# Study of the nonleptonic charmless $B$ ${\to}$ $SS$ decays
with the QCD factorization approach

Lili Chen  Institute of Particle and Nuclear Physics, Henan Normal University,
Xinxiang 453007, China  Mengfei Zhao  Institute of Particle and Nuclear
Physics, Henan Normal University, Xinxiang 453007, China  Liting Wang
Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang
453007, China  Yueyang Kang  Institute of Particle and Nuclear Physics, Henan
Normal University, Xinxiang 453007, China  Qin Chang  Institute of Particle
and Nuclear Physics, Henan Normal University, Xinxiang 453007, China
Institute of Physics, Henan Academy of Sciences, Zhengzhou 455004, China
Junfeng Sun  Institute of Particle and Nuclear Physics, Henan Normal
University, Xinxiang 453007, China

###### Abstract

Inspired by the brilliant prospects of the ongoing $B$ meson experiments, the
hadronic charmless $B$ ${\to}$ $SS$ decays are studied by considering the
next-to-leading (NLO) contributions with the QCD factorization approach, where
$S$ denotes the scalar mesons $K_{0}^{\ast}(1430)$ and $a_{0}(1450)$.
Branching ratios and $CP$ violating asymmetries are estimated with the updated
values of hadronic parameters obtained from a covariant light-front quark
model, for two scenarios where the scalar mesons are the $1^{3}P_{0}$ and
$2^{3}P_{0}$ states. It is found that the NLO contributions are very important
for the $B$ ${\to}$ $SS$ decays; For the $B$ ${\to}$
$a_{0}(1450)K_{0}^{\ast}(1430)$ and $B_{s}$ ${\to}$
$K_{0}^{\ast}(1430)\overline{K}_{0}^{\ast}(1430)$ decays, branching ratios can
reach up to the order of ${\cal O}(10^{-5})$ by assuming that the scalar
mesons are the $1P$ states, and should first be investigated in the future
experiments.

##  I Introduction

According to the traditional quark model, the $P$-wave triplet states of the
quark-antiquark system have the quantum number $J^{P}$ $=$ $0^{+}$, and are
called the scalar mesons. The scalar mesons mostly appear as the hadronic
resonances, and have large decay widths. There will exist several resonances
and decay channels within a short mass interval. The overlaps between
resonances and background make it considerably difficult to resolve the scalar
mesons. In addition, the di-boson combinations can also have the quantum
number $J^{P}$ $=$ $0^{+}$. In contrast to the ground pseudoscalar and vector
mesons, the identification of the scalar is long-standing puzzle. To
understand the internal structure of the scalar mesons is one of the most
interesting topics in hadron physics. Generally, the scalar mesons have been
identified as the ordinary quark-antiquark $q\bar{q}$ states, tetraquark
$q\bar{q}q\bar{q}$ states, meson-meson molecular states or even those
supplemented with a scalar glueball. There are many candidates with $J^{PC}$
$=$ $0^{++}$ below 2 GeV, which cannot be accommodated in one $SU(3)$ flavor
nonet satisfactorily. From the mass spectrum of those scalar mesons and their
chromatic as well as electromagnetic decays, a prospective picture (scenario
2, hereafter this text will be abbreviated as S2) suggests that the isovector
$a_{0}(1450)$, isodoublet $K^{\ast}_{0}(1430)$, isoscalar $f_{0}(1710)$ and
$f_{0}(1370)$ above 1 GeV can be assigned to be a conventional $SU(3)$
$q\bar{q}$ scalar nonet with the spectroscopy symbol of $1^{3}P_{0}$ pdg2022 ,
while the scalar mesons $a_{0}(980)$, $K_{0}^{\ast}(700)$ (or ${\kappa}$),
$f_{0}(980)$ and $f_{0}(500)$ (or ${\sigma}$) below 1 GeV form the
unconventional $q\bar{q}q\bar{q}$ exotic nonet pdg2022 ; PhysRevD.15.267 ;
PhysRevD.15.281 . Of course, the above assignments is tentative. In
alternative schemes, the scalar mesons with mass below 1 GeV are interpreted
as the lowest lying $q\bar{q}$ states, while the scalars $a_{0}(1450)$,
$K^{\ast}_{0}(1430)$, $f_{0}(1710)$ and $f_{0}(1370)$ are regarded as the
radial excited states with the spectroscopy symbol of $2^{3}P_{0}$ (scenario
1, namely S1).

It is widely known that the $B$ mesons have rich decay modes. The light scalar
mesons can be produced in the $B$ meson decays. The $B$ meson hadronic decays
involving the final scalar mesons provides another efficient way to
investigate the features and the possible inner structures of the scalar
mesons. Experimentally, some of the $B$ ${\to}$ $SP$, $SV$, $SS$, $SX$ decays
(where the symbols of $S$, $P$, $V$ and $X$ denote the light scalar mesons,
pseudoscalar mesons, vector mesons and other particles, respectively), such as
the $B$ ${\to}$ $K^{\ast}_{0}(1430)^{+}{\pi}^{-}$,
$K^{\ast}_{0}(1430)^{+}{\omega}$,
$K^{\ast}_{0}(1430)^{0}\overline{K}^{\ast}_{0}(1430)^{0}$,
$K^{\ast}_{0}(1430)^{0}{\pi}^{+}{\gamma}$ decays, have been measured by Belle,
BaBar and LHCb groups pdg2022 . With the running of high-luminosity Belle-II
and LHCb experiments and the coming CEPC, FCC-ee and HL-LHC experiments, more
and more data on the $B$ meson decays will be available in the future, more
and more $B$ ${\to}$ $SP$, $SV$, $SS$, $SX$ decays can be discovered and
investigated, and the measurement precision will be higher and higher, which
lays a solid experimental foundation to carefully study the scalar mesons and
distinguish theoretical models. Phenomenologically, many of the $B$ ${\to}$
$SP$, $SV$, $SS$, $SX$ decays have been studied extensively with various
theoretical models. For example, the study of the $B$ ${\to}$ $SP$, $SV$
decays with the QCD factorization (QCDF) approach PhysRevD.71.054020 ;
PhysRevD.73.014017 ; PhysRevD.77.014034 ; PhysRevD.79.094005 ;
PhysRevD.82.034014 ; PhysRevD.85.074010 ; AHEP.2013.175287 ;
PhysRevD.87.114001 ; PhysRevD.91.074022 ; CPC.40.013101 ; PhysRevD.99.076010 ;
PhysRevD.105.016002 , the $B$ ${\to}$ $SP$, $SV$, $SS$ decays with the
perturbative QCD (PQCD) approach PhysRevD.74.114010 ; EPJC.50.877 ; CPC.33.508
; CPC.34.157 ; CPC.34.528 ; PhysRevD.81.074014 ; PhysRevD.82.034036 ;
PhysRevD.82.114016 ; EPJC.67.163 ; CTP.53.540 ; CTP.56.1063 ;
PhysRevD.83.054001 ; EPL.97.11001 ; PhysRevD.88.094003 ; CPC.37.043103 ;
JPG.46.095001 ; EPJC.82.59 ; JPG.40.025002 ; JPG.43.045001 ; EPJC.79.960 ;
PhysRevD.102.116007 ; CTP.73.045201 ; EPJC.82.177 , the semileptonic $B$
${\to}$ $SX$ decays with the sum rules and other approaches PhysRevD.76.074017
; PhysRevD.78.014006 ; PhysRevD.79.014013 ; PhysRevD.80.016009 ;
PhysRevD.82.034016 ; PhysRevD.83.025024 ; PhysRevD.91.074007 ; EPJC.78.909 ;
PhysRevD.105.116027 , and so on. It is easy to imagine that various scenarios,
such as S1 and S2, will inevitably give different theoretical predictions on
the $B$ meson decays. It is noteworthy that some studies have shown that
branching ratios for the charmless $B$ ${\to}$ $SS$ decays with the PQCD
approach can be very large, for example, ${\cal
B}(B_{s}{\to}K^{\ast}_{0}(1430)K^{\ast}_{0}(1430))$ ${\sim}$ ${\cal
O}(10^{-4})$ JPG.40.025002 , ${\cal B}(B{\to}K^{\ast}_{0}(1430){\sigma})$
${\sim}$ ${\cal O}(10^{-4})$ EPJC.79.960 , ${\cal
B}(B_{s}{\to}{\sigma}{\sigma})$ ${\sim}$ ${\cal O}(10^{-4})$, ${\cal
B}(B_{s}{\to}{\sigma}f_{0}(980))$ ${\sim}$ ${\cal O}(10^{-4})$, ${\cal
B}(B_{s}{\to}f_{0}(980)f_{0}(980))$ ${\sim}$ ${\cal O}(10^{-4})$, ${\cal
B}(B{\to}{\sigma}{\sigma})$ ${\sim}$ ${\cal O}(10^{-5})$ EPJC.82.177 , ${\cal
B}(B{\to}a_{0}(980)a_{0}(980))$ ${\sim}$ ${\cal O}(10^{-5})$ JPG.40.025002 .
And the more striking phenomena are that branching ratios for the pure
annihilation $B$ ${\to}$ $SS$ decays, which might be very small by intuition,
are very large with the PQCD approach, for example, ${\cal
B}(B_{s}{\to}a_{0}(980)a_{0}(980))$ ${\sim}$ ${\cal O}(10^{-5})$, ${\cal
B}(B_{s}{\to}a_{0}(1450)a_{0}(1450))$ ${\sim}$ ${\cal O}(10^{-5})$, ${\cal
B}(B_{d}{\to}{\kappa}^{+}{\kappa}^{-})$ ${\sim}$ ${\cal O}(10^{-6})$, ${\cal
B}(B_{d}{\to}K^{\ast}_{0}(1430)^{+}K^{\ast}_{0}(1430)^{-})$ ${\sim}$ ${\cal
O}(10^{-6})$ CTP.73.045201 . So the study of the $B$ ${\to}$ $SS$ decays is
very promising and tempting both theoretically and experimentally. In order to
deepen our understanding on the properties of the light scalar mesons and
provide the ongoing and coming experimental analysis with additional
theoretical references, in this paper, we will study the nonleptonic charmless
$B$ ${\to}$ $SS$ decays with the QCDF approach, by considering scenarios S1
and S2 for the scalar mesons, where $S$ $=$ $K_{0}^{\ast}(1430)$ and
$a_{0}(1450)$.

This paper is organized as follows. In Section II, the theoretical framework
are briefly reviewed, the next-to-leading order effective coefficients for the
$B$ ${\to}$ $SS$ decays and the weak annihilation amplitudes are given with
the QCDF approach. In Section III, the values of the nonperturbative input
parameters are fixed. The numerical results and our comments are presented in
Section IV. Finally, We conclude with a summary in Section V. The decay
amplitudes are displayed in the Appendix.

##  II theoretical framework

###  II.1 The effective Hamiltonian

The low-energy effective Hamiltonian for the charmless nonleptonic $B$ ${\to}$
$SS$ decays is written as RevModPhys.68.1125 ,

$\displaystyle{\cal H}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\sum\limits_{q=d,s}\Big{\\{}V_{ub}V_{uq}^{\ast}\Big{[}C_{1}({\mu})O_{1}({\mu})+C_{2}({\mu})O_{2}({\mu})\Big{]}$
(1) $\displaystyle-$ $\displaystyle
V_{tb}V_{tq}^{\ast}\Big{[}\sum\limits_{i=3}^{10}C_{i}({\mu})O_{i}({\mu})+C_{7{\gamma}}({\mu})O_{7{\gamma}}({\mu})+C_{8g}({\mu})O_{8g}({\mu})\Big{]}\Big{\\}}+{\rm
h.c.},$

where the Fermi constant $G_{F}$ and the Cabibbo-Kobayashi-Maskawa (CKM)
matrix elements $V_{ij}$ have been well determined experimentally pdg2022 .
The Wilson coefficients $C_{i}$, which summarize the short-distance physical
contributions, are in principle computable with the perturbative theory order
by order at the scale of ${\mu}$ $=$ $m_{W}$, and can be evaluated to the
energy scale of the $B$ meson decays ${\mu}$ ${\sim}$ $m_{b}$ with the
renormalization group equation (RGE) RevModPhys.68.1125 , where $m_{W}$ and
$m_{b}$ are the mass of the gauge boson $W$ of the weak interactions and the
heavy $b$ quark mass, respectively. The remaining theoretical work is to
calculate the hadronic matrix elements (HMEs),
${\langle}S_{1}S_{2}{|}O_{i}{|}B{\rangle}$, where the local four-quark
effective operators $O_{i}$ are sandwiched between the initial $B$ meson and
the final scalar mesons.

In order to generate the essential strong phases for $CP$ violations in the
hadronic $B$ meson decays, and cancel the unphysical ${\mu}$-dependence of
decay amplitude ${\cal A}$ $=$ ${\langle}S_{1}S_{2}{|}{\cal H}_{\rm
eff}{|}B{\rangle}$ originating from the Wilson coefficients, the high order
radiative corrections to HMEs are necessary and should be taken into
consideration. However, the perturbative contributions embedded in HMEs became
entangled with the nonperturbative contributions, which makes the theoretical
calculations extremely complicated. How to properly and reasonably evaluate
HMEs of the hadronic $B$ meson decays has been an academic focus.

