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Properties of the $η_q$ leading-twist distribution amplitude and its effects to the $B/D^+ \toη^{(\prime)}\ell^+ ν_\ell$ decays

The $η^{(\prime)}$-mesons in the quark-flavor basis are mixtures of two mesonic states $|η_{q}\rangle=|\bar u u+\bar d d\rangle/\sqrt 2$ and $|η_{s}\rangle=|\bar s s\rangle$. In the previous work, we have made a detailed study on the $η_{s}$ leading-twist distribution amplitude. As a sequential work, in the present paper, we fix the $η_q$ leading-twist distribution amplitude by using the light-cone harmonic oscillator model for its wave function and by using the QCD sum rules within the QCD background field to calculate its moments. The input parameters of $η_q$ leading-twist distribution amplitude $ϕ_{2;η_q}$ at an initial scale $μ_0\sim 1$ GeV are then fixed by using those moments. The sum rules for the $0_{\rm th}$-order moment can also be used to fix the magnitude of $η_q$ decay constant, which gives $f_{η_q}=0.141\pm0.005$ GeV. As an application of the present derived $ϕ_{2;η_q}$, we calculate the transition form factors $B(D)^+ \toη^{(\prime)}$ by using the QCD light-cone sum rules up to twist-4 accuracy and by including the next-to-leading order QCD corrections to the twist-2 part, and then fix the related CKM matrix element and the decay width for the semi-leptonic decays $B(D)^+ \toη^{(\prime)}\ell^+ ν_\ell$.