$a_{0}(980)$-meson twist-2 distribution amplitude within the QCD sum rules and investigation of $D\to a_{0}(980)(\to\eta\pi)e^{+}\nu_{e}$

In the past few decades, especially for the discovery of resonance $a_{0}(980)$ state Ammar:1968zur , nature of scalar mesons below 1 GeV is a long-standing puzzle, which becomes one of hot topics in hadron physics. In the quark model scenario, the composition of $a_{0}(980)$ has turned out to be mysterious. Its intriguing internal structure allows tests of various hypotheses, such as quark-antiquark Achasov:1987ts ; Cheng:2005nb , tetraquark states Jaffe:1976ig ; Alford:2000mm ; Humanic:2022hpq ; Brito:2004tv ; Klempt:2007cp ; Alexandrou:2017itd , two-meson molecule bound states Weinstein:1982gc ; Branz:2007xp ; Dai:2012kf ; Dai:2014lza ; Sekihara:2014qxa and hybrid states Ishida:1995 . However, until now there is no definite conclusion of which scenario is correct or have no general agreement on the inner structure of $a_{0}(980)$. Among the hadronic decay processes including $a_{0}(980)$ state, semileptonic decay of charmed meson provides a simpler decay mechanism and final-state interactions, which can give an ideal platform for studying the meson’s properties.

From the experimental side, BESIII collaboration observed the charmless hadronic decay processes involving $a_{0}(980)$-meson, i.e. $D^{0}\to a_{0}(980)^{-}e^{+}\nu_{e}$ and $D^{+}\to a_{0}(980)^{0}e^{+}\nu_{e}$ with the significance up to $6.4\sigma$ and $2.9\sigma$, respectively BESIII:2018sjg . Experimental facilities have reported the most precise results on the semileptonic decay of $D_{(s)}$ to pseudoscalar and vector mesons. From theory point of view, these channels are straightforward to study because the internal stucture/quark content of the meson is a typical quark-antiquark system. But the quark structure of the scalar meson below 1 GeV has varied explanations, cf. the review “Scalar mesons below 1 GeV” of Particle Data Group (PDG) ParticleDataGroup:2022pth . Wang et al. conclude the ratio $(R)$ provides a model independent way to distinguish the quark components of the light scalar meson, i.e. the four-quark and two-quark pictures Wang:2009azc . And this value has not confirmed exactly from experiments. Nowadays, the two-quark picture are researched by some theoretical group, such as covariant confined quark model (CCQM) Soni:2020sgn , light-cone sum rule (LCSR) Cheng:2017fkw ; Huang:2021owr , AdS/QCD Momeni:2022gqb , perturbative QCD (pQCD) Rui:2018mxc and Bethe-Salpeter equation Santowsky:2021ugd . So it is also meaningful to reconsider the two-quark scenario in detail and to compare the experimental observable, which is also the starting point of this paper.

The important physical quantity for semileptonic decays $D\to a_{0}(980)\ell\bar{\nu}_{\ell}$ is the transition form factors (TFFs). As is known that the analogous semileptonic TFFs are also the essential ingredients for the indirect search of new physics beyond the Standard Model Aslam:2009cv . Therefore, an accurate TFFs is crucial for the semileptonic decay process. The LCSR approach is one of an effective method in dealing with heavy to light decays, which makes the operator production expansion (OPE) into the increasing twist light-cone distribution amplitudes (LCDAs), i.e twist-2, 3, 4 LCDAs. The LCSR is suitable in the large and middle recoil region. In this paper, we will adopt the LCSR to recalculate the $D\to a_{0}(980)$ TFFs and the experimental observable. One of the key quantities that characterize the TFFs is the twist-2 LCDA, which describes the dominant momentum fraction distribution for each part of a meson. In general, the $a_{0}(980)$-meson twist-2 LCDA $\phi_{2;a_{0}}(x,\mu)$ can be expanded as a Gegenbauer polynomial series

