Probing isovector scalar mesons in the charmless three-body $B$ decays

It is known that the scalar mesons with the masses below and near $1\,\mathrm{GeV}$, saying the isoscalar mesons $\sigma/f_{0}(500)$ and $f_{0}(980)$, the isovector $a_{0}(980)$ and the isodoublet $\kappa$, form a $SU(3)$ flavor nonet, meanwhile, the mesons heavier than $1\,\mathrm{GeV}$ with including $f_{0}(1370),f_{0}(1500)$, $a_{0}(1450)$ and $K^{\ast}_{0}(1430)$ make up another nonet. The underlying assignment of the heavier nonet is almost accepted as the quark-antiquark configuration replenished with some possible gluon content JaffeIG ; CloseZU ; AchasovHM ; AchasovFH , while the inner nature of scalar mesons in the lighter nonet is still not clear WeinsteinGC ; WeinsteinGD ; Agaev:2017cfz , even though the compact tetraquark state Alford:2000mm ; Maiani:2004uc ; Maiani:2007iw and the $K\bar{K}$ bound state Weinstein:1990gu are the most favorite two candidates nowadays. This is easy to understand from the views of spectral analysis at low energy because the scalar meson in $q\bar{q}$ configuration has a unit of orbital angular momentum which increases their masses, in contrast, it is not necessary to introduce the orbital angular momentum when the scalar meson is being in $q^{2}\bar{q}^{2}$ configuration ChengNB . The case becomes different in the weak decays like $B\to f_{0}(980)l\nu$ with large recoiling, where the conventional $q\bar{q}$ assignment can be expected to be dominated in the energetic $f_{0}(980)$ since the possibility to form a tetra-quark state is power suppressed with comparing to the state of quark pair Cheng:2019tgh , meanwhile, the final state interaction (FSI) is weak too. But this argument encounters challenge from the perturbative QCD (PQCD) calculation of $B\to a_{0}(980)K$ decays Shen:2006ms , where the theoretical predictions of branching fractions are much larger than that of the measured upper limits. We would like to comment that their calculation is carried out in the static $a_{0}(980)$ approximation while the experiment measurement is actually fulfilled by the $\pi\eta$ invariant mass spectral. It is apparent that the salient property of scalar mesons, say, the large decay width which cause a strong overlap between resonances and background, and subsequently influence the PQCD prediction.

The width effect of intermediate resonant states have been studied in three-body $B$ decays with a large number of channels by variable theoretical approaches based on QCD, due to the significant physics to understand the hadron structures and also the matter-antimatter asymmetry. We here highlight some developments in this research filed in the order of different theoretical approaches.

PQCD

A global analysis of three-body charmless decays in the type of $B\to V\left[\to P_{1}P_{2}\right]P_{3}$¹¹1Here $V,P$ denote the vector and pseudoscalar meson, respectively, and $S$ indicates the scalar meson in the following. In the fit, only the $P_{1}P_{2}=\pi\pi,\pi K,K\bar{K}$ channels are considered due to the experiment precision. is performed Li:2021cnd to determine the lowest several gegenbauer moments of two-meson system, which are the nonperturbative inputs describing the non-asymptotic QCD correction effect in the light-cone distribution amplitudes (LCDAs). In Ref.Rui:2021kbn , the factorization formulas of PQCD is expanded in the four-body $B$ decay to two $\left[K\pi\right]_{S,P}$ pairs with the invariant mass around the $K^{\ast}(892)$ resonance, some further observations like the triple-product asymmetries and the $S$-wave induced direct CP asymmetries are presented with the interference between different helicity amplitudes. Motivated by the measurement of significant derivations from the simple phase-space model in the channels $B\to K\bar{K}P_{1}$ and $B_{(s)}\to D_{(s)}P_{1}P_{2}$ at B factories and LHC, the virtual contribution clarified by the experiment collaborations is understood theoretically by the Breit-Wigner-tail (BWT) effects from the corresponding intermediate resonant states, say $\rho,\omega$ and $D_{(s)}^{\ast}$, respectively Wang:2020nel ; Chai:2021kie .

