A}_{\bar{B}^{0}\to K_{0}^{*0}\bar{K}^{*0}}=$
$\displaystyle
A_{K^{*}K_{0}^{*}}[\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{K_{0}^{*}K^{*}}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$ (108)
$\displaystyle{\cal A}_{\bar{B}^{0}\to K_{0}^{*-}K^{*+}}=$ $\displaystyle
A_{K_{0}^{*}K^{*}}[\delta_{pu}\beta_{1}+\beta_{4}^{p}+\beta_{4,EW}^{p}]+B_{K^{*}K_{0}^{*}}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$
(109) $\displaystyle{\cal A}_{\bar{B}^{0}\to K_{0}^{*+}K^{*-}}=$
$\displaystyle
A_{K^{*}K_{0}^{*}}[\delta_{pu}\beta_{1}+\beta_{4}^{p}+\beta_{4,EW}^{p}]+B_{K_{0}^{*}K^{*}}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$
(110) $\displaystyle{\cal A}_{\bar{B}^{0}\to K_{0}^{*-}\rho^{+}}=$
$\displaystyle A_{\rho
K_{0}^{*}}[\delta_{pu}\alpha_{1}+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}-\frac{1}{2}\beta_{3,EW}^{p}]\,,$
(111) $\displaystyle{\cal A}_{\bar{B}^{0}\to\bar{K}_{0}^{*0}\rho^{0}}=$
$\displaystyle\frac{1}{\sqrt{2}}A_{\rho
K_{0}^{*}}[-\alpha_{4}^{p}+\frac{1}{2}\alpha_{4,EW}^{p}-\beta_{3}^{p}+\frac{1}{2}\beta_{3,EW}^{p}]$
$\displaystyle+\frac{1}{\sqrt{2}}A_{K_{0}^{*}\rho}[\delta_{pu}\alpha_{2}+\frac{3}{2}\alpha_{3,EW}^{p}]\,,$
(112) $\displaystyle{\cal A}_{\bar{B}^{0}\to\bar{K}_{0}^{*0}\omega}=$
$\displaystyle\frac{1}{\sqrt{2}}A_{\omega
K_{0}^{*}}[\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}-\frac{1}{2}\beta_{3,EW}^{p}]$
$\displaystyle+\frac{1}{\sqrt{2}}A_{K_{0}^{*}\omega}[\delta_{pu}\alpha_{2}+2\alpha_{3}^{p}+\frac{1}{2}\alpha_{3,EW}^{p}]\,,$
(113) $\displaystyle{\cal A}_{\bar{B}^{0}\to\bar{K}_{0}^{*0}\phi}=$
$\displaystyle
A_{K_{0}^{*}\phi}[\alpha_{3}^{p}+\alpha_{4}^{p}-\frac{1}{2}\alpha_{3,EW}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}-\frac{1}{2}\beta_{3,EW}^{p}]\,,$
(114) $\displaystyle{\cal A}_{\bar{B}_{s}^{0}\to\bar{K}_{0}^{*0}K^{*0}}=$
$\displaystyle
A_{K^{*}K_{0}^{*}}[\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{K_{0}^{*}K^{*}}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$ (115)
$\displaystyle{\cal A}_{\bar{B}_{s}^{0}\to K_{0}^{*0}\bar{K}^{*0}}=$
$\displaystyle
A_{K_{0}^{*}K^{*}}[\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{K^{*}K_{0}^{*}}[b_{4}^{p}-\frac{1}{2}b_{4,EW}^{p}]\,,$ (116)
$\displaystyle{\cal A}_{\bar{B}_{s}^{0}\to K_{0}^{*-}K^{*+}}=$ $\displaystyle
A_{K^{*}K_{0}^{*}}[\delta_{pu}\alpha_{1}+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{K_{0}^{*}K^{*}}[\delta_{pu}b_{1}+b_{4}^{p}+b_{4,EW}^{p}]\,,$
(117) $\displaystyle{\cal A}_{\bar{B}_{s}^{0}\to K_{0}^{*+}K^{*-}}=$
$\displaystyle+A_{K_{0}^{*}K^{*}}[\delta_{pu}\alpha_{1}+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{4}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}]$
$\displaystyle+B_{K^{*}K_{0}^{*}}[\delta_{pu}b_{1}+b_{4}^{p}+b_{4,EW}^{p}]\,,$
(118) $\displaystyle{\cal A}_{\bar{B}_{s}\to K_{0}^{*+}\rho^{-}}=$