where $\bar{x}=(1-x)$ and $\xi=(2x-1)$. The zeroth-order Gegenbauer moment $a_{2;a_{0}}^{0}(\mu)$ is equals to zero, and the even Gegenbauer coefficients are highly suppressed, which exactly equal to zero under the approximation that $m_{1}\simeq m_{2}$ ($m_{1,2}$ are masses of two constituent quarks), and the LCDA of the scalar meson is then dominated by the odd Gegenabuer moments. In contrast, the odd Gegenbauer moments vanish for $\pi$ and $\rho$ mesons. Thus, the behavior of $\phi_{2;a_{0}}(x,\mu)$ tends to antisymmetric form under the exchange $u\to(1-u)$ in isospin symmetry. Currently, the $a_{0}(980)$-meson twist-2 LCDA is mainly coming from QCDSR by Cheng et al. Cheng:2005nb , which gives the first two nonzero $\xi$-moments ²²2The $\xi$-moments can be related with Gegenbauer moments directly, which the formulae can be found in Ref. Cheng:2005nb . In our previous work Zhong:2022lmn , based on the pionic leading-twist DA, we analyzed in detail the influence of different numbers of $\xi$-moments included in the fitting, and found that when the order of $\xi$-moments is not more than ten, the change of the number of $\xi$-moments has an obvious impact on the fitting result. When the order of $\xi$-moment is more than ten, the change of the number of $\xi$-moments has a very small impact on the fitting results. Thus, the higher-order $\xi$-moments should be given in order to get more accuracy $\phi_{2;a_{0}}(x,\mu)$ behavior and get more accuracy predictions for the processes involving $a_{0}(980)$-meson.

To achieve this target, the QCDSR under the framework of background field theory (BFT) is one of the effective way Novikov:1983gd ; Hubschmid:1982pa ; Govaerts:1983ka ; Govaerts:1984bk ; Ambjorn:1982bp ; Ambjorn:1982en ; Reinders:1984sr ; Elias:1987ac ; Huang:1986wm ; Huang:1989gv . In this approach, the quark and gluon fields are composed by background fields and quantum fluctuations around them. 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-range and short-range dynamics of OPE. At present, the BFT has been applied to calculate the LCDAs of pseudoscalar and vector/axial vector mesons Fu:2016yzx ; Hu:2021zmy ; Hu:2021lkl ; Zhong:2014jla ; Fu:2018vap ; Zhong:2016kuv ; Huang:2004tp ; Zhong:2011rg . Therefore, we will study the scalar $a_{0}(980)$-meson twist-2 LCDA by using BFT. To get a more accurate behavior of $\phi_{2;a_{0}}(x,\mu)$, a research scheme is used which combine the LCHO model and nonperturbative QCD sum rule for $\xi$-moments. Specifically, a new QCD sum rule formula will be used due to the zeroth-order $\xi$-moments can not be normalized in the whole Borel parameters region. In this paper, we will calculate the first five-order nonzero $\xi$-moments and determine the LCHO model parameters with the least squares method. This scheme is used in the pion case Zhong:2021epq , and subsequently for the kaon leading-twist DA Zhong:2022ecl and $a_{1}(1260)$-meson longitudinal twist-2 DA Hu:2021lkl .

The remaining parts of this paper are organized as follows. In Sec. II, we calculate the $a_{0}(980)$-meson twist-2 LCDA moments and introduce the LCHO model. A new improved model is proposed and the model parameters shall be obtained by fitting moment with the least square method. Section III gives the numerical results and discussions, which include the transition form factor, decay width, decay branching ratio of semileptonic decays $D\to a_{0}(980)\ell\bar{\nu}_{\ell}$. Meanwhile, the forward-backward asymmetries, the $q^{2}$ differential flat terms and lepton polarization asymmetry of the semileptonic decay $D\to a_{0}(980)\ell\bar{\nu}_{\ell}$ are also given. Section IV is for a brief summary.

The $a_{0}(980)$-meson have three types of states, one is $a_{0}(980)^{0}$-meson with $(\bar{u}u-\bar{d}d)/\sqrt{2}$ component, the other is $a_{0}(980)^{-}$-meson with $(\bar{u}d)$ component, and the third one is $a_{0}(980)^{+}$-meson with $(\bar{d}u)$. Based on the basic procedure of QCD sum rules, one can adopt the following two-point correlator to derive the sum rules for $a_{0}(980)$-meson twist-2 LCDA moments $\langle\xi^{n}_{2;a_{0}}\rangle|_{\mu}$, which can be read off,

where $z^{2}=0$ and $n$ take odd numbers. The currents are taken as $J^{V}_{n}(x)=\bar{q}_{1}(x)/\!\!\!z(iz\cdot\tensor{D})^{n}q_{2}(x)$ and $J^{S}_{0}(0)=\bar{q}_{1}(0)q_{2}(0)$, which are mainly come from definitions of the scalar $a_{0}(980)$-meson twist-2 LCDA $\phi_{2;a_{0}}(x,\mu)$ and twist-3 LCDA $\phi_{3,a_{0}}^{p}(x,\mu)$, which can be written as Cheng:2005nb

