Table Of ContentKEKPreprint2016-59
CHIBA-EP-221
Quark confinement to be caused by Abelian or
non-Abelian dual superconductivity in the SU(3)
Yang-Mills theory
7
1
0
Akihiro Shibata
2 ∗
ComputingResearchCenter,HighEnergyAcceleratorResearchOrganization(KEK)
n
a E-mail: [email protected]
J
Kei-IchiKondo
3
1 DepartmentofPhysics,GraduateSchoolofScience,ChibaUniversity,Chiba263-8522,Japan
E-mail: [email protected]
]
t
a SeikouKato
l
- OyamaNationalCollegeofTechnology,Oyama,Tochigi323-0806,Japan
p
E-mail:[email protected]
e
h
ToruShinohara
[
DepartmentofPhysics,GraduateSchoolofScience,ChibaUniversity,Chiba263-8522,Japan
2
E-mail:[email protected]
v
2
Thedualsuperconductivityisapromisingmechanismforquarkconfinement.Wehavepresented
4
4 a newformulationoftheYang-Millstheoryonthe lattice thatenablesusto changetheoriginal
2 non-Abeliangaugefield into the new field variables such that one of them called the restricted
0
field gives the dominant contribution to quark confinementin the gauge independentway. We
.
1
have pointed out that the SU(3) Yang-Mills theory has another reformulation using new field
0
variables(minimaloption),inadditiontothewayadoptedbyCho,FaddeevandNiemi(maximal
7
1 option). Inthepastlatticeconferences,wehaveshownthenumericalevidencesthatsupportthe
:
v non-Abeliandual superconductivityusing the minimaloption for the SU(3) Yang-Millstheory.
i This result should be compared with Abelian dual superconductivity obtained in the maximal
X
optionwhichisagaugeinvariantextensionoftheconventionalAbelianprojectionmethodinthe
r
a maximalAbeliangauge.
Inthistalk, wefocusondiscriminatingbetweentwo reformulations,i.e.,maximalandminimal
optionsoftheSU(3)Yang-Millstheoryfromtheviewpointofdualsuperconductivityforquark
confinement. We investigatetheconfinement/deconfinementphasetransitionsatfinitetempera-
tureinbothoptions,whicharecomparedwiththeoriginalYang-Millstheory. Forthispurpose,
we measure the distribution of Polyakov-loopsand the Polyakov-loop average, the correlation
functionofthePolyakovloopsandthedistributionofthechromoelectricfluxconnectingaquark
andantiquarkinbothconfinementanddeconfinementphases.
34thannualInternationalSymposiumonLatticeFieldTheory
24-30July2016
UniversityofSouthampton,UK
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
QuarkconfinementtobecausedbyAbelianornon-Abeliandualsuperconductivity AkihiroShibata
1. Introduction
The dual superconductivity is a promising mechanism for quark confinement [1]. In order to
establish this picture, we have presented a new formulation of the Yang-Mills theory on the lat-
tice that enables us to decompose the original Yang-Mills (YM) gauge link valiableUx,m into the
gauge link variable Vx,m corresponding to its maximal stability subgroup of the gauge group and
the remainder Xx,m , Ux,m =Xx,m Vx,m , where the restricted field Vx,m could be the dominant mode
for quark confinement. (For a review see [2]). For the SU(3) YM theory, wehave two options of
thedecomposition: theminimalandmaximaloptions. Intheminimaloption, especially, themaxi-
malstability groupisnon-AbelianU(2)andtherestricted fieldcontains thenon-Abelian magnetic
monopole. In the preceding works, we have shown numerical evidences of the non-Abelian dual
superconductivity using the minimal option for the SU(3) YM theory on a lattice: the restricted
field and the extracted non-Abelian magnetic monopole dominantly reproduces the string tension
inthelinearpotentialoftheSU(3)YMtheory[5],andtheSU(3)YMvacuumisthetypeIdualsu-
perconductor detectedbythechromoelectric fluxtubeandthemagneticmonopolecurrentinduced
aroundit,whichisanovelfeatureobtainedbyoursimulations[6]. Wehavefurtherinvestigatedthe
confinement/deconfinement phase transition in view of this non-Abelian dual superconductivity
picture[7][8][9][10]. Theseresults should becompared withthemaximaloption whichisadopted
first by Cho, Faddeev, and Niemi [12]. The maximal stablity group is AbelianU(1) U(1) and
×
the restricted fieldinvolves only theAbelian magnetic monopole [13][14]. Thisisnothing but the
gaugeinvariant extension oftheAbelianprojection inthemaximalAbeliangauge[15][16].
