The asymptotically-free gauge theories

TL;DR

This paper classifies all asymptotically-free gauge theories with purely fermionic matter in four dimensions, providing a systematic way to identify such theories based on Lie algebra representation properties.

Contribution

It introduces a systematic classification method for asymptotically-free gauge theories using Lie algebra representation theory, including comprehensive tables for irreducible representations.

Findings

Finite number of asymptotically-free representations per Lie algebra

Maximum of two non-zero Dynkin labels in these representations

No Dynkin label exceeds four in the classified representations

Abstract

We show how to classify the asymptotically-free gauge theories in four spacetime dimensions, focussing here on the case of purely fermionic matter. The classification depends on the fact (which we prove) that both the dimension and Dynkin index of irreducible representations of a simple Lie algebra are strictly increasing functions of each Dynkin label. This implies not only that the number of asymptotically-free representations of any one semisimple Lie algebra is finite, but also that they can be written down in a systematic fashion using tables for the asymptotically-free irreducible representations of simple Lie algebras, which we supply. These tables show that at most two out of a possible ten Dynkin labels can be non-zero and that no Dynkin label can exceed four. The extension to bosonic matter or supersymmetric theories is straightforward.