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

1. May 27, 2012

### 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:

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

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

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

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:

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

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

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

where the ellipses denote terms of higher order in $1/g$, 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 $\beta$-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?

Therefore this is a nice example where you can explicitly appreciate the regularization scheme dependence of the $\beta-$function.