Comments on the temperature dependence of the gauge topology

1.9 MB·

by  Edward Shuryak

Comments on the temperature dependence of the gauge topology
Edward Shuryak
Department of Physics and Astronomy, Stony Brook University, Stony Brook NY 11794-3800, USA
Recent efforts in lattice evaluation of the topological susceptibility had shown that at high tem-
peraturesitisgivenbywell-separatedinstantons(eveninQCDwithlightfermions,wherethoseare
highlysuppressed). RecentdevelopmentofthesemiclassicaltheorysuggestthatbelowT 2.5T ,
max c
∼
wherePolyakovlinehasvaluesbetweenoneandzero,thetopologyensemblecanberepresentedbya
plasmaofinstantonconstituents(calledinstanton-dyonsorinstanton-monopoles). Ithasbeenshown
that such ensemble undergoes deconfinement and chiral transitions, semi-qualitatively reproducing
thelatticeresults. Thereareongoingeffortstolocatethemonthelattice,oruse(flavor-dependent)
periodicity phases of the deformed versions of QCD on the lattice and semiclassically, in order to
test this theory. We here propose another possibly useful tool: the topological susceptibility of a
sub-lattice.
7
1
I. INTRODUCTION discuss how different versions of the topological suscep-
0
tibility can help us to tell if this is indeed the case.
2
Recently the field of lattice gauge topology has been
n
a re-activated, due to two independent developments.
J One is several recent extensive lattice studies of the

7 topologicalsusceptibilityχ(T)inawiderangeoftemper-
2 atures T, from zero to about 2 GeV. (Its motivation is

partlyarelationtoaxionmodelsofthedarkmatter.) In
]
t Fig.1(upper) from [3] one can see the continuum extrap-
a
olated value of χ1/4 versus T/T (the lower red shaded
l c
- region) compared for T/T > 2.5 with the dilute instan-
p c
ton gas approximation (DIGA). The upper gray region
e
h corresponds to the results of ref. [1]. Not shown in this
[ plotareresultsfromthework[2],whichimpressivelyfol-

lowed χ(T) to T as large as 2GeV, also with the conclu-
1
sion that DIGA is correct at high enough T. Ref.[1] also CONF12
v
9 had measured higher moment of the topological charge
8 fluctuations, b2. TFhiegudraeta5:frComonttihniusuwmorekxtsrhaopwonlatiendthreesults for χ1/4 measured from gluonic definition of topo-
80 DlowIGerApvaarltueof–Ffoigll.o1lowgaiilncsgoalfscrhohomawrgEethaantdf(omr 2lTχ/cdToicssc(>)θ1)/24–.5fistettehemde separatelyt (l0e.0f20t)and thejointcontinuum exaa t==r 00a..00p87o2047l affmmtion
0 to be reached. Acocfepχt1ti/n4gatnhdes(emvca2locuχnucdmliuscs∼i)o1n/4s,(wrieghdti)s.cuInssthe right pan-0e.0l2, our resultsba2(rTe=0c)ompared waCi ht=Ph 0T.0t5h7e2 fmcon-
. tinuum extrapolated results obtained in Ref. [36]. The solid orange line corresponds to a
1 belowwhy thebehaviorchangesbelowthistemperature, -0.04
0 andwhatisthecorpraecrttiadlestcwriop-tloioonpoDftIGheAtocpaolclouglaytbioenlowwith µ = πbT2,-0.K06 = 1.9 and αs(µ = 1.5 GeV) = 0.336,
7 it. whilethe band is obtained from the variation of αs by 1σ around this central value as well as
1 variation of the scale µ by a factor two (see text).-0.08
Another recent development are works devoted to bDIGA
: -0.1 2
v of the ensemble of the instanton constituents, called
i -0.12
X instanton-monopoles or instanton-dyons. As shown in
A similar analysis was performed for-0(.1m4 2χ )1/4. In this case the low
thepioneeringpapersbyKraanandvanBaal[4]andLee l disc
r 1 2 3 4
a and LuLee:1998bba,nadn ihnsigtahnttoenmcopnesrisattsuorfeNre(gniuomnsbewrere defined as 165 MeV TT/T 240 MeV and
c ≤ ≤c
of colors) of those.2T4h0eyMsheaVre<uniTt topol5og0i4caMl cheaVrg,eroefspectively. From the fits we get b = 1.96(22)
