Double-winding Wilson loops in SU(N) lattice Yang-Mills gauge theory
CCHIBA-EP-244, KEK Preprint 2019-48
Double-winding Wilson loops in SU ( N ) lattice Yang-Mills gauge theory Seikou Kato, ∗ Akihiro Shibata, † and Kei-Ichi Kondo ‡ Oyama National College of Technology, Oyama 323-0806, Japan Computing Research Center, High Energy Accelerator Research Organization (KEK), Oho 1-1, Tsukuba 305-0801, Japan Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan
We study double-winding Wilson loops in SU ( N ) lattice Yang-Mills gauge theory by using bothstrong coupling expansions and numerical simulations. First, we examine how the area law falloffof a “coplanar” double-winding Wilson loop average depends on the number of color N . Indeed,we find that a coplanar double-winding Wilson loop average obeys a novel “max-of-areas law” for N = 3 and the sum-of-areas law for N ≥
4, although we reconfirm the difference-of-areas law for N = 2. Second, we examine a “shifted” double-winding Wilson loop, where the two constituentloops are displaced from one another in a transverse direction. We evaluate its average by changingthe distance of a transverse direction and we find that the long distance behavior does not dependon the number of color N , while the short distance behavior depends strongly on N . PACS numbers: 12.38.Aw, 21.65.Qr
I. INTRODUCTION
What is the true mechanism for quark confinement isnot yet confirmed and still under the debate, althoughmore than 50 years have passed since quark model wasproposed by Gell-Mann [1] in the beginning of 1960s. Inthe 1970s, however, the dual superconductor picture wasalready proposed by Nambu, ’t Hooft and Mandelstam[2] as a mechanism for quark confinement. In fact, va-lidity of the dual superconductor picture was confirmedfor U (1) pure gauge theory [3], Georgi-Glashow model[4] and N = 2 supersymmetric Yang-Mills theory [5],although it is not yet confirmed for the ordinary non-supersymmetric Yang-Mills theory [6] and quantum chro-modynamics (QCD). Therefore, the dual superconductorpicture is now regarded as one of the most promising sce-narios for quark confinement, although this does not denythe existence of the other mechanics for quark confine-ment. See e.g., [7–9] for reviews.In order to establish the dual superconductor scenario,the most difficult issue to be resolved first of all is to guar-antee the existence of magnetic monopoles in the purenon-Abelian Yang-Mills gauge theory, which is differentfrom the ’t Hooft–Polyakov magnetic monopole [10] inthe gauge-scalar model. This issue was circumvented byusing the method called the Abelian projection proposedby ’t Hooft [11]. The Abelian projection is a gauge fix-ing which explicitly breaks the original gauge group intoits maximal torus subgroup where color symmetry is alsobroken. By the Abelian projection, magnetic monopolesof the Abelian type [12, 13] are indeed realized, but theresulting theory is distinct from the original gauge theorywith the non-Abelian gauge group. To avoid the gaugeartifact, we must find a procedure which enables one ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] to define magnetic monopoles in a gauge-invariant way.This issue was solved recently for the Yang-Mills theorywith the gauge group SU ( N ) and any semi-simple com-pact gauge group [14], by using the non-Abelian Stokestheorem for the Wilson loop operator and the new re-formulation of the Yang-Mills theory based on the newfield variables obtained by change of variables throughthe gauge covariant field decomposition of the Cho-Duan-Ge-Faddeev-Niemi-Shabanov [15–22]. See [9] for a recentreview.However, these achievements do not necessarily meansthat the dual superconductivity is the unique scenario forunderstanding quark confinement. Recently, Greensiteand H¨ollwieser [23] introduced a “double-winding” Wil-son loop operator in lattice gauge theory [24] to examinepossible mechanisms for quark confinement. The double-winding Wilson loop operator W ( C = C × C ) is a path-ordered product of (gauge) link variables U (cid:96) ∈ SU ( N )along a closed contour C which is composed of two loops C and C , W ( C ) ≡ tr[ (cid:89) (cid:96) ∈ C U (cid:96) ] , C = C × C . (1)See Fig.1. A more general “shifted” double-winding loopis introduced in such a way that the two loops C and C lie in planes parallel to the x − t plane, but are displacedfrom one another in the transverse direction, e.g., z bydistance R , and are connected by lines running parallelto the z -axis. In the non-shifted case R = 0, the twoloops C and C lie in the same plane, which we call coplanar . We denote by S and S the minimal areasbounded by loops C and C , respectively. Note thatthe double-winding Wilson loop operator is defined in agauge invariant manner, irrespective of shifted R (cid:54) = 0 orcoplanar R = 0.In [23], they investigated the area ( S and S ) depen-dence of the expectation value (cid:104) W ( C = C × C ) (cid:105) of adouble-winding Wilson loop operator W ( C = C × C )for the SU (2) gauge group. Consequently, it has beenshown in a numerical way that both the original SU (2) a r X i v : . [ h e p - l a t ] A ug C S S C xtz R FIG. 1: The double-winding Wilson loops. (left) A “shifted”double-winding Wilson loop W ( C = C × C ) composed of thetwo loops C and C which lie in planes parallel to the x − t plane, but are displaced from one another in the z -directionby distance R . (right) a “coplanar” double-winding Wilsonloop W ( C = C × C ) as the limit R = 0 of the “shifted”double-winding Wilson loop. (cid:62)(cid:62)(cid:1006) (cid:62) (cid:18445) (cid:631)(cid:62)(cid:631)(cid:62) (cid:631)(cid:62) FIG. 2: The setting up of a coplanar double-winding Wilsonloop. lattice gauge theory and center vortex model obey the difference-of-areas ( S − S ) law, while the Abelian-projected model obeys the sum-of-areas ( S + S ) law.In the coplanar case R = 0, a double-winding loop hasbeen set up as given in Fig.2. In order to discriminatedifference-of-areas and sum-of-areas laws, it is efficient tomeasure the L -dependence of a coplanar double-windingWilson loop average (cid:104) W ( C = C × C ) (cid:105) , with the otherlengths L , L , and δL being fixed. For simplicity, we set δL = 0. Then S (= L × L ) and S (= L × L ) are theminimal areas of rectangular loops C and C , respec-tively. We assume S ≥ S for definiteness hereafter. If (cid:104) W ( C × C ) (cid:105) obeys the difference-of-areas law: (cid:104) W ( C × C ) (cid:105) (cid:39) exp[ − σ | S − S | ]= exp[ − σL ( L − L )] , (2)then ln (cid:104) W ( C × C ) (cid:105) must linearly increase in L as L increases. On the other hand, if (cid:104) W ( C × C ) (cid:105) obeys thesum-of-areas law: (cid:104) W ( C × C ) (cid:105) (cid:39) exp[ − σ (cid:48) ( S + S )]= exp[ − σ (cid:48) L ( L + L )] , (3)then ln (cid:104) W ( C × C ) (cid:105) must linearly decrease in L as L increases. W ( C x C ) L1 L2=1L2=2L2=3L2=4L2=5L2=6L2=7L2=8L2=9L2=10 W ( C x C ) L1 L2=1L2=2L2=3L2=4L2=5L2=6L2=7L2=8L2=9L2=10 W ( C x C ) L1 L2=1L2=2L2=3L2=4
