Quantum Chromodynamics PDF
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W.Greiner·S.Schramm·E.Stein QUANTUMCHROMODYNAMICS Greiner Greiner QuantumMechanics ClassicalMechanics AnIntroduction 4thEdition SystemsofParticles andHamiltonianDynamics Greiner QuantumMechanics Greiner SpecialChapters ClassicalMechanics PointParticlesandRelativity Greiner·Müller QuantumMechanics Greiner Symmetries 2ndEdition ClassicalElectrodynamics Greiner Greiner·Neise·Stöcker RelativisticQuantumMechanics Thermodynamics WaveEquations 3rdEdition andStatisticalMechanics Greiner·Reinhardt FieldQuantization Greiner·Reinhardt QuantumElectrodynamics 3rdEdition Greiner·Schramm·Stein QuantumChromodynamics 3ndEdition Greiner·Maruhn NuclearModels Greiner·Müller GaugeTheoryofWeakInteractions 3rdEdition · WalterGreiner StefanSchramm EckartStein QUANTUM CHROMODYNAMICS WithaForewordby D.A.Bromley ThirdRevisedandEnlargedEdition With165Figures, and66WorkedExamplesandExercises 123 Prof.Dr.Dr.h.c.mult.WalterGreiner Dr.EckartStein FrankfurtInstituteforAdvancedStudies(FIAS) PhysicsDepartment JohannWolfgangGoethe-Universität MaharishiUniversityofManagement Max-von-Laue-Str.1 6063NPVlodrop 60438FrankfurtamMain TheNetherlands Germany email:[email protected] email:greiner@fias.uni-frankfurt.de Prof.Dr.StefanSchramm CenterforScientificComputing(CSC) JohannWolfgangGoethe-Universität Max-von-Laue-Str.1 60438FrankfurtamMain Germany email:[email protected] TitleoftheoriginalGermanedition:TheoretischePhysik,EinLehr-undÜbungsbuch,Band10: Quantenchromodynamik(cid:1)cVerlagHarriDeutsch,Thun,1984,1989 ISBN-10 3-540-48534-1 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-48534-6 SpringerBerlinHeidelbergNewYork ISBN-10 3-540-66610-9 2ndEd.SpringerBerlinHeidelbergNewYork LibraryofCongressControlNumber:2006938282 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned, specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmi- crofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublicationorpartsthereofispermitted onlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermission forusemustalwaysbeobtainedfromSpringer.ViolationsareliableforprosecutionundertheGermanCopyright Law. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)cSpringer-VerlagBerlinHeidelberg1994,2002,2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,evenin theabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulations andthereforefreeforgeneraluse. Typesetting:LE-TEXJelonek,Schmidt&VöcklerGbR Production:LE-TEXJelonek,Schmidt&VöcklerGbR Cover:eStudioCalamarS.L.,Spain SPIN11864288 56/3100YL-543210 Printedonacid-freepaper ForewordtoEarlierSeriesEditions More than a generation of German-speaking students around the world have workedtheirwaytoanunderstandingandappreciationofthepowerandbeauty of modern theoretical physics – with mathematics, the most fundamental of sciences–usingWalterGreiner’stextbooksastheirguide. The idea of developing a coherent, complete presentation of an entire field ofscienceinaseriesofcloselyrelatedtextbooksisnotanewone.Manyolder physicists remember with real pleasure their sense of adventure and discovery astheyworkedtheirwaysthroughtheclassicseriesbySommerfeld,byPlanck and by Landau and Lifshitz. From the students’ viewpoint, there are a great manyobviousadvantagestobegainedthroughuseofconsistentnotation,logi- calorderingoftopicsandcoherenceofpresentation;beyondthis,thecomplete coverageofthescienceprovidesauniqueopportunityfortheauthortoconvey hispersonalenthusiasmandloveforhissubject. Thepresentfivevolumeset,TheoreticalPhysics,isinfactonlythatpartof thecompletesetoftextbooksdevelopedbyGreinerandhisstudentsthatpresents the quantum theory. I have long urged him to make the remaining volumes on classicalmechanicsanddynamics,onelectromagnetism,onnuclearandparticle physics,andonspecialtopicsavailabletoanEnglish-speakingaudienceaswell, andwecanhopeforthesecompanionvolumescoveringalloftheoreticalphysics