the instanton accorindintghteolcoewrtatinemfrap≤cetrioantsuνrei,irF(=eigg11u/i1or..e2nN)6.,.cTbhw2ealhsigaiFhltfIeubGnluc.ietni1obn:atnodh(fUiTesp,thtphoeegirreget)shhuelTrtowhtfeeitahmtfitothppetoeoCtlrhhoeaPgTstimcupaarrlleldeesiscuttrlisaoectntegicpaiettoTsibpna=i,lcii0tntyg(h−dχ0ea.t10a/2ru42es)iv-nagenrdastvuhiseriaDlIeGxApapnrseiodnic(tisoene
such that Ni=c1νi =su1lt.ing b = 2.22(27). ThRe−ef.t[w32o]fofirtmsortflehwudeceetttraueiealmst).ipmoenraapttacurhraeemdTe/teTarct.bT,, ffrr=oomm2[[133]]5.(LMoweeVr) Ttoheohbigthaeirnorder
The main rational for the instanton to get dis- 2
(cid:80) a continuum estimate, shown as a red band in Fig. 5. It is clearly evident
assembled into those constituents is the fact that the
1/4
meanPolyakovlinethbealtowthceertcaoinnTtinduevuimatesesfrtoimma1,tefosrcf-or χ from a gluonic observable and that for
t
ingallobjectstoin(tmerp2lχoldaitsec)to1/a4ncooniznercoid‘heo.loTnohimsismyovissatlly-hdiugehtolythendoiffne-rternitvbieahalviaonrodfχm(Ta),kwehsichusshocwosn,afitdleeasnttintthheaetxploredrange,amuch
ues’ofA0. InQCDthweithcopnhtyisnicuaulmquaerxkstrtahpisohlaaptmpioieldnnesriadstepreenldieanbceleoneITvI.ewnithVtArhesRopeuIcOgthtUoSDthIDGeAEcFpurIeNtdoiIcffTtioIneOsff.NeTcShtisOsreFasurTletHoifmEco-ursemightbeaffected
T 2.5T [9]. byresidualsystematiceffTecOtsP,iOnpLaOrtiGcuIlaCrAweLhSavUeSreCsuEltsPaTttIhBreIeLdIiTffeYrentlatticespacingsonlyin
Polyakov ≈ c portant. A joint fit was also performed according to Eq. 7, allowing the pa-
We suggest that deviation of the DIGA fromasuresgcieopn-whichgoesuptoT 300MeV,whiletheoscillationtemperaturederivingfromourdata
rameters a , a , b , b ainsodftbheotrdoerbofefedwiffGeeVres,nsto∼tfhoatrw(emare2χstrongl)y1r/e4lyiangnodnaχn1e/x4traspoinlactieon. Themaindifficulty
tibilities below this temperatu2re is4not2a m4ere coinc6i- In order to see howlthodsisectheoretical idteas can match
inapproachinghighertemperaturesisgivenbythecriticalslowingdownobservedforsmalllattice
dence, and that atthTe c<ut2-.o5ffT etffheectinsstinanttohnessearqeudains-titielsatatirceecmleeaasrulyremdieffntesr,eonnte. nWeeedscfiornsstidtoerceladriftyhraened dis-
c spacingsandbythefactthattopologicalfluctuationsbecomeveryrareathighT.
assembled into instdainffteorne-dnytonrsa.ngInesth1e6se5cMomemVents wTe 2ti1n0guMisheVva,rio1u9s5exMisteiVng defiTnition3o0f0χ.MAesVweawnidll see,
≤Recent≤resultsfromdifferentlatticeinvest≤igations≤haveshowncontrastingresults. Whilethere-
230 MeV T 504 MeV. We checked different fit ans¨atze, setting some of the
sultsofRef.[33]areinqualitativeagreementwithours,otherstudieshavereportedamuchbetter
≤ ≤
parameters a and b to zaegrroeemaenndtwaithlstoheiinnsctalnutodningagsporerdiectxiocnluevdeniningthtehreegiNonrig=hta6bodveaTta[.34–36].
i i τ c
Allsuchtrialsresultedinχ2Ipneprardticeuglarre,etheofauftrheoersdoofmReff.o[r34t]hmeanfiatgteintogdepterromcineedχu(Tre) ftoor Tbeup to a few GeVs and
about one or less. TherefoorbeserwveeassliompepcloynsaisvteentrwagithedDIGovAesrhoartlllysafutecrhTcfi.Tthreemsauinltdsiffienrenecaeschconsistinareweighting
procedureadoptedinRef.[34],basedonthelowestlyingDiracoperatoreigenvalues,andinanew
temperature region and used the spread in the fit to estimate the final error on
strategytoapproachthehighTregime(seealsoRef.[37]).Thisapproachavoidsdirectsimulationsat
our continuumextrapolatihoignh.T,Wanediaslbsaosedchinestcekadeodntshimautlattihonesaetrfirxoerdteosptoilomgya,twehdichtahreisuswedatyodeterminetherelative
is about the same as the swteaitghist,tiinctahleepartrhoinrteogfratl,hoefthreeptorpeolsoegnictaalsteicvtoersfiQts=. W1weitmhraestpcehctetdothetopologicalsector
±
0,sincethosearebelievedtobetheonlyonesrelevantathighT. Ofcoursesuchastrategyassumes
the fit results for the three different regions at T = 194 MeV and T = 277 MeV
rightfromthebeginningthattheinstantongasisdiluteenoughandnon-interacting.
