Phenomenology of renormalons and the OPE from lattice regularization: the gluon condensate and the heavy quark pole mass

0.17 MB·English

by  Gunnar S. Bali

Phenomenology of renormalons and the OPE from lattice
regularization: the gluon condensate and the heavy quark
pole mass
Gunnar S. Bali ,† and Antonio Pineda
∗ ∗∗
5
1
InstitutfürTheoretischePhysik,UniversitätRegensburg,D-93040Regensburg,Germany
0 ∗
†TataInstituteofFundamentalResearch,HomiBhabhaRoad,Mumbai400005,India
2
∗∗GrupdeFísicaTeòricaandIFAE,UniversitatAutònomadeBarcelona,E-08193Bellaterra,Barcelona,Spain
n
a
J Abstract. Westudytheoperatorproductexpansionoftheplaquette(gluoncondensate)andtheself-energyofaninfinitely
heavyquark.Wefirstcomputetheirperturbativeexpansionstoordera 35anda 20,respectively,inthelatticescheme.Inboth
1
caseswereachtheasymptoticregimewheretherenormalonbehaviorsetsin.Subtractingtheperturbativeseries,weobtain
3
theleadingnon-perturbativecorrectionsoftheirrespectiveoperatorproductexpansions.Inthefirstcaseweobtainthegluon

condensateandinthesecondthebindingenergyoftheheavyquarkintheinfinitemasslimit.Theresultsarefullyconsistent
]
h withtheexpectationsfromrenormalonsandtheoperatorproductexpansion.
p
Keywords: Renormalons,operatorproductexpansion,latticeQCD,gluoncondensate,heavyquarkeffectivetheory
- PACS: 12.38.Gc,12.38.Bx,11.55.Hx,12.38.Cy,11.15.Bt
p
e
h
[ INTRODUCTION

1
Theoperatorproductexpansion(OPE)[1]isafundamentaltoolfortheoreticalanalysesinquantumfieldtheories.Its
v
6 validity is only provenrigorously within perturbationtheory, to arbitrary finite orders [2]. The use of the OPE in a
8 non-perturbativeframeworkwasinitiatedbytheITEPgroup[3](seealsothediscussioninRef.[4]),whopostulated
0 thattheOPEofacorrelatorcouldbeapproximatedbythefollowingseries:
0
0 correlator(Q) (cid:229) 1 C (a ) O , (1)
2. ≃ d Qd d h di
0
wherethe expectationvaluesof localoperatorsO aresuppressedbyinversepowersofa largeexternalmomentum
5 d
1 Q L QCD, according to their dimensionality d. The Wilson coefficientsCd(a ) encode the physics at momentum
: sca≫leslargerthanQ.Thesearewellapproximatedbyperturbativeexpansionsinthestrongcouplingparametera :
v
Xi Cd(a ) (cid:229) cna n+1. (2)
≃
n 0
r ≥
a
The large-distance physics is described by the matrix elements O that usually have to be determined non-
d
perturbatively: O L d . h i
h di∼ QCD
Itcanhardlybeoveremphasizedthat(exceptfordirectpredictionsofnon-perturbativelatticesimulations,e.g.,on
light hadron masses) all QCD predictions are based on factorizations that are generalizations of the above generic
OPE.
ThereexistsomemajorquestionsrelatedtotheOPEthathavetobeaddressed:1
• AretheperturbativeexpansionsofWilsoncoefficientsasymptoticseries?
1 Therearealsosomeimportantpointsthatwedonotaddresshere,including:
Wedonotconsiderambiguitiesassociatedtoshort-distancenon-perturbativeeffects,whichwouldgiverisetosingularitiesfurtherawayfromthe
originoftheBorelplanethanthosewestudyhere.
Wetakethevalidity oftheOPEinpureperturbation theoryforgranted.Thisassumptionissolidincaseswithasinglelargescale,Q2,andin
Euclideanspacetime.
Wewillnotdiscussthevalidityofthe(non-perturbative) OPEfortimelikedistancesthatcanoccurinMinkowskispacetime,anissuerelatedto
possibleviolationsofquark-hadronduality.
• Ifso:aretheassociatedambiguitiesoftheasymptoticbehaviorconsistentwiththeOPE,i.e.withthepositionsof
theexpectedrenormalons[5]intheBorelplane?
• IstheOPEvalidbeyondperturbationtheory?
• Whatistherealsizeofthefirstnon-perturbativecorrectionwithinagivenOPEexpansion?
• Isthisvaluestronglyaffectedbyambiguitiesassociatedtorenormalons?
