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Flux-tubes in three-dimensional lattice gauge theories

arXiv:hep-lat/9303008v1 17 Mar 1993SFU HEP-101-93TRIUMF 93-15hep-lat/yymmnnMarch 1993Flux-tubes in three-dimensional lattice gauge theoriesHoward D. TrottierDepartment of Physics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6 ∗andTRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., Canada V6T 2A3R. M. WoloshynTRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., Canada V6T 2A3Measurements of flux-tubes generated by sources in different representationsof SU(2) and U(1) lattice gauge theory in three dimensions are reported.

Heavy“quarks” are considered in three representations of SU(2): fundamental (j = 1/2),adjoint (j = 1), and quartet (j = 3/2).Wilson loops are used to introduce astatic quark-antiquark (QjQj) pair.Several attributes of the fields generated bythe QjQj pair are measured. In particular, the first direct lattice measurements ofthe flux-tube cross-section Aj as a function of representation are made.

It is foundthat Aj ≈constant, to within about 10% (a rough estimate of the overall qualityof the data). The results are consistent with a connection between the string ten-sion σj and cross-section suggested by a simplified model of flux-tube formation,σj = g2j(j + 1)/(2Aj) [where g is the gauge coupling], given that the string tensionscales like the Casimir j(j + 1), as observed in previous lattice studies in both threeand four dimensions (and confirmed here up to the quartet representation).

Theseresults can be used discriminate among phenomenological models of the physics un-derlying confinement. Flux-tube measurements are also made in compact QED3,which exhibits electric confinement due to magnetic monopole condensation.

Singly-and doubly-charged Wilson loops are considered. The string tension is found to scalelike the squared-charge, and the flux-tube cross-section is found is be independent ofthe charge, to a good approximation.

The results of these three-dimensional SU(2)and U(1) simulations taken together lend some support, albeit indirectly, to a con-jecture that the dual superconductor mechanism underlies confinement in compactgauge theories in both three and four dimensions.1

I. INTRODUCTIONFlux-tube formation provides an attractive description of confinement in quantum chro-modynamics (QCD). In a simplified flux-tube picture of a very heavy quark-antiquark pair,color-electric field lines running between the quarks are assumed to be “squeezed” into acylinder whose cross-section is independent of the quark separation R (ignoring end effects),resulting in linear confinement.Consider heavy “quarks” in an arbitrary representation of the gauge group, which wetake to be SU(2) for convenience.

The color-electric field Ej between quarks in the j-threpresentation is determined, in an Abelian approximation, by Gauss’ Law [1] EjAj = gQj,where the quark color “charge” Qj is related to the group Casimir, Q2j = j(j + 1), g is thegauge coupling, and Aj is the cross-section of the flux tube. In this simplified model, Ej isassumed to be constant across the flux-tube cross-section.

In the case of non-fundamentalrepresentation sources, R is also assumed to be below the threshold for fission of the tube.The interaction energy of the system V intj(R) = 12E2j AjR = σjR, and thus the string tensionσj is given byσj = g2j(j + 1)2Aj. (1)In more a detailed flux-tube model, the color-field Ej may vary in magnitude across the flux-tube cross-section; the cross-section in Eq.

(1) is then defined by Aj ≡(R EjdAj)2/R E2j dAj;it is also possible to relax the constraint that Aj is independent of R.If the general features of the flux-tube model of confinement are consistent with QCD,then the connection Eq. (1) between σj and Aj should hold.

However, within the con-text of the flux-tube picture, Aj is an unknown function of the quark representation. Thecross-section is determined by the fundamental dynamics of QCD.

Flux-tube formation hasbeen observed in lattice QCD simulations in four dimensions [2,3,4,5,6,7] and as well as inthree dimensions [8]. However, previous flux-tube measurements have only been made forfundamental representation sources.This paper presents results of the first direct lattice measurements of Aj for static quarksin different representations of the gauge group.

This work is aimed in part at establishingthe connection Eq. (1) between the string tension and flux-tube cross-section.

