#  $K_{0}^{\ast}(1430)$ Twist-2 Distribution Amplitude and $B_{s},D_{s}\to
K_{0}^{\ast}(1430)$ Transition Form Factors

Dong Huang  School of Physics and Mechatronic Engineering, Guizhou Minzu
University, Guiyang 550025, P.R. China  Tao Zhong111Corresponding author
<EMAIL_ADDRESS>School of Physics and Mechatronic Engineering, Guizhou
Minzu University, Guiyang 550025, P.R. China  Hai-Bing Fu<EMAIL_ADDRESS>School of Physics and Mechatronic Engineering, Guizhou Minzu University,
Guiyang 550025, P.R. China  Department of Physics, Chongqing University,
Chongqing 401331, P.R. China  Zai-Hui Wu  School of Physics and Mechatronic
Engineering, Guizhou Minzu University, Guiyang 550025, P.R. China  Xing-Gang
Wu<EMAIL_ADDRESS>Department of Physics, Chongqing University, Chongqing
401331, P.R. China  Hong Tong  School of Physics and Mechatronic Engineering,
Guizhou Minzu University, Guiyang 550025, P.R. China

###### Abstract

Based on the scenario that the $K_{0}^{\ast}(1430)$ is viewed as the ground
state of $s\bar{q}$ or $q\bar{s}$, we study the $K_{0}^{\ast}(1430)$ leading-
twist distribution amplitude (DA) $\phi_{2;K_{0}^{\ast}}(x,\mu)$ with the QCD
sum rules in the framework of background field theory. A more reasonable sum
rule formula for $\xi$-moments $\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}$ is
suggested, which eliminates the influence brought by the fact that the sum
rule of $\langle\xi^{0}_{p}\rangle_{3;K_{0}^{\ast}}$ cannot be normalized in
whole Borel region. More accurate values of the first ten $\xi$-moments,
$\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}(n=1,2,\cdots,10)$, are evaluated. A
new light-cone harmonic oscillator (LCHO) model for $K_{0}^{\ast}(1430)$
leading-twist DA is established for the first times. By fitting the resulted
values of $\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}(n=1,2,\cdots,10)$ via the
least square method, the behavior of $K_{0}^{\ast}(1430)$ leading-twist DA
described with LCHO model is determined. Further, by adopting the light-cone
QCD sum rules, we calculate the $B_{s},D_{s}\to K_{0}^{\ast}(1430)$ transition
form factors and branching fractions of the semileptonic decays
$B_{s},D_{s}\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$. The corresponding numerical
results can be used to extract the Cabibbo-Kobayashi-Maskawa matrix elements
by combining the relative experimental data in the future.

###### pacs:

12.38.-t, 12.38.Bx, 14.40.Aq

##  I INTRODUCTION

Currently, the exclusive determinations of the Cabibbo-Kobayashi-Maskawa (CKM)
matrix elements $|V_{ub}|$ and $|V_{cd}|$ are mainly dominated by the
semileptonic $B\to\pi\ell\bar{\nu}$ and $D\to\pi\ell\nu$ decays, respectively
Workman:2022ynf . The study of the semileptonic $B_{s},D_{s}\to
K_{0}^{\ast}(1430)\ell\nu_{\ell}$ decays can provide a new choice and
supplement for extracting these two matrix elements.

In the theoretical studies on the semileptonic $B_{s},D_{s}\to
K_{0}^{\ast}(1430)$ decays, the most challenging parts are the calculations of
the transition form factors (TFFs). These TFFs are mainly dominated by the
short-distance dynamics in the large recoil region and the soft dynamics in
the region of small recoil. Although the momentum dependence of the $B_{s}\to
K_{0}^{\ast}(1430)$ TFFs calculated by Faustov and Galkin in the relativistic
quark model (RQM) is determined in the whole accessible kinematical range
Faustov:2013ima , the commonly used research methods such as perterbative QCD
(pQCD) factorization, QCD sum rules (QCDSRs), light-cone QCD sum rules (LCSRs)
and Lattice QCD (LQCD) theory, etc., are usually only applicable to calculate
the TFFs in specific $q^{2}$ regions. The corresponding results should be
extrapolated to the whole kinematic region by adopting appropriate
parametrization form. Usually, the LQCD works well in the small recoil region.
However, there are currently no LQCD calculations of the $B_{s},D_{s}\to
K_{0}^{\ast}(1430)$ TFFs in the literature. The pQCD factorization method is
usually applicable near the large recoil point, and which has been used to
calculate $B_{s}\to K_{0}^{\ast}(1430)$ TFFs in Refs. Li:2008tk ; Zhang:2010af
; Chen:2021oul . The QCDSR and LCSR estimations for TFFs are effective in low
and intermediate $q^{2}$ regions. In Ref. Yang:2005bv , the TFFs
$f_{+}^{B_{s}\to K_{0}^{\ast}}(q^{2})$ and $f_{+}^{D_{s}\to
K_{0}^{\ast}}(q^{2})$ are calculated by adopting the QCDSRs with three-point
correlation function (correlator). In which, the TFFs are parameterized as
vacuum condensates with different dimension. The three-point QCDSRs is then
used to calculate TFFs $f^{B_{s}\to K_{0}^{\ast}}_{\pm,T}(q^{2})$ and further
analyze the rare semileptonic $B_{s}\to K_{0}^{\ast}(1430)\ell^{+}\ell^{-}$
decays Ghahramany:2009zz . More researches on the $B_{s}\to
K_{0}^{\ast}(1430)$ TFFs are performed in the framework of LCSRs. Different
from the three-point QCDSRs, the TFFs calculated within LCSRs are
parameterized as the initial or final meson distribution amplitudes (DAs)
arranged with different twist structures. Starting from different correlator,
the LCSRs of $B_{s}\to K_{0}^{\ast}(1430)$ TFFs are expressed as the
convolution integrals of the $B_{s}$ meson DAs Khosravi:2022fzo , the
$K_{0}^{\ast}(1430)$ twist-2, 3 DAs Wang:2008da ; Wang:2014vra ; Wang:2014upa
, only $K_{0}^{\ast}(1430)$ twist-2 DA Sun:2010nv  or twist-3 DAs Han:2013zg ,
respectively. In addition, there are some other studies on the $D_{s}\to
K_{0}^{*}(1430)$ TFFs in the literature. For example, Ref. Cheng:2002ai
extracts the TFF $f_{+}^{D_{s}\to K_{0}^{\ast}}(0)$ from the data of the
hadronic $D$ decay in the generalized factorization model.

In this paper, we will calculate the $B_{s},D_{s}\to K_{0}^{\ast}(1430)$ TFFs
within the LCSR method by adopting the traditional current correlator.
Specifically, we will adopt the calculation technology for the operator
product expansion (OPE) in Refs. Duplancic:2008ix ; Duplancic:2008tk . That
is, the usual suppression by the powers of the Borel parameter for the higher
twist contributions is transferred as exponential suppression through the
integration by parts.

Usually, the LCSRs of the $B_{s},D_{s}\to K_{0}^{\ast}(1430)$ TFFs are
dominated by the contributions proportional to the $K_{0}^{\ast}(1430)$
leading-twist DA $\phi_{2;K_{0}^{\ast}}(x,\mu)$. Then
$\phi_{2;K_{0}^{\ast}}(x,\mu)$ is the mainly error source of these TFFs. In
order to obtain accurate predictions for the semileptonic $B_{s},D_{s}\to
K_{0}^{*}(1430)$ decays, one should determine the accurate behavior of the
$K_{0}^{\ast}(1430)$ leading-twist DA. Before this, however, we should clarify
our understanding of quark content of scalar $K_{0}^{\ast}(1430)$ meson.
Different from the scalar mesons below $1\ {\rm GeV}$ $-$ for which there is
no general agreement on the multiple candidates such as conventional
$q\bar{q}$ states Cheng:2005nb , meson-meson molecular states Weinstein:1982gc
, tetraquark states Jaffe:1976ig ; Jaffe:1976ih ; Wang:2010pn  and so on $-$
the $K_{0}^{\ast}(1430)$ is predominantly viewed as the $s\bar{q}$ or
$q\bar{s}$ state in almost all the relative researches. The only controversy
about $K_{0}^{\ast}(1430)$ lies between the following two scenarios: Scenario
1 (S1), the $K_{0}^{\ast}(1430)$ is assumed to be the excited state
corresponding to the ground state below $1\ {\rm GeV}$; Scenario 2 (S2), the
$K_{0}^{\ast}(1430)$ is viewed as the ground state while the nonet scalars
below $1\ {\rm GeV}$ may be considered as four-quark bound states. Ref.
Du:2004ki  calculates the masses and decay constants of $I=1/2$ scalar mesons
with QCDSRs. Its result favors that $K_{0}^{\ast}(1430)$ is the lowest scalar
stats of $s\bar{q}$ or $q\bar{s}$, i.e., S2. Later, based on S2, Refs.
Lu:2006fr ; Han:2013zg  study the $K_{0}^{\ast}(1430)$ twist-3 DAs with
QCDSRs. Besides, the research on the two-body decays of $B_{(s)}$ containing
$K_{0}^{\ast}(1430)$ in Ref. Chen:2021oul  also supports that
$K_{0}^{\ast}(1430)$ should be described as the lowest-lying p-wave state
rather than the first excited one. Therefore, our research work on
$K_{0}^{\ast}(1430)$ in this paper will also take S2 as the starting point. In
numerical analysis, only the data corresponding to S2 in literature will be
adopted for consistency.

The $\phi_{2;K_{0}^{\ast}}(x,\mu)$ has been investigated in Ref. Cheng:2005nb
with QCDSRs, and in Ref. Chen:2021oul  with light-front (LF) approach. In Ref.
Cheng:2005nb , the first two nonzero $\xi$-moments and Gegenbauer moments are
calculated and substituted into the truncation form of the Gegenbauer
expansion series (TF model) of $\phi_{2;K_{0}^{\ast}}(x,\mu)$ to predict the
behavior of the $K_{0}^{\ast}(1430)$ leading-twist DA. However, our recent
analysis on the commonly used phenomenological model of the pionic leading-
twist DA shows that the simple TF model is far from sufficient to describe the
accurate behavior of DA Zhong:2022lmn . Therefore, we will re study the
$K_{0}^{\ast}(1430)$ leading-twist DA with the QCDSRs in the framework of
background field theory (BFT) Huang:1989gv . In order to obtain more accurate
behavior of $\phi_{2;K_{0}^{\ast}}(u,\mu)$, a research scheme suggested in our
previous study on the pionic leading-twist DA Zhong:2021epq  will be adopted.
This scheme has been used to researches on the kaon leading-twist DA
Zhong:2022ecl  and the $a_{1}(1260)$ meson longitudinal twist-2 DA Hu:2021lkl
. Specifically, we will construct a light-cone harmonic oscillator (LCHO)
model based on the Brodsky-Huang-Lepage (BHL) description BHL  for
$\phi_{2;K_{0}^{\ast}}(u,\mu)$. The model parameters will be determined by
fitting the first ten $\xi$-moments with the least squares method. More
accurate values of those $\xi$-moments will be evaluated by using a new sum
rule formula suggested in Ref. Zhong:2021epq .

This paper is organized as follows. In Sec. II, the sum rules for the
$\xi$-moments of the $K_{0}^{\ast}(1430)$ leading-twist DA and the LCSRs of
the $B_{s},D_{s}\to K_{0}^{\ast}(1430)$ TFFs $f_{\pm,T}(q^{2})$ are derived, a
new LCHO model for the $K_{0}^{\ast}(1430)$ leading-twist DA is established at
the first time. In Sec. III, we provide the relevant numerical results. The
final section is reserved for a summary.

