Quark confinement in a random background Gross-Neveu model

We study an extended Gross-Neveu model with Nf 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 Nf limit. Quark-antiquark 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 functions at high energies, although there is no quark liberation.