Exact $\beta$-function of Yang-Mills theory in 2+1 dimensions

TL;DR

This paper derives an exact expression for the $eta$-function of 2+1 dimensional Yang-Mills theory, showing it has a fixed sign and no accessible infrared fixed points, with implications for understanding its renormalization behavior.

Contribution

It provides the first exact $eta$-function for 2+1 dimensional Yang-Mills theory, establishing its universal sign and fixed points behavior across schemes.

Findings

$eta$-function is exactly $-1$ in the chosen scheme.

Yang-Mills $eta$-function has the same sign for all positive couplings.

Infrared fixed points are unreachable in practice.

Abstract

To set the stage, I discuss the $\beta$-function of the massless O(N) model in three dimensions, which can be calculated exactly in the large N limit. Then, I consider SU(N) Yang-Mills theory in 2+1 space-time dimensions. Relating the $\beta$-function to the expectation value of the action in lattice gauge theory, and the latter to the trace of the energy-momentum tensor, I show that $\frac{d \ln g^2/\mu}{d\ln \mu}=-1$ for all $g$ and all N in one particular renormalization scheme. As a consequence, I find that the Yang-Mills $\beta$-function in three dimensions must have the same sign for all finite and positive bare coupling parameters in any renormalization scheme, and all non-trivial infrared fixed points are unreachable in practice.