As well, theresults obtained here go beyond the flux-tube model, providing important new informationabout the dynamics underlying confinement.For example, if confinement is due to a bulk property of the QCD vacuum, such as avacuum pressure, then Aj is expected to increase with representation (a natural “mechani-cal” response of a “medium” to the injection of more intense fields). Consequently, in sucha scenario, the string tension σj is expected to increase with representation less rapidly thanthe quark Casimir.

This situation is realized in a wide class of phenomenological models,including bag models [1,9], models based on a description of the QCD vacuum as a color-dielectric medium [10], and some models of confinement based on vacuum condensation[11].The MIT bag [9] is typical of this general class of models. In the case of the heavyquark-antiquark (QjQj) system described above, the magnitude of the color-electric field isdetermined by a balance between the pressure generated by the field and an external “bag”2

pressure B [1], 12E2j = B. It follows that Aj ∝Qj, and the resulting string tension σj alsoscales as the square root of the Casimir, σj ∝(j(j + 1))1/2.In fact, the phenomenological models discussed above are not compatible with latticesimulations, which have shown that the string tension actually scales to an excellent ap-proximation like the Casimir of the representation.

This has been observed for both SU(2)and SU(3) gauge groups [12]. In the context of the flux-tube model, this suggests that thecross-section Aj is independent of representation.Direct measurements of the flux-tube cross-section for heavy quark-antiquark (QjQj)sources are obtained here in three representations of three-dimensional SU(2) lattice gaugetheory: fundamental (j = 1/2), adjoint (j = 1), and quartet (j = 3/2).

A more thoroughcheck of scaling and finite volume effects is achieved by working in three dimensions thanwould be obtained (with the same computing power) in four dimensions. We think that ourresults are relevant to the problem of confinement in four-dimensional QCD.

In particular,previous lattice studies have shown that the string tension scales like the Casimir of therepresentation in three dimensions [13,14], as well as in four dimensions [12]. Moreover, theflux-tube picture of confinement is qualitatively the same in both three and four dimensions.It is therefore reasonable to expect that the qualitative features of flux-tubes reported herewould be reproduced in four dimensions.

Of course, such a calculation can and should bedone.We find Aj ≈constant for the three representations, to within about 10% (a roughestimate of the overall quality of our data). This is consistent with the flux-tube picture,given that the string tension scales like the Casimir of the representation (which is confirmedhere up to the quartet representation).

Several additional qualitative features of the flux-tube picture are also verified.These results suggest a connection between confinement in QCD and the physics ofa dual superconductor. Indeed, if a multiply-charged monople would be inserted into anordinary (type II) superconductor, all the quanta of magnetic flux would be carried by asingle flux-tube, whose diameter is fixed by the penetration depth [15].

A pair of monopolesof opposite sign would therefore be confined, with a string tension that would scale like thesquared-charge.It is well-known that dual superconductivity (magnetic monopole condensation) resultsin confinement of electric charges in compact QED in three-dimensions (QED3) [16]. Asimple extension of the analytical calculation of Ref.

[17] in the Villain approximation tothe Wilson action, to include Wilson loops for multiply-charged sources, demonstrates thatthe string tension scales like the squared-charge.We have performed lattice simulations of singly- and doubly-charged Wilson loops incompact QED3, and our results confirm the expected scaling properties of the string tensionand flux-tube cross-section. The potential is found to scale like the squared-charge to withina few percent, and the flux-tubes in the two cases have the same cross-section to withinabout 10%.

The results of our three-dimensional SU(2) and U(1) simulations taken togetherlend some support, albeit indirectly, to the dual superconductor picture of confinement infour-dimensional QCD [18].3

II. METHODTo begin with, we consider the three-dimensional SU(2) lattice theory.

Wilson loopsare used to introduce static QjQj sources. Lattice measurements of the color-electric and-magnetic fields generated by these sources are obtained from correlators F µνjof plaquetteswith a Wilson loopF µνj (x) ≡−βa3DWj12Tr Uµν(x)E⟨Wj⟩−D12Tr UµνE,(2)where Uµν(x) is the plaquette located at x (measured relative to the center of the Wilsonloop), and Wj is the normalized trace of the Wilson loop in the j-th representation:Wj ≡12j + 1TrYl∈LDj[Ul].