##  II THEORETICAL FRAMEWORK

###  II.1 Sum rules for the $\xi$-moments of $\phi_{2;K_{0}^{\ast}}(x,\mu)$

The leading-twist DA $\phi_{2;K_{0}^{\ast}}(x,\mu)$ of the scalar
$K_{0}^{\ast}(1430)^{+}$ meson with quark content $u\bar{s}$ are given by
Cheng:2005nb

$\displaystyle\langle
0\left|\bar{s}(z)\gamma_{\mu}u(-z)\right|K_{0}^{\ast+}\rangle$
$\displaystyle\quad\quad\quad\quad=\bar{f}_{K_{0}^{\ast}}p_{\mu}\int^{1}_{0}due^{i(2u-1)p\cdot
z}\phi_{2;K_{0}^{\ast}}(u,\mu),$ (1)

where $z^{2}=0$, $\bar{f}_{K_{0}^{\ast}}$ and $m_{K_{0}^{\ast}}$ are the decay
constant and mass of $K_{0}^{\ast}(1430)$ meson, respectively. Expanding both
sides of Eq. (1) into series of $z$, one can get

$\displaystyle\langle
0|\bar{s}(0)\gamma_{\mu}(iz\cdot\tensor{D})^{n}u(0)|K_{0}^{\ast+}\rangle=p_{\mu}(p\cdot
z)^{n}\bar{f}_{K_{0}^{\ast}}\langle\xi^{n}\rangle_{2;K_{0}^{\ast}},$ (2)

where
${D_{\mu}}={\partial_{\mu}}-i{g_{s}}{T^{A}}\mathcal{A}_{\mu}^{A}(x)(A=1,...,8)$
is the fundamental representation of the gauge covariant derivative, and

$\displaystyle\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}=\int^{1}_{0}du(2u-1)^{n}\phi_{2;K_{0}^{\ast}}(u).$
(3)

is the $n$th $\xi$-moment. Then, in order to derive the sum rules of the
$\xi$-moments $\langle{\xi^{n}}\rangle_{2;K_{0}^{\ast}}$, we introduce the
following correlator,

$\displaystyle\Pi_{2;K_{0}^{\ast}}(z,q)$ $\displaystyle=i\int
d^{4}x{e^{iq\cdot x}}\langle
0|T\\{{J_{n}}(x),\hat{J}_{0}^{\dagger}(0)\\}|0\rangle$ $\displaystyle={(z\cdot
q)^{n+1}}{I_{2;K_{0}^{\ast}}}(q^{2})$ (4)

with the interpolating currents

$\displaystyle J_{n}(x)$
$\displaystyle=\bar{s}(x)\not{z}(iz\cdot\tensor{D})^{n}u(x),$
$\displaystyle\hat{J}_{0}^{\dagger}(0)$ $\displaystyle=\bar{u}(0)s(0).$ (5)

We now perform the OPE for the correlator (4) in the deep Euclidean region.
The calculation is carried out in the framework of BFT Huang:1989gv . By
decomposing quark and gluon fields into classical background fields describing
nonperturbative effects and quantum fields describing perturbative effects,
BFT can provide clear physical images for the separation of long- and short-
range dynamics in OPE. Based on the basic assumptions and Feynman rules of BFT
Huang:1989gv , the correlator (4) can be rewritten as

$\displaystyle{\Pi_{2;K_{0}^{\ast}}}(z\cdot q)$
$\displaystyle=i\int{{d^{4}}x{e^{iq\cdot x}}}$
$\displaystyle\times\Big{\\{}-{\rm Tr}\langle
0|S_{F}^{s}(0,x)\not{z}(iz\cdot\tensor{D})^{n}S_{F}^{u}(x,0)|0\rangle$
$\displaystyle+{\rm Tr}\langle
0|\bar{s}(x)s(0)\not{z}(iz\cdot\tensor{D})^{n}S_{F}^{u}(x,0)|0\rangle$
$\displaystyle+{\rm Tr}\langle
0|S_{F}^{s}(0,x)\not{z}(iz\cdot\tensor{D})^{n}\bar{u}(0)u(x)|0\rangle$
$\displaystyle+\cdots\Big{\\}},$ (6)

where $\rm{Tr}$ indicates trace for the $\gamma$-matrix and color matrix,
$S_{F}^{s}(0,x)$ indicate the $s$-quark propagator from $x$ to $0$,
$S_{F}^{u}(x,0)$ stands for the $u$-quark propagator from $0$ to $x$,
$\not{z}(iz\cdot\tensor{D})^{n}$ are the vertex operators from current
$J_{n}(x)$, respectively. The expressions up to dimension-six of the quark
propagator, the vertex operator, and the vacuum matrix elements such as
$\langle 0|\bar{s}(x)s(0)\rangle$ and $\langle 0|\bar{u}(0)u(x)\rangle$ have
been derived and given in Refs. Zhong:2011rg ; Zhong:2014jla ; Zhong:2021epq ;
Hu:2021zmy . By substituting those corresponding formula into Eq. (6), the OPE
of correlator (4), $I_{2;K_{0}^{\ast}}^{\rm qcd}(q^{2})$, can be obtained.

By inserting a complete set of hadronic states into correlator (4) in physical
region, whose hadronic representation can be read as

$\displaystyle{\rm Im}I_{2;K_{0}^{\ast}}^{\rm had}(s)$ $\displaystyle=\pi
m_{K_{0}^{\ast}}\delta(s-m_{K_{0}^{\ast}}^{2})\bar{f}_{K_{0}^{\ast}}^{2}\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}\langle\xi^{0}_{p}\rangle_{3;K_{0}^{\ast}}$
$\displaystyle+{\rm Im}I_{2;K_{0}^{\ast}}^{\rm
pert}(s)\theta(s-s_{K_{0}^{\ast}}),$ (7)

where $s_{K_{0}^{\ast}}$ is the continuum threshold,
$\langle\xi^{0}_{p}\rangle_{3;K_{0}^{\ast}}$ is the zeroth $\xi$-moment of
$K_{0}^{\ast}(1430)^{+}$ two-particle twist-3 DA
$\phi_{3;K_{0}^{\ast}}^{p}(u,\mu)$. In the calculation of Eq. (7), the matrix
element formula in Eq. (2), $\langle
K_{0}^{\ast+}(p)|\hat{J}_{0}^{\dagger}(0)|0\rangle=\bar{f}_{K_{0}^{\ast}}m_{K_{0}^{\ast}}\langle\xi^{0}_{p}\rangle_{3;K_{0}^{\ast}}$
and the quark-hadron duality have been used.

Substituting the resulted OPE and hadronic representation of correlator (4),
i.e., $I_{2;K_{0}^{\ast}}^{\rm qcd}(q^{2})$ and ${\rm
Im}I_{2;K_{0}^{\ast}}^{\rm had}(s)$, into the following dispersion relation
after Borel transformation,

$\displaystyle\frac{1}{\pi}\frac{1}{M^{2}}\int_{m_{s}^{2}}dse^{-s/M^{2}}{\rm
Im}I_{2;K_{0}^{\ast}}^{\rm had}(s)={\hat{L}}_{M}I_{2;K_{0}^{\ast}}^{\rm
qcd}(q^{2}),$ (8)

the sum rules of
$\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}\times\langle\xi^{n}_{p}\rangle_{3;K_{0}^{\ast}}$
reads:

$\displaystyle\frac{\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}\langle\xi^{0}_{p}\rangle_{3;K_{0}^{\ast}}m_{K_{0}^{\ast}}\bar{f}^{2}_{K_{0}^{\ast}}}{M^{2}e^{m_{K_{0}^{\ast}}^{2}/M^{2}}}=\frac{1}{\pi}\frac{1}{M^{2}}\int^{s_{K_{0}^{\ast}}}_{m_{s}^{2}}dse^{-s/M^{2}}{\rm
Im}I_{2;K_{0}^{\ast}}^{\rm
pert}(s)+\Big{(}1+\frac{m_{s}m_{u}}{2M^{2}}+\frac{2n+1}{2}\frac{m_{s}^{2}}{M^{2}}\Big{)}\frac{\langle\bar{s}s\rangle}{M^{2}}+\Big{(}-1-\frac{m_{s}m_{u}}{2M^{2}}$
$\displaystyle+\frac{m_{s}^{2}}{M^{2}}\Big{)}\frac{(-1)^{n}\langle\bar{u}u\rangle}{M^{2}}+\hat{I}_{\langle
G^{2}\rangle}(M^{2})+\hat{I}_{\langle
G^{2}\rangle}^{m_{s}^{3}}(M^{2})+\Big{(}-\frac{2n}{3}-\frac{8n-9}{36}\frac{m_{s}m_{u}}{M^{2}}\Big{)}\frac{\langle
g_{s}\bar{s}\sigma
TGs\rangle}{(M^{2})^{2}}+\Big{[}\frac{2n}{3}\Big{(}1-\frac{m_{s}^{2}}{M^{2}}\Big{)}+\frac{8n-9}{36}\frac{m_{s}m_{u}}{M^{2}}$
$\displaystyle+\frac{m_{s}^{2}}{4M^{2}}\Big{]}\frac{(-1)^{n}\langle
g_{s}\bar{u}\sigma
TGu\rangle}{(M^{2})^{2}}+\frac{2(n+3)}{81}m_{u}\frac{\langle
g_{s}\bar{s}s\rangle^{2}}{(M^{2})^{3}}+\frac{-2(n+3)}{81}m_{s}\Big{(}1-\frac{m_{s}^{2}}{M^{2}}\Big{)}\frac{(-1)^{n}\langle
g_{s}\bar{u}u\rangle^{2}}{(M^{2})^{3}}+\hat{I}_{\langle
G^{3}\rangle}(M^{2})+\hat{I}_{\langle G^{3}\rangle}^{m_{s}^{3}}(M^{2})$
$\displaystyle+\hat{I}_{\langle q^{4}\rangle}(M^{2})+\hat{I}_{\langle
q^{4}\rangle}^{m_{s}^{3}}(M^{2}),$ (9)

where the imaginary part of the perturbative contribution

$\displaystyle{\rm Im}I_{2;K_{0}^{\ast}}^{\rm pert}(s)$
$\displaystyle=\frac{3}{16\pi(n+1)(n+2)}\Big{\\{}m_{s}\Big{[}\Big{(}1-\frac{2m_{s}^{2}}{s}\Big{)}^{n+1}\Big{(}2(n+1)\Big{(}1-\frac{2m_{s}^{2}}{s}\Big{)}+1\Big{)}+(-1)^{n}\Big{]}$
$\displaystyle-
m_{u}\Big{[}\Big{(}1-\frac{2m_{s}^{2}}{s}\Big{)}^{n+1}\Big{(}-2(n+1)\Big{(}1-\frac{m_{s}^{2}}{s}\Big{)}+2n+3\Big{)}+(-1)^{n}(2n+3)\Big{]}\Big{\\}},$
(10)

$\hat{I}_{\langle G^{2}\rangle}(M^{2})$, $\hat{I}_{\langle
G^{3}\rangle}(M^{2})$, $\hat{I}_{\langle q^{4}\rangle}(M^{2})$ are the
contributions proportional to the double-gluon condensate
$\langle\alpha_{s}G^{2}\rangle$, triple-gluon condensate $\langle
g_{s}^{3}fG^{3}\rangle$ and four-quark condensate $\langle
g_{s}^{2}\bar{u}u\rangle^{2}$, and $\hat{I}^{m_{s}^{3}}_{\langle
G^{2}\rangle}(M^{2})$, $\hat{I}^{m_{s}^{3}}_{\langle G^{3}\rangle}(M^{2})$,
$\hat{I}^{m_{s}^{3}}_{\langle q^{4}\rangle}(M^{2})$ are corresponding
$\mathcal{O}(m_{s}^{3})$-corrections, respectively. The specific expressions
for those terms are exhibited in Appendix for convenience. In Eq. (9), in
addition, $M$ is the Borel parameter, $m_{u}$ and $m_{s}$ are the current
quark masses of $u$ and $s$ quarks, $\langle\bar{u}u\rangle$ and
$\langle\bar{s}s\rangle$ are double-quark condensates with
$\langle\bar{s}s\rangle/\langle\bar{u}u\rangle=\kappa$, $\langle
g_{s}\bar{u}\sigma TGu\rangle$ and $\langle g_{s}\bar{s}\sigma TGs\rangle$ are
quark-gluon mixed condensates, $\langle g_{s}\bar{u}u\rangle^{2}$ and $\langle
g_{s}\bar{s}s\rangle^{2}$ are four-quark condensates. In the calculation of
OPE, the $SU_{f}(3)$ breaking effect is considered. Specifically, the full $s$
quark mass effect in the perterbative part is preserved; the $s$ quark mass
corrections proportional to $m_{s}^{\leq 3}$ for condensate terms are
calculated owing to $m_{s}\sim 0.1{\rm GeV}$, while $m_{u}^{2}\sim 0$ is
adopted due to smallness.