QCDF

The QCD factorization (QCDF) formula of amplitudes in three-body $B$ decays Klein:2017xti is parameterised in a new way where the contributions from valence $u$ and $c$ quark are separated, and a new source of CP violation can be generated via the strong phase with the opening of $D\bar{D}$ threshold in the high invariant mass region Mannel:2020abt . Motived by the NNLO $\alpha_{s}(m_{b})$ correction and the finite width effect, three-body $B$ decay is studied from the point of view of factorisation for the heavy-to-heavy $B\to D\rho\left[\to\pi\pi\right],DK^{\ast}\left[\to K\pi\right]$ decays in the kinematics with small invariant mass of dimeson system Huber:2020pqb . Very recently, a novel observation named the forward-backward asymmetry induced $CP$ asymmetry (FBI-CPA) is introduced in the three-body heavy meson decays, the estimation based on the generalized factorization approach implies that the FBI-CPA in the channel $D^{\pm}\to K^{+}K^{-}\pi^{\pm}$ is about a milli, which is at the same order of current experiment measurement capability Zhang:2021zhr . In Refs. Cheng:2020iwk ; Cheng:2022vbw , the finite-width effects of intermediate resonant states in three-body $B/D$ decays is expressed by a correlation parameter $\eta_{R}$ and the evaluation is carried out in QCDF.

LCSRs

The width effect of intermediate resonant $\rho$ and its radial excited states is discussed in detail in the $P$-wave $B\to\pi\pi$ transition form factors from the $B$ meson light-cone sum rules (LCSRs) approach Cheng:2017smj , revealing the sizeable effects from width and background ($20\%-30\%$) to the conventional treatment in the single narrow-width approximation for the LCSRs prediction of the $B\to\rho$ transition form factors. This result is confirmed by the other independent LCSRs with dipion distribution amplitudes (DAs) where the hadronic dipion state has a small invariant mass and simultaneously a large recoil Hambrock:2015aor ; Cheng:2017sfk . The further studies are carried out for the $P$-wave $B\to K\pi$ form factors with the isodouble intermediate resonances $K_{0}^{\ast}$ Descotes-Genon:2019bud and the $B_{s}\to K\bar{K}$ form factors with the isoscalar scalar intermediate resonances $f_{0}(980)$ and $f_{0}(1450)$ Cheng:2019tgh .

The above considerations mainly focus on the $P$-wave and isoscalar $S$-wave contributions from the intermediate resonant states, while the study of isovector scalar intermediate resonance is still missing. In this paper we will demonstrate this issue in the framework of PQCD approach. The motivations of this study is twofold. Firstly, we perform the PQCD prediction of $B\to a_{0}(980)\left[\to\eta\pi\right]K$ decays go beyond the single pole approximation, trying to explain the measurement status. Secondly, we consider the roles of $a_{0}(1450)$ and $a_{0}(1950)$ in the $B\to{\bar{K}}KK$ decays inspired by the recent measurements of charm meson decays where $a_{0}(1450)$ and $a_{0}(1950)$ are observed in the $K\bar{K}$ invariant mass spectral CLEO:2008msk ; LHCb:2015lnk ; BaBar:2015kii , supplementing to the $B\to\eta\pi K$ decays observed firstly at Crystal Barrel Collaboration long time ago CrystalBarrel:1994arw ; CrystalBarrel:1995dzq . The study would be executed in parallel by taking two different scenarios of $a_{0}$ states, where the first one says that $a_{0}(980)$ is the lowest lying $q\bar{q}$ state and $a_{0}(1450)$ is the first excited state, and the second one states that $a_{0}(1450)$ and $a_{0}(1950)$ are the lowest lying $q\bar{q}$ state and the first excited state, respectively. Our calculations show that the $q{\bar{q}}$ configuration of $a_{0}(980)$ is not be excluded by the available measurements in $B$ decays, which confirms the statements we made above. Predictions in this work would help us to probe the inner structure of energetic isovector scalar mesons. For examples, (a) the channel $B^{0}\to a_{0}^{-}(980)\left[\to\pi^{-}\eta\right]\pi^{+}$ has the largest branching fraction under the $q{\bar{q}}$ configuration of $a_{0}(980)$, (b) the branching fractions of channels $B^{0}\to a_{0}^{\pm}(1450)\left[\to K^{\pm}{\bar{K}}^{0},\pi^{\pm}\eta\right]\pi^{\mp}$ obtained in the second scenario is about three times larger in magnitude than that obtained in the first scenario, even though the result obtained from two scenarios are close to each other in the most channels with the intermediate state $a_{0}(1450)$, (c) the branching fractions of channels $B^{+}\to a_{0}^{+}(1950)\left[K^{+}{\bar{K}}^{0}/\pi^{+}\eta\right]K^{0}$ are linear dependent on the partial width $\Gamma_{a_{0}(1950)\to KK/\pi\eta}$ in the second scenario.

