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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 $λϕ^{4}$ 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 $ω\propto|k|^{\frac{2}{3}}$ . Perhaps they are the "preons" at strong coupling, whose bound states form massless particles over long distances.