respectively to obtain our final continuum estimate for χ , shown in right panel
Goodquantitiestocheckforthedilutenessthypothesisaretheb coefficientsappearinginEq.(2),
n
of Fig. 5.
whicharefixedbythefactthat,foranon-interactinginstantongas,onehasF(θ) (1 cosθ). Our
∝ −
Comparing our continureusumltsfifotrbfo2rareχrtepworittehdinthFeig.c6o:netviennuifuthmesetaxtitstricaaplounlcaetrteadintrieessaureltsstilllarge,oneseesthat
deviationsfrom the dilute gas hypothesiscould be still appreciablefor T up to 2 3 T . This is
from Ref. [36], we find that the exponent b from our fit is about factor three − c
incontrastwiththepuregaugecase[38–41],wherecorrectionsbecomenegligibleshortlyafterT .
c
larger than the corresponTdhaintcgonfibrmrsetpheonrotne-dtrivtiahlietyroef.ferTmihoenicpcoonstrsiibbutlieonsraenadscolanimfsoforrftuhtuirsestudies,whichshould
furthercheckexistingresults. Variousimprovementsarepossible,especiallytodealwiththecritical
slowingdownoftopologicalmodesbymeansofimprovedalgorithms[42–46].
12
2
fewexistingdefinitionsarenotatallidentical,andshould
lead to very different results.
We start with the canonical definition of the topolog-
ical susceptibility χ , following standard rout of
canonical
thestatisticalmechanics. Thevacuumorthermalensem-
blewithnonzerotheta-angleθisdefinedbyanadditional
term in action, which adds to the partition function an
extrafactoreiθQ, whereQisthetotaltopologicalcharge
of the volume V under consideration. Since iθ play the
4
sameroleasachemicalpotential,thetopologicalsuscep-
tibilitycanthusbegiventhesamestandarddefinitionas
any other susceptibility, namely
1 ∂2
χcanonical =(V Z) ∂(iθ)2Z(θ) ) (1) FIG. 2: An example of the susceptibility in sub-box χsublat
4 (cid:18) (cid:19) as a function of the fraction of the total box, x/L. The thin
paraboliclinecorrespondstorandomlyplacedinstantonsand
The volume V4 in this definition should be the one of a antiinstantons, the dots are for the interacting instanton liq-
subsystem, of even much larger heat bath. Grand canon- uid, from [8]. Strong screening of the topological charge in
ical ensemble with the chemical potential implies that this model is evident. Thin lines show different fits, from
there is free exchange of particles through the bound- which the value of m was extracted.
η(cid:48)
aries of the subsystem. Large heat bath ensures that the
values of variables like T and µ = ıθ are fixed, without
any fluctuations.
The standard lattice definition III. INSTANTONS AND THE HIGH-T REGION
<Q2 > Westartdiscussingthedifferencesbetweenvariousdef-
χ = (2)
lat
V4 initionsofχusingthecontextoftheinstantonensemble,
FIG.17. PseudoscalarcorreilnatowrhKicPh(l4χ)(upperwpaansel)oraingdinscaalllayrgilnutornoicdcuocrreedla.torKS(l4)(lowerpanel)asafunctionofthe
may look identical to the canonical onetlehgneigvstuhernlf4acaoebf,tonhoveteso,unbbvtuhotleuvmoelulm4×eLoLfe3t,thferuotsmorsu(tSssa.uhrubTtlrahywtiaskimt&heaVQnesrCbthaDaatrsiicnnhtoththe1e9p9rce5hs)e.inrScacelroelefinmsicnrigete,inmwinpiglti,ehKst((hlm4a)tagtsohsee-sctoorraelcaotnosrtdanept.enTdhseornelsyulotns
is in fact quite different, due to the factwetrheaotbtianintehdifsorcNasce=3anledsms)uq=umardk=s.10InMetVhiasncdamses=an1y50cMoneVfi.gTuhreatuipopnerwsoitlihdlninoenszcoerrroespondtoarandomsystem
thesystemwithV4 isthewholelattice.oIftinissttaonptoonlso,gwihcialelltyheothetrospoolildolginiceaslhocwhsatrhgeepaQramheatrsizaQtionqudaisrckusszeedriontmexotd(tehse.daTshhederlein-eintheupperpanelshows