(3)Dj[Ul] denotes an appropriate irreducible representation of the link Ul, and L the closedloop. β ≡4/(g2a), where the coupling constant g has dimensions of (mass)1/2 in threedimensions.In the continuum limit, the trace of a 1 × 1 plaquette is by construction independentof representation (up to overall normalizations).

As in several previous lattice calculationsof higher representation Wilson loops (cf. Refs.

[12,13,14]), we use the action expressed interms of links in the fundamental representation to perform simulations at arbitrary β. Thetrace of the plaquette Uµν in the fundamental representation is also used to compute thecorrelators of Eq. (2).In the continuum limit the correlator F µνjcorresponds to the expectation value of thesquare of the Euclidean field strength F aµν = ∂µAaν −∂νAaµ + gǫabcAbµAcν,limβ→∞F µνj= 12DXa(F aµν)2EQjQj −12DXa(F aµν)2E0(4)where the expectation value ⟨.

. .⟩QjQj is taken in a state with external sources in the j-threpresentation, and ⟨.

. .⟩0 is the vacuum expectation value.To compute the energy density, the Euclidean 3-axis is identified with a temporal sideof the Wilson loop, and the 1-axis with a radial side.

We separate contributions to thetotal energy density Etot corresponding to the two spatial components of the color-electricfield (in the directions parallel and perpendicular to the line joining the quarks), and thecolor-magnetic field (a scalar in three dimensions):Etotj (x) = E∥j (x) + E⊥j (x) + EBj (x),(5)whereE∥j (x) ≡−F 13j (x),E⊥j (x) ≡−F 23j (x),(6)EBj (x) ≡F 12j (x).4

Notice the relative minus sign between the electric and magnetic components of the Eu-clidean energy density.Previous flux measurements for fundamental representation sources have been made infour dimensions [2,3,4,5,6,7], and in three dimensions [8]. Following Haymaker and Wosiek[5], we achieve a significant enhancement in the signal to noise for the correlators by replacingEq.

(2) with:F µνj (x) ≈−βa3DWj12Tr {Uµν(x) −Uµν(xR)}E⟨Wj⟩,(7)where xR is a reference point chosen far enough from the Wilson loop that the factorization⟨WjU(xR)⟩≃⟨Wj⟩⟨U⟩is satisfied. As in Ref.

[5], we find that this happens well withinthe lattice volume. We verified explicitly that the right-hand side of Eq.

(7) is insensitive,within our statistical errors, to variations in xR over a wide range (when measurements aremade for x in a region around the Wilson loop of sufficient size to suit our purposes). Theresults presented here were obtained with xR taken at a distance of half the lattice size fromthe center of the Wilson loop, in the direction transverse to the plane of the loop.

Theadvantage to using Eq. (7) is that the fluctuations in the product WjUµν, due mainly tothe Wilson loop, tend to cancel in the vacuum subtraction when computed configuration byconfiguration.Another reduction in the statistical errors is readily achieved by performing some linkintegrations analytically, following the multihit procedure introduced by Parisi, Petronzioand Rapuano [19].

Consider a link variable Ul which appears linearly in the observableof interest. The simplest analytical integration over Ul takes account of nearest neighborcouplings in the action:Z[dUl]Dj[Ul]eβTr(UlK†l )/2 = I2j+1(βkl)I1(βkl) Dj[Vl]Z[dUl]eβTr(UlK†l )/2(8)where Kl is the sum of the four “staples” coupling to the link of interest Ul, andklVl ≡Kl,det Vl = 1.

