Confinement

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Winter '09

1/16 - Wilson loops
  • Kristan Jensen
  • Abstract - I will present a brief introduction to this quarter's journal club on confinement. This introduction will have two parts; the first will give an overview to the rest of the quarter, while the second will introduce Wilson loops as a parameter to measure the existence and properties of confinement.
  • Refs -
    • Kaku's Intro. to QFT Chapter 15
    • Kogut Rev.Mod.Phys. 55, 775 (1983)


1/23 - Wilson loops in the strong coupling limit
  • Chris Vermilion
  • Abstract - I will start by reminding everyone what a Wilson loop is, and why we (or at least some of us) care. Lacking the gumption to calculate them without a few crutches, I will break off to discuss how to create gauge theories on a lattice. I will show how to calculate Wilson loops on said lattice. Finding this intractable in general, I will then take the strong-coupling limit, and show that in this limit, gauge theories are confining. Finally, I will wave my arms around and make vague statements with varying degrees of justification regarding the continuum limit of the calculation I will have just performed, and whether we are in fact wasting our time.

My treatment follows, more or less, Wilson's PRD article, D10, 2445. A more modern, but to my mind less understandable treatment is given in Kaku's field theory text, chapter 15. Wilson and Kogut have between them several nice review articles on lattice gauge theory.

  • Refs -
    • Wilson, "Confinement of Quarks", Phys. Rev. D10, 2445 (1974)
    • Kaku, above
    • Wilson and Kogut, "The renormalization group and the \epsilon expansion", Phys. Rep. C12, 75, (1974)
    • Wilson, "Quarks on a Lattice, or the colored string model", Phys. Rep. C23, 331 (1976)
    • Kogut and Susskind, "New ideas about confinement", ibid., 348


1/30 - 't Hooft loops
  • Jon Walsh
  • Abstract - I will start with a discussion of the Abelian Higgs model and the Meissner effect, which exhibits a confinement of magnetic field in the Higgs phase. I will use this to argue that there is an analogous order parameter for confinement, then go on to construct it for an SU(N) gauge theory in 3d. This is the 't Hooft loop, and I will develop a relationship between it and the Wilson loop. I will then discuss the 't Hooft loop in 4d, and use the relationship with the Wilson loop to exhibit the value of the order parameter for confinement.
  • Refs -
    • 't Hooft, Nucl. Phys. B138, 1, (1978)
    • 't Hooft, Physica Scripta 25, 133, (1982)
    • Yaffe, Phys. Rev. D21, #6, 1574, (1980)
    • Greensite, hep-lat/0301023
    • Greensite and Alkofer, hep-ph/0610365


2/6 - Wilson and 't Hooft loops; electric- and magnetic-flux free energies
  • Steve Paik
  • Abstract - We will continue our discussion of useful observables for confinement. We will define, in SU(N) lattice gauge theory, the Wilson and 't Hooft loop operators and explain how they should be physically interpreted. By imposing periodic boundary conditions, we can trap stable flux in a box and define electric- and magnetic-flux free energies. We will have four observables in all that can be used to probe the phase of a theory at zero temperature. We will list the criteria for confinement.
  • Refs -
    • Yaffe, Phys. Rev. D21, 1574 (1980)
    • 't Hooft, Nucl. Phys. B153, 141 (1979)


2/13 - Migdal-Kadanoff relations
  • Brian Mattern
  • Abstract - Chris showed us that we see confinement in SU(3) gauge theory on the lattice in the strong coupling regime. Also, we know that the continuum limit lies in the weak coupling regime. Using Mean Field techniques, one can show that a first order deconfining phase transition separates the two regimes in large dimension. Furthermore, 2-d gauge systems reduce to 1-d spin systems, which can be solved and contain no phase transition. There must be some critical dimension above which we have a phase transition and at or below which we don't. I will discuss a procedure due to Migdal and Kadanoff, by which one can derive approximate RG equations for couplings in gauge and spin systems on the lattice that provide evidence that 4 is the critical dimension for SU(N) leading one to believe that confinement should survive in the continuum limit.
  • Refs -
    • Kadanoff, Ann. Phys. 100, 359 (1976)
    • Creutz. Quarks, gluons and lattices. Ch. 14 and 17


2/20 - Hamiltonian lattice gauge theory
  • Alan Jamison
  • Abstract - We will review how to get a Hamiltonian from a one-particle QM path integral, follow a similar procedure for lattice gauge theories, consider the ramifications of gauge invariance, and then use our Hamiltonian to do some perturbation theory.
  • Refs -
    • The original paper by Kogut and Susskind:
      • PRD Vol. 11 p395-408
    • Standard books of the lattice:
      • Lattice Gauge Theories: An Introduction by Rothe (Ch 11)
      • Quarks, Gluons, Lattices, and Cryptic Derivations by Creutz
    • Kogut's two big review articles:
      • Rev. Mod. Phys. Vol 51 p659-713
      • Rev. Mod. Phys. Vol 55 p775-836


2/27 - Confinement in 3d Georgi-Glashow model
  • Amy Nicholson
  • Abstract - I'll introduce instantons in QFT as a possible mechanism for confinement. In particular, I'll use magnetic monopole configurations in the 2+1-d Georgi-Glashow model (SU(2) Yang-Mills plus adjoint Higgs) to derive the mass gap and show area law behavior, all at weak coupling.
  • Refs -
    • A.M. Polyakov, Nucl. Phys. B 120, 429 (1977)
    • Also, many reviews on confinement, in particular:
    • Plus! many reviews on instantons/solitons/magnetic monopoles. Coleman's always good for this.


3/6 - Theories on R^3 x S^1
  • Can Kozcaz
  • Abstract - I will be talking about Mithat's paper in which he introduced a new mechanism for confinement and mass gap in QCD-like theories on R^3xS^1. This mechanism is based on a new type of bound state of topological excitations, and he calls this new state a 'magnetic bion'.
  • Refs -


3/13 - Evidence for confinement from the lattice
  • Apparently there isn't any.
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