atoruswithnoboundaries,sinceperiodaicslibghotulynmdaorreysocpohnis-ticatefdoprear,amtheetrifzeartimoni)o.nNicotedethteerqmuailnitaantivteidsiffzeerreonceifbQetw=een0t,hesodatthaefor topological andnumber
ditions of the fields are imposed. Bothflueclteucattrioincs.QE and gauge ensemble include only configurations w(cid:54) ith Q = 0
mthaegtnoeptoicloQgiMcalcchhaarrggeeoQf tmhiusstvobleuminetemgaweunrhsg-evtlreae.bluDTeeh(dzmee.c,roxor),re=alantmdor/((42π272a)xn)hLdKaes1tt(hamunuxsso)bχfivislriaosttuth=,sepf(0hoey.ruscislciiadmleaipnnlt)iecrpiptrryoe,ptaaftgoiaoctnuo.rsToohfnealoTsccaall>atrerTpmacrs,tiiswcltehheaerncedonφtriibsutthioenηfr−omη′amsiixnignlge
Another definition χsublat has been ipnrsotapnotsoendlobcayteVdeart-the ctehnetrere, wishinleothqeusaerckoncdotnedrmenissattheeaconndtritbhuetiocnhifrroaml styhme smcreeternyingcloud. One can easily
baaschot and myself [8]. Since it wascdhoencketmhaatntyheyienatergsral ofrethmeacionrsreulantobrroiskoefno.rdIenr mth2i,ssocaχse themtoinpothloegcihciarlalolbimjeitc.tsWe also observe that the
ago,letusremindit. Weproposedtocusctretehneinlgaltetnicgtehinistgoivenbcyantheexmisatssoonflythaesη′s.omecluπsterstwopit∼hthetotaltopological
two subsystems, a and b, with subvolumeDseVta4aile+dVn4ubm=erVic4al studciheasrogfetoQpo=log0ic.alTchhaergseimscpreleenstingofinththoeseinaterreactthinegininssttaannttoonn-model were performed in
anddefinethecorrespondingsusceptibi(lSithiuersybaky&thVeesrbaamareschota1n9t9i5i)n.sTtahnetaounthmoroslevceruifileesd.tThahteceomnspelmeteblsecrmeeanidnegotafktehsopsleacheaisfoneofthequarkmasses
expression above. The simplest arrangegmoeesnttoizsertooacnudttbhyatthebsecreenendinisgcluenssgethdisbyconIlsgisetnenfrtitwzithantdhemη′ymsealsfs.[1T1h].eyWalesodaoddnroestsedthequestionhowthe
two planes normal to one of the coordηin′amtaesss,cparnobdeuceixntrgacteddfrisocmustsoptohloegmicahlecrheaargsetflhuecytuaarteionnso.tTrehleevmaanintfidoeratiospnooltogtoicsatludy the limiting value of
Q2 /V forlargevolumes,butdetermineitsdependenceonV forsmallvolumesV <1fm4. Inthiscase,onehasto
two “slices” with ⟨worr⟩yaboutpossiblesurfascuesecffeepcttsib. iIltitisy.thereforebesttoconsiderthetopologicalchargeinasegmentH(l )=l L3
ofthetorusL4 (ahypercubeWwhitihlepχerliaotdi=cb0o,utnhdearsyucbolnadtittiicoensd).efiTnhiisticoonnswtrouuctlidonleeandsutreosthatthesurf4acea4re×aof
Va =L3x,Vb =L3x¯,x¯=LH(xl )isindepend(e3n)tofitsavnoolunm-zee.rUosivnaglutheeχeffective=me0s,onbeacctaiuonseintthroediuncsetdaanbtoovnes,waenedxpect(inthechirallimit)
4 4 − 4 sublat (cid:54)
the antiinstantons may happen to be located in different
N 1
Timhpisordteafinntitaidovnanneteadgsessoomveer[1t4h]eeχxtra. wOonrek,isbtuhtaittnhoawsttwhoe saubbsovKroPblu(elmd4)eb≡sy,⟨QIs¯e(mle4)aF2y⟩ig“=3leLa.k3”!TVhthe"roqmuuηga′hrk1tsh−,eecb−remoauηt′nle4dda.bryy! I and (228)
lat # $
sub-volumes do have a boundary, and they do not have Note, that for the particular geometry of sub-box pro-
a requirement that Q = Q = 0. As we will discuss posed, by moving a plane6a9nd changing x one changes
E M
below, quarks and Dirac strings can “leak” through it. the sub-volumes Va,Vb but not the area of the surface
4 4
Notealso,thatinthissettingoneobtainsnotanumber A3 separating them. Since the leakage is expected to be
but the function χa(x), which can be used to define the proportional to this area, χsublat A3, not volume, it
∼
“screening length of the topological charge”, known also should become x-independent at large x.
