May 2001
On Casimir scaling in QCD

Vladimir Shevchenko and Yuri Simonov Institute of Theoretical and Experimental Physics,
117218 B.Cheremushkinskaya, 25, Moscow, Russia.

Recent lattice calculations have confirmed that QCD static potential for sources in different representations of the gauge group is proportional to eigenvalue of the corresponding quadratic Casimir operator with an accuracy of a few percents. We review the present theoretical status of the ”Casimir scaling” phenomenon and stress its importance for analysis of nonperturbative QCD vacuum models and other field theories. It is argued that Casimir scaling strongly advocates the property of Gaussian dominance and we propose different lattice tests to improve our understanding of these phenomena.

PACS numbers: 12.38.Aw, 12.38.Gc

I Introduction

There is little doubt nowadays that QCD is the true theory of strong interactions and in perturbative domain agreement between theory and experiment is impressive. At the same time for nonperturbative (NP) phenomena an exact analytic formalism is still lacking and one is dealing with a set of models or approaches, which are not derived directly and rigorously from the QCD Lagrangian without any assumptions and most of these models are introduced ad hoc.

Comparison of model predictions with experiment allows to select models describing one or another set of phenomena, e.g. constituent quark model works reasonably well for hadron spectrum, while parton model does the job for high- events and instanton model is suitable for describing chiral effects. However these models are not well connected to each other and may contradict other phenomena, e.g instanton gas model lacks confinement. In addition there is a class of specific models with a special purpose to explain vacuum structure of QCD with its property of confinement: magnetic monopoles with abelian dominance hypothesis, center vortices model etc.

Somewhat apart from particular microscopic models a nonperturbative approach is being developed for the last ten years [1] (see also review [2] and references therein), so called field correlator method. It is based on QCD and expresses all observables ultimately in terms of gauge-invariant field strength correlators. The crucial assumption of the method which makes it workable is that one can restrict oneself to a few lowest field correlators; taking only two-point Gaussian correlator one has Gaussian stochastic model of QCD vacuum. Extracting Gaussian correlator from lattice data [3] one is able to predict confinement property of QCD and calculate quantitatively an impressive amount of data on hadron spectra and other properties [2, 4, 5].

It is clear that QCD vacuum structures derived from different models should not necessarily be compatible with each other and also with GSM, while they can describe equally well some features of the vacuum dynamics, for example actual value of string tension. Taking gross features of the vacuum one should distinguish two different scenarios: one, based on an ensemble of classical solutions, i.e. of coherent lumps of fields may be called a coherent picture of the vacuum. Here belong instanton model, effective abelian models, central vortex models and some other models. On the opposite side there is Gaussian stochastic model where stochastic properties of the vacuum are taken to their extreme, so that all higher field correlators are suppressed and the picture of vacuum quantum correlations is close to that of the Gaussian white noise. It would be of great importance to understand first of all what the real QCD vacuum is, to which of these two different pictures it belongs.

The most important theoretical tool to treat NP QCD available up to now is numerical simulation of the theory on the lattice. Confinement and other NP phenomena have been extensively studied in this approach. Data obtained in this way can be compared with results of real experiments as well as with theoretical predictions, based on particular NP models. The latter comparison is of special interest. Since the only essential dynamical input of the lattice theory is QCD Lagrangian such comparison is important to distinguish between models, which are in agreement with QCD (at least in lattice formulation) and those, which are not.

The property of confinement in QCD has two important facets. From observational point of view it exhibits itself as absence of particles, carrying nonzero color charge as asymptotic states in Nature (in particular, massless vectors and light spinors). On the lattice there is another criteria of confinement - area law: there exists constant force (equal to 15 tons in SI units) between static fundamental color charge and anticharge at large distances (if no dynamical charges can be created from the vacuum to screen it). Most of NP models which can be compared with lattice measurements are designed to describe confinement of colour charge and anticharge in the fundamental representation of the gauge group , i.e. the area law for the fundamental rectangular Wilson loop and hence linearly rising potential between static quark and antiquark. There is a lot of information about static fundamental potential obtained from lattice simulations both in quenched and unquenched approximation. Being of great value, these data cannot answer the questions posed above and one needs another type of lattice experiments which were done recently with great accuracy [6, 7, 8]. These are the investigation of interaction between static charges in higher representations. In this way one can extract information about field content of NP QCD vacuum, which is not available if only fundamental charges are considered.

The studies in this direction have a long history (see [7] and references therein). Recently two sets of accurate lattice measurements of the static potentials for fundamental and higher representations of the gauge group have been presented by G.Bali [6, 7] and independently by S.Deldar [8]. The data strongly support the so called Casimir scaling hypothesis [9] which states the proportionality of the static NP potentials for different representations to eigenvalue of the quadratic Casimir operator for the corresponding representation. Actually, the level of violation of Casimir scaling (CS) for the string tension is found to be at the range in the measurements [8] and at the level of a few percents in the results [6], while extrapolation to the continuum limit performed in [7] remarkably indicated the violation of CS law not exceeding statistical errors, of the order of . All that give a hint that CS is indeed a fundamental property of QCD (at least on the lattice). Surprisingly, however, the analysis of [10, 11] has shown that to incorporate such simple feature in some natural way is not an easy task for most popular QCD vacuum models. Let us make a comment about general reason for that. In most of coherent QCD vacuum models the corresponding field configurations are endowed with nontrivial colour structure, whose gauge-invariant meaning is often not transparent. The CS, on the other hand, means that at least for the particularly chosen NP quantity – static potential – only the simplest invariant of the given representation, eigenvalue of the quadratic Casimir operator, i.e. just square of the (nonabelian) charge characterizes the potential. Thus strongly interacting theory with complicated vacuum paradoxically demonstrates the same scaling law of the potential as it would be in perturbation theory at tree level. In terms of the confining field configuration ensemble it implies either delicate (and as we will try to advocate in what follows, rather unnatural) fine tuning of the contributions coming from more complicated color structures (higher powers of the quadratic Casimir and higher Casimirs) or actual smallness of such structures. The phenomenon of CS seen from different NP QCD models points of view will be discussed in Section III of the present paper.

