Progress on Carolons

0.45 MB·

by  Pierre van Baal

9
0
0 Progress on Calorons
2

n
a
J

9
1

]
h
p Pierre van Baal∗
- Instituut-LorentzforTheoreticalPhysics
p
e UniversityofLeiden,P.O.Box9506
h NL-2300RALeiden,TheNetherlands
[
E-mail: [email protected]

1
v
3 Theprogressoncalorons(finitetemperatureinstantons)issketched. Inparticularthereissome
5
interestforconfiningtemperatures,wheretheholonomyisnon-trivial.
8
2
.
1
0
9
0
:
v
i
X
r
a
8thConferenceQuarkConfinementandtheHadronSpectrum
September1-62008
Mainz,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
ProgressonCalorons PierrevanBaal
1. Introduction
There has been a revised interest in studying instantons at finite temperature T, so-called
calorons[1,2],becausenewexplicitsolutionscouldbeobtainedwherethePolyakovloopatspatial
infinity(theso-calledholonomy)isnon-trivial. Theyrevealmoreclearlythemonopoleconstituent
nature of these calorons [3]. Non-trivial holonomy is therefore expected to play arole inthe con-
fined phase (i.e. for T <T ) where the trace of the Polyakov loop fluctuates around small values.
c
Theproperties ofinstantons aretherefore directly coupledtotheorderparameterforthedeconfin-
ingphasetransition.
At finite temperature A plays in some sense the role of a Higgs field in the adjoint rep-
0
resentation, which explains why magnetic monopoles occur as constituents of calorons. Since
A is not necessarily static it is better to consider the Polyakov loop as the analog of the Higgs
0
b
field, P(t,~x)=Pexp A (t+s,~x)ds , which transforms under a periodic gauge transformation
0 0
g(x) to g(x)P(x)g−1(cid:16)(xR), like an adjoin(cid:17)t Higgs field. Here b = 1/kT is the period in the imag-
inary time direction, under which the gauge field is assumed to be periodic. Finite action re-
quires the Polyakov loop at spatial infinity to be constant. For SU(n) gauge theory this gives
P¥ =lim|~x|→¥ P(0,~x)=g†exp(2p idiag(m 1,m 2,...,m n))g,wheregbringsP¥ toitsdiagonalform,
withneigenvaluesbeingorderedaccordingto(cid:229) n m =0andm ≤m ≤...≤m ≤m ≡1+m .
i=1 i 1 2 n n+1 1
Inthealgebraicgauge,whereA (x)istransformedtozeroatspatialinfinity,thegaugefieldssatisfy
0
theboundary condition Am (t+b ,~x)=P¥ Am (t,~x)P¥−1.
Caloron solutions are such that the total magnetic charge vanishes. A single caloron with
topological charge one contains n−1 monopoles with a unit magnetic charge in the i-th U(1)
subgroup, which are compensated by the n-th monopole of so-called type (1,1,...,1), having a
magneticchargeineachofthesesubgroups[4]. Attopologicalchargekthereareknconstituents, k
monopolesofeachofthentypes. Monopolesoftype jhaveamass8p 2n /b ,withn ≡m −m .
j j j+1 j
Thesumrule(cid:229) n n =1guarantees thecorrectaction,8p 2k.
j=1 j
Priortotheir explicit construction, calorons withnon-trivial holonomy wereconsidered irrel-
evant [2], because the one-loop correction gives rise to an infinite action barrier. However, the
infinity simply arises due to the integration over the finite energy density induced by the pertur-
bative fluctuations in the background of a non-trivial Polyakov loop [5]. The calculation of the
non-perturbative contribution wasperformed in[6]. Whenadded tothisperturbative contribution,
withminimaatcenterelements,theseminimaturnunstablefordecreasingtemperaturerightaround
theexpected valueofT . Thislendssomesupport tomonopole constituents being therelevant de-
c
grees of freedom which drive the transition from a phase in which the center symmetry is broken
at high temperatures to one in which the center symmetry is restored atlow temperatures. Lattice
