Renormalization of composite operators

TL;DR

This paper develops a geometric framework for understanding the scale evolution of composite operators in scalar field theory, introducing a connection on the RG trajectory that captures operator mixing and scaling regimes.

Contribution

It introduces a novel geometric approach using a connection on the RG trajectory to analyze operator mixing and scaling in scalar field theories.

Findings

Operator mixing can be described as a linear transformation.

The connection on the RG trajectory governs the scale evolution of operators.

Eigenvalue analysis reveals different scaling regimes and relevant operators.

Abstract

The blocked composite operators are defined in the one-component Euclidean scalar field theory, and shown to generate a linear transformation of the operators, the operator mixing. This transformation allows us to introduce the parallel transport of the operators along the RG trajectory. The connection on this one-dimensional manifold governs the scale evolution of the operator mixing. It is shown that the solution of the eigenvalue problem of the connection gives the various scaling regimes and the relevant operators there. The relation to perturbative renormalization is also discussed in the framework of the $\phi^3$ theory in dimension $d=6$.