FIG. 3: L dependence of a coplanar double-winding Wilsonloop average (cid:104) W ( C = C × C ) (cid:105) (top panel) for the origi-nal SU (2) field, [reproduced from Fig.7.(a) in [23]], (middlepanel) for center vortex [reproduced from Fig.7.(c) in [23]],(bottom panel) for Abelian degree of freedom, [reproducedfrom Fig.8.(c) in [23]]. The numerical evidences were obtained as given inFig.3 which summarizes their results for L dependenceof ln (cid:104) W ( C × C ) (cid:105) with the other lengths being fixed,e.g., L = 10, L = 1, δL = 0, based on numerical sim-ulations performed on a lattice of size 20 at β = 2 . SU (2)gauge field and center vortex lead to the difference-of-areas law, while Abelian-projected configurations lead tothe sum-of-areas law.From a physical point of view, a double-winding Wil- FIG. 4: (top panel) Interactions between flux tubes generatedby two pairs of a quark and an antiquark, leading to the sum-of-areas law [reproduced from Fig.3 in [23]]. (bottom panel)W boson neutralizes the widely separated positive and nega-tive charges, leading to the difference-of-areas law in SU (2),[reproduced from Fig.11 in [23]]. son loop can be interpreted as a probe for studying inter-actions between two pairs of a particle and an antiparti-cle. Then differences among three cases are understoodas follows. In the Abelian model, a particle and an an-tiparticle in a pair are respectively connected by the elec-tric flux with the length of L and L , as indicated in thetop panel of Fig.4. The total energy of flux tubes shiftedby R > σ (cid:48) ( L + L ), where σ (cid:48) is a string tension,if the flux-flux interactions are neglected. This argumentwill give a reason why the Abelian model gives the sum-of-areas law. Moreover, they argue that even in the limit R → SU (2) gauge theory, they argue that the “ W bosons” play the crucial role, since they are off-diagonalcomponents of the SU (2) gauge field which are not in-cluded in the Abelian model. W bosons have chargedcomponents W −− and W ++ with respect to the Abelian U (1) group. They explain that charged off-diagonal com-ponents W −− and W ++ of the SU (2) gauge field neu-tralize respectively positive and negative static charges.Consequently, flux tubes exist only for connecting twopositive charges and two negative static charges, whichleads to difference-of-areas law. See the bottom panel ofFig.4.In the vortex picture, if a vortex pierces the minimalarea of a loop, it will multiply the holonomy around theloop by a factor −
1. Therefore, if a vortex pierces twoloops C and C simultaneously, it gives a trivial ef-fect. The non-trivial result is obtained only if a vortexpieces the non-overlapping region S − S . This leads todifference-of-areas law.Quite recently, Matsudo and Kondo [25] have inves-tigated a double-winding, a triple-winding, and generalmultiple-winding Wilson loops in the continuum SU ( N )Yang-Mills theory. They have found that a coplanardouble-winding SU (3) Wilson loop average follows a novel area law which is neither difference-of-areas law norsum-of-areas law, and that sum-of-areas law is allowed for SU ( N ) ( N ≥ SU ( N ) lattice Yang-Mills gauge theory by using both strong coupling expan-sion and numerical simulations.In this paper, we show that the “coplanar” double-winding Wilson loop average has the N dependent arealaw falloff: “max-of-areas law” for N = 3 and sum-of-areas law for N ≥
4, which add a new result to theknown difference-of-areas law for an N = 2 “coplanar”double-winding Wilson loop average. Moreover, we inves-tigate the behavior of a “shifted” double-winding Wilsonloop average as a function of the distance in a transversedirection and find that the long distance behavior doesnot depend on the number of color N , while the shortdistance behavior depends on N .This article is organized as follows. In section II, weexamine how the area law falloff of a “coplanar” double-winding Wilson loop average depends on the number ofcolor N . In section III, we examine a “shifted” double-winding Wilson loop, where the two constituent loopsare displaced from one another in a transverse direction,especially evaluate its average by changing the distanceof a transverse direction. The final section IV is devotedto conclusion and discussion. We also discuss the validityof the Abelian operator studied in [23]. Recently, thereare numerical evidences that the dual superconductor for SU (2) and SU (3) lattice Yang-Mills theory is type I [26],although they explain sum-of-areas law on the basis oftype II superconductor. We should study the interactionbetween two flux tubes in the limit R →
0, in case oftype I superconductor.
II. A “COPLANAR” DOUBLE-WINDINGWILSON LOOP
First of all, we consider the coplanar case R = 0 of adouble-winding Wilson loop in the SU ( N ) lattice Yang-Mills gauge theory, as indicated in Fig.2. For simplicity,we set δL = 0. Let S (= L × L ) and S (= L × L )be the minimal areas of rectangular loops C and C , re-spectively. We assume S ≥ S for definiteness hereafter. n n + ˆ µn + ˆ ν U † n +ˆ ν,µ U n +ˆ µ,ν U † n,ν U n,µ U n,µν n ˆ µ ˆ ν + S g = ! n,µ<ν g FIG. 5: (top panel) a plaquette variable U n,µν , (bottompanel) a plaquette action. A. strong coupling expansion
Let S g be a plaquette action for the SU ( N ) latticeYang-Mills theory: S g := (cid:88) n,µ (cid:54) = ν g tr( U n,µ U n +ˆ µ,ν U † n +ˆ ν,µ U † n,µ )= (cid:88) n,µ<ν g tr( U n,µν + U † n,µν ) , (4)where the link field U n,µ satisfies U n +ˆ µ, − µ = U † n,µ .This action reproduces the ordinary Yang-Mills action − (cid:82) d D x (cid:80) µ<ν tr( F µν ) up to constant in the naive con-tinuum limit (lattice spacing (cid:15) → U n,µν and the plaque-tte action are given in Fig.5.Note that the standard Wilson action S W is definedby S W = (cid:88) n,µ<ν β (cid:26) ) tr[ U n,µν + U † n,µν ] − (cid:27) , (5)see e.g., [29]. The difference of the constant term in theaction is physically insignificant and we drop it in thestrong coupling analysis. By comparing S g and S W , wecan find β = 2 N/g . (6)We define a partition function Z by Z := (cid:90) (cid:89) n,µ dU n,µ e S g , (7)where dU n,µ is the invariant integration measure of SU ( N ). Then the expectation value (cid:104) W ( C ) (cid:105) of an oper-ator W ( C ) is defined by (cid:104) W ( C ) (cid:105) := (cid:82) (cid:81) n,µ dU n,µ e S g W ( C ) Z . (8) (cid:8)(cid:1)(cid:5)(cid:3)(cid:2)(cid:4) (cid:1)
