$a_0(980)$-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 this paper, moments of $a_0(980)$-meson twist-2 light-cone distribution amplitudes were deeply researched by using QCD sum rules approach within background field theory. Up to 9th-order accuracy, we present $⟨{\xi }_{2;a_0}^n⟩{\mathrm{\mid }}_{{\mu }_0}$ at the initial scale ${\mu }_0=1\mathrm{G}\mathrm{e}\mathrm{V}$, i.e. $⟨{\xi }_{2;a_0}^1⟩{\mathrm{\mid }}_{{\mu }_0}=-0.307(43)$, $⟨{\xi }_{2;a_0}^3⟩{\mathrm{\mid }}_{{\mu }_0}=-0.181(34)$, $⟨{\xi }_{2;a_0}^5⟩{\mathrm{\mid }}_{{\mu }_0}=-0.078(28)$, $⟨{\xi }_{2;a_0}^7⟩{\mathrm{\mid }}_{{\mu }_0}=-0.049(26)$, $⟨{\xi }_{2;a_0}^9⟩{\mathrm{\mid }}_{{\mu }_0}=-0.036(24)$, respectively. An improved light-cone harmonic oscillator model for $a_0(980)$-meson twist-2 light-cone distribution amplitudes is adopted, where its parameters are fixed by using the least squares method based on the $⟨{\xi }_{2;a_0}^n⟩{\mathrm{\mid }}_{{\mu }_0}$, and their goodness of fit reach to $95.4\mathrm{\%}$. Then, we calculate the $D\to a_0(980)$ transition form factors within the light-cone sum rules approach, and at largest recoil point, we obtain $f_{+}^{D\to a_0}(0)=1.058_{-0.035}^{+0.068}$ and $f_{-}^{D\to a_0}(0)=0.764_{-0.036}^{+0.044}$. As a further application, the branching fractions of the $D\to a_0(980)\mathrm{\ell }{\bar{\nu }}_{\mathrm{\ell }}$ semileptonic decays are given. Taking the decay $a_0(980)\to \eta \pi$ into consideration, we obtain $\mathcal{B}(D^0\to a_0(980{)}^{-}(\to \eta {\pi }^{-})e^{+}{\nu }_e)=(1.330_{-0.134}^{+0.216})\times {10}^{-4}$, $\mathcal{B}(D^{+}\to a_0(980{)}^0(\to \eta {\pi }^0)e^{+}{\nu }_e)=(1.675_{-0.169}^{+0.272})\times {10}^{-4}$, which are consistent with the BESIII collaboration and PDG data within errors. Finally, we present the angle observables of forward-backward asymmetries, $q^2$-differential flat terms and lepton polarization asymmetry of the semileptonic decay $D\to a_0(980)\mathrm{\ell }{\bar{\nu }}_{\mathrm{\ell }}$.