Higgs phenomena in asymptotically free gauge theories

We examine in detail the possibility of using the Higgs mechanism to remove the catastrophic infrared singularities in non-Abelian gauge theories which are asymptotically free. Our investigation encompasses theories based on SU(N) or O(N) with scalars in one vector, two vector, M vector, adjoint, tensor, and adjoint plus one vector representations. We find that for these theories an S matrix, in the perturbative sense, and asymptotic freedom can not coexist. We show that a wide class of Yukawa couplings can be ignored in studying the large-momentum properties of the scalar couplings.