FIG. 6: A set of plaquettes tiling the areas S and S which gives the non-trivial contribution to a coplanar double-winding Wilson loop average (cid:104) W ( C × C ) (cid:105) for SU (2). (toppanel) the leading contribution, (bottom panel) a higher ordercontribution. In order to evaluate the expectation value in eq.(8),we perform the strong coupling expansion: For the largebare coupling constant g , we can expand the weight e S g into the power-series of 1 /g , e S g = (cid:89) n,µ<ν (cid:40) ∞ (cid:88) k =0 k ! (cid:18) g (cid:19) k [tr( U n,µν ) + tr( U † n,µν )] k (cid:41) , (9)and perform the group integration over each link variable U n,µ according to the measure dU n,µ . In Appendix A, wesummarize the formulas needed for the strong couplingexpansion and for the SU ( N ) group integration. SU (2) First, we study the case of SU (2) gauge group. Fora coplanar double-winding Wilson loop, there is a singlelink variable U (cid:96) for a link (cid:96) ∈ C − C and there is adouble link variable U (cid:96) U (cid:96) for a link (cid:96) ∈ C , as shown inthe top diagram of Fig.6.We list some of explicit SU (2) group integration for-mula as (cid:90) dU , (10a) (cid:90) dU U ab = 0 , (cid:90) dU U † ab = 0 , (10b) (cid:90) dU U ab U † kl = 12 δ al δ bk , (10c) (cid:90) dU U a b U a b = 12! (cid:15) a a (cid:15) b b = (cid:90) dU U † a b U † a b , (10d) (cid:90) dU U a b U a b · · · U a M b M = 0 , M (cid:54) = 0 (mod 2) , (10e) (cid:90) dU U ab U cd U † ij U † kl = 1(2 −
1) [ δ aj δ bi δ cl δ dk + δ al δ bk δ cj δ di −
12 ( δ aj δ bk δ cl δ di + δ al δ bi δ cj δ dk )] + (cid:18) (cid:19) (cid:15) ac (cid:15) bd (cid:15) ik (cid:15) jl . (10f)For a single link variable U (cid:96) (resp. U † (cid:96) ) for (cid:96) ∈ C − C ,we need at least one additional link variable with an op-posite direction U † (cid:96) (resp. U (cid:96) ) to obtain non-vanishingresult after integration in eq.(8) according to the in-tegration formulas (10c) for the SU (2) group integra-tions. Such link variables are supplied from the expansioneq.(9) of e S g . Since the number of plaquettes which arebrought down from e S g must be equal to the power of1 /g in the expansion eq.(9), the leading contribution to (cid:104) W ( C × C ) (cid:105) comes from a set of plaquettes tiling theminimal area S − S with the least number of plaquettes.See the top diagram of Fig.6. For double link variables U (cid:96) U (cid:96) for (cid:96) ∈ C , on the other hand, we do not need ad-ditional link variables coming from the expansion of e S g to obtain the non-vanishing result due to the integration(10d), giving the g -independent contribution.For the SU (2) gauge group, therefore, the leading con-tribution to (cid:104) W ( C × C ) (cid:105) in the strong coupling expan-sion comes from the term in which a set of plaquettestiles the surface with the area S − S , as shown in thetop diagram of Fig.6. Therefore, group integrations givethe result (cid:104) W ( C × C ) (cid:105) leading = − (cid:18) g (cid:19) S − S = − e − σ ( S − S ) , (11)where σ = log(2 g ). This result was first obtained byGreensite and H¨ollwieser in [23]. We reconfirmed thedifference-of-areas law of coplanar double-winding Wil-son loops for SU (2). The bottom diagram of Fig.6 showsone of higher-order contributions in the strong couplingexpansion for SU (2). This diagram gives non-vanishingcontribution due to the integration formula (10f). SU ( N ) , ( N ≥ ) Next, we study the case of SU ( N ) ( N ≥
3) gaugegroups. We list some of explicit SU ( N ) ( N ≥
3) group integration formula as (cid:90) dU , (12a) (cid:90) dU U ab = 0 , (12b) (cid:90) dU U ab U † kl = 1 N δ al δ bk , (12c) (cid:90) dU U a b U a b · · · U a M b M = 0 , M (cid:54) = 0 (mod N ) , (12d) (cid:90) dU U a b U a b · · · U a N b N = 1 N ! (cid:15) a a ··· a N (cid:15) b b ··· b N , (12e) (cid:90) dU U ab U cd U † ij U † kl = 1( N −
1) [ δ aj δ bi δ cl δ dk + δ al δ bk δ cj δ di − N ( δ aj δ bk δ cl δ di + δ al δ bi δ cj δ dk )] . (12f)Notice that the SU ( N ) case is different from the SU (2)case. For a double link variable U (cid:96) U (cid:96) for a link (cid:96) ∈ C ,we need additional N − U (cid:96) ) N − with thesame direction to be brought down from the expansion of e S g in eq.(8) to obtain the non-vanishing result after theintegration according to the integration formulas (12e)for the SU ( N ) group integrations. See the top diagramof Fig.7. For a single link variable U (cid:96) (resp. U † (cid:96) ) for alink (cid:96) ∈ C − C , on the other hand, we need at leastone additional link variable with the opposite direction U † (cid:96) (resp. U (cid:96) ) to obtain non-vanishing result after inte-gration in eq.(8) according to the integration formulas(12c) for the SU (2) group integrations. Therefore, thecontribution from the top diagram of Fig.7 is given by p N (cid:18) g N (cid:19) ( N − S +( S − S ) , (13)where the coefficient p N is calculated by collecting thenumerical factors coming from link integrations and thepower-series expansions of e S g .We have another contribution from the bottom dia-gram of Fig.7. For a double link variable U (cid:96) U (cid:96) with thesame direction for a link (cid:96) ∈ C , we have additional 2link variables ( U † (cid:96) )( U † (cid:96) ) with the opposite directions tobe brought down from the expansion of e S g in eq.(8) toobtain the non-vanishing result after the integration ac-cording to the integration formulas (12f) for the SU ( N )group integrations. For a single link variable U (cid:96) (resp. U † (cid:96) ) for a link (cid:96) ∈ C − C , on the other hand, we need atleast one additional link variable with an opposite direc-tion U † (cid:96) (resp. U (cid:96) ) to obtain non-vanishing result afterintegration in eq.(8) according to the integration formu-las (12c) for the SU ( N ) group integrations. Therefore,the contribution from the bottom diagram of Fig.7 is (cid:2)(cid:9)(cid:10)(cid:1)(cid:6)(cid:4)(cid:3)(cid:5) (cid:1) (cid:8)(cid:1)(cid:5)(cid:3)(cid:2)(cid:4) (cid:1) FIG. 7: A set of plaquettes tiling the areas S and S whichgives the leading contribution to a coplanar double-windingWilson loop average (cid:104) W ( C × C ) (cid:105) for SU ( N ) ( N ≥ S (= L × L ) and S (= L × L ) are respectively the minimalareas bounded by rectangular loops C and C with S ≥ S .For N = 3, the diagram of the top panel gives the leadingcontribution and that of the bottom panel gives the next-to-leading contribution. For N = 4, the two diagrams giveidentical contributions. For N >