From one side, one can apply the OPE for the correlator, e.g. Eq. (2) in deep Euclidean region $q^{2}\ll 0$. The calculation is carried out in the framework of BFT Huang:1989gv . Based on the basic assumptions and Feynman rules of BFT, the correlator can be expanded into three terms including the quark propagators and the vertex operators,

where “Tr” indicates trace for the $\gamma$-matrix and color matrix, $S^{q_{1}}_{F}(0,x)$ and $S^{q_{2}}_{F}(x,0)$ indicate the light-quark propagator from $x$ to 0 and from 0 to $x$. The $/\!\!\!z(iz\cdot\tensor{D})^{n}$ are the vertex operators from current $J_{n}(x)$. The expressions up to dimension-six of the quark propagator, the vertex operator, and the vacuum matrix elements such as $\langle 0|\bar{q}_{1}(x)q_{1}(0)|0\rangle$ and $\langle 0|\bar{q}_{2}(x)q_{2}(0)|0\rangle$ have been derived in our previous works Zhong:2011rg ; Zhong:2014jla ; Zhong:2021epq ; Hu:2021zmy . By substituting those corresponding formula into Eq. (5), the OPE of correlator (2), $I^{\rm QCD}_{2;a_{0}}(q^{2})$, can be obtained. Meanwhlie, we take $q_{1}=q_{2}=q$ stand for the light $u$, $d$-quark. On the other hand, one can insert a complete set of $a_{0}(980)$-meson intermediated hadronic states with the same $J^{P}$ quantum number into the correlator and obtain the hadronic expression

with $m_{q}=m_{q_{1}}=m_{q_{2}}$ is the light quark mass, where the slight difference between mass of $u$-quark and $d$-quark is ignored in this paper. Meanwhile, the $s_{a_{0}}$ stands for the continuum threshold. Here the definitions of moments for $a_{0}(980)$-meson twist-2, 3 LCDAs are used, which have the following formula

where $m_{a_{0}}$ and $\bar{f}_{a_{0}}$ stand for $a_{0}(980)$-meson mass and its decay constant, respectively. By considering the dispersion relation and performing Borel transformation, we can get the sum rule for $\langle\xi_{2;a_{0}}^{n}\rangle|_{\mu}\langle\xi_{3;a_{0}}^{p;0}\rangle|_{\mu}$,

with $\tilde{\psi}(n)=\psi(\frac{n+1}{2})-\psi(\frac{n}{2})+(-1)^{n}\ln 4$. For a more thorough considering the sum rule for $\langle\xi^{n}_{2;a_{0}}\rangle|_{\mu}$, it might be convenient to calculate $\langle\xi^{p;0}_{3;a_{0}}\rangle|_{\mu}$. To achieve this target, one can use the correlator $\Pi^{(0,0)}_{3;a_{0}}(z,q)=i\int d^{4}xe^{iq\cdot x}\langle 0|T\{J^{S}_{0}(x),J^{S{\dagger}}_{0}(0)\}|0\rangle$ for the $a_{0}(980)$-meson twist-3 LCDA 0th moment. Followed by the basic procedure of the QCDSR, the expression of the 0th moment of scalar meson $a_{0}(980)$-meson twist-3 LCDA can be obtained,

Here, the light quark is taken as $q=(u,d)$ and corresponding vacuum condensates are given in the next Section. Since higher-order and higher-dimensional corrections are difficult to calculate completely, the moments $\langle\xi^{n}_{2;a_{0}}\rangle|_{\mu}$ of the sum rule (9) cannot be normalized in the intire Borel parameter region. To get a more accurate moments $\langle\xi^{n}_{2;a_{0}}\rangle|_{\mu}$ for sum rule, we can use the following expression

This method can eliminate the systematic errors caused by many factors. The discussion for the pion and kaon cases can be found in our previous work Zhong:2021epq ; Zhong:2022ecl .

