Center dominance, Casimir scaling, and confinement
in lattice gauge theory ^{*}^{*}*Based on talks presented by Š. Olejník at the 31st International Symposium on the Theory of Elementary Particles, Buckow, Germany, September 2–6, 1997, and at the 7th Workshop on Lattice Field Theory, Vienna, Austria, September 18–20, 1997. Work supported in part by the Slovak GAS, Grant No. 2/4111/97.
L. Del Debbio , M. Faber , J. Greensite , Š. Olejník
Dept. of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK
Institut für Kernphysik, Technische Universität Wien, A–1040 Vienna, Austria
The Niels Bohr Institute, DK–2100 Copenhagen Ø, Denmark
Institute of Physics, Slovak Academy of Sciences, SK–842 28 Bratislava, Slovakia
Abstract: We present numerical evidence that supports the theory of quark confinement based on center vortex condensation. We introduce a special gauge (“maximal center gauge”) and center projection, suitable for identification of center vortices. Main focus is then put on the connection of vortices in center projection to “confiners” in full, unprojected gaugefield configurations. Topics briefly discussed include: the relation between vortices and monopoles, first results for SU(3), and the problem of Casimir scaling.
1 Introduction
The most popular model of colour confinement in QCD relies on the idea of “dual superconductivity”, due to ‘t Hooft and Mandelstam. A realization of the idea is the abelianprojection theory of ‘t Hooft [1]: he suggested to fix to an “abelian projection” gauge, reducing the SU() gauge symmetry to U(1), and identifying abelian gauge fields (with respect to the residual symmetry) and magnetic monopoles. Abelian electric charges then become confined due to monopole condensation. In 1987, Kronfeld et al. [2] suggested testing ‘t Hooft’s theory in lattice simulations, in a special gauge that makes SU() link variables as diagonal as possible. If one computes various physical observables using the diagonal parts of the links only, one observes “abelian” dominance [3]: the expectation values of the physical quantities in the full nonabelian theory (often) coincide with the ones in the abelian theory obtained by the abelian projection in the maximal abelian gauge. Much evidence has been obtained for the abelianprojection picture and the model of dual superconductivity, but there remain problems to be solved. We have underlined its inability to explain approximate Casimir scaling of the linear potential between higherrepresentation colour sources at intermediate length scales [4, 5].
Another picture of confinement was quite popular before the advent of dual superconductivity, namely the vortex condensation theory, proposed, in various forms, by many authors [6]. According to this model, the QCD vacuum is filled with vortices, having the topology of tubes (in 3 Euclidean dimensions) or surfaces (in 4 dimensions) of finite thickness, which carry magnetic flux quantized in terms of elements of the center of the gauge group. Center vortices are assumed to condense in the QCD vacuum. The arealaw falloff of large Wilson loops comes from fluctuations in the number of center vortices linked to the loops.
The vortex condensation theory apparently suffers from the same “Casimirscaling disease” as the abelian projection does: there seems no way of accommodating the existence of a linear potential between adjoint sources to the idea of vortices dominating the QCD vacuum. There exists a simple solution to this controversy [7], and we will discuss it at the end of this paper.
With perhaps only one exception [8], the ideas behind the vortexcondensation picture have not in the past been subjected to lattice tests. The aim of our investigation is to study the vortex theory in numerical Monte Carlo simulations, by methods and approaches inspired to some extent by earlier work of many authors in the abelian projection theory.
2 Maximal center gauge, projection and dominance
The maximal abelian gauge underscores the role of the largest abelian subgroup of the gauge group. In much the same way, one can choose a gauge condition in which the gauge group center is given prominent importance. In SU(2) lattice gauge theory we proposed [5, 9] to fix to maximal center gauge (MCG) by making link variables as close as possible to its center elements . There are many (in fact infinitely many) ways how to do it; we implemented two simple choices:
1. The indirect maximal center gauge (IMCG) [5, 9, 10, 11]: We first fix to maximal abelian gauge (MAG) in the usual way, by maximizing the quantity
(1) 
then extract from their diagonal parts , and use the remnant U(1) symmetry to bring as close as possible to the center elements by maximizing
(2) 
2. The direct maximal center gauge (DMCG) [12, 10]: To fix this gauge one looks directly for the maximum of
(3) 