4, the diagram of the bottompanel gives the leading contribution and that of the top panelgives the next-to-leading contribution. given by q N (cid:18) g N (cid:19) S +( S − S ) = q N (cid:18) g N (cid:19) S + S , (14)where the coefficient q N is calculated in the similar wayto p N .For the SU ( N ) ( N ≥ e S g is equal to the power of1 /g , these two contributions can be written as (cid:104) W ( C × C ) (cid:105) = p N (cid:18) g N (cid:19) ( N − S + S − S + q N (cid:18) g N (cid:19) S + S + · · · , (15)where coefficients p N , q N are determined by expansioncoefficients of the power series expansion of e S g and SU ( N ) group integrations for link variables. Which con-tribution becomes dominant is naively determined bycomparing the power index of g N , which depends onthe number of color N .For N ≥
4, we find that the second term in eq.(15)gives the dominant contribution in the strong couplingexpansion for (cid:104) W ( C × C ) (cid:105) , since the inequality holds, S + S ≤ ( N − S + S − S for N ≥
4. Thus weconclude that the sum-of-areas law of a coplanar double-winding Wilson loop is allowed for N ≥
4. This resultis consistent with the result obtained by Matsudo andKondo in [25].From the top panel of Fig.7, we can easily find thatthe coefficient p N should be calculated for each numberof color N , because type of diagrams are different withthe number of color N . On the other hands, we canobtain general formula for the coefficient q N , since thediagram of the bottom panel of Fig.7 is common to allnumbers of color N . The result is q N = − N S (cid:40)(cid:20) N ( N − (cid:21) S − − (cid:20) N ( N + 1) (cid:21) S − (cid:41) , (16)for S ≥ SU (2), SU (3)and SU (4) in more detail. SU (2) For the number of color N = 2, eq. (15) re-duces to (cid:104) W ( C × C ) (cid:105) = 2 p (cid:18) g (cid:19) S − S + 2 q (cid:18) g (cid:19) S + S + · · · , (17)where p = − , (18) q = − S (cid:40)(cid:20) (cid:21) S − − (cid:20) (cid:21) S − (cid:41) , ( S ≥ . (19)The factor 2 in front of p and q arises from the non-oriented nature of the plaquettes for SU (2), which is tobe compared with (11). SU (3) For the number of color N = 3, eq. (15) re-duces to (cid:104) W ( C × C ) (cid:105) = p (cid:18) g (cid:19) S + q (cid:18) g (cid:19) S + S + · · · , (20)where p = − , (21) q = − S (cid:40)(cid:20) (cid:21) S − − (cid:20) (cid:21) S − (cid:41) , ( S ≥ . (22) - < W ( C ) > L1 FIG. 8: L -dependence of a coplanar double-winding Wilsonloop average −(cid:104) W ( C × C ) (cid:105) from the strong coupling expan-sion in SU (2) lattice gauge theory. We plot eq.(17) times − L = 1 ∼
10 for L = 10, L = 1 and 1 /g N = 2 . / The coefficient q is obtained from eq.(16). See Ap-pendix C for the calculation of p .From this result, we find that the first term in eq.(20)gives the dominant contribution to (cid:104) W ( C × C ) (cid:105) forsufficiently large areas S and S , which is neitherdifference-of-areas law nor sum-of-areas law for the area-law falloff of the coplanar double-winding Wilson loopaverage. We call this area-law falloff “max-of-areas law”(or max( S , S ) law). This result is also consistent withthe result obtained by Matsudo and Kondo in [25]. SU (4) For the number of color N = 4, eq. (15) re-duces to (cid:104) W ( C × C ) (cid:105) = p (cid:18) g (cid:19) S + S + q (cid:18) g (cid:19) S + S + · · · , (23)where p = − (cid:20) (cid:21) S − , (24) q = − S (cid:40)(cid:20) (cid:21) S − − (cid:20) (cid:21) S − (cid:41) , ( S ≥ . (25)In this case, both terms in eq.(23) behave as sum-of-areaslaw. L dependence of the (cid:104) W ( C × C ) (cid:105) From the above discussions, we can understand the L dependence of the coplanar double-winding Wilson loopaverage (cid:104) W ( C × C ) (cid:105) in SU ( N ) lattice Yang-Mills gaugetheory for fixed L , L , and gauge coupling g .For SU (2) gauge group, we plot eq.(17) in Fig.8, whichshows the difference-of-areas law behavior of a coplanardouble-winding Wilson loop for N = 2. -0.0001-5e-05 0 5e-05 0.0001 1 2 3 4 5 6 7 8 < W ( C ) > L1 -1e-47-5e-48 0 5e-48 1e-47 1 2 3 4 5 6 7 8 < W ( C ) > L1 FIG. 9: L -dependence of a coplanar double-winding Wilsonloop average (cid:104) W ( C × C ) (cid:105) from the strong coupling expansionin SU (3) lattice gauge theory. We plot eq.(20) versus L =1 ∼ /g N = 6 . /
18. (top panel) L = 10, L = 1.(bottom panel) L = 10, L = 10. On the other hand, we plot eq.(20) in Fig.9. For SU (3)gauge group, as the coplanar double-winding Wilson loopaverage follows the max-of-areas law, it is expected thatthere are no L -dependence of (cid:104) W ( C × C ) (cid:105) for efficientlylarge areas S and S . In fact, we can see that the plotsflatten at L ∼ L ∼
1) in top (resp. bottom)panel in Fig.9.
B. Numerical simulation
We examine the L -dependence of (cid:104) W ( C × C ) (cid:105) thatwe discussed above. SU (2): We generate the configurations of SU (2) linkvariables { U n,µ } , using the (pseudo-)heat-bath methodfor the standard Wilson action. The numerical simula-tions are performed on the 24 lattice at β (= 2 N/g ) =2 .
5. We thermalize 3000 sweeps, and in particular, wehave used 100 configurations for calculating the expec-tation value of coplanar double-winding Wilson loops (cid:104) W ( C × C ) (cid:105) .Fig.10 shows the obtained plot for the −(cid:104) W ( C × C ) (cid:105) for various value of L , when we choose parameters L = 10, L = 3. The results of numerical simulations areconsistent with analytical results in Fig.8. Thus we re- - < W ( C ) > L1 ’L2=3’ FIG. 10: L -dependence of a coplanar double-winding Wil-son loop average −(cid:104) W ( C × C ) (cid:105) in the SU (2) lattice gaugetheory obtained from numerical simulations on a lattice ofsize 24 at β = 2 . L = 10, and L = 3. confirm the difference-of-areas law for SU (2). Note thatwe can also confirm (cid:104) W ( C × C ) (cid:105) (cid:39) − / S = S from Fig.8. SU (3): We also generate the configurations of SU (3)link variables { U n,µ } , using the (pseudo-)heat-bathmethod for the standard Wilson action. The numericalsimulations are performed on the 24 lattice at β = 6 . (cid:104) W ( C × C ) (cid:105) , where we have used APE smearing method( N = 12, α = 0 .
