General Properties of Multiscalar RG Flows in $d=4-\varepsilon$

TL;DR

This paper investigates the properties of multiscalar renormalization group flows in four minus epsilon dimensions, establishing bounds on fixed points and exploring the structure of scalar conformal field theories with quartic interactions.

Contribution

It proves a lower bound on the fixed-point value of the scalar function A in scalar theories, linking it to the dimension of the order parameter, and analyzes conditions for saturation involving marginal deformations.

Findings

Fixed points are characterized by a scalar function A with a lower bound.

Saturation of the bound occurs at coincident fixed points with the same symmetry.

Several general results and known fixed points of scalar CFTs are reviewed.

Abstract

Fixed points of scalar field theories with quartic interactions in $d=4-\varepsilon$ dimensions are considered in full generality. For such theories it is known that there exists a scalar function $A$ of the couplings through which the leading-order beta-function can be expressed as a gradient. It is here proved that the fixed-point value of $A$ is bounded from below by a simple expression linear in the dimension of the vector order parameter, $N$. Saturation of the bound requires a marginal deformation, and is shown to arise when fixed points with the same global symmetry coincide in coupling space. Several general results about scalar CFTs are discussed, and a review of known fixed points is given.