Isospin-breaking contribution to the model-independent axion-photon-photon coupling in $U(3)$ chiral theory

We pursue the calculation of the model-independent component of the axion-photon-photon coupling in the $U(3)$ chiral perturbation theory up to next-to-leading order, with the emphasis on the isospin breaking effect. The mixing of the ${\pi }^0$-$\eta$-${\eta }^{\mathrm{\prime }}$-axion system is revised as well by working out the complete linear isospin-breaking terms. Our calculation shows that the isospin-breaking correction to the axion-photon-photon coupling amounts to more than 15%, comparing with the result in the isospin limit.