The Exact Renormalization Group and Approximate Solutions

We investigate the structure of Polchinski’s formulation of the flow equations for the continuum Wilson effective action. Reinterpretations in terms of I.R. cutoff Green’s functions are given. A promising nonperturbative approximation scheme is derived by carefully taking the sharp cutoff limit and expanding in “irrelevancy” of operators. We illustrate with two simple models of four-dimensional λφ 4 theory: the cactus approximation, and a model incorporating the first irrelevant correction to the renormalized coupling. The qualitative and quantitative behaviour give confidence in a fuller use of this method for obtaining accurate results.