In particular, the sum rules (9) is regarded as that for
$\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}\times\langle\xi^{n}_{p}\rangle_{3;K_{0}^{\ast}}$
instead of $\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}$ in this work due to
dependence of $\langle\xi^{n}_{p}\rangle_{3;K_{0}^{\ast}}$ on the Borel
parameter as suggested in Ref. Zhong:2021epq . This assumption can be
confirmed by the sum rule of $\langle\xi^{n}_{p}\rangle_{3;K_{0}^{\ast}}$
derived from the correlator $i\int d^{4}xe^{iq\cdot x}\langle
0|\hat{J}_{0}(0)\hat{J}_{0}^{\dagger}(0)|0\rangle$. Following the above sum
rule calculation procedure performed for correlator (4), one can easy obtain,

$\displaystyle\frac{m_{K_{0}^{\ast}}^{2}\bar{f}_{K_{0}^{\ast}}^{2}\langle\xi^{0}_{p}\rangle_{3;K_{0}^{\ast}}^{2}}{M^{2}e^{m_{K_{0}^{\ast}}^{2}/M^{2}}}=\frac{1}{\pi}\frac{1}{M^{2}}\int^{s_{K_{0}^{\ast}}}_{m_{s}^{2}}dse^{-s/M^{2}}{\rm
Im}I^{\rm
pert}_{3;K_{0}^{\ast}}(s)+\Big{(}\frac{m_{s}}{2}+m_{u}\Big{)}\frac{\langle\bar{s}s\rangle}{M^{2}}+\Big{(}\frac{m_{u}}{2}+m_{s}-\frac{m_{s}^{3}}{M^{2}}\Big{)}\frac{\langle\bar{u}u\rangle}{M^{2}}+\frac{1}{24\pi}\Big{(}3-\frac{4m_{s}^{2}}{M^{2}}\Big{)}$
$\displaystyle\times\frac{\langle\alpha_{s}G^{2}\rangle}{M^{2}}+\frac{m_{u}}{2}\frac{\langle
g_{s}\bar{s}\sigma
TGs\rangle}{(M^{2})^{2}}+\frac{m_{s}}{2}\Big{(}1-\frac{3m_{s}^{2}}{2M^{2}}\Big{)}\frac{\langle
g_{s}\bar{u}\sigma TGu\rangle}{(M^{2})^{2}}-\frac{4}{27}\frac{\langle
g_{s}\bar{s}s\rangle^{2}}{(M^{2})^{2}}-\frac{4}{27}\Big{(}1-\frac{5m_{s}^{2}}{4M^{2}}\Big{)}\frac{\langle
g_{s}\bar{u}u\rangle^{2}}{(M^{2})^{2}}-\frac{1}{96\pi^{2}}\frac{m_{s}^{2}}{M^{2}}\frac{\langle
g_{s}^{3}fG^{3}\rangle}{(M^{2})^{2}}$
$\displaystyle+\frac{1}{972\pi^{2}}\Big{\\{}-12\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}+70+\frac{m_{s}^{2}}{M^{2}}\Big{[}15\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}+93\Big{]}\Big{\\}}\frac{(2+\kappa^{2})\langle
g_{s}^{2}\bar{u}u\rangle^{2}}{(M^{2})^{2}},$ (11)

with

$\displaystyle{\rm Im}I^{\rm
pert}_{3;K_{0}^{\ast}}(s)=\frac{3s}{16\pi}\Big{\\{}\Big{(}1-\frac{2m_{s}^{2}}{s}\Big{)}\Big{[}\frac{2m_{s}^{2}}{s}\Big{(}1-\frac{m_{s}^{2}}{s}\Big{)}+1\Big{]}+1\Big{\\}}-\frac{3m_{s}^{2}}{16\pi}\Big{\\{}\Big{(}1-\frac{2m_{s}^{2}}{s}\Big{)}\Big{[}2\Big{(}1-\frac{m_{s}^{2}}{s}\Big{)}+1\Big{]}+1\Big{\\}}.$
(12)

As discussed above and suggested in Ref. Zhong:2021epq , a better sum rules
for $\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}$ is suggested as

$\displaystyle\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}=\frac{\Big{(}\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}\times\langle\xi^{0}_{p}\rangle_{3;K_{0}^{\ast}}\Big{)}|_{\rm
From\
Eq.~{}\eqref{SRxinxi0}}}{\sqrt{\langle\xi^{0}_{p}\rangle_{3;K_{0}^{\ast}}^{2}}|_{\rm
From\ Eq.~{}\eqref{SRxi0xi0}}}$ (13)

###  II.2 LCHO model for ${\phi_{2;K_{0}^{\ast}}}(x,\mu)$ based on BHL
prescription

The $K_{0}^{\ast}(1430)$ leading-twist DA, ${\phi_{2;K_{0}^{\ast}}}(x,\mu)$,
describes the momentum fraction distribution of partons in
$K_{0}^{\ast}(1430)$ meson for the lowest Fock state. The
${\phi_{2;K_{0}^{\ast}}}(x,\mu)$ is a universal nonperturbative objects, and
which should be studied with nonperturbative QCD. However, we usually can only
use the method of combining nonperturbative QCD and phenomenological model to
study ${\phi_{2;K_{0}^{\ast}}}(x,\mu)$ due to the difficulty of
nonperturbative QCD. In Sec. II.1, we have calculated the $n$th $\xi$-moment
with nonperturbative QCDSR method. In this subsection, we will construct a
LCHO model to describe the overall behavior of
${\phi_{2;K_{0}^{\ast}}}(x,\mu)$ based on BHL prescription BHL .

The starting point of BHL prescription is the assumption that there is a
connection between the equal-times wave function (WF) in the rest frame and
the light-cone WF. Through this assumption, one can map the approximate bound
state solution in the quark model for meson in the rest frame to the light-
cone frame by equating the off-shell propagator in the two frames, thus
obtaining the LCHO model of the meson WFs Huang:1994dy . So far, LCHO has been
widely used in the study of various meson WFs or DAs and has been continuously
improved Zhong:2014jla ; Zhong:2021epq ; Zhong:2022ecl ; Hu:2021lkl ;
Huang:1994dy ; Cao:1997hw ; Huang:2004fn ; Wu:2005kq ; Huang:2006wt ;
Wu:2011gf ; Wu:2012kw ; Huang:2013gra ; Huang:2013yya ; Zhong:2014fma ;
Zhong:2015nxa ; Zhong:2016kuv ; Zhang:2017rwz ; Zhong:2018exo ; Zhang:2021wnv
. Formally, the WF of $K_{0}^{\ast}(1430)$ leading-twist DA can be expressed
as

$\displaystyle\Psi_{2;K_{0}^{\ast}}(x,\mathbf{k}_{\perp})=\chi_{2;K_{0}^{\ast}}(x,\mathbf{k}_{\perp})\Psi^{R}_{2;K_{0}^{\ast}}(x,\mathbf{k}_{\perp}),$
(14)

where $\mathbf{k}_{\perp}$ is the transverse momentum,
$\chi_{2;K_{0}^{\ast}}(x,\mathbf{k}_{\perp})$ stands for the spin-space WF
coming from the Wigner-Melosh rotation. As a scalar meson, the spin WF of
$K_{0}^{\ast}(1430)$ should be Zhong:2022ecl

$\displaystyle\chi_{2;K_{0}^{\ast}}(x,\mathbf{k}_{\perp})=\frac{\widetilde{m}}{\sqrt{\mathbf{k}_{\perp}^{2}+\widetilde{m}^{2}}},$
(15)

where $\widetilde{m}=\hat{m}_{q}x+\hat{m}_{s}\bar{x}$ with $q=u/d$ and
$\bar{x}=1-x$. $\hat{m}_{q}$ and $\hat{m}_{s}$ are the corresponding
constituent quark masses of $K_{0}^{\ast}(1430)$, and we take
$\hat{m}_{s}=370\ {\rm MeV}$ and $\hat{m}_{q}=250\ {\rm MeV}$ as discussed in
Ref. Zhong:2022ecl . The $\Psi_{2;K_{0}^{*}}^{R}(x,{{\rm{k}}_{\bot}})$ is the
spatial wave function, and which can be divided into the $x$-dependent part,
i.e., ${\varphi_{2;K_{0}^{*}}}(x)$, dominating WF’s longitudinal distribution,
and the $\mathbf{k}_{\perp}$-dependent part arising from harmonic oscillator
solution for meson in the rest frame. Then, the
$\Psi_{2;K_{0}^{\ast}}^{R}(x,\mathbf{k}_{\perp})$ can be written as:

$\displaystyle\Psi_{2;K_{0}^{\ast}}^{R}(x,\mathbf{k}_{\perp})$
$\displaystyle=A_{2;K_{0}^{\ast}}\varphi_{2;K_{0}^{\ast}}(x)$
$\displaystyle\times\exp\Big{[}-\frac{1}{8\beta_{2;K_{0}^{\ast}}^{2}}\Big{(}\frac{\mathbf{k}_{\perp}^{2}+\hat{m}_{s}^{2}}{x}+\frac{\mathbf{k}_{\perp}^{2}+\hat{m}_{q}^{2}}{\bar{x}}\Big{)}\Big{]}$
(16)

with

$\displaystyle\varphi_{2;K_{0}^{\ast}}(x)$
$\displaystyle=(x\bar{x})^{\alpha_{2;K_{0}^{\ast}}}\Big{[}C_{1}^{3/2}(2x-1)+\hat{B}_{2;K_{0}^{\ast}}C_{2}^{3/2}(2x-1)\Big{]},$
(17)

where ${A_{2;K_{0}^{\ast}}}$ is the normalization constant,
${\beta_{2;K_{0}^{\ast}}}$ is the harmomous parameters that dominates the WF’s
transverse distribution, $C_{n}^{3/2}(2x-1)$ is the Gegenbauer polynomial.
Considering the $SU_{f}(3)$ breaking effect of $K_{0}^{\ast}(1430)$, we
introduce a term proportional to $C_{2}^{3/2}(2x-1)$. We take
$\hat{B}_{2;K_{0}^{\ast}}\simeq-0.24$ in order to make the undetermined model
parameters as few as possible. The value of $\hat{B}_{2;K_{0}^{\ast}}$ is
taken by referring to the ratio of the second and first $\xi$-moments
calculated in Sec. III.1, i.e.,
$\langle\xi^{2}\rangle_{2;K_{0}^{\ast}}/\langle\xi^{1}\rangle_{2;K_{0}^{\ast}}$,
and whose rationality can be judged by the goodness of fit.

There is a relationship between the $K_{0}^{\ast}(1430)$ leading-twist DA and
its WF,

$\displaystyle\phi_{2;K_{0}^{\ast}}(x,\mu)=\int_{|\mathbf{k}_{\perp}|^{2}\leq\mu^{2}}\frac{d^{2}\mathbf{k}_{\perp}}{16\pi^{3}}\Psi_{2;K_{0}^{\ast}}(x,\mathbf{k}_{\perp}).$
(18)

Substituting the WF formula (14) with Eqs. (15), (16) and (17) into (18) and
after integrating over the transverse momentum $\mathbf{k}_{\perp}$, the
$K_{0}^{\ast}(1430)$ leading-twist DA reads

$\displaystyle\phi_{2;K_{0}^{\ast}}(x,\mu)$
$\displaystyle=\frac{A_{2;K_{0}^{\ast}}\beta_{2;K_{0}^{\ast}}\widetilde{m}}{4\sqrt{2}\pi^{3/2}}\sqrt{x\bar{x}}\varphi_{2;K_{0}^{\ast}}(x)$
$\displaystyle\times\exp\Big{[}-\frac{\hat{m}_{q}^{2}x+\hat{m}_{s}^{2}\bar{x}-\widetilde{m}^{2}}{8\beta_{2;K_{0}^{\ast}}^{2}x\bar{x}}\Big{]}$
$\displaystyle\times\Big{\\{}{\rm
Erf}\Big{(}\sqrt{\frac{\widetilde{m}^{2}+\mu^{2}}{8\beta_{2;K_{0}^{\ast}}^{2}x\bar{x}}}\Big{)}-{\rm
Erf}\Big{(}\sqrt{\frac{\widetilde{m}^{2}}{8\beta_{2;K_{0}^{\ast}}^{2}x\bar{x}}}\Big{)}\Big{\\}}$
(19)

with the error function ${\rm Erf}(x)=2\int^{x}_{0}dte^{-t^{2}}/\sqrt{\pi}$.

It needs to be clear that the LCHO models for WF
$\Psi_{2;K_{0}^{\ast}}(x,\mathbf{k}_{\perp})$ and DA
$\phi_{2;K_{0}^{\ast}}(x,\mu)$ established above are the same for
$K_{0}^{\ast}(1430)^{+}$ and $K_{0}^{\ast}(1430)^{0}$ due to the isospin
symmetry between the $u$ and $d$ quarks. The leading-twist WF and DA of
$\overline{K}_{0}^{\ast}(1430)^{0}$ and $K_{0}^{\ast}(1430)^{-}$ can be
obtained by replacing $x$ with $\bar{x}$ in Eqs. (14) and (19).

Now, there are three unknown model parameters such as $A_{2;K_{0}^{\ast}}$,
$\beta_{2;K_{0}^{\ast}}$ and $\alpha_{2;K_{0}^{\ast}}$. In order to
definitively describe the behavior of $K_{0}^{\ast}(1430)$ leading-twist DA
with the LCHO model (19), these three parameters can be determined by fitting
the $\xi$-moments with the least squares method as fitting parameters. For
specific fitting procedure, one can refer to Refs. Zhong:2021epq ;
Zhong:2022ecl .

###  II.3 $B_{s},D_{s}\to K_{0}^{\ast}$ TFFs within LCSRs

In order to uniformly express the derivation process of the LCSRs for
$B_{s},D_{s}\to K_{0}^{\ast}(1430)$ TFFs, we introduce the following vacuum-
to-$K_{0}^{\ast}(1430)$ correlators

$\displaystyle\Pi_{\mu}(p,q)$ $\displaystyle=i\int d^{4}xe^{iq\cdot x}\langle
K_{0}^{\ast}(p)|T\\{\bar{q}_{2}(x)\gamma_{\mu}\gamma_{5}Q(x)$
$\displaystyle\times\bar{Q}(0)i\gamma_{5}q_{1}(0)\\}|0\rangle$
$\displaystyle=F(q^{2},(p+q)^{2})p_{\mu}+\widetilde{F}(p^{2},(p+q)^{2})q_{\mu},$
$\displaystyle\widetilde{\Pi}_{\mu}(p,q)$ $\displaystyle=i\int
d^{4}xe^{iq\cdot x}\langle
K_{0}^{\ast}(p)|T\\{\bar{q}_{2}(x)\sigma_{\mu\nu}\gamma_{5}q^{\nu}Q(x)$
$\displaystyle\times\bar{Q}(0)i\gamma_{5}q_{1}(0)\\}|0\rangle$
$\displaystyle=F^{T}(p^{2},(p+q)^{2})[p_{\mu}q^{2}-q_{\mu}(p\cdot q)].$ (20)

In Eq. (20), the light quark $q_{1}=s,q_{2}=u$ and the heavy quark $Q=b$ is
for $H_{q_{1}}(=B_{s})\to K_{0}^{\ast}(1430)$ decay; the light quark
$q_{1}=s,q_{2}=d$ and the heavy quark $Q=c$ is for $H_{q_{1}}(=D_{s})\to
K_{0}^{\ast}(1430)$ decay, respectively.

