Four Fermi Theory in Four Dimensions is Renormalisable

TL;DR

This paper proves the renormalisability of four-dimensional four-fermion quantum field theories with many flavors, providing exact beta functions and discussing their universality, scaling, and implications for model building.

Contribution

It establishes the renormalisability of four-fermion theories in four dimensions and derives exact large-Nf beta functions, extending understanding of their quantum properties.

Findings

Four-fermion theories are renormalisable in four dimensions.

Exact large-Nf beta functions are derived.

Extended interactions remain well-defined and predictive.

Abstract

We demonstrate the renormalisability of quantum field theories in four dimensions with elementary self-interacting Dirac fermions and to leading order in the limit of many fermion flavours $N_{\rm f}$. Starting from the underlying divergence structure and using Gross-Neveu-type interactions as a template, we explain why extended four-fermion theories including higher-derivative interactions are well-defined, renormalisable, and predictive with only a few free parameters. We also provide the exact large-$N_{\rm f}$ leading beta functions of couplings and discuss quantum scaling dimensions, universality, $1/N_{\rm f}$ corrections, and extensions to other types of four fermion interactions. Implications for effective theory and model building are indicated.