as the η mass. In this case one gets an idea what is a Supposenowweallowsmallbutnon-zeroquarkmasses
(cid:48)
“largeenoughbox”,sinceform(η )x 1thedependence m (for simplicity, the same for N quark flavors). Quark
(cid:48) f
(cid:29)
on x disappears. “veto” on Q = 0 configurations such as individual in-
(cid:54)
3
dozenpapersonthat. Thosewillnotbediscussedbelow,
for a brief review of some of them see [6].
When the mean Polyakov line < P > is between
¯
II I I¯ 1 to 0, gauge topology is expected to be described
by an ensemble of the instanton-dyons, with different
temperature-dependentactionsandnon-integer[15]topo-
logical charges, driven by <P(T)>.
In the simplest case of the SU(2) gauge theory there
are two types of dyons, selfdual M with S /S =ν and
M 0
FIG. 3: Some configurations which produce no contributions L with SL/S0 = ν¯, plus anti-selfdual anti-dyons. The
to the χ , but contribute to χ parameter ν is related to the Polyakov line by < P >=
lat sublat
cos(πν). In the SU(2) gauge theory there are two M-
I = LM L M typedyons,relatedtocomplex-conjugatedeigenvaluesof
< P >. In this case ν¯ = 1 2ν and < P >= (1/3)+
stantons is now lifted. Since in a dilute ensemble the (2/3)cos(2πν). −
instantons can be considered to be non-interacting, one
Plugging in the lattice input < P(T) > one can plot
should use the Poisson distribution, and therefore
the dyon action: see example in Fig.4 (for QCD). This
plot can be used to identify the region in which the
χ(T) n(T) (Λ)b Nf(mf) (4) instanton-dyon theory is semiclassical. The semiclassical
T4 ∼ T4 ∼ T T density depends on the action by n S2exp( S): the
1 ∼ −
(cid:89) poweroftheactionrepresenthalfofbosoniczeromodes.
where b = 11Nc/3+2N /3. So, the high-T limit corre- This formula has a maximum at S = 2, and we take
f
sponds to very small χ , decreasing as relatively large it as the lowest possible action at which it makes sense.
lat
powerofT,timesarathersmallproductofquarkmasses. The left side of Fig.4 indicate the lowest temperature at
(WewilldiscussSU(2)gaugetheory,SU(3)gaugetheory which this condition is fulfilled Tmin 80MeV. The
∼
andQCDwith3dynamicalquarks: the(inverse)powers right side – high T – shows that while the action of the
of the temperature in those cases are 22/3=7.66, 11 and Ldyongrows,thatofM decreases,soTmax 370MeV.
∼
6, respectively.) The phase transition is indicated as a transition from a
symmetric to asymmetric phase. These considerations
of course refer to simple non-interacting dyons. We use
20 �(�) them simply to convey the range of T <T <T in
min max
which this approach is expected to work.
15 Thesemiclassicalformulaeforthedensityofinstanton-
� dyons are higher than instantons, because they have
smaller actions. This is the generic reason why the χ(T)
10
at T < 2.5T gets larger than the DIGA prediction.
c
Another generic reason is that M-type dyons have no
5 � quarkzeromodesandthusarenotsuppressedbyfermion
masses.
¯
0 PropersItudIiesofthedyonensIemble–suchI¯as[10]from
which we borrowed Fig.6 – include their mutual interac-
100 150 200 250 300 350 400 450
tion as well as back reaction to the holonomy potential,
determining its value ν from the global minimum of the
FIG. 4: The temperature (MeV) dependence of the actions
S/(cid:126) of L and M type dyons, in SU(3) QCD.
