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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.
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.