In this talk, we focus on discriminating between two reformulations, i.e., maximal and min-
imal options of the SU(3) YM theory from the viewpoint of dual superconductivity for quark
confinement. For this purpose, we measure string tension for the restricted non-Abelian field of
both minimalandmaximaloption incomparison withthestring tension fortheoriginal YMfield.
We also investigate the dual Meissner effect by measuring the distribution of the chromoelectric
flux connecting a quark and an antiquark and the induced magnetic-monopole current around the
fluxtube.
2. Gaugelinkdecompositions
LetUx,m =Xx,m Vx,m beadecomposition oftheYMlinkvariableUx,m ,wheretheYMfieldand
the decomposed new variables are transformed by full SU(3) gauge transformation W such that
x
Vx,m istransformed asagaugelinkvariableandXx,m asasitevariable [11]:
Ux,m −→Ux′,m =W xUx,m W †x+m , (2.1a)
Vx,m −→Vx′,m =W xVx,m W †x+m , Xx,m −→Xx′,n =W xXx,m W †x. (2.1b)
For the SU(3) YM theory, we have two options discriminated by its stability group, so called the
minimalandmaximaloptions.
2.1 Minimaloption
TheminimaloptionisobtainedforthestabilitysubgaugegroupH˜ =U(2)=SU(2) U(1)
× ⊂
SU(3). By introducing the color field h = x (l 8/2)x † SU(3)/U(2) with l 8 being the last
x
∈
1
QuarkconfinementtobecausedbyAbelianornon-Abeliandualsuperconductivity AkihiroShibata
diagonal Gell-Mann matrix and x an SU(3) group element, the decomposition is given by the
definingequation:
1
e
Dm [V]hx:= e Vx,m hx+m −hxVx,m =0, (2.2a)
g :=ei2p q/3ex(cid:2)p( ia0h i(cid:229) 3 (cid:3)a(j)u(j)). (2.2b)
x − x x− j=1 x x
(j)
Here, the variable g is an undetermined parameter from Eq.(2.2a), u ’s are su(2)-Lie algebra
x x
valued,and qhasanintegervalue. Thesedefiningequationscanbesolvedexactly,andthesolution
isgivenby
Xx,m =L†x,m det(Lx,m )1/3g−x1, Vx,m =Xx†,m Ux,m , (2.3a)
Lx,m :=bLx,m L†x,bm −1/2Lx,m , (2.3b)
5 2
bLx,m :=(cid:0)31+√3(cid:1)(hx+Ux,m hx+m Ux†,m )+8hxUx,m hx+m Ux†,m . (2.3c)
Note that the above defining equations correspond to the continuum version: Dm [V ]h(x)=0 and
tr(h(x)Xm (x)) =0, respectively. Inthe naive continuum limit, wehave reproduced the decompo-
sitionAm (x)=Vm (x)+Xm (x)inthecontinuum theory[3].