It should be said that many ideas connected with the meaning of CS and extensively discussed in the present paper were formulated already in the original papers [9], [12], [13]. Nevertheless we find it useful to collect them here all together with new developments since the results from [7] demonstrating much higher level of accuracy of the phenomenon are calling for reconsideration of the theoretical background behind it.

The aim of the present paper is to draw possible consequences from the data on Casimir scaling constructing the most logical scheme of the QCD vacuum structure, respecting CS. The paper is organized in the following way: Section II is devoted to definitions and general discussion of the lattice data on CS, analysis of different field theoretical models from CS point of view is presented in Section III, Section IV contains our proposals of additional observables aimed to clarify the CS behavior and structure of the QCD vacuum, and Section V presents conclusion and outlook.

Ii General formalism

The average of the Wilson loop for the rectangular contour (which we choose in the ”34” plane) for the dimension representation of the gauge group is given by


Here the normalized trace is defined as , fundamental generators are normalized according to . The representations labelled by are characterized by hermitian generators which satisfy the commutation relations . One of the main characteristics of the representation is an eigenvalue of quadratic Casimir operator , which is defined according to


Our normalisation is such that for adjoint representation of , it differs by factor from normalisation of [14]. Since any simple algebra of rank has exactly primitive Casimir–Racah operators [15, 16] of order , it is possible to express those of higher order in terms of the primitive ones. In the case of the primitive Casimir operators are given by and . The higher rank Casimir operators are defined as


where the totally symmetric tensors on are expressed in terms of and (see, e.g. [14] and references therein). Following the notations from [6] we introduce Casimir ratio , where the fundamental Casimir equals to for . Generally, for representations labelled by Dynkin coordinates the eigenvalue of is to be calculated according to the following expression (see, e.g. [14]):


Notice the positivity of the quadratic Casimir charge (it is not true, for example, for the cubic one).

The static potential between sources at the distance in the representation is defined as:


We are not considering the possibility of screening of the potential by dynamical degrees of freedom at the moment, see discussion below. The l.h.s. of (5) can be decomposed into two pieces of different physical meaning: actual -dependent potential and self-energy part not depending on , which explicitly contains ultraviolet cutoff scale . For the data extrapolated to the continuum limit (see [7] for details) one assumes the proper subtraction procedure to be applied and we will denote by only the former part of the potential in this case. Notice, however that the numbers shown in Table 1 of the present paper are based on raw lattice data from [6] where no any special subtraction have been performed (a comparison shows that with the discussed accuracy the self-energy part satisfies CS as well).

In the confinement phase lattice data are well described by the sum of perturbative Coulomb part, confining linear and constant terms:


where all coefficients are -dependent. The Coulomb term is now known up to two loops in the continuum limit [17] and up to one loop on the lattice [18] and is proportional to in both cases. It is not trivial that perturbative interaction is proportional to at two loops, and it is not clear what will happen with more loops taken into account. The CS hypothesis [9] states, that the NP confining potential is also proportional to the first power of the quadratic Casimir , i.e. for the string tensions one should have .

As it was already discussed in the Introduction, this scaling law is in very good agreement with the results found in [8, 6, 7]. Earlier lattice calculations of static potential between sources in adjoint representation [19] are in general agreement with [6], however, deviations from scaling at the level of are found in [20], in particular, the value of is closer to 2 than to in [20] (see further references and discussion in [7]).

The lattice data can be understood in terms of bounds on higher Casimir terms in the potential (6) [10, 11]. Namely, one can write quite general decomposition for the expression (6) of the following form:


where the dots denote terms proportional to higher powers of as well as to higher Casimir ratios. It is worth mentioning, that since there are two independent Casimirs in case, the series in (7) is in fact Taylor expansion of a function depending on two variables (i.e. quadratic and qubic Casimirs) with –dependent coefficients. Notice also, that for self–adjoint representations (those having form in Dynkin coordinates) the cubic Casimir vanishes (see, e.g. [14]), so the potential depends only on .

The form of Taylor expansion (7) is strongly motivated by perturbation theory series as well as by cluster expansion (22). However it is not in one-to-one correspondence with any of them. For example, the lowest quadratic correlator contributes only to , while higher ones contribute to as well as to higher terms. Neither the expression (7) has a meaning of perturbative series in .

The terms in the expansion (7) are representation independent. One can parametrize the first CS violating term in the same way as the total potential as , where we omit possible CS–violating contribution to the Coulomb potential since at small distances its magnitude should in principle be small due to asymptotic freedom and independent analysis confirms this. Here measure the –contribution of the CS violating terms to the total potential (their artificial dependence on could come from higher powers of , and should disappear if all terms in the expansion (7) are taken into account). Notice, that we do not need to specify the coordinate dependence of . Some results based on the data from [6] are presented in Table 1. The author of [6] used anisotropic lattice with the spatial unit Fm and anisotropy . Standard fit was performed for the data in the whole range of all measured , since no fingerprint of screening is seen up to the largest distances explored in [6]. Errors shown in Table 1 include statistical and systematical ones.

8 -3.5 1.2 -2.5 2.8 0.004 19 / 43
6 -6.4 1.2 1.0 2.6 0.007 26 / 42
15a -5.2 0.6 -0.6 1.1 0.003 39 / 42
10 -4.9 0.5 0.2 1.0 0.003 22 / 41
Table 1: The Casimir–scaling and Casimir–violating string tensions and shifts [11]. Based on the lattice data from [6]. All quantities with the hats are scaled according to

Several comments are in order. First of all it is seen that the CS behaviour holds with a good accuracy, of the order of a percent with the reasonable . The ratio is less than for all cases presented in Table 1. The value of the constant term is found to be compatible with zero within the error bars for all considered , while it is not the case for . The value of for sextet, for example, is found to deviate from zero at the level of approximately five standard deviations. We have not found any strong systematic dependence of on , which presumably confirms the validity of the expansion (7) and shows, that the omitted higher terms do not change the picture in a crucial way. The negative sign of the correction is probably related in Euclidean metric with the fact, that the fourth order contribution is proportional to while the Gaussian term is multiplied by .

