Four Fermi Theory in Four Dimensions is Renormalisable

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.