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The $\eta^{(\prime)}$-mesons in the quark-flavor basis are mixtures of twomesonic states $|\eta_{q}\rangle=|\bar u u+\bar d d\rangle/\sqrt 2$ and$|\eta_{s}\rangle=|\bar s s\rangle$. In the previous work, we have made adetailed study on the $\eta_{s}$ leading-twist distribution amplitude. As asequential work, in the present paper, we fix the $\eta_q$ leading-twistdistribution amplitude by using the light-cone harmonic oscillator model forits wave function and by using the QCD sum rules within the QCD backgroundfield to calculate its moments. The input parameters of $\eta_q$ leading-twistdistribution amplitude $\phi_{2;\eta_q}$ at an initial scale $\mu_0\sim 1$ GeVare then fixed by using those moments. The sum rules for the $0_{\rm th}$-ordermoment can also be used to fix the magnitude of $\eta_q$ decay constant, whichgives $f_{\eta_q}=0.141\pm0.005$ GeV. As an application of the present derived$\phi_{2;\eta_q}$, we calculate the transition form factors $B(D)^+\to\eta^{(\prime)}$ by using the QCD light-cone sum rules up to twist-4accuracy and by including the next-to-leading order QCD corrections to thetwist-2 part, and then fix the related CKM matrix element and the decay widthfor the semi-leptonic decays $B(D)^+ \to\eta^{(\prime)}\ell^+ \nu_\ell$.

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