Coulomb Energy, Remnant Symmetry, and the Phases of Non-Abelian Gauge Theories

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by  Jeff Greensite

CoulombEnergy, Remnant Symmetry, and thePhases ofNon-AbelianGaugeTheories
Jeff Greensite
PhysicsandAstronomyDept., San FranciscoStateUniversity, San Francisco, CA94117, USA
∗
Sˇtefan Olejn´ık
Institute of Physics, Slovak Academy of Sciences, SK–845 11 Bratislava, Slovakia†
Daniel Zwanziger
Physics Department, New York University, New York, NY 10003, USA‡
(Dated:February1,2008)
Weshowthattheconfiningpropertyoftheone-gluonpropagator,inCoulombgauge,islinkedtotheunbroken
realizationofaremnantgaugesymmetrywhichexistsinthisgauge. Anorderparameterfortheremnantgauge
symmetryisintroduced,anditsbehaviorisinvestigatedinavarietyofmodelsvianumericalsimulations. We
4 findthatthecolor-Coulombpotential,associatedwiththegluonpropagator,growslinearlywithdistancebothin
0 theconfinedand surprisingly inthehigh-temperaturedeconfinedphaseofpureYang-Millstheory.Wealso
− −
0 findaremnantsymmetry-breakingtransitioninSU(2)gauge-HiggstheorywhichcompletelyisolatestheHiggs
2 fromthe(pseudo)confinementregionofthephasediagram. Thistransitionexistsdespitetheabsence,pointed
outlongagobyFradkinandShenker,ofagenuinethermodynamicphasetransitionseparatingthetworegions.
n
a
J
I. INTRODUCTION SU(2) gauge-Higgstheory, with the Higgs field in the
1 •
fundamentalrepresentation;

1 It is well known that the phase transition from a confined
compactQED .
4
v toanunconfinedphaseinanon-abeliangaugetheoryisasso- •
3 ciatedwiththebreakingofglobalcentersymmetry. Ifanon- Inthe first, second, andfourththeoriesthe actionhasa non-
0 trivialcentersymmetryexistsandisunbroken,thenPolyakov trivial global center symmetry, which may be broken spon-
0
lineexpectationvaluesarezero,andin consequencethefree taneously in some range of couplings or temperature. In
1
energy of a static color charge is infinite in an infinite vol- the third case center symmetry is explicitly broken, and the
0
4 ume. But the confined phase may be associated with other asymptoticstringtensionvanishesatallcouplings. Bystudy-
0 global symmetries as well. In this article we will focus on ingthesedifferentcases,wecanexploretowhatextentrem-
/ the remnant gauge symmetry which is found after imposing nant symmetry breaking is correlated with center symmetry
t
a Coulomb gauge. What is interesting about this symmetry is breaking,andwhethertheconfiningCoulombpotentialisal-
l
- thatitsunbrokenrealizationimpliestheexistenceofaconfin- waysassociatedwithaconfiningstaticquarkpotential.
p ing color-Coulomb potential, and this in turn is a necessary Thisarticleisorganizedasfollows:InsectionII,below,we
e
(butnotsufficient)conditionforconfinement. introduce the remnantsymmetry order parameter, and relate
h
: Thecolor-Coulombpotentialarisesfromtheenergyofthe ittotheconfiningpropertiesofthe color-Coulombpotential.
v longitudinalcolor-electricfieldinCoulombgauge,andcorre- Wealsoexaminescalingofs coul withb . SectionIIIconcerns
i
X spondsdiagrammaticallyto instantaneousdressed one-gluon the gauge-adjoint Higgs model, where we find perfect cor-
exchangebetweenstaticsources. Wehavepreviouslystudied respondencebetweenremnantsymmetryand center symme-
r
a thispotentialnumericallyinpurelatticeYang–Millstheoryat try breaking. Butthis correspondenceis lostalreadyin pure
zero temperature, and found that the potential rises linearly Yang–Millstheoryathightemperature,studiedinsectionIV,
withcolorchargeseparation[1,2],albeitwithastringtension wherewefindthatCoulombconfinementandunbrokenrem-
s whichissignificantlyhigher(byaboutafactorofthree nant symmetry persist in the deconfined phase. A possible
coul
[2])thanthestringtensions ofthestaticquarkpotential. In explanationofthisphenomenonisdiscussed.InsectionVwe
thisarticleweintroduceanorderparameterforremnantgauge review the Gribov confinement scenario in Coulomb gauge
symmetrybreakingandstudyitsbehavior,aswellasthebe- andshowthatitaccordswiththevortexdominancescenario,
haviorofthecolor-Coulombpotential,in by gauge transforming from the maximal center gauge to
the minimal Coulomb gauge. In section VI we present our
SU(2) gauge-Higgstheory, with the Higgs field in the
results for the gauge-fundamental Higgs model, where the
•
adjointrepresentation;
gauge-Higgs interaction breaks global center symmetry ex-
plicitly. Inthiscasewefindveryclearnumericalevidenceof
pureSU(2)gaugetheory,intheconfinedandthehigh-
• a remnant symmetry-breaking transition which is unaccom-
temperaturedeconfinedphase;
panied by a true thermodynamic phase transition, and also
argue for the existence of such a transition from a lattice
strong-coupling analysis. This result is in complete accord
with the earlier work of Langfeld [3, 4], which found rem-
Electronicaddress:[email protected]
∗
†Electronicaddress:[email protected] nantsymmetrybreakinginLandaugauge,anditimpliesthat
‡Electronicaddress:[email protected] a sharp distinction can be made between the Higgs and the
2
(pseudo)confiningregionsofthegauge-Higgscouplingplane. = (cid:229) hY bq|Y nihY n|Y aqie−(En−E0)T
Thisdistinctionexistsdespitethefact,pointedoutbyFradkin n
andShenker[5],thattheseregionsarecontinuouslyconnected = G e mqT Lba(x,1,T) (2.4)
−
h i
intheusualsenseofthermodynamics. Theremnantsymme-
trybreakingintheadjointHiggstheoryisre-examinedinsec- whereindexnreferstosummationoverchargedenergyeigen-
tionVII,wherewedrawsomeconclusionsaboutthemeasure states,mq ¥ istheheavyquarkmass,G isanumericalfac-
→
ofabelianconfigurationsinthefundamentalmodularregion. tor,andL(x,t1,t2)isaWilsonline
OurmethodsareappliedtocompactQED insectionVIII.