###  II.2 The QCDF decay amplitudes

The QCDF approach PhysRevLett.83.1914 ; NPB.591.313 ; NPB.606.245 ; PLB.488.46
; PLB.509.263 ; PhysRevD.64.014036  is one of many QCD-inspired
phenomenological remedies to deal with HMEs. Based on the power counting rules
in the heavy quark limits and an expansion series in the strong coupling
${\alpha}_{s}$ assisted by the collinear approximation, the long- and short-
distance contributions are factorized. The nonperturbative contributions in
HMEs are either power suppressed by $1/m_{b}$ or incorporated into the
hadronic transition form factors and mesonic distribution amplitudes (DAs). Up
to the leading power corrections of order $1/m_{b}$, the QCDF factorization
formula for HMEs concerned is written as NPB.591.313 ,

$\displaystyle{\langle}S_{1}S_{2}{|}O_{i}({\mu}){|}B{\rangle}$
$\displaystyle=$
$\displaystyle\sum\limits_{j}F_{j}^{B{\to}S_{1}}\,f_{S_{2}}{\int}dy\,{\cal
T}_{ij}^{I}(y)\,\phi_{S_{2}}(y)+(S_{1}{\leftrightarrow}S_{2})$ (2)
$\displaystyle+$ $\displaystyle
f_{B}\,f_{S_{1}}\,f_{S_{2}}{\int}dx\,dy\,dz\,{\cal
T}_{i}^{II}(x,y,z)\,{\phi}_{S_{1}}(x)\,{\phi}_{S_{2}}(y)\,{\phi}_{B}(z),$

where $x$, $y$ and $z$ are the longitudinal momentum fractions of the valence
quarks. The form factors $F_{j}^{B{\to}S}$, the decay constants $f_{B}$ and
$f_{S}$, the mesonic light cone DAs ${\phi}_{B}$ and ${\phi}_{S}$, all of them
are the nonperturbative parameters. These parameters are regarded to be
universal and process-independent, and can be obtained from the experimental
data, lattice QCD simulation, QCD sum rules, or by comparison with other
exclusive processes. ${\cal T}^{I}$ and ${\cal T}^{II}$ are the hard-
scattering functions describing the local interactions among quarks and gluons
at the $B$ meson decay scale. They are, in principle, perturbatively
calculable to all orders in ${\alpha}_{s}$ at the leading power order of
$1/m_{b}$. At the leading order (LO) ${\alpha}_{s}^{0}$, ${\cal T}^{I}$ $=$
$1$ and ${\cal T}^{II}$ $=$ $0$. The convolution integrals of ${\cal T}^{I}$
and ${\phi}_{S}$ result in the decay constant of the emission scalar mesons.
One can return from the QCDF formula Eq.(2) to the naive factorization (NF)
approximations ZPC.34.103 ; NPBPS.11.325 , i.e., the four-quark HMEs can be
written as the product of two diquark HMEs, and the diquark HMEs can be
replaced by HMEs of the corresponding hadronic currents and then further
parameterized by hadronic transition form factors and decay constants. Beyond
the order ${\alpha}_{s}^{0}$, the radiative corrections to HMEs make ${\cal
T}^{I,II}$ no longer trivial, and some information about the $CP$-violating
strong phases and ${\mu}$-dependence of HMEs can be retrieved naturally.

With the QCDF approach, the amplitudes for the concerned $B$ ${\to}$ $SS$
decays can be generally written as,

$\displaystyle{\cal A}\,=\,{\langle}S_{1}S_{2}{|}{\cal H}_{\rm
eff}{|}B{\rangle}\,=\,\frac{G_{F}}{\sqrt{2}}\sum\limits_{i}{\lambda}_{i}\sum\limits_{j=1}^{10}a_{j}{\langle}S_{1}S_{2}{|}O_{j}{|}B{\rangle}_{\rm
NF},$ (3)

where the parameter ${\lambda}_{i}$ is the product of the CKM elements; the
coefficient $a_{j}$ including the nonfactorizable contributions beyond the
leading order of ${\alpha}_{s}$ is the combinations of the Wilson
coefficients; HMEs ${\langle}S_{1}S_{2}{|}O_{j}{|}B{\rangle}_{\rm NF}$ are
defined and evaluated with the NF approximation.

###  II.3 The QCDF coefficients

To simplify the decay amplitude expressions, we will use the notations in
Refs. NPB.675.333  and write the QCDF coefficients as follows.

$\displaystyle{\alpha}_{1}(S_{1}\,S_{2})$ $\displaystyle=$ $\displaystyle
a_{1}(S_{1}\,S_{2}),$ (4) $\displaystyle{\alpha}_{2}(S_{1}\,S_{2})$
$\displaystyle=$ $\displaystyle a_{2}(S_{1}\,S_{2}),$ (5)
$\displaystyle{\alpha}_{3}^{p}(S_{1}\,S_{2})$ $\displaystyle=$ $\displaystyle
a_{3}^{p}(S_{1}\,S_{2})+a_{5}^{p}(S_{1}\,S_{2}),$ (6)
$\displaystyle{\alpha}_{4}^{p}(S_{1}\,S_{2})$ $\displaystyle=$ $\displaystyle
a_{4}^{p}(S_{1}\,S_{2})+\bar{\gamma}_{\chi}^{S_{2}}\,a_{6}^{p}(S_{1}\,S_{2}),$
(7) $\displaystyle{\alpha}_{3,EW}^{p}(S_{1}\,S_{2})$ $\displaystyle=$
$\displaystyle a_{9}^{p}(S_{1}\,S_{2})+a_{7}^{p}(S_{1}\,S_{2}),$ (8)
$\displaystyle{\alpha}_{4,EW}^{p}(S_{1}\,S_{2})$ $\displaystyle=$
$\displaystyle
a_{10}^{p}(S_{1}\,S_{2})+\bar{\gamma}_{\chi}^{S_{2}}\,a_{8}^{p}(S_{1}\,S_{2}),$
(9)

where $S_{1}$ denotes the recoiled scalar meson which absorbs the light
spectator quark of the initial $B$ mesons, and $S_{2}$ denotes the emitted
scalar meson. The ratio $\bar{\gamma}_{\chi}^{S}$ is defined as

$\bar{\gamma}_{\chi}^{S}({\mu})\,=\,{\gamma}_{\chi}^{S}({\mu})\,\bar{\mu}_{S}^{-1}({\mu})\,=\,\frac{2\,m_{S}}{\overline{m}_{b}({\mu})},$
(10)
${\gamma}_{\chi}^{S}({\mu})\,=\,\frac{2\,m_{S}^{2}}{\overline{m}_{b}({\mu})\,[\overline{m}_{1}({\mu})-\overline{m}_{2}({\mu})]},$
(11)
$\bar{\mu}_{S}({\mu})\,=\,\frac{m_{S}}{\overline{m}_{1}({\mu})-\overline{m}_{2}({\mu})},$
(12)

where $m_{S}$ is the mass of the emission scalar meson, and the
${\mu}$-dependent $\overline{m}_{i}$ is the $\overline{\rm MS}$ running quark
mass and can be evaluated with RGE. $\overline{m}_{1}$ and $\overline{m}_{2}$
correspond to the two valence quarks in a scalar meson.

Up to the next-to-leading order (NLO) in the coupling ${\alpha}_{s}$, the
general form of the QCDF coefficients $a_{i}^{p}$ is expressed as,

$\displaystyle a_{i}^{p}(S_{1}\,S_{2})$ $\displaystyle=$
$\displaystyle\Big{(}C_{i}+\frac{C_{i{\pm}1}}{N_{c}}\Big{)}\,N_{i}(S_{2})+P_{i}^{p}(S_{2})$
(13) $\displaystyle+$
$\displaystyle\frac{C_{i{\pm}1}}{N_{c}}\,\frac{C_{F}\,{\alpha}_{s}}{4{\pi}}\Big{[}V_{i}(S_{2})+\frac{4{\pi}^{2}}{N_{c}}\,H_{i}(S_{1}\,S_{2})\Big{]},$

where the superscript $p$ is to be omitted for $i$ $=$ $1$ and $2$, and the
upper (lower) signs apply when $i$ is odd (even). $C_{i}$ is the Wilson
coefficients, the color factor $C_{F}$ $=$ $(N_{c}^{2}-1)/(2\,N_{c})$ and the
color number $N_{c}$ $=$ $3$. Due to the relations between the scalar and
vector decay constants for the scalar meson (see ), the factor $N_{i}(S_{2})$
is

$N_{i}(S_{2})\,=\,\bigg{\\{}\begin{array}[]{lcl}1&{}{}\hfil&\text{for}~{}~{}i\,=\,6,8;\\\
\bar{\mu}_{S}^{-1}&&\text{others}.\end{array}$ (14)

In Eq.(13), the terms proportional to $N_{i}(S_{2})$ are the LO contributions.
It is obvious that except for the coefficients of $a_{6,8}$, the LO
contributions are proportional to the mass difference ${\Delta}\overline{m}$
$=$ $\overline{m}_{1}$ $-$ $\overline{m}_{2}$. For the scalar mesons
consisting of the light quarks, a common sense is that the mass difference
${\Delta}\overline{m}$ is usually very small. So, it is easy to picture that
the LO contributions are suppressed by the chiral factors, and that the NLO
contributions would be necessary and important for the $B$ ${\to}$ $SS$
decays. The terms proportional to ${\alpha}_{s}$ are the NLO contributions,
including the vertex corrections $V_{i}(S_{2})$, penguin contributions
$P_{i}^{p}(S_{2})$, and hard spectator scattering amplitudes
$H_{i}(S_{1}\,S_{2})$. When the emission $S_{2}$ meson can be decoupled from
the $B$-$S_{1}$ system, corresponding to the first line in Eq.(2),
$V_{i}(S_{2})$ and $P_{i}^{p}(S_{2})$ are written as the convolution integrals
of hard scattering kernels $T^{I}(y)$ and mesonic DAs ${\phi}_{S_{2}}(y)$.
When the initial $B$ meson is entangled with the final states by the hard
spectator scattering interactions, $H_{i}(S_{1}\,S_{2})$ are written as the
convolution integrals of hard scattering kernels $T^{II}$ and all
participating mesonic DAs, corresponding to the second line in Eq.(2). For the
$B$ ${\to}$ $SS$ decays, the explicit expressions of $V_{i}(S_{2})$,
$P_{i}^{p}(S_{2})$ and $H_{i}(S_{1}\,S_{2})$ have been shown in our previous
paper PhysRevD.105.016002  by using the replacements of the Gegenbauer moments
$a_{i}^{M_{j}}$ $\to$ $b_{i}^{S_{j}}$, the chiral factor
${\gamma}_{\chi}^{M_{i}}$ $\to$ $\bar{\gamma}_{\chi}^{S_{i}}$, and DAs
${\phi}_{M_{i}}$ $\to$ ${\phi}_{S_{i}}$. For example, by integrating out the
momentum fraction, $H_{i}(S_{1}\,S_{2})$ can be expressed as the functions of
the Gegenbauer moments embedded in the mesonic DAs.

$\\!\\!\\!\\!\\!\\!\\!\\!H_{i}(S_{1}\,S_{2})=\left\\{\begin{array}[]{lcl}0,&{}\hfil&\text{for~{}}i\,=\,6,8;\\\
\displaystyle-\frac{B_{S_{1}\,S_{2}}}{A_{S_{1}\,S_{2}}}\frac{m_{B}}{{\lambda}_{B}}\Big{[}9\sum\limits_{m=0}^{3}b_{m}^{S_{1}}\sum\limits_{j=0}^{3}(-1)^{j}b_{j}^{S_{2}}-3\,\bar{\gamma}_{\chi}^{S_{1}}X_{H}\sum\limits_{k=0}^{3}b_{k}^{S_{2}}\Big{]},&{}\hfil&\text{for~{}}i\,=\,5,7;\\\
\displaystyle~{}\frac{B_{S_{1}\,S_{2}}}{A_{S_{1}\,S_{2}}}\frac{m_{B}}{{\lambda}_{B}}\Big{[}9\sum\limits_{m=0}^{3}b_{m}^{S_{1}}\sum\limits_{j=0}^{3}b_{j}^{S_{2}}-3\,\bar{\gamma}_{\chi}^{S_{1}}X_{H}\sum\limits_{k=0}^{3}(-1)^{k}b_{k}^{S_{2}}\Big{]},&{}\hfil&\text{others}\end{array}\right.$
(15)

with the common factors are

$A_{S_{1}\,S_{2}}\,=\,i\,\frac{G_{F}}{\sqrt{2}}\,U_{0}^{B\,S_{1}}(m_{S_{2}}^{2})\,\bar{f}_{S_{2}}\,(m_{B}^{2}-m_{S_{1}}^{2}),$
(16)
$B_{S_{1}\,S_{2}}\,=\,i\,\frac{G_{F}}{\sqrt{2}}\,f_{B}\,\bar{f}_{S_{1}}\,\bar{f}_{S_{2}},$
(17)
$\frac{m_{B}}{{\lambda}_{B}}\,=\,{\int}_{0}^{1}\,dz\,\frac{{\phi}_{B}(z)}{z},$
(18) $X_{H}\,=\,{\int}_{0}^{1}\,\frac{dx}{1-x},$ (19)

where $U_{0}^{B\,S_{1}}$ is the form factors, $f_{B}$ is the decay constant
for the $B$ meson, $\bar{f}_{S_{i}}$ is the scalar decay constant for the
scalar mesons, the quantity ${\lambda}_{B}$ is used to parameterize our
ignorance about the $B$ mesonic DAs, and the phenomenological parameter
$X_{H}$ is introduced to regularize the end point singularities.

