Bose Condensation and $Z_N$ Symmetry Breaking in the Mixed Model of Induced QCD

The mixed model of the large $N$ induced QCD, with $N_f\ll N$ flavors of heavy fermions in fundamental representation, is solved in the local limit. The $Z_N$ symmetry is broken spontaneously in the large $N$ limit, evading the Elitzur “no-go” theorem. As a result of this symmetry breaking, there is the Bose condensate of the eigenvalues of the scalar field, proportional to $\frac{N_f}{N}$. This condensate leads to the mass unit, which goes to zero as fractional power of $\frac{N_f}{N}$, thus defining the new kind of the local limit of this lattice theory. There is a strong coupling region below this mass scale, which revives the hopes of induction of realistic QCD.