In this paper we summarize and discuss our recent results [6–10], which address these questions for the case of
the plaquette and the energy of an infinitely heavy quark in the pure gluodynamics approximation to QCD. Both
analyses utilize lattice regularization.Contrary to, e.g., dimensionalregulation,lattice regularizationcan be defined
non-perturbatively. Using a lattice scheme rather than the MS scheme, we can, not only expand observables in
perturbationtheory,butalsoevaluatethemnon-perturbatively.Anotheradvantageofthischoiceisthatitenablesusto
usenumericalstochasticperturbationtheory[11–13]toobtainperturbativeexpansioncoefficients.Thisallowsusto
realizemuchhigherordersthanwouldhavebeenpossiblewithdiagrammatictechniques.Adisadvantageofthelattice
schemeisthat,atleastinourdiscretization,latticeperturbativeexpansionsconvergeslowerthanexpansionsintheMS
coupling.Thismeansthatwehavetogotocomparativelyhigherorderstobecomesensitivetotheasymptoticbehavior.
ManyoftheresultsobtainedinalatticeschemeeitherdirectlyapplytotheMSschemetooorcansubsequentlyeasily
(andinsomecasesexactly)beconvertedintothisscheme.
InourstudiesweusedtheWilsongaugeaction[14].Wedefinethevacuumexpectationvalueofagenericoperator
Bofengineeringdimensionzeroas
1
B W BW = [dUx,m ]e−S[U]B[U] (3)
h i≡h | | i Z
Z
LwEithisthaeEpuacrltiidtieoannfsupnaccteitoinmZel=attRic[edUwxi,tmh]lea−ttSi[cUe] sapnadcminegaasuarned[dUUx,xm,m ]=eiA(cid:213) m (xx∈+L aE/,2m)dUSx,Um .(3|W)iisdaegnaoutegsetlhinekv.acuumstate,
≈ ∈
THE PLAQUETTE: OPEINPERTURBATION THEORY
ForthecaseoftheplaquettewehaveB P,where
→
hPi= N14x(cid:229) L EhPxi, Px=1−316m(cid:229)>n Tr Ux,mn +Ux†,mn , (4)
∈ (cid:0) (cid:1)
andUx,mn denotestheorientedproductofgaugelinksenclosinganelementarysquare(plaquette)inthem -n planeof
thelattice.Fordetailsonthenotationandsimulationset-upseeRef.[9].
P willdependonthelatticeextentNa,thespacingaanda =g2/(4p ) a (a 1)(notethata isthebarelattice
−
couhpliingand its naturalscale is of ordera 1). We first computethis expect≡ationvalue in strict perturbationtheory.
−
Inotherwords,weTaylorexpandinpowersofgbeforeaveragingoverthegaugeconfigurations(whichwedousing
NSPT[11–13]).Theoutcomeisapowerseriesina :
P pert(N) 1 [dUx,m ]e−S[U]P[U] = (cid:229) pn(N)a n+1.
h i ≡ Z
Z (cid:12)NSPT n 0
(cid:12) ≥
(cid:12)
Thedimensionlesscoefficients pn(N)arefunctionsofthelinear(cid:12)latticesizeN.Weemphasizethattheydonotdepend
onthelatticespacingaoronthephysicallatticeextentNaalonebutonlyontheratioN=(Na)/a.
Weareinterestedinthelarge-N(i.e.infinitevolume)limit.Inthissituation
1 1
(5)
a ≫ Na
anditmakessensetofactorizethecontributionsofthedifferentscaleswithintheOPEframework.Thehardmodes,
ofscale 1/a,determinetheWilsoncoefficients,whereasthesoftmodes,ofscale 1/(Na),canbedescribedbyex-
∼ ∼
pectationvaluesoflocalgaugeinvariantoperators.Therearenosuchoperatorsofdimensiontwo.Therenormalization
groupinvariantdefinitionofthegluoncondensate
2 b (a ) a
hG2i=−b W a Gcmn Gcmn W = W [1+O(a )]p Gcmn Gcmn W (6)
0(cid:28) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) D (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) E
istheonlylocalgaugeinvariantexpectationvalueofanoperatorofdimensiona 4inpuregluodynamics.Inthepurely
−
perturbativecasediscussedhere,thisonlydependsonthesoftscale1/(Na),i.e.onthelatticeextent.Ondimensional
grounds,theperturbativegluoncondensate G2 isproportionalto1/(Na)4,andthelogarithmic(Na)-dependence
soft
isencodedina [1/(Na)].Therefore, h i
p 2a4 G2 = 1 (cid:229) f a n+1[1/(Na)], (7)
36 h isoft −N4 n
n 0
≥
andtheperturbativeexpansionoftheplaquetteonafinitevolumeofN4 sitescanbewrittenas
p 2 1
P (N)=P (a ) 1 + C (a )a4 G2 +O , (8)
h ipert pert h i 36 G h isoft N6
(cid:18) (cid:19)
where
P (a )= (cid:229) p a n+1 (9)
pert n
n 0
≥
and p aretheinfinitevolumecoefficientsthatweareinterestedin.Theconstantpre-factorp 2/36ischosensuchthat
n
theWilsoncoefficient,whichonlydependsona ,isnormalizedtounityfora =0.Itcanbeexpandedina :
C (a )=1+(cid:229) c a k+1. (10)
G k
k 0
≥
SinceouractionisproportionaltotheplaquetteP,C isfixedbytheconformaltraceanomaly[15,16]:
G
2pb (a ) b a b a 2 b a 3
C 1(a )= =1+ 1 + 2 + 3 +O(a 4). (11)
G− − b a 2 b 4p b 4p b 4p
0 0 0 0
(cid:16) (cid:17) (cid:16) (cid:17)
Theb -functioncoefficients2b areknowninthelatticeschemefor j 3(seeEq.(25)ofRef.[9]).
