The $SU(\infty)$ twisted gradient flow running coupling

TL;DR

This paper measures the running of the $SU( {infty})$ 't Hooft coupling using a step scaling analysis of the TEK model, confirming the expected large N behavior and matching perturbative results at weak coupling.

Contribution

It introduces a novel method to determine the running coupling in the large N limit via the TEK model and twisted boundary conditions, extending the understanding of non-perturbative gauge dynamics.

Findings

Successful determination of the running coupling over a wide scale range.

Agreement with two-loop perturbative predictions at weak coupling.

Validation of the finite volume effect conjecture for SU(N) gauge theories.

Abstract

We measure the running of the $SU(\infty)$ 't Hooft coupling by performing a step scaling analysis of the Twisted Eguchi-Kawai (TEK) model, the SU($N$) gauge theory on a single site lattice with twisted boundary conditions. The computation relies on the conjecture that finite volume effects for SU(N) gauge theories defined on a 4-dimensional twisted torus are controlled by an effective size parameter $\tilde l = l \sqrt{N}$, with $l$ the torus period. We set the scale for the running coupling in terms of $\tilde l$ and use the gradient flow to define a renormalized 't Hooft coupling $\lambda(\tilde l)$. In the TEK model, this idea allows the determination of the running of the coupling through a step scaling procedure that uses the rank of the group as a size parameter. The continuum renormalized coupling constant is extracted in the zero lattice spacing limit, which in the TEK model corresponds to the large $N$ limit taken at fixed value of $\lambda(\tilde l)$. The coupling constant is thus expected to coincide with that of the ordinary pure gauge theory at $N =\infty$. The idea is shown to work and permits us to follow the evolution of the coupling over a wide range of scales. At weak coupling we find a remarkable agreement with the perturbative two-loop formula for the running coupling.