High-quality grand unified theories with three generations

We extend the unitary groups beyond the SU(5)and SU(6)to determine possible grand unified theories that give rise to three-generational Standard Model fermions without simple repetitions.By demanding asymptotic free theories at short distances,we find gauge groups of SU(7),SU(8),and SU(9),together with their anomaly-free irreducible representations,are such candidates.Two additional gauge groups of SU(10)and SU(11)can also achieve the generational structure without asymptotic freedom.We also deduce that these models can solve the Peccei-Quinn(PQ)quality problem,which is intrinsic in the axion models,with the leading PQ-breaking operators determined from the symmetry requirement.