Double-winding Wilson loops and monopole confinement mechanisms
aa r X i v : . [ h e p - l a t ] D ec Double-winding Wilson loops and monopole confinement mechanisms
Jeff Greensite and Roman H¨ollwieser
2, 3 Physics and Astronomy Department, San Francisco State University,San Francisco, CA 94132, USA Department of Physics, New Mexico State University,Las Cruces, NM, 88003-0001, USA Institute of Atomic and Subatomic Physics, Vienna University of Technology,Operngasse 9, 1040 Vienna, Austria (Dated: July 9, 2018)We consider “double-winding” Wilson loops in SU(2) gauge theory. These are contours which wind oncearound a loop C and once around a loop C , where the two co-planar loops share one point in common, andwhere C lies entirely in (or is displaced slightly from) the minimal area of C . We discuss the expectation valueof such double-winding loops in abelian confinement pictures, where the spatial distribution of confining abelianfields is controlled by either a monopole Coulomb gas, a caloron ensemble, or a dual abelian Higgs model, andargue that in such models an exponential falloff in the sum of areas A + A is expected. In contrast, in a centervortex model of confinement, the behavior is an exponential falloff in the difference of areas A − A . Wecompute such double-winding loops by lattice Monte Carlo simulation, and find that the area law falloff followsa difference-in-areas law. The conclusion is that even if confining gluonic field fluctuations are, in some gauge,mainly abelian in character, the spatial distribution of those abelian fields cannot be the distribution predictedby the simple monopole gas, caloron ensemble, or dual abelian Higgs actions, which have been used in the pastto explain the area law falloff of Wilson loops. I. INTRODUCTION
Magnetic monopole confinement mechanisms, in eitherthe monopole plasma [1, 2] or (closely related) dual super-conductor incarnations [3, 4], provide a durable image ofthe mechanism underlying quark confinement in non-abeliangauge theories. The more recent notion that long-range fieldfluctuations in QCD are dominated by caloron gas ensem-bles [5], [6], fits nicely into the framework of the earliermonopole plasma conjectures. In view of the ongoing inter-est in monopole/caloron confinement mechanisms [7–11], itis reasonable to examine those conjectured mechanisms crit-ically, at least as they pertain to pure SU(N) gauge theoriesdefined in either three or four Euclidean dimensions, with no“small” dimension imposed by compactification [12].The mechanisms we are discussing have this point in com-mon: there is some choice of gauge in which the large scalequantum fluctuations responsible for disordering Wilson loopsare essentially abelian, and are found primarily in the gaugefields associated with the Cartan subalgebra of the gaugegroup. In a caloron ensemble, for example, while the dyoncores may be essentially non-abelian, there exists a gaugein which the long range field which diverges from the dyoncores, and which is responsible for confinement in this picture,lies entirely in the Cartan subalgebra. For the SU(2) gaugegroup, which is sufficient for our purposes, let this abelianfield be the A m color component. Then if all we are inter-ested in is the area law falloff and corresponding string ten-sion extracted from large Wilson loops, and not in perimeterlaw or short-range contributions from small Wilson loops, we can make the “abelian dominance” approximation W ( C ) = h Tr P exp (cid:20) i I C dx m A a m s a (cid:21) i≈ h Tr exp (cid:20) i I C dx m A m s (cid:21) i = h exp (cid:20) i I C dx m A m (cid:21) i = h exp (cid:20) i Z S d s mn f mn (cid:21) i , (1)where f mn is the corresponding abelian field strength. Now anexpectation value is the average of an observable over a verylarge number of samples drawn from some probability distri-bution. So the expectation value of the abelian Wilson loop isthe average taken over a very large number of sample abelianconfigurations A m ( x ) (or corresponding field strengths) drawnfrom some probability distribution P [ A m ( x )] (or P [ f mn ( x )] ).The question we are concerned with is: what do typical con-figurations drawn from the abelian field distribution look like?Do they resemble what is predicted by monopole plasma,caloron gas, and dual superconductor models?To be clear, we do not challenge the notion that, in somegauge, most of the confining fluctuations are