The SU(∞) twisted gradient flow running coupling

Abstract

Abstract:We measure the running of the SU(∞) '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 parameterl = l √ N , with l the torus period. We set the scale for the running coupling in terms ofl and use the gradient flow to define a renormalized 't Hooft coupling λ(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 λ(l). The coupling constant is thus expected to coincide with that of the ordinary pure gauge theory at N = ∞. 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.

Perturbative contributions to Wilson loops in twisted lattice boxes and reduced models

“…This applies rather well to the 2+1 dimensional case both in perturbation theory and non-perturbatively [39,49,65] and to the non-perturbative calculation of the twisted gradient flow running coupling in SU(∞) [66].…”

A comparison of updating algorithms for large N reduced models

“…There is now strong numerical evidence that its results for various lattice observables coincide with those of SU(N ) gauge theories extrapolated to N → ∞ [13]. Furthermore, the tests have also extended to physical quantities in the continuum limit such as the string tension [14] or the renormalized running coupling [15].…”

“…In particular we have focused on the TEK model, which is relevant in the context of the large N expansion of gauge theories and has recently been used to compute many interesting properties of SU(∞) gauge theories [14,15,17,18].…”

“…We will consider the renormalized coupling λ TGF [36] as defined for the TEK model in ref. [15]. We will tune b for various values of N , such that λ TGF 23, and hence the physical volume (a √ N ) is kept approximately constant.…”

Prospects for large N gauge theories on the lattice

“…H f v Y 9 l a 8 P K Z U / g D 7 / M H n q W P K g = = < / l a t e x i t > u u Figure 10: The continuum extrapolated SU(∞) step-scaling function compared to the one-loop and two-loop predictions [47].…”