\end{aligned}$$ The matrices acting in flavor space, $\lambda_a,\ a=0,1,\ldots ,8,$ are normalized such that $\mbox{tr} (\lambda_a \lambda_b )=2\delta_{ab}$; $\lambda_0=\sqrt{\frac{2}{3}}\, 1$, and $\lambda_k,\ k=1,2,\ldots ,8$ are the standard $SU(3)$ Gell-Mann matrices; $P_{L,R}=(1\mp\gamma_5)/2$ are chiral projectors and the determinant is over flavor indices. The large $N_c$ behaviour of the model is reflected in the dimensionful coupling constants, $[G]=M^{-2},\, [\kappa ]=M^{-5},\, [g_1]=[g_2]=M^{-8}$, which count as $G\sim 1/N_c$, $\kappa\sim 1/N_c^{N_f}$, and $g_1,g_2\sim 1/N_c^4$ or less. As a result the NJL interaction (\[L4q\]) dominates over ${\cal L}_{6q}$ at large $N_c$, as one would expect, because Zweig’s rule is exact at $N_c = 
\infty$. Let us note that the $8q$-interaction ${\cal L}_{8q}^{(1)}$ breaks Zweig’s rule as well.

Since the coupling constants $G, \kappa, g_1, g_2$ are dimensionful, the model is not renormalizable. We use the cut-off $\Lambda$ to render quark loops finite. The global chiral $SU(3)_L\times SU(3)_R$ symmetry of the Lagrangian (\[efflag\]) at $m=0$ is spontaneously broken to the $SU(3)$ group, showing the dynamical instability of the fully symmetric solutions of the theory. In addition, the current quark mass $m$, being a diagonal matrix in flavor space with elements $\mbox{diag} (m_u, m_d, m_s)$, explicitly breaks this symmetry down, retaining only the reduced $SU(2)_I\times U(1)_Y$ symmetries of 