$\eta$ and ${\eta }^{\prime }$ in a coupled Schwinger-Dyson and Bethe-Salpeter approach

Extending our earlier treatments of ${\pi }^0,{\eta }_c,$ and ${\eta }_b,$ we study the $\eta -{\eta }^{\prime }$ system and its $\gamma \gamma$ decays using a model which is a leading version of the consistently coupled Schwinger-Dyson (SD) and Bethe-Salpeter (BS) approaches. The electromagnetic interactions are incorporated through a (generalized) impulse approximation consistent with this bound-state approach, so that the Ward-Takahashi identities of QED are preserved when quarks are dynamically dressed. To overcome some of the limitations due to the ladder approximation, we introduce a minimal extension to the bound-state approach employed, so that the ${\mathrm{U}}_A\left(1\right)$ problem is avoided. Pointing out which of our predictions hold in the coupled SD-BS approach in general, and which are the consequences of the specific, chosen model, we present the results for the axial-current decay constants of ${\eta }_8$, ${\eta }_0$, and of their physical combinations $\eta$ and ${\eta }^{\prime }$, the results for the $\gamma \gamma$-decay constants of ${\eta }_0$ and ${\eta }_8$, for the two-photon decay widths of $\eta$ and ${\eta }^{\prime }$, and for the mixing-independent $R$ ratio constructed from them.