Generalized quark-antiquark potential in AdS/CFT

0.13 MB·English

by  Valentina Forini

FortschrittederPhysik,31January2012
ICCUB–12–004
kcl-mth–12–01
Generalized quark-antiquark potential in AdS/CFT
ValentinaForini1, andNadavDrukker2,
∗ ∗∗
1 InstituteofCosmosSciencesandEstructuraiConstituentsdelaMateria
FacultatdeF´ısica,UniversitatdeBarcelona,Av.Diagonal647,08028Barcelona,Spain
2
1 2 DepartmentofMathematics,King’sCollegeLondon
0 TheStrand,WC2R2LS,London,UK
2

n
a Keywords AdS/CFTcorrespondence,supersymmetricgaugetheory.
J
InthistalkwepresentafamilyofWilsonloopoperatorswhichcontinuouslyinterpolatesbetweenthe1/2
0
BPS line and the antiparallel lines, and can be thought of as calculating a generalization of the quark–
3 antiquarkpotentialforthegaugetheoryonS3×R. Weevaluatethefirsttwoordersoftheseloopsper-

turbativelybothinthegaugeandstringtheory. Weobtainanalyticalexpressionsinasystematicexpansion
]
h aroundthe1/2BPSconfiguration,andcommentonpossibleall-looppatternsfortheseWilsonloops.
t
-
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e
h
[
1 Overview

1
v Oneofthemostfundamentalobservablesinaquantumfieldtheoryisthepotentialbetweenchargedparti-
8
cles,whichinagaugetheoryiscapturedbyalongrectangularWilsonloop,orapairofantiparallellines
5
representingthe trajectories of infinitely heavy quarks. Such quark-antiquarkpotentialcan be also con-
2
6 sideredinthemaximallysupersymmetric = 4SYMtheory,where“quarks”aremodeledbyinfinitely
N
. massiveW-bosonsarisingfromaHiggsmechanism[1].
1
0 TheexpectationvalueofthisobservablewascalculatedveryearlyaftertheintroductionoftheAdS/CFT
2 correspondenceby the effective action of a string ending along the curve on the four-dimensionalAdS
1 boundary,and is in fact a seminalexampleof the duality itself. In this contextof a conformalfield the-
v: ory the potential is fixed to be Coulomb-like and the whole dynamical content is in the corresponding
i coefficient,forwhichtheweakandstrongcoupling(’tHooftcouplingλ)previouslyobtainedresultsread
X
ar Vqq¯(λ,L)=−L1 c(λ), c(λ)=4λ√πλhπ1− 12λπ+2(cid:16)al1n+2λπ −γE1+1(cid:17),+O(λ2)i, λλ≪11. (1)
4K(12)2 h √λ O(cid:16)(√λ)2(cid:17)i ≫
Above,Listhedistancebetweenthelines,Kisthecompleteellipticintegralofthefirstkindandtheweak-
couplingexpansionisthefield-theoreticalcalculationof[2,3,4]. Onthestringtheory(strongcoupling)
side, the question of evaluating the first quantum string correction a to the classical result of [1] 1 is
1
a hard mathematical problem. The absence of parameters in the problem (the only one, L, being fixed
by conformal invariance) precludes considering special scaling limits in which nice results in σ-model
perturbation theory have been obtained for some relevant string solutions (see, for example, [6, 7] and
referencetherein).Thecoefficienta waspresentedformallyin[8,9],evaluatednumericallyin[10]tobe
1
a =1.33459andsimplifiedfurtherin[11]toananalyticone–dimensionalintegralrepresentation.
1
∗ Correspondingauthor [email protected]
∗∗ [email protected]
1 ThisisactuallytheAdS5×S5counterpartoftheso-called“Lu¨scherterm”,whichinflatspaceisacoulombictermproportional
tothenumberoftransversedimensions[5].
