On the pattern of Casimir scaling violation in gluodynamics

Results of lattice analysis indicate that the static potential in SU(3) gauge theory is proportional to eigenvalue of quadratic Casimir operator for the corresponding representation with a good accuracy. We discuss the mechanism of possible deviations from the Casimir scaling in gluodynamics in terms of correlators of path-ordered gauge-invariant operators defined on the worldsheet of the confining string.


Introduction
Physics of confinement in QCD has been attracting considerable attention since the Yang-Mills theory with dynamical quarks was established as a theory of strong interactions.Despite significant theoretical progress the operational framework for the exact calculations beyond perturbation theory has not been found, leaving aside numerical simulations of the theory on the lattice.In particular, one has no clear understanding of the relation between confinement and mass gap property.It is commonly believed that confining theory such as gluodynamics (i.e.QCD with no quarks) exhibits the mass gap, e.g. the lightest excitation over the physical vacuum is a massive particle.The latter one is indicated by the lattice calculations to be scalar 0 ++ glueball with the mass m ≈ 1.5 Gev in the case of N = 3.On the other hand, if one studies confining potential between static sources in pure Yang-Mills theory, the physics of area law is governed by the formation of the confining string with nonzero tension √ σ ≈ 0.4 Gev and this process is not directly related to the details of glueball physics.The dynamical reasons for that are well known -corresponding effective interaction is suppressed by relatively large mass of the glueballs in the units of √ σ and also by 1/N-factor.Put in simple terms, even if one acquired full knowledge of the glueball spectrum in gluodynamics, it would not help to compute generic Wilson loop.It is worth mentioning that in the abelian Higgs model with electrically charged condensate the mass of the photon (which plays the role of the lightest glueball in this theory) enters monopole-antimonopole string tension in rather nontrivial way and in combination with other quantities, giving the simplest example of the same nature.
The standard approach of quantum field theory describes interactions in terms of particle exchanges.In confining theories the applicability of this program is limited, in some sense: string creation, which is one of the most important manifestations of confinement can be understood neither in terms of gluon exchanges nor as a result of glueball interactions. 1evertheless it is desirable to establish some connection between different model approaches or at least to understand, where such relations are to be expected.The aim of the present paper is to make a step in this direction.It will be shown that the certain linear combination of the static potentials arising from the Wilson loops in different representations gets contributions from the exchange by colorless states.Our actual analysis will be presented for adjoint and fundamental loops which are of the main phenomenological interest but we believe that the mechanism is quite general.The paper is organized as follows: theoretical and lattice status of static potentials in different representations is briefly discussed in the Section 2, the Section 3 presents the details and results of our analysis on more technical level, and short conclusion can be found in the Section 4.However, one is not to expect CS to be exact law in the case under consideration.As a matter of principle, CS should be violated at large distances because of screening.Qualitative picture for asymptotically large Wilson loops corresponds to the string tension σ D equal to fundamental string tension σ = σ f und , if the Wilson loop in the representation D nontrivially transforms under the center of the gauge group, or zero otherwise, in obvious contradiction with CS.On the other hand, the actual value of critical distance where effects of screening become important may be as large as 1.4 Fm [7] and one still has large enough "CS window", where the corrections to CS due to screening are exponentially small.As to perturbation theory, CS law for static potential is known to hold up to two loops in continuum theory [10,11], however there is no any general theorem that CS should be exact property of perturbative potential and analysis of diagrams provides qualitative reasons in favor of CS violation at the three-loop level.It is also worth mentioning that CS is violated in the strong coupling expansion at the order 1/g 8 [12] while it holds at lower orders.
Another line of research confronts CS with alternative theoretical predictions for D-dependence of the static potential.The only law different from CS which has not yet been definitely ruled out in lattice simulations is the so called sin-formula, motivated by supersymmetric Yang-Mills theory analysis [13,14].Moreover, the authors of [15] claim that their numerical data for asymptotic string tension in d = 3 + 1 nonsupersymmetric SU(N) theory with N = 4, 6 favor sin-formula and disagree with CS (see also [16]).On the contrary, authors of [17] claim their results to be in agreement with both CS and sin-formula predictions within two standard deviations in d = 3 + 1 Yang-Mills theory and definitely deviate from both in d = 2 + 1 case.It should be noted that in all studied cases the numerical difference between CS law and sin-formula is actually quite small that plagues the discrimination between the two predictions.
Unfortunately, as such, the sin-formula gives no nontrivial testable predictions for physically interesting case of the gauge group SU(3).On the other hand, there are theoretical reasons to believe [18] that it receives non-universal corrections.This seems to be a good motivation to take the predictions like CS or sin-formula as the first approximations and study the pattern of possible deviations from these regimes.Needless to say that despite numerically the two predictions are surprisingly close, the qualitative physical picture behind them is drastically different.In any case, data from [1,2,3,4] as well as from [15,16,17] undoubtedly indicate that CS as the first approximation for static potential works with percent-level accuracy at not too large distances.
The starting point is the general expression for the Wilson loop where the generators T a D correspond to irreducible representation of dimension D. The normalized trace Tr D is defined as Tr D 1D = 1 D Tr 1D = 1, fundamental and adjoint generators are normalized according to The eigenvalue C D of quadratic Casimir operator C D is defined as follows

