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

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.

Comments: 22 pages, 7 figures