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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 - p Copyrightlinewillbeprovidedbythepublisher 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]. Copyrightlinewillbeprovidedbythepublisher 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 Copyrightlinewillbeprovidedbythepublisher fdpheaderwillbeprovidedbythepublisher 3 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. Copyrightlinewillbeprovidedbythepublisher 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. Copyrightlinewillbeprovidedbythepublisher fdpheaderwillbeprovidedbythepublisher 5 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. Copyrightlinewillbeprovidedbythepublisher 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). 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[24] N.Drukker,S.Giombi,R.RicciandD.Trancanelli,JHEP0805,017(2008)[arXiv:0711.3226[hep-th]]. [25] M. Beccaria, G. V. Dunne, V. Forini, M. Pawellek and A. A. Tseytlin, J. Phys. A 43, 165402 (2010) [arXiv:1001.4018[hep-th]]. 15 Thisstatementistrueassumingthecancellation forthestraightlinedoesnotrequireintegration. Otherwise,therewillbe boundarytermsinthedisconnectedgraphs,whichcanberegardedasconnectedones. Copyrightlinewillbeprovidedbythepublisher