C
(2) We will confine our attention to adjoint representation in the present paper.The corresponding Casimir ratio , where the fundamental Casimir C f und = (N 2 − 1)/2N and adjoint one C adj = N equal to 4/3 and 3, respectively, for the case of SU(3).The static potential is formally defined as where µ stays for renormalization scale. 3The physical potential is given by where both terms in the r.h.s.diverge if µ → ∞ while V phys D (R) is well defined in this limit (see, e.g.[19]).It is known that perturbative series for the static potential in Yang-Mills theory does not exponentiate, contrary to abelian case [20].Nevertheless it is convenient to exponentiate Wilson loop (1) as (see review [21] and references therein) where each ∆ (m) D (R, T ) is equal to well defined linear combinations of gauge-invariant field correlators (see (14), (15) below).We take (4) as a nonperturbative definition of the potential in the present paper and problems of actual renormalization will not be considered, correspondingly the argument µ is omitted below.It is worth remembering however that all terms in the series defined by ( 6) contribute to the potential and are to be renormalized accordingly.
In perturbation theory each ∆ (m) D (R, T ) carries at least a factor g m which is small, the full series however is divergent due to factorial growth of the coefficients.On the other hand, nonperturbatively there are no general reasons to conclude that there exists some hierarchy between correlators with different m.The general terminology calls the ensemble of the correlators It can be argued that the vacuum of Yang-Mills theory is stochastic and not coherent [22,23,24] and Casimir scaling is one of the strongest arguments in favor of that [5,6,7].The ultimate form of stochasticity is given by Gaussian dominance scenario when one keeps only the lowest nontrivial irreducible correlator in (6) corresponding to m = 2.The area law for the Wilson loop in the confining regime is given in the Gaussian vacuum in the following form: where σ (2) g and typical gluon correlation length T g is of the order of 0.2 Fm for pure N = 3 gluodynamics (see, e.g.[25,26]).As it is clear from (7), Gaussian string tension σ (2) D is always proportional to the eigenvalue of quadratic Casimir operator for the representation D and therefore Gaussian dominance implies CS (not vice versa, however).Therefore to address the question of possible deviations from CS law, one has to go beyond bilocal approximation.This will be done in the next, somewhat more technical section.

