[Paper Review] The string tension for Large N gauge theory from smeared Wilson loops

Antonio González-Arroyo, M. Okawa|arXiv (Cornell University)|Dec 16, 2012

Black Holes and Theoretical Physics2 references3 citations

TL;DR

This paper proposes a method to extract the string tension in large-N SU(N) gauge theory using smeared Creutz ratios from lattice simulations, achieving smooth continuum scaling and linear extrapolation in 1/N². The results match precisely with those from the Twisted Eguchi-Kawai reduced model, providing strong evidence for continuum volume independence and the validity of twisted reduction in large-N gauge theories.

ABSTRACT

Using smeared Creutz ratios we extract the string tension for SU(N) pure gauge theory and $N$=3,4,5,6,8. We employ these results to extrapolate to large N. The same methodology is applied to the single-site Twisted Eguchi Kawai model. The corresponding string tension matches perfectly within errors with the extrapolated one, providing strong evidence in favour of the twisted reduction framework. Interesting results are also obtained on the behaviour of Creutz ratios for large sizes.

Motivation & Objective

• To compute the continuum string tension in large-N SU(N) gauge theory using lattice simulations with smeared Wilson loops.
• To test the validity of the twisted reduction framework by comparing results from full lattice gauge theory with those from the single-site Twisted Eguchi-Kawai (TEK) model.
• To establish a non-perturbative renormalization scheme based on continuum Creutz ratios to define a physical scale.
• To investigate the scaling behavior of Creutz ratios and their deviation from Nambu-Goto string predictions at finite N.
• To determine the leading and subleading behavior of the effective string tension function and compare with perturbative and Nambu-Goto expectations.

Proposed method

• Use smeared Wilson loops to compute Creutz ratios on 32⁴ lattices for SU(N) with N = 3, 4, 5, 6, 8, and up to N = 1369 in the TEK model.
• Extract the continuum limit of Creutz ratios by fitting to the form χ(T,R) = a²(b)F̃(t,r) + a⁴(b)H̃(t,r) + ..., where t = Ta(b), r = Ra(b), and F̃(t,r) is the continuum function.
• Define a physical scale r̄ via the condition r̄²F̃(r̄,r̄) = 1.65, analogous to the Sommer scale, to enable non-perturbative renormalization.
• Fit the continuum function F̃(r,r) for each N to a quadratic polynomial in r̄²/r² to extract σ(∞) and γ(1).
• Compare the string tension and subleading correction γ(z) from full SU(N) simulations with those from the TEK model to test reduction.
• Use effective string theory and perturbative QCD to interpret deviations from Nambu-Goto predictions.

Experimental results

Research questions

• RQ1 Does the string tension in SU(N) gauge theory extrapolate linearly in 1/N² toward the large-N limit?
• RQ2 To what extent do the Creutz ratios from full lattice simulations scale smoothly to the continuum limit?
• RQ3 How well does the Twisted Eguchi-Kawai model reproduce the string tension of the full large-N gauge theory?
• RQ4 What is the behavior of the subleading correction γ(z) in the effective string tension, and how does it compare to Nambu-Goto and perturbative predictions?
• RQ5 Can a non-perturbative renormalization scheme based on Creutz ratios be consistently defined and used to extract physical observables?

Key findings

• The string tension σ(∞) is extracted as 1.105(10) in units of r̄², with a smooth continuum limit and linear extrapolation in 1/N².
• The subleading correction γ(1) = 0.272(5) is significantly larger than the Nambu-Goto prediction γ_NG(1) ≈ 0.16, indicating deviation from the Nambu-Goto model.
• The data for γ(z) are well described by a parametrization γ(z) = γ(1)(1 + τ(z−1)²/(2z)) with τ = 0.31(6), close to the perturbative value of 0.39.
• The string tension from the Twisted Eguchi-Kawai model matches the extrapolated value from full SU(N) simulations within errors, providing strong evidence for continuum reduction.
• The continuum function F̃(r,t) is well-defined and exhibits scaling behavior consistent with effective string theory, supporting the use of a non-perturbative scale r̄.
• The results suggest that the effective string description at finite N includes contributions beyond the Nambu-Goto action, possibly from one-gluon exchange effects.