Four-dimensional conformal field theory models with rational correlation functions

Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in four-dimensional Minkowski spacetime. We consider in detail a model of a neutral scalar field ϕ of dimension two. It depends on a positive real parameter c, an analogue of the Virasoro central charge, and admits for all (finite) c an infinite number of conserved symmetric tensor currents. The operator product algebra of ϕ is shown to coincide with a simpler one, generated by a bilocal scalar field V(x₁,x₂) of dimension (1,1). The modes of V together with the unit operator span an infinite-dimensional Lie algebra _V whose vacuum (i.e. zero-energy lowest-weight) representations only depend on the central charge c. Wightman positivity (i.e. unitarity of the representations of _V) is proven to be equivalent to c∊.