abelian in char-acter. This certainly appears to be true in, e.g., maximalabelian gauge, which forces most of the A -field into the Car-tan subalgebra. Nor will we venture an opinion on whethercalorons, say, are somehow important to vacuum structureat near-zero temperature. Our study has a more specific fo-cus: assuming that the long range fluctuations which disorderlarge Wilson loops are mainly abelian in some gauge, whichis an assumption common to monopole, caloron, and dual su-perconductor pictures of confinement, how are those abelianfluctuations distributed in typical vacuum configurations? Ar-guments in favor of these monopole-related pictures derive afinite string tension using models which predict a specific spa-tial distribution of the confining abelian field. The purpose ofthis article is to subject a qualitative feature of those predicteddistributions to a numerical test, using an observable to be de-scribed below.Let us first illustrate how the probability distribution P [ A m ( x )] can be formally defined, using maximal abeliangauge on the lattice as an example. In this gauge we decom-pose link variables U m into an abelian (or “diagonal”) part u m ,defined by U m ( x ) = a + iaaa · sss u m ( x ) = q a + a h a + ia s i = (cid:20) e i q m ( x ) e − i q m ( x ) (cid:21) , (2)and an “off-diagonal” part C m , where U m ( x ) = C m u m ( x )= (cid:16) − | c m ( x ) | (cid:17) / c m ( x ) − c ∗ m ( x ) (cid:16) − | c m ( x ) | (cid:17) / × (cid:20) e i q m ( x ) e − i q m ( x ) (cid:21) . (3)Then the probability distribution for the abelian (or “photon”)field is obtained by integrating out the off-diagonal (or “W”)fields, which are charged under the remaining U(1) symmetry: P [ u m ( x )] = Z Z DC m DcDc exp [ − ( S W + S g f )] , (4)where S W is the Wilson action, S g f are the gauge-fixing termsrelevant to maximal abelian gauge, and c , c are the ghostfields. A restriction to the first Gribov region is understood.Monte Carlo simulations in maximal abelian gauge, followedby an abelian projection U m → u m , are drawing the abelianconfigurations u m from precisely the above probability dis-tribution. It should emphasized that the W-bosons are gone.Whatever contribution they make to the probability distribu-tion of the abelian fields is fully taken into account in (4), andthey have no further role to play when computing observables,such as abelian-projected Wilson loops, that depend only on u m ( x ) .In monopole/dyon gas and dual superconductor theories,the probability function is supplied indirectly for the abelianfield strengths f mn . The idea is that the abelian field strengthsare functions of some other set of variables { v } , such asmonopole/dyon moduli or dual gauge fields, and a probabilitydistribution P [ { v } ] is supplied. Then typical abelian vacuumfluctuations are obtained by drawing { v } from the given prob-ability distribution, and computing f mn from that.In section II we will review in more detail what monopolegas, dyon ensemble, and dual superconductor models have to say about the distribution of abelian fields in the vacuum. Wewill then, in section III, introduce a gauge-invariant observ-able, called the “double-winding” Wilson loop, and argue thatthis observable has a qualitatively different behavior accord-ing the monopole/dyon/dual-superconductor distributions, ascompared to the predictions of the center vortex theory of con-finement. The actual behavior of this observable can be deter-mined by lattice Monte Carlo simulations, which we reportin section IV. The effects of W-bosons are discussed in sec-tion V. Some of the arguments presented below are actuallyquite old, but we feel that those arguments are clarified andstrengthened by consideration of the gauge-invariant double-winding Wilson loop operators, and their numerical evalua-tion. We conclude in section VI. II. ABELIAN FIELDS AND ABELIAN MODELS
We consider several specific proposals for abelian field dis-tributions.
A. Monopole plasma in D = Euclidean dimensions
The classic example is Polyakov’s demonstration [1] thatcompact QED in D = Z mon = ¥ (cid:229) N = x N N ! (cid:229) { r n } (cid:229) { m n = ± } exp " − p g a (cid:229) i = j m i m j D ( r i − r j ) , (5)where a is the lattice spacing, D ( r ) is the inverse of the latticeLaplacian in a subspace orthogonal to the zero modes, and x = exp (cid:20) − p g a D ( ) (cid:21) (6)is the fugacity, with D ( ) ≈ .