sometimeinthefuture. WhatmakesGreiner’svolumesofparticularvaluetothestudentandprofes- sor alike is their completeness. Greiner avoids the all too common “it follows that... ”whichconcealsseveralpagesofmathematicalmanipulationandcon- foundsthestudent.Hedoesnothesitatetoincludeexperimentaldatatoillumi- nateorillustrateatheoreticalpoint andthesedata,likethe theoreticalcontent, havebeenkeptuptodateandtopicalthroughfrequentrevisionandexpansionof thelecturenotesuponwhichthesevolumesarebased. Moreover,Greinergreatlyincreasesthevalueofhispresentationbyinclud- ing something like one hundred completely worked examples in each volume. Nothing is of greater importance to the student than seeing, in detail, how the theoreticalconceptsandtoolsunderstudyareappliedtoactualproblemsofinter- esttoaworkingphysicist.And,finally,Greineraddsbriefbiographicalsketches toeachchaptercoveringthepeopleresponsibleforthedevelopmentofthethe- oretical ideas and/or the experimental data presented. It was Auguste Comte (1798–1857) in his Positive Philosophy who noted, “To understand a science it is necessary to know its history”. This is all too often forgotten in modern VI ForewordtoEarlierSeriesEditions physicsteachingandthebridgesthatGreinerbuildstothepioneeringfiguresof ourscienceuponwhoseworkwebuildarewelcomeones. Greiner’s lectures, which underlie these volumes, are internationally noted fortheirclarity,theircompletenessandfortheeffortthathehasdevotedtomak- ingphysicsanintegralwhole;hisenthusiasmforhisscienceiscontagiousand shinesthroughalmosteverypage. ThesevolumesrepresentonlyapartofauniqueandHerculeanefforttomake alloftheoreticalphysicsaccessibletotheinterestedstudent.Beyondthat,they areofenormousvaluetotheprofessionalphysicistandtoallothersworkingwith quantumphenomena.Againandagainthereaderwillfindthat,afterdippinginto aparticularvolumetoreviewaspecifictopic,hewillendupbrowsing,caught up by often fascinating new insights and developments with which he had not previouslybeenfamiliar. HavingusedanumberofGreiner’svolumesintheiroriginalGermaninmy teaching and researchat Yale, I welcome these new and revised English trans- lations and would recommend them enthusiastically to anyone searching for acoherentoverviewofphysics. YaleUniversity D.AllanBromley NewHaven,CT,USA HenryFordIIProfessorofPhysics 1989 PrefacetotheThirdEdition The theory of strong interactions, quantum chromodynamics (QCD), was for- mulated more than 30 years ago and has been ever since avery active field of research. Its continuing importance may be estimated by the Nobel prize in physics for the year 2004, which was awarded to Gross, Wilczek, and Politzer fortheirdiscoveryofasymptoticfreedom,oneofthekeyfeaturesofQCD.The underlying equations of motion for the gauge degrees of freedom provided by QCD are nonlinear and minimally coupled to fermions with global and local SU(3)charges.ThisleadstospectacularproblemscomparedwiththoseofQED sincethegaugebosonsthemselvesinteractwitheachother.Ontheotherhand, itisexactlytheself-interactionofthegluonswhichleadstoasymptoticfreedom andthepossibilitytocalcuatequark–gluoninteractionatsmalldistancesinthe frameworkofperturbationtheory.Wediscoveroneofthemostcomplicatedbut most beautiful gauge theories which poses extremely challenging problems on moderntheoreticalandexperimentalphysicstoday. Quantumchromodynamicsisthequantumfieldtheorythatallowsustocal- culate the propagation and interaction of colored quarks and gluons at small distances.Today’sexperimentsdonotallowthesecoloredobjectstobedetected directly;insteadonedealswithcolorlesshadrons:mesonsandbaryonsseenfar awayfromtheactualinteractionpoint.Thehadronizationitselfisacomplicated process and not yet understood from first principles. Therefore one may won- derhowthesignatureofquarkandgluoninteractionscanbetracedthroughthe