Gross-Neveu and Thirring models. Covariant Gaussian analysis

Two-dimensional fermionic theories, the SU(N) Gross-Neveu and the massive Thirring models, are analyzed in the covariant Gaussian approximation. In the Gross-Neveu model we find three phases (renormalizations). In one of them the results coincide with the leading order in 1/N expansion. In the other two phases the gap equation has no solution and there are no fermionic excitations in the spectrum of the theory. It is argued that those renormalizations are relevant for N=1,2. The massive Thirring model is found to possess a line of ultraviolet fixed points. In the limit ${m}_{b}$\ensuremath{\rightarrow}0 the axial symmetry is not broken. The 2\ensuremath{\rightarrow}2 S-matrix element for the nonasymptotically free phase is calculated and it qualitatively agrees with the exact expression. We also find an asymptotically free phase with vanishing bare coupling.