Recently the continuum limit extrapolation of the lattice data on static potentials for different representations was performed in [7]. The actual ratio of the potentials measured in [7] demonstrated the deviation from CS (i.e. from a constant equal to ) at the level of statistical errors not exceeding a few percents.

It is important to stress at the same time that CS property is not universal at all possible distances. The area law implies linear asymptotics of the potential at large . There is however an important restriction: even in quenched approximation with no dynamical quarks included one should take into account the effects of string breaking. The conventional picture of this phenomenon suggests that in case of charges in higher representations the parts of the potential corresponding to the octet components of the sources are to be screened by dynamical gluons from the vacuum and so called gluelump states are formed eventually. In particular, zero triality (-ality in general case) representations are screened completely. For example, in case of adjoint charges one has with the singlet component. Screening for higher representations might require more dynamical gluons. In case of nonzero triality representations the noncompensated triplet is always present (for example, ) which makes asymptotic string tension to be equal to the fundamental one eventually.

It is obvious that in the region where the string is broken the expressions of the form (6), (7) have no sense. The modification of (6) due to this effect was considered in [13, 21] in the strong coupling expansion, resulting in the following expression for adjoint Wilson loop in large approximation


It follows from (8) that the potential (5) has two different regimes at and ( is the critical distance, where the second term in (8) starts to dominate). Namely, one has in limit:

The potential becomes flat after one reaches the critical distance. The estimates of the gluelump mass show [22] that the actual value of is rather large, of the order of for the gauge group in the adjoint case. In other words, the lightest gluelump is heavy in the units of . At the distances actually explored in [6, 7, 8] the second term in (8) contributes at the level of lattice statistical errors. This makes the ”CS region” rather wide, at least for not very large - from the smallest distances where tree level perturbation theory works well to the onset of screening.

It is also worth mentioning that in the limit the force is discontinuous according to (8) at the point , switching from the constant value equal to at to zero value at . This dynamical screening seems to be different from the center vortex kinematical screening scenario where the potential is becoming flat with the distance smothly (see below) even in the limit.

Let us show how (8) appears in dynamical background perturbation theory [22]. We concentrate on the adjoint case for simplicity, generalization to other representations can be done in the same way. To this end we split gluon field as , where represent the confining background and – the valence gluon field. Referring the reader to [22] for details, one gets the valence Green’s function in the background Feynman gauge in the form and the result of integration over valence gluons at the lowest order yields in the partition function the factor . The averaging in (5), (22) turns out to be


In a similar way quark loops are accounted for by the factor instead of in (9). The highest terms in expansion can be calculated systematically.

The next step is the standard loop expansion of the determinant augmented by the world–line (Feynman–Schwinger) formalism :


where the curly brackets stand for the path integral over contours, forming the loop and the corresponding proper time integration. Expanding the exponent in (10) and keeping only the first two terms one has


where double brackets denote the connected correlator. The final step is the representation of the last term in (11) as a Green’s function of two gluelumps, so one finally arrives to the asymptotic expression (8).

Numerically, estimating as 1.2 GeV for the adjoint source [22], one gets for the critical distance Fm. Therefore as it was already stressed the preasymptotic region (where the confining potential is linearly rising for all ) is rather wide which allows to establish CS up to the distances where signal disappears into noise on the lattice. Additional measurements at larger distances could hopefully shed some light on the string breaking and physics of gluelumps and establish representation dependent bounds of the CS regions.

Let us conclude this section with the remark about large limit. First of all, the ”CS region” expands up to infinity in this case, as it is clearly seen from (8). It was also advocated in [21] that the ratio is equal to 2 if the limit is taken (which is in line with both flux counting rule and CS). However we do not see clear arguments in continuum theory why CS should become exact if in general case for all higher representations. It would be very interesting to establish in future lattice measurements the relative deviation from CS as a function of , at least for (see also [23]). Notice, that the standard large factorisation arguments are not applicable to the cumulants entering (22) since they are color-irreducible. In particular, the proper understanding of Gaussian dominance property in large framework has not yet been reached.

Iii NP QCD vacuum models and Casimir scaling

As it is mentioned in the Introduction, up to the authors knowledge there is no example of NP QCD vacuum model (with except of Gaussian stochastic model, see below) which does not violate CS in ”naive” sense. All considered microscopic models need some kind of special adjustment in order to achieve approximate CS behavior of the static potential. We illustrate this statement taking as examples three popular pictures of the QCD vacuum - instanton liquid [25], see [26] for review; condensate of abelian-projected monopoles [27], see [28, 29] for review and center vortex scenario [30], [31, 32]. We will briefly discuss the confining string picture, Georgi-Glashow model, MIT bag model and dimensional reduction scenario.

To begin, let us make an important remark about the role of gauge invariance. As it was already mentioned in the Introduction, the property of CS is essentially nonabelian and therefore to correctly reproduce it a NP QCD vacuum model should take into account nonabelian degrees of freedom in a proper gauge-invariant way (roughly speaking, the sum in expressions like (2) must be taken over all generators). However, it is not sufficient since even manifest gauge-invariance of a model does not guarantee it will demonstrate CS law. Nevertheless we find it instructive to separate the discussion of the models where some particular gauge-fixing procedure plays crucial role from those where it does not. We will see that pattern of CS violation is different in many respects for these two groups of models.

iii.1 Models with the preferred choice of gauge

As the first example of a model, based on NP field configurations we mention here the model of dilute instanton ensemble [25, 26]. Strictly speaking this model is not expected to give reasonable predictions for the static NP potential since it lacks confinement and hence some essential part of NP physics. It is instructive however to investigate CS in this framework since it clearly illustrates the pattern which is common for all set of models based on ensemble of classical field configurations.

