Probing $|V_{cs}|$ and lepton flavor universality through $D\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$ decays

As the lightest particle containing the $c$ quark, the exclusive semileptonic decay processes of $D$ meson are highly valuable in enhancing our understanding of weak and strong interactions within the framework of the Standard Model (SM). The Cabibbo-Kobayashi-Maskawa (CKM) matrix elements describe the flavor-changing transitions involving quarks that can be determined by semileptonic decays. The binding effect of strong interaction is limited to a hadronic current, which can be parametrized by the form factors, which are viewed as one of the most important precision tests of the SM CLEO:2008bkh . From this point, the semileptonic decay of charm mesons plays an important role in the determination of CKM matrix elements Cabibbo-favored $|V_{cs}|$, and Cabibbo-suppressed $|V_{cd}|$.

Experimentally, the charmed $D$-meson semileptonic decay processes have been measured by the BESIII Collaboration Liu:2019tsi ; Zhang:2019tcs ; Yang:2018qdx ; BESIII:2016gbw ; BESIII:2021mfl ; BESIII:2015tql ; BESIII:2021pvy ; BESIII:2015kin and the CLEO Collaboration CLEO:2011ab ; CLEO:2004arv ; CLEO:2009dyb ; CLEO:2009svp ; CLEO:2005rxg , etc. In which, for $D\to P,V+\ell\nu_{\ell}$ ($P$ and $V$ stand for pseudoscalar and vector mesons, respectively), its discussion is quite mature now, and the experimental and theoretical groups are still trying to improve the accuracy of the relevant calculations. The precision of the measurement has been continuously improved in recent years. However, there are few experimental studies on scalar mesons. It is known that only BESIII BESIII:2018qmf ; BESIII:2018sjg ; BESIII:2023wgr ; BESIII:2021drk ; BESIII:2023opt ; BESIII:2021tfk and CLEO CLEO:2009ugx ; CLEO:2009dyb have studied the semileptonic decays of $D_{(s)}$ to $a_{0}(980),f_{0}(500)$ or $f_{0}(980)$, which all involve only a $u$ and $d$ quark, without considering scalar mesons containing an s quark. Particularly, for $D$-meson semileptonic decays into one scalar meson, the BESIII Collaboration reported the $D^{0(+)}\to a_{0}(980)^{-(0)}e^{+}\nu_{e}$ decays with a significance of $6.4\sigma$ and $2.9\sigma$, respectively, by utilizing the $e^{+}e^{-}$ collision data sample of $2.93~{}\rm{fb}^{-1}$ collected at a center-of-mass energy of $3.773~{}\rm{GeV}$ with the order of the absolute branching fractions of $10^{-4}$ BESIII:2018sjg . Recently, the BESIII Collaboration measured a branching ratio of $10^{-3}$ for $D^{+}_{s}\to f_{0}(980)e^{+}\nu_{e}$ based on a data with integrated luminosity of $7.33~{}\rm{fb}^{-1}$ at $4.128$ and $4.226~{}\rm{GeV}$ BESIII:2023wgr . Presently, for scalar mesons like $a_{0}(980)$ and $f_{0}(980)$, despite the remarkable progress made by the quark model in explaining the majority of hadronic states over the past decades, their internal structures have remained a subject of intense theoretical and experimental scrutiny and controversy. As for semileptonic decays $D\to K_{0}^{*}(1430)\ell\nu_{\ell}$, the FOCUS Collaboration presented the ratio $\Gamma(D^{+}\to\bar{K}^{\ast}_{0}(1430)^{0}\mu^{+}\nu)/\Gamma(D^{+}\to K^{-}\pi^{+}\mu^{+}\nu)<0.64\%$, which is from the discussion about the hadronic mass spectrum analysis of $D^{+}\to K^{-}\pi^{+}\mu^{+}\nu$ FOCUS:2005iqy . Till now, there is no experimental result for $D\to K_{0}^{*}(1430)\ell\nu_{\ell}$ directly. But the observables for the decay processes with $K_{0}^{\ast}(1430)$ finial state also can provide valuable insights for further understanding the internal nature of scalar meson. So, it is meaningful to have a deep look into the semileptonic $D\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$ with $c\to s\ell\nu_{\ell}$ transition, which is the main purpose of this work.

