$$\eta ^{(\prime )}$$ η ( ′ ) -meson twist-2 distribution amplitude within QCD sum rule approach and its application to the semi-leptonic decay $$ D_s^+ \rightarrow \eta ^{(\prime )}\ell ^+ \nu _\ell $$ D s + → η ( ′ ) ℓ + ν ℓ

In this paper, we make a detailed discussion on the $$\eta $$ and $$\eta ^\prime $$ -meson leading-twist light-cone distribution amplitude $$\phi _{2;\eta ^{(\prime )}}(u,\mu )$$ by using the QCD sum rules approach under the background field theory. Taking both the non-perturbative condensates up to dimension-six and the next-to-leading order (NLO) QCD corrections to the perturbative part, its first three moments $$\langle \xi ^n_{2;\eta ^{(\prime )}}\rangle |_{\mu _0} $$ with $$n = (2, 4, 6)$$ can be determined, where the initial scale $$\mu _0$$ is set as the usual choice of 1 GeV. Numerically, we obtain $$\langle \xi _{2;\eta }^2\rangle |_{\mu _0} =0.231_{-0.013}^{+0.010}$$ , $$\langle \xi _{2;\eta }^4 \rangle |_{\mu _0} =0.109_{ - 0.007}^{ + 0.007}$$ , and $$\langle \xi _{2;\eta }^6 \rangle |_{\mu _0} =0.066_{-0.006}^{+0.006}$$ for $$\eta $$ -meson, $$\langle \xi _{2;\eta '}^2\rangle |_{\mu _0} =0.211_{-0.017}^{+0.015}$$ , $$\langle \xi _{2;\eta '}^4 \rangle |_{\mu _0} =0.093_{ - 0.009}^{ + 0.009}$$ , and $$\langle \xi _{2;\eta '}^6 \rangle |_{\mu _0} =0.054_{-0.008}^{+0.008}$$ for $$\eta '$$ -meson. Next, we calculate the $$D_s\rightarrow \eta ^{(\prime )}$$ transition form factors (TFFs) $$f^{\eta ^{(\prime )}}_{+}(q^2)$$ within QCD light-cone sum rules approach up to NLO level. The values at large recoil region are $$f^{\eta }_+(0) = 0.476_{-0.036}^{+0.040}$$ and $$f^{\eta '}_+(0) = 0.544_{-0.042}^{+0.046}$$ . After extrapolating TFFs to the allowable physical regions within the series expansion, we obtain the branching fractions of the semi-leptonic decay, i.e. $$D_s^+\rightarrow \eta ^{(\prime )}\ell ^+ \nu _\ell $$ , i.e. $${{\mathcal {B}}}(D_s^+ \rightarrow \eta ^{(\prime )} e^+\nu _e)=2.346_{-0.331}^{+0.418}(0.792_{-0.118}^{+0.141})\times 10^{-2}$$ and $${{\mathcal {B}}}(D_s^+ \rightarrow \eta ^{(\prime )} \mu ^+\nu _\mu )=2.320_{-0.327}^{+0.413}(0.773_{-0.115}^{+0.138})\times 10^{-2}$$ for $$\ell = (e, \mu )$$ channels respectively. And in addition to that, the mixing angle for $$\eta -\eta '$$ with $$\varphi $$ and ratio for the different decay channels $${{\mathcal {R}}}_{\eta '/\eta }^\ell $$ are given, which show good agreement with the recent BESIII measurements.