$\eta$-${\eta }^{\mathrm{\prime }}$ mixing and its application in the $B^{+}\mathrm{/}{D}^{+}\mathrm{/}{D}_s^{+}\to {\eta }^{(\mathrm{\prime })}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ decays

In this paper, we take into account the intrinsic charm and gluonic contents into the $\eta -{\eta }^{\mathrm{\prime }}$ mixing scheme and formulate the tetramixing $\eta -{\eta }^{\mathrm{\prime }}-G-{\eta }_c$ to study the mixing properties of ${\eta }^{(\mathrm{\prime })}$ mesons. Using the newly derived mixing parameters, we calculate the transition form factors (TFFs) of $B^{+}\mathrm{/}{D}^{+}\mathrm{/}{D}_s^{+}\to {\eta }^{(\mathrm{\prime })}$ within the QCD light-cone sum rules up to next-to-leading order QCD corrections and twist-4 contributions. Using the extrapolated TFFs, we then calculate the decay widths and branching fractions of the semi-leptonic decays $B^{+}\mathrm{/}{D}^{+}\mathrm{/}{D}_s^{+}\to {\eta }^{(\mathrm{\prime })}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$. Our results are consistent with the recent Belle and BES-III measurements within reasonable errors.