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A precise calculation of the fundamental string tension in SU(N) gauge theories in 2+1 dimensions

Barak Bringoltz, Michael Teper

TL;DR

The paper tests the Karabali–Nair prediction for the fundamental string tension in SU(N) gauge theories in 2+1 dimensions by performing a high-precision lattice calculation with controlled systematic errors. It analyzes winding flux tubes to extract string energies, using Luscher corrections and a Nambu–Goto–inspired universal form, and then performs continuum and large-N extrapolations. The results show KN is within ~3% for finite N and remains not exact at N → ∞, with the infinite-N lattice value of $\sqrt{\sigma}/(g^2 N)$ about 0.98–1.2% below the KN value $1/\sqrt{8\pi} \approx 0.199471$, a statistically significant discrepancy. This provides strong empirical support for KN while clarifying its limitations in the strict large-N limit, and demonstrates careful control of lattice systematics in a challenging 2+1D setting.

Abstract

We use lattice techniques to calculate the continuum string tensions of SU(N) gauge theories in 2+1 dimensions. We attempt to control all systematic errors at a level that allows us to perform a precise test of the analytic prediction of Karabali, Kim and Nair. We find that their prediction is within 3% of our values for all N and that the discrepancy decreases with increasing N. When we extrapolate our results to N=oo we find that there remains a discrepancy of ~ 1%, which is a convincing ~6 sigma effect. Thus, while the Karabali-Nair analysis is remarkably accurate at N=oo, it is not exact.

A precise calculation of the fundamental string tension in SU(N) gauge theories in 2+1 dimensions

TL;DR

The paper tests the Karabali–Nair prediction for the fundamental string tension in SU(N) gauge theories in 2+1 dimensions by performing a high-precision lattice calculation with controlled systematic errors. It analyzes winding flux tubes to extract string energies, using Luscher corrections and a Nambu–Goto–inspired universal form, and then performs continuum and large-N extrapolations. The results show KN is within ~3% for finite N and remains not exact at N → ∞, with the infinite-N lattice value of about 0.98–1.2% below the KN value , a statistically significant discrepancy. This provides strong empirical support for KN while clarifying its limitations in the strict large-N limit, and demonstrates careful control of lattice systematics in a challenging 2+1D setting.

Abstract

We use lattice techniques to calculate the continuum string tensions of SU(N) gauge theories in 2+1 dimensions. We attempt to control all systematic errors at a level that allows us to perform a precise test of the analytic prediction of Karabali, Kim and Nair. We find that their prediction is within 3% of our values for all N and that the discrepancy decreases with increasing N. When we extrapolate our results to N=oo we find that there remains a discrepancy of ~ 1%, which is a convincing ~6 sigma effect. Thus, while the Karabali-Nair analysis is remarkably accurate at N=oo, it is not exact.

Paper Structure

This paper contains 9 sections, 18 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Lattice setup
  3. Methodology
  4. Extracting string masses from correlation functions
  5. Extracting string tensions from string energies
  6. Extrapolation to the continuum limit
  7. Extrapolation to $N=\infty$
  8. Results
  9. Summary

Figures (2)

  • Figure 1: The dimensionless quantity $\beta_{\rm MF} \frac{a\sqrt{\sigma}}{2N^2} \stackrel{a\to 0}{\longrightarrow} \frac{\sqrt{\sigma}}{g^2N}$ as a function of the improved inverse coupling coupling $1/\beta_{\rm MF}$ for $N=4,6$. The error bars at $1/\beta_{\rm MF}=0$ denote the result of the continuum extrapolation, while the horizontal bars denote the values predicted by Karabali, Kim, and Nair KN.
  • Figure 2: The ratio $r$ between the prediction of Eq. (\ref{['KN_sigma']}) and our data, as a function of $1/N^2$. The error bar at $1/N^2=0$ denotes the linear extrapolation to the $N=\infty$ limit.