4
As in the previous work of ref. [2], we also investigate L(x,t1,t2)=U0(x,t1)U0(x,t1+1)...U0(x,t2). (2.5)
numerically the relevance of center vortices to the existence
Inaconfiningtheory,theexcitationenergyD E =E E
of a confining Coulomb potential, particularly in the high- ofanystate Y containinga singlestatic quarkmnustbne−infi0-
temperaturedeconfinedphaseofpureYang–Millstheory,and n
nite. ItfollowsthatGba(T)=0,andthereforethattheexpec-
in the (pseudo)confined phase of gauge-fundamental Higgs
tationvalueofanytimelikeWilsonlineL(x,1,T)mustvanish.
theory. In section V it is shown that thin center vortices lie
Now,undertheremnantgaugesymmetry,Ltransformsas
on the Gribov Horizon, which may be relevant to their dra-
matic effect on the Coulomb potential. Vortex removal, by L(x,t ,t ) g(t )L(x,t ,t )g†(t +1). (2.6)
1 2 1 1 2 2
thedeForcrand–D’Eliaprocedure[6],isfoundineverycase →
toconvertaconfiningCoulombpotentialtoanon-confining, If T =t t +1 is less than the lattice extension n in the
2 1 t
asymptoticallyflatpotential. timedirec−tion,thennotonlyLbutalsoTr[L]isnon-invariant
InsectionIX, weprovidea2-waytranslationbetweenthe undertheremnantgaugesymmetry.1 Iftheremnantsymme-
temporal gauge (A0 =0) and the minimal Coulomb gauge. try is unbroken, then L must vanish. The confining phase
This allows our measurements, which are made by gauge- is therefore a phase ohf uinbroken remnant gauge symmetry;
fixingto theminimalCoulombgauge,tobeequivalentlyde- i.e.unbrokenremnantsymmetryis anecessaryconditionfor
scribedintemporalgauge.Weshowthatthestateinwhichour confinement.
measurements are made is, in temporal gauge, a quark-pair There are several issues which require some further com-
stateofthetypeintroducedbyLavelleandMcMullan[7]. It ment. First of all, why is unbroken remnant symmetry not
hascorrectgauge-transformationproperties,althoughitdoes alsoasufficientconditionforconfinement?Theansweristhat
not make use of Wilson lines running between the sources. inanunconfinedphase,wherethereexistfiniteenergystates
SectionXcontainssomeconcludingremarks. containing a single static charge in an infinite volume, there
isstillthepossibilitythatthesefiniteenergystateshavevan-
ishingoverlapwith Y a asdefinedin eq.(2.3). Thus L =0
II. COULOMBENERGYANDREMNANTSYMMETRY q h i
andunbrokenremnantsymmetrycouldbefound,inprinciple,
alsointheabsenceofconfinement. Secondly,itisobviously
Onthelattice,minimalCoulombgaugeconsistsoffixingto impossible to insert a single charge in a finite volume with,
theconfigurationonthegaugeorbitmaximizingthequantity e.g.,periodicboundaryconditions;electricfieldlinesstarting
fromthestaticchargemustendonsomeothercharge,regard-
3
(cid:229) (cid:229) lessofwhetherornotthetheoryisinaconfiningphase.How,
R= Tr[U (x,t)]. (2.1)
k
then, is this factreflectedin the remnantsymmetry-breaking
x,tk=1
criterion?Thesamequestioncanberaisedinconnectionwith
MaximizingRdoesnotfixthegaugecompletely,sincethere Polyakov lines, and the answer is the same: Strictly speak-
isstillthefreedomtoperformtime-dependentgaugetransfor- ing, spontaneoussymmetrybreakingcannotoccurin a finite
mations volume, so L = 0 always, consistent with the absence of
h i
an isolated static charge. Nevertheless (again like Polyakov
Uk(x,t) g(t)Uk(x,t)g†(t) (k=1,2,3), lines),itispossibletoconstructanorderparameterwhichde-
→
U (x,t) g(t)U (x,t)g†(t+1). (2.2) tects the infinite-volumetransition via finite-volumecalcula-
0 0
→ tionswhicharesubsequentlyextrapolatedtoinfinitevolume.2
Tounderstandtheroleofthisremnantgaugesymmetry,con- Wewillconstructsuchanoperatorbelow.Finally,thereisthe
sider a state Y a obtained by operating on the lattice Yang– questionofElitzur’stheorem. Althoughtheremnantsymme-
q
Mills vacuumstate Y 0 (i.e. the groundstate ofthe Coulomb try is globalon a time slice, it is local in the time direction,
gaugetransfermatrix)withaheavyquarkoperatorqa and accordingto the theoremlocalsymmetriescannotbreak
spontaneously.Sohowcouldweeverhave L =0,eveninan
Y a[x;A]=qa(x)Y [A]. (2.3) h i6
q 0
EvolvethisstateforEuclideantimeT <n,wheren isthelat-
t t
ticeextensioninthetimedirection. Dividingoutthevacuum
factorexp[ E T]andtakingtheinner-productwithY b[x;A], 1IfT =nt,thenLisaPolyakovline,whosetraceisinvariantundergauge
− 0 q transformations,butnon-invariantunderglobalcentertransformations.
wehave 2Inthecaseofspontaneouscentersymmetrybreaking, theacceptedorder
parameteronafinitelatticeistheabsolutevalueofthespatialaverageof
Gba(T) = hY bq|e−(H−E0)T|Y aqi Polyakovlines.
3
infinitevolume? Theansweristhatinfacttheaveragevalue as shown in [9]. We see from these expressions that
ofL(x,t,T+t)doesindeedvanishonaninfinitelattice,inac- the Coulomb interaction energy between two charged static
cordancewiththeElitzurtheorem,providingtheaveragingis sources is given by instantaneous (dressed) one-gluon ex-
doneoverallspatialxandalltimest. Ona timeslice, how- change.
ever, the symmetry is only global, and it is possible in any NowconsideraphysicalstateinCoulombgaugecontaining
given configurationthat the average value of L(x,t,T+t) is massivequark-antiquarksources
finite on an infinite lattice, when averagedoverall x at fixed
timet. Thisiswhatwewillmeanbythephrase“spontaneous Y qq =q(0)q(R)Y 0 (2.12)
| i | i
breakingoftheremnantsymmetry,”anditinvolvesnoactual
which is invariantunder the remnantsymmetry. The excita-
violationoftheElitzurtheorem.
tionenergyis
Withthesepointsinmind,weproposetoconstructanorder
parameterfor remnantsymmetrybreakingfrom the timelike
E = Y H Y Y H Y
linkvariableaveragedoverspatialvolumeatfixedtime. Let h qq| | qqi−h 0| | 0i
= V (R)+E , (2.13)
1 (cid:229) coul se
U(t)= U (x,t) (2.7)
0
V3 x whereEseisanR-independentconstant,ontheorderofthein-
verselatticespacing,tobespecifiedbelow.TheR-dependence
whereV3=nxnynzisetheisthe3-volumeofalatticetime-slice. of E can only come from the expectation value of the non-
1/2
Ifremnantsymmetryisunbroken,thenU(t)=0+O(1/V )
3 localquark-quarkpartoftheHamiltonian
inanythermalizedlatticeconfiguration. Theorderparameter
Qisdefinedtobe,forSU(2), e H = 1 d3xd3yr a(x)Kab(x,y;A)r b(y). (2.14)
qq 2 q q
Q= 1 (cid:229)nt 1Tr[U(t)U†(t)] . (2.8) Z
ntt=1(cid:28)q2 (cid:29) Thusthe R-dependentpieceVcoul(R) can be identified as the
Coulombpotentialduetothesestaticsources. Moreover,the
Then Q is positivedefinite on a fieniteelattice, and on general
samekernelK(x,y;A)appearsinH andintheinstantaneous
grounds qq
partD(x)ofthe(dressed)one-gluonpropagator A A . This
0 0
h i
b c=0 inthesymmetricphase, yieldstheformula,
Q=c+ where
√V c>0 inthebrokenphase.
