Mixing interactions and effects in the NJL-model

The flavor-dependent quark-antiquark contact interactions, induced by vacuum polarization and recently derived for flavor U(3) Nambu-Jona-Lasinio model, are articulated with the resulting mixing effects emerging from flavor symmetry breaking in view. The formal effects of the explicit mixing interactions, $G_{i\neq j}$, are detailed firstly for the meson mixing problem without the inclusion of 't Hooft interactions induced by instantons. Secondly, it is shown that these mixings, in the scalar channel of quark-antiquark interactions, might give rise to quark mixing in the gap equations. Sixth order quark-antiquark interactions from vacuum polarization, that break $U_A(1)$ symmetry, also contribute.