$\displaystyle
A_{K_{0}^{*}\rho}[\delta_{pu}\alpha_{1}+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}-\frac{1}{2}\beta_{3,EW}^{p}]\,,$
(119) $\displaystyle{\cal A}_{\bar{B}_{s}\to K_{0}^{*0}\rho^{0}}=$
$\displaystyle\frac{1}{\sqrt{2}}A_{K_{0}^{*}\rho}[\delta_{pu}\alpha_{2}-\alpha_{4}^{p}+\frac{3}{2}\alpha_{3,EW}^{p}+\frac{1}{2}\alpha_{4,EW}^{p}-\beta_{3}^{p}+\frac{1}{2}\beta_{3,EW}^{p}]\,,$
(120) $\displaystyle{\cal A}_{\bar{B}_{s}\to K_{0}^{*0}\omega}=$
$\displaystyle\frac{1}{\sqrt{2}}A_{K_{0}^{*}\omega}[\delta_{pu}\alpha_{2}+2\alpha_{3}^{p}+\alpha_{4}^{p}+\frac{1}{2}\alpha_{3,EW}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}-\frac{1}{2}\beta_{3,EW}^{p}]\,,$
(121) $\displaystyle{\cal A}_{\bar{B}_{s}\to K_{0}^{*0}\phi}=$ $\displaystyle
A_{\phi
K_{0}^{*}}[\alpha_{4}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}+\beta_{3}^{p}-\frac{1}{2}\beta_{3,EW}^{p}]+A_{K_{0}^{*}\phi}[\alpha_{3}^{p}-\frac{1}{2}\alpha_{3,EW}^{p}]\,.$
(122)

## References

  * [1] R. L. Jaffe, Phys. Rev. D 15 (1977) 267.
  * [2] R. L. Jaffe, Phys. Rev. D 15 (1977) 281.
  * [3] N. Mathur et al., Phys. Rev. D 76 (2007) 114505.
  * [4] S. Prelovsek, T. Draper, C. B. Lang, M. Limmer, K. F. Liu, N. Mathur and D. Mohler, Phys. Rev. D 82 (2010) 094507.
  * [5] F. E. Close and N. A. Tornqvist, J. Phys. G 28 (2002) R249.
  * [6] K. Abe et al. [Belle Collaboration], Phys. Rev. D 65 (2002) 092005.
  * [7] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D 70 (2004) 092001.
  * [8] T. Abe et al. [Belle Collaboration], Phys. Rev. Lett. 96 (2006) 251803.
  * [9] P. Chang et al. [Belle Collaboration], Phys. Lett. B 599 (2004) 148.
  * [10] M. Prim et al. [Belle Collaboration], Phys. Rev. D 88 (2013) no.7, 072004.
  * [11] M. Prim et al. [Belle Collaboration], Phys. Rev. D 81 (2010) 071101.
  * [12] J. P. Lees et al. [BABAR Collaboration], Phys. Rev. D 85 (2012) 112010.
  * [13] J. P. Lees et al. [BABAR Collaboration], Phys. Rev. D 83 (2011) 112010.
  * [14] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D 79 (2009) 052005.
  * [15] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D 78 (2008) 012004.
  * [16] B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. 101 (2008) 161801.
  * [17] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D 78 (2008) 092008.
  * [18] J. P. Lee et al. [BaBar Collaboration], Phys. Rev. D 85 (2012) 072005.
  * [19] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D 76 (2007) 071103.
  * [20] J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D 83 (2011) 112010.
  * [21] R. Aaij et al. [LHCb Collaboration], JHEP 1906 (2019) 114.
  * [22] R. Aaij et al. [LHCb Collaboration], JHEP 1907 (2019) 032.
  * [23] H. Y. Cheng and C. K. Chua, Phys. Rev. D 102 (2020) no.5, 053006.
  * [24] H. Y. Cheng, C. K. Chua and Z. Q. Zhang, Phys. Rev. D 94 (2016) no.9, 094015.