3 (cid:26)
(2.9) V (x)+E = C D(0) D(x) , (2.15)
coul se r
IfQextrapolatestoanon-zerovalueasV ¥ ,thentherem- | | −
3→ (cid:16) (cid:17)
nantsymmetryinCoulombgaugeisspontaneouslybroken. whereC (=3/4 for the SU(2) gauge group) is the Casimir
r
Next, we maketheconnectionbetweenunbrokenremnant factorinthefundamentalrepresentation.
symmetry and the existence of a confining Coulomb poten- The correlator of two Wilson lines can be expressed in
tial. WefirstrecallthattheHamiltonianoperatorinCoulomb terms of the Hamiltonian operator and the state Y as fol-
qq
gaugehastheformH=Hglue+Hcoulwhere,inthecontinuum lows:
1 1
Hglue= 12 d3x(J−2~Etr,aJ ·~Etr,aJ−2 +~Ba·~Ba) G(R,T) = h21Tr[L†(x,0,T)L(y,0,T)]i
Z
1 1
Hcoul = 21 d3xd3yJ−2r a(x)JKab(x,y;A)r b(y)J−2 = Y qq e−(H−E0)T Y qq , (2.16)
Z h | | i
1 1 ab
Kab(x,y;A) = ( (cid:209) 2) where R= x y, and L is now the timelike Wilson line in
(cid:209) D(A) − (cid:209) D(A) | − |
(cid:20) · · (cid:21)xy thecontinuumtheory.Wehave
r a = r a gfabcAbEtr,c
q− k k G(R,T)=(cid:229) Y Y 2e D EnT (2.17)
J = det[ (cid:209) D(A)] (2.10) h n| qqi −
− · n
and the factors of J arise from operator ordering consider- (cid:12)(cid:12) (cid:12)(cid:12)
andwedefinethelogarithmicderivative
ations[8]. Itis understoodthat A is identicallytransversein
Coulombgauge,A=Atr.NotethatJ commuteswithallquan- d
titiesexceptEtr,andinparticularwithr (x)andKab(x,y;A). V(R,T)= log[G(R,T)]. (2.18)
q −dT
Theexpectation-valueofK(x,y;A)istheinstantaneouspiece
ofthe A A gluonpropagator,i.e. ItiseasytoseethattheCoulombenergyisobtainedatT 0,
h 0 0i i.e. →
Aa(x)Ab(y) = D(x y)d abd (x y )
h 0 0 i − 0− 0
+ non-instantaneous E = Vcoul(R)+Ese
1 1 a,b = V(R,0). (2.19)
D(x y)d ab = ( (cid:209) 2) ,
− *(cid:20)(cid:209) ·D[A] − (cid:209) ·D[A](cid:21)x,y+ The minimum energy of a state containing two static quark
(2.11) antiquarkcharges, which in a confiningtheory would be the
4
energyofthefluxtubegroundstate,isobtainedintheopposite 1. V(R,T) increaseslinearly with R at largeR and all T,
T ¥ limit atanycoupling.
→
Emin = V(R)+Es′e 2. The associated string tension s (T) converges (from
above)to theusualasymptoticstringtensions , atany
= limV(R,T) (2.20)
T ¥ givenb ,asT increases.
→
andinthislimitV(R)isusualstaticquarkpotential. 3. At weaker couplings, the Coulomb string tension
The idea of using the correlator of timelike Wilson lines s s (0) is substantiallygreater(byabouta factor
coul
to compute the static quark potential, in Coulomb gauge on ofthre≡e)thantheasymptoticstringtension.
thelattice,wasputforwardsomeyearsagobyMarinarietal.
[10]. Theseauthorsalsonotedthattheremnantsymmetryin 4. Removing center vortices from lattice configurations
Coulombgaugeisunbrokenintheconfiningphase. (by the de Forcrand–D’Elia procedure [6]) sends
We nowrecall an inequalityfirst pointedoutby oneof us s (T) 0 at all T, including the Coulomb string ten-
(D.Z.)in ref.[11]. With a latticeregularization,Ese andEs′e, sions →coul →0atT =0.
arefiniteconstants. Ina confiningtheory,bothofthesecon-
In the following sections we extend the investigation to
stantsarenegligiblecomparedtoV(R),atsufficientlylargeR.
models including scalar matter fields, and to SU(2) gauge
ButsinceE E,itfollowsthat
min≤ theoryacrossthehightemperaturedeconfinementtransition.3
First, however,wewouldliketoremarkonthescalingprop-
V(R) V (R) (2.21)
≤ coul ertiesofs s (0)inpureSU(2)gaugetheoryatzerotem-
coul
≈
asymptotically. The intriguingimplication is that if confine- perature. TheratiooftheCoulomb(T =0)totheasymptotic
mentexistsatall,thenitexistsalreadyatthelevelofdressed (T ¥ )stringtensions,reportedinref.[2],variessomewhat
one-gluonexchangeinCoulombgauge. Butwealsoseethat wit→h b in the range(b [2.2,2.5])ofcouplingsinvestigated.
becausetheCoulombpotentialisonlyanupperboundonthe The ratio s (0)/s tend∈s to rise in this interval, as shown in
staticpotential,aconfiningCoulombpotentialisanecessary Fig. 1. However, it is known that s does not quite conform
butnota sufficientconditionfortheexistenceofa confining to the two-loop scaling formula associated with asymptotic
staticquarkpotential. freedom, in the range of b we have studied, and it is always
On the lattice, the continuumlogarithmicderivativein eq. possiblethatscalingsetsinatdifferentvaluesofb fordiffer-
(2.18)isreplacedby entphysicalquantities. Whatwefindisthatwhenourvalues
fors (0)aredividedbytheasymptoticfreedomexpression
1 G(R,T)
V(R,T)= log (2.22)
a G(R,T+a) 6p 2 102/121 6p 2
(cid:20) (cid:21) F(b )= b exp b (2.25)
where a is the lattice spacing. In particular, in lattice units (cid:18) 11 (cid:19) (cid:18)− 11 (cid:19)
a=1, relevant to the SU(2) string tension, the ratio s (0)/F(b ) is
virtually constant as seen in Fig. 2. This fact suggests that
V(R,0) = log[G(R,1)]
− scalingaccordingtoasymptoticfreedommaysetinearlierfor
= −log 12Tr[U0(x,1)U0†(y,1)] , (2.23) ttohteicCsoturilnogmtbensstiroinngs t.ensions coul ≈s (0)thanfortheasymp-
hD Ei
and at large b , where the lattice logarithmic derivative ap-
proximatesthecontinuum,V(R,0)providesanestimateofthe
CoulombpotentialVcoul(R)(uptoanadditiveconstantEse). A. DivergentconstantinD(R)
Ineq.(2.23)therelationbetweentheconfiningpropertyof
the Coulombpotential, and the unbrokenrealizationof rem- We would also like to remark, at this stage, on a subtlety
nantsymmetry,ismanifest. ForifQ 0 atinfinitevolume, inidentifyingthecolorCoulombpotentialwithdressedone-
→
thenalso gluonexchange.4 The energyexpectationvalue of the static
qqstate,E(R)=V (R)+E ,isfiniteforfinitequarksepa-
lim Tr[U (x,t)U†(y,t)] = lim Tr[U(t)U†(t)] coul se
R→¥ h 0 0 i = V03→¥ h i(2.24) rthatiisoqnuRanatnitdyfi(n=itVel(aRtt,i0c)e)snpuamcinergicaa;lilny.faScotwitemhiagvhetcsaelecmulantaetd-
e e
ural, from eq. (2.15), to identify the R-dependent Coulomb
inwhichcasethepotentialV(R,0)risestoinfinityasR ¥ .