253 in lattice units. The num-ber of monopoles together with their positions and chargesconstitute the variables { v } from which the field strength isdetermined and, in continuum notation, f i j = e i jk Z d r ′ ( r − r ′ ) k | r − r ′ | r ( r ′ ) , (7)where r ( r ) = N (cid:229) i = m i d ( r − r i ) . (8)This distribution of abelian field strength results in an area lawfor Wilson loops. Essentially the same result is derived in thecontinuum, for the D = B. Monopole plasma in D = Euclidean dimensions
A straightforward generalization of the monopole plasma to D = { v } are the divergencelessinteger-valued monopole currents k m ( n ) which exist on linksof the dual lattice, and the partition function is Z = ( (cid:213) s , m ¥ (cid:229) k m ( s )= − ¥ )( (cid:213) s d ¶ ′ m k m ( s ) , ) exp ( − S [ k ]) , (9)where S [ k ] = (cid:229) s m k m ( s ) k m ( s ) +
12 4 p g (cid:229) s , s ′ k m ( s ) D ( s − s ′ ) k m ( s ′ ) . (10)Again D ( s − s ′ ) is the inverse of the lattice Laplacian ( − (cid:209) L ) on a space orthogonal to the zero modes, and m is a monopolemass. Shiba and Suzuki [14] have made an effort to showthat this form of monopole action describes the distribution ofmonopole currents found in abelian-projected configurationsin maximal abelian gauge. The abelian field strength at a pla-quette, due to the monopole currents, is [13] f mn ( x ) = pe mnab ¶ ′ a (cid:229) y D ( x − y ) k b ( y ) , (11)where ¶ ′ a denotes the backward lattice derivative ¶ / ¶ x a .We are not aware of an analytical result, along the lines ofPolyakov’s discussion in D = . The phase in whichmonopole currents percolate, at sufficiently large g , corre-sponds to the strong-coupling phase of QED , and Wilsonloops in that phase certainly follow an area law. C. Dyon Ensemble
In a remarkable paper, Diakonov and Petrov [5] de-rived analytically a confining quark-antiquark potential fromPolyakov line correlators, and an area law for spacelike Wil-son loops, from dyon-antidyon configurations in D = The abelian field strength is, in this case, controlled by vari-ables v m ( x ) , which appear in the partition function for the Except to note in passing the critical comments in [17]. dyon ensemble. For SU(N) gauge theory this partition func-tion has the form Z = Z D c † D c Dv Dw exp Z d x (cid:26) T p (cid:0) ¶ i c † m ¶ i c m + ¶ i v m ¶ i w m (cid:1) + f (cid:20) ( − pm m + v m ) ¶ F ¶ w m + c † m ¶ F ¶ w m ¶ w n c n (cid:21)(cid:27) , (12)where the subscripts ( m = , .., N ) label the dyon type. Foran explanation of the terms in this expression, see [5]. Theabelian magnetic field B i = e i jk f jk due to the m -th dyon typeis given by [ B i ( xxx )] m = − T ¶ i v m ( xxx ) , (13)where T is temperature. Note that this expression for B i doesnot include Dirac strings, which have no effect on Wilsonloops, but which are important in showing that (cid:209) · B = , representing amonopole-antimonopole sheet along the minimal surface ofthe loop. The analysis provides a demonstration of the area-law falloff of a spacelike Wilson loop in the dyon ensemble,and an explicit calculation of the string tension in group rep-resentations of N -ality k .An alternate dyon ensemble, in which non-interactingdyons are distributed with a uniform positional probability inthe volume, was advocated in [10]. Here also the dyon fielddiverging from a dyon core is spherically symmetric aroundthe center of the dyon. This distribution does not appear tobe amenable to analytic methods, and results for Wilson loopsmust be obtained numerically. For this reason, we will notbe able to draw strong conclusions about this ensemble (see,however, remarks in section V). D. Dual Superconductivity
In this case the variables { v } which determine the abelianfield strength f mn are the dual gauge potentials C m ( xxx ) , whosedistribution is controlled by a dual abelian Higgs model (for areview, see [18]), with Lagrangian density L = ( ¶ m C n − ¶ n C m )( ¶ m C n − ¶ n C m ) + | ¶ m f − igC m f | + l ( | f | − m ) , (14)and f mn = e mnab ¶ a C b . (15)The massive phase of this theory corresponds to the existenceof a monopole condensate.Confinement in the dual abelian Higgs model is derivedfrom the existence of Abrikosov vortices in the dual the-ory, connecting sources of opposite abelian electric (rather FIG. 1. A double winding loop, which runs once around contour C ,and once around the coplanar loop C . than magnetic) charge. String tension is the energy of theAbrikosov vortex per unit length. It is also worth noting thatthere is a close connection between the monopole plasma,compact QED and the dual abelian Higgs model in a certainlimit [13, 20, 21].Since this article is concerned with only abelian models ofconfinement, the non-abelian dual models reviewed in [22] areoutside the scope of our discussion. III. DOUBLE-WINDING WILSON LOOPS