The instanton ensemble is characterized by averaged instanton density , where is number of instantons and antiinstantons and - 4-volume, and mean instanton radius . Phenomenologically one chooses and , in this case the ”diluteness parameter” or packing fraction is much less than unity, justifying the small density expansion in the model.

The static potential in the dilute instanton gas was calculated in [33], see also [34]. In the approximation used in [33, 34] each instanton contribution to Wilson loop is calculated independently, the potential at the leading order in density is given in this case by:


where the instanton size distribution is such that and the numbers of instantons and antiinstantons are assumed to be equal: . The potential is normalised according to the condition . In case of static quarks belonging to higher representations eq. (12) should be generalised [34], with the following result:


and the function is given by some cumbersome double integral whose exact form can be found in [34]. Here is the dimension of the representation (not to be confused with the Casimir ratio ) and sum over goes over all multiplets for decomposition of the given representation with the corresponding weights. One finds and also .

At small the functions , while at large the functions tend to -dependent constants [34]. Numerically one finds at small distances


where and numerical coefficients for are given by

instead of Casimir scaling results . Similar situation takes place for the large distance asymptotics of the instanton–induced potentail. It violates CS at the level of 20% . (see [10])

One-instanton approximation used in (12), (13) was recently examined on the lattice [35] in order to study the instanton density dependence as well as CS of the instanton-induced heavy quark potential. It was found that at the distance the deviation of the dilute gas formula (12) from the exact expression (5) is about for fundamental charges and for adjoint charges at the density , while it increases up to for fundamental charges and for adjoint charges at the phenomenological density . Moreover, linear density dependence of the potential extracted from the Wilson loop (12) breaks down for the densities at the distance . The ratio of the potentials obtained according to (12) is found in [35] at different distances and densities (to be compared with the Casimir ratio ) and it goes as at ; at ; at . The ratio of the potentials extracted from (5) has been found in [35] to be equal to 2 with an accuracy of a few percents for all explored distances.

All these results make it clear that instanton ensemble cannot be a natural framework for explanation of CS. It also demonstrates that if one is considering field configurations on the lattice which by some reasons are mostly made of instantons (as it happens, for example, after the cooling procedure), CS does not hold in such vacuum. It would be interesting to observe directly on the lattice the CS breaking in this case.

Let us mention another interesting problem in this respect – the pattern of CS in SUSY QCD. One should expect the role of instantons in vacuum dynamics of and even supersymmetric Yang-Mills theory to be much more transparent than in real world QCD. On general grounds one expects the simpler structure of perturbation theory and hence the absence of higher Casimirs in the perturbatively induced potential. On the other hand, coherent NP vacuum made mostly of instantons and antiinstantons should produce significant violations of CS for the NP part of the static potential.

Another aspect is related to elementary -strings, which can be formed between static sources for the gauge groups with . Such strings are stable and characterized by the string tension (see, e.g. [36]). Recent lattice studies in this direction [23, 24, 37] are of much interest. They are aimed to test CS in a different setup from that of [6, 8] and compare the results with alternative predictions, e.g. with the so called MQCD conjecture [38], see also [36]. This project is far from being complete and one needs more accurate data to make reliable conclusions, In particular, MQCD predicts , which gives for the ratios for , respectively. On the other hand CS hypothesis predicts that this ratio should scale as . It would be very interesting to test these predictions in further lattice calculations (see [24] and references therein).

Coming back to the real QCD vacuum, one can guess two possible explanations of the fact that instantons do not destroy CS. Either instantons are strongly suppressed in the real (hot) QCD vacuum (as it was observed in different respect in [39]) while they are recovered by the cooling procedure. Or else instanton medium is dense and strongly differs from dilute instanton gas, in such a way that higher cumulant components of such collectivized instantons are suppressed. Interesting to note, that linear confinement missing in the dilute gas, would be recovered in this case.

Now we come to another NP QCD vacuum model, based on the center dominance idea. An important role of the center of the gauge group was stressed by ’tHooft in [30]. A center vortices model tries to understand confinement in terms of interaction of the current forming the Wilson loop with the topologically nontrivial field configurations charged with respect to the center of the gauge group. A center vortex is a topologically stable field configuration with the topology of the surface, which carries (chromo)magnetic flux and interacts with the external (chromo)electric current in the following way:


where and represents the linking number (in four dimensions) between Wilson contour and vortex surface . The idea of center dominance implies that dominant contribution to the string tension comes from the quantum fluctuations in the number of center vortices linking the loop. The model predicts the following static potential (the reader interested in the details of the center vortex model is referred to [31, 32] and references therein):


where the function is defined as and is the set of generators from the Cartan subalgebra. The function represents the corresponding solid angle and depends both on the Wilson contour and on the position of the vortex center . The parameter determines the probability for the vortex to cross with the contour. The expression (16) is derived in the so called fat center vortex model, where each vortex has finite thickness. This parameter is crucial for potential to be nonzero at intermediate distances in the case of zero triality representations: in the limit of vanishing thickness adjoint (and all zero -ality representations) loop has no interactions with center vortices. So the introduction of thickness leaves a window for CS, which is essentially perimeter-type effect in this model [32]. The price to pay, however, is the contradiction between (16) and (8), in particular, the potential in the adjoint representation is not linear (and even not a convex [40]) function of the distance, despite at some interval of distances it can be approximated by linear term and its flattening is going smoothly with the distance in this model, to be compared with (8).

The exact value of the CS violation is strongly model dependent in this approach which makes it difficult to put stringent bounds on the parameters of fat vortices model based on CS. For example, in the case the deviation of the center vortices induced potential from CS behaviour is about for (adjoint) and for at large distances [32], but presumably it can be made much smaller by the proper adjustment of parameters.