The decomposition is uniquely obtained as the solution (2.3) of Eqs.(2.2), if color fields h
x
{ }
are obtained. To determine the configuration of color fields, we use the reduction condition to
formulate the new theory written by new variables (Xx,m ,Vx,m ) which is equipollent to the original
YMtheory. Here,weusethereduction functional:
Fred[hx]=(cid:229) x,m tr (Dem [Ux,m ]hx)†(Dem [Ux,m ]hx) , (2.4)
(cid:8) (cid:9)
andthencolorfields h areobtained byminimizingthefunctional (2.4).
x
{ }
2.2 Maximaloption
The maximal option is obtained for the stability subgroup of the maximal tarus group H˜ =
U(1) U(1) SU(3), and the resulting decomposition is the gauge-invariant extension of the
× ⊂
Abelian projection inthemaximalAbelian A˛iC´lC´‘A˛jgauge. Byintroducing the colorfieldn(3) =
x (l 3/2)x †,n(8)=x (l 8/2)x † SU(3)/U(2)withl 3,l 8 beingthetwodiagonal Gell-Mann ma-
∈
tricesandx anSU(3)groupelement,thedecomposition isgivenbythedefiningequation:
1
e (j) (j) (j)
Dm [V]nx := e Vx,m nx+m −nx Vx,m =0 j=3,8, (2.5a)
g :=ei2p q/3exph( ia3n(3) ia(8)n(8)i). (2.5b)
x x x x x
− −
The variable g is an undetermined parameter from Eq.(2.5a), and q has an integer value. These
x
definingequations canbesolvedexactly,andthesolution isgivenby
Xx,m =Kx†,m det(Kx,m )1/3g−x1, Vx,m =Xx†,m Ux,m , (2.6a)
Kx,m :=bKx,m Kx†,bm −1/2Kx,m , (2.6b)
Kbx,m :=1(cid:0)+6(n(x3)(cid:1)Ux,m nx(3+)m Ux†,m )+6(nx(8)Ux,m nx(8+)m Ux†,m ) (2.6c)
2
QuarkconfinementtobecausedbyAbelianornon-Abeliandualsuperconductivity AkihiroShibata
Yang-Mills field V field minimal option V field maximal option
0 0.0 .015 bbbbbbbbbbbbbbb===============556666666665666..99.............543322110998872005050505505055 - 0000 ....01123 bbb==bbbbbbbbbbbb=============556..66666666566699.............572433221109988055050505055050 - 0000 ....01123 bbbbbbbbbbbbbbb===============556666666665666..99.............543322110998872005050505505055
-0.05 -0.2 -0.2
-0.3 -0.3
-0.1
-0.05 0 0.05 0.1 0.15 0.2 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
Figure 1: Distribution of the space-averagedPolyakovloops: The space-averagedPolyakov loop eq(3.2)
for each configurationare plotted in the complexplane. (a) original(YM) field, (b) restricted field in the
minimaloption,(c)restrictedfieldinthemaximaloption.
Note that the above defining equations correspond to the continuum version: Dm [V]n(j)(x) = 0
and tr(n(j)(x)Xm (x)) = 0, respectively. In the naive continuum limit, we have reproduced the
decomposition Am (x)=Vm (x)+Xm (x)inthecontinuum theory.(See[3][12].)
Todeterminetheconfiguration ofcolorfields,weusethereductionconditiontoformulatethe
new theory written by new variables (Xx,m ,Vx,m ) which is equipollent to the original YM theory.
Here,weusethereduction functional:
Fred[n(x3),n(x8)]=(cid:229) x,m (cid:229) j=3,8tr (Dem [Ux,m ]n(xj))†(Dem [Ux,m ]n(xj)) , (2.7)
n o
(3) (8)
and then color fields n ,n are obtained by minimizing the functional (2.7). It should be
x x
noticedthatthemaximnaloptiongoivesthegaugeinvariantextensionoftheAbelianprojectioninthe
maximalAbeliangauge.
3. Lattice Data
We generate the YM gauge field configurations (link variables) Ux,m using the standard
{ }
Wilson action on the lattice with the size of L3 N = 243 6. We prepare 500 data sets at
T
× ×
b := 2N /g2 (N = 3) = 5.8,5.9,6.0,6.1,6.2,6.3 every 500 sweeps after 10000 thermalization.
c c
The temperature is controlled by changing the parameter b . We obtain two types of decomposed
gauge link variablesUx,m =Xx,m Vx,m for each gauge link by using the formula Eqs.(2.3) and (2.6)
(3) (8)
givenintheprevioussection, afterthecolor-field configuration h and n ,n areobtained
x x x
{ }
bysolvingthereductionconditionasminimizingthefunctional(2.4)and(n2.7),respeoctively. Inthe
measurement ofthe Wilson loop average defined below, weapply the APEsmearing technique to
reducenoises.