In addition, according to many practical application of the QCDF approach in
the two-body hadronic $B$ decays, such as Refs. PhysRevD.71.054020 ;
PhysRevD.73.014017 ; PhysRevD.77.014034 ; PhysRevD.79.094005 ;
PhysRevD.82.034014 ; PhysRevD.85.074010 ; AHEP.2013.175287 ;
PhysRevD.87.114001 ; PhysRevD.91.074022 ; CPC.40.013101 ; PhysRevD.99.076010 ;
PhysRevD.105.016002 ; NPB.591.313 ; NPB.606.245 ; NPB.675.333 ;
PhysRevD.65.074001 ; PhysRevD.65.094025 ; PhysRevD.67.014023 ; NPB.774.64 ;
PhysRevD.80.114008 ; PhysRevD.80.114026 ; PLB.702.408 ; PhysRevD.86.054016 ;
PhysRevD.88.014043 ; PhysRevD.90.054019 ; PLB.740.56 ; PLB.743.444 ;
PhysRevD.91.074026 ; JPG.43.105004 ; EPJC.77.415 , it was shown that the weak
annihilation (WA) contributions are important and worth of consideration,
although they are formally power suppressed relative to the LO contributions
based on the QCDF power counting rules in the heavy quark limits. The QCDF
coefficients of the WA amplitudes for the $B$ ${\to}$ $SS$ decays have the
same expression as those in Eq.(55) of Ref. NPB.675.333 , i.e.,

${\beta}_{i}^{p}\,=\,-\frac{B_{S_{1}\,S_{2}}}{A_{S_{1}\,S_{2}}}\,b_{i}^{p},$
(20) $\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle\frac{C_{F}}{N_{c}^{2}}\,C_{1}\,A_{1}^{i},\qquad\,b_{2}\,=\,\frac{C_{F}}{N_{c}^{2}}\,C_{2}\,A_{1}^{i},$
(21) $\displaystyle b_{3}^{p}$ $\displaystyle=$
$\displaystyle\frac{C_{F}}{N_{c}^{2}}\,\big{[}C_{3}\,A_{1}^{i}+C_{5}\,(A_{3}^{i}+A_{3}^{f})+N_{c}\,C_{6}\,A_{3}^{f}\big{]},$
(22) $\displaystyle b_{4}^{p}$ $\displaystyle=$
$\displaystyle\frac{C_{F}}{N_{c}^{2}}\,\big{[}C_{4}\,A_{1}^{i}+C_{6}\,A_{2}^{i}\big{]},$
(23) $\displaystyle b_{3,EW}^{p}$ $\displaystyle=$
$\displaystyle\frac{C_{F}}{N_{c}^{2}}\,\big{[}C_{9}\,A_{1}^{i}+C_{7}\,(A_{3}^{i}+A_{3}^{f})+N_{c}\,C_{8}\,A_{3}^{f}\big{]},$
(24) $\displaystyle b_{4,EW}^{p}$ $\displaystyle=$
$\displaystyle\frac{C_{F}}{N_{c}^{2}}\,\big{[}C_{10}\,A_{1}^{i}+C_{8}\,A_{2}^{i}\big{]},$
(25)

and the building blocks are respectively written as the functions of the
Gegenbauer moments.

$\displaystyle A_{1}^{i}$ $\displaystyle\approx$ $\displaystyle
2\,{\pi}\,{\alpha}_{s}\Big{\\{}9\,\Big{[}b_{0}^{S_{1}}\Big{(}b_{0}^{S_{2}}\,(X_{A}-4+\frac{{\pi}^{2}}{3})+b_{2}^{S_{2}}\,(6\,X_{A}-\frac{107}{3}+2\,{\pi}^{2})$
(26) $\displaystyle\qquad\qquad\quad\
+b_{1}^{S_{2}}\,(3\,X_{A}+4-{\pi}^{2})+b_{3}^{S_{2}}\,(10\,X_{A}+\frac{23}{18}-\frac{10}{3}\,{\pi}^{2})\Big{)}$
$\displaystyle-
b_{1}^{S_{1}}\,\Big{(}b_{0}^{S_{2}}\,(X_{A}+29-3\,{\pi}^{2})+b_{2}^{S_{2}}\,(6\,X_{A}+754-78\,{\pi}^{2})$
$\displaystyle\qquad+b_{1}^{S_{2}}\,(3\,X_{A}-213+21\,{\pi}^{2})+b_{3}^{S_{2}}\,(10\,X_{A}-\frac{12625}{6}+210\,{\pi}^{2})\Big{)}$
$\displaystyle+b_{2}^{S_{1}}\,\Big{(}b_{0}^{S_{2}}\,(X_{A}-119+12\,{\pi}^{2})+b_{2}^{S_{2}}\,(6\,X_{A}-9609+972\,{\pi}^{2})$
$\displaystyle\qquad+b_{1}^{S_{2}}\,(3\,X_{A}+1534-156\,{\pi}^{2})+b_{3}^{S_{2}}\,(10\,X_{A}+\frac{118933}{3}-4020\,{\pi}^{2})\Big{)}$
$\displaystyle-
b_{3}^{S_{1}}\,\Big{(}b_{0}^{S_{2}}\,(X_{A}+\frac{2956}{9}-\frac{100}{3}\,{\pi}^{2})+b_{2}^{S_{2}}\,(6\,X_{A}+\frac{198332}{3}-6700\,{\pi}^{2})$
$\displaystyle\qquad+b_{1}^{S_{2}}\,(3\,X_{A}-\frac{20743}{3}+700\,{\pi}^{2})+b_{3}^{S_{2}}\,(10\,X_{A}-\frac{3585910}{9}+\frac{121100}{3}\,{\pi}^{2})\Big{)}\Big{]}$
$\displaystyle-\bar{\gamma}_{\chi}^{S_{1}}\,\bar{\gamma}_{\chi}^{S_{2}}\,X_{A}^{2}\Big{\\}},$
$\displaystyle A_{2}^{i}$ $\displaystyle\approx$ $\displaystyle
2\,{\pi}\,{\alpha}_{s}\Big{\\{}9\,\Big{[}b_{0}^{S_{2}}\Big{(}b_{0}^{S_{1}}\,(X_{A}-4+\frac{{\pi}^{2}}{3})+b_{2}^{S_{1}}\,(6\,X_{A}-\frac{107}{3}+2\,{\pi}^{2})$
(27) $\displaystyle\qquad\qquad\quad\
-b_{1}^{S_{1}}\,(3\,X_{A}+4-{\pi}^{2})-b_{3}^{S_{1}}\,(10\,X_{A}+\frac{23}{18}-\frac{10}{3}\,{\pi}^{2})\Big{)}$
$\displaystyle+b_{1}^{S_{2}}\,\Big{(}b_{0}^{S_{1}}\,(X_{A}+29-3\,{\pi}^{2})+b_{2}^{S_{1}}\,(6\,X_{A}+754-78\,{\pi}^{2})$
$\displaystyle\qquad-
b_{1}^{S_{1}}\,(3\,X_{A}-213+21\,{\pi}^{2})-b_{3}^{S_{1}}\,(10\,X_{A}-\frac{12625}{6}+210\,{\pi}^{2})\Big{)}$
$\displaystyle+b_{2}^{S_{2}}\,\Big{(}b_{0}^{S_{1}}\,(X_{A}-119+12\,{\pi}^{2})+b_{2}^{S_{1}}\,(6\,X_{A}-9609+972\,{\pi}^{2})$
$\displaystyle\qquad-
b_{1}^{S_{1}}\,(3\,X_{A}+1534-156\,{\pi}^{2})-b_{3}^{S_{1}}\,(10\,X_{A}+\frac{118933}{3}-4020\,{\pi}^{2})\Big{)}$
$\displaystyle+b_{3}^{S_{2}}\,\Big{(}b_{0}^{S_{1}}\,(X_{A}+\frac{2956}{9}-\frac{100}{3}\,{\pi}^{2})+b_{2}^{S_{1}}\,(6\,X_{A}+\frac{198332}{3}-6700\,{\pi}^{2})$
$\displaystyle\qquad-
b_{1}^{S_{1}}\,(3\,X_{A}-\frac{20743}{3}+700\,{\pi}^{2})-b_{3}^{S_{1}}\,(10\,X_{A}-\frac{3585910}{9}+\frac{121100}{3}\,{\pi}^{2})\Big{)}\Big{]}$
$\displaystyle-\bar{\gamma}_{\chi}^{S_{1}}\,\bar{\gamma}_{\chi}^{S_{2}}\,X_{A}^{2}\Big{\\}},$
$\displaystyle A_{3}^{i}$ $\displaystyle\approx$
$\displaystyle-6\,{\pi}\,{\alpha}_{s}\Big{\\{}\bar{\gamma}_{\chi}^{S_{1}}\,\Big{[}b_{0}^{S_{2}}\,(X_{A}^{2}-2\,X_{A}+\frac{{\pi}^{2}}{3})+6\,b_{2}^{S_{2}}\,(X_{A}^{2}-\frac{16}{3}\,X_{A}+\frac{15}{2}+\frac{{\pi}^{2}}{3})$
(28) $\displaystyle\qquad\qquad\
+3\,b_{1}^{S_{2}}\,(X_{A}^{2}-4\,X_{A}+4+\frac{{\pi}^{2}}{3})+10\,b_{3}^{S_{2}}\,(X_{A}^{2}-\frac{13}{9}\,X_{A}+\frac{191}{18}+\frac{{\pi}^{2}}{3})\Big{]}$
$\displaystyle\qquad\quad+\bar{\gamma}_{\chi}^{S_{2}}\,\Big{[}b_{0}^{S_{1}}\,(X_{A}^{2}-2\,X_{A}+\frac{{\pi}^{2}}{3})+6\,b_{2}^{S_{1}}\,(X_{A}^{2}-\frac{16}{3}\,X_{A}+\frac{15}{2}+\frac{{\pi}^{2}}{3})$
$\displaystyle\qquad\qquad\
-3\,b_{1}^{S_{1}}\,(X_{A}^{2}-4\,X_{A}+4+\frac{{\pi}^{2}}{3})-10\,b_{3}^{S_{1}}\,(X_{A}^{2}-\frac{13}{9}\,X_{A}+\frac{191}{18}+\frac{{\pi}^{2}}{3})\Big{]}\Big{\\}},$
$A_{1}^{f}\,=\,A_{2}^{f}\,=\,0,$ (29) $\displaystyle A_{3}^{f}$
$\displaystyle\approx$
$\displaystyle-6\,{\pi}\,{\alpha}_{s}\,X_{A}\,\Big{\\{}\bar{\gamma}_{\chi}^{S_{1}}\,\Big{[}b_{0}^{S_{2}}\,(2\,X_{A}-1)+b_{2}^{S_{2}}\,(12\,X_{A}-31)$
(30)
$\displaystyle\qquad\qquad\qquad\quad+b_{1}^{S_{2}}\,(6\,X_{A}+11)+b_{3}^{S_{2}}\,(20\,X_{A}-\frac{187}{3})\Big{]}$
$\displaystyle\qquad\ \qquad\ \
-\bar{\gamma}_{\chi}^{S_{2}}\,\Big{[}b_{0}^{S_{1}}\,(2\,X_{A}-1)+b_{2}^{S_{1}}\,(12\,X_{A}-31)$
$\displaystyle\qquad\qquad\qquad\quad-
b_{1}^{S_{1}}\,(6\,X_{A}+11)-b_{3}^{S_{1}}\,(20\,X_{A}-\frac{187}{3})\Big{]}\Big{\\}},$

where $X_{A}$ has the similar definition and function as the the parameter
$X_{H}$ in Eq.(19) to regularize the end point divergence appearing in the
weak annihilation topologies. With the QCDF approach, $X_{H}$ and $X_{A}$ are
usually parameterized as

$X_{H}\,=\,{\ln}\Big{(}\frac{m_{B}}{{\Lambda}_{h}}\Big{)}\,(1+{\rho}_{H}\,e^{i\,{\phi}_{H}}),$
(31)
$X_{A}\,=\,{\ln}\Big{(}\frac{m_{B}}{{\Lambda}_{h}}\Big{)}\,(1+{\rho}_{A}\,e^{i\,{\phi}_{A}}),$
(32)

with ${\Lambda}_{h}$ $=$ $0.5$ GeV NPB.675.333 , and ${\rho}_{H,A}$ and
${\phi}_{H,A}$ are the undetermined parameters. Theoretically, $X_{H}$ and
$X_{A}$ are respectively related to the contributions from hard spectator
scattering and weak annihilations, and their physical implication and
significations are in nature different. What’s more, these parameters should
depend on the specific process and hadrons, because they actually originate
from the convolution integrals of hard scattering functions and hadronic DAs.
In the practical application of the QCDF approach, $X_{H}$ and $X_{A}$ are
usually and approximately regarded as the universal quantities to reduce the
number of phenomenological model parameters. Here, we will consider two
special cases. One case (C1) is to use the minimal parameters as possible, for
example, ${\rho}_{H}$ $=$ ${\rho}_{A}$ $=$ $1$ and ${\phi}_{H}$ $=$
${\phi}_{A}$ $=$ $-55^{\circ}$ NPB.675.333 . The other case (C2) is that the
factorizable and nonfactorizable WA contributions are treated independently,
and two quantities $X_{A}^{f}$ and $X_{A}^{i}$ are introduced to replace
$X_{A}$. A global fit on the $B$ ${\to}$ $PP$ decays with an approximation
$X_{H}$ ${\approx}$ $X_{A}^{i}$ gives $({\rho}_{A}^{i},\,{\phi}_{A}^{i})$ $=$
$(2.98,\,-105^{\circ})$ and $({\rho}_{A}^{f},\,{\phi}_{A}^{f})$ $=$
$(1.18,\,-40^{\circ})$ PLB.740.56 .