Lattice measurements of the potential for different representations of were performed in center vortex model in [40]. In is instructive to consider normalised Casimir ratio


In the theory with exact CS one expects . If we take for some fixed point, for example such that (this point corresponds to the edge of the region, where quasilinear asymptotics of begins), we get from the results [40] for some of the representations:

These numbers clearly demonstrate the level of the CS violation in the fat center vortex model. It was suggested in [32] that the model can be tuned in order to explain properly the CS at intermediate distances, in particular, by adjusting the vortex profile function. At the same time, the initial root for CS violation lies in this case in the lack of the gauge-invariant formulation of the model. It would be interesting to understand in this respect the gauge-invariant contents if any of the notion of center dominance.

It is of interest to discuss the situation which takes place in another popular confinement scenario - dual Meissner effect in the abelian-projected formulation of QCD [27, 53]. We refer the reader to the review papers [29, 28], where the details of this approach are extensively discussed. It is worth noting that at the present moment the dual Meissner confinement scenario is often considered as the most reliable microscopic picture of confinement and the method of abelian projections (in line with the abelian dominance hypothesis [54]) as the most adequate language for its description. As a warm-up example, let us consider first Georgi-Glashow model. The model contains nonabelian vector field with two components acquiring nonzero mass in the Higgs phase, while the third component remains massless and provides dominant contribution to the Wilson loop. Existence of monopoles in this model leads to area law [57]. The CS property of Georgi-Glashow theory in comparison with abelian-projected QCD was discussed in [58]. In the former case one has


The average for large contours is dominated by or for even and odd, respectively. It is clear therefore that e.g. zero charge part of diagonal component of adjoint external current remains unconfined and hence the model is CS violating.

In terms of gauge-invariant correlators given by where one defines abelian field strength , where is adjoint scalar Higgs field and - the mass of components of full nonabelian gauge field, the vacuum of Georgi-Glashow model made of ’t Hooft-Polyakov monopole-antimonopole pairs is coherent and not stochastic. Indeed, nonperturbatively generated mass scale characterizing exponential falloff of the correlators in -space enters string tension in highly nontrivial nonlinear way, while extracting confining part of the two-point correlator one gets simply [63]. It is worth saying that nonperturbative physics of Georgi Glashow model is investigated on the lattice in details (see, e.g. [58] and references therein) and conclusions of theoretical analysis performed in [57] have been supported.

We have learned from this example that Higgs mechanism may be a source of CS breaking since it makes some gauge degrees of freedom massive without destroying of gauge invariance, while other ones remain massless. This pattern is supposed to occur in QCD at finite density and we will briefly discuss it at the end of Section IV. Now we are coming back to abelian-projected QCD. Application of the method of abelian projections crucially depends on adopted gauge fixing procedure, so one could expect problems with the explanation of CS in this framework. Indeed it happens to be the case [55]. Abelian dominance hypothesis [54] in QCD states, roughly speaking, infrared dominance of diagonal components of the full nonabelian gauge field and hence suggests to omit nondiagonal parts in (1) in QCD just as it is done in (18). Doing so, one immediately obtaines deconfinement of adjoint charges at all distances in plain contradiction with the lattice data [58]. It indicates that the dynamics of long-range QCD is not like that of duplicate compact QED (in contrast with Georgy-Glashow model, which behaves as compact QED at the considered limit) and color gauge symmetry is not broken by some composite Higgs field (as it effectively happens in abelian dominance scenario in QCD).

There exist a few approaches in the literature aimed to cure this problem in the framework of abelian projection. All of them try to take into account the nonabelian degrees of freedom in more or less sophisticated manner. In particular, the reader is referred to [56] where the component of adjoint source is effectively dressed with a virtual cloud of charged degrees of freedom and hence is able to interact with neutral ”photon”. Adjoint string tension arises in this picture at intermediate distances due to interaction of diagonal abelian projected gluons with the part of the adjoint source doubly charged with respect to the Cartan subgroup. If one naively omits the corresponding Faddeev–Popov determinant it gives , i.e. just the square of the abelian double charge. It is expected that loop expansion of the determinant produces terms, correcting the above behaviour to the Casimir scaling ratio, but it has never been shown explicitly.

Another approach was discussed in [59]. The results of [59] are based on the so called Weyl symmetric formulation of effective dual abelian Higgs model (DAHM) with the direct summation over root vectors. The following prescription for the external current interaction term was adopted (see also [54])


with subsequent recipe . The above ansatz is different from (18) since it implicitely assumes some kind of averaging over different color components of external current and consequently produces (contrary to (18)) area law for Wilson loop in all representations. It was found in this framework in [59] that CS favours different values of Ginzburg-Landau parameter for different representations, lying in the Type II range . For flux tubes become noninteracting and one has flux counting rule , where are Dynkin weights of the representation (see [59] and references therein).

The formalism proposed in [60] deals with some lattice motivated reformulation of QCD, where a special kind of averaging over different choices of subgroups of the original is implemented instead of having one fixed choice. The situation with CS in this model if realistic values of Ginzburg-Landau parameter are taken (e.g. from fitting of QCD string profile by DAHM induced formulas [61]), is still unclear. We refer the reader to the review [28] where this set of problems is discussed.

Concluding the discussion of abelian projected theories let us mention that it could be of interest to calculate the potential in approaches advocated in [56, 59, 60] at all distances and for different representations of and directly compare it with the results of [7, 8]. As it is clear from the above discussion the problem to reproduce CS is still far from being closed for abelian projected theories.

All the examples discussed above clearly demonstrates crucial importance to count properly all relevant nonabelian degrees of freedom. Any NP QCD vacuum picture analysed so far lacking manifest gauge-invariance had encountered internal difficulties explaining CS. In the next section we will consider another class of models where the nonabelian gauge invariance is kept intact from the beginning.

iii.2 Gauge-invariant models

We start with MIT bag model [41] and refer the reader to papers [42] where the problem of the CS in this model was addressed. It was advocated the following relation between string tensions in the MIT bag model: . It is easy to see, that this relation is in contradiction with the data from [7] at the level of a few dozen standard deviations. Qualitatively, it comes from the fact, that bag model misses the string, a crucial ingredient of modern picture of NP QCD, based on lattice results. The string in the static charge–anticharge case has two ends, with the factor attached to each one, and fixed, representation–independent radius (see below), so the total contribution to the energy in proportional to the charge squared, i.e. . At the same time, in the MIT bag model the charge enters the energy linearly.