IV. THE INSTANTON-DYONS I = LM L M
Semiclassical theory of instantons, incorporating a
nonzero Polyakov line VEV and thus a nonzero mean
value of the gauge field < A >= 0, lead in 1998 to the
4
(cid:54)
discovery of the instanton-dyons [4, 5]. It is nearly two
decades since these papers, but only recently a heavy FIG.5: Someconfigurationswhichproduceverydifferentcon-
work on building a semiclassical theory of their ensem- tributions to the χlat and χsublat
blewasintensified. Lastyearalonehasproducedabouta
INSTANTON-DYONENSEMBLEWITHTWODYNAMICAL … PHYSICAL REVIEWD93,054029(2016)
of M dyons N and v describing the correction to the S,oneofthemajorparametersofthemodelcontrollingthe
M 0
pointthatweretheminimum.Misa3times3matrixwith dilutenessoftheensemble.Wealsoindicateatthetopthe
M MT containing the coefficients for the fit. corresponding temperature, relative to the critical temper-
T¼hisexpressionwasfittedtofreeenergyvaluesof53 atureT ,chosenasS 7.5.Itshouldbenotedthatthisisa
¼ c ¼
125pointsfromacube,containingfivepointsaroundthe choice,andisdoneinordertosetascale.Therealinputis
minimumineachdirection.Theresultingvaluesoftheten theactionSorthecouplingconstantg.Thetemperatureis
parameters fitted are used as follows: (i) v and its found from the running coupling constant:
0
uncertainties give the values of densities and holonomy
attheminimum,plottedasresultsbelow;(ii)thediagonal 8π2 T 11 2
componentofMintheholonomydirectionwasconverted SðTÞ¼g2 T ¼b·ln Λ ; b¼ 3 Nc−3NF: ð20Þ
into the value of the Debye mass Md. An additional ð Þ ! "
requirement of the procedure, to make the ensemble
Thistoptemperaturescaleisapproximateandshouldonly
approximatelyself-consistent,isthattheDebyemassfrom
be used for qualitative comparison to other models and
the fit should be within 0.5 of the used input Debye
mass value. " lattice data.
To obtain the chiral properties—such as the Dirac
eigenvalue distributions and its dependence on dyon A. Parametrization A for T
ij
number and volume—we only used the “dominant” con-
The results in this subsection are for
figurationsforeachactionS,definedasfollows.SinceN
M
is always an integer, we use the value closest to that
oanbatalyinzeeddfarsomextphleaifnite.dTihneSeeigce.nIvVaAlu.edistributionsarethen Tij¼v¯c0exp − 11.2þðv¯r=2Þ2 : ð21Þ
! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"
VI. PHYSICAL RESULTS Minimizing the free energy gives the dominating param-
etersforaspecificactionSortemperatureT.Thisisdone
An accurate gauge-independent determination of the for Λ 4 and −Log c0 −2.60. This gives the holon-
hoppingmatrixelementEq.(3)is,ingeneral,notatrivial omy, t¼he density, Figð. 9Þ,¼and Debye mass, Fig. 12. The
procedure. While zero modes for a single dyon are well dominating configurations have been analyzed using the
known,combiningapairofLandL¯dyonsisnotassimpleas methodsdescribedinSec.IVinordertoobtainthechiral
itisforinstantons:thecomplicationiscausedbymagnetic condensate, which is shown together with the Polyakov 4
charges andtheDirac stringsassociatedwith them,trans- loopinFig.10andisalsocomparedtothegapinFig.11.
portingsingularmagneticfluxtotheircenters.Ideallythose free eWneerogbys.erAvesaosnmeooctahntrasenesitfiroonmtowtharidssfithgeuzreer,otehxepreecti-s no The crucial observation is that only twisted L-type dyon
are invisible pure-gaugeartifacts, whosedirection isirrel- symatmionetrvicalupehaosfe,thaendPotlhyaekreovarleooaplwPayassmteomrepeMratudreyons has physical – anti-periodic – quark zero modes. There-
evant:butitisnotsoforsimpleconfigurationslikethesum thadnecLre.aseAs.lsWoetahlesodoebcsoenrvfienaemnoennzteraonvdalucehiorfalthterachnisriatlions fore, if quarks are massless, those can only exist inside
ansatz.“Combinggaugefactors,”whichappearinthezero beccoomndeenisnatQeaCsDte-mlipkeerattuhreeodreiecsreajusesst.Tahsismisoaotmhorceroabssru-opvters, “neutral clusters”, such as L¯L molecules. We however
modewavefunction,complicate the calculation, although
hapchpaenngien,gthaoturgohuignhlsyomtheewsaaymsetiltlesmmpoeortha.tuItrse.inflection will not discuss the molecular component here, as any
numerically their effect is relatively small: see more in
point (change of curvature) is found around S 7.5,
topologically neutral objects are irrelevant for the topo-
AppendixAof[17].Currentlyweareworkingonsolving ¼
theDiracequationfor“streamline”configurationsdefined logical susceptibility.
in[12],butthisworkisnotyetfinished.
