$$\eta ^{(\prime )}$$ 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

Abstract In this paper, we make a detailed discussion on the $$\eta $$ η and $$\eta ^\prime $$ η ′ -meson leading-twist light-cone distribution amplitude $$\phi _{2;\eta ^{(\prime )}}(u,\mu )$$ ϕ 2 ; η ( ′ ) ( u , μ ) 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 $$\langle \xi ^n_{2;\eta ^{(\prime )}}\rangle |_{\mu _0} $$ ⟨ ξ 2 ; η ( ′ ) n ⟩ | μ 0 with $$n = (2, 4, 6)$$ n = ( 2 , 4 , 6 ) can be determined, where the initial scale $$\mu _0$$ μ 0 is set as the usual choice of 1 GeV. Numerically, we obtain $$\langle \xi _{2;\eta }^2\rangle |_{\mu _0} =0.231_{-0.013}^{+0.010}$$ ⟨ ξ 2 ; η 2 ⟩ | μ 0 = 0 . 231 - 0.013 + 0.010 , $$\langle \xi _{2;\eta }^4 \rangle |_{\mu _0} =0.109_{ - 0.007}^{ + 0.007}$$ ⟨ ξ 2 ; η 4 ⟩ | μ 0 = 0 . 109 - 0.007 + 0.007 , and $$\langle \xi _{2;\eta }^6 \rangle |_{\mu _0} =0.066_{-0.006}^{+0.006}$$ ⟨ ξ 2 ; η 6 ⟩ | μ 0 = 0 . 066 - 0.006 + 0.006 for $$\eta $$ η -meson, $$\langle \xi _{2;\eta '}^2\rangle |_{\mu _0} =0.211_{-0.017}^{+0.015}$$ ⟨ ξ 2 ; η ′ 2 ⟩ | μ 0 = 0 . 211 - 0.017 + 0.015 , $$\langle \xi _{2;\eta '}^4 \rangle |_{\mu _0} =0.093_{ - 0.009}^{ + 0.009}$$ ⟨ ξ 2 ; η ′ 4 ⟩ | μ 0 = 0 . 093 - 0.009 + 0.009 , and $$\langle \xi _{2;\eta '}^6 \rangle |_{\mu _0} =0.054_{-0.008}^{+0.008}$$ ⟨ ξ 2 ; η ′ 6 ⟩ | μ 0 = 0 . 054 - 0.008 + 0.008 for $$\eta '$$ η ′ -meson. Next, we calculate the $$D_s\rightarrow \eta ^{(\prime )}$$ D s → η ( ′ ) transition form factors (TFFs) $$f^{\eta ^{(\prime )}}_{+}(q^2)$$ f + η ( ′ ) ( q 2 ) within QCD light-cone sum rules approach up to NLO level. The values at large recoil region are $$f^{\eta }_+(0) = 0.476_{-0.036}^{+0.040}$$ f + η ( 0 ) = 0 . 476 - 0.036 + 0.040 and $$f^{\eta '}_+(0) = 0.544_{-0.042}^{+0.046}$$ 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. $$D_s^+\rightarrow \eta ^{(\prime )}\ell ^+ \nu _\ell $$ D s + → η ( ′ ) ℓ + ν ℓ , i.e. $${{\mathcal {B}}}(D_s^+ \rightarrow \eta ^{(\prime )} e^+\nu _e)=2.346_{-0.331}^{+0.418}(0.792_{-0.118}^{+0.141})\times 10^{-2}$$ B ( D s + → η ( ′ ) e + ν e ) = 2 . 346 - 0.331 + 0.418 ( 0 . 792 - 0.118 + 0.141 ) × 10 - 2 and $${{\mathcal {B}}}(D_s^+ \rightarrow \eta ^{(\prime )} \mu ^+\nu _\mu )=2.320_{-0.327}^{+0.413}(0.773_{-0.115}^{+0.138})\times 10^{-2}$$ B ( D s + → η ( ′ ) μ + ν μ ) = 2 . 320 - 0.327 + 0.413 ( 0 . 773 - 0.115 + 0.138 ) × 10 - 2 for $$\ell = (e, \mu )$$ ℓ = ( e , μ ) channels respectively. And in addition to that, the mixing angle for $$\eta -\eta '$$ η - η ′ with $$\varphi $$ φ and ratio for the different decay channels $${{\mathcal {R}}}_{\eta '/\eta }^\ell $$ R η ′ / η ℓ are given, which show good agreement with the recent BESIII measurements.