(a)  (b) 
In both cases, after gauge fixing one is left with the remnant gauge symmetry.^{†}^{†}†Qualitatively, the same physical results are obtained in both gauges (cf. [9, 12]).
The next step is center projection, i.e. replacing full link matrices (in a particular MCG) by center elements , which are defined to be
(4) 
and to compute various physical quantities of interest, e.g. Wilson loops and Creutz ratios, using the links.
Figure 1a compares Creutz ratios at computed from full lattice configurations and from centerprojected configurations. We clearly see center dominance: Creutz ratios computed from links agree with full Creutz ratios at large enough distances, the asymptotic values of the string tension almost coincide. On the contrary, Creutz ratios computed from links with the variable factored out (dotted line in Figure 1a) show no string tension at all. Another interesting observation is that the Creutz ratios computed from centerprojected configurations almost do not depend on ; center projection removes Coulombic contributions.
In Figure 1b we plot Creutz ratios vs. , extracted from centerprojected configurations in DMCG. The straight line is the asymptotic freedom prediction with the value of , which very well agrees with “stateoftheart” asymptotic string tension computations [13].
Our data show that the center variables are crucial parts of the links in MCG, in particular they carry most of the information on the string tension. This phenomenon of center dominance gives rise to a whole series of questions on the role and nature of center vortex configurations in the QCD vacuum. We will list a few of those questions here and sketch our tentative answers.
3 Questions and answers
3.1 Vortices and confinement?

Question 1: Has center dominance anything to do with confinement? What, if any, is the relation of vortices seen after center projection to “confiners” in full, unprojected configurations?



(a)  (b) 
To answer the question, we first introduce the notion of a P(rojection)vortex. The excitations of a lattice gauge theory with nonzero action are “thin” vortices, having the topology of a surface, one lattice spacing thick. We will call such vortices in center projected link configurations Pvortices. A plaquette is pierced by a Pvortex if, after maximal center gauge fixing and center projection, the corresponding projected plaquette has the value of .
However, we want to emphasize that center projection, and abelian projection as well, represents an uncontrollable truncation of full lattice configurations. Therefore we will not base our following arguments on measurements in center projected configurations. Instead, we will use center projection mainly for selecting subensembles of configurations on which physical quantities are evaluated. In particular, we will compute , Wilson loops evaluated on such a subensemble of configurations that precisely Pvortices, in the corresponding centerprojected configurations, pierce the minimal area of the loop. Though the data set is selected in center projection, the Wilson loops themselves are evaluated using the full, unprojected link variables.
From the computed vortexlimited Wilson loops one can determine Creutz ratios . A simple test of a relation of Pvortices to confinement is then the following: if the presence/absence of Pvortices is irrelevant for confinement, then we would expect for large loops. The result of the test is shown in Figure 2a. The string tension vanishes if Pvortices are excluded from Wilson loops; it also vanishes if only odd numbers of Pvortices are excluded (Fig. 2b). The presence/absence of Pvortices seems strongly correlated with the presence/absence of “confiners” in unprojected field configurations.
However, the true “confiners” do not necessarily have to be any sort of vortices. The natural question is, whether the objects identified using center projection tend to carry magnetic flux. A simple argument [9, 10] leads to the expectation that for large loops.
Figure 3 shows our data for and at . They are consistent with the expectation and thus indicate that the confining gauge field configurations are center vortices. However, the values of are reached for relatively large loop areas. The objects corresponding to Pvortices then appear to be rather “thick” vortices. In Section 3.4 will this fact be related to a simple explanation of approximate Casimir scaling.



