Four Fermi Theory in Four Dimensions is Renormalisable

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.

Figures (2)

• Figure 1: Divergent subdiagrams appearing at leading order in the large-$N_\mathrm{f}$ expansion. Only the index contraction with a trace over the propagators in the loop is implied in each.
• Figure 2: Large-$N_\mathrm{f}$ leading contributions to the 8F vertex in the extended GN theory. Shaded blobs with label $\Sigma$ represent an infinite series of large-$N_\mathrm{f}$ fermion bubbles given in \ref{['eq:4Fsum']}.