We first calculate the correlator (20) in QCD. At $q^{2}\ll m_{Q}^{2}$ and
$(p+q)^{2}\ll m_{Q}^{2}$ with the heavy quark mass $m_{Q}$, the heavy quark
propagating in the correlator is highly virtual and the distances are near the
light-cone Duplancic:2008ix . Thus one can contract the heavy quark fields,
and the following light-cone expansion of the heavy quark propagator,

$\displaystyle\langle 0|Q_{\alpha}^{i}(x)\bar{Q}_{\beta}^{j}(0)|0\rangle$
$\displaystyle\quad\quad=i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik\cdot
x}\Big{[}\delta^{ij}\frac{\not{k}+m_{Q}}{k^{2}-m_{Q}^{2}}+\cdots\Big{]}_{\alpha\beta},$
(21)

enters the correlator. The vacuum-to-$K_{0}^{\ast}(1430)$ matrix element can
be expanded in terms of the $K_{0}^{\ast}(1430)$ light-cone DA’s of growing
twist. That is,

$\displaystyle\langle
K_{0}^{\ast}(p)|\bar{q}_{2\alpha}^{i}(x)q_{1\beta}^{j}(0)|0\rangle$
$\displaystyle=\frac{\delta^{ji}}{12}\bar{f}_{K_{0}^{\ast}}\int^{1}_{0}due^{iup\cdot
x}\Big{\\{}\not{p}\phi_{2;K_{0}^{\ast}}(u,\mu)+m_{K_{0}^{\ast}}\phi_{3;K_{0}^{\ast}}^{s}(u,\mu)$
$\displaystyle-\frac{1}{6}m_{K_{0}^{\ast}}\sigma_{\mu\nu}p^{\mu}x^{\nu}\phi_{3;K_{0}^{\ast}}^{\sigma}(u,\mu)\Big{\\}}_{\beta\alpha}+\cdots$
(22)

with the $K_{0}^{\ast}(1430)$ two-particle twist-3 DA
$\phi_{3;K_{0}^{\ast}}^{\sigma}(u,\mu)$. In which, the Fock components of the
$K_{0}^{\ast}(1430)$ with multiplicities larger than two as well as the twists
higher than 3 are neglected due to that only the free propagator is retained
in Eq. (21). This truncation is reasonable and has to be done because we
almost know nothing about those components and their contributions are usually
small. Then the OPE for the invariant amplitudes $F$, $\widetilde{F}$ and
$F^{T}$ can be obtained as

$\displaystyle F_{\rm qcd}(q^{2},(p+q^{2}))$
$\displaystyle\quad\quad=i\bar{f}_{K_{0}^{\ast}}m_{Q}\int^{1}_{0}\frac{du}{m_{Q}^{2}-(up+q)^{2}}\Big{\\{}\phi_{2;K_{0}^{\ast}}(u,\mu)$
$\displaystyle\quad\quad-\frac{m_{K_{0}^{\ast}}}{m_{Q}}u\phi^{p}_{3;K_{0}^{\ast}}(u,\mu)-\frac{m_{K_{0}^{\ast}}}{6m_{Q}}\Big{[}2+\frac{m_{Q}^{2}+q^{2}-u^{2}p^{2}}{m_{Q}^{2}-(up+q)^{2}}\Big{]}$
$\displaystyle\quad\quad\times\phi_{3;K_{0}^{\ast}}^{\sigma}(u,\mu)\Big{\\}},$
$\displaystyle\widetilde{F}_{\rm qcd}(q^{2},(p+q)^{2})$
$\displaystyle\quad\quad=i\bar{f}_{K_{0}^{\ast}}\int^{1}_{0}\frac{du}{m_{Q}^{2}-(up+q)^{2}}\Big{\\{}-m_{K_{0}^{\ast}}\phi_{3;K_{0}^{\ast}}^{p}(u,\mu)$
$\displaystyle\quad\quad-\frac{m_{K_{0}^{\ast}}}{6}\Big{[}1-\frac{m_{Q}^{2}-q^{2}+u^{2}p^{2}}{m_{Q}^{2}-(up+q)^{2}}\Big{]}\frac{\phi_{3;K_{0}^{\ast}}^{\sigma}(u,\mu)}{u}\Big{\\}},$
$\displaystyle F^{T}_{\rm qcd}(q^{2},(p+q^{2}))$
$\displaystyle\quad\quad=\bar{f}_{K_{0}^{\ast}}\int^{1}_{0}\frac{du}{m_{Q}^{2}-(up+q)^{2}}\Big{\\{}\phi_{2;K_{0}^{\ast}}(u,\mu)$
$\displaystyle\quad\quad-\frac{m_{K_{0}^{\ast}}m_{Q}}{3[m_{Q}^{2}-(up+q)^{2}]}\phi_{3;K_{0}^{\ast}}^{\sigma}(u,\mu)\Big{\\}},$
(23)

respectively.

One can also insert a complete set of hadronic states between the currents in
correlator (20) to obtain the hadronic representations of the invariant
amplitudes. In which, the TFFs $f_{\pm,T}(q^{2})$ enters the correlator (20)
via the hadronic matrix elements for the interpolating currents indicating the
weak transition of $Q$ to $q_{2}$. They can be parameterized in terms of the
TFFs $f_{\pm,T}(q^{2})$ as

$\displaystyle\langle
K_{0}^{\ast}(p)|\bar{q}_{2}\gamma_{\mu}\gamma_{5}Q|H_{q_{1}}(p+q)\rangle$
$\displaystyle\quad\quad\quad\quad=-2if_{+}(q^{2})p_{\mu}-i[f_{+}(q^{2})+f_{-}(q^{2})]q_{\mu},$
$\displaystyle\langle
K_{0}^{\ast}(p)|\bar{q}_{2}\sigma_{\mu\nu}\gamma_{5}q^{\nu}Q|H_{q_{1}}(p+q)\rangle$
$\displaystyle\quad\quad\quad\quad=[2p_{\mu}q^{2}-2q_{\mu}(p\cdot
q)]\frac{-f_{T}(q^{2})}{m_{H_{q_{1}}}+m_{K_{0}^{\ast}}}.$ (24)

Otherwise, the vacuum-to-meson matrix element for the interpolating current
representing the $H_{q_{1}}$ channel can be given by

$\displaystyle\langle
H_{q_{1}}|\bar{Q}i\gamma_{5}q_{1}|0\rangle=\frac{m_{H_{q_{1}}}^{2}f_{H_{q_{1}}}}{m_{Q}+m_{q_{1}}}$
(25)

with the heavy meson mass $m_{H_{q_{1}}}$ and decay constant $f_{H_{q_{1}}}$.
Then the hadronic representations of the invariant amplitudes $F$,
$\widetilde{F}$ and $F^{T}$ can be written as

$\displaystyle F_{\rm had}(p^{2},(p+q)^{2})$
$\displaystyle\quad\quad=\frac{-2im_{H_{q_{1}}}^{2}f_{H_{q_{1}}}f_{+}(q^{2})}{(m_{Q}+m_{q_{1}})[m_{H_{q_{1}}}^{2}-(p+q)^{2}]}+\cdots,$
$\displaystyle\widetilde{F}_{\rm had}(p^{2},(p+q)^{2})$
$\displaystyle\quad\quad=\frac{-im_{H_{q_{1}}}^{2}f_{H_{q_{1}}}[f_{+}(q^{2})+f_{-}(q^{2})]}{(m_{Q}+m_{q_{1}})[m_{H_{q_{1}}}^{2}-(p+q)^{2}]}+\cdots,$
$\displaystyle F_{\rm had}^{T}(p^{2},(p+q)^{2})$
$\displaystyle\quad\quad=\frac{-2m_{H_{q_{1}}}^{2}f_{H_{q_{1}}}f_{T}(q^{2})}{(m_{Q}+m_{q_{1}})(m_{H_{q_{1}}}+m_{K_{0}^{\ast}})[m_{H_{q_{1}}}^{2}-(p+q)^{2}]}$
$\displaystyle\quad\quad+\cdots,$ (26)

respectively. In Eq. (26), the ground state heavy meson contributions have
been isolated, and the ellipses indicate the contributions from the excited
states, the continuum states and possible subtraction terms.

Without losing generality, we take the invariant amplitude
$F(q^{2},(p+q)^{2})$ as an example to illustrate the subsequent calculation
procedure. One can write a general dispersion relation for
$F(q^{2},(p+q)^{2})$ and further apply the Borel transformation with respect
to the momentum squared $(p+q)^{2}$ of the heavy meson Belyaev:1993wp ,

$\displaystyle
F(q^{2},M^{2})=\int^{\infty}_{t_{min}}\rho(q^{2},s)e^{-s/M^{2}}ds$ (27)

with $t_{min}=(m_{Q}+m_{q_{1}})^{2}$. In which, the spectral density is given
by

$\displaystyle\rho(q^{2},s)$ $\displaystyle=\frac{1}{\pi}{\rm Im}F_{\rm
had}(q^{2},s)$
$\displaystyle=\delta(s-m_{H_{q_{1}}}^{2})\frac{-2im_{H_{q_{1}}}^{2}f_{H_{q_{1}}}f_{+}(q^{2})}{m_{Q}+m_{q_{1}}}$
$\displaystyle+\frac{1}{\pi}{\rm Im}F_{\rm
qcd}(q^{2},s)\theta(s-s_{H_{q_{1}}}),$ (28)

where the contributions of the excited states and continuum states in $F_{\rm
had}(q^{2},s)$ have been parameterized as ${\rm Im}F_{\rm qcd}(q^{2},s)/\pi$
and been delimited by the effective threshold parameter $s_{H_{q_{1}}}$ with
the quark-hadronic duality approximation, and the possible subtractions will
be got rid of due to Borel transformation in Eq. (27). Substituting Eq. (28)
into Eq. (27), one can get

$\displaystyle F(q^{2},M^{2})$
$\displaystyle=\frac{-2im_{H_{q_{1}}}^{2}f_{H_{q_{1}}}f_{+}(q^{2})}{m_{Q}+m_{q_{1}}}e^{-m_{H_{q_{1}}}^{2}/M^{2}}$
$\displaystyle+\frac{1}{\pi}\int^{\infty}_{s_{H_{q_{1}}}}{\rm Im}F_{\rm
qcd}(q^{2},s)e^{-s/M^{2}}ds.$ (29)

On the other hand, the invariant amplitude after Borel transformation can also
be written as Belyaev:1993wp

$\displaystyle F(q^{2},M^{2})$
$\displaystyle=\frac{1}{\pi}\int^{\infty}_{t_{min}}{\rm Im}F_{\rm
qcd}(q^{2},s)e^{-s/M^{2}}ds.$ (30)

Finally, equating Eqs. (29) with (30), the LCSR of TFF $f_{+}(q^{2})$ can be
obtained as

$\displaystyle f_{+}(q^{2})$
$\displaystyle=\frac{(m_{Q}+m_{q_{1}})\bar{f}_{K_{0}^{\ast}}e^{m_{H_{q_{1}}}^{2}/M^{2}}}{2m_{H_{q_{1}}}^{2}f_{H_{q_{1}}}}$
$\displaystyle\times\int^{\widetilde{u}_{0}}_{u_{0}}due^{-(m_{Q}^{2}-\bar{u}q^{2}+u\bar{u}m_{K_{0}^{\ast}}^{2})/(uM^{2})}$
$\displaystyle\times\Big{\\{}-m_{Q}\frac{\phi_{2;K_{0}^{\ast}}(u,\mu)}{u}+m_{K_{0}^{\ast}}\phi_{3;K_{0}^{\ast}}^{p}(u,\mu)$
$\displaystyle+m_{K_{0}^{\ast}}\Big{[}\frac{2}{u}+\frac{4um_{Q}^{2}m_{K_{0}^{\ast}}^{2}}{(m_{Q}^{2}-q^{2}+u^{2}m_{K_{0}^{\ast}}^{2})^{2}}$
$\displaystyle-\frac{m_{Q}^{2}+q^{2}-u^{2}m_{K_{0}^{\ast}}^{2}}{m_{Q}^{2}-q^{2}+u^{2}m_{K_{0}^{\ast}}^{2}}\frac{d}{du}\Big{]}\frac{\phi_{3;K_{0}^{\ast}}^{\sigma}(u,\mu)}{6}\Big{\\}}.$
(31)

