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QCD string tension at strong coupling: log(g^2) vs. g^2

  1. May 27, 2012 #1
    Hi everyone,

    I am trying to understand some things about confinement in lattice QCD. It has been very difficult so far to find a clear book chapter or review article, so I had to resort to the original literature in many cases, and I came across some apparently incompatible statements about the string tension in QCD.

    I am very confused by the following: many books present the computation of the string tension (that I am going to denote with [itex]\tau[/itex]) in the Euclidean path integral formulation, which gives:

    \tau = \frac{\log g^2}{a^2},

    where [itex]a[/itex] is the Lattice spacing and [itex]g[/itex] is the YM coupling constant. Since [itex]\tau[/itex] is a physical quantity, it must be independent of the regulator [itex]a[/itex], and as a consequence [itex]g[/itex] becomes a function of the regulator [itex]a[/itex].

    In other words, we can compute the strong-coupling limit of the [itex]\beta-[/itex] function

    \beta(g) = - a \frac{dg(a)}{da},

    by just asking that the derivative of [itex]\tau[/itex] with respect to [itex]a[/itex] vanishes. The minus sign comes about because [itex]a[/itex] is related to the UV cutoff by [itex]a \propto 1/\Lambda[/itex].

    It turns out that the Hamiltonian version of Lattice QCD gives a completely different result for [itex]\tau[/itex] as a function of the coupling constant:

    \tau = \frac{3}{8} \frac{g^2}{a^2}.

    (this formula can be found, for example, in the book by Kogut & Stephanov "The Phases of Quantum Chromodynamics", eq. (6.49)).

    Of course these two results for [itex]\tau[/itex] give two different [itex]\beta-[/itex]functions at strong coupling: the first gives

    \beta(g) = - g \log(g^2) + \ldots

    where the ellipses denote terms of higher order in [itex]1/g[/itex], while the second gives

    \beta(g) = -g + \ldots

    This latter result can also be found in the literature, for example DOI 10.1016/0370-2693(81)90369-5 (sorry, I do not yet have clearance for links in posts :)

    I know that the [itex]\beta[/itex]-function depends on the regularization scheme, so I should not expect a precise matching between two, but I still feel uneasy about it. I have the strong feeling I am missing something in this story, so I would like to ask you: are things really like that? Are the [itex]\beta[/itex]-functions for Euclidean and Hamiltonian Lattice QCD really different, so that one grows as [itex]g[/itex] and the other as [itex] g \log g^2 [/itex] in the strong coupling limit?

    Sorry for the long post and thank you in advance for any answers/comments.
  2. jcsd
  3. May 29, 2012 #2
    In case anyone is interested, I asked an expert in Lattice QCD working in my department, and he confirmed that what I said in the previous post is correct.
    Therefore this is a nice example where you can explicitly appreciate the regularization scheme dependence of the [itex]\beta-[/itex]function.
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