j
≤
CombiningEqs.(7),(8)and(10)gives
P (N)= (cid:229) p fn(N) a n+1 (12)
h ipert n− N4
n 0(cid:20) (cid:21)
≥
= (cid:229) p a n+1 1 1+(cid:229) c a k+1(a 1) (cid:229) f a n+1((Na) 1)+O 1 ,
n 0 n −N4 k 0 k − !×n 0 n − (cid:18)N6(cid:19)
≥ ≥ ≥
where f (N) is a polynomialin powersof ln(N). Fitting this equationto the perturbativelattice results, the first 35
n
coefficientsp weredeterminedinRef.[9].Theresultswereconfrontedwiththeexpectationsfromrenormalons:
n
plnattn→=¥ NPlatt 2bp0d nG (Gn(+1+1+dbd)b) 1+20.n08+9d3b1...+(n+d5b0)5(n±+3d3b 1)+O n13 , (13)
(cid:18) (cid:19) (cid:20) − (cid:18) (cid:19)(cid:21)
pn = b 0 1+db+db(1−ds1)+db 1−3ds1+d2b(s1+2s2) +O 1 , (14)
np 2p d n n2 n3 n4
n−1 ( (cid:2) (cid:3) (cid:18) (cid:19))
whereb=b /(2b 2),s =(b 2 b b )/(4bb 4)ands =(b 3 2b b b +b 2b )/(16b2b 6)aredefinedsothat
1 0 1 1 − 0 2 0 2 1 − 0 1 2 0 3 0
1 1 t b
a= exp bln +s bt s b2t2+ with t= 0a . (15)
L −t − 2 1 − 2 ··· 2p
latt (cid:20) (cid:21)
InFig.1wecomparetheinfinitevolumeratiosp /(np )totheexpectationEq.(14):theasymptoticbehaviorofthe
n n 1
perturbativeseriesduetorenormalonsisreachedaroun−dordersn 27 30,proving,forthefirsttime,theexistence
∼ −
of the renormalonin the plaquette.Note that incorporatingfinite volumeeffectsis compulsoryto see this behavior,
sincetherearenoinfraredrenormalonsonafinitelattice.Toparameterizefinitesizeeffectswemadeuseofthepurely
perturbativeOPE Eq.(8).Thebehaviorseen inFig. 1,althoughcomputedfromperturbativeexpansioncoefficients,
goesbeyondthepurelyperturbativeOPEsinceitpredictsthepositionofanon-perturbativeobjectintheBorelplane.
2 Wedefinetheb -functionasb (a )=da /dlnm = b0/(2p )a 2 b1/(8p 2)a 3 ,i.e.b0=11.
− − −···
¥
N =
1.2
N = 28
6 ¥
(1/N ) N =
1.0
LO
NLO
)
1 0.8
- NNLO
n
p
NNNLO
n
0.6
(
/
n
p
0.4
0.2
0
10 15 20 25 30 35
n
FIGURE1. Theratios pn/(npn 1)compared withtheleadingorder (LO),next-to-leadingorder (NLO),NNLOandNNNLO
predictions of the1/n-expansion E−q.(14).Onlythe“N =¥ ”extrapolation includesthesystematic uncertainties. Wealsoshow
finite volume data for N =28, and the result from the alternative N ¥ extrapolation including some 1/N6 corrections. The
→
symbolshavebeenshiftedslightlyhorizontally.
THE PLAQUETTE: OPEBEYONDPERTURBATION THEORY
SinceinNSPTweTaylorexpandinpowersofgbeforeaveragingoverthegaugevariables,nomassgapisgenerated.