(9)A further variance reduction has been developed by Mawhinney [14], which takes accountof effective next-to-nearest neighbor interactions with the link of interest1Z[dUl]Dj[Ul]e−βS = I2j+1(βk′l)I1(βk′l)I2j+1(βkl)I1(βkl) Dj[V ′l Vl]Z[dUl]e−βS,(10)where S is the action [only next-to-nearest neighbor couplings to Ul are relevant in Eq. (10)],and1Mawhinney derived next-to-nearest neighbor variance reductions for fundamental and adjointrepresentations by employing an axial gauge-fixing [14].

We have generalized his result to arbitraryrepresentations without gauge fixing.5

k′lV ′l ≡Xˆl⊥Uˆl⊥(x)Xˆµ̸=ˆl⊥Ulˆµ(x + ˆl⊥) U†ˆl⊥(x),det V ′l ≡1. (11)ˆl⊥are four unit vectors perpendicular to ˆl.

The oriented plaquette Ulˆµ(x + ˆl⊥) is computedwith the link Ul(x + ˆl⊥) appearing first on the left [x is the position of the base of the linkUl in Eq. (10)].

The sum over unit vectors ˆl⊥and ˆµ in Eq. (11) is taken over both paralleland antiparallel orientations with respect to a set of fixed basis vectors.

An integration overthe four links Ul(x + ˆl⊥) is implicit in Eq. (10).The second order variance reduction of Eq.

(10) cannot be applied to the links in thecorners of the Wilson loop, since some links would then appear more than once in theintegrand. Likewise, the first-order variance reduction Eq.

(8) can only be applied to onelink in a corner. Further restrictions apply to calculations of the plaquette correlators.For Wilson loops with less than six links on a side, we use the first-order variancereduction Eq.

(8) for all links in the loop, except for one link at each corner, where novariance reduction is used. A plaquette correlator can be measured simultaneously providedthat all sides of the plaquette are at least one node from the sides of the Wilson loop.For Wilson loops of size 6×6 or larger, we minimize the variance by using a combinationof first- and second-order variance reductions.

Equation (8) is appplied to the first link andto the second-to-last link on each (oriented) side of the Wilson loop; Eq. (10) is applied toall other links, except the last link on each side, where no variance reduction is used.

Inthis case, a correlator can be measured simultaneously only if all sides of the plaquette areat least two nodes from the sides of the Wilson loop. [This variance reduction scheme canalso be applied to Wilson loops as small as 4 × 4 if measurements of correlators near thecenter of the loop are not desired.

This scheme is significantly more effective than the oneemployed in Ref. [14], which uses only second-order variance reductions.

]For large β, kl ∼4 and k′l ∼12; Eq. (10) then provides an estimate of the reduction vredin the variance of a Wilson loop using the above scheme, compared to the variance whenonly “unreduced” links are used (cf.

Refs. [20,14])vred ≈ I2j+1(12β)I1(12β)!

(2T+2R−12) I2j+1(4β)I1(4β)!(2T+2R−4). (12)For example, the variance reduction for a Wilson loop of size 6 × 6 in the quartet represen-tation at β = 10 is estimated to be a factor of ≈90.

Our numerical results are consistentwith Eq. (12).The trace of an element of the group in the j-th representation can be expressed in termsof its trace in the fundamental representation using trigonometric relations among the groupcharacters.

In the case of the adjoint and quartet representations [21]:W3/2 = 2W 31/2 −W1/2,W1 =4W 21/2 −1/3. (13)Hence one need only compute the Wilson loop in the fundamental representation, using the“unreduced” links Ul, or the “reduced” elements Vl, V ′l of Eqs.

(8) and (10), as the case maybe. The Wilson loops in higher representations then follow from Eq.

(13). The Bessel func-tions for the analytical integrations are tabulated separately for the three representations.6

III. RESULTS AND DISCUSSIONOur main results were obtained on a 323 lattice at β = 10 (which is well within thescaling region for the string tension on a lattice of this size [14]).