→
Conversely,ifQ>0,thenthelimitineq.(2.24)isfinite,and
V(R,0)isasymptoticallyflat. SinceV (R) V(R,0)isan
coul
≈
upper bound on the static quark potential, we see again that 3AnoteongaugefixingtoCoulombgauge: Inthisinvestigationwegener-
unbrokenremnantsymmetryisanecessarybutnotsufficient ateeightrandomgaugecopiesofeachlatticeconfiguration,andcarryout
conditionforconfinement. gauge-fixing oneachconfiguration byover-relaxation for250iterations.
Thebestcopyofeightcopiesisthenchosen,andtheover-relaxationpro-
V(R,T)hasbeencomputednumericallyforpureSU(2)lat-
cedure is continued until the average value ofthe gauge-fixed links has
tice gaugetheoryata rangeoflatticecouplingsb inref.[2]. changed,inthelast10iterations,bylessthan2 10 7.
Werecalltheessentialresults: 4WethankM.Polikarpovforahelpfuldiscussion×ont−hispoint.
5
subtractionatR=R whichremovestheinfrareddivergence,
String tensions extracted from V(R,0) 0
i.e.
3.5
b=0
b=p /12
V (R) = C D(R) D(R )
coul r 0
3 − −
= E(R(cid:16)) E(R ) (cid:17)
mpt − 0
asy 2.5 Ese = Cr D(0) D(R0)
s0)/ −
( = E((cid:16)R ) (cid:17) (2.26)
s 0
2
Defined in this way, V (R) crosses zero at the subtraction
coul
point, and E contains the ultraviolet, but not the infrared,
se
1.5
contributionstothequark-antiquarkself-energies.
2.2 2.3 2.4 2.5
b Asa checkofthecancellation(ornon-cancellation)ofin-
frareddivergences,consideracoloredstateconsisting,e.g.,of
FIG.1:Theratios (0)/s atvariousb ,fromfitswhicheitherinclude twostaticquarks,ratherthanaquarkandantiquark. Theen-
(b=p /12) or do not include (b=0) the Lu¨scher term. Thisratio ergyofsuchastatecouldbeextractedfromanLLcorrelator,
shouldequaltheratioofs coul/s inthecontinuumlimit.(Datapoints whichis zeroif remnantsymmetryis unbroken. The energy
comefromnumericalsimulationsonlatticesizes164,204,204,and
is therefore infinite, and according to our previous analysis
324atb =2.2,2.3,2.4,and2.5respectively.)
would be proportionalto D(0)+D(R). In this case, the di-
vergentconstantin D(R)addsto, ratherthan subtractsfrom,
String tensions extracted from V(R,0) thedivergentconstantinD(0),andtheresultingsumisdiver-
gent, as it should be. The argumentcan be readily general-
14000 b=0 izedtobaryonicstatesinSU(N)gaugetheoriescomposedof
b=p /12
12000 staticcharges.Theenergyofacolorsingletstate,withcharges
at pointsx ,x ,...,x is obtainedfromthe logarithmictime
1 2 N
10000
derivativeofthecorrelator
)
bF( 8000
s(0)/ 6000 G({xi},T)=e i1...iNe j1...jNhLi1j1(x1,0,T)...LiNjN(xN,0,T)i.
(2.27)
4000
The order T contributions to G( x ,T) are terms propor-
i
{ }
2000 tionaltoD(0)andD(x x ),m=n,withdifferingsigns.On
m n
theotherhand,foracol−orsinglet6 state,theoperatoree L...L
0
2.2 2.3 2.4 2.5 is a T-independentconstant in the x =x =...=x coin-
1 2 N
b
cidencelimit. Fromthisitisclearthatthepropagatorscom-
pletelycancelinthecoincidencelimit,andanyconstantterms
FIG.2: Theratios (0)/F(b )vs.b ,againfromfitswith(b=p /12)
inthepropagatorscancelingeneral. Thismeansthattheen-
orwithout(b=0)theLu¨scherterm.F(b )istheasymptoticfreedom
ergyofacolorsingletbaryonicstateisfinite. Conversely,the
expressiongiveninthetext;aconstantratioimpliesscalingaccord-
divergentconstantsdo notcancel in color non-singletstates,
ingtothetwo-loopbetafunction.
sotheirenergiesareinfinite.
interaction energy as V (R)= C D(R). That cannot be
coul r
quite right, however. The reason−is that 21CrD(0) is the en- III. SU(2)GAUGE-ADJOINTHIGGSTHEORY
ergyofanisolated quarkstateinaninfinitevolume;itisthe
energywewouldextractfromthelogarithmictimederivative
ThelatticeactionforSU(2)gaugetheorywithaHiggsfield
of G(T)= 1TrL(0,0,T) . In thecase ofunbrokenremnant
h2 i intheadjointrepresentationofthegaugegroupis
symmetrywehaveG(T)=0,andtherefore 1C D(0)=¥ in
2 r
an infinite volume, even thoughthe lattice spacing a is non- S = b (cid:229) 1Tr[UUU†U†] (3.1)
zero. Thisinfinity,whichhasanon-perturbative,infraredori- 2
plaq
gin, shouldnotbeconfusedwiththeusualultravioletcontri- g
bution to the quark self-energy, which is only infinite in the + 4(cid:229) f a(x)f b(x+µ)Tr[s aUµ(x)s bUµ†(x)]
continuumlimit. SinceE(R)isfinite,theinfrareddivergence x,µ
inCrD(0)mustbe cancelled, in eq. (2.15),bya correspond- withtheradially“frozen”,threeb-componentHiggsfieldf sub-
ing divergentconstantcontained inC D(R). In other words,
r
jecttotherestriction
only the difference D(0) D(R)is finite, andV (R), if fi-
coul
−
nite, differs from C D(R) by an infinite constant. In order
r
− 3
thatVcoul(R)andEsebeseparatelyfiniteandwell-defined,we (cid:229) f a(x)f a(x)=1. (3.2)
may relate them to the gluon propagator with an (arbitrary)
a=1
6
¥ Compact QED4
SU(2) with adjoint Higgs, 124 lattice
Higgs 0.8 gg==230.0.0
5 Q > 0 g=0.5
4
3
0.6
g 2 Heisenberg
Spin Model
1 Q 0.4
Confined
Q = 0
0.2
00 1 2 ... ¥
b 0
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
FIG.3: PhasediagramoftheSU(2)adjointHiggsmodel. Thepla-
b
quetteenergyEpandtheremnantsymmetryorderparameterQlocate
thesametransitionlinebetweentheconfinedandHiggsphases.
