Integrability in QCD and beyond

arXiv:hep-th/0407232v22Sep2004February1,200818:57WSPC/TrimSize:9.75inx6.5inforProceedingsBBGKINTEGRABILITYINQCDANDBEYONDA.V.BELITSKYDepartmentofPhysicsandAstronomy,ArizonaStateUniversityTempe,AZ85287-1504,USAandDepartmentofPhysics,UniversityofMarylandatCollegeParkCollegePark,MD20742-4111,USAV.M.BRAUNInstitutf¨urTheoretischePhysik,Universit¨atRegensburgD-93040Regensburg,GermanyA.S.GORSKYInstituteofTheoreticalandExperimentalPhysicsB.Cheremushkinskaya25,Moscow,117259RussiaG.P.KORCHEMSKYLaboratoiredePhysiqueTh´eorique∗,Universit´eParisSud91405OrsayC´edex,FranceYang–Millstheoriesinfourspace-timedimensionspossessahiddensymmetrywhichdoesnotexhibititselfasasymmetryofclassicalLagrangians,butisonlyrevealedonthequantumlevel.ItturnsoutthattheeffectiveYang–MillsdynamicsinseveralimportantlimitsisdescribedbycompletelyintegrablesystemsthatprovetoberelatedtothecelebratedHeisenbergspinchainanditsgeneralizations.Inthisreview,weexplainthegeneralphenomenonofcompleteintegrabilityanditsrealizationinseveraldifferentsituations.Asaprimeexample,weconsiderinsomedetailthescaledependenceofcomposite(Wilson)operatorsinQCDandsuper-Yang–Mills(SYM)theories.High-energy(Regge)behaviorofscatteringamplitudesinQCDisalsodiscussedandprovidesonewithanotherrealizationofthesamephenomenonthatdiffers,however,fromthefirstexampleinessentialdetails.Asathirdexample,weaddressthelow-energyeffectiveactioninaN=2SYMtheorywhich,contrarytotheprevioustwocases,correspondstoaclassicalintegrablemodel.Finally,weincludeashortoverviewofrecentattemptstousegauge/stringdualityinordertorelateintegrabilityofYang–Millsdynamicswiththehiddensymmetryofastringtheoryonacurvedbackground.∗Unit´emixtederechercheduCNRS(UMR8627).264February1,200818:57WSPC/TrimSize:9.75inx6.5inforProceedingsBBGKIntegrabilityinQCDandbeyond265TableofContents1Introduction2662Light-conedominatedprocessesinQCD2742.1Conformalsymmetryandthecollinearsubgroup.......2762.2Hamiltonianapproachtooperatorrenormalization......2792.3Completeintegrabilityandnon-compactHeisenbergmagnets2852.4Semiclassicalexpansion.....................2892.5Breakdownofintegrability....................2912.6Openspinchains.........................2962.7Quasipartonicandnon-quasipartonicoperators........2972.8Conformalsymmetrybreaking.................3003Dilatationoperatorsinsupersymmetricgaugetheories3013.1SYMtheoriesinlight-conesuperspace.............3023.2Superconformalsymmetryonthelight-cone..........3053.3One-loopdilatationoperator..................3083.4DilatationoperatorasaSL(2|N)Heisenbergspinchain...3114HighEnergyScattering3154.1TheBFKLHamiltonian.....................3164.2Multireggeoncompoundstates.................3205SpinchainsandN=2superYang-Milltheories3276Gauge/stringcorrespondence3346.1Derivationofthestringinthethermodynamicallimit....3376.2Semiclassicalstringmotionandintegrablemodels.......3386.3Openstringpictureforanomalousdimensions.........3427Conclusion343References345February1,200818:57WSPC/TrimSize:9.75inx6.5inforProceedingsBBGK266A.Belitsky,V.Braun,A.GorskyandG.KorchemskyWehadtheprivilegeandpleasureofknowingIanKoganasaclosefriend.Inourinformalchatsitwasclearthat,whateverhewasdoing,therealwayswasasinglecentralproblemthatkepthismindconstantlybusy–theprob-lemofconfinement.Allotherprojectshepursued–rangingfromtheQuan-tumHallEffecttostringsandquantumgravity–heconsideredfromthisgeneralperspective.Heviewedtheissueofintegrabilityasaunifyingconceptwhichcouldpotentiallyexplainmanyunsolvedproblemsofstronglycoupledtheoriesandwasveryenthusiasticaboutthisnewdevelopmentconcerningfour-dimensionalgaugetheories.Ifitwerenotforhisuntimelydeath,hewoulddefinitelyhavehadaprofoundimpactonthewholefieldwithhisusualpassionfornewconceptsandtrends.1.IntroductionUnderstandingthedynamicsofgaugetheoriesatstrongcouplingisoneoftheoutstandingproblemsinquantumfieldtheory.Ithasbeenanticipated,sincetheclassicworkbyWilson[1],thattheYang–MillsdynamicsatstrongcouplingcanbereformulatedasaneffectivetheoryofFaradaylines(gaugefieldflux)which,initsturn,canbereinterpretedasastring-liketheory.Anaturalframeworkfordiscussingthestringyrepresentationisofferedbythemulticolorlimit.Inthislimit,onecanexpressobservablesasasumoversurfaces(strongcouplingexpansion`alaWilson)butexplicitrealizationofthisprogramingaugetheoriesisstilllackingduetoabsenceofanoperationalgauge/stringcorrespondence.Itisexpectedthatinthemulticolorlimit,thepathintegraloverallgaugefieldconfigurationswillbedominatedbyasaddlepoint–theso-called“masterfield”.Polyakovhassuggested[2]thatthemasterfieldcanbefoundasasolutiontoatwo-dimensionalsigmamodeldefinedonanontrivialbackground(Lobachevskispace-time).Healsodemonstratedthatsuchmodelspossessaninfinitesetofconservedcurrentsand,therefore,canbesolvedexactly.Ithaslongbeenspeculatedthatsimilarnontrivialalgebraicstructuresand,inparticular,integrabilitycanberevealedinfour-dimensionalquantumgaugetheoriesaswell.Inthelastfewyearsseveralplausibleindicationsthatthisactuallyoccurshaveappearedandseveralintriguingconnectionshavebeenestablishedbetweenquantumgaugefieldtheoriesandintegrablelatticespinmo