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 oneand 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, our closer inspection shows, at least for all simple gauge groups up to and including rank 8, that Clifford-like solutions prove to be inconsistent with the requirements of oneand two-loop finiteness of the gauge coupling and of absence of gauge anomalies. PACS : 11.10.Gh, 11.30.Pb ∗ E-mail: [email protected] † E-mail: [email protected]