Wilson loops and plaquettecorrelators were calculated in the three representations j = 1/2, 1, and 3/2 for all loops ofsizes T × R from 3 × 4 to 8 × 8 (these observables were measured in groups in severalseparate runs). Some additional data was taken at β = 14 in order to check for scaling ofthe physical flux-tube dimensions.

A standard heat-bath algorithm was employed. Morethan 10,000 sweeps were typically used for thermalization.

2,000 measurements were made,taking 20 sweeps between measurements. The resulting integrated autocorrelation times τintfor the Wilson loops generally satisfy τint <∼1, consistent with the results of a systematicstudy made in Ref.

[14]. Estimates of the statistical errors were obtained using the jackknifemethod.

However, measurements of different observables (and of a given observable in thethree representations) tend to be strongly correlated, since many Wilson loops and plaquettecorrelators were measured simultaneously on a given lattice.The quartet representation is much more difficult to measure than the two lower repre-sentations, due to the exponential suppression of the Wilson loop with the QjQj potential,which is found to scale with the Casimir of the representation. Energy density measurementsin the quartet case obtained from loops larger than about 6×6 are of poor quality, althoughthese data are consistent with conclusions drawn from results obtained from smaller loops.Representative data for Wilson loops in the three representations are shown in Fig.

1.Earlier studies have shown that the potentials scale with the Casimir of the representation atessentially all lengths scales R [13,14]. This is made evident in Fig.

1, where the logarithmsof the Wilson loops are scaled by a ratio of Casimirs,cj ≡3/4j(j + 1). (14)The quantity −ln⟨Wj(T, R)⟩/T, which extrapolates to the QjQj potential Vj(R) in thelimit T →∞, is found to scale as j(j + 1) to within a few tenths of a percent at all Tand R considered here.

A simple extrapolation of the data using Vj(R) ≈ln[⟨Wj(Tmax −1, R)⟩/⟨Wj(Tmax, R)⟩], where Tmax is the largest T value in the data set, gives agreement toa few tenths of a percent with the results of a careful statistical analysis of fundamental andadjoint Wilson loops reported in Ref. [14].Several attributes of the plaquette correlators were measured.

To begin with, results forthe fundamental and adjoint representations are presented. The correlators were measuredover a range of distances x⊥from the center of the Wilson loop, in the direction normalto the plane of the loop.

Results for the T × R = 8 × 6 loop are shown in Fig. 2.

Thecross-sections of the fundamental and adjoint representation flux-tubes are indistinguishablewithin statistical errors. This is true for all Wilson loops that were considered.

For example,the T evolution of E∥j for R = 6 Wilson loops is illustrated in Fig. 3.

As observed in Refs. [4,6], the plaquette correlators are more sensitive to higher states than the Wilson loop.

Ourdata are consistent with a one-excited-state parameterization given in Ref. [6].Figure 2 demonstrates that the component of the color-electric field parallel to the linejoining the charges dominates the energy, as assumed in the flux-tube model.

The magneticenergy turns out to be negative, which has also been observed in four-dimensional SU(2)7

lattice theory [5]. The formation of a well-defined flux-tube is demonstrated by measure-ments of E∥j in the plane of the Wilson loop.

Figure 4 shows E∥j for the T × R = 6 × 8 loopas a function of the longitudinal distance x∥of the plaquette centroid from the center ofthe loop. Notice the approximate symmetry of the energy density about the center of theloop.

The formation of the flux-tube is further illustrated in Fig. 5, where E∥j is shown as afunction of the radial separation R of the Wilson loop (for fixed T = 6)A stringent test of energy density calculations using Eq.

(7) is provided a sum rulederived by Michael [22]a2 X⃗xEtotj (⃗x) = Vj(R). (15)The analogous sum rule in four-dimensional SU(2) was studied in detail by Haymaker andWoseik [5].The flux-tube picture suggests a related sum rule that is much simpler tomeasure.

