Axion-meson mixing in light of recent lattice $\eta$-${\eta }^{\mathrm{\prime }}$ simulations and their two-photon couplings within $U(3)$ chiral theory

We study the mixing of the QCD/QCD-like axion and light-flavor mesons ${\pi }^0,\eta ,{\eta }^{\mathrm{\prime }}$ within the framework of $U(3)$ chiral perturbation theory up to next-to-leading order in this work. The axion-meson mixing formulas are calculated order by order in the $U(3)$ $\delta$-expansion scheme, namely the joint expansions of the momentum, light-quark masses and $1\mathrm{/}{N}_C$. We provide axion-meson mixing relations in terms of the ${\pi }^0$-$\eta$-${\eta }^{\mathrm{\prime }}$ mixing parameters and their masses. The recent lattice simulations on the $\eta$-${\eta }^{\mathrm{\prime }}$ systems turn out to be able to offer valuable inputs to constrain the unknown low-energy constants. The relation of the mass and decay constant of the axion is then further explored based on our updated calculations. The two-photon couplings of the light-flavor mesons, together with the axion, are also investigated in the $U(3)$ chiral theory up to next-to-leading order in the $\delta$-counting scheme.