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2 V.ForiniandN.Drukker:Generalizedquark-antiquarkpotentialinAdS/CFT
Itis hard to guesshow to connectthe two regimesof (1). Itis temptingto thinkaboutthe chanceof
exploitingtheintegrabilityoftheunderlyingAdS/CFTsystemanddescribecorrectlytheinterpolationof
c(λ)betweenthetworegimesof(1),asinthebynowmostfamousexampleofsmoothinterpolationfora
non-protectedquantity-thecuspanomalyof =4SYM[12].
N
Our proposal [13] for addressing the problem relies on the introduction of extra parameters in the
initialsetup. Theydonotmaketheperturbativeorsupergravitycalculationanyharderandallow,infact,to
interpolatebetweenprotected,muchsimpler,operatorsandthedesiredobservable. Thefirstdeformation
parameter(indicatedbelowwithθ)allowsforthetwolinestocoupletotwodifferentscalarfields,andwas
alreadyintroducedin[1]. InthegeneralexpressionoftheMaldacena-Wilsonloop
1
W = Tr exp (iA x˙µ+Φ ΘI x˙ )ds , (2)
µ I
N P (cid:20)I | | (cid:21)
weallowtwodifferentvaluesofΘ~ ofrelativeangleθ onthetwolongedgesoftherectangle. Forθ = 0
the two lines couple to the same scalar field, say Φ . When θ = π/2 the two lines coupleto Φ Φ ,
1 1 2
±
whichareorthogonaltoeach-other. Thenforθ = π theycoupletothefieldΦ , butwithoppositesigns,
2
whichmeansthatthelinesareeffectivelyparallel,ratherthanantiparallel. Inthatcasethetwolinesshare
eightsuperchargesandthecorrelatoristrivial. Theotherdeformationparameter(indicatedbelowwithφ)
is geometric,and a way to illustrate it is to replacethe theoryonR4 with the theoryonS3 R (related
bytheexponentialmap). Weconsiderapairofantiparallellinesseparatedbyanangleπ φ×onS3. For
−
φ = 0 the two linesare antipodaland mutuallyBPS, while forφ π the linesgetveryclose together.
→
“Zoomingin”tothevicinityofthelinesbyaconformaltransformationwegetasituationverysimilarto
theoriginalantiparallellinesinflatspace. AnequivalentpictureisthatofacuspintheplaneinR4. For
φ=0thecuspdisappearsandthesystemisthatofasingleinfinitestraightline.
IntheS3 RpicturetheexpectationvalueoftheWilsonloopcalculatestheeffectivepotentialV(φ,θ,λ)
betweenag×eneralizedquark-antiquarkpair. InthecaseofacuspinR4 theloopsuffersfromlogarithmic
divergences[14]. Theexpectationvaluesoftheloopinthetwopicturesarerespectively
W exp TV(φ,θ,λ) , W exp log(R/ǫ)V(φ,θ,λ) . (3)
cusp
h i≈ h− i h i≈ h− i
The logarithmicdivergenceis exactly the same as the linear time divergence,and the cutoffsof the two
calculationsarerelatedbylog(R/ǫ) T.
∼
TheeffectivepotentialV(φ,θ,λ)dependsonthe’tHooftcouplingλ=g2N (wedonotconsidernon-
planar corrections) and it can be expanded at weak coupling and at strong coupling in the two relevant
asymptoticexpansions
V(φ,θ,λ)=P√4πλ∞n=1∞l(cid:0)=106λ(cid:16)π2√4(cid:1)πλn(cid:17)VlV(nA()ld()φS,(φθ),,θ), λλ≪≫11. (4)
P

Below,wewillpresenttheevaluationofthefirsttwotermsofbothregimes,adoptingthepictureofacusp
in R4 at weak couplingand the S3 R picture at strong coupling. In particular, at strong couplingthe
×
coefficientsintheperturbativeexpansionsarecomplicatedfunctionsoftheanglesφandθwhicharegiven
onlyimplicitly(attheclassicallevel)orinintegralform(one–loop). Weconsiderthereforetheexpansion
ofthesefunctionsaroundφ = θ = 0. Thisisanexpansionaroundthe1/2BPSline(relatedtothecircle
via conformaltransformation),one of the mostsimple observablesin the theory. As a consequence,we
obtainhereanalyticresultsatbothweakandstrongcoupling.