The structures of light scalar mesons are a long standing puzzle. Currently, the internal structure of scalar mesons is considered to be in the form of $q\bar{q}$ states Cheng:2005nb , $qq\bar{q}\bar{q}$ states Jaffe:1976ig , molecular states Weinstein:1990gu , glueball states Weinstein:1982gc , or hybrid states Weinstein:1983gd . Among them, the view that scalar mesons above 1 GeV are dominantly composed of as $q\bar{q}$ states is more widely accepted by the public Aliev:2007rq ; Du:2004ki ; Aslam:2009cv ; Sun:2010nv ; Wang:2014vra ; Wang:2014upa ; Khosravi:2022fzo ; Khosravi:2024zaj ; Yang:2005bv ; Agaev:2018fvz . The controversy surrounding the internal structure of scalar mesons below 1 GeV has always been relatively prominent. Recently, from a survey of the accumulated experimental data, two viable and publicly acceptable theoretical schemes for studying scalar mesons have been proposed Cheng:2005nb . For one picture (P1), the light scalar mesons $f_{0}(980),a_{0}(980),K_{0}^{*}(700)$, etc., are seen as the ground $q\bar{q}$ states, and nonet mesons near $1.5~{}\rm{GeV}$ are interpreted as the first excited states. For another picture (P2), $f_{0}(1370),a_{0}(1450),K_{0}^{\ast}(1430)$, etc., are treated as lowest-lying $P$-wave $q\bar{q}$ states and nonet mesons below $1~{}\rm{GeV}$ are viewed as four-quark bound states Lee:2022jjn ; Humanic:2022hpq ; Brito:2004tv ; Alexandrou:2017itd ; Klempt:2007cp . As early as 2002, the Ref. Dosch:2002rh studies the semileptonic decay $D\to\kappa\ell\nu_{\ell}$ in three point QCD sum rule (3PSR), where the $\kappa$ (also known as $K_{0}^{*}(700)$) is treated as $q\bar{q}$ bound states. This falls under the first picture, and recently, the BESIII collaboration have study the charmed meson semileptonic decay into scalar states below 1 GeV BESIII:2024zvp ; BESIII:2024lnh ; BESIII:2023wgr , which give the result $f^{+}(0)$ at large recoil region. In this context, if we discuss the semi-leptonic decay of the $D$ meson into the first excited state $K_{0}^{*}(1430)$ using the QCD sum rule method, we need to clearly understand that there will be interference between the light-cone distribution amplitudes (LCDA) of $K_{0}^{*}(700)$ and $K_{0}^{*}(1430)$. Specifically, when calculating the $\xi$-moment of the $K_{0}^{*}(1430)$ LCDA, the contribution from the ground state $K_{0}^{*}(700)$ cannot be ignored. This point is evident from the hadronic expression. In addition, from another perspective, the recent review by the Particle Data Group (PDG) mentions that scalar mesons below 1 GeV are recognized as four-quark states dominantly ParticleDataGroup:2024cfk . Based on this view, some research groups can provide more reasonable explanations for its properties, corresponding quantum numbers, low masses and mass level inside the light nonets of these scalar mesons below 1 GeV Weinstein:1990gu ; Weinstein:1982gc . Meanwhile, if consider to study the scalar mesons mass spectrum and their strong decay and electromagnetic decays, the tetraquark picture is also reasonable Jaffe:1976ig ; Alford:2000mm . Within this case, due to the different internal structures of $K_{0}^{*}(700)$ and $K_{0}^{*}(1430)$, the interference issue mentioned earlier regarding the LCDA will not occur. Thus, in this paper, we mainly focus on P2 scenario mentioned above for describing scalar mesons, and treat $K_{0}^{*}(1430)$ as a $s\bar{q}$ or $q\bar{s}$ ground state for relevant calculations of $D\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$. At the same time, we will also simply discuss the interference of the two state since $K_{0}^{*}(700)$ is taken as $q\bar{q}$ state from P1 scenario.

Furthermore, the $D\to K_{0}^{\ast}(1430)$ transition form factors (TFFs) are crucial components in this decay within the SM, which can be calculated by various nonperturbative methods, such as 3PSR Yang:2005bv , the covariant light-front approach (CLF) Cheng:2003sm , and generalized factorization model (GFM) Cheng:2002ai . As an transitional SVZSR, the 3PSR approach has been proved to be quite success, which have achieved good results in the TFFs for $D$-meson semileptonic decays Colangelo:2001cv ; Du:2003ja ; Ball:1993tp ; Ball:1991bs . The light cone sum rules (LCSR) method was developed since 1980s Balitsky:1989ry ; Chernyak:1990ag , is an effective combination of SVZSR technique and hard exclusive process theory, and is regarded as an advanced tool for dealing with exclusive heavy-to-light processes Cheng:2017bzz ; Tian:2023vbh ; Gao:2019lta ; Duplancic:2008ix . Both methods have their respective advantages. For instance, in 3PSR, the non-perturbative dynamics are parameterized as vacuum condensates. These vacuum condensates of each dimension are the trivial and general parameters which are independence from the processes. By determining two Borel parameters $M_{1}^{2}$ and $M_{2}^{2}$, as well as two threshold $s_{1}^{0}$ and $s_{2}^{0}$, which are made by double Borel transformation and double dispersion relation in 3PSR, the good predictions for the TFFs can be obtained. On the other hand, the advantage of the LCSR lies in the less parameters in the hadronic expression that have single Borel parameter $M^{2}$ and and one threshold $s_{0}$. The range of these two parameters can be obtained by standard criteria in LCSR Hu:2021zmy . With the LCDA of the final-state meson determined, reliable TFF results can also be achieved. Till now, 3PSR and LCSR still have some weakness in determine the parameters. Therefore, we should deeply study different processes and gradually reduce the error of parameters in the sum rule approach to obtain more accurate results. In our previous work, the $D_{s}\to K_{0}^{\ast}(1430)$ with $c\to d$ transition has been studied by the LCSR approach Huang:2022xny , which has provided us with positive feedback. So the Cabibbo-favored channel $D\to K_{0}^{\ast}(1430)$ with a $c\to s$ transition can also be researched by the LCSR approach.