Wilson loops in the adjoint representation, which have zero N -ality, do not have an asymptotic area-law falloff. The mech-anisms summarized above comply with this behavior, sincethe abelian projection of an adjoint loop contains a componentwhich is neutral with respect to the abelian subgroup, and thisfact is sometimes taken as evidence that the mechanisms insection II are consistent with the dependence of string tensionon N -ality. It is therefore useful to consider a different op-erator, which we believe is a better probe of the mechanismsunder discussion.Let C and C be two co-planar loops, with C lying entirelyin the minimal area of C , which share a point xxx in common.Consider a Wilson loop in SU(2) gauge theory which windsonce around C and once, winding with the same orientation,around C , as indicated in Fig. 1. It will also be useful to con-sider Wilson loop contours in which C lies mainly in a planedisplaced in a transverse direction from the plane of C by adistance d z comparable to a correlation length in the gaugetheory. Such a contour is indicated in Fig. 2. We will referto both of these cases as “double-winding” Wilson loops. Inboth cases we imagine that the extension of loops C , C ismuch larger than a correlation length, so in the latter exam-ple the displacement of loop C from the plane of C is smallcompared to the size of the loops. Let A , A be the minimalareas of loops C , C respectively. What predictions can bemade about the expectation value W ( C ) of a double-windingWilson loop, as a function of areas A and A ? FIG. 2. A “shifted” double winding loop, in which contours C and C lie in planes parallel to the x − y plane, but are displaced from oneanother in the transverse direction by distance d z , and are connectedby lines running parallel to the z -axis. A. Sum of areas behavior
In all of the models summarized in the previous section, theanswer for the displaced loops in Fig. 2 is simply W ( C ) = exp [ − s ( A + A ) − m P ] , (16)where P is a perimeter term, equal to the sum of the lengthsof C and C . The argument goes as follows. Begin withthe assumption that the large scale fluctuations are abelian incharacter, so that (1) holds, and the distribution of f mn is givenby any of the models discussed. Then W ( C ) = (cid:28) Tr P exp (cid:20) i I C dx m A a m s a (cid:21)(cid:29) ≈ (cid:28) exp (cid:20) i I C dx m A m (cid:21)(cid:29) = (cid:28) exp (cid:20) i I C dx m A m (cid:21) exp (cid:20) i I C dx m A m (cid:21)(cid:29) . (17)If loops C and C are sufficiently far apart, then the expecta-tion value of the product is approximately the product of theexpectation values, i.e. W ( C ) ≈ (cid:28) exp (cid:20) i I C dx m A m (cid:21)(cid:29) (cid:28) exp (cid:20) i I C dx m A m (cid:21)(cid:29) ≈ exp [ − s ( A + A )] , (18)which we refer to as a “sum-of-areas falloff.” Physically, ina monopole plasma, the setup can be interpreted as insertingtwo independent current loops into the the plasma. Monopoles(or monopole currents) will respond by forming a monopole-antimonopole layer at the minimal surface of each loop. Theargument in the case of the dual superconductor is similar; weimagine that loops C and C are rectangular and oriented par-allel to the x − t plane, but displaced along the z -axis. In atime slice, this setup represents a pair of positive charges, adistance d z apart, interacting with a pair of negative charges,also a distance d z apart, and two electric flux form, as seen inFig. 3. The energy is s ( L + L ) , where L , L are the lengthsof the two flux tubes, and this implies, from the usual rela-tionship between Wilson loops and static potentials, a sum-of-areas falloff for the Wilson loop.Now, what happens as d z →
0? We would argue that thislimit does not really change the sum-of-areas behavior. For a
FIG. 3. A timeslice of shifted rectangular timelike loops can be inter-preted as representing two static particles on one side, and two staticantiparticles on the other. In the dual abelian Higgs model, the pairsof ± charges are connected by a pair of electric flux tubes. dual abelian Higgs model with couplings corresponding to aType II (dual) superconductor, electric flux tubes tend to repel.So as the two positive and two negative charges converge, wewould still expect to find two electric flux tubes separated byroughly the vortex width, and the sum-of-areas rule does notchange qualitatively. It has also been suggested [23] that therelevant dual abelian Higgs model is weakly Type I, near thecrossover from Type I to Type II behavior. In a Type I dualsuperconductor the electric flux tubes would attract, and pre-sumably merge. The energy per unit length of the merged fluxtubes would then be somewhat less than the sum of energiesper unit length of two flux tubes of minimal electric flux. Thedouble-winding Wilson loop falloff would then be a little lessthan sum-of-areas, but this slight difference would not affectour argument in any essential way.In the case of a D = x − y plane, with C at z = C at z = d z . Then, bythe standard manipulations introduced by Polyakov, we have h W ( C ) i = Z mon Z D c ( r ) exp h − g p Z d r (cid:16) ( ¶ m ( c − h S ( C ) ) − M cos c ( r ) (cid:17)i , (19)where − ¶ h S ( C ) = pd ′ ( z ) q S ( x , y ) + pd ′ ( z − d z ) q S ( x , y ) , (20)and q S ( ) ( x , y ) = x , y lie in the minimal area of C ( C ) ,and is zero otherwise. Assuming d z ≫ / M , an approximatesaddlepoint solution is the superposition c = sign z · ( e − M | z | ) q S ( x , y )+ sign ( z − d z ) · ( e − M | z − d z | ) q S ( x , y ) . (21)As d z → S , S to be dis-placed from one another in the z -direction, except near theloop boundaries. If we take this displacement to be d ≫ / M ,then (21) with d z → d is still an approximate solution for largeloops, where the areas of S , S are still nearly minimal, andnearly parallel to