1) as a noise reduction technique. See[27] for the detail.Fig.11 shows the obtained plot for the (cid:104) W ( C × C ) (cid:105) forvarious value of L , when we choose parameters L = 10, L = 4 , ,
8. The results of numerical simulations areconsistent with analytical results in Fig.9. For example,we can see that the plots flatten at L ∼ L = 8,which means that there are no L -dependence of (cid:104) W ( C × C ) (cid:105) . Thus, we numerically confirm the max-of-areas lawfor SU (3). III. A ”SHIFTED” DOUBLE-WINDINGWILSON LOOPS
Finally, we consider the shifted case R (cid:54) = 0 of a double-winding Wilson loop in the SU ( N ) lattice Yang-Millsgauge theory, as indicated in Fig.12. Contours C and C lie in planes parallel to the x - t plane, but are displacedfrom one another in the z direction by distance R . Justlike the previous section, for simplicity, let C ( C ) bea rectangular loop of length L , L ( L , L ), and S ( ≡ L × L ), S ( ≡ L × L ) be the minimal areas of contour C , C respectively. -0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0 2 4 6 8 10 < W ( C ) > L1 ’L2=4’’L2=6’’L2=8’ FIG. 11: L -dependence of a coplanar double-winding Wil-son loop average (cid:104) W ( C × C ) (cid:105) in SU (3) lattice gauge theoryobtained from numerical simulations on a lattice of size 24 at β = 6 . L = 10, and L = 4 , , LL L L R FIG. 12: The setting up of a shifted double-winding Wilsonloop operator W ( C × C ) R (cid:54) =0 . A. strong coupling expansion
First, we study the shifted double-winding Wilson loopbased on the strong coupling expansion.One of the diagrams which gives a leading contribu-tion in the strong coupling expansion is given by a set ofplaquettes tiling the two minimal surfaces S and S , asshown in Fig.13. The results of a group integration forthe links U (cid:96) ’s on both surfaces become N (1 /g N ) S + S FIG. 13: One of diagrams which also contributes to a shifteddouble-winding Wilson loop average (cid:104) W ( C × C ) (cid:105) R (cid:54) =0 in thestrong coupling expansion of the SU ( N ) lattice Yang-Millstheory. = ! g N " R ( L + L ) × FIG. 14: Another diagram which contributes to a shifteddouble-winding Wilson loop average (cid:104) W ( C × C ) (cid:105) R (cid:54) =0 in thestrong coupling expansion of the SU ( N ) lattice Yang-Millstheory. for N ≥
3, and 2 N (1 /g N ) S + S for N = 2, respectively.The difference of factor 2 in front of N for N = 2 arisesfrom the non-oriented nature of the plaquettes to con-clude the N = 2 result:4 (cid:18) g (cid:19) S + S . (26)Another type of diagram which also gives a leadingcontribution in the strong coupling expansion is given bya set of plaquettes tiling the minimal surface S − S and the four sides with the area 2 R ( L + L ) of a cuboidwith a height R , whose bottom is a rectangular of size L × L , as shown in the upper panel of Fig.14. Aftergroup integrations for the links on the side surfaces givinga factor (1 /g N ) R ( L + L ) , this diagram is equivalent toa coplanar double-winding Wilson loop, as shown in thelower panel of Fig.14. The expectation value of this typeof a coplanar double-winding Wilson loop is already cal-culated in the previous subsection, and the results areeq.(17) for SU (2), eq.(20) for SU (3), and eq.(23) for SU (4), respectively. Consequently, the diagram of Fig.14yields the contribution for N = 2: (cid:18) g (cid:19) R ( L + L ) (cid:40) p (cid:18) g (cid:19) S − S + 2 q (cid:18) g (cid:19) S + S (cid:41) . (27)To summarize the above discussion, the expecta-tion value of the shifted double-winding loop (cid:104) W ( C × C ) (cid:105) R (cid:54) =0 from diagrams as shown in Fig.13 and Fig.14becomes for N = 2, SU (2) : (cid:104) W ( C × C ) (cid:105) R (cid:54) =0 =4 (cid:18) g (cid:19) S + S + 2 p (cid:18) g (cid:19) S − S +2 R ( L + L ) + 2 q (cid:18) g (cid:19) S + S +2 R ( L + L ) + · · · . (28)Note that the R → (cid:2)(cid:2) (cid:1) (cid:1)(cid:1) (cid:1)(cid:2) (cid:2) xtz LL R FIG. 15: A shifted double-winding Wilson loop as a probefor interactions between two flux tubes.
R < R C R > R C FIG. 16: Lowest order diagrams giving the dominant con-tribution to a shifted double-winding Wilson loop average (cid:104) W ( C × C ) (cid:105) R (cid:54) =0 . The dominant diagram switches at a cer-tain value R c of R from left to right. second and third terms in eq.(28) from the diagram ofFig.14 reproduce the coplanar result eq.(17) in the limit R →
0. This is because the first term in eq.(28) comingfrom the diagram of Fig.13 does not have in the limit R → R (cid:54) = 0.For SU (2) gauge group, especially, we perform the de-tailed study on the R -dependence of a shifted double-winding Wilson loop average (cid:104) W ( C × C ) (cid:105) R (cid:54) =0 . In whatfollows, we rewrite L into T , T := L . (29)Let us imagine T direction be time t -axis, L and L di-rection be spatial x -axis, and R direction be also space z -axis as seen in top side in Fig.15. As is explained in[23], the shifted double-winding Wilson loop at a fixedtime can be interpreted as a tetra-quark system consist-ing of two static quarks and two static antiquarks. Thepairs of quark-antiquarks are connected by a pair of colorflux tubes, as seen in the bottom side in Fig.15. We studyhow interactions between the two color flux tubes change,when the distance R is varied.0 -0.01-0.008-0.006-0.004-0.002 0 0.002 0 1 2 3 4 5 6 7 8 < W ( C ) > R FIG. 17: R -dependence of of a shifted double-winding Wilsonloop average (cid:104) W ( C × C ) (cid:105) in the SU (2) lattice gauge the-ory obtained from the strong coupling expansion for 1 / g =2 . / L = 5, L = 1 and L = 3. We find that the second term in eq.(28) dominates for
R < R C := L L /T , and the first term in eq.(28) dom-inates for R > R C , because the comparison of the twoexponents of these terms for S = LT and S = L T reads S − S + 2 R ( L + L ) < S + S = ⇒ R ( L + L ) < S = ⇒ R ( L + T ) < L T = ⇒ R < L L /T := R C , (30)where we have neglected the third (higher order) term ineq.(28) for the naive estimate of R C . This means thatthe left diagram of Fig.16 dominates for R < R C , and theright diagram of Fig.16 dominates for R > R C . There-fore, the dominant diagram switches from left to rightat a certain value R C of R as R increases, just like theminimal surface spanned by a soap film.In Fig.17, we plot the R -dependence eq.(28) of a shifteddouble-winding Wilson loop average (cid:104) W ( C × C ) (cid:105) forfixed L , L , and L in the SU (2) lattice gauge the-ory. The second and third terms in eq.(28) have R -dependence, but the first term in eq.(28) does not dependon R . Therefore, the plot gets flattened for R ≥ R C ∼ N . In fact, SU (3) and SU (4) cases are given as follows. SU (3) : (cid:104) W ( C × C ) (cid:105) R (cid:54) =0 =3 (cid:18) g (cid:19) S + S + p (cid:18) g (cid:19) S +2 R ( L + L ) + q (cid:18) g (cid:19) S + S +2 R ( L + L ) + · · · , (31) SU (4) : (cid:104) W ( C × C ) (cid:105) R (cid:54) =0 -0.001-0.0005 0 0.0005 0.001 0 1 2 3 4 5 6 7 8 < W ( C ) > R FIG. 18: R -dependence of of a shifted double-winding Wilsonloop average (cid:104) W ( C × C ) (cid:105) in the SU (3) lattice gauge the-ory obtained from the strong coupling expansion for 1 / g =6 . / L = 5, L = 1 and L = 3. =4 (cid:18) g (cid:19) S + S + p (cid:18) g (cid:19) S + S +2 R ( L + L ) + q (cid:18) g (cid:19) S + S +2 R ( L + L ) + · · · . (32)In Fig.18, we also plot the R -dependence eq.(31) of ashifted double-winding Wilson loop average (cid:104) W ( C × C ) (cid:105) for fixed L , L , and L in the SU (3) lattice gauge theory.In general, (cid:104) W ( C × C ) (cid:105) R (cid:54) =0 for N ≥ (cid:104) W ( C × C ) (cid:105) R (cid:54) =0 = N (cid:18) g N (cid:19) S + S + (cid:18) g N (cid:19) R ( L + L ) × (cid:40) p N (cid:18) g N (cid:19) ( N − S + S − S + q N (cid:18) g N (cid:19) S + S (cid:41) + · · · . (33) B. Numerical simulation
Next, we examine the R -dependence of (cid:104) W ( C × C ) (cid:105) based on numerical simulations on a lattice. SU (2): In order to calculate the shifted double-winding Wilson loop average, we use the same gaugefield configurations as those used in calculating the copla-nar double-winding Wilson loop. However, we have usedAPE smearing method ( N = 5, α = 0 .