As one of the most important nonperturbative parameters, $K_{0}^{\ast}(1430)$-meson LCDAs, including long-distance dynamics at a lower energy scale, are critical to the behavior of TFFs. Thus, a detailed investigation of LCDAs is conducive to enhancing the precision of the calculation of TFFs. Theoretically, the QCDSR Cheng:2005nb and CLF approach Chen:2021oul present the $K_{0}^{\ast}(1430)$-meson leading-twist LCDAs. As we know, the $K_{0}^{\ast}(1430)$-meson leading-twist LCDA can be expanded with a series of Gegenbauer coefficients (also called Gegenbauer moments), which can be calculated by the QCD sum rule approach. One often takes the first few order Gegenbauer moments, which lead to the truncated form (TF) to avoid false oscillation from the higher order moments, such as the pion and kaon twist-2 LCDA up to second order calculated by QCDSR LatticeParton:2022zqc , and $\rho$, $K^{*}$, $\phi$-meson longitudinal and transverse leading-twist LCDAs up to second order from QCDSR Ball:2007zt . Recently, the lattice QCD proved the effectiveness of the TF from LatticeParton:2022zqc ; Hua:2020gnw . Meanwhile, we have calculated the first ten-order $\xi$-moments of $K_{0}^{\ast}(1430)$-meson leading-twist LCDA by using the QCDSRs within the framework of background field theory (BFT) Huang:2022xny . Thus, in this paper, we will take the first three order Gegenbauer moments to make the TF, which is called scenario 1 (S1). On the other hand, one will take the nature light cone harmonic oscillator (LCHO) model to describe the behavior of $K_{0}^{\ast}(1430)$-meson leading-twist LCDA, which is considered as scenario 2 (S2) of our study. The model-dependent parameters can be fixed by the first ten-order $\xi$ moments at scale $\mu_{k}=1.4~{}{\rm GeV}$. By comparing the observables of semileptonic decay $D\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$ under the two different forms of $K_{0}^{\ast}(1430)$ meson leading-twist LCDA, it is helpful to see which one has a better behavior in these semileptonic decays. This will not only test the SM but also test the accuracy of our determination of LCDA parameters. Particularly, in order to make the behavior of TFFs for $D\to K_{0}^{\ast}(1430)$ more precise, we need to consider the contribution of twist-3 LCDAs $\phi_{3;K_{0}^{*}}^{p,\sigma}(x,\mu)$. The twist-3 LCDAs based on Gegenbauer series expansion are also calculated within the framework of the background field theory in our early previous work Han:2013zg , which will be reused here.

To express the full spectrum of $D\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$ decays, one can start with the simple form for differential decay width with respect to the squared momentum transfer $q^{2}$ and the angle $\cos\theta_{\ell}$ between the direction of flight of $K_{0}^{*}(1430)$ and $\ell$ in the center of mass frame of $\ell\nu_{\ell}$, which have the following form:

	$\displaystyle\frac{d^{2}\Gamma}{dq^{2}d\cos\theta_{\ell}}$	$\displaystyle=\frac{1}{32(2\pi)^{3}m_{D}^{2}}|{\bf q}|\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)$
		$\displaystyle\times|{\cal M}(D\to K_{0}^{\ast}(1430)\ell\nu_{\ell})|^{2},$		(1)

in which, the symbol ${\bf q}$ stands for the three-momentum of the $\ell\nu_{\ell}$ pair in the $D$-meson rest frame. To write the amplitude ${\cal M}(D\to K_{0}^{\ast}(1430)\ell\nu_{\ell})$ explicitly, we decompose the nonvanishing hadronic matrix elements of the quark operators in the effective Hamiltonian i.e. $\mathcal{H}_{\rm{eff}}=\frac{G_{F}}{\sqrt{2}}V_{cs}\bar{c}\gamma_{\mu}(1-\gamma_{5})s\bar{\ell}\gamma^{\mu}(1-\gamma_{5})\nu_{\ell}$, in terms of the Lorentz invariant hadronic form factors $f_{+}(q^{2})$ and $f_{0}(q^{2})$ with the definition,

	$\displaystyle\langle K^{*}_{0}(p)|\bar{s}\gamma_{\mu}\gamma_{5}c|D(p+q)\rangle=\bigg{[}(2p+q)_{\mu}-\frac{m_{D}^{2}-m_{K_{0}^{*}}^{2}}{q^{2}}q_{\mu}\bigg{]}$
	$\displaystyle\qquad\qquad\times f_{+}(q^{2})+\frac{m_{D}^{2}-m_{K_{0}^{*}}^{2}}{q^{2}}q_{\mu}f_{0}(q^{2}).$		(2)

The full differential decay rate for the $D\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$ semileptonic decay can be expressed as $d^{2}\Gamma/(dq^{2}d\cos\theta_{\ell})=a_{\theta_{\ell}}(q^{2})+b_{\theta_{\ell}}(q^{2})\cos\theta_{\ell}+c_{\theta_{\ell}}(q^{2})\cos^{2}\theta_{\ell}$, with the angular coefficient functions as Becirevic:2016hea

	$\displaystyle a_{\theta_{\ell}}(q^{2})$	$\displaystyle=\mathcal{N}_{\rm{ew}}\lambda^{3/2}\bigg{(}1-\frac{m_{\ell}^{2}}{q^{2}}\bigg{)}^{2}\bigg{[}|f_{+}(q^{2})|^{2}+\frac{1}{\lambda}\frac{m_{\ell}^{2}}{q^{2}}$
		$\displaystyle\times\bigg{(}1-\frac{m_{K_{0}^{*}}^{2}}{m_{D}^{2}}\Bigg{)}^{2}|f_{0}(q^{2})|^{2}\bigg{]},$
	$\displaystyle b_{\theta_{\ell}}(q^{2})$	$\displaystyle=2\mathcal{N}_{\rm{ew}}\lambda\bigg{(}1-\frac{m_{\ell}^{2}}{q^{2}}\bigg{)}^{2}\frac{m_{\ell}^{2}}{q^{2}}\bigg{(}1-\frac{m_{K_{0}^{*}}^{2}}{m_{D}^{2}}\bigg{)}$
		$\displaystyle\times{\rm Re}[f_{+}(q^{2})f_{0}^{*}(q^{2})],$
	$\displaystyle c_{\theta_{\ell}}(q^{2})$	$\displaystyle=-\mathcal{N}_{\rm{ew}}\lambda^{3/2}\bigg{(}1-\frac{m_{\ell}^{2}}{q^{2}}\bigg{)}^{3}|f_{+}(q^{2})|^{2},$		(3)

