Radiatively induced spontaneous symmetry breaking for asymptotically free gauge theories

We investigate the possibility that the symmetry of asymptotically free theories whose effective scalar quartic coupling constants (in the sense of the renormalization group) take on negative values is dynamically spontaneously broken. We find this phenomenon to be common for theories whose scalars transform as the irreducible second-rank tensor representations of $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ and $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$. The symmetry-breaking pattern is determined by the tree approximation in some cases, by the one-loop approximation in other cases, but for none of these models does it allow for an asymptotically free theory whose $S$-matrix elements are defined. Our dynamical mechanism does provide a natural explanation of superstrong spontaneous symmetry breaking.