Similarly,

$\displaystyle f_{+}(q^{2})$
$\displaystyle+f_{-}(q^{2})=\frac{(m_{Q}+m_{q_{1}})\bar{f}_{K_{0}^{\ast}}m_{K_{0}^{\ast}}e^{m_{H_{q_{1}}}^{2}/M^{2}}}{m_{H_{q_{1}}}^{2}f_{H_{q_{1}}}}$
$\displaystyle\times\int^{\widetilde{u}_{0}}_{u_{0}}due^{-(m_{Q}^{2}-\bar{u}q^{2}+u\bar{u}m_{K_{0}^{\ast}}^{2})/(uM^{2})}$
$\displaystyle\times\Big{[}\frac{\phi_{3;K_{0}^{\ast}}^{p}(u,\mu)}{u}+\frac{1}{u}\frac{d}{du}\frac{\phi_{3;K_{0}^{\ast}}^{\sigma}(u,\mu)}{6}\Big{]},$
(32) $\displaystyle f_{T}(q^{2})$
$\displaystyle=\frac{(m_{H_{q_{1}}}+m_{K_{0}^{\ast}})(m_{Q}+m_{q_{1}})\bar{f}_{K_{0}^{\ast}}e^{m_{H_{q_{1}}}^{2}/M^{2}}}{m_{H_{q_{1}}}^{2}f_{H_{q_{1}}}}$
$\displaystyle\times\int^{\widetilde{u}_{0}}_{u_{0}}due^{-(m_{Q}^{2}-\bar{u}q^{2}+u\bar{u}m_{K_{0}^{\ast}}^{2})/(uM^{2})}$
$\displaystyle\times\Big{\\{}-\frac{\phi_{2;K_{0}^{\ast}}(u,\mu)}{2u}+\frac{m_{Q}m_{K_{0}^{\ast}}}{m_{Q}^{2}-q^{2}+u^{2}m_{K_{0}^{\ast}}^{2}}$
$\displaystyle\times\Big{[}\frac{2um_{K_{0}^{\ast}}^{2}}{m_{Q}^{2}-q^{2}+u^{2}m_{K_{0}^{\ast}}^{2}}-\frac{d}{du}\Big{]}\frac{\phi_{3;K_{0}^{\ast}}^{\sigma}(u,\mu)}{6}\Big{\\}}$
(33)

with

$\displaystyle u_{0}$
$\displaystyle=\Big{[}\sqrt{(q^{2}-s_{H_{q_{1}}}+m_{K_{0}^{\ast}}^{2})^{2}+4m_{K_{0}^{\ast}}^{2}(m_{Q}^{2}-q^{2})}$
$\displaystyle+q^{2}-s_{H_{q_{1}}}+m_{K_{0}^{\ast}}^{2}\Big{]}/(2m_{K_{0}^{\ast}}^{2}),$
(34) $\displaystyle\widetilde{u}_{0}$
$\displaystyle=\Big{[}\sqrt{(q^{2}-t_{min}+m_{K_{0}^{\ast}}^{2})^{2}+4m_{K_{0}^{\ast}}^{2}(m_{Q}^{2}-q^{2})}$
$\displaystyle+q^{2}-t_{min}+m_{K_{0}^{\ast}}^{2}\Big{]}/(2m_{K_{0}^{\ast}}^{2}).$
(35)

In particular, the upper limit of the integral of variable $u$ in LCSRs (31),
(32) and (33) is $\widetilde{u}_{0}$ instead of 1, because the lower limit of
the integral variable $s$ in the dispersion relationship (27), $t_{min}$, is
larger than, but not equal to, $m_{Q}^{2}$.

##  III NUMERICAL ANALYSIS

###  III.1 $\xi$-moments and behavior of $\phi_{2;K_{0}^{\ast}}(u,\mu)$

Now we can calculate the values of the $\xi$-moments of $K_{0}^{\ast}(1430)$
leading-twist DA. In calculation, we take the mass of $K_{0}^{\ast}(1430)$ as
$m_{K_{0}^{\ast}}=1.425_{-0.050}^{+0.050}$ $\rm{GeV}$, the $u$ and $s$ current
quark mass are adopted as $m_{u}=2.16_{-0.26}^{+0.49}$ $\rm{MeV}$ and
$m_{s}=93_{-5}^{+11}$ $\rm{MeV}$ at scale $\mu=2$ $\rm{GeV}$, respectively
Workman:2022ynf . For the decay constant of $K_{0}^{\ast}(1430)$, we take the
value from QCD sum rule calculation, that is,
$f_{K_{0}^{\ast}}=427_{-85}^{+85}$ $\rm{MeV}$ at $\mu\sim 1\ {\rm GeV}$
Du:2004ki . The values of the scale dependent vacuum condensates are:
$\langle\bar{u}u\rangle=(-2.417_{-0.114}^{+0.227})\times
10^{-2}~{}\rm{GeV}^{3}$, $\langle\bar{s}s\rangle=\kappa\langle\bar{u}u\rangle$
with $\kappa=0.74\pm 0.03$, $\langle g_{s}\bar{u}\sigma
TGu\rangle=(-1.934^{+0.188}_{-0.103})\times 10^{-2}~{}\rm{GeV}^{5}$, $\langle
g_{s}\bar{s}\sigma TGs\rangle=\kappa\langle g_{s}\bar{u}\sigma TGu\rangle$,
$\langle g_{s}\bar{u}u\rangle^{2}=(2.082_{-0.679}^{+0.734})\times
10^{-3}~{}{\rm GeV}^{6}$, $\langle g_{s}\bar{s}s\rangle^{2}=\kappa^{2}\langle
g_{s}\bar{u}u\rangle^{2}$ and $\langle
g_{s}^{2}\bar{u}u\rangle^{2}=(7.420_{-2.483}^{+2.614})\times 10^{-3}~{}{\rm
GeV}^{6}$ at $\mu=2~{}{\rm GeV}$; the values of the scale independent gluon
condensates are: $\langle\alpha_{s}G^{2}\rangle=0.038_{-0.011}^{+0.011}~{}{\rm
GeV}^{4}$ and $\langle g_{s}^{3}fG^{3}\rangle\simeq 0.045~{}{\rm GeV}^{6}$
Zhong:2021epq ; Zhong:2014jla ; Colangelo:2000dp ; Narison:2014ska ;
Narison:2014wqa . In addition, we take the scale $\mu=M$ in sum rules (13) as
usual, and the continuum threshold $s_{K_{0}^{\ast}}\simeq 8~{}{\rm GeV}^{2}$
by requiring that there is a reasonable Borel window to normalize
$\langle\xi^{0}_{p}\rangle_{3;K_{0}^{\ast}}$ in Eq. (11) as suggested in Ref.
Zhong:2021epq . The scale evolutions of $\xi$-moments are realized by that of
the Gegenbauer moments via the relationship between both of them. For the
scale evolutions of Gegenbauer moments, one can refer to Ref. Cheng:2005nb .

Substituting the above inputs into Eq. (13), the $\xi$-moments
$\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}$, continuum state contributions and
dimension-six term contributions versus Borel parameter $M^{2}$ can be
obtained. We will evaluate the values of the first ten $\xi$-moments in this
work. In order to estimate these values, one should determine the appropriate
Borel windows. We require the continuum state contributions are not more than
$25\%$, $35\%$, $45\%$, $60\%$, $80\%$ for odd $\xi$-moments
$\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}(n=1,3,5,7,9)$, and are less than
$20\%$, $25\%$, $30\%$, $35\%$, $35\%$ for even $\xi$-moments
$\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}(n=2,4,6,8,10)$, respectively, to get
the upper limits of the corresponding Borel windows. In which, the continuum
state contributions for $\langle\xi^{7}\rangle_{2;K_{0}^{\ast}}$ and
$\langle\xi^{9}\rangle_{2;K_{0}^{\ast}}$ are large. The reason is that the
seventh and ninth $\xi$-moments are very close to zero, rather than that the
continuum contributions play a dominant role in
$\langle\xi^{7}\rangle_{2;K_{0}^{\ast}}$ and
$\langle\xi^{9}\rangle_{2;K_{0}^{\ast}}$. On the other hand, the dimension-six
term contributions for all the first ten $\xi$-moments are far less than $5\%$
in a very wide Borel parameter region, then the basic criteria that require
dimension-six term contributions to be as small as possible are automatically
satisfied. Reasonably, we directly fix the lengths of all Borel windows as
$1~{}{\rm GeV}^{2}$.

Figure 1: $K_{0}^{\ast}(1430)$ leading-twist DA $\xi$-moments $\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}(n=1,2,\cdots,10)$ versus the Borel parameter $M^{2}$, where the shaded bands indicate the Borel windows, and all input parameters are set to be their central values. Table 1: Our predictions for the first ten $\xi$-moments $\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}(n=1,2,\cdots,10)$ of $K_{0}^{\ast}(1430)$ leading-twist DA with the scale $\mu=1,1.4$ and $3~{}{\rm GeV}$. $\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}$ |  $1$ $\rm{GeV}$ |  $1.4$ $\rm{GeV}$ |  $3$ $\rm{GeV}$
---|---|---|---
$\langle\xi^{1}\rangle_{2;K_{0}^{\ast}}$ | $-0.235_{-0.069}^{+0.059}$ | $-0.190_{-0.056}^{+0.047}$ | $-0.141_{-0.041}^{+0.035}$
$\langle\xi^{2}\rangle_{2;K_{0}^{\ast}}$ | $~{}~{}0.056_{-0.011}^{+0.015}$ | $~{}~{}0.043_{-0.009}^{+0.012}$ | $~{}~{}0.030_{-0.006}^{+0.008}$
$\langle\xi^{3}\rangle_{2;K_{0}^{\ast}}$ | $-0.145_{-0.037}^{+0.028}$ | $-0.114_{-0.029}^{+0.023}$ | $-0.081_{-0.021}^{+0.017}$
$\langle\xi^{4}\rangle_{2;K_{0}^{\ast}}$ | $~{}~{}0.031_{-0.006}^{+0.008}$ | $~{}~{}0.024_{-0.005}^{+0.006}$ | $~{}~{}0.017_{-0.003}^{+0.004}$
$\langle\xi^{5}\rangle_{2;K_{0}^{\ast}}$ | $-0.081_{-0.023}^{+0.019}$ | $-0.064_{-0.018}^{+0.015}$ | $-0.046_{-0.013}^{+0.011}$
$\langle\xi^{6}\rangle_{2;K_{0}^{\ast}}$ | $~{}~{}0.017_{-0.004}^{+0.005}$ | $~{}~{}0.013_{-0.003}^{+0.004}$ | $~{}~{}0.010_{-0.002}^{+0.003}$
$\langle\xi^{7}\rangle_{2;K_{0}^{\ast}}$ | $-0.042_{-0.019}^{+0.018}$ | $-0.035_{-0.014}^{+0.013}$ | $-0.026_{-0.010}^{+0.009}$
$\langle\xi^{8}\rangle_{2;K_{0}^{\ast}}$ | $~{}~{}0.013_{-0.003}^{+0.004}$ | $~{}~{}0.010_{-0.003}^{+0.003}$ | $~{}~{}0.007_{-0.002}^{+0.002}$
$\langle\xi^{9}\rangle_{2;K_{0}^{\ast}}$ | $-0.029_{-0.017}^{+0.016}$ | $-0.024_{-0.013}^{+0.011}$ | $-0.018_{-0.008}^{+0.008}$
$\langle\xi^{10}\rangle_{2;K_{0}^{\ast}}$ | $~{}~{}0.007_{-0.003}^{+0.003}$ | $~{}~{}0.006_{-0.002}^{+0.002}$ | $~{}~{}0.004_{-0.001}^{+0.002}$
Table 2: Our predictions for the first three $\xi$-moments and Gegenbauer moments of $K_{0}^{\ast}(1430)$ leading-twist DA at scale $\mu=1~{}{\rm GeV}$, compared to other theoretical predictions. |  $\langle\xi^{1}\rangle_{2;K_{0}^{\ast}}$ |  $\langle\xi^{2}\rangle_{2;K_{0}^{\ast}}$ |  $\langle\xi^{3}\rangle_{2;K_{0}^{\ast}}$ |  $a_{1}^{2;K_{0}^{\ast}}$ |  $a_{2}^{2;K_{0}^{\ast}}$ |  $a_{3}^{2;K_{0}^{\ast}}$
---|---|---|---|---|---|---
BFTSR (This Work) | $-0.235_{-0.069}^{+0.059}$ | $0.056_{-0.011}^{+0.015}$ | $-0.145_{-0.037}^{+0.028}$ | $-0.392_{-0.115}^{+0.096}$ | $0.164_{-0.033}^{+0.045}$ | $-0.231_{-0.037}^{+0.018}$
QCD SR Cheng:2005nb  | $-0.35^{+0.08}_{-0.08}$ | $-$ | $-0.23^{+0.06}_{-0.06}$ | $-0.57^{+0.13}_{-0.13}$ | $-$ | $-0.42^{+0.22}_{-0.22}$
LF Holographic Chen:2021oul  | $-0.078^{+0.018}_{-0.018}$ | $-0.010^{+0.001}_{-0.001}$ | $\sim-0.034$ | $-0.130^{+0.030}_{-0.030}$ | $-0.030^{+0.002}_{-0.002}$ | $-0.005^{+0.001}_{-0.001}$