the x − y plane. In either case we have twomonopole-antimonopole sheets where the x , y coordinates of S , S coincide, and one sheet where x , y lies in S , but not in S . The result is a sum-of-areas falloff for the double-windingWilson loop. However, at d z = C and C , that was found in [24]. For x , y ∈ the minimal area of C , and d ≫ / M but small compared to the extension of theloop, the solution is c = q ( z ) ( e − M ( z − d ) )+ q ( − z )[ ( e − M ( z + d ) ) − p ] , (22)while for x , y ∈ the minimal area between C and C , the so-lution is the standard Polyakov soliton for a single-windingloop c = sign z · ( e − M | z | ) . (23)In both cases x , y are far from the loop perimeters. The resultis again a sum-of-areas falloff.For a monopole plasma in D = at stronglattice couplings, and the result is essentially a sum-of-areasfalloff.The Diakonov-Petrov calculation of spacelike Wilson loopsin D = + D = d z is greater than the thick-ness of this soliton. B. Difference of areas behavior
In the center vortex picture of confinement, and also instrong coupling lattice gauge theory, the behavior of thedouble-winding loops, whether co-planar or slightly shifted,is W ( C ) = a exp [ − s | A − A | ] . (24)The same difference-of-areas law is obtained in SU(3) puregauge theory, in the vortex picture and from strong-couplingexpansions, for a Wilson loop which winds twice around loop C and once around the co-planar loop C in the directionsindicated in Fig. 1. For simplicity, however, we will restrictour discussion to SU(2).It is assumed that the loops are so large that the thicknessof center vortices can be neglected. For co-planar loops, ifa vortex pierces the minimal area of loop A , it will multi-ply the holonomy around loop C by −
1, and also multiplythe holonomy around C by −
1, producing no effect whateveron the double-winding loop (unless the vortex crosses a loopperimeter, which can only result in a perimeter-law contribu-tion). So the vortex crossing can only produce an effect if itpierces the minimal area of C but not the minimal area of C (difference of areas A − A ). This supplies an overall factor of − C by d z in the transverse directiondoes not make any difference to the argument, providing thescales of A and A are so large compared to d z that a vortexpiercing the smaller area A is guaranteed to also pierce thelarger area A .The double-winding loop is also easily computed in strong-coupling SU(2) lattice gauge theory, with the result W ( C ) = −
12 exp [ − s | A − A | ] s = − log (cid:20) I ( b ) I ( b ) (cid:21) , (25)which is again a difference-of-areas law. A small shift d z inthe loop C will not affect this answer. The center vortexmodel does not pick up the same overall sign, but a modelwhich only considers center vortex contributions to large Wil-son loops is not complete enough to pick up either the perime-ter law behavior or any overall constant, but only the area-lawfalloff.Clearly the strong-coupling expansion and center vortexmodel, which both argue for a difference-in-area falloff for thedouble-winding Wilson loops, are in conflict with the predic-tions of monopole/dyon plasmas and the dual abelian Higgsmodel. So the next question is which prediction is actuallycorrect, away from the strong coupling limit. This is a ques-tion which can be answered by lattice Monte Carlo simula-tions. IV. SUM OR DIFFERENCE OF AREAS?
We will begin with a trivial example: the case where loops C = C = C are identical, so that the difference in areas iszero. We can then make use of an SU(2) group identityTr [ U ( C ) U ( C )] = − + Tr A U ( C ) , (26)where the trace on the right-hand side is in the adjoint repre-sentation. Since, apart from very small loops, h Tr A U ( C ) i ≪ W ( C ) ≈ − , (27)which is obviously consistent with difference-in-area behav-ior. For center-projected loops, the result is W ( C ) = C , which is again a trivial example of thedifference-in-area law. However, if we wish to test this lawin less trivial circumstances, where the difference in areas is The difference in sign compared to the unprojected result can be attributedto the neglect, in center projection, of fluctuations which make h Tr A U ( C ) i fall with a perimeter law. L δ L +2 FIG. 4. A coplanar, double winding contour. The trace of a Wilsonloop around this contour, divided by 2, will be denoted W ( L , d L ) . non-zero and the loop holonomy does not contain a singlet, itis necessary to consider contours with C = C .Consider the double-winding loop shown in Fig. 4, where C , C are coplanar, C is a square loop of length L , and C isa loop with sides of length L + d L , L + d L , L + d L , L + d L .We will denote the double-winding Wilson loop around thiscontour as W ( L , d L ) . Given that a single-winding planar loophas the behavior W ( C ) = exp [ − s Area − m Perimeter ] , a sum-of-areas falloff for the double-winding loop would give us W ( L , d L ) = a exp [ − AL − BL ] sum of areas , (28)while a difference-of-areas behavior gives W ( L , d L ) = a exp [ − BL ] difference of areas , (29)where A = s , B = sd L + m . (30)Because the expectation value of the double-winding loopsturns out to be negative, we will redefine W ( C ) for double-winding loops to be W ( C ) = − h Tr U ( C ) i , (31)where U ( C ) is the Wilson loop holonomy. We will also con-sider center projected and