Deviations from CS law
We are to study the general structure of CS violating terms in the cluster expansion.All calculations are performed for the simplest case of flat rectangular contour C = R × T in Euclidean space and its minimal surface.It does not mean any loss of generality since all expressions below can be trivially generalized to the case of arbitrary contour, whose minimal surface has the disk topology.In the case under study it is natural to adopt the so called contour gauge condition, introduced in [27,28] as generalization of Fock-Schwinger fixed point gauge [29].It is defined by the condition that the phase factor orthogonal to the temporal axis is always equal to unity, i.e.

Φ( x, y)
where z 4 = x 4 = y 4 and A(x) = A a (x)T a D .Without loss of generality one may also choose Φ(x 4 , y 4 ) = P exp The Wilson loop (1) takes especially simple form in this gauge where | z| = R and the potential A a 4 ( z, t) is expressible in terms of the electric components of the field strength The contour gauges are well suited for studies of general properties of the gauge theories, but present considerable difficulties in perturbative calculations.Expanding (10) one gets, in the matrix form Each average in the r.h.s. of ( 12) is proportional to the unit matrix in color space due to triviality of the vacuum quantum numbers, so the whole expression is, namely With ( 8), ( 9) and ( 11) at hands one can easily rewrite every Tr D K D (R, T ) in manifestly gauge-invariant form (see [21] and references therein).
We will omit arguments R and T in the correlators Tr D K (n) D (R, T ) below for simplicity of notation.We also do not discuss the actual dependence of each correlator on its arguments in the present paper.Lattice data suggest to fit the correlator as a sum of hard perturbative part and soft nonperturbative one, the latter being responsible for confinement.Notice that the series (12) is not strictly speaking perturbative series in the sense that each term in (12) is given in perturbation theory by its own perturbative subseries.It is of no importance for our analysis which will proceed in terms of general expression and can be applied to whatever dynamics if it provides the set of correlators which obey Eq. ( 13).The perturbation theory itself can be regarded as a good example.D also vanishes while the relation between higher correlators is to be read off from the Taylor expansion of (6):

It is now crucial to determine the relationship between Tr
and the terms of higher orders are built in full analogy with (14): where (...) stays for the sum of products of the correlators Tr D K (p) D of lower orders, p < m.The expression (15) does not represent the Green's function of any physical colorless state, instead, it describes the propagation of some gluelump state, which is not a physical excitation in gluodynamics.It is also worth saying that ( 14), (15) should be considered as a definition, valid beyond perturbation theory (each correlator is given by the corresponding perturbative series in the latter).
The potential V D (R) is given up to an additive constant by We are going to study now the structure of special linear combination of potentials, measuring the deviations from Casimir scaling We confine our attention to the case of fundamental and adjoint representations in this paper, in this particular case (17) takes the form As a next step, we are exploiting the identity We need to compute the quantity |W f (C)| 2 in terms of correlators (13).It can be done as follows.The average of the product is given by where the trace Tr 12 = Tr f 1 Tr f 2 acts on the color indices corresponding to the first and the second loop independently.Global gauge invariance dictates the following form of the corresponding averages (see also [30]): The matrices 1f , 1 and ê can be written in index notation as where α 1,2 , β 1,2 are fundamental indices running from 1 to N. The algebra they obey encodes the effects of path ordering.In our case since the orientations of minimal surfaces corresponding to the first and the second multiplier in (20) are antiparallel, the matrix ê should be multiplied from the right with respect to indices carrying subscript "2" but from the left with respect to indices carrying subscript "1".It results in the following algebra: and the traces are given by It is convenient to represent (20) in the following form: Comparing ( 25) with ( 19), ( 20) and ( 21) one concludes that This is exact expression for adjoint Wilson loop average, written in terms of fundamental correlators and traces and as such it is suitable for our analysis.We are to study the difference where the first term in the curly brackets in ( 27) is equal to log W a (C) = A where A is defined by (26), while the second term is given by formal series To proceed further one can exploit the identity It is now straightforward to rewrite (26) as: Till now we have not made any approximations yet and the expressions ( 28), (30) are valid for arbitrary contour C and any contour gauge.It is worth reminding that despite each term in ( 28), ( 30) is gauge-invariant, its actual value depends on the profile of the contours used in (8), ( 9) but this dependence is cancelled after summation of all terms.Likewise the property of path-dependence of Gaussian correlator (found in [31] to be rather strong) shed no light on the problem of CS since gauge-invariant path-dependent bilocal correlator Tr D F µν (x, x 0 )F ρσ (y, x 0 ) where F µν (x, x 0 ) = Φ(x 0 , x)F µν (x)[Φ(x 0 , x)] † and Φ(x 0 , x) are the phase factors (chosen to be unit matrices in our analysis) is always proportional to C D for any path, taken in Φ(x 0 , x).
We can now use the Euclidean rotational 4-invariance of our problem which dictates W f (C) = W † f (C) or, at the level of correlators, Let us now look at the general structure of terms in δV a/f .It is straightforward to notice that each δ∆ can be shown to be exactly cancelled in (27) (this cancellation happens in the terms corresponding to k = 1 in ( 28) and ( 30)).To illustrate the point, consider the lowest nontrivial order of cluster expansion violating CS, which corresponds to m = 4.One gets This expression is gauge-invariant and the fields A 4 (x) in the correlators TrK (n) are functionals of the field strengths F i4 (x) as given by (11).
The l.h.s. of expression (32) vanishes identically if CS law is exact.Posed differently, it measures the deviation from CS in terms of integral moments of particular correlators, entering the r.h.s. of (32).The coefficient in front of the second term in the r.h.s. of (32) (equal to 5/4 in the case of SU( 3)) provides exact cancellation of reducible contributions from the first term, so both sides of equality (32) have area law asymptotics in confinement regime.In large N limit they both vanish, as it should be.The same pattern holds in higher orders of cluster expansion.Casimir scaling violating contributions are given at fixed m by the irreducible correlators of colorless operators Tr f K It is to be compared with the expressions ( 14), (15) where the potential itself is written in terms of correlators Tr f K (m) f .