##  III input parameters

There are many input parameters in the numerical calculations. These
parameters generally fall into two categories. One has been well determined
experimentally or theoretically and listed explicitly in Ref. pdg2022 , such
as the Fermi coupling constant $G_{F}$, Wilson coefficients, the CKM elements,
and hadron mass as well. Their central values in Ref. pdg2022  will be
regarded as the default inputs unless otherwise specified. The other is the
nonperturbative parameters, such as the decay constants, mesonic transition
form factors, and hadronic DAs, which lead to produce the main theoretical
errors. The choice of these parameters requires certain caution.

###  III.1 The CKM elements

The Wolfenstein parameterization is traditionally and commonly used for the
unitary CKM matrix, due to the obvious power series in the Wolfenstein
parameter ${\lambda}$ among the CKM elements. The values of the four
Wolfenstein parameters are pdg2022 ,

$\displaystyle
A=0.790^{+0.017}_{-0.012},\quad{\lambda}=0.22650{\pm}0.00048,\quad\bar{\rho}=0.141^{+0.016}_{-0.017},\quad\bar{\eta}=0.357{\pm}0.011.$
(33)

###  III.2 The decay constants

The lattice QCD results of the isospin averages of the $B$ meson decay
constants are pdg2022 ,

$\displaystyle f_{B_{u,d}}$ $\displaystyle=$ $\displaystyle
190.0{\pm}1.3\,\text{MeV},$ (34) $\displaystyle f_{B_{s}}$ $\displaystyle=$
$\displaystyle 230.3{\pm}1.3\,\text{MeV}.$ (35)

There are two kinds of definitions of the decay constants for the scalar
mesons, i.e.,

${\langle}S(p)\,{|}\bar{q}_{1}\,{\gamma}^{\mu}\,q_{2}\,{|}\,0{\rangle}\,=\,f_{S}\,p^{\mu},$
(36)
${\langle}S(p)\,{|}\bar{q}_{1}\,q_{2}\,{|}\,0{\rangle}\,=\,m_{S}\,\bar{f}_{S}({\mu}).$
(37)

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

$f_{S}\,=\,\bar{f}_{S}({\mu})\,\bar{\mu}_{S}^{-1}({\mu}).$ (38)

Clearly, the vector decay constant $f_{S}$ is proportional to the running mass
difference, ${\Delta}\overline{m}$, between the two valence quarks resided in
the scalar mesons. $f_{S}$ for the light scalars should be seriously
suppressed by the small ${\Delta}\overline{m}$, especially for the
electrically neutral scalar mesons owing to charge conjugation invariance or
conservation of vector current. For example, $f_{S}$ will vanish for the
$a_{0}^{0}$ meson. At the same time the scalar decay constants $\bar{f}_{S}$
remain finite. Here a preferable solution scheme is to use the scalar decay
constants $\bar{f}_{S}$. This is one of the main reasons for the factors in
Eq.(14). In addition, the scalar mesons and its antiparticles have the same
scalar decay constants, $\bar{f}_{S}$ $=$ $\bar{f}_{\bar{S}}$. It means that
the vector decay constants $f_{S}$ $=$ $-f_{\bar{S}}$ from Eq.(12) and
Eq.(38), which results in the $f_{S}$ $=$ $0$ for the $a_{0}^{0}$ meson.

Table 1: The scalar decay constants $\bar{f}_{S}$ (in units of MeV) at the scale of ${\mu}$ $=$ $1$ GeV. Here the theoretical errors come mainly from the Gaussian parameter ${\beta}$ responsible for mesonic wave functions. | this work | Refs. PhysRevD.73.014017  | Ref. PLB.619.105  | Ref. EPJA.49.78
---|---|---|---|---
scenarios | S1 | S2 | S1 | S2 | S2 | S2
$\bar{f}_{K_{0}^{\ast}(1430)}$ | $234^{+85}_{-87}$ | $542^{+180}_{-190}$ | $-300{\pm}30$ | $445{\pm}50$ | $427{\pm}85$ | $358{\pm}1$
$\bar{f}_{a_{0}(1450)}$ | $256^{+56}_{-54}$ | $456^{+57}_{-56}$ | $-280{\pm}30$ | $460{\pm}50$ |  | $375{\pm}2$

Experimentally, these decay constants can be extracted from the purely
leptonic decays of the scalar mesons. It is widely known that the scalar
mesons usually appear as resonances and decay dominantly through the strong
interactions, and the occurrence probability of the leptonic decays of the
scalar mesons should in principle be very small. The leptonic decays of the
scalar mesons have not been discovered by now. The experimental data on the
decay constants of the scalar mesons is still unavailable. The theoretical
values of the scalar decay constants $\bar{f}_{S}$ corresponding to the S1 and
S2 scenarios are listed in Table 1. It is clearly seen that for the S2
scenario, the central values of the decay constants $\bar{f}_{S}$ obtained
with the covariant light-front quark model (CLFQM) in this paper are generally
in agreement with those from the QCD sum rules PhysRevD.73.014017 ;
PLB.619.105  and light-cone sum rules EPJA.49.78  within an error range. Of
course, the errors arising from the Gaussian parameter ${\beta}$ responsible
for mesonic wave functions are still very large due to the inadequate data and
our insufficient understanding on the scalar mesons for the moment, especially
for $\bar{f}_{K_{0}^{\ast}(1430)}$ with the S2 scenario. What’s more, the
values of $\bar{f}_{S}$ with the S2 scenario are about twice as larger as
those with the S1 scenario, which will inevitably bring the obviously
hierarchical relations with the branching ratios with these two different
scenarios, because the decay amplitudes are directly proportional to the decay
constants. A significant difference between branching ratios might be used to
distinguish whether these scalar mesons are the $1P$ or $2P$ states.

###  III.3 Hadronic transition form factors

The form factors of $B$ ${\to}$ $S$ transitions are defined as
PhysRevD.71.054020 ; PhysRevD.73.014017 ; PhysRevD.77.014034 ,

${\langle}S(k)\,{|}\bar{q}\,{\gamma}_{\mu}\,{\gamma}_{5}\,b\,{|}B(p){\rangle}\,=\,-i\,\Big{[}\Big{(}P_{\mu}-\frac{m_{B}^{2}-m_{S}^{2}}{q^{2}}\,q_{\mu}\Big{)}\,U_{1}(q^{2})+\frac{m_{B}^{2}-m_{S}^{2}}{q^{2}}\,q_{\mu}\,U_{0}(q^{2})\Big{]},$
(39)

where $P_{\mu}$ $=$ $p_{\mu}$ $+$ $k_{\mu}$ and $q_{\mu}$ $=$ $p_{\mu}$ $-$
$k_{\mu}$. $U_{0}(q^{2})$ and $U_{1}(q^{2})$ respectively denote longitudinal
and transverse form factors. To regulate the singularities at the pole $q^{2}$
$=$ $0$, the relations $U_{0}(0)$ $=$ $U_{1}(0)$ is required. The values of
$U_{0,1}(0)$ can be obtained by fit the dependence of the form factors on
$q^{2}$ with the 3-parameter formula PhysRevD.73.014017 ; PhysRevD.77.014034 ,

$U_{i}(q^{2})\,=\,\frac{U_{i}(0)}{1-a\,(q^{2}/m_{B}^{2})+b\,(q^{2}/m_{B}^{2})^{2}}.$
(40)

Table 2: Form factors for the $B$ ${\to}$ $S$ transitions obtained from CLFQM, considering the S1 and S2 scenarios for the scalar mesons. | transition | $U_{1}(0)$ | $a$ | $b$ | $U_{0}(0)$ | $a$ | $b$
---|---|---|---|---|---|---|---
S1 |  this work |  $B$ ${\to}$ $K_{0}^{\ast}(1430)$ | $0.18{\pm}0.01$ | $1.03$ | $0.15$ | $0.18{\pm}0.01$ | $-0.23$ | $0.29$
$B$ ${\to}$ $a_{0}(1450)$ | $0.19{\pm}0.01$ | $1.01$ | $0.16$ | $0.19{\pm}0.01$ | $-0.17$ | $0.30$
$B_{s}$ ${\to}$ $K_{0}^{\ast}(1430)$ | $0.23{\pm}0.02$ | $0.92$ | $0.29$ | $0.23{\pm}0.02$ | $-0.23$ | $0.36$
Ref. PhysRevD.73.014017  |  $B$ ${\to}$ $K_{0}^{\ast}(1430)$ | $0.21$ | $1.59$ | $0.91$ | $0.21$ | $0.59$ | $0.09$
$B$ ${\to}$ $a_{0}(1450)$ | $0.21$ | $1.66$ | $1.00$ | $0.21$ | $0.73$ | $0.09$
S2 |  this work |  $B$ ${\to}$ $K_{0}^{\ast}(1430)$ | $0.29{\pm}0.02$ | $1.27$ | $0.33$ | $0.29{\pm}0.02$ | $0.16$ | $0.11$
$B$ ${\to}$ $a_{0}(1450)$ | $0.29{\pm}0.02$ | $1.33$ | $0.38$ | $0.29{\pm}0.02$ | $0.32$ | $0.06$
$B_{s}$ ${\to}$ $K_{0}^{\ast}(1430)$ | $0.28{\pm}0.02$ | $1.58$ | $0.84$ | $0.28{\pm}0.02$ | $0.55$ | $0.20$
Ref. PhysRevD.73.014017  |  $B$ ${\to}$ $K_{0}^{\ast}(1430)$ | $0.26$ | $1.52$ | $0.64$ | $0.26$ | $0.44$ | $0.05$
$B$ ${\to}$ $a_{0}(1450)$ | $0.26$ | $1.57$ | $0.70$ | $0.26$ | $0.55$ | $0.03$

The form factors obtained from CLFQM are listed in Table 2. It is clearly seen
that (1) the central values of $U_{0,1}(0)$ in this work are very close to
those of Ref. PhysRevD.73.014017 . They are slightly larger (smaller) than
those given in Ref. PhysRevD.73.014017  for the S2 (S1) scenario. The
differences come mainly from the quark running mass and the Gaussian parameter
${\beta}$ as well. (2) For the S2 scenario, the $SU(3)$ flavor symmetry among
the central values of $U_{0,1}(0)$ seems to be held well. (3) For the $B$
${\to}$ $K_{0}^{\ast}(1430)$, $a_{0}(1450)$ transition form factors, the
differences between the S1 and S2 scenarios given in Ref. PhysRevD.73.014017
are less obvious than those obtained in this work. Here the ratio of
$U_{0,1}^{B{\to}K_{0}^{\ast},a_{0}}(0)$ between the S1 and S2 scenarios is
approximately 2/3, which will result in the ratio of branching ratio
proportional to the square of the form factors is approximately 1/2. The
bigger difference of the ratio, the easier the measurement becomes, and the
more helpful it is to distinguish whether these scalar mesons are the $1P$ or
$2P$ states from the semileptonic $B$ ${\to}$ $S{\ell}{\nu}$ decays in the
future experiments, and to check the different theoretical predictions.