In non-perturbativeMonte-Carlo (MC) lattice simulations an additional scale, L QCD 1/ae−2p /(b0a ), is generated
dynamically(seealsoEq.(15)).However,wecanalwaystuneN anda suchthat ∼
1 1
L . (16)
QCD
a ≫ Na ≫
In this small-volume situation we encounter a double expansion in powers of a/(Na) and aL [or, equivalently,
QCD
(Na)L a/(Na)]. The constructionof the OPE is completely analogousto that of the previoussection and we
QCD
obtain3 ×
1 p 2
P MC= [dUx,m ]e−S[U]P[U] =Ppert(a ) 1 + CG(a )a4 G2 MC+O(a6). (17)
h i Z h i 36 h i
Z (cid:12)MC
(cid:12)
Inthelastequalitywehavefactoredoutthehard(cid:12)scale1/afromthescales1/(Na)andL ,whichareencodedin
(cid:12) QCD
G2 .Exploitingtheright-mostinequalityofEq.(16),wecanexpand G2 asfollows:
MC MC
h i h i
G2 = G2 1+O[L 2 (Na)2] . (18)
h iMC h isoft QCD
(cid:8) (cid:9)
3 In the last equality, we approximate the Wilson coefficients by their perturbative expansions, neglecting the possibility of non-perturbative
contributionsassociatedtothehardscale1/a.Thesewouldbesuppressedbyfactors exp( 2p /a )andthereforewouldbesub-leading,relative
∼ −
tothegluoncondensate.
Hence,anon-perturbativesmall-volumesimulationwouldyieldthesameexpressionasNSPT,uptonon-perturbative
corrections that can be made arbitrarily small by reducing a and therefore Na, keeping N fixed. In other words,
pNSPT(N)=pMC(N)uptonon-perturbativecorrections.
n n
Wecanalsoconsiderthelimit
1 1
L . (19)
QCD
a ≫ ≫ Na
Thisisthestandardsituationrealizedinnon-perturbativelatticesimulations.AgaintheOPEcanbeconstructedasin
theprevioussection,Eq.(17)holds,andthe p -andc -valuesarestillthesame.Thedifferenceisthatnow
n n
1
G2 = G2 1+O , (20)
h iMC h iNP" L 2QCD(Na)2!#
where G2 L 4 istheso-callednon-perturbativegluoncondensateintroducedinRef.[3].Fromnowonwewill
h iNP∼ QCD
callthisquantitysimplythe“gluoncondensate” G2 .Wearenowintheposition
h i
• todeterminethegluoncondensateand
• tocheckthevalidityoftheOPE(atlowordersinthea2 scaleexpansion)forthecaseoftheplaquette.
Inordertodosoweproceedasfollows.Theperturbativeseriesisdivergentduetorenormalonsandother,sub-leading,
instabilities.4Thismakesanydeterminationof G2 ambiguous,unlesswedefinepreciselyhowtotruncateorhowto
approximatetheperturbativeseries.Areasonabhlediefinitionthatisconsistentwith G2 L 4 canonlybegivenif
h i∼ QCD
theasymptoticbehavioroftheperturbativeseriesisundercontrol.Thishasonlybeenachievedrecently[9],wherethe
perturbativeexpansionoftheplaquettewascomputeduptoO(a 35),seetheprevioussection.Theobservedasymptotic
behaviorwas in full compliancewith renormalonexpectations,with successive contributionsstarting to divergefor
ordersarounda 27–a 30 withintherangeofcouplingsa typicallyemployedinpresent-daylatticesimulations.
Extracting the gluon condensate from the average plaquette was pioneered in Refs. [17–20] and many attempts
followedduringthenextdecades,see,e.g.,Refs.[21–30].Thesesufferedfrominsufficientlyhighperturbativeorders
and,insomecases,alsofinitevolumeeffects.Thefailuretomakecontacttotheasymptoticregimepreventedareliable
latticedeterminationof G2 .ThisproblemwassolvedinRef.[10],whichwenowsummarize.
h i
Truncatingtheinfinitesumattheorderoftheminimalcontributionprovidesonedefinitionoftheperturbativeseries.
VaryingthetruncationorderwillresultinchangesofsizeL 4 a4,wherethedimensiond=4isfixedbythatofthe
QCD
gluoncondensate.Weapproximatetheasymptoticseriesbythetruncatedsum
n
S (a ) S (a ), where S (a )= (cid:229) p a j+1. (21)
P ≡ n0 n j
j=0
n0≡n0(a )istheorderforwhich pn0a n0+1isminimal.Wethenobtainthegluoncondensatefromtherelation
36C 1(a )
G2 = G− [ P (a ) S (a )]+O(a2L 2 ). (22)
h i p 2a4(a ) h iMC − P QCD
C 1(a )isproportionaltotheb -function,andthefirstfewtermsareknown,seeEq.(11).ThecorrectionstoC =1
G− G
aresmall.However,theO(a 2)andO(a 3)termsareofsimilarsizes.Wewillaccountforthisuncertaintyinourerror
budget.