FIG.5: RemnantsymmetryorderparameterQvs.b atthreevalues
ofg inthegauge-adjointHiggsmodel.
SU(2) with adjoint Higgs, 124 lattice
g=20.0 rem,whichstatesthatalocalsymmetrycannotbreaksponta-
0.8 g=3.0
g=0.5 neously,thischaracterizationisalittlemisleading. However,
aswehavediscussedabove,thereexistsinCoulombgaugea
0.6
remnantgaugesymmetrywhichisglobalonatimeslice,and
which can break spontaneously on a time slice in the sense
EP 0.4
describedin the previoussection. We havethereforestudied
thephasediagramoftheadjointHiggstheoryviatwoobserv-
0.2
ables:(i)theplaquetteenergy
0
E = 1Tr[UUU†U†] (3.4)
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 p 2
b D E
and (ii) the remnant symmetry breaking order parameter Q
FIG.4: PlaquetteenergyEp vs.b atthreevaluesofg inthegauge- definedin eq. (2.8). Whatwe find is thatthe transitionlines
adjointHiggsmodel. (Fig. 3) detected by each of these two parameters coincide;
the common line location agrees with the earlier results of
Brower et al. based on the plaquette energy alone. In Fig.
ThistheorywasfirststudiednumericallybyBroweretal.[12]. 4 we plot E vs. b at three values of g . The existence of a
p
InadditiontoSU(2)gaugesymmetry,theactionisalsoinvari- phasetransitionforthetwolargervaluesofg isclearlyvisible;
antunderglobalZ2 centersymmetry there is no transitionapparentat the smallest g value. Fig. 5
isthe correspondingplotof Q vs. b atthe same threevalues
U0(x,t0)→−U0(x,t0) (3.3) ofg . Atb ,g valueswhereEpshowsatransition,thetransition
in Q is even more evident. Conversely, where no transition
forsomechoiceoft=t ,withallotherfieldsunchanged.The
0 is seen in E , at g =0.5, neither is there a transition in Q.
existenceof thisapparentlyinnocentglobalsymmetryin the p
1/2
action has profound consequences; in its absence the static Finally,inFig.6,QisplottedagainstV3− ,andweshowthe
quarkpotentialisasymptoticallyflat,andtherecanbenotruly extrapolation of Q to a small value (consistent with zero) at
confined phase. The significance of center symmetry to the infinitevolume,forcouplingsintheconfinedphase.
confinementpropertyingeneralisreviewedinref.[13].
Since the action (3.2) is symmetric under global center
transformations, a transition from the Higgs phase to a dis- IV. HIGH-TEMPERATUREDECONFINEMENT
tinct confinementphase is possible. The Higgs phase is the
phase of spontaneously broken center symmetry, while con- WehaveseenthatintheSU(2)-adjointHiggsmodel,things
finement correspondsto the symmetric phase. This division gomuchasonemighthaveexpecteda priori: remnantsym-
oftheb g phasediagramintotwoseparatephaseswasveri- metrybreakingcoincideswithZ centersymmetrybreaking,
2
−
fiednumericallylongago,intheMonteCarloinvestigationof andinconsequencethepresenceofaconfiningCoulombpo-
ref. [12], which mapped out the approximatelocation of the tential is correlated with the presence of a confining static
transitionline. quarkpotential.Onemightthenguessthatremnantandcenter
TheHiggsphaseoftheadjointHiggsmodelisoftenchar- symmetrybreakingalwaysgotogether.Thisappearsnottobe
acterized as a spontaneousbreakingof the local gaugesym- true,aswehavediscoveredinourinvestigationofpureSU(2)
metry,fromSU(2)downtoU(1). InviewoftheElitzurtheo- latticegaugetheoryathightemperature.
7
SU(2) with adjoint Higgs, confined phase V(R,0)=-log[G(R,1)], b =2.3
0.4 5
b =2.2, g=0.5 123 * 2
b =1.2, g=3.0 163 * 2
0.3 ba=+1b.(cid:215)x0, g=20.0 4 222048333 *** 222
a+b(cid:215)x +c(cid:215)x 2 323 * 2
3
0)
Q 0.2 R,
V(
2
0.1
1
0 0
0 0.01 0.02 0.03 0.04 0.05 0 2 4 6 8 10 12 14 16
x ” V -1/2 R
3
FIG.6: PlotofQvs.rootinverse3-volume,andextrapolationofQ FIG.8:V(R,0)inthedeconfinedphase,atb =2.3withnt=2lattice
toinfinitevolume,atseveralcouplingsintheconfinedphaseofthe spacingsinthetimedirection,andspacevolumesrangingfrom123
gauge-adjointHiggsmodel. to323.
Pure SU(2), b =2.3, nt=2 Recall that in the continuum, the instantaneous part of
0.6 the timelike gluon propagator, which is proportional to the
Coulomb interaction energy between static charges, is given
0.5
by eq. (2.11). Note that this is the expectation value of an
0.4 operatorwhichdependsonlyonthespacecomponentsofthe
0.4
vectorpotentialatafixedtime. Onthelattice,thistranslates
Q 0.3 0.3 into an operatorwhich dependsonly on spacelike links on a
0.2 time-slice. However, we know that spacelike links, at fixed
0.2
time,areaconfiningensemble,inthesensethatspacelikeWil-
0.1
sonloopshaveanarealawfalloffeveninthehigh-temperature
0.1 0
0 0.005 0.01 0.015 deconfinementphase. Iftheconfiningpropertyofthespace-
0 likelinksonatimesliceisnotremovedbythedeconfinement
0 0.01 0.02 0.03 0.04 0.05
transition,thenitisperhapslesssurprisingthattheconfining
V -1/2
3 property of the (latticized) operator in eq. (2.11), which de-
pendsonly on spacelike links on a timeslice, survivesin the
FIG. 7: The Q parameter vs. root inverse 3-volume in the high-
temperaturedeconfined phase, pureSU(2)gauge theoryatb =2.3 deconfinementregime.
