$1/N_c$ Nambu -- Jona-Lasinio model: $\pi^0$, $\eta$ and $\eta'$ mesons

We continue to study the properties of the light pseudoscalar nonet within the combined framework of Nambu -- Jona-Lasinio model and $1/N_c$ expansion, assuming that current quark masses count of order $\mathcal O(1/N_c)$. The masses, mixing angles and decay constants of the $\pi^0$, $\eta$ and $\eta'$ are calculated. The role of the $U(1)_A$ anomaly is emphasized. It is shown that the gluon anomaly suppresses the leading order effects that might otherwise be substantial for the $\eta\to 3\pi$ amplitude. A detailed comparison with the known results of $1/N_c$ chiral perturbation theory is made.