FollowingEq.(22),wesubtractthetruncatedsumS (a )calculatedfromthecoefficientsp ofRef.[9]fromtheMC
P n
dataon P (a )ofRef.[31]intherangeb [5.8,6.65](b =6/g2),wherea(b )isgivenbythephenomenological
MC
paramethriziationofRef.[32](x=b 6) ∈
−
a=r exp 1.6804 1.7331x+0.7849x2 0.4428x3 , (23)
0
− − −
wherer 0.5fm.Thiscorrespondsto(a(cid:0)/r )4 [3.1 10 5,5.5 10 3],coveringmo(cid:1)rethantwoordersofmagnitude.
0 0 − −
≈ ∈ × ×
4 Theleadingrenormalonislocatedatu=d/2=2intheBorelplane,whilethefirstinstanton-anti-instanton contribution occursatu=b0=
11Nc/3=11 2.
≫
0.020
N=32 3.18(a/r )4
0
4.0 N=16 3.26(a/r )4- 4.38(a/r )6
0 0
0.015
3.5
> >
<2G 3.0 <2G 0.010
4 0 4
r a
2.5
0.005
2.0
0
5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 0 0.002 0.004 0.006
b a4/r4
0
FIGURE 2. Leftpanel: Eq. (22) evaluated using the N =16 and N =32 MC data of Ref. [31]. TheN =32 outer error bars
includetheerrorofSP(a ).Theerrorbandisourpredictionfor OG ,Eq.(24).
Rightpanel:Eq.(22)timesa4vs.a4(a )/r4fromEq.(23).ThehlineairfitwithslopeEq.(24)istothea4<0.0013r4pointsonly.
0 0
Multiplyingthisdifferenceby36r4/(p 2C a4)givesr4 O plushigherordernon-perturbativeterms.Weshowthis
0 G 0h Gi
combinationintheleftpanelofFig.2.ThesmallererrorbarsrepresenttheerrorsoftheMCdata,theoutererrorbars
(notplottedforN=16)thetotaluncertainty,includingthatofS .Thispartoftheerroriscorrelatedbetweendifferent
P
b -values(seethediscussioninRef[10]).TheMCdatawereobtainedonvolumesN4=164 andN4=324.Towards
largeb -valuesthephysicalvolumes[Na(b )]4willbecomesmall,resultingintransitionsintothedeconfinedphase.For
b <6.3wefindnosignificantdifferencesbetweentheN=16andN=32results.Intheanalysiswerestrictourselves
to the more precise N =32 data and, to keep finite size effects undercontrol, to b 6.65. We also limit ourselves
tob 5.8toavoidlargeO(a2)corrections.Atverylargeb -valuesnotonlydoesthe≤parametrizationEq.(23)break
down≥butobtainingmeaningfulresultsbecomeschallengingnumerically:theindividualerrorsbothof P (a )and
MC
ofS (a )somewhatdecreasewithincreasingb .However,therearestrongcancellationsbetweenthesehtwioterms,in
P
particularatlargeb -values,sincethisdifferencedecreaseswitha 4 L 4 exp(16p 2b /33)ondimensionalgrounds
− ∼ latt
while P dependsonlylogarithmicallyona.
MC
ThehdaitaintheleftpanelofFig.2showanapproximatelyconstantbehavior.5Thisindicatesthat,aftersubtracting
S (a ) fromthecorrespondingMC values P (a ), the remainderscaleslike a4. Thiscan beseen moreexplicitly
P MC
in the right panel of Fig. 2, where we ploththiis difference in lattice units against a4. The result is consistent with a
linearbehaviorbutasmallcurvatureseemstobepresentthatcanbeparametrizedasana6-correction.Theright-most
point(b =5.8)correspondstoa 1 1.45GeVwhileb =6.65correspondstoa 1 5.3GeV.Notethata2-termsare
− −
≃ ≃
clearlyruledout.
Wenowdeterminethegluoncondensate.Weobtainthecentralvalueanditsstatisticalerror G2 =3.177(36)r 4
fromaveragingtheN=32datafor6.0 b 6.65.We nowestimatethesystematicuncertainthies.iDifferentinfini0−te
≤ ≤
volumeextrapolationsofthep (N)data[9]resultinchangesofthepredictionofabout6%.Another6%errorisdueto
n
includingana6-termornotandvaryingthefitrange.Nextthereisascaleerrorofabout2.5%,translatinga4intounits
of r . The uncertaintyof the perturbativelydeterminedWilson coefficientC is of a similar size. This is estimated
0 G
as the difference between evaluating Eq. (11) to O(a 2) and to O(a 3). Adding all these sources of uncertainty in
quadratureandusingthepuregluodynamicsvalue[33]L =0.602(48)r 1yields
MS 0−
G2 =3.18(29)r 4=24.2(8.0)L 4 . (24)
h i 0− MS
ThegluoncondensateEq.(6)isindependentoftherenormalizationscale.However, G2 wasobtainedemploying
h i
oneparticularprescriptionintermsoftheobservableandourchoiceofhowtotruncatetheperturbativeserieswithina
givenrenormalizationscheme.Different(reasonable)prescriptionscaninprinciplegivedifferentresults.Onemayfor
5 Notethatn0increasesfrom26to27atb =5.85,from27to28atb =6.1andfrom28to29atb =6.55.Thisquantizationofn0explainsthe
visiblejumpatb =6.1.