Asatemporalsolution,weusetwoparametrizationsof
thehoppingmatrixelement.Weperformsimulationswith V. THE TOPOLOGICAL SCREENING AND
bothsets.Theparametrizationsthemselvesareexplainedin THE η(cid:48) MASS
the Appendix. The physical results are, respectively, split
upintotwosections,oneforeachchoiceofTij.Sincethe AtT <Tc thechiralsymmetryisbroken. Howexactly
overall constant c0 is unknown, values of c0 have been it happens from the point of view of topological object
chosen, such that the transition happens around S 7.5.
has been worked out in the instanton liquid model, see
¼
We are actively trying to obtain c from numerical simu-
0 [7] for a review. The nature of the topological objects
lations. While the different T ’s behave similar for large
ij involved –an instanton or only its L-type constituent –
distances, the behavior is different around zero. This also
is unimportant: any one with a fermionic zero mode is
meansthattheconstantc canbedifferentinthetwocases.
0 generatingthecorresponding’tHooftvertex. AsT T
Forthese results c0 waschosen suchthatthe densityof L → c
dyons did not become too small, while having a smooth RASMUSLARSENandEDWARDSHURYAK quarks trPaHvYeSlItChArLouRgEhVIEloWngDer93a,n05d40l2o9n(g2e0r16c)hains of alter-
PolyakovloopthatwenttozerointherangeofS 5–10. FIG.9. ParametrizationA:ThedensityoftheM(bluecircles) nating topological objects. The length of a chain scales
The plots below have two scales, on their bott¼om and andL(redsquares)dyonsasafunctionofactionS 8π2=g2or as V4, not as its dimension (V4)1/4, which in the ther-
top.Theformeroneshowsthe“instantonaction”parameter temperatureT=T . ¼ modynamical limit become infinite. That is why Dirac
c
eigenvalues reach zero and quark condensate is formed.
As a result, pions get massless and one can use chiral
perturbation theory to describe χ(T) at T <T . This is
054029-7 c
all well known and we do not need to describe it.
There are however some issues related to χ and
sublet
the topological screening length m we would like to
η(cid:48)
commenton. Letusstartwiththefollowing(wellknown)
puzzle: its numerical value 1/m = 1/.958GeV
η(cid:48) ∼
0.2fm) is several times smaller compared to the typical
distancebetweenthetopologicalobjects,e.g. L-dyonsat
T ,whichisabout1fm. Onemaywanderifindeedthe
FIG.10. ParametrizationA:ThePolyakovloopP(bluecircles) min
FIG.an6d: t(hUecphpirearlcpolnodte)nsTatheeΣd(reendssiqtuiearseso)fasLaafunndctiMonotfyapcetiodnyonsF,IG.12q.uaPrakra-minetdriuzacteiodnAin:tDeerbaycetimoansscMadnasgeanfuenrcatitoensoof strong corre-
vliegrhstuSSsq¼utha87reπ.5k2a=fogcflr2taitvoohreonrtceshSm.irpa=el(rcLao8tounπwrde2ee/nTrsg=a2pTte(lc.To.Σt)A),isciTsnlechaaSerleUdrmisb(eey2a)i0sn.Q2s.ePCeTnohDleayrabwolkuaionctkdvh ltiwnaoectionSlian¼tip8orπno2s=ggri2neossrsid,teemanspuedrcahtsuorechwTa=eiTnccs.a.nTohnelycamlceunlatitoionnthfoartdityoinnsdeaerde
<P c>o¼nastnandtqliunaerckorcroesnpdoenndssattoethΣe,uvpepresruslimthiteosfaΣmeunvdaerriathbele S. The Debyemass, Fig.12, as compared to lattice results
worked out in the instanton liquid, even in its simplest
assumptionthattheentireeigenvaluedistributionbelongstothe [20],isseentobearound66%toolarge.Thiscouldbedueto
almost-zero-modezone,i.e.themaximumofΣ2. thechofiocermof,wsoerekiχngsuwbiltehtaahlarredacdoyre,sohroitwcnouildnsFiginga.l2th.at
thecorrecAt vnaoluteheforrcthoemsmizeenotf trheefecorsretiosstlihgehtllyimlairtgeor.f large number
Now let us go back to the topological susceptibilities.
