Connection between the renormalization groups of St\"uckelberg-Petermann and Wilson

TL;DR

This paper explores the connection between the Stueckelberg-Petermann and Wilson renormalization groups, introducing an UV-cutoff in causal perturbation theory and deriving a flow equation that links both approaches.

Contribution

It establishes a formal relationship between the Stueckelberg-Petermann and Wilson renormalization groups within causal perturbation theory by introducing an UV-cutoff and deriving a flow equation.

Findings

The flow of the effective potential $V_\Lambda$ aligns with Wilson's renormalization group.

Operators restricted to local interactions can be approximated by a subset of the Stueckelberg-Petermann group.

A comparison between causal perturbation theory and functional integral formalism is achieved.

Abstract

The Stueckelberg-Petermann renormalization group is the group of finite renormalizations of the S-matrix in the framework of causal perturbation theory. The renormalization group in the sense of Wilson relies usually on a functional integral formalism, it describes the dependence of the theory on a UV-cutoff $\Lambda$; a widespread procedure is to construct the theory by solving Polchinski's flow equation for the effective potential. To clarify the connection between these different approaches we proceed as follows: in the framework of causal perturbation theory we introduce an UV-cutoff $\Lambda$, define an effective potential $V_\Lambda$, prove a pertinent flow equation and compare with the corresponding terms in the functional integral formalism. The flow of $V_\Lambda$ is a version of Wilson's renormalization group. The restriction of these operators to local interactions can be approximated by a subfamily of the Stueckelberg-Petermann renormalization group.