The first ten $\xi$-moments of $K_{0}^{\ast}(1430)$ leading-twist DA versus
the Borel parameter and the corresponding Borel windows are shown in Fig. 1,
where all inputs are taken to be their central values. By taking all error
sources into account, the values of the first ten $\xi$-moments at the scale
$\mu=1~{}{\rm GeV}$, $1.4~{}{\rm GeV}$ and $3~{}{\rm GeV}$ can be obtained,
and which are exhibited in Table 1. In Table 1, the values at $\mu=1.4~{}{\rm
GeV}$ and $3~{}{\rm GeV}$ will be used for subsequent calculations of the
$D_{s}\to K_{0}^{\ast}(1430)$ and $B_{s}\to K_{0}^{\ast}(1430)$ TFFs,
respectively. Our values for the first three $\xi$-moments and Gegenbauer
moments at scale $\mu=1~{}{\rm GeV}$ are also exhibited in Table 2, the other
theoretical predictions such as by traditional QCDSRs Cheng:2005nb  and LF
approach Chen:2021oul  are displayed for comparison. By only adopting the
central values, we can further get

$\displaystyle\frac{\langle\xi^{2}\rangle_{2;K_{0}^{\ast}}}{\langle\xi^{1}\rangle_{2;K_{0}^{\ast}}}=-0.238,\quad\frac{a_{2}^{2;K_{0}^{\ast}}}{a_{1}^{2;K_{0}^{\ast}}}=-0.418,$
(36)

at $\mu=1~{}{\rm GeV}$. This implies that there is obvious $SU_{f}(3)$
breaking effect.

Table 3: Fitted LCHO model parameters and the corresponding goodness of fit for the $K_{0}^{\ast}(1430)$ leading-twist DA with the scale $\mu=1,1.4$ and $3~{}{\rm GeV}$, respectively. $\mu[\rm GeV]$ | $A_{2;K_{0}^{\ast}}[{\rm GeV}^{-1}]$ | $\alpha_{2;K_{0}^{\ast}}$ | $\beta_{2;K_{0}^{\ast}}[{\rm GeV}]$ |  $\chi_{\rm min}^{2}$ |  $P_{\chi_{\rm min}^{2}}$
---|---|---|---|---|---
$1$ | $-1015$ | $0.903$ | $1.593$ | $3.12581$ | $0.873125$
$1.4$ | $-477$ | $0.774$ | $1.494$ | $2.84837$ | $0.898666$
$3$ | $-186$ | $0.601$ | $1.498$ | $2.80795$ | $0.902182$
Figure 2: Our prediction for the behavior of $K_{0}^{\ast}(1430)$ leading-
twist DA at scale $\mu=1~{}{\rm GeV}$. We also present other predictions from
the traditional QCDSRs Cheng:2005nb  and LF approach Chen:2021oul  for
comparison.

Then, we can determine the behavior of $K_{0}^{\ast}(1430)$ leading-twist DA
by fitting the values exhibited in Table 2 with the LCHO model shown in Eq.
(19) via the least squares method. Following the fitting procedure introduced
in detailed in Refs. Zhong:2021epq ; Zhong:2022ecl , the model parameters and
the corresponding goodness of fit are determined and displayed in Table 3. The
curve of our LCHO model for $K_{0}^{\ast}(1430)$ leading-twist DA at
$\mu=1~{}{\rm GeV}$ are shown in Fig. 2. As a comparison, the curves from the
traditional QCDSRs Cheng:2005nb  and LF approach Chen:2021oul  are also shown
in this figure. In Refs Cheng:2005nb ; Chen:2021oul , the TF model for
$K_{0}^{\ast}(1430)$ leading-twist DA is adopted, which is,

$\displaystyle\phi_{2;K_{0}^{\ast}}^{{\rm
TF},\mathcal{N}}(x,\mu)=6x(1-x)\sum_{n=1}^{\mathcal{N}}a_{n}^{K_{0}^{\ast}}C_{n}^{3/2}(2x-1).$
(37)

Here, in order to be consistent with our matrix element definition of DA,
i.e., Eq. (1), so as to facilitate comparison, we have got rid off the decay
constant $\bar{f}_{K_{0}^{\ast}}$ in “$\Phi_{K_{0}^{\ast}}(x)$” in Ref.
Cheng:2005nb , and the constant $\bar{\mu}_{K_{0}^{\ast}}$ in
“$\Phi_{K_{0}^{\ast}}(x)$” in Ref. Chen:2021oul , respectively. In Fig. 2, the
TF models corresponding to the truncations up to $\mathcal{N}=1$ (I) and $3$
(II), respectively, are shown. From Fig. 2, one can also find that, our DA
$\phi_{2;K_{0}^{\ast}}(x,1~{}{\rm GeV})$ takes the maximum value at $x=0.16$
($\phi_{2;K_{0}^{\ast}}(x=0.16,1~{}{\rm GeV})=1.05$), the minimum value at
$x=0.73$ ($\phi_{2;K_{0}^{\ast}}(x=0.73,1~{}{\rm GeV})=-0.68$), while the zero
point is taken at $x\simeq 0.44$, respectively, due to the $SU_{f}(3)$
breaking effect of $K_{0}^{\ast}(1430)$.

###  III.2 TFFs and branching fractions for the semileptonic $B_{s},D_{s}\to
K_{0}^{\ast}(1430)$ decays

In order to calculate the TFFs of the semileptonic $B_{s},D_{s}\to
K_{0}^{\ast}(1430)$ decays with Eqs. (31), (32) and (33), we adopt the heavy
meson masses $m_{\overline{B}_{s}^{0}}=5366.92(10)~{}{\rm MeV}$ and
$m_{D_{s}^{+}}=1968.35(7)~{}{\rm MeV}$, decay constants
$f_{B_{s}}=266(19)~{}{\rm MeV}$ and $f_{D_{s}}=256.0(42)~{}{\rm MeV}$, the
$\overline{MS}$ heavy $b$ and $c$ quark masses
$\bar{m}_{b}(\bar{m}_{b})=4.18_{-0.03}^{+0.04}~{}{\rm GeV}$ and
$\bar{m}_{c}(\bar{m}_{c})=1.27\pm 0.02~{}{\rm GeV}$ Workman:2022ynf ,
respectively. For the effective threshold parameters, we take $s_{B_{s}}=38\pm
1~{}{\rm GeV}$, $s_{D_{s}}\simeq 6.5\pm 0.5~{}{\rm GeV}$. The Borel windows
are taken as $13~{}{\rm GeV}^{2}\leq M^{2}\leq 17~{}{\rm GeV}^{2}$. The
$K_{0}^{\ast}(1430)$ twist-3 DA $\phi_{3;K_{0}^{\ast}}^{s}(x,\mu)$ and
$\phi_{3;K_{0}^{\ast}}^{\sigma}(x,\mu)$ adopted in this work are the TF model
obtained in Ref. Han:2013zg  with QCDSRs in the framework of BFT, which read,

$\displaystyle\phi_{3;K_{0}^{\ast}}^{p}(x,\mu)$
$\displaystyle=1+a_{1,p}^{3;K_{0}^{\ast}}(\mu)C_{1}^{1/2}(2x-1)$
$\displaystyle+a_{2,p}^{3;K_{0}^{\ast}}(\mu)C_{2}^{1/2}(2x-1),$
$\displaystyle\phi_{3;K_{0}^{\ast}}^{\sigma}(x,\mu)$
$\displaystyle=6x\bar{x}[1+a_{1,\sigma}^{3;K_{0}^{\ast}}(\mu)C_{1}^{3/2}(2x-1)$
$\displaystyle+a_{2,\sigma}^{3;K_{0}^{\ast}}(\mu)C_{2}^{3/2}(2x-1)],$ (38)

with $a_{1,p}^{3;K_{0}^{\ast}}(\mu)=0.0126\pm 0.0017$,
$a_{2,p}^{3;K_{0}^{\ast}}(\mu)=0.187\pm 0.024$,
$a_{1,\sigma}^{3;K_{0}^{\ast}}(\mu)=0.0342\pm 0.0126$ and
$a_{2,\sigma}^{3;K_{0}^{\ast}}(\mu)=0.0235\pm 0.0045$ at $\mu=1~{}{\rm GeV}$.

Table 4: TFFs of the semileptonic $B_{s}\to K_{0}^{\ast}(1430)$ decays at large recoil point. Method | $f_{+}^{B_{s}\to K_{0}^{\ast}}(0)$ | $f_{-}^{B_{s}\to K_{0}^{\ast}}(0)$ | $f_{T}^{B_{s}\to K_{0}^{\ast}}(0)$
---|---|---|---
This work | $0.39_{-0.08}^{+0.08}$ | $-0.24_{-0.05}^{+0.05}$ | $0.44_{-0.10}^{+0.09}$
RQM Faustov:2013ima  | $0.27^{+0.03}_{-0.03}$ | $-0.62^{+0.06}_{-0.06}$ |
pQCD Li:2008tk  | $0.56^{+0.16}_{-0.13}$ | $-$ | $0.72^{+0.22}_{-0.17}$
pQCD Zhang:2010af  | $0.56^{+0.07}_{-0.09}$ | $-$ | $-$
pQCD Chen:2021oul  | $0.28^{+0.02}_{-0.02}$ | $-$ | $-$
QCDSR Yang:2005bv  | $0.24^{+0.10}_{-0.10}$ | $-$ | $-$
QCDSR Ghahramany:2009zz  | $0.25^{+0.05}_{-0.05}$ | $-0.17^{+0.04}_{-0.04}$ | $0.21^{+0.04}_{-0.04}$
LCSR Khosravi:2022fzo (I) | $0.28^{+0.11}_{-0.09}$ | $-0.23^{+0.09}_{-0.19}$ | $0.32^{+0.13}_{-0.11}$
LCSR Khosravi:2022fzo (II) | $0.41^{+0.14}_{-0.12}$ | $-0.37^{+0.11}_{-0.14}$ | $0.50^{+0.18}_{-0.14}$
LCSR Khosravi:2022fzo (III) | $0.51^{+0.16}_{-0.12}$ | $-0.51^{+0.12}_{-0.16}$ | $0.64^{+0.22}_{-0.15}$
LCSR Wang:2008da  | $0.42^{+0.13}_{-0.07}$ | $-0.35$ | $0.52^{+0.18}_{-0.08}$
LCSR Wang:2014upa ; Wang:2014vra  | $0.548^{+0.062}_{-0.062}$ | $-0.240^{+0.058}_{-0.058}$ | $0.575^{+0.098}_{-0.098}$
LCSR Sun:2010nv  | $0.44$ | $-0.44$ | $-$
LCSR Han:2013zg  | $0.39^{+0.04}_{-0.04}$ | $-0.25^{+0.05}_{-0.05}$ | $0.41^{+0.04}_{-0.04}$
Table 5: TFFs of the semileptonic $D_{s}\to K_{0}^{\ast}(1430)$ decays at large recoil point. Method | $f_{+}^{D_{s}\to K_{0}^{\ast}}(0)$ | $f_{-}^{D_{s}\to K_{0}^{\ast}}(0)$ | $f_{T}^{D_{s}\to K_{0}^{\ast}}(0)$
---|---|---|---
This work | $0.67_{-0.14}^{+0.13}$ | $0.20_{-0.10}^{+0.11}$ | $0.63_{-0.14}^{+0.15}$
QCDSR Yang:2005bv  | $0.51^{+0.20}_{-0.20}$ | $-$ | $-$
Fit to Data Cheng:2002ai  | $1.02^{+0.07}_{-0.07}$ | $-$ |

With the above inputs and the $K_{0}^{\ast}(1430)$ leading-twist DA determined
in previous subsection, the dependence of the TFFs for semileptonic
$B_{s},D_{s}\to K_{0}^{\ast}(1430)$ decays on $q^{2}$ in allowable $q^{2}$
regions of LCSRs can be obtained. In particular, the values of those TFFs at
the large recoil point are exhibited in Table 4 and 5. As a comparison, the
corresponding values estimated with RQM Faustov:2013ima , pQCD Li:2008tk ;
Zhang:2010af ; Chen:2021oul , three-point QCDSR Yang:2005bv ;
Ghahramany:2009zz , LCSR Khosravi:2022fzo ; Wang:2008da ; Wang:2014upa ;
Wang:2014vra ; Sun:2010nv ; Han:2013zg , etc., are also shown in Table 4 and
5.