Focussingonthefirstcoefficientsofthisexpansion,wearguebelowhowtheyshouldreceivecontribu-
tionsonlyfroma subsetofgraphsinperturbationtheory– themostconnectedgraphs. Atvariancewith
thecaseofthecircularWilsonloop,whereintheFeynmangaugeonlyladderdiagramscontributeandall
interactinggraphscombinetovanish[3,15,16],wefindhereanobservablewhichgetscontributionsonly
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fromthemostinteractinggraphs. Tooursurprise,fromtheexplicitcalculationofthe2–loopgraphs,we
find that the result of these internally–connectedgraphs is simpler than the internally–disconnectedone
anddoesnotinvolvepolylogarithms. Since summingupladdergraphsis rathereasy2, itwouldbe very
interestingtoexplorethe3–loopgraphsandseewhetherasimilarpatternpersistsandperhapslearnhow
tocalculatethemostconnectedgraphstoallorders.
Intherestofthetalkwepresentasummaryofourresultsatweakandstrongcoupling(Section2),the
explicitanalyticexpressionsoftheexpansionaroundtheBPSconfigurationandashortdiscussiononhow
the relevantcoefficientscan be evaluatedvia insertionsof localoperatorsinto the loop (Section3). The
results obtained are suggestive of the framework in which an efficient description of the weak-to-strong
couplinginterpolationfortheseWilsonloopsmighttakeplace. Certainly,theyrepresentasetofanalytic
datatobeofreferenceifanall-loopcalculationwilleveremerge.
2 Results atweakand atstrong coupling
Atweakcoupling,weworkwiththecuspinR4 [18]andallowforanextraangleθ in = 4SYM.For
thepotentialV(φ,θ)uptotwo-loopswefound3 N
cosθ cosφ
V(1)(φ,θ)= 2 − φ;
− sinφ
V(2)(φ,θ) =V(2)(φ,θ)+V(2)(φ,θ),
lad int
(cosθ cosφ)2 π2 i (5)
V(2)(φ,θ)= 4 − Li e2iφ ζ(3) iφ Li e2iφ + + φ3 ,
lad − sin2φ (cid:20) 3 − − (cid:18) 2 6 (cid:19) 3 (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
4 cosθ cosφ
V(2)(φ,θ)= − (π φ)(π+φ)φ,
int 3 sinφ −
whereV(2)iswrittenasasumofthecontributionofladder4andinteractinggraphs.
The analytic expressions (5) undergo various checks. In the BPS case [21], where φ = θ, then
±
V(1) =V(2) =0asexpected. Atlargeimaginaryangle,theprefactorofthelineartermmatchesindeeda
quarteroftheperturbativeexpansionofthecuspanomalousdimension[22]. Formulas(5)alsoreproduce
(and generalize)the antiparallellines resultof [2]. Taking the φ π limit and specializingto the case
→
θ =0,theresultingexpressionmatchestheonein[2]withthereplacementL π φ. Itisinterestingto
→ −
noticethatthecomplicatedinteractinggraphsresultinacontributionmuchsimplerthantheoneduetothe
2–loopladdergraphandwithoutpolylogarithmicfunctions5. Indeeditisproportionaltothe1–loopresult
witharatiowhichisjustisapolynomialinφ.
