Highly nonlinear wave solutions in a dual to the chiral model

We consider a two-dimensional scalar field theory with a nilpotent current algebra, which is dual to the Principal Chiral Model. The quantum theory is renormalizable and not asymptotically free: the theory is strongly coupled at short distances (encountering a Landau pole). We suggest it can serve as a toy model for $\lambda {\varphi }^4$ theory in four dimensions, just as the principal chiral model is a useful toy model for Yang-Mills theory. We find some classical wave solutions that survive the strong coupling limit and quantize them by the collective variable method. They describe excitations with an unusual dispersion relation $\omega \propto \mathrm{\mid }k{\mathrm{\mid }}^{\frac{2}{3}}$ . Perhaps they are the “preons” at strong coupling, whose bound states form massless particles over long distances.