We consider the SU(2) lattice gauge model. We propose a new gauge invariant
definition of center projection, which we call the Simple Center Projection. We
demonstrate the center dominance, i.e., the coincidence of the projected
potential with the full potential up to the mass renormalization term at low
energies. We also consider the center vortices and the center monopoles ({\em
nexuses}). It turns out that the behavior of such objects qualitatively
coincides with the behavior of the vortices and monopoles in the Maximal Center
gauge. The connection of the condensation of nexuses with the dual
superconductor theory is discussed. Numerically the procedure of extracting the
center vortices proposed in this paper is much simpler than the usual Maximal
Center Projection.Comment: 11 pages LaTeX, four figure

A.I. Veselov

Alexandrou

Ambjørn

B.L.G. Bakker

Bakker

Bornyakov

Chernodub

de Forcrand

Del Debbio

Faber

Greensite

Ivanenko

M.A. Zubkov

Ogilvie

Polikarpov

Volovik

Zubkov

English

arXiv

Crossref

Nuclear Physics B - Proceedings Supplements

The simple center projection of SU(2) gauge theory

Bakker, B.L.G.

Veselov, A.I.

Zubkov, M.A.

DSpace at VU

the Simple Centern Projection of SU (2) Gauge Theory

We consider the SU(2) lattice gauge model. We propose a new gauge invariant definition of center projection, which we call the Simple Center Projection. We demonstrate the center dominance, i.e., the coincidence of the projected potential with the full potential up to the mass renormalization term at low energies. We also consider the center vortices and the center monopoles (nexuses). It turns out that the behavior of such objects qualitatively coincides with the behavior of the vortices and monopoles in the Maximal Center gauge. The connection of the condensation of nexuses with the dual superconductor theory is discussed. Numerically the procedure of extracting the center vortices proposed in this Letter is much simpler than the usual Maximal Center Projection. © 2001 Published by Elsevier Science B.V