in which $G_{F}$ is the Fermi constant, $m_{\ell}$ and $\theta_{\ell}$ are lepton mass and helicity angle, $\mathcal{N}_{\rm{ew}}=G^{2}_{F}|V_{cs}|^{2}m_{D}^{3}/256\pi^{3}$ and $\lambda\equiv\lambda(1,m_{K_{0}^{*}}^{2}/m_{D}^{2},q^{2}/m_{D}^{2})$ with $\lambda(a,b,c)\equiv a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$. With $f_{0}(q^{2})=f_{+}(q^{2})+q^{2}/(m_{D}^{2}-m_{K_{0}^{*}}^{2})f_{-}(q^{2})$, $f_{\pm}(q^{2})$ are the $D\to K_{0}^{\ast}(1430)$ TFFs. After integrating over the helicity angle, $\theta_{\ell}\in[-1,1]$, the differential decay width of $D\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$ over $q^{2}$ with respect to kinematic variables $q^{2}$ can be written as

	$\displaystyle\frac{d\Gamma}{dq^{2}}$	$\displaystyle=\frac{G^{2}_{F}|V_{cs}|^{2}m_{D}^{3}}{192\pi^{3}}\lambda^{3/2}\bigg{(}1-\frac{m_{\ell}^{2}}{q^{2}}\bigg{)}^{2}\bigg{\{}\bigg{(}1+\frac{m_{\ell}^{2}}{2q^{2}}\bigg{)}$
		$\displaystyle\times|f_{+}(q^{2})|^{2}+\frac{1}{\lambda}\frac{3m_{\ell}^{2}}{2q^{2}}\bigg{(}1-\frac{m^{2}_{K_{0}^{*}}}{m_{D}^{2}}\bigg{)}^{2}|f_{0}(q^{2})|^{2}\bigg{\}}.$		(4)

Furthermore, one can calculate three independent observables from three angular coefficient functions $a_{\theta_{\ell}}(q^{2}),b_{\theta_{\ell}}(q^{2}),c_{\theta_{\ell}}(q^{2})$, $i.e.$, forward-backward asymmetries, lepton polarization asymmetries, and flat terms. These observables are very sensitive to beyond the Standard Model (BSM) physics. So, one can extract results of three observables from $D\to K_{0}^{*}\ell\nu_{\ell}$ through the relationship between these observables and TFF, $i.e.$, Cui:2022zwm

	$\displaystyle\mathcal{A}_{\rm{FB}}(q^{2})=\Big{[}\frac{1}{2}b_{\theta_{\ell}}(q^{2})\Big{]}:\Big{[}a_{\theta_{\ell}}(q^{2})+\frac{1}{3}c_{\theta_{\ell}}(q^{2})\Big{]},$
	$\displaystyle\mathcal{A}_{\lambda_{\ell}}(q^{2})=1-\frac{2}{3}\Big{\{}\Big{[}3\Big{(}a_{\theta_{\ell}}(q^{2})+c_{\theta_{\ell}}(q^{2})\Big{)}$
	$\displaystyle\hskip 34.14322pt+\frac{2m_{\ell}^{2}}{q^{2}-m_{\ell}^{2}}c_{\theta_{\ell}}(q^{2})\Big{]}:\Big{[}a_{\theta_{\ell}}(q^{2})+\frac{1}{3}c_{\theta_{\ell}}(q^{2})\Big{]}\Big{\}},$
	$\displaystyle\mathcal{F}_{\rm{H}}(q^{2})=\Big{[}a_{\theta_{\ell}}(q^{2})+c_{\theta_{\ell}}(q^{2})\Big{]}:\Big{[}a_{\theta_{\ell}}(q^{2})+\frac{1}{3}c_{\theta_{\ell}}(q^{2})\Big{]}.$		(5)

The above observables are functions of the ratio of TFFs. So next step, we calculate the TFFs by using LCSR and star with the the following correlator:

$\displaystyle\Pi_{\mu}(p,q)$

$\displaystyle=i\int d^{4}xe^{iq\cdot x}\langle K_{0}^{*}(p)|{\rm T}\{j_{2\mu}(x),j_{1}(0)\}|0\rangle$