(a)  (b) 
Now we are ready to formulate

Answer 1: We found evidence that center dominance in maximal center gauge is a reflection of the presence of thick center vortices in the unprojected configurations. Those vortices are identified as thin Pvortices in center projection.
3.2 Vortices and/or monopoles?

Question 2: If the vacuum is dominated, at long wavelengths, by vortex configurations, then how do we explain the numerical successes of abelian projection in maximal abelian gauge?
We believe that the question is answered in the following way:

(Probable) Answer 2: A center vortex configuration, transformed to maximal abelian gauge and then abelianprojected, will appear as a chain of monopoles alternating with antimonopoles. These monopoles essentially arise because of the projection; they are condensed because the long vortices from which they emerge are condensed.
The support for the answer was given in much detail in [10]. It is clear, however, that this question deserves further study.
3.3 Su(3)?

Question 3: In nature quarks appear in three colours. Do the observed phenomena survive transition from SU(2) to SU(3)?
The maximal center gauge in SU(3) gauge theory is defined as the gauge which brings link variables as close as possible to elements of its center . This can be achieved e.g. by maximizing the quantity
(5) 
Fixing to the maximal center gauge in SU(3) gauge theory turns out to be much more difficult and CPUtime consuming than in the case of SU(2). Therefore our simulations have until now been restricted to small lattice sizes and to strong coupling.
Our strong coupling results for the SU(2) and SU(3) lattice gauge theory are compared in Figure 4. In SU(2) Monte Carlo data agree with the strong coupling expansion up to almost . Figure 4b shows centerprojected Wilson loops in SU(3) together with results of strongcoupling expansion to leading and nexttoleading order. The agreement extends up to .



(a)  (b) 

(Partial) Answer 3: The situation at strong coupling looks much the same in SU(2) and SU(3): in both cases full Wilson loops are well reproduced by those constructed from center elements alone in MCG. Thus, center dominance is seen in SU(3) gauge theory at strong coupling.
3.4 Casimir scaling?

Question 4: Is there any way of accommodating the approximate Casimir scaling of higherrepresentation potentials to the vortex dominated QCD vacuum?
At the first sight, there is not. The adjoint representation transforms trivially under the gauge group center, large adjoint Wilson loops are unaffected by center vortices. As a result, the adjoint string tension vanishes.
There is however a loophole in the above statements: Adjoint loops are unaffected, unless the core of the vortex happens to overlap with the perimeter of the loop. If, then, the vortex thickness is quite large, on the order or exceeding the typical diameter of lowlying hadrons – and our data seem to indicate the presence of rather “thick” center vortices, – the Wilson loops can be influenced by vortices up to relatively large loop sizes.
A phenomenological model of the “thick” center vortex core has been worked out in a recent paper of three of us [7]. We cannot discuss the model in detail here, we just mention that it leads to potentials between colour sources that show approximate Casimir scaling at small and intermediate distances, and colour screening of integerrepresentation sources at large distances.

Answer 4: The Casimir scaling of the string tensions of higherrepresentation Wilson loops is an effect due to the finite (and large) thickness of the center vortex cores (see also [14]).
4 Conclusions
We subjected the picture of quark confinement based on center vortices to simple tests on the lattice. We proposed a method of localizing center vortices in thermalized lattice configurations, and found vortices to be responsible for the asymptotic string tension in SU(2) lattice gauge theory. The same holds also for SU(3) at strong coupling.
Further, we posed the lattice a lot of simple questions on the nature of vortex configurations. The tests indicate that the “confiners” in QCD are center vortices, and monopoles appear along vortices as artifacts of abelian projection. It is tempting to believe that monopole condensation might be just a manifestation of the underlying vortex condensation.
Finally, the vortices observed in our simulations possess a thick core; the thickness of the core is the cause for approximate Casimir scaling of potentials at intermediate distances. This solves a longstanding problem of the center vortex theory.
The “spaghetti vacuum” picture, believed to be dead for more than a decade, returns to the stage, in quite a good health.
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