1) as a noise re-duction technique. Fig.19 gives the plots obtained for the (cid:104) W ( C × C ) (cid:105) for various values of R where we have fixed L = 5, T (= L ) = 2, L = 3. We see that the behaviorof data in Fig.19 is consistent with the analytical resultgiven in Fig.17. SU (3): Similarly, Fig.20 shows the obtained plot forthe (cid:104) W ( C × C ) (cid:105) for various value of R for SU (3) case,when we choose parameters L = 8, T (= L ) = 8, L =1 -0.2-0.15-0.1-0.05 0 0.05 0.1 0 1 2 3 4 5 6 7 8 < W ( C ) > R ’SU2YM’ FIG. 19: R -dependence of a shifted double-winding Wilsonloop average (cid:104) W ( C × C ) (cid:105) in the SU (2) lattice gauge theoryobtained from numerical simulations on a lattice of size 24 at β = 2 . L = 5, T (= L ) = 2, and L = 3. -0.2-0.1 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 8 9 < W ( C ) > R ’L1=1’’L1=2’’L1=3’’L1=4’’L1=5’’L1=6’ FIG. 20: R -dependence of a shifted double-winding Wilsonloop average (cid:104) W ( C × C ) (cid:105) in the SU (3) lattice gauge theoryobtained from numerical simulations on a lattice of size 24 at β = 6 . L = 8, T (= L ) = 8, and L = 1 ∼ ∼
6. We see that the data in Fig.20 also consistentwith the analytical result given in Fig.18 for sufficientlylarge areas S and S . IV. CONCLUSION AND DISCUSSION
In this paper, we have studied the double-winding Wil-son loops in SU ( N ) lattice Yang-Mills gauge theory byusing both strong coupling expansion and numerical sim-ulation. First of all, we have examined how the area law falloffof a “coplanar” double-winding Wilson loop average de-pends on the number of color N , by changing the size ofminimal area S of loop C . We have reconfirmed thedifference-of-areas law for N = 2, and have found newresults that “max-of-areas law” for N = 3 and sum-of-areas law for N ≥ N , but theshort distance behavior depends on N .It should be remarked that this “shifted” double-winding Wilson loop may contain an information aboutinteractions between two color flux tubes. For this pur-pose, we need to accumulate more data on the fine lat-tices with more larger size.Originally, one of reasons why Greensite andH¨ollwieser considered the double-winding Wilson loopsseems to be that they want to evaluate monopole con-finement mechanism in lattice SU (2) gauge theory. Theyhave considered an operator which simply replaces SU (2)link variable U n,µ with the Abelian variable u n,µ as an“Abelian” double-winding Wilson loop, and have shownthat the expectation value of such a naive operator obeysthe sum-of-areas law. But, it is known that such naive op-erator should work only for a single-winding Wilson loopin the fundamental representation. Recently, Matsudoand his collaborators [28] have given the explicit expres-sion for the Abelian operator which reproduces the fullWilson loop average in higher representations, which issuggested by the gauge-covariant field decomposition andthe non-Abelian Stokes theorem (NAST) for the Wilsonloop operator. Similarly, we hope that a correct form ofthe Abelian operator for a double-winding Wilson loopcan be found in the similar way. When we change theline integral to the surface integral, our considerations ofthe diagrams which give the leading contribution to thestrong coupling expansion seems to be useful to constructthe NAST for a double-winding Wilson loop. These re-sults will be discussed in a forthcoming paper. Acknowledgments
This work was supported by Grant-in-Aid for Sci-entific Research, JSPS KAKENHI Grant Number (C)No.19K03840 and No.15K05042.2
Appendix A: SU ( N ) group integrals and useful formulae In order to perform the strong coupling expansion in the lattice gauge theory, we must calculate the followingintegrations for the polynomials of group matrix elements over each links: I = (cid:90) dU U i j · · · U i n j n ( U − ) k l · · · ( U − ) k m l m , (A.1)where U ij ( i, j = 1 , , · · · , N ) denotes a matrix element of a matrix U ∈ SU ( N ) belonging to the SU ( N ) group withthe property U − = U † , and dU is an invariant measure (Haar measure) on the compact group which is left-invariant (cid:90) dU f ( U ) = (cid:90) dU f ( V U ) ( ∀ V ∈ SU ( N )) , (A.2)and right-invariant (cid:90) dU f ( U ) = (cid:90) dU f ( U V ) ( ∀ V ∈ SU ( N )) . (A.3)We can normalize the measure such that (cid:90) dU = 1 . (A.4)By using properties of the invariant measure, Creutz has shown that eq.(A.1) can be evaluated by the followingformula [29, 30]: I = ( ∂ j i · · · ∂ j n i n · cof( ∂ ) l k · · · cof( ∂ ) l m k m ) ∞ (cid:88) i =0 · · · ( N − i !( i + 1)! · · · ( i + N − | J | i | J =0 , (A.5)where J is a source variable and is an arbitrary N × N matrix, | J | = det( J ), ∂ ji ≡ ∂/∂J ji , and cof( ∂ ) is a cofactorof ∂ , respectively.We list some of explicit results from the above formula as (cid:90) dU , (A.6) (cid:90) dU U ab = 0 , (A.7) (cid:90) dU U ab U † kl = 1 N δ al δ bk , (A.8) (cid:90) dU U a b U a b · · · U a N b N = 1 N ! (cid:15) a a ··· a N (cid:15) b b ··· b N , (A.9) (cid:90) dU U a b U a b · · · U a M b M = 0 , M (cid:54) = 0 (mod N ) , (A.10) (cid:90) dU U ab U cd U † ij U † kl = 1( N − (cid:20) δ aj δ bi δ cl δ dk + δ al δ bk δ cj δ di − N ( δ aj δ bk δ cl δ di + δ al δ bi δ cj δ dk ) (cid:21) . (A.11)The last eq.(A.11) consist for N >