(6)

where $j_{2\mu}(x)=\bar{q}_{2}(x)\gamma_{\mu}\gamma_{5}c(x)$ and $j_{1}(0)=\bar{c}(0)i\gamma_{5}q_{1}(0)$. $q_{1,2}$ present the light quark, $q_{2}=s$, $q_{1}=d$ is for $D^{+}\to K_{0}^{\ast 0}$, and $q_{1}=u$ is for $D^{0}\to K_{0}^{\ast+}$, where $m_{d}\sim m_{u}\sim 0$; the results of TFFs are almost the same. In the timelike $q^{2}$ region, we can insert a complete intermediate state with $D$ meson quantum numbers into the correlator (6) and separate the pole term of the lowest $D$ meson to obtain the hadronic representation. By using the hadronic dispersion relations, the $D\to K_{0}^{\ast}$ matrix element also can be derived. In the spacelike $q^{2}$ region, we carry out the operator product expansion (OPE) near the light cone $x^{2}\rightsquigarrow 0$, where the light cone expansion of the $c$-quark propagator retains only the two-particle, while twists higher than three are neglected, as the contributions from the remaining components are small and can reasonably be neglected. After separating all the continuum states and excited states by introducing a effective threshold parameter $s_{D}$ and using the Borel transformation, the TFF can be obtained, which has a similar expression to Ref. Huang:2022xny , in which, we repeat the corresponding calculation process and get the same results. For the brevity of this short essay, we will not provide specific expressions here.

Then, $K_{0}^{\ast}(1430)$-meson twist-2 and twist-3 LCDAs are the main nonperturbative uncertainty for TFFs. For scalar meson LCDAs, under the premise of following to the principle of conformal symmetry hidden in the QCD Lagrangian, they can be systematically expanded into a series of Gegenbauer polynomials with increasing conformal spin Wang:2008da . The first twist-2 LCDA $\phi_{2;K_{0}^{*}}^{\rm{(S1)}}(x,\mu)$ based on the TF model can be expressed in the following formulations:

$\displaystyle\phi_{2;K_{0}^{*}}^{\rm{(S1)}}(x,\mu)$

$\displaystyle=6x\bar{x}\sum_{n=0}^{\mathcal{N}=3}a_{n}^{2;K_{0}^{\ast}}(\mu)C^{3/2}_{n}(\xi).$

(7)

Here, $C_{n}^{3/2}(\xi)$ is the Gegenbauer polynomial with $\xi=2x-1$, and the zeroth order Gegenbauer moment $a_{0}^{2;K_{0}^{\ast}}(\mu)$ is equal to 0 for scalar mesons and the behavior of the LCDA is mainly determined by the odd Gegenbauer moment.

In addition, we constructed the twist-2 LCDA by using the LCHO model. The wave functions (WF) of the $K_{0}^{\ast}(1430)$ are obtained based on the Brodsky-Huang-Lepage (BHL) description, which postulates that a correlation exists between the equal-time WF in the rest frame and the light cone WF. The leading-twist WF can be expressed as $\Psi_{2;K_{0}^{*}}(x,\mathbf{k}_{\perp})=\chi_{2;K_{0}^{\ast}}(x,\mathbf{k}_{\perp})\Psi^{R}_{2;K_{0}^{\ast}}(x,\mathbf{k}_{\perp})$. After determining the scalar meson $K_{0}^{\ast}(1430)$ total spin-space WF $\chi_{2;K_{0}^{\ast}}(x,\mathbf{k}_{\perp})$ and spatial WF $\Psi^{R}_{2;K_{0}^{\ast}}(x,\mathbf{k}_{\perp})$ Zhong:2022ecl , we have the following expression by using the relation between $K_{0}^{\ast}(1430)$ leading-twist LCDA and WF, $\phi_{2;K_{0}^{*}}(x,\mu)=\int_{|\mathbf{k}_{\perp}|^{2}\leq\mu^{2}}\frac{d^{2}\mathbf{k}_{\perp}}{16\pi^{3}}\Psi_{2;K_{0}^{*}}(x,\mathbf{k}_{\perp})$, to integrate over the transverse momentum $\mathbf{k}_{\perp}$, and get the following expression:

	$\displaystyle\phi_{2;K_{0}^{*}}^{\rm{(S2)}}(x,\mu)=\frac{A_{2;K_{0}^{*}}\beta_{2;K_{0}^{*}}\tilde{m}}{4\sqrt{2}\pi^{3/2}}\sqrt{x\bar{x}}\varphi_{2;K_{0}^{*}}(x)$
	$\displaystyle\qquad\times\exp\left[-\frac{\hat{m}_{q}^{2}x+\hat{m}_{s}^{2}\bar{x}-\tilde{m}^{2}}{8\beta^{2}_{2;K_{0}^{*}}x\bar{x}}\right]$
	$\displaystyle\qquad\times\left\{{\rm Erf}\left(\sqrt{\frac{\tilde{m}^{2}+\mu^{2}}{8\beta^{2}_{2;K_{0}^{*}}x\bar{x}}}\right)-{\rm Erf}\left(\sqrt{\frac{\tilde{m}^{2}}{8\beta^{2}_{2;K_{0}^{*}}x\bar{x}}}\right)\right\},$		(8)