If the interaction energy is dominated by a constant color-electric field along theline joining the charges (as expected in the limit of quark separations much greater than theflux-tube thickness), then the integral of the energy density along one transverse “slice” ofthe flux-tube should equal the string tension [cf. σj = limR→∞(Vj(R) −Vj(R −a))/a]:aX|x⊥|E∥j (x⊥, x∥= fixed) ≈σj,(16)where the sum is taken over positive and negative distances x⊥from the plane of the Wilsonloop.Our results are in good agreement with Eq.

(16). Figure 6 shows the left-hand-side ofthis equation for the T × R = 8 × 6 loop in the fundamental and adjoint representations,using a variable cutoffx∗⊥on the sum.

The right-hand-side of Eq. (16) is illustrated by thedashed line in Fig.

6 (our estimates of the string tension in the two representations agreewith Ref. [14] to within a few percent).

These results again demonstrate that the flux-tubecross-sections for the fundamental and adjoint repesentations are indistinguishable withinstatistical errors.Figures 2–6 also demonstrate that the local energy densities scale to a good approxima-tion like the Casimir of the representation throughout the flux-tube. This is a very strongtest of the validity of the flux-tube model for Aj = constant.

Since the magnitude of thecolor-electric field varies across the flux-tube cross-section (cf. Fig.

2), a proper determi-nation of the numerical value of Aj should be made in terms of expectation values of thecolor-field, as described below Eq. (1) [some prescription for defining the Abelian projectionof the color-field would also be required].

However, a rough estimate Aj ≈8a inferred fromFig. 6 is consistent with Eq.

(1), given the estimate of the string tension cjσj ≈0.14g4 (cf.Ref. [14]).The cross-section is also found to be approximately independent of R. A similar con-clusion was reached in four dimensions in Ref.

[5]. On the other hand, the cross-section inthe strong coupling limit in four dimensions is found to increase logarithmically with R [24].Within statistical errors the range in R considered here is not sufficient to rule out such aweak dependence on the radial separation.Our measurements of the quartet representation (j = 3/2) correlators are consistent withthe above results.

We compare data in the three representations taken from the T ×R = 5×68

Wilson loop: energy density profiles transverse to the flux-tube are shown in Fig. 7, andthe sum rules Eq.

(16) in Fig. 8.

Data obtained from larger Wilson loops are consistentwith these results although, as mentioned above, the quartet data for larger loops are ofpoor quality, due to an exponential suppression of the Wilson loop with the Casimir of therepresentation.We checked for scaling of the physical flux-tube dimensions by running at β = 14. Thesum rule Eq.

(16) for the fundamental representation is compared at the two values of β inFig. 9.

The cutoffx∗⊥on the sum is expressed here in units of the physical coupling constantg. The data at β = 10 are for a 6 × 6 Wilson loop, while the data at β = 14 are for an8 × 8 loop.

These Wilson loops have roughly the same dimensions in physical units (T andR measured in units of 1/g2).These results show good evidence for scaling in the energy density and flux-tube cross-section (scaling is also observed in our adjoint and quartet representation data). However,the factorization assumed in Eq.

(7) breaks down in the β = 14 data at the largest values ofthe cutoffx∗⊥shown in Fig. 9 (x∗⊥∼9.5 in lattice units, to be compared with xR = 16); thesum is found to diverge linearly with x∗⊥at large cutoffs.

A similar behavior was observedin four-dimensional lattice calculations in Ref. [5], where a correction for this effect wasproposed.Nevertheless, scaling of the cross-section is clearly supported by data in theregion x∗⊥<∼1.8/g2.As described in the Introduction, the results of our SU(2) simulation suggest a connectionbetween confinement in QCD and the physics of a dual superconductor.

In this connection,we have calculated Wilson loops Wn in compact QED3 for singly- and doubly-charged sources(n = 1, 2):Wn ≡ReYl∈L(Ul)n,(17)where the phase Ul for the link l defines the singly-charged representation (i.e., Ul is thephase used to compute the Wilson action).The string tension is expected to scale likethe squared-charge, as demonstrated by an extension of the Villain approximation used inRef. [17] to include multiply-charged Wilson loops.