Furthermore, $a_{0}(980)$-meson twist-2 DA $\phi_{2;a_{0}}(x,\mu)$ describes the momentum fraction distribution of partons in $a_{0}(980)$-meson for the lowest Fock state. Due to the $\phi_{2;a_{0}}(x,\mu)$ is the universal nonperturbative physical quantity, it should be researched by the nonperturbative QCD approach. Normally, one can study the $\phi_{2;a_{0}}(x,\mu)$ based on the combination of nonperturbative QCD and phenomenological model. Meanwhile, conformal expansion of LCDAs Gegenbauer polynomials makes the higher-order Gegenbauer moments unreliable. To improve this situation, the LCHO is adopted to determine $a_{0}(980)$-meson twist-2 LCDA. Referring to the LCHO model of the pion leading-twist WF raised in Refs. Wu:2010zc ; Wu:2011gf , one can start with the Brodsky-Huang-Lepage (BHL) prescription, which assuming there is a connection between the equal-times wave function in the rest frame and the light-cone wave function BHL . it can be expressed as:

where $\textbf{k}_{\bot}$ is transverse momentum. Furthermore, $\lambda_{1}$ and $\lambda_{2}$ are the helicities of the two constituent quark. $\chi_{2;a_{0}}^{\lambda_{1}\lambda_{2}}(x,\textbf{k}_{\bot})$ stands for the spin-space WF that comes from the Wigner-Melosh rotation. The different forms of $\lambda_{1}\lambda_{2}$ are shown in Table 1, which can also been seen in Refs. Huang:1994dy ; Cao:1997hw ; Huang:2004su ; Wu:2005kq . The spin-space WF, e.g. $\sum_{\lambda_{1}\lambda_{2}}\chi_{2;a_{0}}^{\lambda_{1}\lambda_{2}}(x,\textbf{k}_{\bot})={\hat{m}_{q}}^{2}/(\textbf{k}^{2}_{\bot}+{\hat{m}_{q}}^{2})^{1/2}$. Then, by combing the spatial WF $\Psi^{R}_{2;a_{0}}(x,\textbf{k}_{\bot})=A_{2;a_{0}}\varphi_{2;a_{0}}(x)\exp[-(\textbf{k}^{2}_{\bot}+{\hat{m}_{q}}^{2})/(8\beta_{2;a_{0}}^{2}x\bar{x})]$, the $a_{0}(980)$-meson WF will be obtained. Here, the $A_{2;a_{0}}$, $\hat{m}_{q}$ stand for the normalization constant and light quark mass. The final $a_{0}(980)$-meson twist-2 LCDA can be obtained by using the relationship between the twist-2 LCDA and WF of $a_{0}(980)$-meson, e.g. integrated over the squared transverse momentum, which can be expressed as

where ${\rm Erf}(x)=2\int^{x}_{0}e^{-t^{2}}dx/{\sqrt{\pi}}$ is the error function, and $\varphi_{2;a_{0}}(x)=(x\bar{x})^{\alpha_{2;a_{0}}}C_{1}^{3/2}(2x-1)$. Based on the experience of other mesons Huang:2013gra ; Huang:2013yya ; Wu:2012kw ; Zhong:2015nxa ; Zhang:2021wnv ; Hu:2021lkl ; Zhong:2018exo ; Zhong:2022ecl ; Zhong:2021epq , we take the wavefunction parameter $\beta_{2;a_{0}}=0.5$. Whether the value of $\beta_{2;a_{0}}$ is accurate can be judged by goodness of fit $P_{\chi^{2}}$. The free parameters $\alpha_{2;a_{0}}$ and $A_{2;a_{0}}$ can be obtained by fitting the moments $\langle\xi^{n}_{2;a_{0}}\rangle|_{\mu}$ with the least squares method directly.

In deriving the complete expression for $D\to a_{0}(980)$ TFFs, one can take the following correlator

with $J_{n}(x)=\bar{q}_{1}(x)\gamma_{\mu}\gamma_{5}c(x)$, $j_{n}^{\dagger}(0)=\bar{c}i\gamma_{5}q_{2}(0)$. Followed by the standard LCSR approach, one can make the OPE near the light-cone in the space-like region, and insert a complete set of $D$-meson states in the physical region. After performing the Borel transformation, we can get the $D\to a_{0}(980)$ TFFs $f_{\pm}^{D\to a_{0}}(q^{2})$ up to twist-3 accuracy, which can be read off

