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$η^{(\prime)}$-meson twist-2 distribution amplitude within QCD sum rule approach and its application to the semi-leptonic decay $ D_s^+ \toη^{(\prime)}\ell^+ ν_\ell$

In this paper, we make a detailed discussion on the $η$ and $η'$-meson leading-twist light-cone distribution amplitude $ϕ_{2;η^{(\prime)}}(u,μ)$ by using QCD sum rules approach under the background field theory. Taking both the non-perturbative condensates up to dimension-six and NLO QCD corrections to the perturbative part, its first three moments $\langleξ^n_{2;η^{(\prime)}}\rangle|_{μ_0} $ with $n = (2,4,6)$ at initial scale $μ_0 = 1$ GeV can be determined. e.g. $\langleξ_{2;η}^2\rangle|_{μ_0} =0.231_{-0.013}^{+0.010}$, $\langleξ_{2;η}^4 \rangle|_{μ_0} =0.109_{-0.007}^{+0.007}$, and $\langleξ_{2;η}^6 \rangle|_{μ_0} =0.066_{-0.006}^{+0.006}$ for $η$-meson, $\langleξ_{2;η'}^2\rangle|_{μ_0} =0.211_{-0.017}^{+0.015}$, $\langleξ_{2;η'}^4 \rangle|_{μ_0} =0.093_{-0.009}^{+0.009}$, and $\langleξ_{2;η'}^6 \rangle|_{μ_0} =0.054_{-0.008}^{+0.008}$ for $η'$-meson. Next, we calculate $D_s\toη^{(\prime)}$ TFFs $f^{η^{(\prime)}}_+(q^2)$ within QCD light-cone sum rules approach up to NLO level. The values at large recoil region are $f^η_+(0) = 0.476_{-0.036}^{+0.040}$ and $f^{η'}_+(0) = 0.544_{-0.042}^{+0.046}$. After extrapolating TFFs to the allowable physical regions within the series expansion, we obtain the branching fractions of the semi-leptonic decay, i.e. $D_s^+\toη^{(\prime)}\ell^+ ν_\ell$, i.e. ${\cal B}(D_s^+\toη^{(\prime)} e^+ν_e)=2.346_{-0.331}^{+0.418}(0.792_{-0.118}^{+0.141})\times10^{-2}$ and ${\cal B}(D_s^+\toη^{(\prime)} μ^+ν_μ)=2.320_{-0.327}^{+0.413}(0.773_{-0.115}^{+0.138})\times10^{-2}$ for $\ell = (e, μ)$ channels respectively. And in addition to that, the mixing angle for $η-η'$ with $φ$ and ratio for the different decay channels ${\cal R}_{η'/η}^\ell$ are given, which show good agreement with the recent BESIII measurements.