Table 6: Extrapolation parameters for the semileptonic $B_{s},D_{s}\to K_{0}^{\ast}(1430)$ decays. |  | $f_{+}^{B_{s}\to K_{0}^{\ast}}(q^{2})$ | $f_{-}^{B_{s}\to K_{0}^{\ast}}(q^{2})$ | $f_{T}^{B_{s}\to K_{0}^{\ast}}(q^{2})$ | $f_{+}^{D_{s}\to K_{0}^{\ast}}(q^{2})$ | $f_{-}^{D_{s}\to K_{0}^{\ast}}(q^{2})$ | $f_{T}^{D_{s}\to K_{0}^{\ast}}(q^{2})$
---|---|---|---|---|---|---|---
upper limit | $f_{i}(0)$ | $0.47$ | $-0.19$ | $0.53$ | $0.80$ | $0.31$ | $0.78$
| $a_{i}$ | $1.41$ | $1.19$ | $1.36$ | $0.77$ | $5.60$ | $-0.11$
| $b_{i}$ | $0.71$ | $1.59$ | $0.84$ | $0.45$ | $14.22$ | $2.37$
central value | $f_{i}(0)$ | $0.39$ | $-0.24$ | $0.44$ | $0.67$ | $0.20$ | $0.63$
| $a_{i}$ | $1.40$ | $1.19$ | $1.36$ | $0.76$ | $6.62$ | $-0.33$
| $b_{i}$ | $0.72$ | $1.45$ | $0.86$ | $0.51$ | $21.27$ | $3.65$
lower limit | $f_{i}(0)$ | $0.31$ | $-0.29$ | $0.34$ | $0.53$ | $0.10$ | $0.49$
| $a_{i}$ | $1.41$ | $0.18$ | $1.36$ | $0.77$ | $9.47$ | $-0.62$
| $b_{i}$ | $0.73$ | $1.38$ | $0.88$ | $0.60$ | $45.94$ | $5.47$

Figure 3: Behaviors of the TFFs for the semileptonic $B_{s},D_{s}\to
K_{0}^{\ast}(1430)$ decays in whole $q^{2}$ region. The solid lines and shaded
bands indicate the central values and uncertainties of our predictions,
respectively. The dark parts and light parts in shaded bands are for the
direct LCSR calculations and extrapolation results, respectively. We also
shown other theoretical predictions such as RQM Faustov:2013ima , pQCD
Li:2008tk ; Chen:2021oul , QCDSR Yang:2005bv ; Ghahramany:2009zz , LCSR
Khosravi:2022fzo ; Wang:2008da ; Wang:2014vra  for comparison.

In order to calculate the branching fractions of the semileptonic decays
$B_{s},D_{s}\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$, one should extrapolate the
results with LCSRs to the whole $q^{2}$ region, i.e.,
$q^{2}\in[0,(m_{H_{q_{1}}}-m_{K_{0}^{\ast}})^{2}$]. In this work, we adopt the
usual pole model parametrization

$\displaystyle
f_{i}(q^{2})=\frac{f_{i}(0)}{1-a_{i}(q^{2}/m_{H_{q_{1}}}^{2})+b_{i}(q^{2}/m_{H_{q_{1}}}^{2})^{2}},$
(39)

with $i=+,-,T$. The values of the extrapolation parameters $a_{i}$ and $b_{i}$
corresponding to the $B_{s},D_{s}\to K_{0}^{\ast}(1430)$ TFFs are exhibited in
Table 6. The behaviors of the $B_{s},D_{s}\to K_{0}^{\ast}(1430)$ TFFs in
whole $q^{2}$ region can be determined, and which are shown in Fig 3. In Fig
3, the solid lines are our central values and the shaded bands stand for the
uncertainties. In particular, the dark parts are for the LCSR predictions, and
the light parts are for the extrapolation results, respectively. Otherwise,
the corresponding results calculated with RQM Faustov:2013ima , pQCD Li:2008tk
; Chen:2021oul , QCDSR Yang:2005bv ; Ghahramany:2009zz , LCSR Khosravi:2022fzo
; Wang:2008da ; Wang:2014vra  are also shown in Fig 3 for comparison.

Figure 4: Differential decay ratios of the semileptonic decays $\overline{B}_{s}^{0}\to K_{0}^{\ast}(1430)^{+}\ell^{-}\bar{\nu}_{\ell}$ with $\ell=e,\tau$ and $D_{s}^{+}\to K_{0}^{\ast}(1430)^{0}\ell^{+}\nu_{\ell}$ with $\ell=e,\mu$, respectively. The solid and dashed lines are for the central values and the shaded bands are for the corresponding errors. Table 7: Branching fractions $(\times 10^{4})$ of the semileptonic decays $\overline{B}_{s}^{0}\to K_{0}^{\ast}(1430)^{+}\ell^{-}\bar{\nu}_{\ell}$ with $\ell=e,\mu$ and $\tau$, respectvely. |  $\mathcal{B}(\overline{B}_{s}^{0}\to K_{0}^{\ast}(1430)^{+}e^{-}\bar{\nu}_{e})$ |  $\mathcal{B}(\overline{B}_{s}^{0}\to K_{0}^{\ast}(1430)^{+}\mu^{-}\bar{\nu}_{\mu})$ |  $\mathcal{B}(\overline{B}_{s}^{0}\to K_{0}^{\ast}(1430)^{+}\tau^{-}\bar{\nu}_{\tau})$
---|---|---|---
This work |  $1.23_{-0.55}^{+0.80}$ |  $1.23_{-0.47}^{+0.59}$ | $0.56_{-0.25}^{+0.34}$
Ref. Faustov:2013ima  |  $0.71^{+0.14}_{-0.14}$ |  $0.71^{+0.14}_{-0.14}$ |  $0.21^{+0.04}_{-0.04}$
Ref. Li:2008tk  |  $2.45^{+1.77}_{-1.05}$ |  $2.45^{+1.77}_{-1.05}$ |  $1.09^{+0.82}_{-0.47}$
Ref. Khosravi:2022fzo (I) |  $-$ |  $0.99^{+0.89}_{-0.37}$ | $0.49^{+0.33}_{-0.17}$
Ref. Khosravi:2022fzo (II) |  $-$ |  $1.67^{+1.32}_{-0.53}$ | $0.71^{+0.57}_{-0.26}$
Ref. Khosravi:2022fzo (III) |  $-$ |  $1.90^{+1.48}_{-0.63}$ | $0.65^{+0.55}_{-0.24}$
Ref. Wang:2008da  |  $1.3^{+1.3}_{-0.4}$ |  $1.3^{+1.2}_{-0.4}$ |  $0.52^{+0.57}_{-0.18}$
Ref. Wang:2014upa  |  $1.27^{+0.36}_{-0.36}$ |  $1.27^{+0.36}_{-0.36}$ |  $0.54^{+0.16}_{-0.16}$
Table 8: Branching fractions $(\times 10^{4})$ of the semileptonic decays $D_{s}^{+}\to K_{0}^{\ast}(1430)^{0}\ell^{+}\nu_{\ell}$ with $\ell=e$ and $\mu$, respectively. | $\mathcal{B}(D_{s}^{+}\to K_{0}^{\ast}(1430)^{0}e^{+}\nu_{e})$ | $\mathcal{B}(D_{s}^{+}\to K_{0}^{\ast}(1430)^{0}\mu^{+}\nu_{\mu})$
---|---|---
This work |  $0.38^{+0.19}_{-0.15}$ |  $0.32^{+0.16}_{-0.13}$
Ref. Yang:2005bv  |  $0.24^{+0.22}_{-0.15}$ |  $0.24^{+0.22}_{-0.15}$

Then, we can calculate the differential decay widths of the semileptonic
decays $B_{s},D_{s}\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$ with the following
formula Yang:2005bv ; Sun:2010nv

$\displaystyle\frac{d\Gamma}{dq^{2}}(H_{q_{1}}\to
K_{0}^{\ast}\ell\nu_{\ell})=\frac{G_{F}^{2}|V_{Qq_{2}}|^{2}}{192\pi^{3}m_{H_{q_{1}}}^{3}}\frac{q^{2}-m_{\ell}^{2}}{(q^{2})^{2}}\sqrt{\frac{(q^{2}-m_{\ell}^{2})^{2}}{q^{2}}}$
$\displaystyle\times\sqrt{\frac{(m_{H_{q_{1}}}^{2}-m_{K_{0}^{\ast}}^{2}-q^{2})^{2}}{4q^{2}}-m_{K_{0}^{\ast}}^{2}}\Big{\\{}(m_{\ell}^{2}+2q^{2})\Big{[}q^{2}-(m_{H_{q_{1}}}$
$\displaystyle-
m_{K_{0}^{\ast}})^{2}\Big{]}\Big{[}q^{2}-(m_{H_{q_{1}}}+m_{K_{0}^{\ast}})^{2}\Big{]}f_{+}^{2}(q^{2})+3m_{\ell}^{2}(m_{H_{q_{1}}}^{2}$
$\displaystyle-
m_{K_{0}^{\ast}}^{2})^{2}\Big{[}f_{+}(q^{2})+\frac{q^{2}}{m_{H_{q_{1}}}-m_{K_{0}^{\ast}}}f_{-}(q^{2})\Big{]}^{2}\Big{\\}}.$
(40)

In calculation, we take the fermi coupling constant $G_{F}=1.166\times
10^{-5}~{}\rm{GeV^{-2}}$, CKM Matrix elements $|V_{ub}|=(3.82\pm 0.20)\times
10^{-3}$, $|V_{cd}|=0.221\pm 0.004$, $m_{\ell}$ denotes the lepton mass and
$m_{e}=0.511\times 10^{-3}~{}{\rm MeV}$, $m_{\mu}=105.658\times 10^{-3}~{}{\rm
MeV}$ and $m_{\tau}=1776.86\times 10^{-3}~{}{\rm MeV}$ Workman:2022ynf . The
resulted differential decay ratios for semileptonic decays
$\overline{B}_{s}^{0}\to K_{0}^{\ast}(1430)^{+}\ell^{-}\bar{\nu}_{\ell}$ with
$\ell=e,\tau$ and $D_{s}^{+}\to K_{0}^{\ast}(1430)^{0}\ell^{+}\nu_{\ell}$ with
$\ell=e,\mu$ versus $q^{2}$ are shown in Fig. 4, where the solid and dashed
lines are for the central values and the shaded bands are for the
corresponding errors.

Integrating Eq. (40) over $q^{2}$ in the region $m_{\ell}^{2}\leq
q^{2}\leq(m_{H_{q_{1}}}-m_{K_{0}^{\ast}})^{2}$, and using the heavy meson mean
lifetimes $\tau_{\overline{B}_{s}^{0}}=1.520(5)\times 10^{-12}~{}s$ and
$\tau_{D_{s}^{+}}=0.504(4)\times 10^{-12}~{}s$ Workman:2022ynf , the branching
fractions of the semileptonic decays $\overline{B}_{s}^{0}\to
K_{0}^{\ast}(1430)^{+}\ell^{-}\bar{\nu}_{\ell}$ with $\ell=e,\mu,\tau$ and
$D_{s}^{+}\to K_{0}^{\ast}(1430)^{0}\ell^{+}\nu_{\ell}$ with $\ell=e,\mu$ can
be obtained and are exhibited in Table 7 and Table 8, respectively. As a
comparison, the corresponding branching fractions obtained in Refs.
Faustov:2013ima ; Li:2008tk ; Yang:2005bv ; Khosravi:2022fzo ; Wang:2008da ;
Wang:2014upa  are also exhibited in Table 7 and Table 8.

##  IV SUMMARY

The semileptonic $B_{s},D_{s}\to K_{0}^{\ast}(1430)$ decays can provide
another option for testing standard model beyond the semileptonic progresses
with pseudoscalar mesons in the final states. In which, the $B_{s},D_{s}\to
K_{0}^{\ast}(1430)$ TFFs are the key objects, and whose accuracy mainly
depends on the main error source, $\phi_{2;K_{0}^{\ast}}(x,\mu)$, the
$K_{0}^{\ast}(1430)$ leading-twist DA. Motivated by this, we have studied the
$K_{0}^{\ast}(1430)$ leading-twist DA and the semileptonic $B_{s},D_{s}\to
K_{0}^{\ast}(1430)$ decays in detail in this article. Our work is based on the
scenario that the $K_{0}^{\ast}(1430)$ is viewed as the ground state of
$s\bar{q}$ and $q\bar{s}$.

The $K_{0}^{\ast}(1430)$ leading-twist DA has studied following the scheme
proposed in Ref. Zhong:2021epq  at the first time. The $\xi$-moments are
calculated with the QCDSRs in the framework of BFT by taking the $SU_{f}(3)$
symmetry breaking into account. Considering the fact that the zeroth
$\xi$-moment of the $K_{0}^{\ast}(1430)$ twist-3 DA,
$\langle\xi^{0}_{p}\rangle_{3;K_{0}^{\ast}}$, cannot be normalized in whole
Borel region, a more reasonable and accurate sum rule formula for $n$th
$\xi$-moments of $K_{0}^{\ast}(1430)$ leading-twist DA, i.e., Eq. (13), has
been suggested. The values of the first ten $\xi$-moments,
$\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}(n=1,2,\cdots,10)$, have been
calculated and exhibited in Table 1. On the other hand, a new LCHO model has
been established at the first time to describe the behavior of the
$K_{0}^{\ast}(1430)$ leading-twist DA. By fitting the resulted $\xi$-moments
shown in Table 1 with this LCHO model via the least squares method, the
behavior of the $K_{0}^{\ast}(1430)$ leading-twist DA has been determined. The
fitted model parameters at scale $\mu=1~{}{\rm GeV}$, $1.4~{}{\rm GeV}$ and
$3~{}{\rm GeV}$ have been displayed in Table 3, the predicted curve of
$\phi_{2;K_{0}^{\ast}}(x,\mu)$ at scale $\mu=1~{}{\rm GeV}$ is shown in Fig.
2, respectively.