  * [25] H. Y. Cheng and C. K. Chua, Phys. Rev. D 89 (2014) no.7, 074025.
  * [26] H. Y. Cheng and C. K. Chua, Phys. Rev. D 88 (2013) 114014.
  * [27] H. Y. Cheng, C. K. Chua and A. Soni, Phys. Rev. D 76 (2007) 094006.
  * [28] B. El-Bennich, A. Furman, R. Kaminski, L. Lesniak, B. Loiseau and B. Moussallam, Phys. Rev. D 79 (2009) 094005.
  * [29] Y. Li, H. Y. Zhang, Y. Xing, Z. H. Li and C. D. Lu, Phys. Rev. D 91 (2015) no.7, 074022.
  * [30] H. Y. Cheng, C. K. Chua and K. C. Yang, Phys. Rev. D 73 (2006) 014017.
  * [31] H. Y. Cheng, C. K. Chua and K. C. Yang, Phys. Rev. D 77 (2008) 014034.
  * [32] H. Y. Cheng and C. K. Chua, Phys. Rev. D 82 (2010) 034014.
  * [33] H. Y. Cheng, C. K. Chua, K. C. Yang and Z. Q. Zhang, Phys. Rev. D 87 (2013) no.11, 114001.
  * [34] J. J. Qi, X. H. Guo, Z. Y. Wang, Z. H. Zhang and C. Wang, Phys. Rev. D 99 (2019) 076010.
  * [35] N. Wang, Q. Chang, Y. L. Yang and J. F. Sun, J. Phys. G 46 (2019) no.9, 095001.
  * [36] Z. T. Zou, Y. Li and X. Liu, Eur. Phys. J. C 80 (2020) no.6, 517.
  * [37] Z. T. Zou, Y. Li and Q. X. Li, Eur. Phys. J. C 80 (2020) no.5, 394.
  * [38] Z. T. Zou, Y. Li and X. Liu, Eur. Phys. J. C 77 (2017) no.12, 870.
  * [39] Q. X. Li, L. Yang, Z. T. Zou, Y. Li and X. Liu, Eur. Phys. J. C 79 (2019) no.11, 960.
  * [40] W. F. Wang, H. C. Hu, H. N. Li and C. D. Lu, Phys. Rev. D 89 (2014) no.7, 074031.
  * [41] Z. Rui and W. F. Wang, Phys. Rev. D 97 (2018) no.3, 033006.
  * [42] Z. Rui, Y. Q. Li and J. Zhang, Phys. Rev. D 99 (2019) no.9, 093007.
  * [43] X. Liu, Z. Q. Zhang, Z. J. Xiao, Chin. Phys. C 34 (2010) 157.
  * [44] X. Liu, Z. J. Xiao and Z. T. Zou, J. Phys. G 40 (2013) 025002.
  * [45] X. Liu, Z. J. Xiao and Z. T. Zou, Phys. Rev. D 88 (2013) 094003.
  * [46] R. H. Li, C. D. Lu, W. Wang and X. X. Wang, Phys. Rev. D 79 (2009) 014013.
  * [47] Q. X. Li, L. Yang, Z. T. Zou, Y. Li and X. Liu, Eur. Phys. J. C 79 (2019) no.11, 960.
  * [48] Z. T. Zou, Y. Li and X. liu, Phys. Rev. D 95 (2017) no.1, 016011.
  * [49] Z. T. Zou, Y. Li and X. Liu, Eur. Phys. J. C 77 (2017) no.12, 870.
  * [50] Z. Q. Zhang, S. J. Wang, L. Y. Zhang, Chin. Phys. C 37 (2013) 043103.
  * [51] Z. Q. Zhang, EPL 97 (2012) no.1, 11001.
  * [52] Z. Q. Zhang, Phys. Rev. D 82 (2010) 034036.
  * [53] Z. Q. Zhang, Phys. Rev. D 82 (2010) 114016.
  * [54] Z. W. Liu, Z. T. Zou, Y. Li, X. Liu and J. Wang, [arXiv:2111.04235 [hep-ph]].
  * [55] B. El-Bennich, A. Furman, R. Kaminski, L. Lesniak and B. Loiseau, Phys. Rev. D 74 (2006) 114009.