abelian projected double-windingloops, in maximal center and maximal abelian gauges. These,however, will be defined in the usual way, without the addi-tional minus sign.If log W ( L , d L ) is linear in L for fixed d L , then the behav-ior is difference-in-areas. Of course, as L increases at d L > m of the perimeterfalloff.In Fig. 5 we show our results for W ( L , ) (5(a)) and W ( L , ) (5(b)) vs. L , both at b = . lattice volume. For comparison, the resultsobtained from center projection in maximal center gauge are W ( L , ) L1 smeared SU(2)MCG projected (a) d L = W ( L , ) Lsmeared SU(2)MCG projected (b) d L = FIG. 5. Wilson loop expectation values W ( L , d L ) for the double-winding loops of Fig. 4. Note the minus sign convention (31) for full SU(2)loops with smeared links. For comparison, center-projected loop values are also shown. (a) d L =
1; (b) d L = also shown. In center projection the only excitations are thincenter vortices which, as already mentioned, must result in adifference-in-areas falloff, and therefore a linear dependenceof log [ W ( L , d L )] on L . This linear dependence is clearly seenin Fig. 5. The data for the smeared, unprojected links alsohas a linear dependence, albeit with a different slope. Theimportant point is that the data fits a straight line on a log-arithmic scale, indicating a difference-in-areas falloff. For asum-of-areas falloff, one expects the data to fall away fromthe straight line for the larger loops.Of course one may worry that our loops are not largeenough to see a sum-of-areas falloff, and that the behavior ofthe smaller loops is dominated by the perimeter term. To ad-dress this issue, consider the contour shown in Fig. 6, where L , L are fixed and we vary L . We denote the Wilson looparound this contour as W ( C × C ) , where C is the rectan-gular contour of area L × L . In this case the perimeter in-creases, and the sum-of-areas increases, as L is increased.So for a sum-of-areas falloff, W ( C × C ) must decrease as L increases. For a difference-of-areas falloff, there are twocompeting effects. The perimeter increases, but the differenceof areas decreases as L increases. If the area law falloff isthe dominant effect, then W ( C × C ) will actually increase as L increases. For loops composed of smeared links, andfor center-projected loops, this is exactly what happens, as wesee in Fig. 7. This increase of loop expectation value withincreasing L simply cannot occur for the sum-of-areas be-havior. Therefore the area-law falloff is the dominant effect,and the sum-of-areas behavior is definitely ruled out. Two effects can account for the difference in slope. First, for the unpro-jected links, there may still be a perimeter law contribution, although weexpect this to be reduced by smearing. Secondly, while the string tensionfor center-projected Wilson loops in SU(2) gauge theory is known to bevery close to the asymptotic value [26], even for the smallest loops, thisis not the case for unprojected loops, where the string tension (defined byCreutz ratios), only reaches the asymptotic value for relatively large loops(roughly 6 × b = . Another way to illustrate these results is to plot the val-ues of double-winding smeared SU(2) Wilson loops, of fixedperimeter P , vs. the difference in area A − A . This is shownin Fig. 8(b) for contours indicated in Fig. 8(a). Note that thepoints seem to cluster around a universal line, regardless ofperimeter. This is another indication that the perimeter contri-bution for the smeared loops is relatively small, compared tothe area law falloff. L L δ L δ L FIG. 6. Another coplanar double-winding loop. As L increases with L , L fixed, the sum-of-areas law would predict that the magnitude ofthe Wilson loop should decrease. W ( C x C ) L1 L2=1L2=2L2=3L2=4L2=5L2=6L2=7L2=8L2=9L2=10 (a) full SU(2), d L = W ( C x C ) L1 L2=1L2=2L2=3L2=4L2=5L2=6 (b) full SU(2), d L = W ( C x C ) L1 L2=1L2=2L2=3L2=4L2=5L2=6L2=7L2=8L2=9L2=10 (c) center projection, d L = W ( C x C ) L1 L2=1L2=2L2=3L2=4L2=5L2=6L2=7L2=8 (d) center projection, d L = FIG. 7. Data for loop expectation values on the double-winding loop contours of Fig. 6. Both unprojected SU(2) loops on smeared links(subfigures (a) at d L = , L =
10 and (b) at d L = , L = d L = , L = d L = , L =
9) are shown. The fact that Wilson loop values increase in magnitude as the sum of areas increases means that thesum-over-areas law is ruled out. C C (a) W ( C x C ) A - A P=20P=22P=24P=26P=28P=30 (b)
FIG. 8. Wilson loop expectation values W ( C × C ) at fixed perimeter P vs. difference in area (subfigure 8(b)), for the rectangular contoursshown in subfigure 8(a). Two sides of loops C and C overlap on the lattice, although they are drawn as slightly displaced. W ( L , ) L1 MAG projected (a) MAG, contour of Fig. 4, d L = W ( L , ) L1 MAG projected (b) MAG, contour of Fig. 4, d L = W ( C x C ) L1 L2=1L2=2L2=3L2=4 (c) MAG, contour of Fig. 6, d L = W ( C x C ) L1 L2=1L2=2L2=3 (d) MAG, contour of Fig. 6, d L = FIG. 9. Results for abelian-projected loop expectation values in maximal abelian gauge. Subfigures (a) and (b) correspond to Figs. 5(a) and5(b) for unprojected loops, respectively, on contours of the type shown in Fig. 4. The linear dependence of log W ( L , d L ) on L on suggests adifference-of-areas behavior. Subfigures (c) and (d) correspond to Figs. 7(a) and 