Another family of models to be considered here are the so called ”confining string” models. Attempts to build a theory of the confining string have begun more than 30 years ago (see e.g.[43]). Till the present moment however there is no selfconsistent quantum string theory applicable to QCD string, those formation in the confinement phase is clearly seen on the lattice. Moreover, such theory is absent even for abelian Higgs model, where the strings exist as classical solutions (Abrikosov-Nielsen-Olesen strings) (see [62] and references therein). As an example let us consider here the static potential induced by quantum Nambu-Goto string.

The action of the model is proportional to area of a surface bounded by the static sources worldlines. Quantum dynamics of this surface produces additional contribution to the confining potential besides the leading linear term [44]:


where the term will be referred to as the String Vibration (SV) term. Despite the Nambu–Goto string model cannot be rigorously defined in (see e.g. [43]), and, in particular the expansion of the r.h.s. of (20) is valid at large distances only, it is instructive to look whether or not the lattice data support the existence of such term. It is also worth noting, that the dimensionless coefficient is universal and determined by the only two factors: target space dimension and the chosen string model. Having both factors fixed, it cannot be freely adjusted. Assuming , it is easy to see, that the Nambu–Goto SV term violates CS of the potential, as it was noticed already in [9].

It is a nontrivial task to separate the contributions of the discussed sort in the confining potential as it is because these corrections are essentially large distance effect, where they are subleading. But they have to become pronounced in the expression (7) due to scaling violation. Namely, assuming that SV term is the only one violating CS, one obtains


where the dots denote terms, omitted in (20) and is some arbitrary scale, for example Sommer scale. On general grounds one should expect, that string picture works at distances since no other dimensionful parameter enters the Nambu-Goto string action (in particular, string in this model is infinitely thin). We expect therefore, that the data [6, 7] allow to extract the possible contribution from CS-violating SV term at intermediate distances Fm. The l.h.s. of (21) is compatible with zero at the level of from the data [7] while the r.h.s. of (21) is nonzero and is rising with . However the statistical errors of the expression on the l.h.s. at such large distances are too big to make definite conclusion about ruling out the SV term of the discussed form. Additional accurate measurements are needed for this purpose. Notice, that the sign of the Nambu–Goto SV correction is opposite to what was found for CS violation in [6].

One way to combine CS and string dynamics was proposed already in [9] – just to multiply all the potential (20) with , accounting for ” elementary fluxes” attached to the static sources. It should be stressed that it implies different physical mechanisms responsible for creation of the string and its quantum fluctuations and presents actually a model, different from Nambu-Goto string. The situation with direct lattice measurements of SV corrections is not yet clear. While the authors of [45] claim the disagreement between hybrid spectrum and the string picture, but there are also evidences in favor of SV term [46]. The question certainly deserves futher study.

From theoretical point of view it stays to be proved that the QCD confining string and the simplest bosonic Nambu–Goto string model belong to the same universality class, see e.g. recent discussion in [47], and the theoretical background of (20) is not yet clear. There are also arguments against this possibility (see, e.g. [43]). The theory of the QCD confining string – whatever it will be – must explain the observed Casimir scaling of the potential at intermediate distances. There are arguments [47] that the confining string in the strongly coupled continuous Yang-Mills theory posessess two limiting regimes: CS respecting, which occurs if the area of the surface bound by the Wilson loop is smaller than some critical area, and flux-counting regime, which starts to dominate for asymptotically large surfaces. It could happen that strings created by the sources in higher representations would break at smaller distances than those where the quantum fluctuations of the string become visible. In this case CS region will cover all the scales where the string exists as a stable object and no room to detect any significant contribution from SV terms will be left. This problem is absent for stable -strings on (see above), and it is reasonable to expect CS violation at large distances in this case caused just by SV corrections, i.e. by quantum dynamics of the string.

Let us mention two regimes of the gauge theory where the CS property is under theoretical control. The first one is two-dimensional Yang-Mills theory. In case the theory has no propagating degrees of freedom and the static Coulomb potential which is linear in two dimensions is exactly proportional to . Another example is the strong coupling regime, where the partition function of the lattice Yang-Mills theory is dominated by the two–dimensional surfaces with the bare string tension proportional to the quadratic Casimir. It is worth reminding that the original motivation for invention of the CS hypothesis [9] was based on the dimensional reduction scenario [12], which assumes that there exist nontrivial relation between NP observables, e.g. string tensions in and Yang-Mills theories, if proper identification of the couplings is done. QCD in and have indeed much in common while also some important differences. One of them to be mentioned here is the Lorentz structure of NP linear confining potential which is vector in but scalar in . We will briefly discuss the relation of this picture with Gaussian dominance in Section V.

In the gauge–invariant NP background field formalism (see review[2]) the CS property has two possible interpretations. To understand the corresponding physical pictures, let us formulate some basic ideas of field correlator method. The approach decomposes the gluon field into perturbative field and nonperturbative background with the latter to be taken into account by means of gauge-invariant irreducible correlators (cumulants) . In particular, the Wilson loop average in the plane admits following expansion (see [2] for details)


The nonabelian Stokes theorem [48] and the cluster expansion theorem have been used in deriving (22). Here , where is a phase factor,


and are the points on the surface bound by the contour . The double brackets denote irreducible Green’s functions proportional to the unit matrix in the colour space (and therefore only spacial ordering enters (22)). Since (22) is gauge-invariant, it is convenient to make use of generalized contour gauge [49], which is defined by the condition .

