Co-Variant Derivatives And The Renormalisation Group

TL;DR

This paper explores a geometric interpretation of the renormalisation group equations for correlation functions, introducing connections on the coupling space to account for point proximity effects and their impact on the RG flow.

Contribution

It proposes a modification to the RG equation involving covariant derivatives and curvature, linking the connection to operator expansion coefficients.

Findings

Connection relates to operator expansion coefficients

Modification accounts for points approaching each other

Curvature appears in the RG equation

Abstract

The renormalisation group equation for $N$-point correlation functions can be interpreted in a geometrical manner as an equation for Lie transport of amplitudes in the space of couplings. The vector field generating the diffeomorphism has components given by the $\beta$-functions of the theory. It is argued that this simple picture requires modification whenever any one of the points at which the amplitude is evaluated becomes close to any other. This modification requires the introduction of a connection on the space of couplings and new terms appear in the renormalisation group equation involving co-variant derivatives of the $\beta$-function and the curvature associated with the connection. It is shown how the connection is related to the operator expansion co-efficients, but there remains an arbitrariness in its definition.