  * [56] H. Y. Cheng, C. K. Chua and C. W. Hwang, Phys. Rev. D 69 (2004) 074025.
  * [57] X. W. Kang, T. Luo, Y. zhang, L. Y. Dai and C. Wang, Eur. Phys. J. C 78 (2018) no.11, 909.
  * [58] A. Issadykov, M. A. Ivanov and S. K. Sakhiyev, Phys. Rev. D 91 (2015) no.7, 074007.
  * [59] S. Cheng and J. M. Shen, Eur. Phys. J. C 80 (2020) no.6, 554.
  * [60] Y. Li, E. L. Wang and H. Y. Zhang, Adv. High Energy Phys. 2013 (2013) 175287.
  * [61] Y. Li, D. D. Wang and C. D. Lu, Chin. Phys. C 40 (2016) no.1, 013101.
  * [62] N. Ghahramany and R. Khosravi, Phys. Rev. D 80 (2009) 016009.
  * [63] T. M. Aliev, K. Azizi and M. Savci, Phys. Rev. D 76 (2007) 074017.
  * [64] M. Z. Yang, Phys. Rev. D 73 (2006) 034027.
  * [65] H. Y. Han, X. G. Wu, H. B. Fu, Q. L. Zhang and T. Zhong, Eur. Phys. J. A 49 (2013) 78.
  * [66] Z. G. Wang, Eur. Phys. J. C 75 (2015) no.2, 50.
  * [67] G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68 (1996) 1125.
  * [68] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83 (1999) 1914.
  * [69] M. Beneke G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B 606 (2001) 245.
  * [70] M. Beneke and M. Neubert, Nucl. Phys. B 675 (2003) 333.
  * [71] H. Y. Cheng, C. K. Chua, Phys. Rev. D 80 (2009) 114008.
  * [72] H. Y. Cheng, C. K. Chua, Phys. Rev. D 80 (2009) 114026.
  * [73] Q. Chang, J. F. Sun, Y. L. Yang, and X. N. Li Phys. Rev. D 90 (2014) 054019.
  * [74] Q. Chang, X. H. Hu, J. F. Sun, and Y. L. Yang, Phys. Rev. D 91 (2015) 074026.
  * [75] J. F. Sun, Q. Chang, X. H. Hu, and Y. L. Yang, Phys. Lett. B 743 (2015) 444.
  * [76] Q. Chang, J. F. Sun, Y. L. Yang, and X. N. Li, Phys. Lett. B 740 (2015) 56.
  * [77] P. A. Zyla et al. [Particle Data Group], PTEP 2020 (2020) no.8, 083C01.
  * [78] M. V. Terentev, Yad. Fiz. 24 (1976) 207 [Sov. J. Nucl. Phys. 24 (1976) 106].
  * [79] V. B. Berestetsky and M. V. Terentev, Yad. Fiz. 25 (1977) 653 [Sov. J. Nucl. Phys. 25 (1977) 347].
  * [80] W. Jaus, Phys. Rev. D 41 (1990) 3394.
  * [81] W. Jaus and D. Wyler, Phys. Rev. D 41 (1990) 3405.
  * [82] W. Jaus, Phys. Rev. D 60 (1999) 054026.
  * [83] W. Jaus, Phys. Rev. D 67 (2003) 094010.
  * [84] H. M. Choi and C. R. Ji, Phys. Rev. D 89 (2014) no. 3, 033011.
  * [85] Q. Chang, L. T. Wang and X. N. Li, JHEP 1912 (2019) 102.
  * [86] Q. Chang, X. N. Li, X. Q. Li, F. Su and Y. D. Yang, Phys. Rev. D 98 (2018) no.11, 114018.
  * [87] H. M. Choi, H. Y. Ryu and C. R. Ji, Phys. Rev. D 96 (2017) no.5, 056008.
  * [88] H. M. Choi, [arXiv:2108.10544 [hep-ph]].
  * [89] H. M. Choi, arXiv:2102.02015 [hep-ph].
  * [90] Q. Chang, X. N. Li and L. T. Wang, Eur. Phys. J. C 79 (2019) no.5, 422.
  * [91] H. M. Choi and C. R. Ji, Phys. Rev. D 102 (2020) no.3, 036005.
