Semileptonic decays $D_{(s)} \to η^{(\prime)} \ell^+ ν_\ell$ from QCD Light-Cone Sum Rules

In light of recent precision measurements from BESIII, we reanalyze the $D, D_s \to η^{(\prime)}$ transition form factors using QCD light-cone sum rules, incorporating high-twist contributions and well-established next-to-leading-order QCD corrections. Our analysis confirms the chiral enhancement effect arising from twist-3 light-cone distribution amplitudes of the pseudoscalar mesons, and demonstrates a rapid convergence of the operator product expansion. The resulting high-accuracy form factors enable us to determine the optimal $η$-$η^\prime$ mixing parameters from the precise experimental data for the $D, D_s \to η^{(\prime)} \ell^+ ν_\ell$ (with $\ell = e, μ$) differential decay rates. We find that the BESIII data strongly favor a set of mixing parameters, characterized by small decay constants and a large mixing angle, in the quark flavor basis. Notably, the light-cone-sum-rule predictions for the decays $D \to η^{(\prime)} \ell^+ ν_\ell$, induced by weak $c\to d$ current, reach a precision comparable to the BESIII experimental results. Nevertheless, further refined measurements and more accurate form-factor determinations will be essential to scrutinize the potential role of gluonic components in charmed meson semileptonic decays.