Wilsonian renormalisation of CFT correlation functions: field theory

Abstract

We examine the precise connection between the exact renormalisation group with local couplings and the renormalisation of correlation functions of composite operators in scale-invariant theories. A geometric description of theory space allows us to select convenient non-linear parametrisations that serve different purposes. First, we identify normal parameters in which the renormalisation group flows take their simplest form; normal correlators are defined by functional differentiation with respect to these parameters. The renormalised correlation functions are given by the continuum limit of correlators associated to a cutoff-dependent parametrisation, which can be related to the renormalisation group flows. The necessary linear and non-linear counterterms in any arbitrary parametrisation arise in a natural way from a change of coordinates. We show that, in a class of minimal subtraction schemes, the renormalised correlators are exactly equal to normal correlators evaluated at a finite cutoff. To illustrate the formalism and the main results, we compare standard diagrammatic calculations in a scalar free-field theory with the structure of the perturbative solutions to the Polchinski equation close to the Gaussian fixed point.