7(b), respectively, on contours of the type shown in Fig. 6. Incontrast to the unprojected and center projected loops, the expectation values of the abelian projected loops mostly decrease with increasing L , although we see in 9(c) some indication that the data levels out for L > L . P ( A m ) defined in eq. (4), and if these loops would exhibita sum-of-areas behavior, whereas unprojected loops have adifference-of-areas behavior, it would mean that the abeliandominance assumption in eq. (1) is wrong. The W ( L , d L ) re-sults for abelian projection on the contour shown in Fig. 4 aredisplayed in Figs. 9(a) and 9(b). The data clearly indicatesa linear dependence in the logarithmic plot, consistent withthe difference-of-areas law. The abelian-projection results forthe contour of Fig. 6 are shown in Figs. 9(c) and 9(d). In thiscase, in contrast to the full and center-projected loops, the loopvalues initially decrease with increasing L . Since there is acompetition between perimeter law and area falloff for thiscontour, as already mentioned, the result does not necessarilyrule out a difference-of-areas falloff for the abelian projectedloops, and in fact in Fig. 9(c) there is some indication that thedata levels out as L increases, at the larger L values. Thereis no such indication in Fig. 9(d), although we think it is likelythat this data would also level out (and even begin to increase)for sufficiently large loop contours, as in Fig. 9(c). W ( C x C ) s h i f t e d Asmeared SU(2)MCG projectedMAG projected
FIG. 10. Planar loops C and C are parallel, as in Fig. 2, but dis-placed in a transverse direction by one lattice spacing. The two loopsare of equal area A , so the difference in area is zero. We see that W ( C × C ) for the unprojected SU(2) loops levels off at A ≈ Finally we consider loops of the type shown in Fig. 2, where C and C are displaced from one another in a transverse di-rection. Fig. 10 shows the results for W ( C × C ) vs. area A ,where C and C have equal minimal area A = A = A , andare displaced by one lattice spacing. The difference in areas inthis case is zero, and therefore we would expect W ( C × C ) to fall only with a perimeter law, for sufficiently large loops,as area A increases. In fact we clearly see this behavior for thefull SU(2) loops, where the data flattens out at A ≈
8. On theother hand we do not clearly see a leveling off for the abelianprojection loops in this range of loop area.
FIG. 11. For the same situation depicted in Fig. 3, insertion of a pos-itively and negatively charged W boson neutralizes the widely sep-arated positive and negative charges. Then there are only flux tubesbetween the positive static charges and the W −− , and (separately) thenegative static charges and the W ++ , leading to a difference-in-areaslaw. V. THE EFFECT OF W-BOSONS
We draw the obvious conclusion that if confinement canbe attributed, in some gauge, to the quantum fluctuations ofgauge fields in the Cartan subalgebra of the gauge group,then the spatial distribution of the corresponding abelian fieldstrength cannot follow any of the models discussed in sectionII. On the other hand, the models under consideration ne-glect the main feature which makes the underlying theory non-abelian, namely the off-diagonal gluons, also known as “W”-bosons. W-bosons are often ignored on the apparently reason-able grounds that these bosons are very heavy, and thereforecannot have a significant impact on low energy, long-rangephenomena, and in particular cannot affect the spatial distri-bution of confining fields at large scales.In fact it is easy to see how the W-bosons could change thedouble-winding falloff from a sum to a difference-in-areas be-havior. The process is illustrated in Fig. 11, where we see thatW-bosons can neutralize the two pairs of positively and nega-tively charged particles. Granting that point, imagine integrat-ing out those W-fields. This leaves us with a probability distri-bution for the abelian fields alone, as we have discussed in theIntroduction. But then, assuming that the difference-in-areaslaw is obtained, the resulting probability distribution P [ A m ] or P [ f mn ] for abelian fields in the vacuum must be very differ-ent from the distributions implied by the various models sum-marized in section II. This is because those models give thewrong sum-of-areas result. So in fact the W-bosons, despitetheir large mass, must have a dramatic effect on the spatial dis-tribution of abelian field strength at large scales. Clearly onecannot use the abelian field distributions of section II to arguefor an area law for ordinary Wilson loops, and then appeal tosome other distribution when confronted with double-windingloops. The same distribution of abelian fields must be used ineach case. This raises an obvious question: Can we imagine,even in principle, a set of abelian configurations which dom-inate vacuum fluctuations on large scales, and which wouldresult in a difference-of-areas law for double-winding loops?Abelian configurations which can satisfy that conditionwere proposed many years ago in ref. [27], and we recallthem here. Consider the field distribution at a fixed time,and suppose that, rather than being arranged in a monopole1 + + - +- + - +-+- +- + - FIG. 12. An example of monopole-antimonopole magnetic flux or-ganized into center vortices.
