The geometry of RG flows in theory space

TL;DR

This paper explores the geometric properties of RG flows in theory space, revealing how flows focus towards fixed points and analyzing their expansion, shear, and rotation using methods inspired by General Relativity.

Contribution

It introduces a geometric framework for RG flows, analyzing their properties with a metric ansatz and differential equations, providing new insights into flow behavior near fixed points.

Findings

Expansion is negative and inversely related to the beta function norm

RG flows tend to focus towards fixed points

The evolution of flow expansion is characterized and analyzed

Abstract

Renormalisation Group (RG) flows in theory space (the space of couplings) are generated by a vector field -- the $\beta$ function. Using a specific metric ansatz in theory space and certain methods employed largely in the context of General Relativity, we examine the nature of the expansion, shear and rotation of geodesic RG flows. The expansion turns out to be a negative quantity inversely related to the norm of the $\beta$ function. This implies the focusing of the flows towards the fixed points of a given field theory. The evolution equation for the expansion along geodesic RG flows is written down and analysed. We illustrate the results for a scalar field theory with a $j\phi$ coupling and pointers to other areas are briefly mentioned.