First,weinvestigatethedistribution ofasinglePolyakovloopP (x)inbothoptionsaswellas
∗
theoriginal Yang-Millstheory:
(cid:213)NT (cid:213)NT (min) (cid:229) (cid:213)NT (max)
P (x):=tr P U , P (x):=tr P V ,P (x):= tr P V ,
YM (x,t),4 min (x,t),4 max (x,t),4
t=1 ! t=1 ! x t=1 !
(3.1)
Figure1represents thedistribution ofthespace-averaged PolyakovloopP (x)definedby
∗
1 (cid:229) 1 (cid:229) 1 (cid:229)
P = P (x), P = P (x), P = P (x), (3.2)
YM L3 YM min L3 min max L3 max
x x x
3
QuarkconfinementtobecausedbyAbelianornon-Abeliandualsuperconductivity AkihiroShibata
Polyakov loop average Polyakov-loop susceptibility
0.6 0.3
Vmax Vmax
Vmin Vmin
0.5original 0.25 original
0.4 0.2
PL ave 00..23 susceptibility 0 0.1.15
0.05
0.1
0
0
-0.05
5.8 5.9 6 6.1 6.2 6.3 6.4 5.8 5.9 6 6.1 6.2 6.3 6.4
beta beta
Figure2: (Left)ThecombinationplotofthePolyakov-loopaveragesfortheoriginal(YM)field,minimal
option maximal option from bottom to top. (Right) The combination plot of the suseceptability of the
Polyakov-loopfortheoriginalfield,minimaloptionandmaximaloption.
b = 5.8 b = 5.9 b = 6,0 b = 6,1
3 3
10 original (YM) 1.5
minimal 2.5 2.5
8 maximal 2 2 1
1.5 1.5 0.5
6
eV(R/) 4 0 .15 0 .15 -0 .05
2 0 0
-1
-0.5 -0.5
0 -1 -1 -1.5
-2 -1.5 -1.5 -2
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
distance (R/e ) distance (R/e ) distance (R/e ) distance (R/e )
Figure 3: A˛@The two-point correlation functions of the Polyakov loops calculated from the Yang-Mills
fieldandV-fieldsintheminimalandmaximaloptionsareplotedinthesamepanelforvariousvaluesofb
for each configuration in the complex plane. The value of the space-averaged Polyakov loop are
different among theoptions, but allthe distributions equally refrect the expected center symmetry
Z(3)ofSU(3). TheleftpanelofFigure2showsthePolyakov-loop average, P , P , P .
YM min max
h i h i h i
ThePolyakov-loopaverageisanorderparameterofthecentersymmetrybreakingandrestoration,
whichistheconventional orderparameteroftheconfinement/deconfinement phasetransition. The
right panel of Fig.2 shows the susceptibility. Fig.2 shows that three Polyakov-loop average give
the same critical point b =5.9. Therefore, both the minimal and maximal options reproduce the
critical pointintheoriginalYang-Millsfield.
Next,weinvestigatethetwo-pointcorrelationfunctionofthePolyakovloops,whichisrelated
tothemixedstatic-potential ofthesingletandadjointcomposite statesinthefollowingway[18]:
1 N2 1
VU(r= x y)=log( PU(x)PU(y) ), PU(x)PU(y) e−F(S)/T + c − e−F(A)/T. (3.3)
| − | h i h i≃ N N
c c
Figure 3 shows the combination plot eq.(3.3) for various temperature (b ). In the both options we
find that the restricted field (V-field) is dominant for the Polyakov loop correlation function and
reproduces thestaticpotential oftheoriginalYMtheoryinthelongdistance r= x y.
