Renormalization of a Scalar Field Theory in Strong Coupling

The renormalization problem is solved, qualitatively, for the λ₀φ⁶ interaction of a scalar field φ(x) in two space and one time dimensions. The theory is found to be finite after mass renormalization, although perturbation theory predicts there should be φ⁴ and φ⁶ counter-terms also. The renormalized theory is an interacting theory; it is scale-invariant at short distances. The field φ has canonical dimension 1/2 in mass units (but this dimension may change in a more quantitative analysis) while the renormalized form of φ² has a noncanonical dimension, about 1.36. The cutoff dependence of the theory is computed by evaluating the Feynman path integral qualitatively. The analysis reduces the problem to a recursion formula for a function of one variable which acts as a representation of the renormalization group. The method of analysis is applicable to any scalar field theory.