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Non-Abelian magnetic monopole dominance for SU(3) Wilson loop average Kei-Ichi Kondo , Akihiro Shibata†, Toru Shinohara and Seikou Kato ∗ ∗ ∗∗ DepartmentofPhysics,GraduateSchoolofScience,ChibaUniversity,Chiba263-8522,Japan ∗ †ComputingResearchCenter,HighEnergyAcceleratorResearchOrganization,Tsukuba305-0801,Japan 1 ∗∗FukuiNationalCollegeofTechnology,Sabae916-8507,Japan 1 0 Abstract. Weshowthatthenon-Abelianmagneticmonopoledefinedinagauge-invariantwayinSU(3)Yang-Millstheory 2 givesadominantcontributiontoconfinementofthefundamentalquark,insharpcontrasttotheSU(2)case. n Keywords: Quarkconfinement,magneticmonopole,dualsuperconductor,Wilsonloop a PACS: 12.38.Aw,21.65.Qr J 4 2 INTRODUCTION confinement. In this case, the problem is not settled yet. In this talk, we give a theoretical framework for ] The dual superconductorpicture proposed long ago [1] describing non-Abelian dual superconductivity in D- h is believed to be a promising mechanicsfor quarkcon- dimensionalSU(N)Yang-Millstheory,whichshouldbe t - finement. For this mechanism to work, however, mag- compared with the conventionalAbelianU(1)N 1 dual p − e netic monopoles and their condensation are indispens- superconductivityin SU(N)Yang-Millstheory,hypoth- h abletocausethedualMeissnereffectleadingtothelin- esizedbyAbelianprojection.Wedemonstratethatanef- [ earpotentialbetweenquarkandantiquark,namely,area fective low-energy description for quarks in the funda- lawoftheWilsonloopaverage.TheAbelianprojection mentalrepresentation(abbreviatedtorep.hereafter)can 2 v method proposed by ’t Hooft [2] can be used to intro- begivenbyasetofnon-Abelianrestrictedfieldvariables 8 ducesuchmagneticmonopolesintothepureYang-Mills and that non-AbelianU(N 1) magnetic monopoles in 4 theoryevenwithoutmatterfields.Indeed,numericalev- the sense of Goddard–Nuy−ts–Olive–Weinberg [17] are 6 idences supporting the dual superconductor picture re- the mostdominanttopologicalconfigurationsfor quark 0 sulting from such magnetic monopoles have been ac- confinementasconjecturedin[18,19]. . 2 cumulated since 1990 in pure SU(2) Yang-Mills theory 1 [3,4,5].However,theAbelianprojectionmethodexplic- 0 itlybreaksboththelocalgaugesymmetryandtheglobal WILSON LOOPANDGAUGE-INV. 1 colorsymmetry bypartialgaugefixingfromanoriginal MAGNETIC MONOPOLE : v non-Abelian gauge group G = SU(N) to the maximal Xi torus subgroup, H =U(1)N−1. Moreover, the Abelian A version of a non-Abelian Stokes theorem (NAST) dominance [3] and magnetic monopole dominance [4] r for the Wilson loop operator originally invented by Di- a wereobservedonlyinaspecialclassofgauges,e.g.,the akonovandPetrov[6]forG=SU(2)wasprovedtohold maximally Abelian (MA) gauge and Laplacian Abelian [7] and was extended to G=SU(N) [18, 20] in a uni- (LA)gauge,realizingtheideaofAbelianprojection. fied way [20] as a path-integral rep. by making use of For G = SU(2), we have already succeeded to set- a coherent state for the Lie group. For the Lie algebra tle the issue of gauge (in)dependence by introducing a su(N)-valued Yang-Mills field Am (x)=AmA(x)TA with gauge-invariant magnetic monopole in a gauge inde- su(N) generatorsT (A=1, ,N2 1),the NAST en- pendent way, based on another method: a non-Abelian A ··· − ablesonetorewriteanon-AbelianWilsonloopoperator Stokes theorem for the Wilson loop operator [6, 7] and a new reformulation of Yang-Mills theory rewritten in terms of new field variables[8, 9, 10] and [11, 12, 13], WC[A]:=tr Pexp igYM dxm Am (x) /tr(1), (1) (cid:20) (cid:26) IC (cid:27)(cid:21) elaborating the technique proposed by Cho [14] and Duan and Ge [15] independently,and later readdressed intothesurface-integralform: byFaddeevandNiemi[16]. For G=SU(N), N 3, there are no inevitable rea- sonswhydegreesoffr≥eedomassociated with themaxi- WC[A]=Z dm S (g)exp(cid:20)igYMZS :¶ S =CF(cid:21), (2) mal torus subgroupshould be most dominantfor quark where dm S (g):=(cid:213) x S dm (gx), with an invariant mea- thecolorfieldn[18]:ForSU(3), suredm onG normal∈izedas dm (g )=1,g isan ele- x x mofenGt),ofthaegtawuog-efogrrmouFp G:=(mdRAor=e p12reFcmnise(xly),dmrxemp∧. DdRx(ngxi)s p=2(pS1U(U(3()1/)[)SU=(Z2),×U(1)])=p 1(SU(2)×U(1))(9) defined from the one-form A := Am (x)dx , Am (x) = tr{r [g†xAm (x)gx+ig−YM1g†x¶ m gx]},by whileforSU(2) p (SU(2)/U(1))=p (U(1))=Z. (10) Fmn (x)= 2(N−1)/N[Gmn (x)+ig−YM1tr{r g†x[¶ m ,¶ n ]gx}], 2 1 p (3) ForSU(3),themagneticchargeofthenon-Abelianmag- neticmonopoleobeysthequantizationcondition[20]: withthefieldstrengthGmn definedby Gmn (x):=¶ m tr{n(x)An (x)}−¶ n tr{n(x)Am (x)} Qm:=Z d3xk0=2p √3g−YM1n, n∈Z. (11) 2(N 1) + N− ig−YM1tr{n(x)[¶ m n(x),¶ n n(x)]}, (4) The NAST shows that the SU(3) Wilson loop operator in the fundamentalrep. detects the inherentU(2) mag- andanormalizedtracelessfieldn(x)calledthecolorfield netic monopole which is SU(3) gauge invariant. The rep. can be classified by its stability group H˜ of G n(x):= N/[2(N 1)]g [r 111/tr(111)]g†. (5) [18, 20]. For the fundamentalrep. of SU(3), the stabil- − x − x p ity group is U(2). Therefore, the non-Abelian U(2) Herer isdefinedasr := L L usingareferencestate SU(2) U(1) magnetic monopole follows from H˜ ≃= (highestorlowestweights|tatieho|ftherep.) L makinga SU(2)× U(1) ,whiletheAbelianU(1) U(1)mag- rep. of the Wilson loop we consider. Note|thiat tr(r )= netic m1,o2,n3o×pole c8omes from H˜ =U(1) ×U(1) . The 3 8 L L =1followsfromthenormalizationof L . adjoint rep. belongs to the latter case. Th×e former case h | i | i Finally, the Wilson loop operator in the fundamental occurs only when the weight vector of the rep. is or- rep.ofSU(N)reads[20] thogonalto some of root vectors.The fundamentalrep. isindeedthiscase.ForSU(2),suchadifferencedoesnot WC[A]= dm S (g)exp igYM(k,X S )+igYM(j,NS ) , exist andU(1) magnetic monopoles appear, since H˜ is Z { } always U(1) for any rep.. For SU(3), our result is dif- k:=d f = df, j:=d f, f := 2(N 1)/NG, ferent from Abelian projection: two independent U(1) ∗ ∗ − X S := dQ S D −1=d Q S D −1, NS :=pd Q S D −1, (6) magneticmonopolesappearforanyrep.,since ∗ ∗ p (SU(3)/U(1) U(1))=p (U(1) U(1))=Z2. wheretwoconservedcurrents,“magnetic-monopolecur- 2 × 1 × (12) rent” k and “electric current” j, are introduced, D := dd +d distheD-dimensionalLaplacian,andQ isanan- tisymmetric tensor of rank two called the vorticity ten- NUMERICALSIMULATIONS sor:Q Smn (x):= S d2Smn (x(s ))d D(x x(s )),whichhas the support on Rthe surface S (with t−he surface element The SU(3) Yang-Mills theory can be reformulated in dSmn (x(s )))whoseboundaryistheloopC.Incidentally, the continuumandona lattice usingnewvariables.For the last part ig−YM1tr{r g†x[¶ m ,¶ n ]gx} in F correspondsto SU(3),twooptionsarepossible,maximalforH˜ =U(1)2 the Dirac string [23, 24], which is not gauge invariant [25,26]andminimalforH˜ =U(2)[21].Inourreformu- anddoesnotcontributetotheWilsonloopintheend. lation,allthenewvariablesCm ,Xm andnareobtained staFteorLSU=(3()0i,n0,t1h)elfeuanddsatmoentalrep.,thelowest-weight fromAm : h | AmA= (nb ,Cnk,Xnb), (13) ⇒ n(x)=gx(l 8/2)g†x∈SU(3)/[SU(2)×U(1)]≃CP2, oncethecolorfieldnisdeterminedbysolvingthereduc- (7) tioncondition: with the Gell-Mann matrix l := diag.(1,1, 2)/√3, 8 whileforSU(2), L =(0,1)yields − ccc [A,n]:=[n,Dm [A]Dm [A]n]=0, (14) h | n(x)=g (s /2)g† SU(2)/U(1) S2 CP1, (8) On a four-dimensional Euclidean lattice, gauge field x 3 x∈ ≃ ≃ configurations Ux,m are generated by using the stan- with the Pauli matrix s :=diag.(1, 1). The existence dardWilson ac{tiona}ndpseudoheat-bathmethod.Fora 3 − of magnetic monopole can be seen by a nontrivial Ho- given Ux,m ,colorfield nnnx aredeterminedbyimpos- motopyclassofthemapnfromS2 tothetargetspaceof ing a {lattice}version of r{edu}ction condition. Then new ACKNOWLEDGMENTS Potential 244 lattice beta=6.0 1.2 1 ThisworkissupportedbyGrant-in-AidforScientificRe- 0.8 s U e2 = 0.0413 (T=6) search(C)21540256fromJapanSocietyforthePromo- tionofScience(JSPS). eR) 0.6 V( 0.4 s V e2 = 0.038 (T=10) 0.2 s M e2 = 0.0352 (T=10) REFERENCES 0 0 2 4 6 8 10 R/ e 1. Y.Nambu,Phys.Rev.D10,4262(1974). G. ’t Hooft, in: High Energy Physics, edited by A. FIGURE1. SU(3)quark-antiquarkpotential:(fromaboveto Zichichi(EditoriceCompositori,Bologna,1975). below)fullpotentialVf(r),restrictedpartVa(r)andmagnetic– S.Mandelstam,Phys.Report23,245(1976). monopolepartVm(r)atb =6.0on244(e :latticespacing). A.M.Polyakov,Nucl.Phys.B120,429(1977). 2. G.’tHooft,Nucl.Phys.B190[FS3],455(1981). 3. T. Suzuki and I. 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