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 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 $$ \lambda \left(\tilde{l}\right) $$ . 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 $$ \lambda \left(\tilde{l}\right) $$ . 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.
Highlights
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
One would start by measuring the Twisted Gradient Flow (TGF) coupling on a set of SU(N ) TEK lattices, tuning the bare coupling b to obtain the same value of the renormalized coupling u for several values of N :
We have pushed the idea of volume independence to the extreme by determining the scale dependence of the SU(N ) renormalized gauge coupling from a scaling analysis of the single site TEK lattice, where the rank of the gauge group acts as a size parameter
Summary
Before presenting the outcome of the step scaling analysis we need to discuss the lattice definition of the TGF coupling. The dependence on the flow time t of the plaquette (blue) and symmetric (red) definitions compared to the continuum expression (green) is displayed in figure 1(a). (4.7) and (4.8) the lattice kernel, exp(−2tq2), by the continuum one, exp(−2tq2), we obtain the results displayed in figure 1(b) In this case the plaquette definition approximates the continuum result much better than the symmetric one. For comparison we display in figure 1(c) the results that are obtained if the Symanzik improved Square action [20, 21] is used for the flow (details are given in appendix A). These artifacts affect the determination of the TGF running coupling. Most of the results that will be discussed correspond to these improved coupling definitions
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