Here the abbreviation $s(u)=(m_{b}^{2}+u\bar{u}m_{a_{0}}^{2}-\bar{u}q^{2})/u$ is used. The lower limits of the integration is $u_{0}=\{[(s-q^{2}-m_{a_{0}}^{2})^{2}+4m_{a_{0}}^{2}(m_{c}^{2}-q^{2})]^{1/2}-(s-q^{2}-m_{a_{0}}^{2})\}/(2m_{a_{0}}^{2})$. Here, $m_{D}$ and $f_{D}$ are the mass and decay constant of $D$-meson, $m_{c}$ is the $c$-quark mass, $\bar{u}=(1-u)$, $s_{0}$ stands for the continuum threshold. $\phi_{2;a_{0}}$ and $\phi_{3,a_{0}}^{\sigma}$, $\phi_{3;a_{0}}^{p}$ are twist-2 and twist-3 LCDAs, respectively. The twist-3 LCDA can be found in Refs. Han:2013zg ; Lu:2006fr . Here, we have a notation that the analytical expressions for the sum rules of the $D\to a_{0}(980)$ TFFs, i.e. Eqs. (15) and (16) are equivalent with the tree-level LCSR for the exclusive heavy-to-light $B\to S$ TFFs ($S=a_{0}(1450),\,K_{0}^{\ast}(1430),\,f_{0}(1500)$) have been previously constructed in Ref. Wang:2008da , without the surface term coming from the twist-3 LCDA $\Phi_{S}^{\sigma}(u)$ contributions. In this paper, the TFFs analytical expressions are derived based on the pion cases Duplancic:2008ix , where the power of denominator is reduced by taking the derivative of the final state light-cone distribution amplitude.

Then, the explicit expression for the full differential $D\to a_{0}(980)\ell\bar{\nu}_{\ell}$ decay width have the following form

with the angular coefficient functions $a_{\theta_{\ell}}(q^{2})$, $b_{\theta_{\ell}}(q^{2})$ and $c_{\theta_{\ell}}(q^{2})$ are Becirevic:2016hea

Here ${\cal N_{\rm ew}}={G^{2}_{F}|V_{ub}|^{2}m^{3}_{D}}/{256\pi^{3}}$ with $|V_{cd}|$ and $G_{F}$ stand for CKM matrix element and fermi coupling constant. $m_{\ell}$ and $\theta_{\ell}$ stand for the leptonic mass and helicity angle. In the massless lepton limit, the angular functions $b_{\theta_{\ell}}(q^{2})=0$, $a_{\theta_{\ell}}(q^{2})+c_{\theta_{\ell}}(q^{2})=0$. In addition, $\lambda\equiv\lambda(1,{m^{2}_{a_{0}}}/{m^{2}_{D}},{q^{2}}/{m^{2}_{D}})$ with $\lambda(a,b,c)\equiv a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$. The $f_{0}^{D\to a_{0}}(q^{2})$ in the expression can be expressed as $f_{0}^{D\to a_{0}}(q^{2})=f_{+}^{D\to a_{0}}(q^{2})+q^{2}/{(m^{2}_{B}-m^{2}_{D})}f_{-}^{D\to a_{0}}(q^{2})$ Fu:2013wqa . After integrating over the helicity angle $\theta_{\ell}\in[-1,1]$, we can get the expression $q^{2}$-dependence differential $D\to a_{0}(980)\ell\bar{\nu}_{\ell}$ decay width Cheng:2017fkw ; Huang:2021owr

Furthermore, the $D\to a_{0}(980)$ TFFs are also the basic component of indirect search for new physics Beyond the Standard Model (BSM) phenomenologically. So the angular observables which are sensitive to BSM, i.e. the normalized forward-backward asymmetries, the $q^{2}$-differential flat terms and lepton polarization asymmetry ${\cal A}^{D\to a_{0}(980)\ell\bar{\nu}_{\ell}}_{\rm FB}$, ${\cal F}^{D\to a_{0}(980)\ell\bar{\nu}_{\ell}}_{H}$, ${\cal A}^{D\to a_{0}(980)\ell\bar{\nu}_{\ell}}_{\lambda_{\ell}}$ of the semileptonic decay $D\to a_{0}(980)\ell\bar{\nu}_{\ell}$, respectively. The relationships between these observables and TFFs are as follows Cui:2022zwm :