in which $\tilde{m}=\hat{m}_{q}x+\hat{m}_{s}\bar{x}$ with $\hat{m}_{s}=370~{}\rm{MeV}$ and $\hat{m}_{q}=250~{}\rm{MeV}$ are constituent quark, $\bar{x}=(1-x)$ and $\hat{B}_{2;K_{0}^{\ast}}\simeq-0.025$ which is determined by $\langle\xi^{2}_{2;K_{0}^{\ast}}\rangle/\langle\xi^{1}_{2;K_{0}^{\ast}}\rangle$ whose rationality can be judged by the goodness of fit. Obviously, the behavior of $\phi_{2;K_{0}^{*}}^{\rm{(S2)}}(x,\mu)$ is determined by unknown model parameters $A_{2;K_{0}^{\ast}},\alpha_{2;K_{0}^{\ast}}$ and $\beta_{2;K_{0}^{\ast}}$, and the function $\varphi_{2;K_{0}^{*}}(x)=(x\bar{x})^{\alpha_{2;K_{0}^{*}}}\Big{[}C_{1}^{3/2}(\xi)+\hat{B}_{2;K_{0}^{*}}C_{2}^{3/2}(\xi)\Big{]}$. $\varphi_{2;K_{0}^{*}}(x)$ determines the WF’s longitudinal distribution by a factor $(x\bar{x})^{\alpha_{2;K_{0}^{*}}}$, which is close to the asymptotic form $\phi_{2;K_{0}^{*}}^{\rm{(S1)}}(x,\mu\rightarrow\infty)=6x\bar{x}$. Its rationality has been discussed in Ref. Zhong:2021epq . The unknown model parameters $A_{2;K_{0}^{\ast}},\alpha_{2;K_{0}^{\ast}}$, and $\beta_{2;K_{0}^{\ast}}$ can be obtained by fitting the first ten $\xi$ moments of $K_{0}^{\ast}(1430)$ through the least squares method and using the definition $\langle\xi^{n}_{2;K_{0}^{\ast}}\rangle|_{\mu}=\int^{1}_{0}dx\xi^{n}\phi_{2;K_{0}^{*}}(x,\mu)$, The specific fitting process can be found in Refs. Zhong:2022ecl ; Zhong:2021epq . Corresponding $\langle\xi^{n}_{2;K_{0}^{\ast}}\rangle|_{\mu}$ have been calculated based on BFT, which describes the nonperturbative effects through the vacuum expectation values of the background fields and the calculable perturbative effects by quantum fluctuations, the calculation also can be simplified through various gauge conditions. Compared to the traditional SVZ sum rules, the reliability of result $a_{n}^{2;K_{0}^{\ast}}(\mu)$ can be improved to a certain extent on this basis.

Finally, the twist-3 LCDAs $\phi_{3;K_{0}^{*}}^{p}(x,\mu)$ and $\phi_{3;K_{0}^{*}}^{\sigma}(x,\mu)$ for scalar mesons also can be expanded into a series of Gegenbauer polynomials Sun:2010nv ; Lu:2006fr ,

	$\displaystyle\phi_{3;K_{0}^{*}}^{p}(x,\mu)$	$\displaystyle=1+\sum_{n=1}^{\mathcal{N}=2}a_{n,p}^{3;K_{0}^{*}}(\mu)C_{n}^{1/2}(\xi),$		(9)
	$\displaystyle\phi_{3;K_{0}^{*}}^{\sigma}(x,\mu)$	$\displaystyle=6x\bar{x}\Big{[}1+\sum_{n=1}^{\mathcal{N}=2}a_{n,\sigma}^{3;K_{0}^{*}}(\mu)C_{n}^{3/2}(\xi)\Big{]}.$		(10)

where $a_{n,p/\sigma}^{3;K_{0}^{*}}(\mu)$ are determined by the relationship with $\langle\xi^{p(\sigma);n}_{3;K_{0}^{*}}\rangle$, which are also calculated by BFT method.

In order to carry out the next calculation, we adopt following basic input parameters: the quark masses $m_{c}(\bar{m}_{c})=1.27\pm 0.02~{}\rm{GeV}$, $m_{d}=4.67^{+0.48}_{-0.17}~{}\rm{MeV}$, and $m_{u}=2.16\pm 0.07~{}\rm{MeV}$ at $\mu=2~{}\rm{GeV}$, the meson masses $m_{D^{0}}=1864.84\pm 0.05~{}\rm{MeV}$, $m_{D^{+}}=1869.66\pm 0.05~{}\rm{MeV}$, and $m_{K_{0}^{\ast}}=1425\pm 50~{}\rm{MeV}$, the decay constants $f_{K_{0}^{\ast}}=427\pm 85~{}\rm{MeV}$ at $\mu_{0}=1~{}\rm{GeV}$, and $f_{D}=208.4\pm 1.5~{}\rm{MeV}$ Kuberski:2024pms .

Next step, for this part of the undetermined parameters of two twist-2 LCDAs, we calculated the moments $\langle\xi^{n}_{2;K_{0}^{\ast}}\rangle|_{\mu}$ with $n=(1,2,3,\cdots,10)$ in the framework of BFT, and nonperturbative vacuum condensates are up to dimension-six. Then, the Gegenbauer moments $a_{n}^{2;K_{0}^{\ast}}(\mu)$ of twist-2 LCDA $\phi_{2;K_{0}^{*}}^{\rm(S1)}(x,\mu)$ based on truncated form can be determined by the relationship between with $\langle\xi^{n}_{2;K_{0}^{\ast}}\rangle|_{\mu}$. Therefore, we have following results at $\mu_{k}=\sqrt{m_{D}^{2}-m_{c}^{2}}\approx 1.4~{}\rm{GeV}$ Huang:2022xny :

	$\displaystyle a_{1}^{2;K_{0}^{\ast}}(\mu_{k})=-0.408^{+0.087}_{-0.111},$
	$\displaystyle a_{2}^{2;K_{0}^{\ast}}(\mu_{k})=-0.018^{+0.013}_{-0.016},$
	$\displaystyle a_{3}^{2;K_{0}^{\ast}}(\mu_{k})=-0.321^{+0.048}_{-0.072}.$		(11)