NARCIS 

VU Research Portal

VU Research Portalthe Simple Centern Projection of SU (2) Gauge TheoryBakker, B.L.G.; Veselov, A.I.; Zubkov, M.A.published inPhysics Letters B2001DOI (link to publisher)10.1016/S0370-2693(00)01343-5document versionPublisher's PDF, also known as Version of recordLink to publication in VU Research Portalcitation for published version (APA)Bakker, B. L. G., Veselov, A. I., & Zubkov, M. A. (2001). the Simple Centern Projection of SU (2) Gauge Theory.Physics Letters B, 497, 159-164. https://doi.org/10.1016/S0370-2693(00)01343-5General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.E-mail address:vuresearchportal.ub@vu.nlDownload date: 23. May. 20214 January 2001Physics Letters B 497 (2001) 159–164www.elsevier.nl/locate/npeThe simple center projection ofSU(2) gauge theoryB.L.G. Bakkera, A.I. Veselovb, M.A. Zubkovb,∗a Department of Physics and Astronomy, Vrije Universiteit, Amsterdam, The Netherlandsb ITEP, B.Cheremushkinskaya 25, Moscow, 117259, RussiaReceived 18 July 2000; received in revised form 31 October 2000; accepted 21 November 2000Editor: P.V. LandshoffAbstractWe consider theSU(2) lattice gauge model. We propose a new gauge invariant definition of center projection, which we callthe Simple Center Projection. We demonstrate the center dominance, i.e., the coincidence of the projected potential with the fullpotential up to the mass renormalization term at low energies. We also consider the center vortices and the center monopoles(nexuses). It turns out that the behavior of such objects qualitatively coincides with the behavior of the vortices and monopolesin the Maximal Center gauge. The connection of the condensation of nexuses with the dual superconductor theory is discussed.Numerically the procedure of extracting the center vortices proposed in this Letter is much simpler than the usual MaximalCenter Projection. 2001 Published by Elsevier Science B.V.1. IntroductionWe may need to go a long way before we understandhow the confinement mechanism works within non-Abelian gauge theory. A lot of physicists now considerthe Abelian projection as the best way to understandthe confinement mechanism.After Abelian projection one should consider theelement of some Abelian subgroup of the gauge groupinstead of the full gauge group element. Thus, thetheory becomes Abelian and one can look at thepicture of the confinement phenomenon in a simplerway.Abelian projections differ from each other by thechoice of the subgroup of the gauge group and theprojection method. The closeness of a given Abelianprojection to the solution of the confinement problem* Corresponding author.E-mail address:zubkov@vxitep.itep.ru (M.A. Zubkov).is measured as follows. Suppose that the link gaugegroup elementsglink ∈ G are projected onto theelements of the given Abelian subgroupelink ∈E ⊂G.We consider(1)ZC = Tr∏link∈Celink,instead of the Wilson loop and extract the potentialfrom ZC (the projectedpotential). If that potential isclose to the original confining potential at sufficientlylarge distances one can say that the projection is suit-able for the investigation of the confinement mecha-nism.Dealing with theSU(2) gauge group, the CartansubgroupU(1) and the center subgroupZ2 havebeen considered. The most popular projections are theMaximal Abelian and the Maximal Center projections.Those projections are achieved by minimization withrespect to the gauge transformations of the distancebetween the given configuration of the link gaugefield and the Cartan (center) subgroup ofSU(2). In0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(00)01343-5160 B.L.G. Bakker et al. / Physics Letters B 497 (2001) 159–164both cases the potentials are very close to theSU(2)confining potential but unfortunately do not exactlycoincide with it. Greensite et al. [1–3] have arguedthat only central chargesq =±1 are confined in non-Abelian gauge theories.From the work of Bornyakov et al. [4] it is knownthat gauge fixing procedures suffer from a gaugecopies problem. Several gauges have been used so farin practical computations: the direct and indirect [3]center gauges and the Laplacian center gauge [5].Only the first two suffers from the occurrence ofgauge copies, the last one is free from this difficulty.According to Refs. [1,6] this problem disappears forlarge lattices and when the number of Gribov copiesconsidered is increased. Of course, this means that thecomputational effort must also be increased.In this Letter we propose a new center projectionwhich is not connected with partial gauge fixing.Thus, all the objects existing within the projectedtheory have a gauge invariant nature. We call thisprocedure the Simple Center Projection (SCP). Basedon numerical simulations we make the conjecturethat the projected