andnt=2latticespacings. Asacheck,weapplyaprocedurethatisknowntoremove
the confiningpropertiesof lattice configurations. Thisis the
deForcrand–D’Elia[6]methodofcentervortexremoval.The
Monte Carlo simulations of the pure SU(2) gauge theory procedure is to first fix a given thermalized lattice configu-
werecarriedoutonL3 2latticesatb =2.3,whichisinside ration to direct maximal center gauge, i.e. the gauge which
thedeconfinedphase.F×igure7isaplotofQvs.1/√V ,where maximizes
3
it seems that Q tends to zero at large volume (although we
cannotentirelyruleoutasmallnon-zerointerceptatV3 ¥ ). R=(cid:229) 21Tr[Uµ(x)] 2 (4.1)
ThisimpressionisstrengthenedbyourplotofV(R,0)i→nFig. x,µ
(cid:0) (cid:1)
8,whereitisclearthattheCoulombpotentialgoesasymptot-
andcarryoutcenterprojection
icallytoastraightline,asthelatticevolumeincreases. There
is no indicationof the screeningof the static quarkpotential
(as measured by Polyakov line correlators), which occurs at
Z (x)=signTr[U (x)] (4.2)
µ µ
muchsmallerdistances.
The results for the Coulomb potential in the deconfined to locate the vortices. Vortices are then “removed”from the
phasearenotparadoxical;wehavealreadynotedthatremnant originalconfigurationbysetting
symmetrybreakingisanecessarybutnotsufficientcondition
forconfinement,andthattheCoulombpotentialisonlyanup-
ptheerbCoouunlodmonbtphoetestnattiiaclqtouairnkcpreoatseentliianle.aTrhlyuseviteinspifotshsieblsetaftoicr Uµ(x)→Uµ′(x)=Zµ(x)Uµ(x). (4.3)
quarkpotentialisscreened,asevidentlyoccursinthedecon- Ineffectthis proceduresuperimposesa thin Z vortexinside
2
finedphase. Nevertheless,thisresultisalittlesurprising,and thethickSU(2)centervortices. Theeffectofthethinvortex
itwouldbenicetounderstanditalittlebetter. is to cancel out the long range influence of the thick vortex
8
continuum gauge theory, L is convex and bounded in every
V(R,0)=-log[G(R,1)], b =2.3, 163*2 lattice
direction [15]. By simple entropy considerations, the popu-
1
without vortices, fund. lation in a boundedregion of a high-dimensionalspace gets
0.8 concentrated at the boundary. For example inside a sphere
of radius R in a D-dimensional space, the radial density is
0.6 givenbyrD 1drand,forr R,ishighlyconcentratednearthe
−
0) boundaryr=RforDlarge≤. Withlatticediscretizationthedi-
R,
V( 0.4 mensionDofconfigurationspacedivergeslikethevolumeV
ofthelattice. MoreoverontheboundarytheFaddeev–Popov
0.2 operatorM(A)= (cid:209) D(A)hasavanishingeigenvalue,aswe
− ·
willsee. (Atlargevolumeithasahighdensityofsmallpos-
0
itiveeigenvalues.) Thismakesthecolor-Coulombinteraction
0 1 2 3 4 5 6 7 8 kernelK(x,y;A)=M−1(A)( (cid:209) 2)M−1(A)xyoflongrangefor
− |
R typicalconfigurationsAthatdominatethefunctionalintegral.
WethusexpectthatV (x y)= K(x,y;A) islongrange,
coul
FIG. 9: The effect of vortex removal onV(R,0) inthe deconfined althoughthisqualitativear|gu−me|ntishnotprecisieenoughtoes-
phase,b =2.3ona163 2lattice. tablishthatV (R)riseslinearlyatlargeR, assuggestedby
× coul
thenumericaldata.
We shall show that vortex dominance, which is strongly
on Wilson loops. It was found that this procedure not only supported by the data just presented, is consistent with this
removes the Wilson loop area law falloff, but also removes simple confinement scenario in minimal Coulomb gauge.
chiralsymmetrybreaking,andsendseveryconfigurationinto More precisely we shall show that when a center config-
thezerotopologicalchargesector[6]. uration (defined below) is gauge transformed to minimal
Having removed center vortices from the L3 2 lattice, Coulombgaugeitlies onthe boundaryL¶ ofthe fundamen-
×
thereby removing the area-law falloff of spacelike Wilson tal modular region L . According to the confinement sce-
loops, we fix the modified configuration to Coulomb gauge, narioin minimalCoulombgauge,the probabilitymeasureis
and compute timelike link correlators in order to measure dominated by points at or near the boundary L¶ . So center
V(R,0). The effect is quite dramatic. It was found in ref. dominance,whentranslatedintotheminimalCoulombgauge,
[2] that vortex removal in pure gauge theory, in the low- meansdominancebyasubsetofconfigurationsonthebound-
temperature confined phase, removes the confinement prop- aryL¶ . Thisisastrongerconditionthantheconfinementsce-
erty of the Coulomb potential. Now we see, from Fig. 9, narioinminimalCoulombgauge,butconsistentwithit.
that vortex removal also removes the confining property of Proof of assertion. To simplify the kinematics we give
theCoulombpotentialinthehigh-temperaturedeconfinement thecontinuumversionoftheargument.Numericalgaugefix-
phase. Thisis in accordwith theidea thatit isthe confining ingtominimalCoulombgaugecorrespondstominimizingon
propertyof theensembleofspacelike linksatfixedtime (or, eachtime slice the functional,F(A)= A 2, with respectto
perhaps,thepercolationofcentervorticesonanytime slice) local gauge transformations. Here A||2 =|| d3x A2 is the
thatis crucialfor the confiningpropertyof the Coulomben- squareHilbertnormofAa. Atamin|i|mu||m,whichm|ay| berel-
i R
ergy. ative or absolute, the first variation with respect to infinites-
It is interesting to ask whether there is some connection imal gauge transformations d A = D(A)w , vanishes for all
i i
betweenthecentervortexconfinementmechanism,andother w , d A 2=2(A,D(A)w )=2(A,¶ w )=0,whichgivesthe
i i i i
proposals based on the Gribov horizon [14]. While this Coul|o|m||bgaugecondition¶ A =0. Moreoveratarelativeor
i i
questioncannotyetbeanswereddefinitively,thereisonevery absoluteminimumthesecondvariationwithrespecttogauge
intriguingfactthatmayberelevant:whengauge-transformed transformationsd 2 A 2=2(D(A)w ,¶ w ) 0isnon-negative
i i
totheminimalCoulombgauge,thin-vortexconfigurationslie forallw ,whichist|h|e|s|tatementthattheFad≥deev–Popovoper-
ontheGribovhorizon,aswenowexplain. atorM(A)= (cid:209) D(A)isnon-negative.Thesetwoconditions
definetheGr−ibov·regionW ,
W A:¶ A =0and ¶ D(A) 0 . (5.1)
i i i i
≡{ − ≥ }
V. CONFINEMENTSCENARIOINCOULOMBGAUGE ThefundamentalmodularregionL isthesetofabsolutemin-
ANDVORTEXDOMINANCE
imawithrespecttogaugetransformations,
In the confined phase [2], and in the deconfined phase, L A: A gA forallg . (5.2)
Fig. 8, one sees clear evidence of a linearly rising color- ≡{ || ||≤|| || }
Coulomb potential. There is a simple intuitive scenario in Itis includedin theGribovregion,L W . Inthe interiorof
the minimal Coulomb gauge that explains why V (R) is W all eigenvaluesof M(A) are strictly⊂positive l >0 (apart
coul n
long range [14]. In minimal Coulomb gauge the gauge- fromthetrivialnulleigenvalueassociatedwithconstantgauge
fixed configurationsare (3-dimensionally)transverse config- transformations ¶ w =0), and on the boundary W¶ there is
i
urations that lie in the fundamental modular region L . In a non-trivial null eigenvector ¶ D(A)w = 0, and all other
i i 0
9
eigenvaluesarenon-negative.5 W¶ touch. ButnowtheequationD(A)w =0has,ingeneral,
i
We call a center configuration any lattice configuration only one linearly independent (non-trivial) solution, instead
Z(x)forwhicheverylinkvariableisacenterelement,Z(x) of N2 1. Thus abelian dominance is also compatible with
Ziforeverylink(x,ıˆ). Theonlynon-zeroactionexcitatioinso∈f domin−anceofconfigurationsontheboundaryL¶ .
