SU(9) grand unification without domain walls

Abstract

The domain wall problem in grand unified theories due to the unbroken $U(1)_{PQ}$ symmetries are in contradiction with the modern cosmology based on the big bang theory. We construct a SU(9) grand unified model without the domain walls and study the symmetry breaking patterns, the intermediate mass scale and the fermion contents of the model. First, we give some constraints which the fermion representations of the SU(N) groups should satisfy. Then we find that there are two kinds of fermion representations according to the assignment of the hypercharges. One is the representation which allows the trivial charge assignment and the other is the one which allow the nontrivial charge assignment. Next, we search for the possible grand unified model without the domain walls. For the representations with the trivial charge assignment, we find only one model in SU(9) which contains the three generations and for the representations with the nontrivial charge assignment, we find a unique model in SU(9). Hence the study of the details of the SU(9) becomes important. Then we give the possible symmetry breaking patterns and the corresponding projection matrices which can be used to find the fermion contents of the low energy symmetry. Also we discuss the grand unification mass scale for some typical symmetry breaking patterns. Finally, we give a formula to determine the intermediate mass scale of the SU(N) group. Using the limit on the proton life time and the cosmological requirements on the intermediate mass scale, we determine the symmetry breaking patterns and the value of the nontrivial charge assignment by applying the formula stated above.