potential coincides with theSU(2)potential up to the mass renormalization term atsufficiently large distances.Within the SCP we can construct the center vorticesand the center monopoles (also known asnexuses,see Refs. [7] and [8]). The properties of the SCPmonopoles found in our investigations lead us to be-lieve that those monopoles are the objects which playthe role of the Cooper pairs in the dual superconductor.2. Simple center projectionWe considerSU(2) gluodynamics with the Wilsonaction(2)S(U)= β∑plaq(1− 1/2 TrUplaq).The sum runs over all the plaquettes of the lattice. Theplaquette actionUplaq is defined in the standard way.First we consider the plaquette variablezplaq= 1, if Tr Uplaq< 0,(3)zplaq= 0, if Tr Uplaq> 0.We can representz as the sum of a closed formdN 1for N ∈ {0,1} and the form 2m+ q . HereN = Nlink,q ∈ {0,1}, andm ∈ Z.(4)z= dN + 2m+ q.The physical variables depending uponz could beexpressed through(5)signTrUplaq= cos(π(dN + q)).We shall say thatNlink is the center projected linkvariable. There are many different ways to make thisprojection. The Maximal Center Projection (MCP)uses the gauge ambiguity to make all link matricesU as close as possible toeiπN . Thus the 1-formNis fixed for every gauge configuration.There exist several ways to achieve this [3]. Oneway, thedirect way, is to minimize the quantity(6)R =∑x∑µTr[Uµ(x)]Tr[U†µ(x)].In the indirect wayone minimizes the quantity(7)R′ =∑x∑µTr[Uµ(x)σ3U†µ(x)σ3],and extracts fromUµ(x) the diagonal partAµ =exp[iθµ(x)σ3] thus fixing the gauge. Finally the rem-nant Abelian symmetry is used to bringAµ as close toan element ofZ2 as possible by maximizing(8)R′′ =∑x∑µcos2 θµ(x).It is clear that both procedures are complicated andtime consuming, as the number of variables involvedin the case of, e.g.,SU(2) is three. Whatever methodis used, the 1-formN is fixed for every gaugeconfiguration.Now we choose a simpler and more natural proce-dure. Imagine the surfaceΣ formed by the plaquettesdual to the “negative” plaquettes (for whichzplaq= 1).This surface has a boundary. We enlarge the surface byadding a surfaceΣadd so that:(1) the resulting surfaceΣ1 = Σ + Σadd will beclosed;1 We use the formulation of differential forms on the lattice, asdescribed for instance in Ref. [9].B.L.G. Bakker et al. / Physics Letters B 497 (2001) 159–164 161(2) when we eliminate from the surfaceΣadd theplaquettes carrying even numberszplaq= . . . ,−4,−2,2,4, . . . , the area of the remnant surface willbe minimal for the given boundary.SoΣ1 can be represented by the closed formdNon the original lattice for the integer link variableN ∈ {0,1}. AndN is the required center projected linkvariable.Numerically this procedure looks as follows. For thegiven variablez we should choose such aZ2 variableN that[dN]mod2 is as close as possible toz. It meansthat we minimize the functional(9)Q=∑plaq∣∣(z− dN)mod2∣∣,with respect toN .The procedure works in the following way. Weconsider a given linkL and the sum over plaquettes(10)Qlink =∑L∈plaq∣∣(z− dN)mod2∣∣.We minimize this sum with respect to the one linkvalue N . All links are treated in this way and theprocedure is iterated until a global minimum is found.We call this unique and simple procedure theSimple Center Projection. Our projection methodalso finds local minima, Gribov copies, but as theprocedure is much simpler and faster that the methodsuse until now, we can afford to repeat our calculationsto include several Gribov copies.The physical meaning of the projected variablesbecomes clear after considering the continuum limit.Naively, the considered surfaces disappear as the fieldstrength on them tends to infinity. Nevertheless thisfact should be investigated more carefully. In any casefor reasonable sizes of lattices finite volume effectsoccur and the scaling window ends at some valueof β . Thus the direct drop into the continuum limit isimpossible and within the scaling window our surfacesΣ carry large but not infinite field strength. The resultsof the next section give us the reason to believe thatthese surfaces are those which play the crucial role inthe confinement mechanism.Let us also mention that it is reasonable to considerthe analogous construction for which we change theplaquettes into loops of sizes 2× 2, 3× 3, . . . . Theseextended projections solve the problem of the positiveplaquette model, for which our “1× 1” Σ is absent.3. Numerical resultsIn the calculations we report on here, we used as ourstandard lattice one with dimension 244 and for inves-tigations at finite temperature one with size 243× 4.For reasons of comparison we have occasionally usedsmaller lattices, of sizes 123 × 4 and 163 × 4. Someresults were checked on the larger lattice 323× 4. Wehave mentioned the gauge copy problem. We checkedthat