[Paper Review] The spectrum of closed loops of fundamental flux in D=2+1 SU(N) gauge theories

Andreas Athenodorou, Barak Bringoltz+1 more|ArXiv.org|Sep 19, 2007

Black Holes and Theoretical Physics6 references3 citations

TL;DR

This paper investigates the spectrum of closed fundamental flux tubes in 2+1-dimensional SU(N) gauge theories using lattice simulations with N=3 and N=6. By constructing a large basis of smeared Polyakov loop operators and applying variational methods, the authors find that the Nambu-Goto string model describes the energy levels—especially the ground and first excited states—with excellent accuracy even for short strings, significantly outperforming effective string theory predictions like the Lüscher or Lüscher-Weisz terms.

ABSTRACT

We study the closed-string spectrum of SU(N) gauge theories in the fundamental representation in 2+1 dimensions. We calculate the energies of the lowest lying ~ 30 states using a large variety of operators characterised by the quantum numbers of parity and longitudinal momentum. We find that our results for the ground state are very well approximated by the Nambu-Goto (NG) predictions even for short strings. For the excited states, we observe significant deviations from the NG predictions only for very short strings and they decrease rapidly with increasing string length. Finally, we see that Nambu-Goto provides a much better description of our results than the effective string theoretical predictions. We discuss the continuum and large-N limits.

Motivation & Objective

• To determine whether confining flux tubes in 2+1D SU(N) gauge theories can be described by a low-energy effective string theory.
• To test the validity of the Nambu-Goto model versus alternative effective string theories for flux-tube spectra.
• To investigate the role of quantum numbers like transverse parity (P) and longitudinal momentum (q) in the spectrum of closed flux tubes.
• To examine the behavior of the spectrum in the large-N and continuum limits.
• To assess the accuracy of the Nambu-Goto model for short strings, where quantum corrections may be significant.

Proposed method

• Lattice simulations of SU(3) and SU(6) gauge theories on 3D periodic lattices with spacings a ≈ 0.04–0.08 fm and string lengths from 0.65 to 2.60 fm.
• Use of the Wilson plaquette action with β scaling as N² to maintain fixed lattice spacing in the large-N limit.
• Construction of a basis of 80–200 operators using smeared and blocked Polyakov loops with varying transverse parity (P = ±) and longitudinal momentum (q = 0, ±1, ±2, ...).
• Application of the variational method to the transfer matrix e^{-aH} to extract energy levels from correlation matrices.
• Comparison of extracted energy levels with predictions from the Nambu-Goto model, the Lüscher term, and the Lüscher-Weisz effective string model.
• Use of fitting ansätze to analyze excited states and assess the power-law behavior of corrections, with statistical and systematic error control.

Experimental results

Research questions

• RQ1 Does the Nambu-Goto string model accurately describe the energy spectrum of closed fundamental flux tubes in 2+1D SU(N) gauge theories, even for short strings?
• RQ2 How do the quantum numbers of transverse parity (P) and longitudinal momentum (q) affect the degeneracy structure of flux-tube states?
• RQ3 To what extent do effective string theory predictions such as the Lüscher term or Lüscher-Weisz correction match lattice data for excited states?
• RQ4 How does the agreement between lattice data and theoretical models depend on string length and the number of colors N?
• RQ5 Can the Nambu-Goto model be considered a valid effective description of confining flux tubes down to very short lengths?

Key findings

• The ground state energy of the flux tube is well described by the Nambu-Goto prediction even for short strings, with l√σ ≈ 3 (≈0.65 fm).
• Deviations from Nambu-Goto predictions for excited states are small and decrease rapidly with increasing string length, indicating good convergence to the effective string limit.
• The Nambu-Goto model provides a significantly better fit to the lattice data than the Lüscher-Weisz or standard Lüscher term predictions, especially for excited states.
• The observed degeneracy pattern of states with quantum numbers (NR, NL, q, w) matches the Nambu-Goto prediction, confirming the expected level-matching condition NR - NL = q w.
• The data for non-zero longitudinal momentum (q ≠ 0) shows excellent agreement with the Nambu-Goto formula √(E²/σ - (2πq/l√σ)²) as a function of l√σ.
• Fits to excited states suggest that higher-order corrections are small, but statistical errors currently prevent unambiguous determination of the power-law behavior predicted by Lüscher-Weisz.