Toward finite quantum field theories

Abstract

The properties that make theN=4 super Yang-Mills theory free from ultraviolet divergences are (i) a universal coupling for gauge and matter interactions, (ii) anomaly-free representations, (iii) no charge renormalization, and (iv) if masses are explicitly introduced into the theory, then these are required to satisfy the mass-squared supertrace sum rule ∑s=0,1/2(−1)2s+1(2s+1)Ms2=0. FiniteN=2 theories are found to satisfy the above criteria. The missing member in this class of field theories are finite field theories consisting ofN=1 superfields. These theories are discussed in the light of the above finiteness properties. In particular, the representations of all simple classical groups satisfying the anomaly-free and no-charge renormalization conditions for finiteN=1 field theories are discussed. A consequence of these restrictions on the allowed representations is that anN=1 finiteSU (5)-based model of strong and electroweak interactions can contain at most five conventional families of quarks and leptons, a constraint almost compatible with the one deduced from cosmological arguments.