studies, bothusingcooling [7]andchiral fermionzero-modes [8]asfilters, havealsoconclusively
confirmedthatmonopoleconstituents dodynamically occurintheconfinedphase.
2. SomeProperties ofCaloronSolutions
Usingtheclassicalscaleinvariancewecanalwaysarrangeb =1,aswillbeassumedthrough-
out. Aremarkably simpleformulafortheSU(n)actiondensity exists[4],
TrFab2 (x)=¶ a2¶ b2logy (x), y (x)= 21tr(An···A1)−cos(2p t),
2
ProgressonCalorons PierrevanBaal
1 r |~r | cosh(2pn r )sinh(2pn r )
A ≡ m m+1 m m m m ,
m r 0 r sinh(2pn r )cosh(2pn r )
m m+1! m m m m !
with r ≡|~x−~y | and~r ≡~y −~y , where~y is the location of the mth constituent monopole
m m m m m−1 m
withamass8p 2n . Notethattheindexmshouldbeconsideredmodn,suchthate.g. r =r and
m n+1 1
~y =~y (thereisoneexception, m =1+m ). Itissufficientthatonlyoneconstituent location
n+1 1 n+1 1
isfarseparatedfromtheothers,toshowthatonecanneglectthecos(2p t)terminy (x),givingrise
toastaticactiondensityinthislimit[4].
Figure1: ShownarethreechargeoneSU(2)caloronprofilesatt =0withb =1andr =1. Fromleftto
right for m =−m =0 (n =0,n =1), m =−m =0.125 (n =1/4,n =3/4) and m =−m =0.25
2 1 1 2 2 1 1 2 2 1
(n =n =1/2)onequallogarithmicscales,cutoffbelowanactiondensityof1/(2e).
1 2
In Fig. 1 we show how for SU(2) there are two lumps, except that the second lump is absent
for trivial holonomy. Fig. 2 demonstrates for SU(2) and SU(3) that there are indeed n lumps (for
SU(n)) which can be put anywhere. These lumps are constituent monopoles, where one of them
hasawindinginthetemporaldirection (whichcannot beseenfromtheactiondensity).
Figure 2: On the left are shown two charge one SU(2) caloron profiles at t = 0 with b = 1 and
m =−m =0.125, for r =1.6 (bottom) and 0.8 (top) on equal logarithmic scales, cutoff below an ac-
2 1
tion density of 1/(2e2). On the right are shown two charge one SU(3) caloron profiles at t = 0 and
(n ,n ,n )=(1/4,7/20,2/5),implementedby(m ,m ,m )=(−17/60,−1/30,19/60). Thebottomcon-
1 2 3 1 2 3
figurationhasthelocationofthelumpsscaledby8/3.Theyarecutoffat1/(2e).
2.1 FermionZero-Modes
An essential property of calorons is that the chiral fermion zero-modes are localized to con-
stituents of a certain charge only. The latter depends on the choice of boundary condition for
the fermions inthe imaginary time direction (allowing for an arbitrary U(1)phase exp(2p iz)) [9].
This provides an important signature for the dynamical lattice studies, using chiral fermion zero-
modes as a filter [8]. To be precise, the zero-modes are localized to the monopoles of type
3
ProgressonCalorons PierrevanBaal
m provided m < z < m . Denoting the zero-modes by Yˆ (x), we can write Yˆ†(x)Yˆ (x) =
m m+1 z z z
−(2p )−2¶ m2fˆx(z,z),where fˆx(z,z′)isaGreen’sfunctionwhichforz∈[m m,m m+1]satisfies fˆz(z,z)=
p <v (z)|A ···A A ···A |w (z)> /(r y ), where the spinors v and w are defined by
m m−1 1 n m m m m m
v1(z)=−w2(z)=sinh(2p (z−m )r ),andv2(z)=w1(z)=cosh(2p (z−m )r ).
m m m m m m m m
To obtain the finite temperature fermion zero-mode one puts z= 1, whereas for the fermion
2
zero-mode with periodic boundary conditions one takes z=0. From this it is easily seen that in
caseofwellseparated constituents thezero-mode islocalized onlyat~y forwhichz∈[m ,m ].
m m m+1
To be specific, in this limit fˆ(z,z)=p tanh(p r n )/r for SU(2), and more generally fˆ(z,z)=
x m m m x
2p sinh[2p (z−m )r ]sinh[2p (m −z)r ]/(r sinh[2pn r ])−1. We illustrate in Fig. 3 the lo-
m m m+1 m m m m
calization ofthefermionzero-modes forthecaseofSU(3).
Figure3: FortheSU(3)configurationinthelowerrightcornerofFig.2wehavedeterminedontheleftthe
zero-modedensityforfermionswithanti-periodicboundaryconditionsintimeandontherightforperiodic