This article is organized as follows. In section II, the framework of PQCD approach to deal with the resonance contribution in three-body $B$ decays is briefly described in turns of kinematics and dynamics. Section III presents the PQCD predictions of the $B\to a_{0}\left[\to K\bar{K},\eta\pi\right]h$ decays with some discussions. We summary in section IV. The PQCD predictions on $B_{s}$ decays are presented in appendix A, and the factorization formulas of the related quasi-two-body decay amplitudes are listed in appendix B.

Concerning three-body $B$ decays, there are three typical kinematical configurations in the physical Dalitz plot of two independent invariant mass by considering the four-momentum conservation, in which only the kinematical region with collinear decay products can be calculated reliably from the perturbative theory based on the factorization hypothesis Chai:2021kie . The other two kinematical regions with the three energetic decay products and a soft decay product configurations are either in lack of the rigorous factorization proof or beyond the available perturbative picture of heavy meson decays. Collinear decay products means that two energetic hadrons move ahead with collinear momenta while the rest one recoiling back²²2$E_{i}\sim m_{B}/2$ and $E_{j}+E_{k}\sim m_{B}/2$ in the massless approximation of final mesons., corresponding to the intermediate parts of three edges in the Dalitz plot.

The matrix element from vacuum to collinear two meson system sandwiched with certain two quark operators is defined by the dimeson DAs, the chirally even two quark dimeson DA is quoted for example as Polyakov:1998ze

where the indexes $f,f^{\prime}$ respect the (anti-)quark flavor; $a,b$ indicate the electric charge of each meson, $\kappa_{ab}$ is the isospin symmetry coefficient which in the case of dipion system reads $\kappa_{+-/00}=1$ and $\kappa_{+0}=\sqrt{2}$, $p_{R}=k_{1}+k_{2}$ is the total momentum of dimeson state, $\tau=1/2,\tau_{3}/2$ correspond to the isoscalar and isovector dimeson DAs, respectively. The generalized dimeson DA $\Phi_{\parallel}^{ab,ff^{\prime}}$ is characterised by three independent kinematical variables, saying the momentum fraction $z$ carried by the antiquark, the longitudinal momentum fraction carried by one of the mesons $\zeta=p_{1}^{+}/p_{R}^{+}$ and the invariant mass squared $s=p_{R}^{2}$. Besides the conventional Gegenbauer expansion stemmed from the eigenfunction of QCD evolution equation, the partial wave expansion considered in the dimeson system contributes the other Legendre polynomial $C_{l}^{1/2}$. The double expansion of two quark dimeson DA is written as

here the even Gegenbauer index $n$ and the odd partial-wave index $l$ are guaranteed by the $C$ parity. For the expansion coefficients $B_{nl}$, they have the similar scale dependence as the Gegenbauer moments of single pion and rho mesons. In the narrow width approximation in the vicinity of the resonance, dimenson DAs reduce to the DAs of the relative resonance, indicating that the Gegenbuer moments of the intermediate resonance is actually proportional to the expansion coefficient at zero point with the lowest partial wave, says $a_{n}^{R}(\mu)\propto B_{n1}(s=0,\mu)$. In this way, the decay constant of intermediate resonance is proportional to the product of its decay width with the imaginary part of first expansion coefficient at the resonant pole, that is $f_{R}\propto\Gamma_{R}\,{\rm Im}[B_{01}(m_{R}^{2})]$ Cheng:2019hpq .

With this definition, the dimeson DAs are the most general objects to describe the dimeson mass spectrum in hard production processes whose asymptotic formula indicates the information of the deviation from the unstable intermediate resonant meson DAs. After integrating over the momentum fraction of antiquark, the isovector scalar dimeson DA in our interest is normalised to timelike meson form factor as

where the timelike form factor at zero energy point is normalised to unit as ${\it\Gamma}_{M_{1}M_{2}}^{I=1}(0)=1$. When the invariant mass of dimeson system is small, the higher $\mathcal{O}(s)$ terms in the expansion of coefficient $B_{nl}(s,\mu)$ around the resonance pole can be safely neglected due to the large suppression $\mathcal{O}(s/m_{b}^{2})$ in contrast to the energetic dimeson system in $B$ decay, so the relation $B_{n1}(s,\mu)\to a_{n}(\mu)\,{\it\Gamma}_{M_{1}M_{2}}^{I=1}(s)$ can be obtained in the lowest partial wave approximation. This argument induces the basic assumption in PQCD that the energetic dimeson DAs can be deduced from the DAs of resonant meson by replacing the decay constant by the timelike form factor.

