Quark Confinement in a Random Background Gross-Neveu Model

We study an extended Gross-Neveu model with $N_f$ quark flavors and with an additional SU(2) (global color) degree of freedom of the quarks. The four fermion interaction in the color channel is mediated by a random color matrix with fixed strength. The effective potential for the quark condensate and the scalar correlation function is investigated in the large $N_f$-limit. Quark anti-quark thresholds are absent in the scalar correlation function implying that the decay of the scalar meson in free quarks is avoided. The transition from the low energy to the high energy perturbative regime is found to be smooth. Perturbation theory provides a good description of Green's functions at high energies, although there is no quark liberation.