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Large N and confining flux tubes as strings - a view from the lattice

Michael Teper

TL;DR

The work assesses how lattice simulations illuminate the large-N limit of gauge theories and how confining flux tubes can be described by an effective string theory. It shows that confinement persists at large N and that many SU(3) observables approximate SU(∞), with g^2 scaling as 1/N and modest 1/N^2 corrections in several contexts. The main focus on flux tubes reveals that the Nambu-Goto free-string spectrum captures the low-lying excitations remarkably well in both 2+1 and 3+1 dimensions, with universal Luscher-type corrections extending to higher orders in some regimes. Together, these results provide strong evidence for a universal effective string description of confinement, connect lattice QCD to gauge-gravity duality ideas, and identify precise open questions, especially in the meson sector and for massive modes attached to flux tubes.

Abstract

I begin these three lectures by describing some of the useful things that we have learned about large-N gauge theories using lattice simulations. For example that the theory is confining in that limit, that for many quantities SU(3) is close to SU(oo), and that this includes the strongly coupled gluon plasma just above Tc, thus providing some of the justification needed to make use of gauge-gravity duality in analysing QCD at RHIC/LHC temperatures. I then turn, in a more detailed discussion, to recent progress on the problem of what effective string theory describes confining flux tubes. I describe lattice calculations of the energy spectrum of closed loops of confining flux, and some dramatic analytic progress in extending the `universal Luscher correction' to terms that are of higher order in 1/l, where l is the length of the string. Both approaches point increasingly to the Nambu-Goto free string theory as being the appropriate starting point for describing string-like degrees of freedom in SU(N) gauge theories.

Large N and confining flux tubes as strings - a view from the lattice

TL;DR

The work assesses how lattice simulations illuminate the large-N limit of gauge theories and how confining flux tubes can be described by an effective string theory. It shows that confinement persists at large N and that many SU(3) observables approximate SU(∞), with g^2 scaling as 1/N and modest 1/N^2 corrections in several contexts. The main focus on flux tubes reveals that the Nambu-Goto free-string spectrum captures the low-lying excitations remarkably well in both 2+1 and 3+1 dimensions, with universal Luscher-type corrections extending to higher orders in some regimes. Together, these results provide strong evidence for a universal effective string description of confinement, connect lattice QCD to gauge-gravity duality ideas, and identify precise open questions, especially in the meson sector and for massive modes attached to flux tubes.

Abstract

I begin these three lectures by describing some of the useful things that we have learned about large-N gauge theories using lattice simulations. For example that the theory is confining in that limit, that for many quantities SU(3) is close to SU(oo), and that this includes the strongly coupled gluon plasma just above Tc, thus providing some of the justification needed to make use of gauge-gravity duality in analysing QCD at RHIC/LHC temperatures. I then turn, in a more detailed discussion, to recent progress on the problem of what effective string theory describes confining flux tubes. I describe lattice calculations of the energy spectrum of closed loops of confining flux, and some dramatic analytic progress in extending the `universal Luscher correction' to terms that are of higher order in 1/l, where l is the length of the string. Both approaches point increasingly to the Nambu-Goto free string theory as being the appropriate starting point for describing string-like degrees of freedom in SU(N) gauge theories.

Paper Structure

This paper contains 35 sections, 79 equations, 39 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Large $N$
  4. Lattice calculations lat_books
  5. Large $N$ Physics : some basic results from the lattice
  6. Are large-$N$ gauge theories linearly confining?
  7. Is SU(3) close to SU($\infty$)?
  8. Is $QCD_3$ close to $QCD_\infty$?
  9. $g^2\propto 1/N$ for a smooth large $N$ limit?
  10. $D=2+1$
  11. $D=3+1$ : running bare coupling
  12. $D=3+1$ : a running continuum coupling
  13. How hard are large $N$ lattice calculations?
  14. Large $N$ Physics at high $T$
  15. Deconfinement
  16. ...and 20 more sections

Figures (39)

  • Figure 1: Gluon loop insertion using 't Hooft's double line notation.
  • Figure 2: Glueball correlation function with a simple plaquette operator.
  • Figure 3: Glueball correlation function with best blocked operator and cosh fit.
  • Figure 4: Scalar, $\bullet$, and tensor, $\circ$, glueball masses with $O(a^2)$ extrapolations to the continuum limit.
  • Figure 5: Ground state energy of a flux loop winding around a spatial torus of length $l$, in SU(6) and $D=3+1$.
  • ...and 34 more figures