The isovector scalar form factor of $K\bar{K}$ and $\pi\eta$ systems are defined by the local matrix elements sandwiched by two quark operator Donoghue:1990xh ; Albaladejo:2015aca

with the normalization conditions ${\it\Gamma}^{I=1}_{K\bar{K}}(0)=1$ and ${\it\Gamma}^{I=1}_{\pi\eta}(0)=\sqrt{6}/3$. In the single resonance approximation, we insert a $a_{0}$ state in the matrix elements

and ultimately arrive at

Several comments are supplemented in turns to demonstrate this expression.

• •

  The decay constants of scalar meson are defined with the scalar and vector currents,

  	$\displaystyle\langle S|\bar{u}(0)\frac{\tau_{3}}{2}d(0)|0\rangle=m_{S}\bar{f}_{S}\,,$
  	$\displaystyle\langle S(p)|\bar{u}(0)\gamma_{\mu}\frac{\tau_{3}}{2}d(0)|0\rangle=p_{\mu}f_{S}\,.$		(7)

  They are related by the equations of motion $\frac{m_{S}f_{S}}{m_{u}-m_{d}}=\bar{f}_{S}(\mu)$, indicating that the neutral scalar meson can not be produced via the vector current because of the charge conjugation invariance or the conservation of vector current, but the constant $\bar{f}_{S}$ is still finite.

• •

  Under the narrow $a_{0}$ approximation, the matrix element of strong decay is defined by the coupling Wang:2020saq

  $\displaystyle\langle K^{-}K^{0}(\pi^{+}\eta)|a_{0}^{-}\rangle=g_{a_{0}K\bar{K}(\pi\eta)}=\sqrt{\frac{8\pi m^{2}_{a_{0}}\Gamma_{{a_{0}}\to K\bar{K}(\pi\eta)}}{q_{0}}}\,$

  (8)

  with the energy independent partial decay width³³3The partial widths of $a_{0}^{0}\to K\bar{K}$ decays have the relations $\Gamma_{a_{0}^{0}\to K^{+}K^{-}}=\Gamma_{a_{0}^{0}\to K^{0}\bar{K}^{0}}=\Gamma_{a_{0}\to K\bar{K}}/2$. $\Gamma_{{a_{0}}\to K\bar{K}(\pi\eta)}$. In the definition, $q_{0}=q(m^{2}_{a_{0}})$ is the magnitude of daughter meson ($K(\pi)$ or $\bar{K}(\eta)$) momentum

  $\displaystyle q(s)=\frac{1}{2}\sqrt{\left[s-(m_{K(\pi)}+m_{\bar{K}(\eta)})^{2}\right]\left[s-(m_{K(\pi)}-m_{\bar{K}(\eta)})^{2}\right]/s}\,$

  (9)

  at $a_{0}$ mass. We take the renormalized mass of $a_{0}$ rather than the pole mass obtained from $T$-matrix analysis, since the mass and width parameter are strongly distorted with lying just below the opening of $K{\bar{K}}$ channel and hence generating an important cusp-like behaviour in the resonant amplitude Abele:1998qd . Actually, $q=\sqrt{s}\beta(s)$ with $\beta(s)$ being the nondimensional phase space factor of $K\bar{K}(\pi\eta)$ system, which reflects the information of momentum difference described by the variable $\zeta$ mentioned in the dimeson DAs.