for lattices with linear dimensionsL6 24, whichwe have used, 15 copies are sufficient and so we usedeverywhere 15 Gribov copies.3.1. Center dominanceWe consider the following definition of the pro-jected Wilson loop(11)WSCPC =18ZC(3π/4)P(C)/4,whereP(C) is the perimeter of the loopC.It may be interesting for the reader that this expres-sion is very similar to the first term in the character ex-pansion from [6,10]. One can get the above expressionfrom that formula substituting the factor 1/8 insteadof 1/4 and the powerP(C)/4 instead ofP . However,as the authors of Refs. [6,10] derive their expressionfor lattices without gauge fixing, they can derive theirresults with local operators, using the characters of thegauge group to obtain dominance of the fundamentalrepresentation. In our case, we perform gauge fixing,which results in a nonlocal operator. Consequently, wecannot use the same derivation as Refs. [6,10]. As wehave not been able to find a rigorous derivation ofEq. (11), one must consider our results up till now asempirical ones. It should be stressed that these resultswere checked for Wilson loops of sizes up till 6× 6. Itmight be useful to check whether Eq. (11) continues togive good results if larger Wilson loops are consideredand the statistics is improved.The reader can recognize from Fig. 1 thatWCandWSCPC exactly coincide with each other for largeenough sizes of the loop (we represent here− logWC/Area(C) and − logWSCPC /Area(C) as afunction of the loop area for the valuesβ = 2.3and 2.4).WSCPC differs fromZC by the perimeter fac-tor. Thus we understand that the projected potential(extracted fromZC ) differs from the full potential (ex-162 B.L.G. Bakker et al. / Physics Letters B 497 (2001) 159–164Fig. 1.− log(WC)/Area(C) for all Wilson loops,WC , and com-puted from projected ones,WSCPC . β = 2.3 and 2.4.tracted fromWC ) only by the mass renormalizationterm at low energies. Other terms (including the con-fining linear term) are the same. It demonstrates theCenter Dominance.3.2. The center vorticesWe construct the closed two-dimensional centervortices as in [7](12)σ = ∗dN.First, we can expressZC as(13)ZC = exp(iπL(σ,C))whereL is the linking number [7,11].Thus, according to the previous subsection theAharonov–Bohm interaction between the center vor-tices and the charged particle leads to the confinementof the fundamental charge [7,11].Also we investigate other properties of the centervortices for the finite temperature theory (the nonsym-metric lattice 243× 4).The density of the vorticesρ is shown in Fig. 2. Thefractal dimension, defined asD = 1+2A/L, whereAis the number of plaquettes andL is the number oflinks [12] of the vortices, is shown in Fig. 3. A lineLFig. 2. Density of center vortices and monopoles. The triangle pointsat the position of the phase transition.Fig. 3. Fractal dimensionD of the center vortices. The trianglepoints at the position of the phase transition.is counted as belonging to the vortex if at least one ofthe faces of a cube dual to that link contains a plaquettewith charge 1.B.L.G. Bakker et al. / Physics Letters B 497 (2001) 159–164 1633.3. The center monopoles (nexuses)Following [7] we construct the center monopoles(nexuses)(14)j = 12∗d[dN]mod2.We show the density of the nexuses as a functionof β in Fig. 2. It is important to make sure thatthe monopole lines are closed. It follows from theequationδj = 12∗d∗∗d[dN]mod2= 0.We investigated the percolation properties ofjand considered the probability of two points to beconnected by a monopole worldline on a constant-timehypersurface. The dependence of that probability uponβ for a nonsymmetric lattice is shown in Fig. 4. Thepercolation probability is dependent on the lattice size.This is obvious for very small lattices, but it is seen topersists at larger sizes.Here we would like to remark on the role the mono-pole condensate plays as an order parameter. Iva-nenko et al. [13] have shown for Abelian monopoles,obtained after Maximal Abelian projection, that themonopole condensate can be used as an order parame-ter: it vanishes in the deconfined phase, while it takesa finite value in the confining phase. The authors ofRef. [13] made the interesting observation that the be-havior of the condensate near the point of the phasetransition depends on the “thickness” of the monopolelines. It was demonstrated in Ref. [14] that Abelianmonopoles are strongly connected with vortices. Forvortex lines with a thickness of one lattice spacing, thecondensate vanishes smoothly near the critical point,whereas for lines with a thickness of two lattice units,the variation of the condensate near the critical pointis very steep.We see that the monopoles are condensed in theconfinement phase and not condensed in the decon-finement phase, but the phase transition is rathersmooth as is the case for thin Abelian monopoles.If we would investigate, along the lines of Ref. [13]nexuses of larger sizes like 23, 33 and so forth, wethink that we shall obtain a better sensitivity of thecondensatesC(2), C(3) etc. to the phase transition.