###  III.4 Mesonic light cone DAs

The definition of mesonic light cone DAs is PhysRevD.71.054020 ;
PhysRevD.73.014017 ,

$\displaystyle{\langle}S(p)\,{|}\,q_{2{\beta}}(z_{2})\,q_{1{\alpha}}(z_{1})\,{|}0{\rangle}$
(41) $\displaystyle=$
$\displaystyle\frac{1}{4}\,\bar{f}_{S}\,{\int}_{0}^{1}\,dx\,e^{i\,(xp{\cdot}z_{2}+\bar{x}{\cdot}z_{1})}\,\Big{\\{}\not{p}\,{\Phi}_{S}(x)+m_{S}\,\Big{[}{\Phi}_{S}^{s}(x)-{\sigma}_{{\mu}{\nu}}\,p^{\mu}\,z^{\nu}\frac{{\Phi}_{S}^{\sigma}(x)}{6}\Big{]}\Big{\\}}_{{\alpha}{\beta}},$

where the arguments $\bar{x}$ $=$ $1$ $-$ $x$ and $z$ $=$ $z_{2}$ $-$ $z_{1}$.
${\Phi}_{S}$ is the twist-2 light cone DAs. The twist-3 light cone DAs
${\Phi}_{S}^{s,{\sigma}}$ are related by the equations of motion
PhysRevD.73.014017 ,

${\xi}\,{\Phi}_{S}^{s}(x)+\frac{1}{6}\,\frac{d\,{\Phi}_{S}^{\sigma}(x)}{d\,x}\,=\,0,$
(42)

where ${\xi}$ $=$ $x$ $-$ $\bar{x}$ $=$ $2\,x$ $-$ $1$. The twist-2 DAs are
written as PhysRevD.71.054020 ; PhysRevD.73.014017

${\Phi}_{S}(x,\,{\mu})\,=\,6\,x\,\bar{x}\,\Big{\\{}b_{0}^{S}+\sum\limits_{n=1}^{\infty}b_{n}^{S}({\mu})\,C_{n}^{3/2}({\xi})\Big{\\}},$
(43)

where the Gegenbauer moments $b_{i}^{S}$, corresponding to the expansion
coefficients of Gegenbauer polynomials $C_{i}^{3/2}({\xi})$, are hadronic
parameters. The asymptotic forms of the twist-3 DAs are respectively written
as PhysRevD.73.014017 ,

${\Phi}_{S}^{s}(x,\,{\mu})\,=\,1,$ (44)
${\Phi}_{S}^{\sigma}(x,\,{\mu})\,=\,6\,x\,\bar{x}.$ (45)

Table 3: The values of the Gegenbauer moments at the scale of ${\mu}$ $=$ $1$ GeV, considering the S1 and S2 scenarios for the scalar mesons. The results in this work are obtained from CLFQM, and those of Ref. PhysRevD.77.014034  from QCD sum rules. | mesons | $b_{0}^{S}$ | $b_{1}^{S}$ | $b_{2}^{S}$ | $b_{3}^{S}$
---|---|---|---|---|---
S1 | this | $K_{0}^{\ast}(1430)$ | $0.08{\pm}0.01$ | $-0.15{\pm}0.05$ | $0.06{\pm}0.01$ | $-0.09{\pm}0.05$
work | $a_{0}(1450)$ | $0$ | $-0.17{\pm}0.06$ | $0$ | $-0.19{\pm}0.03$
Ref. PhysRevD.77.014034  | $K_{0}^{\ast}(1430)$ | $0$ | $0.58{\pm}0.07$ | $0$ | $-1.20{\pm}0.08$
$a_{0}(1450)$ | $0$ | $0.89{\pm}0.20$ | $0$ | $-1.38{\pm}0.18$
S2 | this | $K_{0}^{\ast}(1430)$ | $0.08{\pm}0.01$ | $-0.13{\pm}0.05$ | $-0.03{\pm}0.00$ | $-0.01{\pm}0.00$
work | $a_{0}(1450)$ | $0$ | $-0.17{\pm}0.03$ | $0$ | $-0.03{\pm}0.01$
Ref. PhysRevD.77.014034  | $K_{0}^{\ast}(1430)$ | $0$ | $-0.57{\pm}0.13$ | $0$ | $-0.42{\pm}0.22$
$a_{0}(1450)$ | $0$ | $-0.58{\pm}0.12$ | $0$ | $-0.49{\pm}0.15$

Figure 1: The twist-2 DAs for the $a_{0}(1450)$ and $K_{0}^{\ast}(1430)$. The
dashed and solid lines correspond to the truncations up to $n$ $=$ $1$ and
$3$, respectively.

The Gegenbauer moments $b_{n}^{S}$ in the twist-2 DAs ${\Phi}_{S}$ are listed
in Table 3. Our comments are (1) for either the S1 or S2 scenarios, the
orbital angular momentum $L$ $=$ $1$ between the two components of the scalar
mesons. Using the parity of Gegenbauer polynomials and the isospin symmetry,
the wave function should in principle be antisymmetric, $(-1)^{L}$, under the
exchange of the longitudinal momentum fractions of the two valence quarks $x$
${\leftrightarrow}$ $\bar{x}$, i.e., the Gegenbauer moments $b_{n}^{S}$ with
even $n$ should be zero. This feature is clearly demonstrated for the
$a_{0}(1450)$ mesons in Table 3 and Fig. 1 (a). (2) For the
$K_{0}^{\ast}(1430)$ mesons, the flavor $SU(3)$ symmetry breaking effects
should be given due consideration. DAs for the $K_{0}^{\ast}(1430)$ mesons
should be asymmetric under $x$ ${\leftrightarrow}$ $\bar{x}$, i.e., the
Gegenbauer moments $b_{n}^{S}$ with both even and odd $n$ are nonzero. This
property is properly illustrated by our results in Table 3 and Fig. 1 (b). (3)
According to the definition of hadronic matrix elements given by Eq.(3.18) and
Eq.(3.20) in Ref. PhysRevD.73.014017 , the Gegenbauer moments $b_{1}^{S}$ in
DAs and the decay constant $\bar{f}_{S}$ are directly interrelated. The
positive values of $b_{1}^{S}$ correspond to the the negative value of
$\bar{f}_{S}$ listed in Table 1 for the S1 scenario, and vice versa for the S2
scenario. In this sense, the positive and negative sign for $b_{1}^{S}$ from
CLFQM in this work and QCD sum rules in Ref. PhysRevD.77.014034  are self-
consistent.

##  IV Numerical results and discussions

In the rest frame of the $B$ meson, the $CP$-averaged branching ratio is
defined as,

${\cal B}\,=\,\frac{{\tau}_{B}}{16{\pi}}\,\frac{p_{\rm
cm}}{m_{B}^{2}}\,\big{\\{}{|}{\cal A}(B{\to}f){|}^{2}+{|}{\cal
A}(\overline{B}{\to}\overline{f}){|}^{2}\big{\\}},$ (46)

where ${\tau}_{B}$ is the $B$ meson lifetime, $p_{\rm cm}$ is the common
momentum of final states.

The direct $CP$ asymmetry is defined as,

$A_{CP}\,=\,\frac{{\Gamma}(\overline{B}{\to}f)-{\Gamma}(B{\to}\overline{f})}{{\Gamma}(\overline{B}{\to}f)+{\Gamma}(B{\to}\overline{f})}.$
(47)

When the final states are common to the neutral $B_{d,s}^{0}$ and
$\overline{B}_{d,s}^{0}$ decays, the $CP$ violating asymmetry is defined as,

$A_{CP}\,=\,A_{CP}^{\rm mix}\,{\sin}(x\,{\Delta}m\,t)-A_{CP}^{\rm
dir}\,{\cos}(x\,{\Delta}m\,t),$ (48) $A_{CP}^{\rm mix}\,=\,\frac{2\,{\cal
I}m({\lambda}_{f})}{1+{|}{\lambda}_{f}{|}^{2}},$ (49) $A_{CP}^{\rm
dir}\,=\,\frac{1-{|}{\lambda}_{f}{|}^{2}}{1+{|}{\lambda}_{f}{|}^{2}},$ (50)
${\lambda}_{f}\,=\,\left\\{\begin{array}[]{ll}\displaystyle\frac{V_{tb}^{\ast}\,V_{td}}{V_{tb}\,V_{td}^{\ast}}\,\frac{{\cal
A}(\overline{B}_{d}^{0}{\to}f)}{{\cal A}(B_{d}^{0}{\to}f)},&\text{~{}~{}for
the $B_{d}^{0}$-$\overline{B}_{d}^{0}$ system},\\\ {}\\\
\displaystyle\frac{V_{tb}^{\ast}\,V_{ts}}{V_{tb}\,V_{ts}^{\ast}}\,\frac{{\cal
A}(\overline{B}_{s}^{0}{\to}f)}{{\cal A}(B_{s}^{0}{\to}f)},&\text{~{}~{}for
the $B_{s}^{0}$-$\overline{B}_{s}^{0}$ system}.\end{array}\right.$ (51)

Table 4: The $CP$-averaged branching ratios (in units of ${10}^{-6}$) for the $B$ ${\to}$ $SS$ decays, where $a_{0}$ ${\equiv}$ $a_{0}(1450)$ and $K_{0}^{\ast}$ ${\equiv}$ $K_{0}^{\ast}(1430)$. The numbers in the “NF” columns corresponds to the LO results. The numbers in the “C1” and “C2” columns are results including the NLO contributions. “C1” corresponds to the case $({\rho}_{A},\,{\phi}_{A})$ $=$ $(1,\,-55^{\circ})$ and $X_{H}$ $=$ $X_{A}$, and “C2” corresponds to $({\rho}_{A}^{i},\,{\phi}_{A}^{i})$ $=$ $(2.98,\,-105^{\circ})$, $({\rho}_{A}^{f},\,{\phi}_{A}^{f})$ $=$ $(1.18,\,-40^{\circ})$ and $X_{H}$ ${\approx}$ $X_{A}^{i}$. The first uncertainties come mainly from the CKM elements. The second uncertainties come from the hadronic parameters including the decay constant, form factors, and the Gegenbauer moments in DAs. decay | class | S1 | S2
---|---|---|---
modes | NF | C1 | C2 | NF | C1 | C2
$B^{-}$ ${\to}$ $a_{0}^{-}a_{0}^{0}$ | T | $0.00^{+0.00+0.00}_{-0.00-0.00}$ | $0.01^{+0.00+0.01}_{-0.00-0.01}$ | $0.03^{+0.00+0.02}_{-0.00-0.01}$ | $0.00^{+0.00+0.00}_{-0.00-0.00}$ | $0.04^{+0.00+0.03}_{-0.00-0.02}$ | $0.10^{+0.01+0.06}_{-0.01-0.05}$
$\bar{B}^{0}$ ${\to}$ $a_{0}^{+}a_{0}^{-}$ | T | $0.04^{+0.00+0.00}_{-0.00-0.00}$ | $0.19^{+0.01+0.12}_{-0.01-0.09}$ | $0.37^{+0.03+0.12}_{-0.02-0.11}$ | $0.31^{+0.02+0.09}_{-0.01-0.08}$ | $0.55^{+0.03+0.17}_{-0.02-0.15}$ | $0.63^{+0.05+0.25}_{-0.04-0.21}$
$\bar{B}_{s}^{0}$ ${\to}$ $K_{0}^{{\ast}+}a_{0}^{-}$ | T | $0.07^{+0.00+0.01}_{-0.00-0.01}$ | $0.14^{+0.01+0.09}_{-0.01-0.07}$ | $0.14^{+0.01+0.04}_{-0.01-0.04}$ | $0.31^{+0.02+0.09}_{-0.01-0.08}$ | $0.51^{+0.03+0.23}_{-0.02-0.19}$ | $0.49^{+0.03+0.30}_{-0.03-0.22}$
$\overline{B}^{0}$ ${\to}$ $a_{0}^{0}a_{0}^{0}$ | C | $0.04^{+0.00+0.00}_{-0.00-0.00}$ | $0.27^{+0.02+0.27}_{-0.01-0.16}$ | $0.63^{+0.05+0.31}_{-0.04-0.25}$ | $0.32^{+0.02+0.10}_{-0.01-0.08}$ | $0.83^{+0.04+0.38}_{-0.04-0.30}$ | $1.17^{+0.09+0.73}_{-0.08-0.53}$
$\overline{B}_{s}^{0}$ ${\to}$ $K_{0}^{{\ast}0}a_{0}^{0}$ | C | $0.03^{+0.00+0.01}_{-0.00-0.01}$ | $0.09^{+0.01+0.06}_{-0.01-0.05}$ | $0.12^{+0.01+0.04}_{-0.01-0.03}$ | $0.16^{+0.01+0.05}_{-0.01-0.04}$ | $0.36^{+0.02+0.18}_{-0.02-0.15}$ | $0.51^{+0.04+0.33}_{-0.03-0.26}$
$B^{-}$ ${\to}$ $a_{0}^{0}K_{0}^{{\ast}-}$ | P | $0.44^{+0.02+0.05}_{-0.01-0.05}$ | $0.98^{+0.05+0.91}_{-0.04-0.66}$ | $0.82^{+0.04+0.26}_{-0.03-0.22}$ | $5.50^{+0.24+4.33}_{-0.17-3.26}$ | $6.65^{+0.30+5.26}_{-0.24-3.97}$ | $5.07^{+0.23+4.03}_{-0.16-3.03}$
$B^{-}$ ${\to}$ $a_{0}^{-}\overline{K}_{0}^{{\ast}0}$ | P | $0.93^{+0.04+0.10}_{-0.03-0.10}$ | $2.05^{+0.09+1.88}_{-0.07-1.36}$ | $1.64^{+0.07+0.49}_{-0.05-0.42}$ | $11.69^{+0.53+9.20}_{-0.40+6.93}$ | $14.13^{+0.64+11.16}_{-0.50+~{}8.42}$ | $10.58^{+0.47+8.37}_{-0.33-6.29}$
$\overline{B}^{0}$ ${\to}$ $a_{0}^{+}K_{0}^{{\ast}0-}$ | P | $0.82^{+0.04+0.09}_{-0.03-0.09}$ | $1.87^{+0.09+1.73}_{-0.07-1.26}$ | $1.70^{+0.08+0.56}_{-0.05-0.47}$ | $10.20^{+0.45+8.04}_{-0.32-6.05}$ | $12.40^{+0.56+9.81}_{-0.44-7.40}$ | $10.42^{+0.46+8.29}_{-0.33-6.23}$
$\overline{B}^{0}$ ${\to}$ $a_{0}^{0}\overline{K}_{0}^{{\ast}0}$ | P | $0.43^{+0.02+0.05}_{-0.01-0.05}$ | $0.97^{+0.04+0.75}_{-0.03-0.50}$ | $0.86^{+0.04+0.27}_{-0.03-0.23}$ | $5.42^{+0.24+4.27}_{-0.17-3.22}$ | $6.56^{+0.30+5.18}_{-0.23-3.91}$ | $5.44^{+0.24+4.30}_{-0.17-3.23}$
$\overline{B}_{s}^{0}$ ${\to}$ $K_{0}^{{\ast}0}\overline{K}_{0}^{{\ast}0}$ | P | $1.37^{+0.06+0.25}_{-0.04-0.23}$ | $1.88^{+0.09+1.70}_{-0.07-1.23}$ | $0.54^{+0.02+0.15}_{-0.02-0.12}$ | $10.91^{+0.48+8.60}_{-0.34-6.48}$ | $14.99^{+0.68+12.14}_{-0.54-~{}9.18}$ | $5.94^{+0.26+5.81}_{-0.19-3.77}$
$\overline{B}_{s}^{0}$ ${\to}$ $K_{0}^{{\ast}+}K_{0}^{{\ast}-}$ | P | $1.29^{+0.06+0.24}_{-0.04-0.22}$ | $1.73^{+0.08+1.57}_{-0.06-1.14}$ | $0.61^{+0.03+0.17}_{-0.02-0.13}$ | $10.27^{+0.45+8.10}_{-0.32-6.10}$ | $13.98^{+0.64+11.39}_{-0.50-~{}8.61}$ | $6.65^{+0.30+6.65}_{-0.21-4.39}$
$B^{-}$ ${\to}$ $K_{0}^{{\ast}0}K_{0}^{{\ast}-}$ | P | $0.03^{+0.00+0.00}_{-0.00-0.00}$ | $0.07^{+0.00+0.07}_{-0.00-0.05}$ | $0.07^{+0.00+0.03}_{-0.00-0.03}$ | $0.38^{+0.02+0.30}_{-0.02-0.23}$ | $0.30^{+0.02+0.24}_{-0.01-0.18}$ | $0.25^{+0.01+0.22}_{-0.01-0.16}$
$\overline{B}^{0}$ ${\to}$ $K_{0}^{{\ast}0}\overline{K}_{0}^{{\ast}0}$ | P | $0.03^{+0.00+0.00}_{-0.00-0.00}$ | $0.06^{+0.00+0.06}_{-0.00-0.04}$ | $0.05^{+0.00+0.02}_{-0.00-0.01}$ | $0.36^{+0.02+0.28}_{-0.02-0.21}$ | $0.29^{+0.02+0.23}_{-0.01-0.17}$ | $0.11^{+0.01+0.12}_{-0.01-0.07}$
$\overline{B}^{0}$ ${\to}$ $K_{0}^{{\ast}+}K_{0}^{{\ast}-}$ | A |  | $0.02^{+0.00+0.02}_{-0.00-0.01}$ | $0.11^{+0.01+0.14}_{-0.01-0.11}$ |  | $0.10^{+0.01+0.11}_{-0.01-0.09}$ | $0.56^{+0.04+0.62}_{-0.04-0.46}$
$\overline{B}_{s}^{0}$ ${\to}$ $a_{0}^{0}a_{0}^{0}$ | A |  | $0.05^{+0.00+0.06}_{+0.00-0.04}$ | $0.38^{+0.02+0.51}_{-0.01-0.29}$ |  | $0.12^{+0.01+0.07}_{-0.01-0.05}$ | $0.66^{+0.03+0.42}_{-0.03-0.29}$
$\overline{B}_{s}^{0}$ ${\to}$ $a_{0}^{+}a_{0}^{-}$ | A |  | $0.10^{+0.00+0.07}_{-0.00-0.06}$ | $0.77^{+0.04+0.60}_{-0.03-0.48}$ |  | $0.23^{+0.01+0.09}_{-0.01-0.08}$ | $1.32^{+0.06+0.53}_{-0.05-0.46}$