• •

  We take the conventional energy-dependent Breit-Wigner denominator for $a_{0}^{\prime}$ and $a_{0}^{\prime\prime}$ mesons⁴⁴4Hereafter we use the abbreviations $a_{0},a_{0}^{\prime}$ and $a_{0}^{\prime\prime}$ to denote $a_{0}(980),a_{0}(1450)$ and $a_{0}(1950)$, respectively.,

  $\displaystyle\mathcal{D}_{a_{0}^{\prime}}=m_{a_{0}^{\prime}}^{2}-s-im_{a_{0}^{\prime}}\,\Gamma^{\rm tot}_{a_{0}^{\prime}}\,\frac{q(s)}{q_{0}}\frac{m_{a_{0}^{\prime}}}{\sqrt{s}}\,,$

  (10)

  where $\Gamma^{\rm tot}_{a_{0}^{\prime}}$ is the total decay widths of resonant state meson $a_{0}^{\prime}$. For the meson $a_{0}(980)$, we consider the Flatt${\rm\acute{e}}$ model Flatte:1976xu

  $\displaystyle\mathcal{D}_{a_{0}}=m_{a_{0}}^{2}-s-i(g_{\pi\eta}^{2}\beta_{\pi\eta}+g_{K{\bar{K}}}^{2}\beta_{K{\bar{K}}})\,,$

  (11)

  the coupling constants $g_{\pi\eta}=0.324$ GeV and $g_{K{\bar{K}}}^{2}/g_{\pi\eta}^{2}=1.03$ is fixed by the isobar model fits Abele:1998qd . Furthermore, we can get $g_{a_{0}\pi\eta}=2.297$ GeV and $g_{a_{0}K{\bar{K}}}=2.331$ GeV with the relations $g_{K{\bar{K}}}=g_{a_{0}K{\bar{K}}}/(4\sqrt{\pi})$ and $g_{\pi\eta}=g_{a_{0}\pi\eta}/(4\sqrt{\pi})$. We mark that, in the $a_{0}\to\pi\eta$ channel, the phase factor $\beta_{K{\bar{K}}}$ could also be pure imaginary number when the invariant mass of $\pi\eta$ state is small than the threshold value of $K{\bar{K}}$ state, the contribution from this region interacts destructively with that from the rest region of $\pi\eta$ invariant mass.

With rearranging the kinematical variable $\zeta$ into the daughter meson momentum $q(s)$ and considering the $SU(3)$ symmetry, the matrix element from vacuum to S-wave $K\bar{K}/\pi\eta$ state can be decomposed as  ChengNB

In the lowest partial-wave accuracy, the twist 2 LCDA is written as Cheng:2013fba

with $B_{0}(\mu)\equiv f_{S}/\bar{f}_{S}(\mu)\gg 1$. It is clear that the even Gegenbauer coefficients $B_{m}$ are suppressed and the odd Gegenabauer moments is dominated in the twist 2 LCDA of scalar meson, this is definitely different from the $\pi$ and $\rho$ mesons in which the odd moments vanish. The twist 3 LCDAs are

The definitions of $B$ meson and light meson wave functions and the models of their LCDAs, as well as the basic procedures of PQCD approach to deal with the so called quasi two-body $B$ decays as a marriage problem, can be found in detail in Ref. Chai:2021kie .

In figure 1, we depict the typical feynman diagrams of the $B\to a_{0}\left[\to K\bar{K}/\pi\eta\right]h$ decays with $h=\pi,K$ in the PQCD approach, in which the symbols $\otimes$ and $\times$ denotes the vertex of weak interaction and the possible attachments of hard gluons, respectively, the rectangle indicates the intermediate resonant states $a_{0}$ and the subsequent strong decays $a_{0}\to K\bar{K}/\pi\eta$. In the $B$ meson rest frame, the explicit definitions of kinematics in the $B(p_{B})\to R(p_{R})\left[\to h_{1}(p_{1})h_{2}(p_{2})\right]\,h_{3}(p_{3})$ decays are considered as follow,

where $k_{B},k_{R}$ and $k_{3}$ are the momenta carried by the antiquark in the meson states with the momentum fractions $x_{B},z$ and $x_{3}$, respectively. The new variable $\xi\equiv s/m^{2}_{B}$ indicates the momentum transfer from $B$ meson to resonant state $R$. The differential branching ratios for the quasi-two-body $B_{(s)}\to a_{0}\left[\to K\bar{K}/\pi\eta\right]h$ decays is written as PDG-2020

in which daughter meson momentum $q(s)$ has been defined in Eq. (9), and $q_{h}(s)$ is the magnitude of momentum for the bachelor meson $h$

The decaying amplitudes is exactly written as a convolution of the hard kernel $H$ with the hadron distribution amplitudes (DAs) $\phi_{B},\phi_{h}$ and $\phi_{K\bar{K},\pi\eta}$

in which $\left[K\bar{K}/\pi\eta\right]_{a_{0}}$ indicates the dimeson system in our interesting, $\mu$ is the factorization scale, $b_{i}$ are the conjugate distances of transversal momenta. We present the expressions of amplitudes ${\mathcal{A}}$ for the considered decaying processes in the appendix B. Under the narrow width approximation

we can extract the branching fractions of two-body decays from the quasi-two-body decays by