n = -1
-1
10
) n = 0
a( n = 1
]n10-2 n = 2
S n = 3

- n = 5

C
n = 7
M -3
>10 n = 9 0
a
n = 11
P
1
< n = 14 a
[10-4 n = 19 a2
3
n = 24 a
n = 29 4
a
0.07 0.1 0.15 0.2 0.3
a
a( )/r
0
FIGURE3. Differences P MC(a ) Sn(a )betweenMCdataandsumstruncatedatordersa n+1 (S 1=0)vs.a(a )/r0.The
lines(cid:181) ajaredrawntoguidheitheeye.− −
instancechoosetotruncatethesumatordersn (a ) n (a )andtheresultwouldstillscalelikeL 4 .Weestimated
0 ± 0 QCD
thisintrinsicambiguityofthedefinitionofthegluonpcondensateinRef.[9]asd hG2i=36/(p 2CGa4)√n0pn0a n0+1,
i.e.as n (a )timesthecontributionoftheminimalterm:
0
p d G2 =27(11)L 4 . (25)
h i MS
Upto1/n -correctionsthisdefinitionisscheme-andscale-independentandcorrespondstothe(ambiguous)imaginary
0
partoftheBorelintegraltimes 2/p .
InQCDwithseaquarkstheOPEoftheaverageplaquetteoroftheAdlerfunctionwillreceiveadditionalcontribu-
tionsfromthechiralcondensatep.Forinstance G2 needstoberedefined,addingterms(cid:181) g (a )myy¯ [34,35].Due
m
h i h i
tothisandtheproblemofsettingaphysicalscaleinpuregluodynamics,itisdifficulttoassesstheprecisenumerical
impactofincludingseaquarksontoourestimates
G2 0.077GeV4, d G2 0.087GeV4, (26)
h i≃ h i≃
whichweobtainusingr 0.5fm[36].WhilethesystematicsofapplyingEqs.(24)–(25)tofullQCDareunknown,
0
≃
ourmainobservationsshouldstillextendtothiscase.WeremarkthatourpredictionofthegluoncondensateEq.(26)
issignificantlybiggerthanvaluesobtainedinone-andtwo-loopsumruleanalyses,rangingfrom0.01GeV4 [3,37]
up to 0.02GeV4 [38, 39]. However,these numberswere not extractedin the asymptotic regime,which for a d =4
renormalonintheMSschemeweexpecttosetinatordersn&7.Moreover,weremarkthatinschemeswithoutahard
ultravioletcut-off,likedimensionalregularization,the extractionof G2 canbecomeobscuredbythepossibilityof
h i
ultravioletrenormalons.Independentoftheseconsiderations,allthesevaluesaresmallerthantheintrinsicprescription
dependenceEq.(25).
Our analysis confirms the validity of the OPE beyond perturbation theory for the case of the plaquette. Our a4-
scaling clearly disfavors suggestions about the existence of dimension two condensates beyond the standard OPE
framework[25, 40–43]. In factwe can also explainwhyana2-contributionto the plaquettewas foundin Ref. [25].
In the log-logplotof Fig. 3 we subtractsums S , truncatedatdifferentfixedordersa n+1, from P . The scaling
n MC
continuouslymovesfrom a0atO(a 0)to a4aroundO(a 30).Notethattruncatingatana -indehpeindentfixedorder
∼ ∼
isinconsistent,explainingwhyinthefigureweneverexactlyobtainana4-slope.Forn 9wereproducethea2-scaling
∼
reportedinRef.[25]forafixedordertruncationatn=7.InviewofFig.3,weconcludethattheobservationofthis
scalingpowerwasaccidental.