Atstrongcoupling,Wilsonloopsaredescribedbymacroscopicstrings[1,23]. Theclassicalsolutions
arefoundin globalLorentzianAdS 6startingfroma time-independentansatz, theboundaryconditions
5
beinglinesseparatedbyπ φontheboundaryofAdSandθonS5. Therelevantsolutions(writtendown
−
inthecaseofθ = 0in[19]andforθ = 0inAppendixC.2of[24])canbefoundforarbitraryvaluesofφ
6
andθasthesolutionsoftranscendentalequations.Theresultforthegeneralizedpotentialisthenfoundin
2 In[17],anintegralequationwaswrittenwhosesolutiongivesthecontributionofladdergraphstoallordersinperturbation
theory.
3 ThecalculationofV(1)atone–looporderwasdonein[19].Theθ=0caseisin[17](seealso[20]),whereexpressionswere
writteninintegralform.Herewehaveextendedtheexpressionstoθ6=0andcomputedtheintegralsinclosedform.
4 AftersubtractingtheexponentiationoftheO(λ)term.
5 Notetheuniformtranscendentalitythree(whene2iφisconsideredrational)ofbothinteractingandladdergraphsatthisorder.
6 ThisistheappropriatestrongcouplingdualofthegaugetheoryonS3×R.
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4 V.ForiniandN.Drukker:Generalizedquark-antiquarkpotentialinAdS/CFT
termsofellipticintegralsKandE7
√λ2 b4+p2 (b2+1)p2
V(0) (φ,θ)= K(k2) E(k2) , (6)
AdS 2π pbp (cid:20) b4+p2 − (cid:21)
wheretheellipticmoduluskandtheparameterbarefunctionsofp,q,whichareinturnrelatedtoφ,θvia
transcendentalequations.
Quadraticfluctuationsaroundtheclassicalsolutioncanbeconsidered,basedontheNambu-Gototype
action in the static gauge. The mass matrix in the resulting quadratic fluctuation Lagrangian, depend-
ing in generalon the two parametersof the problem, becomesdiagonalin the two limiting cases θ = 0
(equivalentlyq = 0)andφ = 0(thelimitp q ). Inparticular,forthesevaluesallthequadratic
∝ → ∞
fluctuationoperators,whichhaveatrivialtimedependence,canbewrittenintheformofone-dimensional
single-gapLame´differentialoperators8. Thelatterpointiscrucial.Itmakesitpossibletotradetheexplicit
evaluationoftheeigenvaluespectrumfortherelevantoperatorswiththeresolutionoftheassociateddif-
ferentialequation(anapproachknownasGelfand-Yaglommethod,seealsotheanalysisin[13]). Relying
ontheknowledgeofthesolutionstotheLame´ spectralproblem,allfluctuationsdeterminantscanbethen
computedanalytically. Theresulting(regularized)effectiveactionΓ ,whichistheratioofdeterminants
reg
includingthecontributionofthetrivialtimedirection = dτ,isthenexpressedasasingleintegral9and
T
definestheone-loopcorrectiontothegeneralizedquark-aRntiquarkpotentialasfollows(e.g. intheθ = 0
case)
V(1) (φ,θ)= Γreg = T lim +∞ dω ln ǫ2ω2det8OFǫ . (7)
AdS T −2T ǫ 0Z 2π det5 ǫdet2 ǫdet ǫ
→ −∞ O0 O1 O2
Theexplicitexpressionsforthe1ddeterminantscanbefoundin[13],herewereportasrepresentativethe
bosoniccontribution
det ǫ = sinh(2K(k22)Z(α2)) , sn(α k2)= √1+k22+ω22 , (8)
O2 ∼−ǫ2ω ω4+(2 4k2)ω2+1 2| 2 k2
−
p
where Z is the Jacobi Zeta function, sn is the Jacobi elliptic sine, k is a rational function of k and ω
2 2
a rational function of k and ω. Above, ǫ is the standard infrared regulator curing the linear divergence
expectedat the boundary,the determinantis takenat leadingorderin a ǫ 0 expansionandan explicit
≃
subtractionoftheremainingdivergences(aregularizationartifact)ismade.
