The Operator Algebra at the Gaussian Fixed-Point

TL;DR

This paper explores the algebraic structure of scalar composite operators at the Gaussian fixed-point in four dimensions using the ERG formalism, revealing how their scaling properties relate to short-distance singularities.

Contribution

It provides a perturbative construction of the $\,\phi^4$ theory with momentum-dependent sources and analyzes the operator algebra at the Gaussian fixed-point using ERG.

Findings

Characterization of multiple products of scalar operators at the Gaussian fixed-point

Relation between scaling properties and short-distance singularities

Framework for perturbative $\,\phi^4$ theory with momentum-dependent sources

Abstract

We consider the multiple products of relevant and marginal scalar composite operators at the Gaussian fixed-point in $D=4$ dimensions. This amounts to perturbative construction of the $\phi^4$ theory where the parameters of the theory are momentum dependent sources. Using the exact renormalization group (ERG) formalism, we show how the scaling properties of the sources are given by the short-distance singularities of the multiple products.