In the confinement phase vacuum is disordered in a sense that averages from the r.h.s. of (22) develop finite correlation length , which for the lowest two-point correlator was found on the lattice to be rather small [3]: Fm in the quenched approximation for (to be compared with the old stochastic proposal of [50]). The Gaussian dominance [1] implies that the dominant contribution to the potential (and most of other observables) comes from the lowest two-point correlator if the integration surface in (22) is taken as minimal. It is easy to see that


so Gaussian approximation provides exact CS. Notice that (24) holds true since parallel transported field strength tensors from (22) obey the same standard commutation relations as bare do.

The CS property of Gaussian vacuum depends neither on the actual coordinate profile of the potential nor on that of the correlator, in particular, it holds for any value of . It is true also for representations where the linear potential is just some kind of intermediate distance behavior and changes profile at larger . The coordinate dependence of the potential, not directly related to CS, can be analysed at the distances small enough not to be affected by screening effects.

The average does not depend on the choice of the surface in (22). At the same time the particular choice of determines the weight of integral contribution of each cumulant to the Wilson loop average, and this weight is obviously -dependent. However this dependence is cancelled out if the whole sum in the r.h.s. of (22) is taken [48]. It is clear therefore that for surfaces strongly different from the minimal one there is some kind of cancellations between higher terms, while for the minimal surface CS might be either a result of cancellations between higher Casimir contributions from higher cumulants or just follow from the Gaussian dominance (terms with in (22) give small contribution). We will discuss below possible ways to check that. In either case, we should stress again that Gaussian dominance is understood throughout this paper in the following sense: if is the minimal surface, in (22) is considerably larger than all other terms. The gauge-invariance of makes the notion of Gaussian dominance gauge-invariant as well.

Iv What can we learn from the CS?

In this section we discuss other consequences of CS and try to argue that some further detalisation of the CS property in lattice calculations might be of much use for our understanding of the strong interaction physics and the mechanism of confinement.

iv.1 Three-point correlators

Only two-body potentials of the meson type have been discussed till the present moment. It is also of interest to study three-body system, which in fundamental and adjoint cases corresponds to baryons and tree-gluon glueballs, respectively. The tree-body potential for fundamental sources is defined by expression (5) where is taken to be


and the contours are formed by static quark trajectories. Indices of the antisymmetric tensors run from 1 to 3. For ajoint sources the corresponding expression is given by


where the tensors determine the colour structure of the state corresponding to the minimal energy and adjoint indices run from 1 to 8. Let us take spacial configuration of the system to be equilaterial triangle with the side . Naively one might assume the same CS law for the 3-body case, i.e. .

However, one has to conclude that the CS hypothesis in 3-body case should fail. The reason lies in different geometry of the 2-body and 3-body systems. Indeed, in case of two static sources the minimal area or the surface bound by the Wilson contour is provided by the plane, one and the same for all representations. In 3-body case in adjoint (and higher) representation the string may go either along Mercedes-star configuration (with the formation of a string junction, Fig.1a) or as triangle configuration (see Fig.1b) and one can easily argue that the energy of the latter variant is lower (notice at the same time that there is no such choice in the case of fundamental sources). Indeed, for the static (i.e. noninteracting and nonvibrating) strings connecting adjoint sources one has for the triangle configuration and for the Mercedes-star case, where CS relation was used. For higher representations one can imagine more complicated geometry of strings, for example with more than one junction. Therefore we do not expect that lattice analysis of the 3-body potentials for higher representations will indicate CS in the same fashion as it does for 2-body potentials. In particular, it is interesting to check the ratio which would indicate different geometry of the strings (and at the same time indirectly support CS law) in the two cases.

iv.2 Stochastic vs coherent

It was shown in the previous section how the gauge-invariant picture of Gaussian dominance explains CS of static potential. It was also stressed that the CS property on the other hand does not imply Gaussian dominance since one can imagine an alternative scenario (to be referred to as ’fine-tuning’ picture) where all higher terms in the expansion (22) strongly contribute to the lowest -proportional term together with Gaussian correlator while all contributions with higher powers of and higher Casimirs effectively cancel each other. Therefore despite CS property seems to be a very strong evidence for Gaussian dominance, the latter stays to be checked explicitly. See [64] in this respect where the quartic field correlator was measured and evidences for its smallness were found. We propose here several types of field correlators which distinguish coherent and stochastic (i.e. Gaussian) pictures. The first one is defined as follows:


The phase factors are defined according to (23) with the straight lines connecting edge points as integration contours (see Fig.2a). The -dependence of (27) is related to the contribution of correlators higher than Gaussian, but respecting CS. Gaussian dominance implies smallness of this contribution. Notice, at the same time that (27) is exactly proportional to the quadratic Casimir by construction. We take all fields in the expression (27) in the fundamental representation for the sake of simplicity. Using standard relations for the phase factors (see, for example, [48]) one gets:


with the explicit contributions from the 3-point correlator to the r.h.s. of (28) (we omit ’s for brevity). One can also define two typical lengths, characterizing the behavior of (27) in different regions of parameter space. It is physically clear that the situation when should be distinguished from the case with the distant reference point . The longitudinal correlation length corresponding to the variation of is to be compared with the transverse one , which parametrize the change of (27) with respect to in both regions in lattice simulations. It is also possible to choose rectangular geometry, it corresponds to the correlator


where all integration contours are again straight lines (see Fig. 2b). The –dependence of (29) is to be explored. The correlators of this form enter different expressions when one quark is heavy and can be considered as a static source. In Gaussian dominance scenario one takes into account -dependence and neglects –dependence of (29) [65].

The second type of correlators directly distinguishing stochastic and coherent pictures is given by multipoint irreducible field strength correlators. The simplest nontrivial one is that of the fourth order (see Fig. 2c, where the choice of contours for four-point correlator with all arguments on straight line is depicted). One has to consider the following expression


where we omit Lorentz indices and phase factors . Notice that in confinement phase where field correlators develop finite correlation length, each term in the r.h.s. of (30) is proportional to at large distances, while their difference and hence the l.h.s. of (30) is linear in .