Table 5: $CP$ asymmetries (in units of $\%$) for the $B$ ${\to}$ $SS$ decays. Other legends are the same as those of Table 4. decay |  |  | S1 | S2
---|---|---|---|---
mode | class |  | C1 | C2 | C1 | C2
$B^{-}$ ${\to}$ $a_{0}^{-}a_{0}^{0}$ | T | $A_{CP}$ | $-2.93^{+0.11+0.65}_{-0.11-0.65}$ | $-2.37^{+0.09+0.53}_{-0.09-0.53}$ | $-4.15^{+0.16+0.22}_{-0.17-0.23}$ | $-3.89^{+0.15+0.21}_{-0.15-0.21}$
$B^{-}$ ${\to}$ $a_{0}^{0}K_{0}^{{\ast}-}$ | P | $A_{CP}$ | $0.99^{+0.04+1.50}_{-0.04-1.78}$ | $4.53^{+0.15+1.13}_{-0.14-1.43}$ | $2.24^{+0.08+0.95}_{-0.07-1.30}$ | $5.19^{+0.17+1.54}_{-0.17-2.04}$
$B^{-}$ ${\to}$ $a_{0}^{-}\overline{K}_{0}^{{\ast}0}$ | P | $A_{CP}$ | $0.18^{+0.01+0.21}_{-0.01-0.18}$ | $1.16^{+0.03+0.15}_{-0.03-0.10}$ | $0.40^{+0.01+0.14}_{-0.01-0.14}$ | $1.68^{+0.05+0.13}_{-0.05-0.13}$
$B^{-}$ ${\to}$ $K_{0}^{{\ast}0}K_{0}^{{\ast}-}$ | P | $A_{CP}$ | $3.40^{+0.13+4.10}_{-0.13-7.52}$ | $-27.35^{+0.94+1.36}_{-0.92-1.66}$ | $-1.94^{+0.08+7.99}_{-0.09-9.52}$ | $-48.17^{+1.49+12.96}_{-1.44-~{}9.18}$
$\overline{B}^{0}$ ${\to}$ $a_{0}^{+}K_{0}^{{\ast}0-}$ | P | $A_{CP}$ | $6.95^{+0.22+0.39}_{-0.21-0.49}$ | $8.24^{+0.26+0.96}_{-0.26-1.01}$ | $7.72^{+0.24+0.43}_{-0.24-0.57}$ | $10.24^{+0.32+0.24}_{-0.32-0.60}$
$\overline{B}^{0}$ ${\to}$ $a_{0}^{0}\overline{K}_{0}^{{\ast}0}$ | P | $A_{CP}$ | $5.88^{+0.18+1.63}_{-0.18-1.40}$ | $5.18^{+0.16+1.97}_{-0.16-1.64}$ | $5.48^{+0.17+1.25}_{-0.17-1.01}$ | $6.85^{+0.21+1.34}_{-0.21-1.15}$
$\overline{B}_{s}^{0}$ ${\to}$ $K_{0}^{{\ast}+}a_{0}^{-}$ | T | $A_{CP}$ | $-34.05^{+1.29+5.08}_{-1.26-4.90}$ | $-25.28^{+1.09+7.00}_{-1.07-7.80}$ | $-39.24^{+1.46+3.95}_{-1.42-3.08}$ | $-30.51^{+1.31+13.36}_{-1.28-15.40}$
$\overline{B}_{s}^{0}$ ${\to}$ $K_{0}^{{\ast}0}a_{0}^{0}$ | C | $A_{CP}$ | $-85.18^{+1.95+9.75}_{-1.77-8.20}$ | $-40.58^{+1.54+13.35}_{-1.42-13.47}$ | $-92.94^{+1.53+5.48}_{-0.38-3.50}$ | $-45.64^{+1.58+24.10}_{-1.46-31.25}$
$\overline{B}^{0}$ ${\to}$ $a_{0}^{+}a_{0}^{-}$ | T | $A_{CP}^{\rm dir}$ | $-81.59^{+2.24+3.10}_{-2.06-2.19}$ | $51.26^{+1.40+4.34}_{-1.42-4.21}$ | $-89.16^{+1.68+3.42}_{-1.48-2.67}$ | $52.09^{+1.46+12.03}_{-1.54-14.57}$
$A_{CP}^{\rm mix}$ | $30.94^{+1.48+8.21}_{-1.53-9.93}$ | $70.48^{+4.49+5.78}_{-4.66-5.72}$ | $-0.35^{+2.16+6.98}_{-2.12-6.94}$ | $76.95^{+3.50+11.07}_{-3.75-11.19}$
$\overline{B}^{0}$ ${\to}$ $a_{0}^{0}a_{0}^{0}$ | C | $A_{CP}^{\rm dir}$ | $-95.63^{+1.35+2.54}_{-1.07-2.26}$ | $31.71^{+0.96+3.72}_{-1.19-4.18}$ | $-99.77^{+0.39+0.90}_{-0.18-0.20}$ | $16.29^{+0.56+12.99}_{-0.60-15.04}$
$A_{CP}^{\rm mix}$ | $27.95^{+3.23+~{}9.76}_{-3.25-11.83}$ | $62.74^{+6.04+9.28}_{-4.61-8.66}$ | $-6.36^{+4.07+8.70}_{-3.90-8.99}$ | $72.93^{+5.38+6.29}_{-5.50-6.49}$
$\overline{B}^{0}$ ${\to}$ $K_{0}^{{\ast}0}\overline{K}_{0}^{{\ast}0}$ | P | $A_{CP}^{\rm dir}$ | $-12.69^{+0.51+3.65}_{-0.50-5.04}$ | $-5.29^{+0.22+2.03}_{-0.21-2.54}$ | $-22.43^{+0.88+2.88}_{-0.86-2.65}$ | $-17.75^{+0.77+17.71}_{-0.76-20.43}$
$A_{CP}^{\rm mix}$ | $6.55^{+0.24+1.16}_{-0.25-1.12}$ | $15.14^{+0.59+2.31}_{-0.60-2.17}$ | $3.98^{+0.15+3.21}_{-0.15-3.96}$ | $32.68^{+1.23+3.81}_{-1.26-6.13}$
$\overline{B}^{0}$ ${\to}$ $K_{0}^{{\ast}+}K_{0}^{{\ast}-}$ | A | $A_{CP}^{\rm dir}$ | $-0.31^{+0.01+1.67}_{-0.01-1.47}$ | $7.84^{+0.36+1.97}_{-0.34-1.65}$ | $1.31^{+0.06+0.28}_{-0.06-0.31}$ | $1.73^{+0.08+1.14}_{-0.07-1.22}$
$A_{CP}^{\rm mix}$ | $-43.94^{+6.48+2.44}_{-6.05-2.77}$ | $-41.56^{+6.61+3.80}_{-6.17-4.73}$ | $-57.28^{+5.61+0.91}_{-5.24-0.97}$ | $-59.00^{+5.48+0.71}_{-5.12-0.68}$
$\overline{B}_{s}^{0}$ ${\to}$ $K_{0}^{{\ast}+}K_{0}^{{\ast}-}$ | P | $A_{CP}^{\rm dir}$ | $10.48^{+0.33+2.19}_{-0.33-2.92}$ | $18.03^{+0.60+1.64}_{-0.58-3.13}$ | $11.08^{+0.34+1.57}_{-0.34-2.48}$ | $3.03^{+0.25+6.44}_{-0.18-6.60}$
$A_{CP}^{\rm mix}$ | $9.83^{+0.31+2.57}_{-0.31-2.58}$ | $-11.42^{+0.35+10.29}_{-0.35-~{}9.36}$ | $8.45^{+0.26+2.41}_{-0.26-2.44}$ | $-7.62^{+0.25+1.79}_{-0.24-1.38}$
$\overline{B}_{s}^{0}$ ${\to}$ $K_{0}^{{\ast}0}\overline{K}_{0}^{{\ast}0}$ | P | $A_{CP}^{\rm dir}$ | $0.90^{+0.02+0.11}_{-0.02-0.11}$ | $1.61^{+0.05+0.11}_{-0.05-0.17}$ | $0.92^{+0.02+0.11}_{-0.02-0.11}$ | $0.93^{+0.03+0.63}_{-0.03-0.60}$
$A_{CP}^{\rm mix}$ | $-0.03^{+0.00+0.14}_{-0.00-0.11}$ | $-0.52^{+0.02+0.48}_{-0.02-0.43}$ | $-0.12^{+0.00+0.14}_{-0.00-0.10}$ | $-1.14^{+0.04+0.10}_{-0.04-0.06}$
$\overline{B}_{s}^{0}$ ${\to}$ $a_{0}a_{0}$ | A | $A_{CP}^{\rm mix}$ | $27.62^{+0.84+1.09}_{-0.84-1.09}$ | $27.62^{+0.84+1.67}_{-0.84-1.67}$ | $27.62^{+0.84+0.32}_{-0.84-0.32}$ | $27.62^{+0.84+0.50}_{-0.84-0.50}$

The numerical results on the $CP$-averaged branching ratios and $CP$
asymmetries for the $B$ ${\to}$ $SS$ decays are listed in Table 4 and 5. Here
we use the symbols T for the color favored tree processes, C for the color-
suppressed tree processes, P for the penguin dominated processes, and A for
the pure annihilation processes. Our comments are as follows.

(1) As we have discussed earlier, the LO contributions are suppressed by the
factor $N_{i}(S_{2})$ in Eq.(13), then the NLO contributions will be very
important for the $B$ ${\to}$ $SS$ decays. It is clearly shown in Table 4 that
for both the S1 and S2 scenarios, the NLO contributions to branching ratios
are generally significant, even a few fold changes to the LO contributions
corresponding to the numbers in the “NF” columns for some processes, such as
C-class $B$ decays where the NLO contributions are proportional to the large
Wilson coefficient $C_{1}$.

