Finite quantum field theories: Clifford algebras for Yukawa couplings?

TL;DR

This paper investigates the conditions for constructing finite quantum field theories in four dimensions, focusing on the role of Clifford algebra structures in Yukawa couplings and their consistency with finiteness and anomaly cancellation.

Contribution

It analyzes the viability of Clifford algebra-based Yukawa couplings in finite quantum field theories, revealing their inconsistency with gauge finiteness and anomaly constraints for simple gauge groups.

Findings

Clifford algebra Yukawa solutions are incompatible with gauge finiteness.

Such solutions fail to satisfy anomaly cancellation conditions.

The study constrains the algebraic structures suitable for finite theories.

Abstract

By imposing on the most general renormalizable quantum field theory the requirement of the absence of ultraviolet-divergent renormalizations of the physical parameters (masses and coupling constants) of the theory, finite quantum field theories in four space-time dimensions may be constructed. Famous prototypes of these form certain well-known classes of supersymmetric finite quantum field theories. Within a perturbative evaluation of the quantum field theories under consideration, the starting point of all such investigations is represented by the conditions for one- and two-loop finiteness of the gauge couplings as well as for one-loop finiteness of the Yukawa couplings. Particularly attractive solutions of the one-loop Yukawa finiteness condition involve Yukawa couplings which are equivalent to generators of Clifford algebras with identity element. However, a closer inspection shows, at least for all simple gauge groups up to and including rank 8, that these Clifford-like solutions prove to be inconsistent with the requirements of one- and two-loop finiteness of the gauge coupling and of absence of gauge anomalies.