The ordered integration goes over the surface , parametrized by according to

It should be clear from the previous discussion that in Gaussian scenario such surface-dependent contribution is assumed to be small in comparison with the two-point one


if the surface is the minimal surface. There exist lattice indications [64] that this is indeed the case. It would be of considerable interest to explicitly estimate (30) in comparison with (31) on the lattice using powerful present day lattice technique and also to study the CS behaviour of (30).

The suggested probes being particular examples of path-dependent gauge-invariant field correlators are more subtle than Wilson loop averages and can be effectively used to discriminate coherent and stochastic pictures in direct way.

iv.3 NP effects at small distances

It was recently suggested on both theoretical [66] and lattice [67] grounds that leading NP contribution to the static potential at small distances linearly depended on , instead of the quadratic dependence naively suggested by conventional OPE:


where the dimension-two coefficient should be distinguished from the asymptotic string tension in (6). From the background perturbation theory point of view described in the previous section, this linear term comes from the interference between perturbative and nonperturbative interactions. In theoretical picture developed in [66] the phenomenon was associated with formation of the so called infinitely thin string between charges. One can address the issue of CS in this approach. It is of interest to understand theoretically and on the lattice in this framework the relation between ”thick confining strings” which provide confinement and infinitely ”thin short strings” violating conventional OPE and presumably contributing to the linear potential at small distances in the light of CS in both the small and the large distance regions, found in [7].

There are arguments to be mentioned [68] that the quasilinear behavior of the potential at small distances observed in [67] could be a manifestation of the phenomenon of freezing of perturbative coupling at large distances. It is interesting therefore to check CS in this setup directly, namely to extract not only , as it was done in [67] but also the same quantity for higher in analogous way. The linear scaling of with would be in line with the proposals of [66] and of course with Gaussian dominance. Hopefully further lattice investigations will shed light on that.

iv.4 Interaction of Wilson loops

Another lattice setup where the CS phenomenon can be successfully checked is the Wilson loop correlators. In the present subsection we start from the disconnected correlators of the form and come to the discussion of the connected ones after that. The case of interaction between loops in different representations is of particular interest. We consider the two-loop correlator in the regime where the sizes of both loops are much smaller than the distance between them. One gets (see Fig.3a)


where does not depend on the choice of representations . The terms denoted by dots in (33) are proportinal to the higher powers of and should be suppressed if Gaussian dominance holds true. Let us mention the results from [69] where some indications were presented that (33) holds true (for adjoint–fundamental correlator). See also [70] where the behavior (33) was obtained in the Gaussian stochastic model framework.

Let us consider now gauge-invariant connected correlators. Lattice simulations demonstrate the phenomenon of confining string formation. The important problem of confinement dynamics is to study the string profile. This question can be addressed for the string attached to fundamental sources and to sources in higher representations as well. One’s first instinct is to say that since for higher representations the string tension is larger the same should in some sense be true for the geometrical characteristics of the string, e.g. its radius. It was already discussed in the previous section that it is in fact not true. To see this, let us consider connected probe of the following form:


where phase factors , connect the plaquette with the loop (compare with (33).

Since in the continuum limit where is lattice link one gets


where is the minimal area surface inside contour . Omitted terms in the r.h.s. of (35) denoted by dots contain higher powers of . It is seen at the same time that the string radius is representation–independent in Gaussian stochastic model (and, generally, for arbitrary CS respecting field ensemble), if the terms proportional to dominate in (35). On the contrary in the absence of CS for (34) the representation dependence of needs not to be factorizable and radius of the string for such coherent ensemble would be representation–dependent.

It is therefore gauge-invariant analysis of the QCD confining string profile for charges in higher representations which could indicate both CS and Gaussian dominance in a way different from just static potential measurements. Notice, that in [69, 70] another (disconnected) type of correlator was used and the results for (35) should not be directly compared to that of those papers.

iv.5 CS for nonstatic potentials and hadron spectra

Till the present moment we have considered only static potentials. From hadron phenomenology point of view, however, not static but dynamical potentials, which take into account the effects of quark motion are of interest. What is the meaning of CS in this case? Let us consider for simplicity gauge-invariant Green’s function for heavy-light spinless system in the gluon field (see, e.g. [4])


There are several different regimes of (36) depending on relative value of the light particle mass and properties of the gluon ensemble, encoded in the weight used in the averaging procedure . In case of large when the nonrelativistic expansion is valid, one has to distinguish two situations: the potential regime and the sum-rule limit. The energy levels get a correction in the latter case which in Gaussian stochastic model is proportional to . Contributions of higher condensates are suppressed in this case by the inverse powers of . In the potential regime the situation becomes less transparent and one is not to expect any exact scaling law with respect to for the spectrum of the system, despite the potential obeys CS. Spin degrees of freedom provide additional complications. In particular, difference in spin structures precludes to identify the spectrum of the heavy-light system where spinor light particle carries fundamental color charge (which models and -mesons) with heavy-light spectrum where vector-like dynamical constituent carries adjoint colour charge (such system might mimic hypothetical hybrid excitations of heavy mesons). Indeed, the only direct fingerprint of CS for the real particles spectrum is presumably the fact that the lowest glueballs are believed to be heavier than the lowest mesons (we do not take chiral effects into account). Let us consider ”hadrons” as bound states of massless and spinless quarks, without perturbative interactions. In this case the mass of a hadron is proportional to the square root of - the only dimensionful parameter of the problem: . Hence one might predict that the equivalent states in glueball and meson spectrum have ratio of masses . Now include spins and perturbative dynamics. Taking into account of Coulomb force changes this ratio, also spin-dependent forces are different since spin of gluon is twice that of quark, and finally quarks have (negative) self-energy corrections (of the order -0.25 GeV per particle [71]) while for gluons selfenergies are forbidden by gauge invariance. All that leads to the masses calculated in potential model in [72] which are in good agreement with lattice and experimental data, namely for center-of-mass of and glueball one finds