center configurations are thin center vortices. Such config-
urationsare invariantunderall globalgaugetransformations
g−1Zi(x)g=Zi(x). Nowapplyanarbitrarylocalgaugetrans- VI. SU(2)GAUGE-FUNDAMENTALHIGGSTHEORY
formation h(x) to the center configuration Z(x) V(x) =
i i
h 1(x)Z(x)h(x+ıˆ). Weshalltakeh(x)tobethega→ugetrans-
− i We now consider SU(2) gauge theory with the radially
formation that brings the center configurationinto the mini-
frozenHiggsfieldinthefundamentalrepresentation. Forthe
mal Coulomb gauge. In general, the transformedconfigura-
SU(2) gauge group, the lattice action can be written in the
tionV(x) is not an element of the center, but it is invariant,
i form[18]
Vi(x)=g′−1(x)Vi(x)g′(x+ıˆ),withrespecttothegaugetrans-
formationg′(x)=h−1(x)gh(x)which,ingeneral,isnolonger S = b (cid:229) 1Tr[UUU†U†]
2
global.
plaq
We giveaninfinitesimalcharacterizationofthe invariance + g (cid:229) 1Tr[f †(x)U (x)f (x+µ)] (6.1)
of the configuration V(x) under the gauge transformations 2 µ
i x,µ
g(x).ThesetofglobalgaugetransformationsformtheSU(N)
′
Lie group and the g′(x) = exp[w (x)] form a representation with f an SU(2) group-valued field. This thbeory cannot be
of this group. Here w (x) is an element of the Lie alge- trulyconfiningfornon-zerog ,sincethematterfieldcanscreen
bra of SU(N). This algebra has N2 1 linearly indepen- anycharge,andthissimplyreflectstheabsenceofanon-trivial
dent elements w n(x), where n = 1,..−.,N2 1, that satisfy globalcentersymmetry.However,atsufficientlysmallg there
[w l(x),w m(x)]= flmnw n(x). Thus the confi−gurationVi(x)= existsa“pseudo-confinement”region,wherethestaticpoten-
exp[Ai(x)], which is the gauge transform of the center con- tial(asmeasuredbythecorrelatorofPolyakovloops)islinear
figurationZi(x)intotheminimalCoulombgauge,isinvariant for some intermediate range of quark separations before the
under local gauge transformations with N2 1 independent onsetofscreening. Atlargeg thereisa Higgsregion,where
generatorsw n.6 Thisisthestatementthatin−continuumnota- thelinearpotentialiscompletelyabsent. Itwasshownmany
tionreadsAi=Ai+e Di(A)w n,orDi(A)w n=0.Itfollowsthat yearsagoby Fradkinand Shenker[5] thatanytwo pointsin
thew nalsosatisfytheweakercondition(cid:209) D(A)w n=0.Here the Higgs and pseudo-confinementregions can be joined by
·
AistherepresentativeinminimalCoulombgaugeofthecen- a path in the b g coupling plane that avoids all thermody-
terconfiguration,andassuchitliesinthefundamentalmod- namicsingularit−ies. Althoughthereexistsalineoffirst-order
ularregionA L (bydefinition)thatmoreoverisincludedin phasetransitionsintheb g plane,thislinehasanendpoint
the Gribov re∈gion L W , so we have A W . However the anddoesnotdividethedia−gramintothermodynamicallysep-
equation (cid:209) D(A)w n ⊂=0 for A W mean∈s that A lies on its aratephases.(ItappearsasthesolidlineinFig.12,below.)In
boundaryW¶ · . WithL W itfoll∈owsthatAliesonthebound- accordwiththeFradkin-Shenkerobservation,numericalsim-
aryL¶ of L , ata poin⊂twherethe two boundariestouch. We ulationssuggestthatthereisonlyonenon-confiningphasein
conclude that the gauge transform of a center configuration thegauge-fundamentalHiggstheory.
liesonL¶ ,asasserted.7 Inapparentcontradictiontothisfact,onecanmakeastrong
Theargumentjustgivenalso appliesto abelianconfigura- casefortheexistenceofaremnantsymmetry-breakingtransi-
tions,namelyaconfigurationthatliesinanabeliansubalgebra tionatsmallb . Wesetb =0sotheactionis
of the Lie algebra. Such a configurationis invariantundera
globalU(1)gaugetransformation. Whenanabelianconfigu- Sf =g (cid:229) 21Tr[f †(x)Uµ(x)f (x+µ)]. (6.2)
ration is gauge-transformedto the minimalCoulomb gauge, x,µ
it is mapped into a point where the two boundaries L¶ and
We shall show that (i) at large g there is spobntaneousbreak-
ingoftheremnantgaugesymmetry,associatedwiththeshort
range of V (R), but (ii) at small g there is a linear rise of
coul
5TheGribovregionW consistsofrelativeminimathatareGribovcopiesof Vcoul(R) at large R. Thus along the line b =0 in the b g
thesetL ofabsolute minima. Ournumerical procedure selects thebest planeweexpectatransitionatsomefinitevalueofg . −
Gribovcopyobtainedfromeightrandomgaugecopies;thereisnoknown (i)Firstconsiderthelimitg ¥ . TheactionSf isa max-
method for finding the absolute minimum. Ontheoretical grounds, one imum when U (x)=f (x)f †(x→+µ) holds on each link, and
expectsthesensitivityofourresultstothechoiceofGribovcopybesmall µ
wheng gets large,U (x) gets frozenat this value. This con-
[16].Thisexpectationissusceptibletonumericalinvestigation,ashasbeen µ
donerecentlyfortheghostpropagator[17]. figurationisagauge-transformofbtheidentityUµ(x)=I,and
6ThegaugeorbitofthecenterconfigurationisdegenerateandhasN2 1 whenfixedtotheminimalCoulombgaugethespatialcompo-
−
fewerdimensionsthanagenericgaugeorbit. nentsgetfixedtotheidentityU(x,t)=f (x,t)f †(x+ıˆ,t)=I,
7Itshouldbenotedthatthenumericalprocedurethatwehaveusedtore- for all x and i. Thus f (x,t) isiindependent of x. We write
movecentervortices is4-dimensional. Theargumentgivenhereapplies
f (x,t)=g(t),andwehaveU (x,t)=g(t)g†(t+1),whichis
inCoulombgaugetocenterconfigurations definedwithin3-dimensional 0
time-slices. also independentof x. This is the gauge analog of all spins
10
aligned,andwe expectspontaneousbreakingoftheremnant
SU(2) with fund. Higgs, b =2.1
gaugesymmetry.Indeed,wehave
1.2
g=0.