2. For N = 2, (cid:90) dU U ab U cd U † ij U † kl = 1( N − (cid:20) δ aj δ bi δ cl δ dk + δ al δ bk δ cj δ di − N ( δ aj δ bk δ cl δ di + δ al δ bi δ cj δ dk ) (cid:21) + (cid:18) N ! (cid:19) (cid:15) ac (cid:15) bd (cid:15) ik (cid:15) jl . (A.12)Following relation can be shown by using property of invariant measure, (cid:90) dU f ( U − ) = (cid:90) dU f ( U ) . (A.13)3From this relation, we also obtain, (cid:90) dU U † ab = 0 , (A.14) (cid:90) dU U † a b U † a b · · · U † a N b N = 1 N ! (cid:15) a a ··· a N (cid:15) b b ··· b N . (A.15)The following more practical formulae are useful to calculate the expectation value of double-winding Wilson loopby using strong coupling expansion. Let X, Y, A, B be elements of SU ( N ) group. From eq.(A.9), we find (cid:90) dU tr( XU Y U ) = X ab Y cd (cid:90) dU U bc U da = δ N, N (cid:15) ca X ab Y cd (cid:15) bd , (A.16) (cid:90) dU tr( XU )tr( Y U ) = X ab Y cd (cid:90) dU U ba U dc = δ N, N (cid:15) ac (cid:15) bd X ab Y cd . (A.17)From eq.(A.8), we find (cid:90) dU tr( XU )tr( Y U † ) = X ab Y lk (cid:90) dU U ba U † kl = X ab Y lk N δ bl δ ak = 1 N tr( XY ) . (A.18)From eq.(A.11), we find for N > (cid:90) dU tr( AU )tr( BU )tr( XU † )tr( Y U † )= A ab B cd X ij Y kl (cid:90) dU U ba U dc U † ji U † lk = A ab B cd X ij Y kl N − (cid:20) δ bi δ aj δ dk δ cl + δ bk δ al δ di δ cj − N ( δ bi δ al δ dk δ cj + δ bk δ aj δ di δ cl ) (cid:21) = 1 N − (cid:20) tr( AX )tr( BY ) + tr( AY )tr( BX ) − N (tr( AXBY ) + tr(
AY BX )) (cid:21) , (A.19) (cid:90) dU tr( AU BU )tr( XU † )tr( Y U † )= A ab B cd X ij Y kl (cid:90) dU U bc U da U † ji U † lk = A ab B cd X ij Y kl N − (cid:20) δ bi δ cj δ dk δ al + δ bk δ cl δ di δ aj − N ( δ bi δ cl δ dk δ aj + δ bk δ cj δ di δ al ) (cid:21) = 1 N − (cid:20) tr( AXBY ) + tr(
AY BX ) − N (tr( AX )tr( BY ) + tr( AY )tr( BX )) (cid:21) , (A.20) (cid:90) dU tr( AU BU )tr( XU † Y U † )= A ab B cd X ij Y kl (cid:90) dU U bc U da U † jk U † li = A ab B cd X ij Y kl N − (cid:20) δ bk δ cj δ di δ al + δ bi δ cl δ dk δ aj − N ( δ bk δ cl δ di δ aj + δ bi δ cj δ dk δ al ) (cid:21) = 1 N − (cid:20) tr( AY )tr( BX ) + tr( AX )tr( BY ) − N (tr( AY BX ) + tr(
AY BX )) (cid:21) . (A.21) Appendix B: Explicit calculation of the coefficient q N In this section, we show explicitly how eq.(16) is obtained. From eq.(8) and eq.(9), a contribution to a coplanardouble-winding Wilson loop average (cid:104) W ( C × C ) (cid:105) from the bottom panel of Fig.7 is expressed as (cid:104) W ( C × C ) (cid:105) q N = (cid:90) (cid:89) (cid:96) ∈ S dU (cid:96) W ( C × C ) · (cid:89) p j ∈ ( S − S ) (cid:20) g tr( U † p j ) (cid:21) · (cid:89) p k ∈ S (cid:40) (cid:20) g tr( U † p k ) (cid:21) (cid:41) , (B.1)4 ! dU X Y U
FIG. B.1: Diagrammatic representation of the integration rule ˜ W eq.(B.4) for the product of two double-plaquettes withthe same clockwise orientation: Integration is performed over the link variables U on the link which is common to twodouble-plaquettes with the same clockwise orientation. By decomposing the path-ordered product of the link variables alongthe loop, the plaquette variables for the single plaquette p and p to the left and right of U is respectively represented bytr( U † p ) := tr( U † X ) and tr( U † p ) := tr( Y U ). Here X and Y represent the products of the link variables along staple-shapedpaths with the same orientations. where U † p j and U † p k denote respectively plaquette variables on ( S − S ) and S areas. Here note that U † p representsthe plaquette variable for the plaquette p with the clockwise orientation.First, integration with respect to the link variables { U (cid:96) } on the ( S − S ) area can be performed with the sametechnique of the strong coupling expansion as that for the fundamental Wilson loop to obtain (cid:104) W ( C × C ) (cid:105) q N = (cid:18) g N (cid:19) S − S (cid:104) W ( C × C ) (cid:105) q N , (B.2)where we have defined (cid:104) W ( C × C ) (cid:105) q N := (cid:90) (cid:89) (cid:96) ∈ S dU (cid:96) W ( C × C ) · (cid:89) p k ∈ S (cid:40) (cid:20) g tr( U † p k ) (cid:21) (cid:41) . (B.3)Next, we perform the integration in eq.(B.3) over the link variables { U (cid:96) } inside of the S area, which excludes thelinks on the loop C = ∂S (the boundary of S ). As shown in Fig.B.1, performing the integration with respect tothe link variables U on the link which is common to two double-plaquettes with the same clockwise orientation usingeq.(A.19), we obtain˜ W := (cid:90) dU (cid:8) tr( U † X ) (cid:9) · { tr( Y U ) } = α { tr( Y X ) } + β tr( Y XY X ) := α W ( D ) + β W ( D × D ) , (B.4)where α = 2 N − , β = − N ( N − . (B.5)Here D represents the loop as the boundary of a 2 × U . Then W ( D ) and W ( D × D ) respectively stand for the single-windingWilson loop and double-winding Wilson loop along the loop D where the Wilson loop means the trace of the productof link variables on the relevant loop.Moreover, we proceed to perform the integration over the link variable for the product of a double-winding loopin ˜ W and an adjacent double-plaquette { tr( V † p ) } . As shown in Fig. B.2, performing the integration of the linkvariable V on the link which is common to the double-winding loop W ( D × D ) (the second term of eq.(B.4)) andthe double-plaquette { tr( V † p ) } adjacent to the common link V by using eq.(A.20) and eq.(A.13), we obtain (cid:90) dV W ( D × D ) · (cid:8) tr( V † p ) (cid:9) = (cid:90) dV tr( AV † AV † ) · { tr( BV ) } = − N ( N − { tr( AB ) } + 2 N − ABAB ) (B.6):= − N ( N − W ( D ) + 2 N − W ( D × D ) , (B.7)5 ! dV A B V = +2
FIG. B.2: Diagrammatic representation of the integration rule eq.(B.7) for the product of a double-winding loop and a double-plaquette with the same clockwise orientation: Integration is performed over the link variable V on the link which is commonto the double-winding loop W ( D × D ) along the loop D (the second term of eq.(B.4)) and the double-plaquette { tr( V † p ) } adjacent to the common link V . Here we have used the decomposition W ( D × D ) := tr( AV † AV † ) and tr( V † p ) := tr( BV ). where D represents the loop as the boundary of a 3 × × V . Then W ( D ) and W ( D × D ) respectively stand for the single-winding Wilson loop and double-winding Wilson loop along the loop D . On the other hand, since the V integralfor the product of the first term of eq.(B.4), i.e., W ( D ) and the double-plaquette variable adjacent to V , namely, (cid:82) dV W ( D ) · (cid:8) tr( V † p ) (cid:9) is the same type as eq.(B.4), we see that the result is again a linear combination of W ( D ) and W ( D × D ). Therefore, defining ˜ W by the result of integration over the common link variable V for the productof ˜ W and the double-plaquette adjacent to the link V , namely, ˜ W := (cid:82) dV ˜ W · (cid:8) tr( V † p ) (cid:9) , we find ˜ W is written asa linear combination of W ( D ) and W ( D × D ).From the above consideration, defining ˜ W n by the result of connecting n adjacent double-plaquettes one afteranother by integrating over the link variables inside the S area, we can conclude that ˜ W n is written as˜ W n = α n W ( D n ) + β n W ( D n × D n ) . (B.8)This statement is proved by the mathematical induction. Indeed, by applying the same procedures as those given ineq.(B.4) and eq.(B.7) to eq.(B.8), we find the relationship˜ W n +1 := (cid:90) dV ˜ W n · (cid:8) tr( V † p ) (cid:9) = (cid:26) α n N − − β n N ( N − (cid:27) W ( D n +1 ) + (cid:26) − α n N ( N −
