Renormalization Group Patterns and C-Theorem in More Than Two Dimensions

TL;DR

This paper explores the extension of the c-theorem and renormalization group irreversibility to higher dimensions, proposing a spectral representation-based c-function and analyzing its properties in various scalar theories.

Contribution

It introduces a spectral representation approach to define a c-function in dimensions greater than two and discusses its validity for perturbative and non-perturbative flows.

Findings

c-function is well-defined for perturbative flows in any dimension

The c-theorem holds for certain scalar theories in 2<d<4

Non-perturbative analysis of O(N) sigma-models suggests potential extensions

Abstract

We elaborate on a previous attempt to prove the irreversibility of the renormalization group flow above two dimensions. This involves the construction of a monotonically decreasing $c$-function using a spectral representation. The missing step of the proof is a good definition of this function at the fixed points. We argue that for all kinds of perturbative flows the $c$-function is well-defined and the $c$-theorem holds in any dimension. We provide examples in multicritical and multicomponent scalar theories for dimension $2<d<4$. We also discuss the non-perturbative flows in the yet unsettled case of the $O(N)$ sigma-model for $2\leq d\leq 4$ and large $N$.