Asymptotic safety: a simple example

TL;DR

This paper demonstrates how the Gross-Neveu model in 2<d<4 dimensions exemplifies Weinberg's asymptotic safety scenario, showing the existence of a non-Gaussian fixed point that ensures the model's validity at arbitrarily high energies.

Contribution

It provides a detailed analysis of asymptotic safety in a fermionic model using the functional renormalization group, including both fermionic and bosonized formulations, and connects to large-Nf expansion results.

Findings

Asymptotic safety is realized at non-Gaussian fixed points in both formulations.

Universal critical exponents are determined quantitatively.

Analytic results are obtained in the large-Nf limit.

Abstract

We use the Gross-Neveu model in 2<d<4 as a simple fermionic example for Weinberg's asymptotic safety scenario: despite being perturbatively nonrenormalizable, the model defines an interacting quantum field theory being valid to arbitrarily high momentum scales owing to the existence of a non-Gaussian fixed point. Using the functional renormalization group, we study the UV behavior of the model in both the purely fermionic as well as a partially bosonized language. We show that asymptotic safety is realized at non-Gaussian fixed points in both formulations, the universal critical exponents of which we determine quantitatively. The partially bosonized formulation allows to make contact to the large-Nf expansion where the model is known to be renormalizable to all-orders. In this limit, the fixed-point action as well as all universal critical exponents can be computed analytically. As asymptotic safety has become an important scenario for quantizing gravity, our description of a well-understood model is meant to provide for an easily accessible and controllable example of modern nonperturbative quantum field theory.