Conclusion
Expression (32) and analogous formulas for higher orders of cluster expansion formalize the statement made in [7] that violations of CS could be a result of exchange by colorless states.We have considered only the case of adjoint and fundamental potentials in the present paper but no reasons for breaking of this mechanism in other representations are seen.The qualitative picture corresponds to the confining string worldsheet populated by irreducible correlators like Tr D K (2) D and correlators of higher orders made of gauge-invariant expectation values of pathdependent field strength operators.On the minimal surface the whole ensemble contributes to the total string tension in such a way that Gaussian contribution is numerically dominant, as CS suggests.In other words, CS indicates that this ensemble is quasi-free in the sense that interactions between such gauge-invariant objects as Tr F F mediated by glueball states and destroying CS are rather weak.Phenomenologically, we expect the r.h.s. of expressions like (32) to be suppressed by some power of inverse glueball mass.The actual smallness of deviations from CS as observed on the lattice simply follows in this case from the fact that even the lightest glueball is rather heavy, of the order of 1.5 GeV.Moreover, one can have a glimpse of general reason why many microscopic confinement models fail to reproduce CS.To this end, the model, apart from explaining string tension formation and area law of the Wilson loop at intermediate distances for all representations ("CS plateau") must also describe the phenomenon of mass gap itself and come out with the correct (rather large) masses of colorless states.One can mention, on the other hand, that reasonable prediction of the instanton liquid model for the mass of the lowest glueball (1.4 ± 0.2) GeV [32] is in some contradiction with the strong violations of CS for the static potential in this model [7].This paradox is possibly resolved by the fact that the instanton-antiinstanton ensemble is coherent and the cluster expansion is not well suited for analysis in this case.

f 2 .
gets contributions from irreducible correlators of gauge-invariant operators of the type Tr f K The contribution from the single correlator of the same order Tr f K (m) f