(Here we use the obvious notationC(n) for the con-densate of vortices of sizen3.)We expect the center monopoles to be the monopo-les that are present in the dual superconductor pictureFig. 4. Probability of connecting two points by a monopoleworldline for lattices 243 × 4. The triangle points at the positionof the phase transition.of confinement. As the phase transition is not very pro-nounced for the thin vortices we considered here, ourresults may be taken as a “proof of principle”. Theymust be substantiated by considering thick vorticeswhich are supposed to be more strongly connected toconfinement [15].The analytical connection of the monopole conden-sation and the formation of the dual superconductorwas considered in [16] for the case ofSU(3) sym-metry. Of course, the results of that paper obtain alsofor the SU(2) theory. It follows from [16] that in thecase both condensation of nexuses and center domi-nance occur, the picture of the dual superconductor inwhich the nexuses play the role of Cooper pairs andthe quarks play the role of the monopoles becomesclear.In particular, one can rewrite the fundamental Wil-son loop as follows〈W(C)〉(15)=π∫−πDH∫DΦ∈CΦ× exp(−Q(dH + π∗A[C])−∑xyΦxe2iHxyΦ+y − V (|Φ|)).164 B.L.G. Bakker et al. / Physics Letters B 497 (2001) 159–164HereH is the electromagnetic field,A[C] is the areaof the surface spanned on the quark loop,Φ is thenexus field,Q is nonlocal effective action,V is aninfinitely deep potential, supporting the infinite valueof the nexus condensate.Thus we have indeed the nonlocal relativistic super-conductor theory, in which the nexuses are the Cooperpairs and the quarks are the monopoles. The con-densation of nexuses gives rise to the formation ofthe quark–antiquark string appearing as the Abrikosovvortex.4. ConclusionsIn this Letter we construct the gauge invariant Cen-ter projection and show that center dominance takesplace. We investigate the properties of the topologicaldefects in the center projected theory, which are shownto be closely connected to the confinement picture.Particularly it occurs that the center monopoles fromthis projection are good candidates for the Cooperpairs in the dual superconductor. The simple numer-ical nature of the Simple Center Projection, the exactCenter Dominance and the properties of the topologi-cal defects existing in the center projected theory giveus the reason to propose SCP as the basic Abelian pro-jection for the considering of the confinement picture.AcknowledgementsWe are grateful to V.G. Bornyakov, M.N. Chern-odub, and M.I. Polikarpov for useful discussions.A.I.V. and M.A.Z. kindly acknowledge the hospitalityof the Department of Physics and Astronomy of theVrije Universiteit, where part of this work was done.This work was partly supported by the grants RFBR99-01-01230, INTAS 96-370.References[1] J. Greensite, Talk at Confinement 2000, Osaka, Japan, March7–10, 2000, hep-lat/0005001.[2] L. Del Debbio, M. Faber, J. Giedt, J. Greensite, S. Olejnik,Phys. Rev. D 58 (1998) 094501.[3] L. Del Debbio, M. Faber, J. Greensite, S. Olejnik, hep-lat/9802003.[4] V.G. Bornyakov, D.A. Komarov, M.I. Polikarpov, A.I.Veselov, Talk at Confinement 2000, Osaka, Japan, March 7–10, 2000, hep-lat/0002017.[5] Ph. de Forcrand, M. D’Elia, Phys. Rev. Lett. 82 (1999) 4582;C. Alexandrou, M.M. D’Elia, Ph. de Forcrand, hep-lat/9907028.[6] M. Faber, J. Greensite, S. Olejnik, D. Yamada, hep-lat/9912002.[7] M.N. Chernodub, M.I. Polikarpov, A.I. Veselov, M.A. Zubkov,Nucl. Phys. Proc. Suppl. 73 (1999) 575, hep-lat/9809158.[8] G.E. Volovik, Pisma Zh. Eksp. Teor. Fiz. 70 (1999) 776, cond-mat/9911374.[9] M.I. Polikarpov, U.J. Wiese, M.A. Zubkov, Phys. Lett. B 309(1993) 133.[10] M.C. Ogilvie, Nucl. Phys. Proc. Suppl. 73 (1999) 542, hep-lat/9809167.[11] M.N. Chernodub, M.I. Polikarpov, A.I. Veselov, Phys. Lett.B 399 (1997) 267, hep-lat/9610007.[12] B.L.G. Bakker, A.I. Veselov, M.A. Zubkov, Phys. Lett. B 471(1999) 214, hep-lat/9902010.[13] T.L. Ivanenko, A.V. Pochinsky, M.I. Polikarpov, Phys. Lett.B 302 (1993) 458.[14] J. Ambjørn, J. Geidt, J. Greensite, hep-lat/9907021.[15] M. Faber, J. Greensite, S. Olejnik, hep-lat/0005017.[16] M.A. Zubkov, hep-lat/0001034.

https://research.vu.nl/en/publications/a48837b0-090a-49bf-b510-a80eb91ada35

The Simple Center Projection of SU(2) Gauge Theory

The Simple Center Projection of SU(2) Gauge Theory

Abstract

Similar works

Full text

Available Versions

Crossref

Crossref

DSpace at VU

NARCIS

VU Research Portal