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 $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ or $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ 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.