Exactly soluble limit of ϕ³field theory with internal Potts symmetry

The authors consider a field theory in which the fields from a (2l+1)-dimensional (complex) irreducible representation of SO(3). This theory contains an additional internal discrete symmetry of the 3-state Potts model. There exists only one quadratic and one cubic invariant. There are many quartic invariants, but they are irrelevant below six dimensions as far as the infra-red behaviour is concerned. It is shown that as l to infinity the model becomes soluble. In contrast to the infinite N limit here it is the wavefunction, rather than the coupling constant, that is renormalised in a nontrivial way, as well as phi 2. The asymptotic behaviour of the infinite l limit is analysed explicitly, and compared with renormalisation group results.