Table Of ContentNon-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. Yotsuyanagi, Phys. Rev. D42,
4257(1990).
variables are introduced by using the lattice version of
4. J.D.Stack,S.D.NeimanandR.Wensley,Phys.Rev.D50,
changeofvariables[22].
3399(1994). H.ShibaandT.Suzuki,Phys.Lett.B333,
Fig. 1 shows the full SU(3) quark-antiquark poten- 461(1994).
tial V(r) obtained from the SU(3) Wilson loop av- 5. K.AmemiyaandH.Suganuma,Phys.Rev.D60,114509
erage W [A] , the restricted part V (r) from the V (1999).V.G.Bornyakov,M.N.Chernodub,F.V.Gubarev,
C a
Wilsonhloopavierage W [V] , andmagnetic–monopole S.M.MorozovandM.I.Polikarpov,Phys.Lett.B559,
C
part Vm(r) from eighYM(k,X S )i. They are gauge invari- 6. 2D1.4D(2ia0k0o3n).ovandV.Petrov,Phys.Lett.B224,131(1989).
h i
ant quantities by construction. These results exhibit in- 7. K.-I.Kondo,Phys.Rev.D58,105016(1998).
fraredV dominanceinthestringtension(85–90%)and 8. K.-I.Kondo,T.MurakamiandT.Shinohara,Prog.Theor.
non-AbelianU(2)magneticmonopoledominanceinthe Phys.115,201(2006).
stringtension(75%)inthegaugeindependentway. 9. K.-I.Kondo,T.MurakamiandT.Shinohara,Eur.Phys.J.
C42,475(2005).
10. K.-I.Kondo,Phys.Rev.D74,125003(2006).
11. S. Kato, K.-I. Kondo, T. Murakami, A. Shibata, T.
CONCLUSION
ShinoharaandS.Ito,Phys.Lett.B632,326(2006).
12. S.Ito,S.Kato,K.-I.Kondo,T.Murakami,A.Shibataand
We have shown: (i) The SU(N) Wilson loop operator T.Shinohara,Phys.Lett.B645,67(2007).
can be rewritten in terms of a pair of gauge-invariant 13. A. Shibata, S. Kato, K.-I. Kondo, T. Murakami, T.
ShinoharaandS.Ito,Phys.Lett.B653,101(2007).
magnetic-monopolecurrentk((D 3)-form)andtheas-
− 14. Y.M.Cho,Phys.Rev.D21,1080(1980);Phys.Rev.D23,
sociated geometric object defined from the Wilson sur-
2415(1981).
faceS boundingtheWilsonloopC,andanotherpairof 15. Y.S.DuanandM.L.Ge,SinicaSci.,11,1072(1979).
anelectriccurrent j (one-formindependentlyofD)and 16. L.FaddeevandA.Niemi,Phys.Rev.Lett.82,1624(1999).
the associated topologicalobject, which follows from a 17. P.Goddard,J.NuytsandD.Olive,Nucl.Phys.B125,
non-Abelian Stokes theorem for the Wilson loop oper- 1(1977).E.J.Weinberg,Nucl.Phys.B167,500(1980)
18. K.-I. Kondo and Y. Taira, Mod. Phys. Lett. A15,
ator [20]. (ii) The SU(N) Yang-Mills theory can be re-
367(2000).Prog.Theor.Phys.104,1189(2000).
formulated in terms of new field variables obtained by
19. K.-I.KondoandY.Taira,Nucl.Phys.Proc.Suppl.83,
changeof variablesfromthe originalYang-Millsgauge
497(2000).
fieldAmA(x)[21], sothatitgivesanoptimaldescription 20. K.-I.Kondo,Phys.Rev.D77,085029(2008).
forthenon-Abelianmagneticmonopoledefinedfromthe 21. K.-I.Kondo,T.ShinoharaandT.Murakami,Prog.Theor.
SU(N)Wilson loopoperatorin the fundamentalrep. of Phys.120,1(2008).
22. K.-I.Kondo,A.Shibata,T.Shinohara,T.Murakami,S.
quarks.(iii) A lattice versionof the reformulatedYang-
KatoandS.Ito,Phys.Lett.B669,107(2008).
Millstheorycanbeconstructed[22].Numericalsimula-
A.Shibata,K.-I.KondoandT.Shinohara,Phys.Lett.
tionsofthelatticeSU(3)Yang-Millstheorygivenumer-
B691,91(2010).
ical evidences that the restricted field variables become 23. K.-I.Kondo,Phys.Rev.D57,7467(1998).
dominant in the infrared for correlation functions and 24. K.-I.Kondo,Phys.Rev.D58,105019(1998).
the string tension(infraredrestricted non-Abeliandom- 25. Y.M.Cho,Phys.Rev.Lett.44,1115(1980).
inance) and that the U(2) magnetic monopole gives a 26. L.FaddeevandA.J.Niemi,Phys.Lett.B449,214(1999).
Phys.Lett.B464,90(1999).
mostdominantcontributiontothestringtensionobtained
27. K.-I. Kondo, A. Shibata, T. Shinohara and S. Kato,
fromSU(3)Wilsonloopaverage(non-Abelianmagnetic
arXiv:1007.2696[hep-th].
monopoledominance).See[27]formoreinformations.
