$eta$-$eta$ Mixing in Large-$N_c$ Chiral Perturbation Theory

We present a calculation of the $eta$-$eta$ mixing in the framework of large-$N_c$ chiral perturbation theory. A general expression for the $eta$-$eta$ mixing at next-to-next-to-leading order (NNLO) is derived, including higher-derivative terms up to fourth order in the four momentum, kinetic and mass terms. In addition, the axial-vector decay constants of the $eta$-$eta$ system are determined at NNLO. The numerical analysis of the results is performed successively at LO, NLO, and NNLO. We investigate the influence of one-loop corrections, OZI-rule-violating parameters, and $mathcal{O}(N_c p^6)$ contact terms.