Light-cone sum rules analysis of the semi-leptonic $D^+_s\to f_0(980)(\toπ^+π^-)e^+ν_e$ decay incorporating $f_0(980)$ mixing state twist-2 distribution amplitudes

Abstract: The isospin-singlet scalar meson $f_0(980)$ is hypothesized to consist of two energy eigenstates, forming a mixture of $\frac{1}{\sqrt{2}}(\bar{u}u + \bar{d}d)$ and $\bar{s}s$. Building on this framework, we apply the QCD sum rules approach in the background field theory to compute the $f_0(980)$ decay constant, yielding $f_{f_0}(\mu_0=1\,\text{GeV}) = 0.386\pm0.009 \, \text{GeV}$. Subsequently, we derive the first two $\xi$-moments of the leading-twist light-cone distribution amplitude $\phi_{2;f_0}$. Using QCD light-cone sum rules, we then calculate the $D^+_s \to f_0(980)$ transition form factor (TFF) $f_+(q^2)$, obtaining $f_+(0) = 0.516^{{+0.027}_{-0.024}$} at the large-recoil point. We extend $f_+(q^2)$ to the full physical region via a simplified series expansion parameterization, enabling calculations of the differential decay widths and the branching fraction $\mathcal{B}(D^+_s \to f_0(980)(\to \pi^{+\pi-)e+\nu_e)} = (1.783^{{+0.227}_{-0.189})} \times 10^{-3}$. Our theoretical predictions align well with the latest BESIII Collaboration measurements within reasonable errors.