Renormalization of a Scalar Field Theory in Strong Coupling

The renormalization problem is solved, qualitatively, for the ${\lambda }_0{\phi }^6$ interaction of a scalar field $\phi \mathrm{}\left(x\right)\mathrm{}$ in two space and one time dimensions. The theory is found to be finite after mass renormalization, although perturbation theory predicts there should be ${\phi }^4$ and ${\phi }^6$ counter-terms also. The renormalized theory is an interacting theory; it is scale-invariant at short distances. The field $\phi$ has canonical dimension $\frac{1}{2}$ in mass units (but this dimension may change in a more quantitative analysis) while the renormalized form of ${\phi }^2$ 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.