THE BINDINGENERGY OFHQET:OPEINPERTURBATION THEORY
The OPE of the plaquette is analogous to the OPE of the vacuum polarization (or the Adler function) in position
space.Aswe alreadymentioned,theOPE conceptofthefactorizationofscalescanalso beappliedto moregeneral
kinematical settings (in particular to cases where some scales are defined in Minkowski spacetime). A prominent
exampleisheavyquarkeffectivetheory(HQET).Inthiscasetheterm O ofEq.(1)isreplacedbyanon-perturbative
1
quantity, the so-called heavy quark binding energy L , that cannot behrepiresented as an expectationvalue of a local
gaugeinvariantoperator.Weconsidertheself-energyofaheavyquarkintheinfinitemasslimit(inotherwords,the
self-energyofastaticsourceinthetripletrepresentation,forothercolorrepresentationsseeRef.[7]),thatisclosely
relatedtoL .Wecomputethisquantityincloseanalogytothecaseoftheplaquetteinlatticeregularization.
Firstwecomputetheself-energyofthestaticquarkinperturbationtheory(againusingNSPT).Weobtainthisfrom
thePolyakovloopinanasymmetricvolumeN3N a4ofspatialandtemporalextentsN aandN a,respectively:
S T S T
1 (cid:229) 1 N(cid:213)T−1
L(N ,N )= Tr U , (27)
S T NS3 x 3 "x4/a=0 x,4#
or,morespecifically,fromitslogarithm
P(N ,N )= lnhL(NS,NT)ipert = (cid:229) c (N ,N )a n+1. (28)
S T n S T
− aNT n 0
≥
Again,Ux,m denotesa gaugelink and x=(x,x4) L E are Euclideanlattice points.We definethe energyof a static
∈
sourceanditsperturbativeexpansioninafinitespatialvolume,
d m(N )= lim P(N ,N )= 1 (cid:229) c (N )a n+1(1/a), where c (N )= lim c (N ,N ), (29)
S S T n S n S n S T
NT→¥ an 0 NT→¥
≥
anditsinfinitevolumelimit
¥
d m= lim P(N ,N )= 1 (cid:229) c a n+1(1/a) with c = lim c (N ). (30)
S T n n n S
NS,NT→¥ an=0 NS→¥
WenowconstructthepurelyperturbativeOPEinafinitevolume.ForlargeN ,wecanwrite[a 1 (N a) 1]:
S − S −
≫
¥
d m(N )= 1 (cid:229) c fn(NS) a n+1(1/a)+O 1 (31)
S a n− N N2
n 0(cid:18) S (cid:19) (cid:18) S(cid:19)
≥
= 1 (cid:229) c a n+1(1/a) 1 (cid:229) f a n+1[1/(N a)]+O 1 .
a n −N a n S N2
n 0 S n 0 (cid:18) S(cid:19)
≥ ≥
Note the similarity between this equationand Eq. (12). f (N )=(cid:229) n f(j)lnj(N ) is again a polynomialin powers
n S j=0 n S
ofln(N ),andthe f(j) areknowncombinationsofb -functioncoefficientsandlowerorderinfinitevolumeexpansion
S n
coefficientsc ,k<n.Themaindifferencewithrespecttothegluoncondensateisthatthepowercorrectionscaleslike
k
1/N ,insteadof1/N4,andthatnowtheWilsoncoefficientistrivial.This1/N scalingalsomeansthattherenormalon
S S S
behaviorwillshowupatlowerordersnoftheperturbativeexpansion.Fittingthec (N )datatothisequation,thefirst
n S
20c coefficientsweredeterminedinRefs.[6–8]andconfrontedwiththerenormalonexpectations:6
n
cnn→=¥ Nm(cid:18)2bp0(cid:19)n G (Gn(+1+1+b)b)"1+(nb+s1b)+(nb+2(cid:0)bs)21(/n2+−bs−2(cid:1)1)+···#. (32)
6 HerewedeviatefromRefs.[6–8]inthedefinitionoftheconstants2,seeEq.(15).
9
unsmeared
8 smeared
LO
7
NLO 1.95
NNLO
) 6
1 1.90
- NNNLO
n
c 5
1.85
n
(
/
n 4 1.80
c
3 1.75
8 10 12 14 16 18
2
1
2 4 6 8 10 12 14 16 18
n
FIGURE 4. Ratios cn/(ncn 1) of the unsmeared (squares) and smeared (circles) triplet static self-energy coefficients cn in
comparison to the theoretical−prediction Eq. (33), truncated at different orders in 1/n. In the inset we magnify the asymptotic
region.
Inthelatticeschemethenumericalvaluesoftheabovecoefficientsreadbs =1.36095381(11)andb2(s2/2 s )=
1 1 − 2
5.34(51). As expected, the above expansion converges much faster in 1/n than Eq. (13). Calculating the ratio of
subsequentperturbativecoefficientsgives
c 1 b b bs 1 1
n = 0 1+ 1 + 2b2s +b(b 1)s +O . (33)
c n 2p n− n2 n3 2 − 1 n4
n−1 (cid:26) (cid:18) (cid:19)(cid:27)
(cid:2) (cid:3)
We remarkthatEq.(14)includestheeffectofthenon-trivialWilsoncoefficientC . Therefore,justsettingd =1in
G
thatequationdoesnotresultinEq.(33)above.
