Wilsonian renormalisation of CFT correlation functions: field theory

Abstract

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.

Functional perturbative RG and CFT data in the $$\epsilon $$ ϵ -expansion

“…A more refined approach to both these shortcomings which also aligns with our discussion of the scheme transformations of Sect. 2.3 can be found in [43] where special "normal" coordinates in the space of all couplings are found in the context of the functional renormalization group (using the Polchinski equation instead of the Wetterich equation, but arguably the conclusions are very similar). The normal coordinates of [43] could be understood as a geometrical generalization of the basis of couplings with well-behaved scaling properties introduced in Sect.…”

“…2.3 can be found in [43] where special "normal" coordinates in the space of all couplings are found in the context of the functional renormalization group (using the Polchinski equation instead of the Wetterich equation, but arguably the conclusions are very similar). The normal coordinates of [43] could be understood as a geometrical generalization of the basis of couplings with well-behaved scaling properties introduced in Sect. 2.3 and their application clearly shows that a consistent renormalization of correlators of all composite operators, thus including in principle all possible OPE coefficients, is possible within the functional renormalization group approach (at least in the vicinity of the Gaussian fixed point).…”

“…A step in this direction has been made for functional-type flows in Ref. [43] in the context of the Polchinski RG equation. In this work, the authors define normal coordinates in the space of couplings which have both geometrical meaning and definite scaling transformations.…”

“…In this work, the authors define normal coordinates in the space of couplings which have both geometrical meaning and definite scaling transformations. In relation with the transformation (2.23), one can follow [43] and argue that all "massive" coefficients can be eliminated by an opportune transformation of the couplings and hence there exists a scheme, or rather a family of schemes, whose only coefficients are the scheme independent ones. This family can be appropriately named "family of minimal subtraction schemes" having the MS scheme as its most famous representative.…”

Connecting holographic Wess-Zumino consistency condition to the holographic anomaly

“…Corresponding to that there is a chart on the theory space with well defined transition maps corresponding to the renormalized sources. This perspective is emphasised in [24].…”