In table 1, we present the PDG averaged value for the masses and total widths of single mesons, as well as the Wolfenstein parameters of CKM matrix. $B_{(s)}$ meson wave function relies on the three independent parameters, saying the mass $m_{B}$, the decay constant $f_{B}$ and the first inverse moment $\omega_{B}$. For the inverse moment $\omega_{B}$, we take the interval $\omega_{B}(1\,{\rm GeV})=0.40\pm 0.04\,{\rm GeV}$ and $\omega_{B_{s}}(1\,{\rm GeV})=0.50\pm 0.05\,{\rm GeV}$ obtained by the QCD sum rules Braun:2003wx with considering smaller uncertainty. The mean lifes of $B$ mesons are also taken from PDG, they are $\tau_{B^{\pm}}=1.638\times 10^{-12}\,{\rm s}$, $\tau_{B^{0}}=1.520\times 10^{-12}\,{\rm s}$ and $\tau_{B_{s}}=1.509\times 10^{-12}\,{\rm s}$.

The PDG value of light meson decay constant follows from the lattice QCD average $f_{K^{+}}/f_{\pi^{+}}=1.193$ Aoki:2016frl . We truncate to the second moments for the Gegenbauer expansion of leading twist LCDAs, and take $a^{\pi}_{1}=0$ and $a^{\pi}_{2}(1\,{\rm GeV})=0.270\pm 0.047$ obtained recently from the LCSR fit Cheng:2020vwr of the pion electromagnetic form factor⁵⁵5This result agrees with the previous LCSRs extractions from spacelike pion electromagnetic form factor Agaev:2005gu , $B\to\pi$ form factor Ball:2005tb ; Duplancic:2008ix ; Khodjamirian:2011ub , and also the QCD sum rule prediction Ball:2006wn , but much larger than the recent lattice QCD evaluation ($a^{\pi}_{2}(1\,{\rm GeV})=0.130$) with the new developed momentum smearing technique RQCD:2019osh .. For the kaon meson, we take the lattice result obtained by using $N_{f}=2+1$ sea quarks and the domain-wall fermions Arthur:2010xf , say, $a^{K}_{1}(1\,{\rm GeV})=0.060\pm 0.004$ and $a^{K}_{2}(1\,{\rm GeV})=0.175\pm 0.065$, which is comparable with the QCD sum rules calculations Khodjamirian:2004ga ; Ball:2006wn and the result from Dyson-Schwinger equations with dynamical chiral spontaneously breaking (DCSB)-improved kernel Shi:2015esa . We takes the chiral masses at $m_{0}^{\pi}=1.913\,{\rm GeV},m_{0}^{K}=1.892\,{\rm GeV}$ with considering the well-known chiral perturbative theory ($\chi{\rm PT}$) relations Leutwyler:1996qg

in which ${\cal R}\equiv 2m_{s}/(m_{u}+m_{d})=24.4\pm 1.5$, ${\cal Q}^{2}\equiv[m_{s}^{2}-(m_{u}+m_{d})^{2}/4]/(m_{d}^{2}-m_{u}^{2})=(22.7\pm 0.8)^{2}$, the current quark masses are $\overline{m}_{s}(1\,{\rm GeV})=0.125\,{\rm GeV}$, $\overline{m}_{d}(1\,{\rm GeV})=0.0065\,{\rm GeV}$ and $\overline{m}_{u}(1\,{\rm GeV})=0.0035\,{\rm GeV}$. For the twist 3 LCDA, we only take into account the asymptotic terms in the numerical analysis.