boundaryconditions.Theyareplottedatequallogarithmicscales,cutoffbelow1/e5.
2.2 CaloronsofHigherCharge
We have been able to use a “mix” of the ADHM and Nahm formalism [10], both in making
powerfulapproximations,likeinthefarfieldlimit(basedonourabilitytoidentifytheexponentially
risingandfallingterms),andforfindingexactsolutionsthroughsolvingthehomogeneousGreen’s
function [11]. We found axially symmetric solutions for arbitrary k, as well as for k=2 two sets
Figure 4: In the middle is shown the action densityin the plane of the constituentsatt =0 for an SU(2)
charge2caloronwithtrP¥ =0,whereallconstituentsstronglyoverlap. Onascaleenhancedbyafactor
10p 2 are shown the densities for the two zero-modes, using either periodic (left) or anti-periodic (right)
boundaryconditionsinthetimedirection.
4
ProgressonCalorons PierrevanBaal
ofnon-trivial solutions forthematching conditions thatinterpolate betweenoverlapping andwell-
separated constituents. Forthis task wecould makeuse ofan existing analytic result for charge-2
monopoles [12],adapting ittothecaseofcarolons. AnexampleisshowninFig.4.
There has also been some progress on constructing the hyperKähler metric which approxi-
mates the metric for an arbitrary number of calorons. They claim that this already gives confine-
ment[13]. Fortheimplications ofcaloronconstituents nearthephasetransition see[14].
Acknowledgments
AgainImanagedtowriteproceedings, butIneededsomewhatmoretime,andIamgratefulto
Matthias Neubert to allow for that and organizing a wonderful conference. Also, there are simply
toomanynames,butIthankeverybody whoworkedwithme.
References
[1] B.J.HarringtonandH.K.Shepard,Phys.Rev.D17(1978)2122;Phys.Rev.D18(1978)2990.
[2] D.J.Gross,R.D.PisarskiandL.G.Yaffe,Rev.Mod.Phys.53(1981)43.
[3] T.C.KraanandP.vanBaal,Phys.Lett.B428(1998)268[hep-th/9802049];Nucl.Phys.B533(1998)
627[hep-th/9805168];K.Lee,Phys.LettB426(1998)323[hep-th/9802012];K.LeeandC.Lu,
Phys.Rev.D58(1998)025011[hep-th/9802108].
[4] T.C.KraanandP.vanBaal,Phys.Lett.B435(1998)389[hep-th/9806034].
[5] N.Weiss,Phys.Rev.D24(1981)475.
[6] D.Diakonov,N.Gromov,V.PetrovandS.Slizovskiy,Phys.Rev.D70(2004)036003
[hep-th/0404042];D.DiakonovandN.Gromov,Phys.Rev.D72(2005)025003[hep-th/0502132].
[7] E.-M.Ilgenfritz,B.V.Martemyanov,M.Müller-Preussker,S.ShcheredinandA.I.Veselov,Phys.Rev.
D66(2002)074503[hep-lat/0206004];F.Bruckmann,E.-M.Ilgenfritz,B.V.MartemyanovandP.van
Baal,Phys.Rev.D70(2004)105013[hep-lat/0408004];P.Gerhold,E.-M.IlgenfritzandM.
Müller-Preussker,Nucl.Phys.B760(2007)1[hep-ph/0607315].
[8] C.GattringerandS.Schaefer,Nucl.Phys.B654(2003)30[hep-lat/0212029];C.GattringerandR.
Pullirsch,Phys.Rev.D69(2004)094510[hep-lat/0402008].
[9] M.GarcíaPérez,A.González-Arroyo,C.PenaandP.vanBaal,Phys.Rev.D60(1999)031901
[hep-th/9905016];M.N.Chernodub,T.C.KraanandP.vanBaal,Nucl.Phys.B(Proc.Suppl.)83-84
(2000)556.
[10] M.F.Atiyah,N.J.Hitchin,V.DrinfeldandYu.I.Manin,Phys.Lett.65A(1978)185;W.Nahm,
“Self-dualmonopolesandcalorons,”in:LectureNotesinPhysics,201(1984)189.
[11] F.BruckmannandP.vanBaal,Nucl.Phys.B645(2002)105[hep-th/0209010];F.Bruckmann,D.
NógrádiandP.vanBaal,Nucl.Phys.B666(2003)197[hep-th/0305063];Nucl.Phys.B698(2004)
233[hep-th/0404210].
[12] H.Panagopoulos,Phys.Rev.D28(1983)380.
[13] D.DiakonovandV.Petrov,Phys.Rev.D76(2007)056001[arXiv:0704.3181[hep-th]];
D.Diakonov,ActaPhys.Polon.B39(2008)3365[arXiv:0807.0902[hep-th]];D.DiakonovandV.
Petrov,“Statisticalphysicsofdyonsandquarkconfinement,”arXiv:0809.2063[hep-th].
[14] F.Bruckmann,“Howinstantonscouldsurvivethephasetransition,”arXiv:0901.0987[hep-ph].
5