As in an ordinary superconductor, theflux-tube cross-section is expected to be independent of source charge.Wilson loops and plaquette correlators were measured on a 323 lattice at β = 2.4. Morethan 10,000 sweeps were used for thermalization, and 1,000 measurements were made (90sweeps were taken between measurements).

Variance reduction methods similar to thoseused in our SU(2) simulations were employed.2Results for the Wilson loops are given in Fig. 10.

Some data for the triply-chargedWilson loop (n = 3) are also shown (useful measurements of the plaquette correlators forn = 3 would require much larger statistics). Estimates of the potential for n = 1 obtainedfrom a simple extrapolation of these data are consistent with results presented in graphicalform in Ref.

[23].2The analytical integrals given in Eqs. (8) and (10) are easily adapted to the U(1) theory.The SU(2) link Dj[Ul] becomes (Ul)n, the Bessel function ratios I2j+1(x)/I1(x) are replaced byIn(x)/I0(x), and det(V ) becomes abs(V ).9

We find that −ln ⟨Wn(R, T)⟩/T, which extrapolates to the potential Vn(R) in the limitT →∞, scales like n2 to within about 2% for all T and R that were considered, in goodagreement with the expected scaling properties of the string tension. However, the deviationfrom n2 scaling is about an order of magnitude larger than the statistical errors in the data.String vibrational modes are known to make a significant contribution to V1 in the rangeof R considered here (lattice simulations [23] are in agreement with theoretical expectations[24]).

Simple arguments [13] suggest that the vibrational term in Vn may scale like n, whichcould account for the small deviation from n2 scaling in the logarithms of the Wilson loops.The energy sum rule analogous to Eq. (16) for the T × R = 5 × 5 loop is shown in Fig.11.

The dashed line shows the n = 1 string tension taken from Ref. [23].

These resultsprovide the first direct evidence from lattice simulations that the flux-tube cross-section incompact QED3 is independent of the source charge, the expected behavior in the case of a(dual) superconducting medium.IV. SUMMARYThe first direct measurements of the flux-tube cross-section as a function of representa-tion in SU(2) lattice gauge theory were made.

We found Aj ≈constant, to within about 10%(a rough estimate of the overall quality of our data) for the three representations j = 1/2,1, and 3/2. Our results are consistent with a connection between the string tension andcross-section suggested by a simplified model of flux-tube formation, σj = g2j(j + 1)/(2Aj),given that the string tension scales like the Casimir j(j + 1), as observed in previous latticestudies in both three and four dimensions (and confirmed here up to the quartet represen-tation).

We also confirmed several additional qualitative features of the flux-tube picture ofcolor-electric confinement. These results can be used discriminate among phenomenologicalmodels of the physics underlying confinement.

For example, many models in which confine-ment is due to a bulk property of the QCD vacuum (such as a vacuum pressure) predict asufficiently rapid increase in Aj with representation as to be incompatible with the resultsobtained from our lattice simulations.We also made flux-tube measurements in compact QED3, which exhibits electric confine-ment due to magnetic monopole condensation. We considered singly- and doubly-chargedWilson loops.

The string tension was found to scale like the squared-charge, and the flux-tube cross-section was found to be independent of the charge, to a good approximation.The results of our three-dimensional SU(2) and U(1) simulations taken together lend somesupport, albeit indirectly, to a conjecture that the dual superconductor mechanism underliesconfinement in compact gauge theories in both three and four dimensions. This conclusionis also supported by the results of a recent study of dual Abrikosov vortices in an Abelianprojection of SU(2) lattice gauge theory in four dimensions [25].

Flux-tube measurementsin four-dimensional SU(2) gauge theory similar to those reported here should be made inorder to further explore this possibility.10

ACKNOWLEDGMENTSThis work was supported in part by the Natural Sciences and Engineering ResearchCouncil of Canada.11

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B240, 189 (1984); C. Michael, Nucl. Phys.