though the transition happens between S 6.5–8. Below of colors N . As famously noted by Witten [12] ,
SuppSose7fithrsetretshueltrseflaucrteuanteoafreorumndioancsonisnt¼antht.e theory. Can in this limitct→he∞η is expected to be light,
sub-lat¼Tthiceeschhiraalvesynmomne-tirnytebgreearkitnogpcoalnogailcsaolbcehaorbgseersv?edYes: B. Parametriz(cid:48)ation B for Tij
theDthirroaucghsttrhiengshcrainnkipngenoeftrthaetegtahpraoruougnhdtzheeroboasunshdoawrnyand The results in this subsectimon2are f1or/N 0 (5)
nothtionggethperrevweitnhtsthae cchoinrafilgcuornadteinosnateshinowFnig.in11.FiAgg5a(irni,ght). η(cid:48) ∼ c →
vSaol,uχetchssoiunnobcklflaiunQtdgewo,offrQuotlmhd.eoitibntfthlaeaictntitohcneonpctoriritniibctsaultotifeomnthpseewrtawittuohrecnufoorvnre−sc,hiiwrnaetleger OofntehsehionusltTdaijna¼tlosvon¯cr0eencs1aeel−mlv¯rtbv=¯h2rle=as2t,tthheesflouccatluleað2dt2ioÞcnosmopfreNss(iVbi)lit=y
condensaate anbd the gap do coincide within errors, at the (g2/32π2) d4xG2 (xþ),satisfiesthefollowinglowenergy
This is in sharp contrast to χ , since lattice configu- µν
ratiosnasmCecoanSnfi¼noemn6.le5yn–t8haapnvodeicnhti.inratlesgyemrmveatlrlayutearseothfeQre.foAretditfhfeirsenptointw,itJhus−ttlhaosegoðfcro0erÞm¼the−(cid:82)0o.t3h8e8r acnhdopicΛffieffiffiffi¼ffiffioffiffifffi3ffiffiffi.Tffi2ffiffi.ffiffiffi discussed in the
one aphlwenaoymseanas,kbwuthayreibnotthhterigdgyeorendtbhyetohreyinacnreyasleatintictheeconp-revious subse<ctiNon,(Vwe)2ob>tain t<heNpa(riVjam)e>ter2s=of4de<nsiNty,(V)> (6)
figurdaetniosintysohfadvyeonisn.teger Q. It is because the lattice, unF-ig. 13, holonomy (Polyakov−loop Fig. 14), andbDebye
like the sublattice, must have zero total magnetic chargmeass,Fwigh.e1r6e,absa=fu1n1ctNionc/o3ft+em2pNerat/ur3e.byFmoriniNmizing the r.h.s.
f c
Q =0. thefreeenergy.ThechiralcondensateFigs.14and15,an→d ∞
magnetic vanishes,whichmeansthequantityN(V)hasinthislimt
Thissimplediscussionnicelyillustratesthedrasticdif-
no fluctuations. This in tern implies, that the isoscalar
ference between the topology on the lattice and the sub-
scalar meson σ must become heavy.
lattices: the former are not canonical but in some way
In the real world QCD with N = 3 these two masses
c
share properties of the microcanonical ensembles, with
have the opposite relation,
fixed charges.
LetusnoreturntoQCD,switchingonthelightquarks. m 958MeV >m 500MeV (7)
η(cid:48) ≈ σ ≈
FIG.11. ParametrizationA:Thegapscaledup15times(blue
circles)andthechiralcondensateΣ(redsquares)asafunctionof
actionS 8π2=g2ortemperatureT=T .Aclearrise/fallisseen
aroundS¼ 7–7.5.WegetacriticaltemcperaturefromS 6.5–8 FIG.13. ParametrizationB:ThedensityoftheM(bluecircles)
forthecon¼densateandS 6.5–8forthegap.Σisscaled¼by0.2. andL(redsquares)dyonsasafunctionofactionS 8π2=g2or
¼ ¼
TheblackconstantlineisdefinedinthecaptionofFig.10. temperatureT=T .
c
054029-8
5
uid study by Schaefer, the m(η ) does indeed decreases
(cid:48)
�(T) with N as in (5). What happens with m (N ) remains
c σ c
unknown. Lattice studies of these issues would be of sig-
nificant interest.
Witten [12] and Veneziano [13] famously related the
topological susceptibility to the η mass. However, their
(cid:48)
argument is for χ in the limit of infinite number of col-
ors, not in physical QCD[16] . A similar expression for
derivedin[8]isbasedonχ (x)inphysical QCDand,
sublat
ironically, corresponds to the limit of small (rather then
instanton- instantons large) volume limit.
dyons
Tmin/Tc 1 Tmax/Tc T/Tc VI. SUMMARY
Thesecommentscanfinallybesummarizedinasketch
shown in Fig.7: below T T 2.5T a dilute in-
FIG.7: Asketchindicatingdifferentformsofthetopological max c
∼ ∼
ensembles as a function of the temperature. stanton gas changes to an ensemble of instanton-dyons.
In this lower region χ = χ , they have different
lat sublat
(cid:54)
T-dependences. If it can be evaluated on the lattic, it
will perhaps reveal the dis-assembly of instantons into
butatsomeN theyshouldbecomeequal,andthencon- the constituents, with non-integer topological charges,
c
tinue to move, up and down. According to instanton liq- directly.
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