InFig.4wecomparethedatatoEq.(33)fortwodifferentlatticediscretizationsofthecovarianttemporalderivative,
whichamountsto“smearing”ornotsmearingtemporalgaugelinks.Thisshouldnotaffecttheinfraredbehaviorand,
indeed,beyondthefirstfewordersthedifferencebecomesinvisible.Theasymptoticbehavioroftheperturbativeseries
duetotherenormalonisconfirmedinfullgloryforn&8,provingtheexistenceoftherenormalonbehaviorinQCD
beyondanyreasonabledoubt.Againtheincorporationoffinitevolumeeffectswasdecisivetoobtainthisresult.
THE BINDINGENERGY OFHQET:OPEBEYONDPERTURBATION THEORY
The methods used for the gluon condensate can also be applied to other observables. We now consider the non-
perturbativeevaluationoftheenergyofastatic-lightmesononthelatticeanditsOPE:
E (a )=E (a )+L +O(aL 2 ). (34)
MC pert QCD
L isthenon-perturbativebindingenergyandE (a )=d m(a )istheself-energyofthestaticsourceinperturbation
pert
theory,i.e.Eq.(30).InHQETthemassoftheBmesonisgivenasm =m +L +O(1/m ),wherem isthe
B b,pert b b,pert
b-quarkpolemassmOS,whichsuffersfromthesamerenormalonambiguityasd m[44–46].
b
0.4
)
(a 0.3
m
Sd
a

- 0.2

)
a(
C
M
E 0.1 MS
2
a
MS
3
latt
0
0 0.05 0.10 0.15 0.20 0.25
a
a( )/r
0
(FMIGSU3)RloEop5s. anadL tr=unacEaMteCd−atathSed mrevssp.eac/tirv0e.mThineimsuamloarSddemrs.ETq.h(e3c6u)rwveassaarlesofictsontoveLrtae+dicnato2/trh02e.MSschemeattwo(MS2)andthree
The perturbativeexpansionof ad m(a )=(cid:229) c a n+1 was obtained in Refs. [6, 7] up to O(a 20), see the previous
n n
section.Itsintrinsicambiguity
d L =√n0cn0a n0+1=0.748(42)L MS=0.450(44)r0−1 (35)
wascomputedinRefs.[7,8].MCdataforthegroundstateenergyE ofastatic-lightmesonwiththeWilsonaction
MC
canbefoundinRefs.[47–49].
Truncatingtheinfinitesumind mattheorderoftheminimalcontributionprovidesonedefinitionoftheperturbative
series.Aswedidfortheplaquette,weapproximatetheasymptoticseriesbythetruncatedsum
Sd m(a )≡ a1n0(cid:229) (a )cja j+1, (36)
j=0
where n0 ≡n0(a ) is the order for which cn0a n0+1 is minimal. While for the gluon condensate we expected an a4-
scaling(seetherightpanelofFig.2),foraL =aEMC aSd m(a )weexpectascalinglinearina.Comfortingenough
thisiswhatwefind,uptotheexpectedaO(aL 2 )di−scretizationcorrections,seeFig.5.Subtractingthepartialsum
QCD
truncatedatordersn (a )=6fromtheb [5.9,6.4]data,weobtainL =1.55(8)r 1fromsuchalinearplusquadratic
0 ∈ 0−
fit,whereweonlygivethestatisticaluncertainty.Theerrorsoftheperturbativecoefficientsarealltiny,whichallows
ustotransformtheexpansionad m(a )intoMS-likeschemesandtocomputeL accordingly.We definetheschemes
MS and MS by truncating a (a 1)=a (1+d a +d a 2+...) exactly at O(a 3) and O(a 4), respectively. The
2 3 MS − 1 2
d are knownfor j 3 [7, 8]. We typicallyfindnMSi(a )=2,3andobtainL 2.17(8)r 1 andL 1.89(8)r 1,
j ≤ 0 MSi ∼ 0− ∼ 0−
respectively,seeFig.5.Weconcludethatthechangesduetotheseresummationsareindeedofthesized L 0.5r 1,
∼ 0−
addingconfidencethatourdefinitionoftheambiguityEq.(35)isneitheragrossoverestimatenoranunderestimate.
Fortheplaquette,whereweexpectnMS 7,wecannotcarryoutasimilaranalysis,duetotheextremelyhighprecision
thatisrequiredtoresolvethediffere0nce∼sbetweenS (a )and P (a ),whichlargelycancelinEq.(22).
P MC
h i