In addition, the optimal model parameters $A_{2;K_{0}^{\ast}},\alpha_{2;K_{0}^{\ast}}$, and $\beta_{2;K_{0}^{\ast}}$ of $\phi_{2;K_{0}^{*}}^{\rm(S2)}(x,\mu)$ constructed by LCHO model are obtained by fitting the first tenth order $\langle\xi^{n}_{2;K_{0}^{\ast}}\rangle|_{\mu_{k}}$-moments with the least squares method at $\mu_{k}=1.4~{}\rm{GeV}$ Huang:2022xny ,

	$\displaystyle A_{2;K_{0}^{\ast}}=-147~{}{\rm GeV}^{-1},$	$\displaystyle\alpha_{2;K_{0}^{\ast}}=0.011,$
	$\displaystyle\beta_{2;K_{0}^{\ast}}=1.091~{}{\rm GeV},$	$\displaystyle P_{\chi^{2}_{\rm min}}=0.923671.$		(12)

The behavior of twist-2 LCDAs in different scenarios at $\mu_{0}=1~{}\rm{GeV}$ are also shown in Fig. 1. Meanwhile, QCDSRs Cheng:2005nb and CLF Chen:2021oul based on Gegenbauer polynomials and truncations with $\mathcal{N}=1~{}\rm{(I)}$ and $\mathcal{N}=3~{}\rm{(II)}$ are also presented in Fig. 1. As can be seen from the Fig. 1, the two scenarios twist-2 LCDAs of our predictions exhibit similar behaviors to the two truncated forms from QCDSRs Cheng:2005nb , respectively.

• •

  For the S1 case, the maximum value at $x=0.096$ with $\phi_{2;K_{0}^{*}}^{\rm(S1)}(x=0.096)=1.147$, the minimum value at $x=0.908$ with $\phi_{2;K_{0}^{*}}^{\rm(S1)}(x=0.908)=-1.081$, and zero point at $x=0.270$, $0.512$, $0.728$, respectively.

• •

  For the S2 case, the maximum value at $x=0.140$ with $\phi_{2;K_{0}^{*}}^{\rm(S2)}(x=0.140)=0.850$, the minimum value at $x=0.860$ with $\phi_{2;K_{0}^{*}}^{\rm(S2)}(x=0.86)=-0.779$, and zero point at $x=0.495$.

• •

  These points indicate that there exist SU_f(3) breaking effect, but the effect is relatively weak in $K_{0}^{\ast}(1430)$-meson twist-2 LCDA.

As for the $K_{0}^{\ast}(1430)$-meson twist-3 LCDAs $\phi_{3;K_{0}^{*}}^{p,\sigma}(x,\mu)$, we present the Gegenbauer moments at $\mu_{k}=1.4~{}\rm{GeV}$ as follows Han:2013zg :

	$\displaystyle a_{1,p}^{3;K_{0}^{\ast}}(\mu_{k})=0.012\pm 0.002,$	$\displaystyle a_{2,p}^{3;K_{0}^{\ast}}(\mu_{k})=0.163\pm 0.021,$
	$\displaystyle a_{1,\sigma}^{\sigma;K_{0}^{\ast}}(\mu_{k})=0.029\pm 0.011,$	$\displaystyle a_{2,\sigma}^{3;K_{0}^{\ast}}(\mu_{k})=0.019\pm 0.004.$		(13)

At the same time, if we consider $K^{*}_{0}(700)$ as the ground state of $q\bar{q}$-state from P1, $K_{0}^{*}(1430)$ will be the first excited $q\bar{q}$-states. Based on the standard procedure of QCD sum rule within background firld theory, we can get the following first three $K_{0}^{*}(700)$ meson twist-2 LCDA moments and Gegenbaure moments at initial scale $\mu_{0}=1~{}{\rm GeV}$,

	$\displaystyle\langle\xi^{1}\rangle_{2;\kappa}=-0.454^{+0.065}_{-0.064},$
	$\displaystyle\langle\xi^{2}\rangle_{2;\kappa}=+0.033^{+0.009}_{-0.010},$
	$\displaystyle\langle\xi^{3}\rangle_{2;\kappa}=-0.304^{+0.047}_{-0.046}.$		(14)
	$\displaystyle a_{1}^{2;\kappa}=-0.757^{+0.109}_{-0.106},$
	$\displaystyle a_{2}^{2;\kappa}=+0.096^{+0.025}_{-0.028},$
	$\displaystyle a_{3}^{2;\kappa}=-0.575^{+0.102}_{-0.100}.$		(15)

Then the first three $K_{0}^{*}(1430)$-meson twist-2 LCDA moments and Gegenbaure moments at initial scale $\mu_{0}=1~{}{\rm GeV}$ will be

	$\displaystyle\langle\xi^{1}\rangle_{2;K_{0}^{*}}=-0.017^{+0.042}_{-0.023},$
	$\displaystyle\langle\xi^{2}\rangle_{2;K_{0}^{*}}=-0.017^{+0.008}_{-0.010},$
	$\displaystyle\langle\xi^{3}\rangle_{2;K_{0}^{*}}=-0.044^{+0.043}_{-0.036}.$		(16)
	$\displaystyle a_{1}^{2;K_{0}^{*}}=-0.028^{+0.070}_{-0.038},$
	$\displaystyle a_{2}^{2;K_{0}^{*}}=-0.050^{+0.023}_{-0.029},$
	$\displaystyle a_{3}^{2;K_{0}^{*}}=-0.192^{+0.132}_{-0.139}.$		(17)

In calculating the above results, we take continuum threshold parameters $s_{\kappa}=2.4~{}{\rm GeV^{2}}$, $s_{K_{0}^{*}}=6~{}{\rm GeV^{2}}$. The Borel window is taken as $M^{2}\in[3,4]~{}{\rm GeV^{2}}$. The results are agree with values from Ref. Cheng:2005nb . The calculation can also be found in our previous work Huang:2022xny . We can find that the $K_{0}^{*}(1430)$-meson twist-2 LCDA moments and Gegenbaure moments changed largely, which indicates the interference effect is very obvious.