processofhadronization. Veryadvancedanalyticalandnumericaltechniqueshavebeendevelopedin ordertoanalyzetheworldofhadronsonthegroundsoffundamentalQCD.To- gether with amuch improved experimental situation we seem to be ready to answer the question wether QCDis the correct theory of strong interactions at allscalesorjustaneffectivehigh-energylineofayetundiscoveredtheory. With the upcoming Large Hadron Collider (LHC) at CERN near Geneva, aproton-proton collider reaching acenter-of-mass energy of 14TeV per col- liding nucleon pair, perturbative QCD will be tested with highest precision in aregimewhereitisexpectedtoworkextremelywell.Thiswillallowprecision testsofQCDastheunderlyingtheoryofquarksandhadrons.Moreover,calcula- tionsperformedusingperturbativeQCDareessentialtodefinethebackground againstwhichallpotentialsignalsofnewphysics–oftheHiggsparticle,super- symmetricparticles,orexoticthingswemaynotyetthinkof–willbegauged. Inthisbook,wetrytogiveaself-consistenttreatmentofQCD,stressingthe practitioners point of view. For pedagogical reasons we review quantum elec- VIII PrefacetotheThirdEdition trodynamics (QED) (Chap.2) after an elementary introduction. In Chap.3 we studyscatteringreactionswithemphasisonlepton–nucleonscatteringandintro- ducethelanguagefordescribingtheinternalstructureofhadrons.AlsotheMIT bagmodelisintroduced,whichservesasanillustrativeandsuccessfulexample forQCD-inspiredmodels. InChap.4thegeneralframeworkofgaugetheoriesisdescribedonthebasis of the famous Standard Model of particle physics. We then concentrate on the gaugetheoryofquark–gluoninteractionandderivetheFeynmanrulesofQCD, whichareveryusefulforpertubativecalculations.Inparticular,weshowexplic- itily how QCD is renormalized and how the often-quoted running coupling is obtained. Chapter5 is devoted to the application of QCD to lepton–hadron scatter- ing and therefore to the state-of-the-art description of the internal structure of hadrons. We start by presenting two derivations of the Dokshitzer–Gribov– Lipatov–Altarelli–Parisi equations. The main focus of this chapter is on the indispensable operator product expansion and its application to deep inelastic lepton–hadron scattering. We show in great detail how to perform this expan- sionandcalculatetheWilsoncoefficients.Furthermore,wediscussperturbative corrections to structure functions and perturbation theory at large orders, i.e. renormalons. After analyzing lepton–hadron scattering we switch in Chap.6 to the case ofhadron–hadronscatteringasdescribedbytheDrell–Yanprocesses.Wethen turntothekinematicalsectorwheretheso-calledleading-logapproximationis nolongersufficient.Thephysicsonthesescalesiscalledsmall-x physics. Chapter7 is devoted to twopromising nonperturbative approaches, namely QCDonthelatticeandtheverypowerfulanalyticaltoolcalledtheQCDsumrule technique.WeshowexplicitelyhowtoformulateQCDonalatticeanddiscuss therelevantalgorithmsneededforpracticalnumericalcalculations,includingthe lattice at finite temperature. This is very important for the physics of hot and dense elementary matter as it appears, for example, in high-energy heavy ion physics.TheQCDsumruletechniqueisexplainedandappliedtothecalculation ofhadronmasses. OurpresentationendswithsomeremarksonthenontrivialQCDvacuumand itsmodificationathightemperatureand/orbaryondensity,includingasketchof currentdevelopmentsconcerningtheso-calledquark–gluonplasmainChap.8. Modernhigh-energyheavyionphysicsisconcernedwiththeseissues. We have tried to give apedagogical introduction to the concepts and tech- niquesofQCD.Inparticular,wehavesuppliedover70examplesandexercises worked out in great detail. Working through these may help the practitioner in perfoming complicated calculations in this challenging field of theoretical physics. In writing this book we profited substantially from anumber of existing textbooks, most notably J.J.R.Aitchison and A.J.G.Hey: ‘Gauge Theories