Clifford algebras in finite quantum field theories II. Reducible Yukawa finiteness condition

Abstract

An arbitrary renormalizable quantum field theory is considered as finite if its dimensionless couplings conspire to yield, at every order of its perturbative expansion, no ultraviolet-divergent renormalizations of the physical parameters of the theory. The "finiteness conditions" resulting from these requirements form highly complicated, non-linear systems of relations. A promising type of solution to the condition for one-loop finiteness of the Yukawa couplings involves Yukawa couplings which are equivalent to the generators of Clifford algebras with identity element. However, our attempt to construct even one finite model based on such Clifford-like Yukawa couplings fails: a Clifford structure of the Yukawa couplings spoils the finiteness of the gauge couplings, at least for every simple gauge group of rank less than or equal to 8.