Finite quantum field theories: Clifford algebras for Yukawa couplings?

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.