(2) The hard spectator scattering amplitudes $H_{i}(S_{1}\,S_{2})$ belong to
the nonfactorizable NLO contributions in Eq.(13) with the QCDF approach. So,
the NLO contributions should be sensitive to the parameter $X_{H}$. And the
parameter $X_{H}$ is closely related to the WA parameter $X_{A}$ in this work.
In Table 4, the differences of branching ratios between the C1 and C2 cases
are still obvious, for example, for the $B$ ${\to}$ $a_{0}a_{0}$ and $B_{s}$
${\to}$ $K_{0}^{\ast}\overline{K}_{0}^{\ast}$ decays, and the A-class decays
as well. Additionally, the parameters $X_{H,A}$ are always accompanied by the
Gegenbauer moments in Eq.(15) and Eqs.(26—30). The smaller uncertainties of
the Gegenbauer moments bring branching ratios with the smaller theoretical
uncertainties, compared with those in Refs. PhysRevD.73.014017 ;
PhysRevD.77.014034 .

(3) As it is well known that the T-class $B$ decays are induced by the
external $W$ boson emission interactions with the factorization approach, and
their amplitudes are proportional to the large Wilson coefficient $C_{1}$ or
${\alpha}_{1}$. These processes should theoretically have a relatively large
branching ratio. It might be a little curious in Table 4 that branching ratios
for the T-class $B$ ${\to}$ $SS$ decays are very small, ${\sim}$ ${\cal
O}(10^{-7})$. Some even are less than the branching ratios of the purely WA
decays. One of the main reasons is that the LO contributions of the T-class
decay amplitudes are seriously suppressed by the factor $N_{i}(S_{2})$ in
Eq.(13), with $N_{i}(a_{0})$ ${\sim}$ $0.002$, and their NLO contributions are
suppressed by both the factor ${\alpha}_{s}/N_{c}$ and the small coefficient
$C_{2}$.

(4) It is obvious in Table 4 that for both the C1 and C2 cases, branching
ratios of the S2 scenario are larger than the corresponding ones of the S1
scenario, because the decay amplitudes are proportional to the product of the
decay constants of the scalar mesons and form factors, and the numerical
values of both the decay constants of the scalar mesons (see Table 1) and form
factors (see Table 2) of the S2 scenario are larger than the corresponding
ones of the S1 scenario. Specifically, the P-class $B$ ${\to}$
$a_{0}\overline{K}_{0}^{\ast}$ and $B_{s}$ ${\to}$
$K_{0}^{\ast}\overline{K}_{0}^{\ast}$ decays where the penguin contributions
are largely enhanced by the CKM elements $V_{tb}V_{ts}^{\ast}$ ${\sim}$ ${\cal
O}({\lambda}^{2})$ relative to the possible tree contributions associated with
$V_{ub}V_{us}^{\ast}$ ${\sim}$ ${\cal O}({\lambda}^{4})$, their branching
ratios can reach up to even ${\cal O}(10^{-5})$ for the S2 scenario. These
flagship decay modes should get priority in the future experimental research
program for searching for the $B$ ${\to}$ $SS$ decays.

(5) Experimentally, more than ten years ago, a hint of the $B^{0}$ ${\to}$
$K_{0}^{{\ast}0}\overline{K}_{0}^{{\ast}0}$ decays with a significance of
$0.8\,{\sigma}$ has been reported by the Belle Collaboration with the
$K^{+}K^{-}{\pi}^{+}{\pi}^{-}$ final states PhysRevD.81.071101 , and branching
ratio ${\cal B}$ $=$ $(3.21^{+2.89+2.31}_{-2.85-2.32}){\times}10^{-6}$ and the
upper limit at the $90\%$ confidence level ${\cal B}$ $<$
$8.4{\times}10^{-6}$. Our results are marginally consistent with data when
considering the large experimental errors. Theoretically, besides the small
Wilson coefficients and the CKM elements $V_{tb}V_{td}^{\ast}$ ${\sim}$ ${\cal
O}({\lambda}^{3})$, the relatively smaller branching ratios might arise from
the Gegenbauer moments, which result in a flatter shape line of the scalar
mesonic DAs, and further leads to a milder overlap among the participating
mesonic DAs, and finally give a more modest decay amplitudes. Experimentally,
it is entirely necessary and desirable to improve the accuracy of measurements
and investigate more and more $B$ ${\to}$ $SS$ decays in the future in order
to verify various theoretical models and explore the properties the scalar
mesons.

(6) The weak annihilation amplitudes are thought to be power suppressed with
the QCDF approach NPB.591.313 ; NPB.606.245 . The purely WA $B$ decays should
in principle have very small branching ratios. The evidences have been
demonstrated in the $B$ ${\to}$ $K^{{\pm}}K^{{\ast}{\mp}}$ and $B_{s}$ ${\to}$
${\pi}{\pi}$ decays theoretically NPB.675.333 ; PhysRevD.65.074001 ;
PhysRevD.65.094025 ; PhysRevD.67.014023 ; PhysRevD.88.014043 ;
PhysRevD.90.054019 ; PLB.740.56 ; PLB.743.444 ; PhysRevD.91.074026  and
experimentally pdg2022 . The similar phenomena or/and patterns also appear in
Table 4 for the A-class $B$ ${\to}$ $SS$ decays, with branching ratios of
${\cal B}$ ${\cal O}(10^{-7})$. The impressive and amazing thing is that in
some cases, branching ratios of the A-class $B$ ${\to}$ $SS$ decays with
appropriate parameters can catch up with or overtake those of the T-class
decays, which is very unlike the hadronic $B$ ${\to}$ $PP$, $PV$ decays. These
typical characteristics for the $B$ ${\to}$ $SS$ decays are closely related
with the properties of the scalar mesons, such as the decay constants, DAs and
so on. Additionally, branching ratios of the A-class $B$ ${\to}$ $SS$ decays
are very sensitive to the parameter $X_{A}$, with both the S1 and S2
scenarios. It is clear that with the topologically dependent parameters, i.e.,
$X_{A}^{i}$ ${\neq}$ $X_{A}^{f}$ for the C2 case, the corresponding branching
ratios are relatively larger, due to the larger value of ${\rho}_{H}^{i}$. Our
understanding of the WA contributions to the nonleptonic $B$ decays with the
QCDF approach is not comprehensive enough. Albeit very challenging, the
experimental measurements on the A-class $B$ ${\to}$ $SS$ decays are
interesting and helpful to explore the underlying dynamical mechanism and the
higher power corrections to HMEs.

(7) In Table 4, the theoretical uncertainties are very large, especially those
from the hadronic parameters. Usually, the ratio of branching ratios are
defined to reduce theoretical uncertainties on one hand, and on the other hand
to check some potential symmetry or conservation quantities, for example, the
observables $R_{K,D}$ for the universality of the inherited electroweak
couplings to all charged leptons. Here, we give some ratios of branching
ratios with the universal parameter $X_{A}$ for the C1 case, for example,

$R_{1}\,=\,\frac{{\cal
B}(B^{-}{\to}a_{0}^{-}\overline{K}_{0}^{{\ast}0})}{2\,{\cal
B}(B^{-}{\to}a_{0}^{0}K_{0}^{{\ast}-})}\approx
1.04^{+0.00+0.16}_{-0.00-0.14}~{}(\text{S1}),\
1.06^{+0.00+0.11}_{-0.00-0.11}~{}(\text{S2});$ (52) $R_{2}\,=\,\frac{{\cal
B}(\overline{B}^{0}{\to}a_{0}^{+}K_{0}^{{\ast}-})}{2\,{\cal
B}(\overline{B}^{0}{\to}a_{0}^{0}\overline{K}_{0}^{{\ast}0})}\approx
0.96^{+0.00+0.15}_{-0.00-0.14}~{}(\text{S1}),\
0.94^{+0.00+0.11}_{-0.00-0.10}~{}(\text{S2});$ (53) $R_{3}\,=\,\frac{{\cal
B}(\overline{B}_{s}^{0}{\to}K_{0}^{{\ast}0}\overline{K}_{0}^{{\ast}0})}{{\cal
B}(\overline{B}_{s}^{0}{\to}K_{0}^{{\ast}+}K_{0}^{{\ast}-})}\approx
1.08^{+0.00+0.02}_{-0.01-0.02}~{}(\text{S1}),\
1.07^{+0.00+0.01}_{-0.00-0.01}~{}(\text{S2}).$ (54)

All these ratios are expected to be $R_{1,2,3}$ $=$ $1$ by applying the
$SU(3)$ flavor symmetry.

(8) It is clear in Table 5 that the $CP$ violating asymmetries depend on the
parameter $X_{A}$ which contains the strong phases. It is known that with the
QCDF approach, the strong phases necessary for the direct $CP$ violation arise
from the NLO contributions, which are the order ${\alpha}_{s}$ or
${\Lambda}_{\rm QCD}/m_{b}$ and suppressed compared with the LO contributions.
However, as noted earlier, the LO contributions are seriously suppressed by
the factor $N_{i}(S_{2})$ in Eq.(13), which will indirectly result in the
larger strong phases from the NLO contributions. These effects will have more
influences on the direct $CP$ violating asymmetries for the T- and C-class $B$
${\to}$ $SS$ decays than the P-class ones, because of the larger Wilson
coefficients for the T- and C-class decays. The larger direct $CP$ asymmetries
for the $B_{d}$ ${\to}$ $a_{0}a_{0}$ and $B_{s}$ ${\to}$ $a_{0}K_{0}^{\ast}$
are expected for both the S1 and S2 scenarios. And by comparison, the absolute
values of the direct $CP$ violating asymmetries in the T- and C-class $B$
${\to}$ $SS$ decays are generally larger than those in the corresponding $B$
${\to}$ $PP$, $PV$ decays NPB.675.333 ; PhysRevD.65.074001 ;
PhysRevD.65.094025 ; PhysRevD.67.014023 ; PhysRevD.88.014043 ;
PhysRevD.90.054019 ; PLB.740.56 ; PLB.743.444 ; PhysRevD.91.074026 . In
addition, to contract with the so-called ${\pi}K$ puzzle, the difference
between the direct $CP$ asymmetries for the $B^{-}$ ${\to}$
$a_{0}^{-}\overline{K}_{0}^{{\ast}0}$ and $\overline{B}_{0}$ ${\to}$
$a_{0}^{+}K_{0}^{{\ast}0-}$ decays is estimated to be,

$\displaystyle{\Delta}A_{CP}$ $\displaystyle=$ $\displaystyle
A_{CP}(B^{-}{\to}a_{0}^{-}\overline{K}_{0}^{{\ast}0})-A_{CP}(\overline{B}_{0}{\to}a_{0}^{+}K_{0}^{{\ast}0-})$
(55) $\displaystyle=$
$\displaystyle(5.95^{+0.18+1.69}_{-0.18-1.36})\%~{}(\text{S1}),\
(5.48^{+0.17+1.21}_{-0.17-0.90})\%~{}(\text{S2}),$

with the universal parameter $X_{A}$ $=$ $X_{H}$ for the C1 case, and

${\Delta}A_{CP}\,=\,(3.71^{+0.11+2.46}_{-0.11-1.95})\%~{}(\text{S1}),\
(5.05^{+0.15+1.66}_{-0.15-1.40})\%~{}(\text{S2}),$ (56)

with the topologically dependent parameters $X_{A}^{i}$ ${\neq}$ $X_{A}^{f}$
for the C2 case. Unfortunately, no data are available on the $CP$ asymmetries
for the $B$ ${\to}$ $SS$ decays at the moment.

##  V Summary

To meet the coming high precision measurements on the $B$ meson decays based
on the huge amount of data, and provide a ready and helpful reference in
clarifying the open questions related to the scalar mesons, the hadronic
charmless $B$ ${\to}$ $SS$ decays are studied with the QCDF approach, where
the symbol $S$ denotes the scalar mesons $K_{0}^{\ast}(1430)$ and
$a_{0}(1450)$. It is found that the LO contributions are proportional to the
mass difference of the two valence quarks embedded in the scalar mesons, and
thereby seriously suppressed. This causes two consequences. (1) The branching
ratios for the $B$ ${\to}$ $a_{0}a_{0}$ and $B_{s}$ ${\to}$
$a_{0}K_{0}^{\ast}$ decays belonging to the T- and C-class are very small,
about the order of ${\cal O}(10^{-7})$. (2) The NLO contributions become
necessary and predominant for the $B$ ${\to}$ $SS$ decays. With the updated
values of hadronic parameters obtained from CLFQM, including the transition
form factors, the decay constants and Gegenbauer moments in mesonic DAs for
the two scenarios where the scalar mesons in question are the $1P$ and $2P$
triplet states, the $CP$-averaged branching ratios and $CP$ violating
asymmetries are given with the universal end-point parameters $X_{A}$ and
topology-dependent parameters $X_{A}^{i}$ ${\neq}$ $X_{A}^{f}$. The numerical
results show that (1) theoretical uncertainties of both the branching ratios
and the direct $CP$ asymmetries come mainly from hadronic parameters. (2)
Branching ratios for the $B_{s}$ ${\to}$ $K_{0}^{\ast}\overline{K}_{0}^{\ast}$
decays and the purely weak annihilation decays $B_{s}$ ${\to}$ $a_{0}a_{0}$
and $B_{d}$ ${\to}$ $K_{0}^{{\ast}+}K_{0}^{{\ast}-}$, and the direct $CP$
asymmetries for the $B_{d}$ ${\to}$ $a_{0}a_{0}$ decays are very sensitive to
the parameter $X_{A}$. (3) For the $B$ ${\to}$ $a_{0}K_{0}^{\ast}$ and $B_{s}$
${\to}$ $K_{0}^{\ast}\overline{K}_{0}^{\ast}$ decays, branching ratios for the
S2 scenario are about one order of magnitude larger than those for the S1
scenario, and can reach up to the order of ${\cal O}(10^{-5})$. These decays
should first be searched and investigated experimentally. (4) Theoretical
uncertainties come mainly from the hadronic parameters. More focus and effort
are needed to improve the theoretical calculation precision. Some ratios of
branching ratios are given based on the $SU(3)$ flavor symmetry. In addition,
there is too little available data to draw any conclusions on whether the
scalar mesons $K_{0}^{\ast}(1430)$ and $a_{0}(1450)$ are the $1P$ or $2P$
states. Hope more and more $B$ ${\to}$ $SS$ decays can be measured with higher
and higher precision at the high-luminosity colliders in the future.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 12275067, 12275068, 12135006, 12105078), Natural Science
Foundation of Henan Province (Grant No. 222300420479), and Excellent Youth
Foundation of Henan Province (Grant No. 212300410010).