  * [92] Q. Chang, X. L. Wang and L. T. Wang, Chin. Phys. C 44 (2020) no.8, 083105.
  * [93] H. M. Choi and C. R. Ji, Phys. Rev. D 95 (2017) no.5, 056002.
  * [94] H. M. Choi and C. R. Ji, Phys. Rev. D 91 (2015) no.1, 014018.
  * [95] C. W. Hwang and R. S. Guo, Phys. Rev. D 82 (2010) 034021.
  * [96] D. S. Du, J. W. Li and M. Z. Yang, Phys. Lett. B 619 (2005) 105.
  * [97] M. Z. Yang, Mod. Phys. Lett. A 21 (2006) 1625-1628.
  * [98] K. Maltman, Phys. Lett. B 462 (1999) 14.
  * [99] P. Ball and R. Zwicky, Phys. Rev. D 71 (2005) 014015.
  * [100] P. Ball and R. Zwicky, Phys. Rev. D 71 (2005) 014029.
  * [101] C. D. Lu, W. Wang and Z. T. Wei, Phys. Rev. D 76 (2007) 014013.
  * [102] R. C. Verma, J. Phys. G 39 (2012) 025005.
  * [103] W. F. Wang and Z. J. Xiao, Phys. Rev. D 86 (2012) 114025.
  * [104] R. H. Li, C. D. Lu and W. Wang, Phys. Rev. D 79 (2009) 034014.
  * [105] N. R. Soni, A. Issadykov, A. N. Gadaria, J. J. Patel and J. N. Pandya, [arXiv:2008.07202 [hep-ph]].
  * [106] G. S. Bali et al. [RQCD], JHEP 08 (2019) 065.
  * [107] P. Ball, V. M. Braun and A. Lenz, JHEP 05 (2006) 004.
  * [108] H. M. Choi and C. R. Ji, Phys. Rev. D 75 (2007) 034019.
  * [109] P. Ball and G. W. Jones, JHEP 03 (2007) 069.
  * [110] A. Bharucha, D. M. Straub and R. Zwicky, JHEP 08 (2016) 098.
  * [111] Q. Chang, S. J. Brodsky and X. Q. Li, Phys. Rev. D 95 (2017) no.9, 094025.
  * [112] C. Wang, M. Z. Zhou and H. Chen, Eur. Phys. J. C 77 (2017) no.4, 219.
  * [113] C. Wang, J. K. He and M. Z. Zhou, Eur. Phys. J. C 79 (2019) no.9, 765.
  * [114] S. Uehara et al. [Belle], Eur. Phys. J. C 53 (2008) 1-14.
  * [115] Y. S. Amhis et al. [HFLAV Collaboration], Eur. Phys. J. C 81 (2021) no.3, 226.
  * [116] W. F. Wang, J. Chai and A. J. Ma, JHEP 03 (2020) 162.
  * [117] Y. L. Shen, W. Wang, J. Zhu, C. D. Lu, Eur. Phys. J. C 50 (2007) 877.
  * [118] C. S. Kim, Y. Li and W. Wang, Phys. Rev. D 81 (2010) 074014.
  * [119] M. Beneke, T. Huber and X. Q. Li, Nucl. Phys. B 832 (2010) 109-151.
  * [120] G. Bell, M. Beneke, T. Huber and X. Q. Li, Phys. Lett. B 750 (2015) 348-355.
  * [121] G. Bell, M. Beneke, T. Huber and X. Q. Li, JHEP 04 (2020) 055.
  * [122] T. Huber, S. Kränkl and X. Q. Li, JHEP 09 (2016) 112.
  * [123] G. Bell, Nucl. Phys. B 795 (2008) 1-26.
  * [124] G. Bell, Nucl. Phys. B 822 (2009) 172-200.
  * [125] M. Beneke and S. Jager, Nucl. Phys. B 751 (2006) 160-185.
  * [126] N. Kivel, JHEP 05 (2007) 019.
  * [127] V. Pilipp, Nucl. Phys. B 794 (2008) 154-188.
  * [128] M. Beneke and S. Jager, Nucl. Phys. B 768 (2007) 51-84.
  * [129] C. S. Kim and Y. W. Yoon, JHEP 11 (2011) 003.