Axion-meson mixing in light of recent lattice $\eta$-$\eta'$ 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 $\pi^0, \eta, \eta'$ 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 $1/N_C$. We provide axion-meson mixing relations in terms of the $\pi^0$-$\eta$-$\eta'$ mixing parameters and their masses. The recent lattice simulations on the $\eta$-$\eta'$ 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.