Static forces in d=2+1 SU(N) gauge theories

TL;DR

This paper presents high-precision lattice calculations of the static force in 2+1 dimensional SU(N) gauge theories, analyzing discretization errors, continuum limits, and the transition between perturbative and string regimes across different N.

Contribution

It provides detailed lattice results for the static force in SU(5), compares with SU(2) and SU(3), and introduces a scale r_s to distinguish short- and long-distance regimes, including large-N estimates.

Findings

The ratio σ r_0^2 ≈ 1.65 - π/24 holds for all N≥2 with 1% accuracy.

The scale r_s decreases relative to r_0 as N increases, with r_s < r_0 in SU(5).

Deviation from Casimir scaling in the k=2 string tension is positive and grows with distance.

Abstract

Using a three-level algorithm we perform a high-precision lattice computation of the static force up to 1fm in the 2+1 dimensional SU(5) gauge theory. Discretization errors and the continuum limit are discussed in detail. By comparison with existing SU(2) and SU(3) data it is found that \sigma r_0^2=1.65-\pi/24 holds at an accuracy of 1% for all N>=2, where r_0 is the Sommer reference scale. The effective central charge c_{eff}(r) is obtained and an intermediate distance r_s is defined via the property c_{eff}(r_s)=\pi/24. It separates in a natural way the short-distance regime governed by perturbation theory from the long-distance regime described by an effective string theory. The ratio r_s/r_0 decreases significantly from SU(2) to SU(3) to SU(5), where r_s < r_0. We give a preliminary estimate of its value in the large-N limit. The static force in the smallest representation of N-ality 2, which tends to the k=2 string tension as r->oo, is also computed up to 0.7fm. The deviation from Casimir scaling is positive and grows from 0.1% to 1% in that range.