1) + 2 β n N − (cid:27) W ( D n +1 × D n +1 ):= α n +1 W ( D n +1 ) + β n +1 W ( D n +1 × D n +1 ) . (B.9)Therefore, we have obtained the recurrence relation which holds for the coefficients α n and β n for n ≥ (cid:18) α n +1 − β n +1 (cid:19) = 2 N − (cid:18) /N /N (cid:19) (cid:18) α n − β n (cid:19) . (B.10)Solving this recurrence relation with the initial condition eq.(B.5), we obtain the explicit form for the coefficients α n and β n : (cid:18) α n − β n (cid:19) = 2 n − (cid:104) N ( N − (cid:105) n − + (cid:104) N ( N +1) (cid:105) n − (cid:104) N ( N − (cid:105) n − − (cid:104) N ( N +1) (cid:105) n − . (B.11)Because the expansion coefficient (cid:16) g (cid:17) is applied to each double-plaquette in eq.(B.3), a factor of n (cid:16) g (cid:17) n isapplied to n double-plaquettes.Finally, we perform the integration over the remaining link variables on the loop C as the boundary of the S area.As shown in Fig.B.3, we express W ( C × C ) as W ( C × C ) := tr( AXAX ). To summarize the above arguments,from eq.(B.2), eq.(B.3), eq.(B.8) and eq.(B.11) etc., (cid:104) W ( C × C ) (cid:105) q N is written by (cid:104) W ( C × C ) (cid:105) q N = (cid:18) g N (cid:19) S − S · (cid:18) g (cid:19) S (cid:90) dAdX tr( AXAX ) (cid:8) x [tr( A † X † )] − y tr( A † X † A † X † ) (cid:9) , (B.12)6 C A X
FIG. B.3: The path-ordered product of the link variables along the loop C is decomposed into A and X to express W ( C × C ) := tr( AXAX ). where x := (cid:20) N ( N − (cid:21) n − + (cid:20) N ( N + 1) (cid:21) n − , y := (cid:20) N ( N − (cid:21) n − − (cid:20) N ( N + 1) (cid:21) n − . (B.13)Using eq.(A.20) and eq.(A.21) to perform X integration, we finally obtain (cid:104) W ( C × C ) (cid:105) q N = q N (cid:18) g N (cid:19) S + S ,q N = − N S (cid:40)(cid:20) N ( N − (cid:21) S − − (cid:20) N ( N + 1) (cid:21) S − (cid:41) , ( S ≥ , (B.14)where S ≥ n ≥ (cid:104) W ( C × C ) (cid:105) q N = 0 when S = 1by using explicit group integration. Appendix C: Explicit calculation of the coefficient p In this section, we show explicitly how eq.(22) is obtained. From eq.(8) and eq.(9), a contribution to a coplanardouble-winding Wilson loop average (cid:104) W ( C × C ) (cid:105) from the top panel of Fig.7 is expressed as (cid:104) W ( C × C ) (cid:105) q = (cid:90) (cid:89) (cid:96) ∈ S dU (cid:96) W ( C × C ) · (cid:89) p j ∈ ( S − S ) (cid:20) g tr( U † p j ) (cid:21) · (cid:89) p k ∈ S (cid:20) g tr( U p k ) (cid:21) , (C.1)where U † p j and U p k stand respectively for plaquette variables on the ( S − S ) and S areas. Here note that U † p and U p respectively represent the plaquette variables for the plaquette p with clockwise and counterclockwise orientations.In this section, we focus on the N = 3 case.First, the integration with respect to the link variables { U (cid:96) } on the ( S − S ) area can be performed with the sametechnique of the strong coupling expansion as that for the fundamental Wilson loop to obtain (cid:104) W ( C × C ) (cid:105) p = (cid:18) g N (cid:19) S − S (cid:104) W ( C × C ) (cid:105) p , (C.2)where we have defined (cid:104) W ( C × C ) (cid:105) p := (cid:90) (cid:89) (cid:96) ∈ S dU (cid:96) W ( C × C ) · (cid:89) p k ∈ S (cid:20) g tr( U p k ) (cid:21) . (C.3)Next, we perform the integration in eq.(C.3) over link variables { U (cid:96) } inside of the S area, which excludes the linkson the loop C as the boundary of the S area. As shown in Fig.C.1, performing the integration over the link variable U using eq.(A.18) for two plaquettes that have a common link U , we obtain (cid:90) dU tr( XU ) · tr( U † Y ) = 1 N tr( XY ) . (C.4)From this observation, we conclude that one factor of 1 /N appears if two plaquettes are connected after commonlinks are integrated. When S plaquettes are connected one after another by using eq.(C.4), a factor of (1 /N ) S − is7 ! dU X YU = X YU † N FIG. C.1: Diagrammatic representation of the integration rule eq.(C.4) for the product of two plaquettes with the samecounterclockwise orientation: Integration is performed over the link variable U on the link which is common to two plaquetteswith the same counterclockwise orientation. The plaquette variables for the plaquette p and p to the left and right of U is respectively represented by tr( U p ) := tr( XU ) and tr( U p ) := tr( U † Y ). Here X and Y represent the products of the linkvariables along staple-shaped paths with the same orientations. applied, and after that only the path ordered product of the link variables on the loop C as the boundary of S isleft unintegrated. Therefore, eq.(C.2) becomes (cid:104) W ( C × C ) (cid:105) p = (cid:18) g N (cid:19) S − S · N (cid:18) g N (cid:19) S (cid:90) [ U ] ∈ C d [ U ] W ( C × C ) · W ( C ) , (C.5)where the integral is only for the link variable on the loop C .As shown in Fig.B.3, by using the decomposition W ( C ) := tr( AX ) and W ( C × C ) := tr( AXAX ), and byrepeatedly using eq.(A.9), we obtain (cid:90) [ U ] ∈ C d [ U ] W ( C × C ) · W ( C ) = (cid:90) dAdX tr( AXAX )tr( AX )= (cid:90) dAdX tr( XAXA )tr( XA )= (cid:90) dAdX ( X ) ab ( A ) bc ( X ) cd ( A ) da · ( X ) pq ( A ) qp = 1 N ! (cid:15) acp (cid:15) bdq (cid:90) dA ( A ) bc ( A ) da ( A ) qp = 1 N ! (cid:15) acp (cid:15) bdq · N ! 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