When treating $D\to K_{0}^{\ast}(1430)$ TFFs, the continuum threshold $s_{0}$ is taken as $4\pm 0.1~{}\rm{GeV}^{2}$, and the Borel window is taken as $M^{2}=17\pm 1~{}{\rm GeV^{2}}$ under the criterion of LCSR. Based on above parameters, we can get the $D\to K_{0}^{\ast}(1430)$ TFFs at a large recoil point, which is presented in Table 1, and the results of 3PSR Yang:2005bv , CLF Cheng:2003sm , and GFM Cheng:2002ai are used for comparison. In different twist-2 LCDAs, the contribution of the LCHO model $\phi_{2;K_{0}^{*}}^{\rm{(S2)}}(x,\mu)$ is greater than the truncated form $\phi_{2;K_{0}^{*}}^{\rm{(S1)}}(x,\mu)$, which accounted for $21.0\%$ and $12.2\%$ of the $f_{+}(0)$ results, and $68.7\%$ and $53.5\%$ of $f_{-}(0)$, respectively. Our results are in a agreement with 3PSR Yang:2005bv . If we consider P1 scenario, the $D\to K_{0}^{*}(700)$ vector TFF at large recoil point will be $f_{+}(0)=0.485^{+0.047}_{-0.026}$. This value is agree with the $0.48\leq f_{+}(0)\leq 0.55$ from Ref. Dosch:2002rh . With the same Borel parameter and threshold, the $D\to K_{0}^{*}(1430)$ vector TFF at large recoil point is $f_{+}(0)=-0.048^{+0.036}_{-0.036}$. The absolute value of this result is very small no more than 10% in comparing with $D\to K_{0}^{*}(700)$. So in this case, the interference between two processes is very evident that need to be further researched deeply.

Due to the mass of $K_{0}^{\ast}(1430)$, the physical allowable region in $D\to K_{0}^{\ast}(1430)\ell\nu_{\ell}$ is not large, $i.e.$, $q^{2}$ is from 0 to $(m_{D}-m_{K_{0}^{\ast}})^{2}\approx 0.193~{}\rm{GeV}^{2}$, and the LCSR method is reliable in lower and intermediate region. So in the next step, we adopt the simplified series expansion (SSE) to extrapolate the $f_{\pm}(q^{2})$ in the whole kinematical region. The TFFs is expanded as

$\displaystyle f_{\pm}(q^{2})=\frac{1}{1-q^{2}/m^{2}_{D}}\sum_{k=0,1,2}\beta_{k}z^{k}(q^{2},t_{0}).$

(18)

The $\beta_{k}$ are real coefficients and the function $z^{k}(q^{2},t_{0})=(\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}})/(\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}})$ with $t_{\pm}=(m_{D}\pm m_{K_{0}^{\ast}})^{2}$ and $t_{0}=t_{+}(1-\sqrt{1-t_{-}/t_{+}})$. Then, the behavior of TFF $f_{\pm}(q^{2})$ in the whole $q^{2}$ region can be determined which are shown in Fig. 2. In which the solid line is our central value, the shadow band is our uncertainty. At the same time, the behavior of $f_{+}(q^{2})$ from 3PSR Yang:2005bv and CLF Cheng:2003sm are also used for comparison. Within the same QCD sum rule appproach, the 3PSR results Yang:2005bv is very important here to make a detailed discussion. From the Fig. 2 (a), we can see that the TFF $f_{+}(q^{2})$ behavior of our predictions within S1 and S2 are agree with the 3PSR within uncertainties for the whole physical $q^{2}$-region especially for the S1 scenario. The results for S1 in this paper are roughly consistent with 3PSR in the large recoil region, while there is some discrepancy in the small recoil region. The reason may lies in the different method will tend to some different behaviors. For the 3PSR, non-perturbative parts of $f_{+}(q^{2})$ are mainly factorized into different dimension discrete vacuum condensates and the double Borel transformations are also made. With the less uncertainties from vacuum condensates, the main errors may comes from the Borel parameters and threshold. The 3PSR is an conventional approach in the QCD sum rule which lead to an accuracy results. In the LCSR approach, the non-perturbative parameters for $f_{+}(q^{2})$ are factorized into different twist LCDAs instead of vacuum condensates. With reasonable LCDA results, we can also achieve good predictions. Therefore, the two methods are actually equivalent in principle. Although there exist some differences in the behavior of $f_{+}(q^{2})$ for the 3PSR and LCSR, the differential decay widths and branching fractions are highly consistent with each other in the following discussion. Additionally, the results of CLF Cheng:2003sm are very close to the lower bound of our $f_{+}^{\rm(S1)}(q^{2})$. Overall, the result of $f_{+}^{\rm(S1)}(q^{2})$ is in better agreement with other groups, and $f_{+}^{\rm(S2)}(q^{2})$ is larger than $f_{+}^{\rm(S1)}(q^{2})$ because of the greater contribution of twist-2 LCDA.