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150B, 196 (1985); N.A. Campbell, I. H. Jorysz, and C. Michael, ibid.

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Phys. B240, 533 (1984).

[14] R. D. Mawhinney, Phys. Rev.

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Phys. B61, 45 (1973).

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Phys. B129, 493 (1977).

[18] G. ’t Hooft, in High Energy Physics, Ed. A. Zichichi (Editrice Compositori, Bologna,1976); S. Mandelstam, Phys.

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Lett. 128B, 418 (1983).

[20] C. Michael, Nucl. Phys.

B259, 58 (1985). [21] See, for example, K. Redlich and H. Satz, Phys.

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B280 [FS18], 13 (1987). [23] A high statistics study of the n = 1 potential was made in R. J. Wensley and J. D. Stack,Phys.

Rev. Lett.

63, 1764 (1989). See also A. Irb¨ack and C. Peterson, Phys.

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See also P. Ceaand L. Cosmai, BARI preprint TH 110/92, for an investigation of the dual Meissner effect.12

FIGURESFIG. 1.

T evolution of Wilson loops in three representations of SU(2) lattice gauge theory:j = 1/2 (◦), j = 1 (✷), and j = 3/2 (△). cj is a ratio of Casimirs, defined in Eq.

(14). Thequantity −ln⟨Wj(T, R)⟩/T extrapolates to the QjQj potential in the limit T →∞.FIG.

2. Energy density profiles transverse to the plane of the T ×R = 8×6 Wilson loop [j = 1/2(◦), and j = 1 (✷)].

These results are an average over plaquettes with centroids at distances ±x⊥transverse to the plane of the loop.FIG. 3.

T evolution of E∥j (x⊥= 0, 2a) for Wilson loops with R = 6 [j = 1/2 (◦), j = 1 (✷)].FIG. 4.

Energy density profile in the plane of the T × R = 6 × 8 Wilson loop [j = 1/2 (◦),j = 1 (✷)]. x∥is the distance of the centroid of the plaquette from the center of the Wilson loop.The radial sides of the Wilson loop are located at x∥= ±4a.

Plaquettes with a side touching theWilson loop cannot be measured using the variance reduction of Eq. (8), and are not shown.FIG.

5. Energy density E∥j (x⊥= 0, 2a) as a function of R, for fixed T = 6 [j = 1/2 (◦), j = 1(✷)].FIG.

6. Energy sum rule Eq.

(16) for the T × R = 8 × 6 Wilson loop [j = 1/2 (◦), j = 1 (✷)].The sum in Eq. (16) is evaluated using a variable cutoffx∗⊥.

The dashed line shows the scaledstring tension cjσj, estimated to about 5% (cf. Ref.

[14]).FIG. 7.

Energy density profile transverse to the plane of the T × R = 5 × 6 Wilson loop, in thefundamental and quartet representations [j = 1/2 (◦), j = 3/2 (△)].FIG. 8.

Energy sum rule Eq. (16) for the T ×R = 5×6 Wilson loop, in all three representations[j = 1/2 (◦), j = 1 (✷), j = 3/2 (△)].

The sum rule improves with increasing T and R (cf. Fig.6).FIG.

9. Scaling of the energy sum rule Eq.

(16) for the fundamental representation. The opendata points were taken at β = 10 (for a 6 × 6 Wilson loop), and the filled points at β = 14 (for an8 × 8 loop).

The cutoffx∗⊥is expressed here in units of the physical coupling constant g.FIG. 10.

T evolution of multiply-charged Wilson loops in compact QED3 [n = 1 (◦), n = 2(✷), and n = 3 (△)].13

FIG. 11.

Energy sum rule Eq. (16) for singly- and doubly-charged sources in compact QED3,for the T × R = 5× 5 Wilson loop [n = 1 (◦), n = 2 (✷)].

The dashed line shows the string tensionσn=1, estimated to about 10% in Ref. [23].14

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