| − |
4
QuarkconfinementtobecausedbyAbelianornon-Abeliandualsuperconductivity AkihiroShibata
Yang-Mills Symbol b=5.8 Minimal option Symbol b=5.8 Maximal option Symbol b=5.8
00..45 EEBBBEyzxyzx 00..45 EEBBBEyzxyzx 00..45 EEBBBEyzxyzx
0.3 0.3 0.3
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
Yang-Mills Symbol b=6.2 Minimal option Symbol b=6.2 Maximal option Symbol b=6.2
00..45 EEEBBBxyzxyz 00..45 EEBBBEyzxyzx 00..45 EEBBBEyzxyzx
0.3 0.3 0.3
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
Figure 4: Chromo flux created by a pair of quark and anti-quark. The flux is measured at the point of
distanceyfromthe1/3dividingpointofquarkandantiquarklyingonthez-axis. Upperandlowerpanels
represent chromo flux in confining phase (b =5.8) and deconfining phase (b =6.2), respectively. Each
panelintheleftmidleandrightrepresentsthemeasurementoftheoriginal(YM)field,theminimalandthe
maximaloptions,respectively.
Finally,westudythedualMeissnereffect. Forthispurpose,wemeasurethechromofluxatfi-
nitetemperaturecreatedbyaquark-antiquark pairwhichisrepresentedbythemaximallyextended
WilsonloopW oftheR T rectangle, i.e.,thefieldstrengthofthechromofluxatthepositionPis
×
measured byusingaplaquette variableU atPastheprobeoperatorforthefieldstrength [17]:
p
tr WLU L† 1 tr(U )tr(W)
r := p h p i, (3.4)
UP tr(W) −3 tr(W)
(cid:10) (cid:0)h i (cid:1)(cid:11) h i
where L is the Schwinger line connecting the source W and the probe U needed to obtain the
p
gauge-invariant result(See[8]). Tomeasurethechromofluxfortherestrictedfieldsoftheminimal
and maximal options, we use the operator that the Schwinger line L and the probe U are made
p
of the restricted fields. Figure 4shows the preliminary measurements of the chromo fluxthe both
options and the original field. We find that the chromo-flux tube appears in the confining phase
(b = 5.8), while in the deconfining phase (b =6.2) the flux tube disappears, that is to say, the
confinement/deconfinement phase transition isunderstood as the dual Meissner effect. Thisis the
caseforbothoptionsandtheoriginalYMfield,althoughthepreciseprofilesofthechromofluxare
different optionbyoption.
4. Summary
By using a new formulation of YM theory, we have investigated possible two types of the
dual superconductivity at finite temperature in the SU(3) YM theory, i.e., the Non-Abelian dual
superconductivity astheminimaloption andthemaximaloption tobecompared withtheconven-
tional Abelian dual superconductivity. Inthemeasurement forbothmaximalandminimaloptions
as well as for the original YM field at finite temperature, we found the restricted V-field domi-
nance for both options. The restricted fields in the both options reproduce the center symmetry
5
QuarkconfinementtobecausedbyAbelianornon-Abeliandualsuperconductivity AkihiroShibata
breaking and restoration of the original Yang-Mills theory, and give the same critical temperature
oftheconfining-deconfining phasetransition. Then,wehaveinvestigated thedualMeissner effect
and found that the chromoelectric flux tube appears in each option in the confining phase, but it
disappears inthedeconfining phase. Thusbothoptions canbeadoptedasthelow-energy effective
descriptionoftheoriginalYang-Millstheoryatleastwithintheinvestigationspresentedinthistalk.
Acknowledgement
Thisworkissupported byGrant-in-Aid forScientificResearch (C)24540252 and15K05042
fromJapanSocietyforthePromotionScience(JSPS),andinpartbyJSPSGrant-in-AidforScien-
tificResearch(S)22224003. Thenumericalcalculations aresupportedbytheLargeScaleSimula-
tionProgramNo.14/15-24(2014-2015) and No.15/16-16(2015-2016)ofHighEnergyAccelerator
ResearchOrganization (KEK).
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