in Particle Physics’, O.Nachtmann: ‘Elementarteilchenphysik’, B.Müller: ‘The Physics of the Quark–Gluon Plasma’, P.Becher, M.Boehm and H.Joos: ‘Eichtheorien’, J.Collins: ‘Renormalization’, R.D.Field: ‘Application of Per- turbative QCD’, and M.Creutz: ‘Quarks, Gluons and Lattices’, and several PrefacetotheThirdEdition IX review articles, especially: L.V.Gribov, E.M.Levin and M.G.Ryskin: ‘Semi- hard processes in QCD’, Phys. Rep. 100 (1983)1, Badelek, Charchula, Krawczyk, Kwiecinski, ‘Small x physics in deep inelastic lepton hadron scat- tering’, Rev. Mod. Phys. 927 (1992), L.S.Reinders, H.Rubinstein, S.Yazaki: ‘HadronpropertiesfromQCDsumrules’,Phys.Rep.127(1985)1. We thank many members of the Institute of Theoretical Physics in Frank- furt who added in one way or another to this book, namely Dr.J.Augustin, M.Bender,C.Best,S.Bernard,A.Bischoff,M.Bleicher,A.Diener,A.Dumi- tru,B.Ehrnsperger,U.Eichmann,S.Graf,N.Hammon,C.Hofmann,A.Jahns, J.Klenner, O.Martin, M.Massoth, G.Plunien, D.Rischke, and A.Scherdin. SpecialthanksgotoA.Steidelwhodrewthefigures,toDr.H.Weberfortechni- calhelp,andtoDr.S.Hofmannwhosupervisedtheeditingprocessofthesecond editionandgavehintsformanyimprovements. In this new edition, typographical errors have been removed, and data and references to the literature have been updated. We thank Dr.S.Scherer for his reliableandefficientassistancethroughouttheeditingprocess. FrankfurtamMain, WalterGreiner November2006 StefanSchramm EckartStein Contents 1. TheIntroductionofQuarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 TheHadronSpectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. ReviewofRelativisticFieldTheory . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 SpinorQuantumElectrodynamics . . . . . . . . . . . . . . . . . . . . . 17 2.1.1 TheFreeDiracEquationandItsSolution . . . . . . . . . 17 2.1.2 DensityandCurrentDensity . . . . . . . . . . . . . . . . . . 20 2.1.3 CovariantNotation . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.4 NormalizationofDiracSpinors . . . . . . . . . . . . . . . . 22 µ 2.1.5 InteractionwithaFour-Potential A . . . . . . . . . . . . . 24 2.1.6 TransitionAmplitudes . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.7 DiscreteSymmetries . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 ScalarQuantumElectrodynamics . . . . . . . . . . . . . . . . . . . . . 26 2.2.1 TheFreeKlein–GordonEquationanditsSolutions . . 26 2.2.2 Interactionofaπ+ withaPotential Aµ . . . . . . . . . . 29 2.2.3 π+K+ Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.4 TheCrossSection . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.5 Spin-1ParticlesandTheirPolarization . . . . . . . . . . . 47 2.2.6 ThePropagatorforVirtualPhotons . . . . . . . . . . . . . . 53 2.3 Fermion–BosonandFermion–FermionScattering . . . . . . . . . 58 2.3.1 TracesandSpinSummations . . . . . . . . . . . . . . . . . . 60 2.3.2 TheStructureoftheFormFactors fromInvarianceConsiderations . . . . . . . . . . . . . . . . 70 3. ScatteringReactionsandtheInternalStructureofBaryons . . . . . 77 3.1 SimpleQuarkModelsCompared . . . . . . . . . . . . . . . . . . . . . 77 3.2 TheDescriptionofScatteringReactions . . . . . . . . . . . . . . . . 80 3.3 TheMITBagModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4. GaugeTheoriesandQuantum-Chromodynamics . . . . . . . . . . . . . 155 4.1 TheStandardModel:ATypicalGaugeTheory . . . . . . . . . . . 155 4.2 TheGaugeTheoryofQuark–QuarkInteractions . . . . . . . . . . 165 4.3 DimensionalRegularization . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.4 TheRenormalizedCouplingConstantofQCD . . . . . . . . . . . 203 4.5 ExtendedExample:AnomaliesinGaugeTheories . . . . . . . . . 220 4.5.1 TheSchwingerModelontheCircle . . . . . . . . . . . . . 220 4.5.2 DiracSea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
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