Determination of the $η$-$η^{\prime}$ Mixing Angle from $F_{η^{(\prime)}γ}(Q^2)$ within the Self-consistent Light-front Quark Model

We study the $\eta$-$\eta^{\prime}$ system in the self-consistent light-front quark model (LFQM) with the quark flavor mixing scheme, where the single mixing angle is constrained by the transition form factors (TFFs) $F_{\eta^{(\prime)}\gamma}(Q^2)$. A combined $\chi^2$ scan of the CLEO and BABAR data yields the best fit value $\theta=41.5^\circ$, in close agreement with the recent LHCb result $(41.6^{+1.0}_{-1.2})^\circ$. Using this mixing angle, we calculate the decay constants, light-cone distribution amplitudes (LCDAs), Gegenbauer moments, $\xi$-moments, transverse momentum moments, electromagnetic interaction radii and two photon decay widths of the $\eta$ and $\eta^{\prime}$ mesons. The predicted $a_2^\eta$ and $\langle\xi^2\rangle_\eta$ are consistent with the latest BABAR results, and the electromagnetic interaction radii also agreeing well with the A2 and BESIII measurements. These results demonstrate that TFFs data provide strong constraints on the light-front description for the nonperturbative properties of the $\eta$-$\eta^{\prime}$ system. Owing to the lack of direct experimental constraints on the real photon value $F_{\eta^{(\prime)}\gamma}(0)$, the predicted decay widths are somewhat lower than the experimental data, which can be further examined with future measurements.