It is possible to see that both the classical and the one-loop strong coupling results, (6) and (7)-(8),
reproduce the known expressions for the antiparallel lines, in [1, 23] and [10, 11] respectively, in the
φ π, θ = 0 limit 10. This happens, as in the weak coupling case, once the replacementof the pole
→
π φ Lisperformed.
− →
Itisstraightforwardtoevaluatetheintegral(7)numericallyforarbitraryvaluesofφ, aswellasinthe
analogcaseofφ=0andarbitraryθ,while,ingeneral,wedonotknowhowtocalculateitanalytically11.
To gain more analytic controlover the form of V(1) we will proceed in a systematic expansionaround
AdS
θ =0andφ=0,towhichthenextsectionisdevoted.
3 Nearstraight-line expansion
In the φ 0 limit the cusp disappears and we are left with an infinite straight line in R4, or a pair of
antipodal→lines on S3 R. In this case the analysis indeed simplifies, and allows for explicit analytic
×
expressionsatweakandatstrongcoupling.
7 Thestandardlineardivergencefortwolinesalongtheboundary,canceledasusualbyaboundaryterm,ishereremoved.
8 Seealso[25].
9 Theintegrationvariableωin(7)istheFourier-transformedτvariable∂τ =−iω.
10 Thislimittranslateintheconditionsp→0, q2 =fixed, k2=1/2ontheparametersrelevantatstrongcoupling.
p
11 Seehowevertheresultsof[11]inthelimitofantiparallellines.
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Atweakcoupling,thefirstfewordersintheexpansionof(5)aroundφ=θ =0read
1
V(1)(φ,θ)=θ2 φ2 (θ2 φ2)2+O((φ,θ)6),
− − 12 −
(9)
2π2 1
V(2)(φ,θ)= (θ2 φ2)+ (π2(θ2 φ2)2+6(θ2 φ2)(3θ2 φ2))+O((φ,θ)6).
− 3 − 18 − − −
Allthetermsareproportionaltoθ2 φ2,andindeedweexpectV(φ,θ,λ) tovanishforθ = φ,which
− ±
areBPSconfigurations[24].
Atstrongcoupling,anexpansionoftheleadingsemiclassicalresultleadsto
1 1
V(0) (φ,θ)= (θ2 φ2) (θ2 φ2) θ2 5φ2 +O((φ,θ)6). (10)
AdS π − − 8π3 − −
(cid:0) (cid:1)
Atone–looporderin σ-modelperturbationtheory,the expansiontranslatesina smallk expansionofall
theellipticfunctionsintheintegrandof(7),andresultsinapowerseriesofregularhyperbolicfunctions.
Anintegrationoverthelogarithmofthisseriescanthenalwaysbeperformed,andgives
3 φ2 53 φ4 223 15 15 φ6
V(1) (φ,0)= + 3ζ(3) + ζ(3) ζ(5) +O(φ8).
AdS 24π2 (cid:18) 8 − (cid:19)16π4 (cid:18) 8 − 2 − 2 (cid:19)64π6
(11)
3 θ2 5 θ4 1 3 15 θ6
V(1) (0,θ)= + 3ζ(3) + + ζ(3) ζ(5) +O(θ8).
AdS −24π2 (cid:18)8 − (cid:19)16π4 (cid:18)8 2 − 2 (cid:19)64π6
Focusnowontheexpansioncoefficientsaroundφ=θ =0,forexamplethefirst(quadratic)one
λ λ2
+ λ 1,
1 ∂2 1 ∂2 16π2 − 384π2 ··· ≪
2∂θ2V(φ,θ,λ)(cid:12)φ=θ=0 =−2∂φ2V(φ,θ,λ)(cid:12)φ=θ=0 =√λ 3 (12)
(cid:12) (cid:12) + λ 1.
