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

[bajo]

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 \tau) in the Euclidean path integral formulation, which gives:

<br /> \tau = \frac{\log g^2}{a^2},<br />

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

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

<br /> \beta(g) = - a \frac{dg(a)}{da},<br />

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

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

<br /> \tau = \frac{3}{8} \frac{g^2}{a^2}.<br />

(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 \tau give two different \beta-functions at strong coupling: the first gives

<br /> \beta(g) = - g \log(g^2) + \ldots<br />

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

<br /> \beta(g) = -g + \ldots<br />

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 \beta-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 \beta-functions for Euclidean and Hamiltonian Lattice QCD really different, so that one grows as g and the other as g \log g^2 in the strong coupling limit?

Sorry for the long post and thank you in advance for any answers/comments.

 

[bajo]

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 \beta-function.