The Dynamical Behaviors in (2+1)-Dimensional Gross-Neveu Model with a Thirring Interaction

Abstract

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 ($α_c,~ β_c,~ N_c$) space. It is observed that the critical behavior is mainly controlled by Gross-Neveu coupling $α_c$ and the region of the broken phase is separated into two parts by the line $α_c=α_c^*(=\frac{8}{π^2})$. The mass function is strongly dependent upon the flavor number N for $α> α_c^*$, while weakly for $α< α_c^*$. For $α> α_c^*$, the critical flavor number $N_c$ increases as Thirring coupling $β$ 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.