The theory of quark confinement

arXiv:hep-ph/9902279v19Feb1999BonnTK98-09Thetheoryofquarkconfinement∗†V.N.GribovLandauInstituteforTheoreticalPhysics,MoscowandResearchInstituteforParticleandNuclearPhysics,BudapestandInstitutf¨urTheoretischeKernphysikderUniversit¨atBonnAbstractThisisthesecondofthetwolastpapersbyV.N.Gribovconcludinghis20yearlongstudyoftheproblemofquarkconfinementinQCD.InthispapertheanalyticstructureofquarkandgluonGreen’sfunctionsisinvestigatedintheframeworkofthetheoryofconfinementbasedonthephenomenonofsupercriticalbindingoflightquarks.Theproblemofunitarityinaconfiningtheoryisdiscussed.Thewrite-upremainedunfinishedandsoitispresentedhere.Theauthorwasplanningtoemphasisethelinkbetweentheelectro-weakandstronginteractions,andinparticulartherˆoleofpions(Goldstonebosons)inconfinement,topresentanexplicitsolutionforboundstates,andtowritedownananalyticmodelforquarkandgluonGreen’sfunctionscorrespondingtoconfinement.1Introduction1Almosttenyearsago[1]Iproposedahypothesisaccordingtowhichquarksandgluonsareconfinedduetotheexistenceoflightquarks(uandd)withComptonwavelengthsmuchlargerthantheradiusofstronginteractiondefinedbyλQCD.Theessenceofthishypothesisisthesupercriticalphenomenonwhichiswell-knowninQEDwhereitisthefollowing.IfthereexistsaheavynucleusofradiusRwithchargeZexceedingacriticalvalueZcr(whichisoftheorderof137),thevacuumofthelightchargedfermions(electrons)becomesunstable∗TheworkwascompletedinBonnundertheHumboldtResearchAwardProgram.†ThetextwaspreparedforpublicationbyYu.Dokshitzer,C.Ewerz,andJ.Nyiri,withhelpfromJ.Bartels,A.Kaidalov,A.MuellerandA.Vainshtein.1TheauthorintendedtorewritethisIntroduction.InthepresentformitcorrespondstotheIntroductiontotheLecturegivenbyV.N.Gribovatthe34thInternationalSchoolofSubnuclearPhysicsinErice,Italy,in1996[7].1duetotheprocessof(e−,e+)paircreation.Thenegativecomponentofthepair,theelectron,fallsintotheheavychargeandthepositivecomponentgoestoinfinity.TheconditionforthisphenomenontooccuristhattheComptonwavelength1/moftheelectronhastobemuchlargerthantheradiusoftheheavycharge:αpZ2−Z2crln(mR)π.(1.1)TheelectronwhichfallsintothecentreformstogetherwiththesourceZanionofchargeZ1=Z−1.IfZ−1Zcr,thisprocesswillcontinueuntilthevacuumbecomesstablei.e.whenZn=Z−nbecomeslessthanthecriticalcharge.FromthepointofviewoftheDiracequationthefallingelectronhasnegativeenergy.Ontheotherhand,thesupercriticalionwithZnZcrisstablebecauseofthePauliprinciple.Indeed,theelectroncannotleavetheion,sinceallnegativeenergystatesoutsideitareoccupied.InQCDthisphenomenoncanoccursince,duetoasymptoticfreedom,thecolourchargecanreachthecriticalvalueforanyobject.Letussupposethattwoheavyquarksarecreatedwithoppositecolours.ThegluonicvacuumpolarisationincreasesthecolourchargesoftheFigure1:quarks(seeFig.1).Whenthesecolourchargesbecomelargeenough,lightantiquarksstarttofallontotheheavyquark.However,theboundstateformedbyalightantiquarkandaheavyquarkwillbeverydifferentfromtheboundstateswhichappearedinQED.InQEDwehadanionwithachargeZ−1.InQCDthecorrespondingstatecanbecolourless.Thedifferenceisduetothefact,thatwhilethelargecolourchargeoftheheavyquarkresultsfromgluonicvacuumpolarisation,theboundstateisformedbytwoparticlesthetotalchargeofwhichequalszero,andthischargecannotbechangedbyvacuumpolarisation.Thisphenomenoncanbecalledquantumscreening.Formallyitmeansthattheheavyquarkqhisunstableandhastodecayintoamesonandalightquarkqlqh→M+ql.(1.2)Thiscase—theconfinementoftheheavyquark—isrelativelysimpleandalmostinde-pendentofthepropertiesofthelightquark[2].Theproblemofthelightquarkismoredifficult.Atfirstsightitisnotclearwhetherthe”fallingintothecentre”andtheformationofasupercriticalstatecanoccurinthesystemofalightquarkandalightantiquark.Itwasshown[3],however,thatindeedthecriticalphenomenonexistsinthelightq¯qsystem.2Moreover,thecriticalcouplingconstantisproventobesufficientlysmall:αcrπ=1−r23!·34≃0.14.(1.3)Thereisanevenmorecomplicatedquestion:whatkindofstatescanbeformedasaresultofthisphenomenon?Iftheappearingnewstateisanormalmeson,the”fallingintothecentre”leadstoql→M+ql(1.4)insteadof(1.2).Equation(1.4)canbesatisfiediftherearenotonlypositiveenergyquarksq+butalsonegativeenergyquarksq−.Thisresult,althoughitlooksverystrange,isnotunexpected.Indeed,wealreadylearnedinQEDthatparticleswhichfallintothecentrecorrespondtothenegativeenergysolutionoftheDiracequation.Toresolvetheproblemofthenegativeenergystates,Diracsupposedthattheyarealloccupiedandthereforenotobservable.ThenecessitytoconsidernowboththepositiveandnegativesolutionsoftheDiracequationmeansonlythatDirac’shypothesisisnotalwaystrue.Iftheinteractionisstrongenough,thereisanotherpossibility:negativeenergystatesmightbeonlypartlyoccupiedandpositiveenergystatesonlypartlyempty.Thisis,however,notincontradictionwiththeabsenceofnegativeenergyparticlesintherealworldifstablesupercriticalboundstates(mesons)existsinceinthiscaseboththepositiveandnegativeenergyquarkswillbeunstable:q+→M+q−q−→M+q−q−¯q−(1.5)and,consequently,unobservable.Fromthepointofviewofthefermionicspectruminthevacuum,(1.5)meansthefollow-ing.Inthecaseofweakinteractions,thefermionicspectrumhasastru