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bajo
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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 [itex]\tau[/itex]) in the Euclidean path integral formulation, which gives:
[tex]
\tau = \frac{\log g^2}{a^2},
[/tex]
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
[tex]
\beta(g) = - a \frac{dg(a)}{da},
[/tex]
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:
[tex]
\tau = \frac{3}{8} \frac{g^2}{a^2}.
[/tex]
(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
[tex]
\beta(g) = - g \log(g^2) + \ldots
[/tex]
where the ellipses denote terms of higher order in [itex]1/g[/itex], while the second gives
[tex]
\beta(g) = -g + \ldots
[/tex]
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.
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:
[tex]
\tau = \frac{\log g^2}{a^2},
[/tex]
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
[tex]
\beta(g) = - a \frac{dg(a)}{da},
[/tex]
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:
[tex]
\tau = \frac{3}{8} \frac{g^2}{a^2}.
[/tex]
(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
[tex]
\beta(g) = - g \log(g^2) + \ldots
[/tex]
where the ellipses denote terms of higher order in [itex]1/g[/itex], while the second gives
[tex]
\beta(g) = -g + \ldots
[/tex]
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.