Construction of Gross-Neveu model using Polchinski flow equation

Abstract

The Gross-Neveu model is a quantum field theory model of Dirac fermions in two dimensions with a quartic interaction term. Like Yang-Mills theory in four dimensions, the model is renormalizable (but not super-renormalizable) and asymptotically free (i.e. its short-distance behaviour is governed by the free theory). We give a new construction of the massive Euclidean Gross-Neveu model in infinite volume based on the renormalization group flow equation. The construction does not involve cluster expansion or discretization of phase-space. We express the Schwinger functions of the Gross-Neveu model in terms of the effective potential and construct the effective potential by solving the flow equation using the Banach fixed point theorem. Since we use crucially the fact that fermionic fields can be represented as bounded operators our construction does not extend to models including bosons. However, it is applicable to other asymptotically free purely fermionic theories such as the symplectic fermion model.