Spontaneous symmetry breaking and global symmetries in unified gauge theories

This work is separated into two parts. Part I is concerned with properties of gauge theories with additional global symmetries. In such theories, it can happen that both the gauge and global symmetry are spontaneously broken, but in such a way that linear combinations of gauge and global generators are left unbroken. We give a general proof that Goldstone bosons are avoided in such cases (the 't Hooft mechanism), and discuss the relation of this result to Goldstone's theorem and the Higgs mechanism. We then discuss a number of models in which this mechanism can be at work, including both toy models based on SU(n) x O(2) and SO(n) x SO(m) groups, and grand unified models based on gauge groups SU(5), SO(10), and E6. In Part II we discuss the renormalization group equations for the coupling constants in E6 grand unified theories. After a short review of the renormalization group analysis and of the E6 theory, we given some group theoretic details related to the fundamental 27-dimensional representation of E6. These mathematical results are used to evaluate the group weight factors that come into the calculation of the E6 renormalization group equations for the coupling consants. We analyze the renormalization group equations for scalar coupling constants in models with n 27-plets of scalars, and for a 27-plet and a 78-plet of scalars. We find that only the cases n = 1 or 2 can be asymptotically free. We also discuss the effect of including fermions in the above models