U˜(t)= 1 (cid:229) U (x,t)=g(t)g†(t+1), (6.3) 1 g=g=0.08.56
V 0 g=1.2
3 x
0.8
andtheorderparameterQdefinedabovehasthevalueQ=1.
Thisismaximalbreakingoftheremnantgaugesymmetry. Q 0.6
(ii)Nowconsidersmallvaluesofg . Weshallcalculatethe
lattice analog of V (R), namely V(R,0)= log[G(R,1)], 0.4
coul
toleadingnon-zeroordering . Thegaugefixing−involvesonly
0.2
thespatiallinkvariablesUi,andwiththeactionSf theintegra-
tion overtheU0 factorizesintoa productoverlink integrals. 0
To evaluate G(R,1)= 1Tr[U (x,1)U†(y,1)] , we first inte- 0 0.01 0.02 0.03 0.04 0.05
grateovertheU (x,t),wh2iththe0result,0toleadiingordering , V3-1/2
0
g 2 FIG.10:PlotofQvs.rootinverse3-volume,andextrapolationofQ
G(R,1)= 1Tr f †(x,2)f (x,1)f †(y,2)f (y,1) . (6.4) toinfinitevolumeinthegauge-fundamentalHiggstheoryatb =2.1
16 2 and various g , above (g =1.2) and below the first-order transition
D h iE
There are now 4 unsaturated f fields. For simplicity we pointaroundg =0.9.
suppose that (x,1) and (y,1) are joined by a principle axis,
which we take to be the 1-axis, and R= x y. The lead-
ing contribution to the f -integration at sm| a−ll g |is obtained is effective in keeping configurations away from the bound-
by saturating each link on the line that runs from (x,1) to aryL¶ ,wherethethinvortexconfigurationsaretobefound,
(y,1) by “bringing down” from the exponent Sf the term andwheretheFaddeev–Popovoperatorisoflongrange. On
1g Tr[f (z,1)U (z,1)f †(z+1,1)], and likewise for the line thecontrary,atsmallg thecouplingtotheHiggsfieldisinef-
f2rom (x,2) to1 (y,2). This gives a factor g 2R. The f - fective,andentropyleadstodominanceofconfigurationson
theboundaryL¶ .
integrations are now effectebd. The remaining integration on
Nowweturntonumericalsimulation.Figure10isacalcu-
thespatiallinkvariablesU isfinitebecauseofthegaugefix-
i lationofQvs.g atb =2.1,whereitisknown(fromthework
ing. We cannot evaluate it explicitly, but this last integra-
tion does not introduce any further g -dependence. We thus of Lang et al. [18]) that the first-order transition is around
obtain G(R,1)= g 2R+2 H(R). Here H(R) is not known, g = 0.9. Below g = 0.9, Q seems to extrapolate to zero at
but it is independent o×f g . This gives V(R,0) = (2R+ infinitevolume,whileabovethetransitionQextrapolatestoa
2)logg log[H(R)]. The asymptotic fall-off in the−correla- non-zeroconstant. Thereappearstobeanactualdiscontinu-
− ityinQacrossthewholelineoffirst-order(thermodynamic)
tor G(R,1) should not be more rapid than exponential, so
phase transitionsin the b g couplingplane. At sufficiently
RliminR→V¥(Rlo,0g)[H(Rs )]=R−,cwRh.eWreeththeu“sCoobutaloinmabl”insetrairnrgisteenastiloanrgies small valuesof b , there is−no thermodynamictransition, and
∼ coul weseenodiscontinuityinQasafunctionofg . Whatwesee
givenby
insteadisthatQ 0overafiniterangeofg ,andthen,beyond
s coul= 2logg +c. (6.5) acriticalvalueg ≈=g cr,Qsmoothlyincreaseswithincreasing
− g . OurresultsforQvs.g atb =0,on84 and164 lattices,are
This is non-zero for small g . To make this argument rigor-
showninFig.11;thesolidlineisthepresumedextrapolation
ousonewouldhavetoshowthattheexpansionofG(R,1)in
toinfinitevolume.Ifweweredealingwithaspinsystem,and
powersofg converges.Howeverthiscalculationdoesstrongly
Qwerethemagnetization,thiswouldclearlyrepresentasec-
suggest that at b =0 the remnantsymmetry is unbrokenfor
ondorderphasetransition. Inthepresentcaseitiscertainlya
smallg whereas,aswehaveseen,itisbrokenatlargeg .8
symmetry-breakingtransition,separatingasymmetricregion
Returning to the SU(2) gauge-fundamental Higgs theory,
withQ=0fromabroken-symmetryregionofQ>0.
we note that the above calculations at large and small g are
On the other hand, despite the existence of a symmetry-
easilyunderstoodintermsoftheCoulomb-gaugeandcenter-
breaking transition, there is no thermodynamic transition of
dominance scenarios. Indeed, for large g (and any b ), the any kindat b =0. At this value of b the free energycan be
gauge-fixedconfigurationsareatornearU(x)=I. Thiscon-
i computedexactly,withtheresult,ina4-volumeV
figurationisaninteriorpointofthefundamentalmodularre-
gionL . Thusatlargeg thecouplingtothefundamentalHiggs 2I (g )
F(g )=4Vlog 1 (6.6)
g
(cid:20) (cid:21)
which is perfectlyanalytic for all g >0. Thuswe have con-
8OnecanmakeasimilarcalculationofV(R,0)forsmallb (strongcoupling) firmedthetheoreticalargumentthattheremustbea remnant
inpureSU(N)gaugetheory.Thisyieldsafinite“Coulomb”stringtension symmetry-breaking transition even at small b , but we have
atsmallb givenbys coul= logb .Thissuggeststhatatleastinthestrong-
− also found that this transition is not accompanied (at small
couplingregionthereisa“Coulombfluxtube”thatconnectstheexternal
sources. b )byathermodynamictransition,definedassomedegreeof