In the present work, we analyze the properties of the transition form factors (TFFs) for the $\gamma\gamma^*\to \eta^{ (\prime)}$ process, employing the $\eta^{ (\prime)}$-meson light-cone distribution amplitude (LCDA) derived within the light-cone sum rule framework. To this end, we adopt the quark-flavor mixing scheme for the $\eta^{ (\prime)}$ meson, and compute the TFFs by systematically incorporating transverse-momentum corrections and contributions beyond the leading Fock state. We utilize light-cone harmonic oscillator models to parameterize the longitudinal and transverse behavior of the leading-twist light-cone wavefunction, for which the corresponding LCDA exhibits a unimodal profile. We further examine the potential contributions of intrinsic charm components to the scaled TFFs $Q^2 F_{ \eta\gamma}(Q^2)$ and $Q^2 F_{ \eta^\prime \gamma}(Q^2)$. Leveraging a range of values for the decay constant $f_{ \eta_{ c_0}}$ and implementing the $\eta$-$\eta'$-$\eta_c$ and $\eta$-$\eta^\prime$-$G$-$\eta_c$ mixing mechanisms accordingly, together with the recently updated mixing angles, we investigate the impact of the intrinsic $c\bar{ c}$ and gluonic component on these observables. In high-$Q^2$ regime, $Q^2 F_{ \eta^\prime\gamma}(Q^2)$ exhibits a marked increase in sensitivity to the charm quark component, whereas $Q^2F_{ \eta\gamma}(Q^2)$ becomes notably stabilized. A detailed discussion of $\chi^2/d.o.f$ and $p$-values indicates that the intrinsic charm quark component is important and yields a substantial, non-negligible contribution across the entire $Q^2$ range.
Categories: hep-ph