Finite SU(3)³ model

We consider N=1 supersymmetric gauge theories based on the group SU(N)_1 x SU(N)_2 x ... x SU(N)_k with matter content (N,N*,1,...,1) + (1,N,N*,..., 1) + >... + (N*,1,1,...,N) as candidates for the unification symmetry of all particles. In particular we examine to which extent such theories can become finite, and find that a necessary condition is that there should be exactly three families. From phenomenological considerations an SU(3)^3 model is singled out. We consider an all-loop and a two-loop finite model based on this gauge group and we study their predictions concerning the third generation quark masses.