##  Appendix A The decay amplitudes for the $B$ ${\to}$ $SS$ decays

Here, the symbols is used to simplify the decay amplitudes.

${\lambda}_{q}\,(\cdots)\,=\,\sum\limits_{p=u,c}V_{pb}\,V^{\ast}_{pq}\,({\cdots})$
(57) $\displaystyle\sqrt{2}\,{\cal A}(B^{-}{\to}a_{0}^{-}a^{0}_{0})$
$\displaystyle=$
$\displaystyle{\lambda}_{d}\,\big{\\{}A_{a_{0}^{-}\,a^{0}_{0}}\,\big{[}{\delta}_{u}^{p}\,{\alpha}_{2}-{\alpha}_{4}^{p}+\frac{3}{2}\,{\alpha}_{3,EW}^{p}+\frac{1}{2}\,{\alpha}_{4,EW}^{p}-{\delta}_{u}^{p}\,{\beta}_{2}-{\beta}_{3}^{p}-{\beta}_{3,EW}^{p}\big{]}$
(58)
$\displaystyle\quad+A_{a_{0}^{0}\,a_{0}^{-}}\,\big{[}{\delta}_{u}^{p}\,{\alpha}_{1}+{\alpha}_{4}^{p}+{\alpha}_{4,EW}^{p}+{\delta}_{u}^{p}\,{\beta}_{2}+{\beta}_{3}^{p}+{\beta}_{3,EW}^{p}\big{]}\big{\\}},$
${\cal
A}(B^{-}{\to}a_{0}^{-}\overline{K}_{0}^{{\ast}0})\,=\,{\lambda}_{s}\,A_{a_{0}\,\overline{K}_{0}^{\ast}}\,\big{[}{\alpha}_{4}^{p}-\frac{1}{2}\,{\alpha}_{4,EW}^{p}+{\delta}_{u}^{p}\,{\beta}_{2}+{\beta}_{3}^{p}+{\beta}_{3,EW}^{p}\big{]},$
(59) $\displaystyle\sqrt{2}\,{\cal A}(B^{-}{\to}a_{0}^{0}K_{0}^{{\ast}-})$
$\displaystyle=$
$\displaystyle{\lambda}_{s}\,\big{\\{}A_{a_{0}\,K_{0}^{\ast}}\,\big{[}{\delta}_{u}^{p}\,{\alpha}_{1}+{\alpha}_{4}^{p}+{\alpha}_{4,EW}^{p}+{\delta}_{u}^{p}\,{\beta}_{2}+{\beta}_{3}^{p}+{\beta}_{3,EW}^{p}\big{]}$
(60)
$\displaystyle\quad+A_{K_{0}^{\ast}\,a_{0}}\,\big{[}{\delta}_{u}^{p}\,{\alpha}_{2}+\frac{3}{2}\,{\alpha}_{4,EW}^{p}\big{]}\big{\\}},$
${\cal
A}(B^{-}{\to}K_{0}^{{\ast}-}K_{0}^{{\ast}0})\,=\,{\lambda}_{d}\,A_{K_{0}^{{\ast}-}\,K_{0}^{{\ast}0}}\,\big{[}{\alpha}_{4}^{p}-\frac{1}{2}\,{\alpha}_{4,EW}^{p}+{\delta}_{u}^{p}\,{\beta}_{2}+{\beta}_{3}^{p}+{\beta}_{3,EW}^{p}\big{]},$
(61) $\displaystyle{\cal A}(\overline{B}^{0}{\to}a_{0}^{+}a_{0}^{-})$
$\displaystyle=$
$\displaystyle{\lambda}_{d}\,\big{\\{}A_{a_{0}^{+}\,a_{0}^{-}}\,\big{[}{\delta}_{u}^{p}\,{\alpha}_{1}+{\alpha}_{4}^{p}+{\alpha}_{4,EW}^{p}+{\beta}_{3}^{p}+{\beta}_{4}^{p}-\frac{1}{2}\,{\beta}_{3,EW}^{p}$
(62)
$\displaystyle\qquad-\frac{1}{2}\,{\beta}_{4,EW}^{p}\big{]}+A_{a_{0}^{-}\,a_{0}^{+}}\,\big{[}{\delta}_{u}^{p}\,{\beta}_{1}+{\beta}_{4}^{p}+{\beta}_{4,EW}^{p}\big{]}\big{\\}},$
$\displaystyle{\cal A}(\overline{B}^{0}{\to}a_{0}^{0}a_{0}^{0})$
$\displaystyle=$
$\displaystyle-{\lambda}_{d}\,A_{a_{0}\,a_{0}}\,\big{[}{\delta}_{u}^{p}\,{\alpha}_{2}-{\alpha}_{4}^{p}+\frac{3}{2}\,{\alpha}_{3}^{p}+\frac{1}{2}\,{\alpha}_{4,EW}^{p}$
(63)
$\displaystyle\qquad-{\delta}_{u}^{p}\,{\beta}_{1}-{\beta}_{3}^{p}-2\,{\beta}_{4}^{p}+\frac{1}{2}\,{\beta}_{3,EW}^{p}-\frac{1}{2}\,{\beta}_{4,EW}^{p}\big{]},$
${\cal
A}(\overline{B}^{0}{\to}a_{0}^{+}K_{0}^{{\ast}-})\,=\,{\lambda}_{s}\,A_{a_{0}\,K_{0}^{\ast}}\,\big{[}{\delta}_{u}^{p}\,{\alpha}_{1}+{\alpha}_{4}^{p}+{\alpha}_{4,EW}^{p}+{\beta}_{3}^{p}-\frac{1}{2}\,{\beta}_{3,EW}^{p}\big{]},$
(64) $\displaystyle{\cal
A}(\overline{B}^{0}{\to}a_{0}^{0}\overline{K}_{0}^{{\ast}0})$ $\displaystyle=$
$\displaystyle{\lambda}_{s}\,\big{\\{}A_{a_{0}\,\bar{K}_{0}^{\ast}}\,\big{[}-{\alpha}_{4}^{p}+\frac{1}{2}\,{\alpha}_{4,EW}^{p}-{\beta}_{3}^{p}+\frac{1}{2}\,{\beta}_{3,EW}^{p}\big{]}$
(65)
$\displaystyle\quad+A_{\bar{K}_{0}^{\ast}\,a_{0}}\,\big{[}{\delta}_{u}^{p}\,{\alpha}_{2}+\frac{3}{2}\,{\alpha}_{3,EW}^{p}\big{]}\big{\\}},$
${\cal
A}(\overline{B}^{0}{\to}K_{0}^{{\ast}+}K_{0}^{{\ast}-})\,=\,{\lambda}_{d}\,\big{\\{}A_{K_{0}^{{\ast}-}\,K_{0}^{{\ast}+}}\,\big{[}{\delta}_{u}^{p}\,{\beta}_{1}+{\beta}_{4}^{p}+{\beta}_{4,EW}^{p}\big{]}+B_{K_{0}^{{\ast}+}\,K_{0}^{{\ast}-}}\,\big{[}b_{4}^{p}-\frac{1}{2}\,b_{4,EW}^{p}\big{]}\big{\\}},$
(66) $\displaystyle{\cal
A}(\overline{B}^{0}{\to}K_{0}^{{\ast}0}\overline{K}_{0}^{{\ast}0})$
$\displaystyle=$
$\displaystyle{\lambda}_{d}\,\big{\\{}A_{\overline{K}_{0}^{\ast}\,K_{0}^{\ast}}\,\big{[}{\alpha}_{4}^{p}-\frac{1}{2}\,{\alpha}_{4,EW}^{p}+{\beta}_{3}^{p}+{\beta}_{4}^{p}-\frac{1}{2}\,{\beta}_{3,EW}^{p}$
(67)
$\displaystyle\quad-\frac{1}{2}\,{\beta}_{4,EW}^{p}\big{]}+B_{K_{0}^{\ast}\,\overline{K}_{0}^{\ast}}\,\big{[}b_{4}^{p}-\frac{1}{2}\,b_{4,EW}^{p}\big{]}\big{\\}},$
${\cal
A}(\overline{B}_{s}^{0}{\to}a_{0}^{+}a_{0}^{-})\,=\,{\lambda}_{s}\,\big{\\{}B_{a_{0}^{+}\,a_{0}^{-}}\,\big{[}b_{4}^{p}-\frac{1}{2}\,b_{4,EW}^{p}\big{]}+B_{a_{0}^{-}\,a_{0}^{+}}\,\big{[}{\delta}_{u}^{p}\,b_{1}+b_{4}^{p}+b_{4,EW}^{p}\big{]}\big{\\}},$
(68) ${\cal
A}(\overline{B}_{s}^{0}{\to}a_{0}^{0}a_{0}^{0})\,=\,{\lambda}_{s}\,B_{a_{0}\,a_{0}}\,\big{[}{\delta}_{u}^{p}\,b_{1}+2\,b_{4}^{p}+\frac{1}{2}\,b_{4,EW}^{p}\big{]},$
(69) ${\cal
A}(\overline{B}_{s}^{0}{\to}K_{0}^{{\ast}+}a_{0}^{-})\,=\,{\lambda}_{d}\,A_{K_{0}^{\ast}\,a_{0}}\,\big{[}{\delta}_{u}^{p}\,{\alpha}_{1}+{\alpha}_{4}^{p}+{\alpha}_{4,EW}^{p}+{\beta}_{3}^{p}-\frac{1}{2}\,{\beta}_{3,EW}^{p}\big{]},$
(70) $\sqrt{2}\,{\cal
A}(\overline{B}_{s}^{0}{\to}K_{0}^{{\ast}0}a_{0}^{0})\,=\,{\lambda}_{d}\,A_{K_{0}^{\ast}\,a_{0}}\,\big{[}{\delta}_{u}^{p}\,{\alpha}_{2}-{\alpha}_{4}^{p}+\frac{3}{2}\,{\alpha}_{3,EW}^{p}+\frac{1}{2}\,{\alpha}_{4,EW}^{p}-{\beta}_{3}^{p}+\frac{1}{2}\,{\beta}_{3,EW}^{p}\big{]},$
(71) $\displaystyle{\cal
A}(\overline{B}_{s}^{0}{\to}K_{0}^{{\ast}0}\overline{K}_{0}^{{\ast}0})$
$\displaystyle=$
$\displaystyle{\lambda}_{s}\,\big{\\{}A_{K_{0}^{{\ast}}\,\bar{K}_{0}^{{\ast}}}\,\big{[}{\alpha}_{4}^{p}-\frac{1}{2}\,{\alpha}_{4,EW}^{p}+{\beta}_{3}^{p}+{\beta}_{4}^{p}-\frac{1}{2}\,{\beta}_{3,EW}^{p}$
(72)
$\displaystyle\quad-\frac{1}{2}\,{\beta}_{4,EW}^{p}+B_{\bar{K}_{0}^{{\ast}}\,K_{0}^{{\ast}}}\,\big{[}b_{4}^{p}-\frac{1}{2}\,b_{4,EW}^{p}\big{]}\big{\\}},$
$\displaystyle{\cal
A}(\overline{B}_{s}^{0}{\to}K_{0}^{{\ast}+}K_{0}^{{\ast}-})$ $\displaystyle=$
$\displaystyle{\lambda}_{s}\,\big{\\{}A_{K_{0}^{{\ast}}\,\bar{K}_{0}^{{\ast}}}\,\big{[}{\delta}_{u}^{p}\,{\alpha}_{1}+{\alpha}_{4}^{p}+{\alpha}_{4,EW}^{p}+{\beta}_{3}^{p}+{\beta}_{4}^{p}-\frac{1}{2}\,{\beta}_{3,EW}^{p}$
(73)
$\displaystyle\quad-\frac{1}{2}\,{\beta}_{4,EW}^{p}+B_{\bar{K}_{0}^{{\ast}}\,K_{0}^{{\ast}}}\,\big{[}{\delta}_{u}^{p}\,b_{1}+b_{4}^{p}+b_{4,EW}^{p}\big{]}\big{\\}}.$

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