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 scaling critical (i.e. renormalizable but not super-renormalizable) and asymptotically free (i.e. its short-distance behavior is governed by the free theory). We give a new construction of the massive Euclidean Gross-Neveu model in infinite volume. The distinctive feature of the construction is that it does not involve cluster expansion, discretization of phase-space or a tree expansion ansatz and is based solely on the renormalization group flow equation. 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. Moreover, we construct a random field in the probability space of the free field such that its moments coincide with the Schwinger functions of the Gross-Neveu model. This is the first construction of a strong coupling between the free and interacting fields for a scaling-critical QFT. 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.