Coulomb gas, the monopoles and antimonopoles are arrangedin monopole-antimonopole chains, with the magnetic flux col-limated, from monopole to antimonopole, along the line ofthe chain. For the SU(2) example that we are discussing, themagnetic flux from monopole to antimonopole precisely cor-responds to the center element −
1. In other words, ratherthan being a monopole plasma, this is a vacuum consistingof center vortices, and the difference-in-area law follows. In D = r of fixed modulus | r | =
1, and Z = Z D r D q m exp " b (cid:229) p cos ( q ( p ))+ l (cid:229) x , m n r ∗ ( x ) e i q m ( x ) r ( x + b m ) + c.c o , (32)with b ≪ l ≫
1. In this case, rewritingthe theory in monopole variables actually obscures the un-derlying physics. The confining field configurations are nolonger Coulombic fields emanating from monopole charges.Rather, the confining configurations are thin Z vortices − afact which is invisible in the monopole formulation. To see this, go to the unitary gauge r =
1, which preserves a residual Z gauge invariance, and make the field decompositionexp [ i q m ( x )] = z m ( x ) exp [ i e q m ( x )] , (33)where z m ( x ) ≡ sign [ cos ( q m ( x ))] , (34)and Z = (cid:213) x , m (cid:229) z m ( x )= ± Z p / − p / d e q m ( x ) p exp " b (cid:229) p Z ( p ) cos ( e q ( p )) + l (cid:229) x , m cos ( e q m ( x )) . (35)This decomposition separates lattice configurations into Z vortex degrees of freedom (the z m ( x ) ), and small non-confining fluctuations around these vortex configurations,strongly peaked at e q =
0. One can easily show, for b ≪ , l ≫
1, that D exp [ in q ( C )] E ≈ h Z n ( C ) i D exp [ in e q ( C )] E , (36)with h Z n ( C ) i = (cid:26) exp [ − s A ( C )] n odd1 n even D exp [ in e q ( C )] E = exp [ − m n P ( C )] , (37)where Z ( C ) is the product of z m ( x ) link variables around theloop C . This establishes that the confining fluctuations, in thiscoupling range, are entirely due to thin vortices identified bythe decomposition (33) in unitary gauge. It is clear that theaddition of a charge-2 matter field has resulted in a qualitativechange in the physics of confinement. Yet the transition froma monopole Coulomb gas mechanism to a vortex dominancemechanism is essentially invisible if the gauge+matter theoryis rewritten in terms of monopole + electric current variables,which in this case tend to obscure, rather than illuminate, thenature of the confining fluctuations.A final remark is that when a caloron ensemble is subjectedto Laplacian center gauge fixing, certain gauge-fixing singu-larities appear, and it has been suggested that these singular-ities should be identified with center vortices [29]. Here wecan only note that, in the center vortex theory of confinement,center vortices are associated with a certain spatial distributionof confining flux; they are not merely singularities of somegauge fixing condition. In the dyon distribution advocated in[10] there is no apparent collimation of abelian fields into vor-tex structures, instead they diverge in a spherically symmetricmanner from the dyon centers. If this is a qualitatively accu-rate representation of the confining fields of a caloron ensem-ble, it is unlikely to be consistent with a center vortex mech-anism. It would be interesting to calculate double-windingloops numerically in the dyon ensembles advocated in [10]and also in the caloron ensembles of [6], where analytical re-2sults are not available. VI. CONCLUSIONS
We have shown that a number of popular models of con-finement due to abelian fields, namely the monopole plasma,dyon gas, and dual superconductor (dual abelian Higgs) mod-els, predict a sum-of-areas falloff for double-winding Wil-son loops which contradicts the difference-of-areas predictionof the center vortex model and strong coupling expansions,and, more importantly, contradicts the results of lattice MonteCarlo simulations. This means that these abelian models donot give the correct spatial distribution of confining abelianvacuum fluctuations. A difference-of-areas result can be ob-tained if one adds in off-diagonal gluons (“W-bosons”) to theabelian models, but this also implies that the spatial distribu-tion of abelian fields in models with W-bosons must be qual-itatively different from the corresponding distributions in amonopole plasma, dyon gas, or dual superconductor. We havesuggested that when W-bosons are added to such models, the result is a theory of center vortices (for some recent develop-ments, see [30, 31].) At least one must consider, in the con-text of models in which the confining fields are dominantlyabelian, what sort of distribution of confining abelian fieldstrength would be compatible with the difference-of-areas re-quirement for double-winding Wilson loops. A center vortexdistribution is one possibility; at present we are not aware ofany alternative.
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