Gluon anomaly and the violation of the Zweig rule

It is well known that the gluon anomaly is responsible for the mass of the $U(1)$ Goldstone boson. It is also manifested in the properties of pseudo-Goldstone states: the gluon anomaly in the next order of the $1/N_c$ expansion induces interactions violating the Zweig rule. A necessary condition for this is the explicit breaking of the chiral symmetry by the masses of light quarks. One of the physical consequences is that the $\eta-\eta'$ mixing does not affect the $\eta \to 3\pi $ amplitude. Another consequence is the suppression of the first $1/N_c$ correction to the $\eta\to\eta'$ mixing angle. The mechanism of such suppression is discussed in detail. The conclusions are based on the $1/N_c$ expansion and the effective meson Lagrangian of the Nambu–Jona-Lasinio model and are compared with the results of the $1/N_c$ chiral perturbation theory.