Concerning the intermediate resonant isovector scalar states $a_{0}$s, the main inputs are the timelike form factor entered in each LCDA and the Gegenbauer moments in the leading twist LCDA. To reveal the timelike form factor described in Eq. (6), we use the QCD sum rules predictions on the decay constants ChengNB , they are $\bar{f}_{a_{0}}(1\,{\rm GeV})=0.365\pm 0.020\,{\rm GeV}$ and $\bar{f}_{a_{0}^{\prime}}(1\,{\rm GeV})=-0.280\pm 0.035\,{\rm GeV}$ obtained in the first scenario where $a_{0}$ is treated as the lowest lying $q\bar{q}$ state and $a_{0}^{\prime}$ as the first excited state, and $\bar{f}_{a_{0}^{\prime}}(1\,{\rm GeV})=0.460\pm 0.050\,{\rm GeV}$ and $\bar{f}_{a_{0}^{\prime\prime}}(1\,{\rm GeV})=0.390\pm 0.040\,{\rm GeV}$ obtained in the second scenario where $a_{0}^{\prime}$ is the lowest lying $q\bar{q}$ state and $a_{0}^{\prime\prime}$ as the first excited state. As shown in Eq. (8), the strong coupling constants $g_{a_{0}K\bar{K}}$ and $g_{a_{0}\pi\eta}$ are decided by the partial decay width, which are fixed by the following considerations

• •

  With the measurements $\left(\Gamma_{a_{0}\to\pi\eta}\times\Gamma_{a_{0}\to\gamma\gamma}\right)/\Gamma^{\rm tot}=0.21\,{\rm keV}$ and $\Gamma_{a_{0}\to\gamma\gamma}=0.30\pm 0.10\,{\rm keV}$ Amsler:1997up , one get $\Gamma_{a_{0}\to\pi\eta}=0.053\pm 0.018\,{\rm GeV}$. We do not use eq. (8) to determine the partial width since it is an approximation expression under the narrow width limit. Furthermore, one can get $\Gamma_{a_{0}\to K\bar{K}}=0.009\pm 0.003\,{\rm GeV}$ with the measurement $\Gamma_{a_{0}\to K\bar{K}}/\Gamma_{a_{0}\to\pi\eta}=0.177$ PDG-2020 .

• •

  The partial decay widths of $a_{0}^{\prime}$ to $K\bar{K}$ and $\pi\eta$ states are decided by the measured branching ratios $\Gamma_{{a_{0}^{\prime}}\to K\bar{K}}/\Gamma_{a_{0}^{\prime}}^{\rm tot}=0.082\pm 0.028$ and $\Gamma_{{a_{0}^{\prime}}\to\pi\eta}/\Gamma_{a_{0}^{\prime}}^{\rm tot}=0.093\pm 0.020$ PDG-2020 .

• •

  For the $a_{0}^{\prime\prime}$ decays, there is no direct measurement and the predictions from different models vary widely. For example, the Extended Linear Sigma Model (eLSM) states that $\Gamma_{a_{0}^{\prime\prime}}\to K\bar{K}=94\pm 54\,{\rm MeV}$ and $\Gamma_{a_{0}^{\prime\prime}}\to\pi\eta=94\pm 16\,{\rm MeV}$ Parganlija:2016yxq , while the $3^{3}P_{0}$ quark model gives the result $0.74\,{\rm MeV}$ and $5.13\,{\rm MeV}$ correspondently Wang:2017pxm . So in our evaluation, we take the largest interval of this variable to account its uncertainty.

• •

  To close the descriptions, we summary the intervals of partial decay widths as

  	$\displaystyle\Gamma_{a_{0}\to K\bar{K}}=0.009\pm 0.003\,{\rm GeV}\,,\quad\quad\Gamma_{a_{0}\to\pi\eta}=0.053\pm 0.018\,{\rm GeV}\,,$
  	$\displaystyle\Gamma_{a_{0}^{\prime}\to K\bar{K}}=0.022\pm 0.008\,{\rm GeV}\,,\quad\quad\Gamma_{a_{0}^{\prime}\to\pi\eta}=0.025\pm 0.006\,{\rm GeV}\,,$
  	$\displaystyle\Gamma_{a_{0}^{\prime\prime}\to K\bar{K}}\in[0,0.150]\,{\rm GeV}\,,\hskip 48.36958pt\Gamma_{{a_{0}^{\prime\prime}}\to\pi\eta}\in[0,0.110]\,{\rm GeV}\,.$		(22)

Concerning the Gegenbauer expansion of scalar mesons, we take into account the first two odd moments $B_{1}$ and $B_{3}$ in the twist 2 LCDAs Cheng:2005nb and the asymptotic terms in the twist 3 LCDAs due to the large theoretical uncertainty of $a_{m}$ and $b_{m}$ Lu:2006fr ; Han:2013zg ; Wang:2014vra . They are

in the first scenario, and

in the second scenario, where the default scale at $1\,{\rm GeV}$ is indicated.