The dynamical behaviors in (2+1)-dimensional Gross-Neveu model with a Thirring interaction

We analyze (2+1)-dimensional Gross-Neveu model with a Thirring interaction, where a vector-vector type four-fermi interaction is on equal terms with a scalar-scalar type one. The Dyson-Schwinger equation for fermion self-energy function is constructed up to next-to-leading order in 1/N expansion. We determine the critical surface which is the boundary between a broken phase and an unbroken one in (\alpha_c,~ \beta_c,~ N_c) space. It is observed that the critical behavior is mainly controlled by Gross-Neveu coupling \alpha_c and the region of the broken phase is separated into two parts by the line \alpha_c=\alpha_c^*(=\frac{8}{\pi^2}). The mass function is strongly dependent upon the flavor number N for \alpha > \alpha_c^*, while weakly for \alpha \alpha_c^*, the critical flavor number N_c increases as Thirring coupling \beta decreases. By driving the CJT effective potential, we show that the broken phase is energetically preferred to the symmetric one. We discuss the gauge dependence of the mass function and the ultra-violet property of the composite operators