Toward finite quantum field theories (Q1071333)

The properties that make the \(N=4\) 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 \[ \sum_{s=0,1/2}(-1)^{2s+1}(2s+1)M^ 2_ s=0. \] Finite \(N=2\) theories are found to satisfy the above criteria. The missing member in this class of field theories are finite field theories consisting of \(N=1\) 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 finite \(N=1\) field theories are discussed. A consequence of these restrictions on the allowed representations is that an \(N=1\) finite SU(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.