Then, we have calculated the $B_{s},D_{s}\to K_{0}^{\ast}(1430)$ TFFs
$f_{\pm,T}(q^{2})$ with LCSR method. The values of those TFFs at the large
recoil point have been given in Table 4 and Table 5. After extrapolating the
LCSR results of $B_{s},D_{s}\to K_{0}^{\ast}(1430)$ TFFs to the whole $q^{2}$
region (the corresponding behaviors have been shown in Fig. 3), the
differential decay ratios and branching fractions of the semileptonic decays
$B_{s},D_{s}\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$ have been obtained and shown
in Fig. 4, Table 7 and Table 8 respectively.

Acknowledgments: This work was supported in part by the National Natural
Science Foundation of China under Grant No.12265009, No.12265010, No.12175025
and No.12147102, the Project of Guizhou Provincial Department of Science and
Technology under Grant No.ZK[2021]024, the Project of Guizhou Provincial
Department of Education under Grant No.KY[2021]030, and by the Chongqing
Graduate Research and Innovation Foundation under Grant No. ydstd1912.

## Appendix: Specific expressions of some condensate terms in sum rules of
$\langle\xi^{n}\rangle_{2;K_{0}^{\ast}}\times\langle\xi^{n}_{p}\rangle_{3;K_{0}^{\ast}}$

$\displaystyle\hat{I}_{\langle G^{2}\rangle}(M^{2})$
$\displaystyle=\frac{\langle\alpha_{s}G^{2}\rangle}{(M^{2})^{2}}((-1)^{n}m_{s}-m_{u})\frac{1}{48\pi}\Big{\\{}-12(-1)^{n}n\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}+6(-1)^{n}(n+2)+\theta(n-1)\Big{[}-4(-1)^{n}n\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}$
$\displaystyle-3\widetilde{\psi}_{3}(n)\Big{]}+\theta(n-2)\Big{[}-(8n+3)\widetilde{\psi}_{2}(n)+(-1)^{n}(4n+9)+7+\frac{6}{n}\Big{]}\Big{\\}},$
(41) $\displaystyle\hat{I}_{\langle G^{3}\rangle}(M^{2})$
$\displaystyle=\frac{\langle
g_{s}^{3}fG^{3}\rangle}{(M^{2})^{3}}((-1)^{n}m_{s}-m_{u})\frac{1}{384\pi^{2}}\Big{\\{}\delta^{n1}\Big{[}-24\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}+84\Big{]}+\theta(n-1)\Big{[}4(-1)^{n}n(3n-5)\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}$
$\displaystyle-2(-1)^{n}(2n^{2}+5n-13)\Big{]}+\theta(n-2)\Big{[}24(-1)^{n}n^{2}\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}+2n(n-4)\widetilde{\psi}_{2}(n)-17(-1)^{n}n^{2}-55(-1)^{n}n$
$\displaystyle-6(-1)^{n}+6\Big{]}+\theta(n-3)\Big{[}-2n(8n-1)\widetilde{\psi}_{1}(n)+\frac{1}{n-1}\Big{(}-19(-1)^{n}n^{3}+16(1+(-1)^{n})n^{2}+3(2+(-1)^{n})n$
$\displaystyle+6\Big{)}\Big{]}\Big{\\}},$ (42) $\displaystyle\hat{I}_{\langle
q^{4}\rangle}(M^{2})$ $\displaystyle=\frac{(2+\kappa^{2})\langle
g_{s}^{2}\bar{u}u\rangle^{2}}{(M^{2})^{3}}((-1)^{n}m_{s}-m_{u})\frac{1}{3888\pi^{2}}\Big{\\{}8(-1)^{n}n\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}-4(-1)^{n}(n+5)+\delta^{n0}\Big{[}-24\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}$
$\displaystyle-148\Big{]}+\delta^{n1}\Big{[}128\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}-692\Big{]}+\theta(n-1)\Big{[}-8\Big{(}6(-1)^{n}n^{2}+(-25+9(-1)^{n})n-6(1+(-1)^{n})\Big{)}\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}$
$\displaystyle-4n\widetilde{\psi}_{3}(n)+2\Big{(}6(-1)^{n}n^{2}+(-47+49(-1)^{n})n+57(-1)^{n}-151-\frac{24}{n}(1+(-1)^{n})\Big{)}\Big{]}+\theta(n-2)\Big{[}-4\Big{(}66(-1)^{n}n^{2}$
$\displaystyle-34(-1)^{n}n+15(1+(-1)^{n})\Big{)}\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}+2\Big{(}-6n^{2}+(-21+50(-1)^{n})n+12(1+(-1)^{n})\Big{)}\widetilde{\psi}_{2}(n)$
$\displaystyle+\frac{1}{n(n-1)}\Big{(}231(-1)^{n}n^{4}+(-94+426(-1)^{n})n^{3}+(116-985(-1)^{n})n^{2}+(98+328(-1)^{n})n-60(1$
$\displaystyle+(-1)^{n})\Big{)}\Big{]}+\theta(n-3)\Big{[}\Big{(}144n^{2}-74n+30(1+(-1)^{n})\Big{)}\widetilde{\psi}_{1}(n)+\frac{1}{n-1}\Big{(}169(-1)^{n}n^{3}-12(12+17(-1)^{n})n^{2}$
$\displaystyle+(10+111(-1)^{n})n-2(63+38(-1)^{n})\Big{)}\Big{]}\Big{\\}},$
(43) $\displaystyle\hat{I}_{\langle G^{2}\rangle}^{m_{s}^{3}}(M^{2})$
$\displaystyle=\frac{\langle\alpha_{s}G^{2}\rangle}{(M^{2})^{3}}m_{s}^{3}\frac{1}{24\pi}\Big{\\{}-2\delta^{n0}-10\delta^{n1}\Big{[}\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}-\frac{17}{5}\Big{]}+\theta(n-1)\Big{[}6n\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}-3(n+3)\Big{]}+\theta(n-2)$
$\displaystyle\times\Big{[}-2n(7n-2)\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}+3(-1)^{n}n\widetilde{\psi}_{2}(n)+9n^{2}+24n-3(-1)^{n}-2\Big{]}+\theta(n-3)\Big{[}(-1)^{n}n(7n-2)\widetilde{\psi}_{1}(n)$
$\displaystyle-\frac{1}{n-1}\Big{(}-7n^{3}+(5+7(-1)^{n})n^{2}+(2+(-1)^{n})n+2(-1)^{n}\Big{)}\Big{]}\Big{\\}},$
(44) $\displaystyle\hat{I}_{\langle G^{3}\rangle}^{m_{s}^{3}}(M^{2})$
$\displaystyle=\frac{\langle
g_{s}^{3}fG^{3}\rangle}{(M^{2})^{4}}m_{s}^{3}\frac{1}{6912\pi^{2}}\Big{\\{}-24\delta^{n0}-288\delta^{n1}+\delta^{n2}\Big{[}2352\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}-10188\Big{]}+\theta(n-2)\Big{[}144n(n-1)$
$\displaystyle\times(2n-5)\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}-9(8n^{3}+21n^{2}-205n+90)\Big{]}+\theta(n-3)\Big{[}24n(n-1)(36n-23)\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}$
$\displaystyle-144(-1)^{n}n(n-1)(n-2)\widetilde{\psi}_{1}(n)-36(-1)^{n}n(n-1)\widetilde{\psi}_{5}(n)-72(-1)^{n}n(n-1)\widetilde{\psi}_{6}(n)+\frac{2}{n-2}\Big{(}-505n^{4}$
$\displaystyle+(281+72(-1)^{n})n^{3}-2(-1274+99(-1)^{n})n^{2}+(-2417+135(-1)^{n})n+90(-1)^{n}+474)\Big{)}\Big{]}+\theta(n-4)$
$\displaystyle\times\Big{[}12(-1)^{n}n(n-1)(36n-23)\widetilde{\psi}_{4}(n)+\frac{1}{n-2}\Big{(}-566n^{4}+(1987-432(-1)^{n})n^{3}+(-1999+492(-1)^{n})n^{2}$
$\displaystyle+(578-354(-1)^{n})n-156(-1)^{n}\Big{)}\Big{]}\Big{\\}},$ (45)
$\displaystyle\hat{I}_{\langle q^{4}\rangle}^{m_{s}^{3}}(M^{2})$
$\displaystyle=\frac{(2+\kappa^{2})\langle
g_{s}^{2}\bar{u}u\rangle^{2}}{(M^{2})^{4}}m_{s}^{3}\frac{1}{23328\pi^{2}}\Big{\\{}\delta^{n0}\Big{[}1368\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}+1512\Big{]}+\delta^{n1}\Big{[}1032\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}-1272\Big{]}+\delta^{n2}$
$\displaystyle\times\Big{[}-3504\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}+13980\Big{]}-48(-1)^{n}n\theta(n-1)+\theta(n-2)\Big{[}-12\Big{(}-8n^{3}+86n^{2}+4(-39+25(-1)^{n})n$
$\displaystyle-9(1+(-1)^{n})\Big{)}\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}+\frac{3}{n(n-1)}\Big{(}-48n^{5}+429n^{4}+(70+204(-1)^{n})n^{3}+(-1473+596(-1)^{n})n^{2}$
$\displaystyle+(622-472(-1)^{n})n+36(1+(-1)^{n})\Big{)}\Big{]}+\theta(n-3)\Big{[}24\Big{(}-76n^{3}+177n^{2}-147n+24(1+(-1)^{n})\Big{)}$
$\displaystyle\times\Big{(}-\ln\frac{M^{2}}{\mu^{2}}\Big{)}-6\Big{(}8(-1)^{n}n^{3}-86(-1)^{n}n^{2}+4(-25+39(-1)^{n})n+9(1+(-1)^{n})\Big{)}\widetilde{\psi}_{1}(n)-\frac{2}{n(n-1)(n-2)}$
$\displaystyle\times\Big{(}-943n^{6}-3(-519+8(-1)^{n})n^{5}+(6041-24(-1)^{n})n^{4}-3(5720-117(-1)^{n})n^{3}+(15359+363(-1)^{n})n^{2}$
$\displaystyle-6(857+259(-1)^{n})n+576(1+(-1)^{n})\Big{)}\Big{]}+\theta(n-4)\Big{[}-12\Big{(}76(-1)^{n}n^{3}-177(-1)^{n}n^{2}+147(-1)^{n}n$
$\displaystyle-24(1+(-1)^{n})\Big{)}\widetilde{\psi}_{4}(n)+\frac{1}{(n-1)(n-2)}\Big{(}1106n^{5}+(-5469+912(-1)^{n})n^{4}+(10358-2580(-1)^{n})n^{3}$
$\displaystyle+(-9669+3294(-1)^{n})n^{2}+(4250-798(-1)^{n})n-12(48+73(-1)^{n})\Big{)}\Big{]}\Big{\\}},$
(46)

where

$\displaystyle\widetilde{\psi}_{1}(n)$
$\displaystyle=\psi\Big{(}\frac{n}{2}\Big{)}-\psi\Big{(}\frac{n-1}{2}\Big{)}-(-1)^{n}\ln
4,$ $\displaystyle\widetilde{\psi}_{2}(n)$
$\displaystyle=\psi\Big{(}\frac{n+1}{2}\Big{)}-\psi\Big{(}\frac{n}{2}\Big{)}+(-1)^{n}\ln
4,$ $\displaystyle\widetilde{\psi}_{3}(n)$
$\displaystyle=\psi\Big{(}\frac{n}{2}+1\Big{)}-\psi\Big{(}\frac{n+1}{2}\Big{)}-(-1)^{n}\ln
4,$ $\displaystyle\widetilde{\psi}_{4}(n)$
$\displaystyle=\psi\Big{(}\frac{n-1}{2}\Big{)}-\psi\Big{(}\frac{n}{2}-1\Big{)}+(-1)^{n}\ln
4,$ $\displaystyle\widetilde{\psi}_{5}(n)$
$\displaystyle=\psi\Big{(}\frac{n+1}{2}\Big{)}-\psi\Big{(}\frac{n}{2}-1\Big{)}-\ln
4,$ $\displaystyle\widetilde{\psi}_{6}(n)$
$\displaystyle=\psi\Big{(}n-2\Big{)}-\psi\Big{(}\frac{n}{2}-1\Big{)}+(-1)^{n}\ln
4$ (47)

with the digamma function $\psi(n)$.

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