(cid:12) (cid:12) 4π2 − 8π2 ··· ≫

Theexpansionaroundthe1/2BPSstraightlinecanbeviewedasadeformationofthestraightlineitself,
andassuchitcanbewrittenintermsofinsertionsoflocaloperatorsintotheWilsonloop. Onecanwrite
thelatterasastraight(φ=0)lineinthex1directionwitharbitraryθ
1 0 ∞
W = Tr exp (iA +Φ )ds exp (iA +Φ cosθ+Φ sinθ)ds , (13)
1 1 1 1 2
N P(cid:20) (cid:16)Z (cid:17) (cid:16)Z0 (cid:17)(cid:21)
−∞
such that it couplesto the scalar Φ for all s < 0 and to the linear combinationΦ cosθ +Φ sinθ for
1 1 2
s>012. Usingthat13
∂2 1 ∂2 1 ∂2
V(0,0)= log W W , (14)
∂θ2 −ln(R/ǫ)∂θ2 h i≈−ln(R/ǫ)∂θ2h i
onefindsforthecoefficientin(12)14
1 ∂2 V = 1 1 ∞ds ∞ds Tr Φ (s )Φ (s )eR−∞∞(iA1+Φ1)ds
2∂θ2 − ln(R/ǫ)2N Z0 1Z0 2D Ph 2 1 2 2 iE (15)
+ 1 1 ∞ds Tr Φ (s )eR−∞∞(iA1+Φ1)ds .
1 1 1
ln(R/ǫ)2N Z0 D Ph iE
12 Wefixedtheparameterizationsuchthat|x˙|=1,sowecanignorethedifferencebetweenxµ(si)andsi.
13 ThefirstidentityisthedefinitionofV.Thesecondfollowsfrom ∂∂θhWi=0andfrom(cid:10)W|φ=θ=0(cid:11)=1.
14 Thevariationwithrespecttoθissomewhatsimplerthanthethevariationwithrespecttoφ,sincethelattermodifiesthepath
oftheloopandiscaptured byinsertions ofthefieldstrengthFµν aswellasitsderivatives intotheloop, whiletheformeronly
introduceslocalscalarfieldinsertions.
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6 V.ForiniandN.Drukker:Generalizedquark-antiquarkpotentialinAdS/CFT
Examiningthe right-handside is suggestive of a pattern expectedto hold for all valuesof the coupling.
OnenoticesthatgraphswhichinvolvepropagatorsbetweentheWilsonloopanditself,andnottheinser-
tions,willvanishduetotheBPSnatureofthestraightline. Atoneandtwo-looporder,onlygraphswith
atmostoneinternallyconnectedcomponentcontribute,astheexplicitexpansionofV(2) andV(2) in(5)
int lad
easilyconfirms.Theinterestingobservationisthatthisargumentshouldapplyalsotohigherordergraphs.
OnlygraphswithonesetofconnectedinternallinesattachedtotheWilsonloopcontributetothisterm15.
Regardingfurtherexpansioncoefficients,theoneofθ4 willinvolveforexamplegraphswithatmosttwo
disconnectedinternalcomponents,and so on. Since by explicitcalculationwe foundthatthe connected
(interacting)graphsat2–looporderhadasimpler(withoutpolylogarithms)functionalformthanthedis-
connected(ladder)ones,itwouldbecertainlyinterestingtoseeifthisstructurepersistsathigherordersin
perturbationtheoryandwhetheritispossibletoguesstheanswerforthemostconnectedgraphsatallloop
order,andreproducethestrongcouplingresultsin(12).
Acknowledgements WethanktheorganizersoftheXVIIEuropeanWorkshoponStringTheoryinPaduaforcreating
anenjoyableandstimulatingmeeting.
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15 Thisstatementistrueassumingthecancellation forthestraightlinedoesnotrequireintegration. Otherwise,therewillbe
boundarytermsinthedisconnectedgraphs,whichcanberegardedasconnectedones.
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