AdSCFT and QCD

arXiv:hep-th/0702205v127Feb2007ADS/CFTandQCDStanleyJ.BrodskyStanfordLinearAcceleratorCenter,StanfordUniversity,Stanford,California94309GuyF.deT´eramond∗UniversidaddeCostaRica,SanJos´e,CostaRica,andStanfordLinearAcceleratorCenter,StanfordUniversity,Stanford,California94309∗E-mail:[e-mail]/CFTcorrespondencebetweenstringtheoryinAdSspaceandconformalfieldtheoriesinphysicalspace-timeleadstoananalytic,semi-classicalmodelforstrongly-coupledQCDwhichhasscaleinvarianceanddimensionalcountingatshortdistancesandcolorconfinementatlargedistances.AlthoughQCDisnotconformallyinvariant,onecanneverthelessusethemathematicalrepresentationoftheconformalgroupinfive-dimensionalanti-deSitterspacetoconstructafirstapproximationtothetheory.TheAdS/CFTcorrespondencealsoprovidesinsightsintotheinherentlynon-perturbativeaspectsofQCD,suchastheorbitalandradialspectraofhadronsandtheformofhadronicwavefunctions.Inparticular,weshowthatthereisanexactcorrespondencebetweenthefifth-dimensionalcoordinateofAdSspacezandaspecificimpactvariableζwhichmeasurestheseparationofthequarkandgluonicconstituentswithinthehadroninordinaryspace-time.Thisconnectionallowsonetocomputetheanalyticformoftheframe-independentlight-frontwavefunctions,thefundamentalentitieswhichencodehadronpropertiesandallowthecomputationofdecayconstants,formfactors,andotherexclusivescatteringamplitudes.Newrelativisticlight-frontequationsinordinaryspace-timearefoundwhichreproducetheresultsobtainedusingthe5-dimensionaltheory.Theeffectivelight-frontequationspossessremarkablealgebraicstructuresandintegrabilityproperties.Sincetheyarecompleteandorthonormal,theAdS/CFTmodelwavefunctionscanalsobeusedasabasisforthediagonalizationofthefulllight-frontQCDHamiltonian,thussystematicallyimprovingtheAdS/CFTapproximation.1.TheConformalApproximationtoQCDQuantumChromodynamics,theYang-MillslocalgaugefieldtheoryofSU(3)Ccolorsymmetryprovidesafun-damentalunderstandingofhadronandnuclearphysicsintermsofquarkandgluondegreesoffreedom.However,becauseofitsstrong-couplingnature,itisdifficulttofindanalyticsolutionstoQCDortomakeprecisepredictionsoutsideofitsperturbativedomain.AnimportantgoalisthustofindaninitialapproximationtoQCDwhichisbothanalyticallytractableandwhichcanbesystematicallyimproved.Forexample,inquantumelectrodynamics,theSchr¨odingerandDiracequationsprovideaccuratefirstapproximationstoatomicboundstateproblemswhichcanthenbesystematicallyimprovedbyusingtheBethe-Salpeterformalismandcorrectingforquantumfluctuations,suchastheLambShiftandvacuumpolarization.OneofthemostsignificanttheoreticaladvancesinrecentyearshasbeentheapplicationoftheAdS/CFTcorre-spondence[1]betweenstringstatesdefinedonthe5-dimensionalAnti–deSitter(AdS)space-timeandconformalfieldtheoriesinphysicalspace-time.TheessentialprincipleunderlyingtheAdS/CFTapproachtoconformalgaugetheo-riesistheisomorphismofthegroupofPoincare’andconformaltransformationsSO(4,2)tothegroupofisometriesofAnti-deSitterspace.TheAdSmetricisds2=R2z2(ημνdxμdxν−dz2),whichisinvariantunderscalechangesofthecoordinateinthefifthdimensionz→λzandxμ→λxμ.Thusonecanmatchscaletransformationsofthetheoryin3+1physicalspace-timetoscaletransformationsinthefifthdimensionz.QCDisnotitselfaconformaltheory;howeverinthedomainwheretheQCDcouplingisapproximatelyconstantandquarkmassescanbeneglected,QCDresemblesastrongly-coupledconformaltheory.AsshownbyPolchinskiandStrassler[2],onecansimulateconfinementbyimposingboundaryconditionsintheholographicvariableatz=z0=1/ΛQCD.ConfinementcanalsobeintroducedbymodifyingtheAdSmetrictomimicaconfiningpotential.Theresultingmodels,althoughadhoc,provideasimplesemi-classicalapproximationtoQCDwhichhasbothcounting-rulebehavior[3,4,5,6]atshortdistancesandconfinementatlargedistances.Thissimpleapproach,whichhasbeendescribedasa“bottom-up”approach,hasbeensuccessfulinobtaininggeneralpropertiesofscatteringamplitudesofhadronicboundstates[2,7,8,9,10,11,12]andthelow-lyinghadronspectra[13,14,15,16,17,18,19,20].Studiesofhadroncouplingsandchiralsymmetrybreaking[16,21,22,23,24],quarkpotentialsinconfiningbackgrounds[25,26]andpomeronphysics[27,28]hasalsobeenaddressedwithinthebottom-upapproachtoholographicQCD,alsoknownasAdS/QCD.Recently,thebehaviorofthespace-likeformfactorsofthepion[29,30,31]andnucleons[32]hasbeendiscussedwithintheframeworkofAdS/QCD.Studiesofgeometricback-reactioncontrollingtheinfraredphysicsaregivenin1refs.[33,34].Itisalsoremarkablethatthedynamicalpropertiesofthequark-gluonplasmaobservedatRHIC[35]canbecomputedwithintheAdS/CFTcorrespondence[36].Incontrasttothesimplebottom-upapproachdescribedabove,theintroductionofadditionalhigherdimensionalbranestotheAdS5×S5backgroundhasbeenusedtostudychiralsymmetrybreaking[37],andrecentlybaryonicpropertiesbyusingD4-D8braneconstructs[38,39,40].ItwasoriginallybelievedthattheAdS/CFTmathematicaltoolwouldonlybeapplicabletostrictlyconformaltheoriessuchasN=4supersymmetry.Inourapproach,wewillapplyAdS/CFTtothelowmomentum,strongcouplingregimeofQCDwherethecouplingisapproximatelyconstant.Theoretical[41]andphenomenological[42]evidenceisinfactaccumulatingthattheQCDcoupli