Clifford Algebras in Finite Quantum Field Theories, I. Irreducible Yukawa Finiteness Condition

TL;DR

This paper investigates the conditions for finiteness in quantum field theories, focusing on Clifford algebra-based Yukawa couplings, and finds conflicts with finiteness and anomaly cancellation constraints.

Contribution

It analyzes Clifford algebra structures in Yukawa couplings and demonstrates their incompatibility with finiteness and anomaly conditions in certain gauge theories.

Findings

Clifford-like Yukawa couplings conflict with two-loop finiteness.

Such couplings are incompatible with gauge anomaly cancellation.

Results apply to simple gauge groups up to rank 8.

Abstract

Finite quantum field theories may be constructed from the most general renormalizable quantum field theory by forbidding, order by order in the perturbative loop expansion, all ultraviolet-divergent renormalizations of the physical parameters of the theory. The relevant finiteness conditions resulting from this requirement relate all dimensionless couplings in the theory. At first sight, Yukawa couplings which are equivalent to the generators of some Clifford algebra with identity element represent a very promising type of solutions of the condition for one-loop finiteness of the Yukawa couplings. However, under few reasonable and simplifying assumptions about their particular structure, these Clifford-like Yukawa couplings prove to be in conflict with the requirements of one- and two-loop finiteness of the gauge coupling and of the absence of gauge anomalies, at least for all simple gauge groups up to and including rank 8.