CLASSICAL SOLUTIONS IN QUANTUM FIELD THEORY Classical solutions play an important role in quantum field theory, high energy physics, and cosmology. Real time soliton solutions give rise to particles, such as magnetic monopoles, and extended structures, such as domain walls and cosmic strings,thathaveimplicationsforthecosmologyoftheearlyuniverse.Imaginary time Euclidean instantons are responsible for important nonperturbative effects, while Euclidean bounce solutions govern transitions between metastable states. Written for advanced graduate students and researchers in elementary par- ticle physics, cosmology, and related fields, this book brings the reader up to the level of current research in the field. The first half of the book discusses the mostimportantclassesofsolitons:kinks,vortices,andmagneticmonopoles.The cosmologicalandobservationalconstraintsonthesearecovered,asaremorefor- mal aspects, including BPS solitons and their connection with supersymmetry. The second half is devoted to Euclidean solutions, with particular emphasis on Yang–Mills instantons and on bounce solutions. erick j. weinberg is a Professor of Physics in the Department of Physics, Columbia University. Since 1996 he has been Editor of Physical Review D. His research interests include the implications of solitons and instantons for high energyphysics,cosmology,andblackholes,aswellasavarietyofothertopicsin quantum field theory. CAMBRIDGEMONOGRAPHSONMATHEMATICALPHYSICS General Editors: P. V. Landshoff, D. R. Nelson, S. Weinberg S.J.AarsethGravitationalN-BodySimulations:ToolsandAlgorithms J.Ambjørn,B.DurhuusandT.JonssonQuantumGeometry:AStatisticalFieldTheoryApproach A.M.AnileRelativisticFluidsandMagneto-fluids: WithApplicationsinAstrophysicsand PlasmaPhysics J.A.deAzc´arragaandJ.M.IzquierdoLieGroups,LieAlgebras,Cohomology andSome Applications inPhysics† O.Babelon,D.BernardandM.TalonIntroductiontoClassicalIntegrableSystems† F.BastianelliandP.vanNieuwenhuizenPathIntegralsandAnomaliesinCurvedSpace V.BelinskiandE.VerdaguerGravitational Solitons J.BernsteinKineticTheoryintheExpandingUniverse G.F.BertschandR.A.BrogliaOscillationsinFiniteQuantumSystems N.D.BirrellandP.C.W.DaviesQuantumFieldsinCurvedSpace† K.Bolejko,A.Krasin´ski,C.HellabyandM-N.C´el´erierStructuresintheUniversebyExact Methods: Formation,Evolution,Interactions D.M.BrinkSemi-Classical MethodsforNucleus-NucleusScattering† M.BurgessClassical CovariantFields E.A.CalzettaandB.-L.B.HuNonequilibriumQuantumFieldTheory S.CarlipQuantumGravityin2+1Dimensions† P.CartierandC.DeWitt-MoretteFunctionalIntegration:ActionandSymmetries J.C.CollinsRenormalization: AnIntroductiontoRenormalization, theRenormalization Group andtheOperator-Product Expansion† P.D.B.CollinsAnIntroduction toReggeTheoryandHighEnergyPhysics† M.CreutzQuarks,GluonsandLattices† P.D.D’EathSupersymmetricQuantumCosmology F.deFeliceandD.BiniClassicalMeasurementsinCurvedSpace-Times F.deFeliceandC.J.SClarkeRelativityonCurvedManifolds B.DeWittSupermanifolds, 2nd edition† P.G.OFreundIntroduction toSupersymmetry† F.G.FriedlanderTheWaveEquationonaCurvedSpace-Time† Y.FrishmanandJ.SonnenscheinNon-PerturbativeFieldTheory:FromTwo-Dimensional ConformalFieldTheorytoQCDinFourDimensions J.A.FuchsAffineLieAlgebrasandQuantumGroups:AnIntroduction, withApplicationsin ConformalFieldTheory† J.FuchsandC.SchweigertSymmetries,LieAlgebrasandRepresentations: AGraduateCourse forPhysicists† Y.FujiiandK.MaedaTheScalar-TensorTheoryofGravitation J.A.H.Futterman,F.A.Handler,R.A.MatznerScatteringfromBlackHoles† A.S.Galperin,E.A.Ivanov,V.I.OrievetskyandE.S.SokatchevHarmonicSuperspace R.GambiniandJ.PullinLoops, Knots,GaugeTheoriesandQuantumGravity† T.GannonMoonshinebeyondtheMonster:TheBridgeConnectingAlgebra,ModularFormsand Physics M.G¨ockelerandT.Schu¨ckerDifferentialGeometry,GaugeTheoriesandGravity† C.G´omez,M.Ruiz-AltabaandG.SierraQuantumGroupsinTwo-Dimensional Physics M.B.Green,J.H.SchwarzandE.Witten SuperstringTheoryVolume1:Introduction† M.B.Green,J.H.SchwarzandE.Witten SuperstringTheoryVolume2:LoopAmplitudes, AnomaliesandPhenomenology† V.N.GribovTheTheoryofComplexAngularMomenta:GribovLecturesonTheoreticalPhysics J.B.GriffithsandJ.Podolsky´ExactSpace-TimesinEinstein’sGeneralRelativity S.W.HawkingandG.F.R.EllisTheLargeScaleStructureofSpace-Time† F.IachelloandA.ArimaTheInteractingBosonModel F.IachelloandP.vanIsackerTheInteractingBoson-FermionModel C.ItzyksonandJ.M.DrouffeStatisticalFieldTheoryVolume1:FromBrownianMotionto Renormalization andLatticeGaugeTheory† C.ItzyksonandJ.M.DrouffeStatisticalFieldTheoryVolume2:StrongCoupling,MonteCarlo Methods, ConformalFieldTheoryandRandomSystems† C.V.JohnsonD-Branes† P.S.JoshiGravitationalCollapseandSpacetimeSingularities J.I.KapustaandC.GaleFinite-TemperatureFieldTheory:PrinciplesandApplications, 2nd edition V.E.Korepin,N.M.BogoliubovandA.G.IzerginQuantumInverseScatteringMethodand CorrelationFunctions† M.LeBellacThermalFieldTheory† Y.MakeenkoMethodsofContemporaryGaugeTheory N.MantonandP.SutcliffeTopological Solitons N.H.MarchLiquidMetals:ConceptsandTheory I.MontvayandG.Mu¨nsterQuantumFieldsonaLattice† L.O’RaifeartaighGroupStructureofGaugeTheories† T.Ort´ınGravityandStrings† A.M.OzoriodeAlmeidaHamiltonianSystems:ChaosandQuantization† L.ParkerandD.J.TomsQuantumFieldTheoryinCurvedSpacetime:QuantizedFieldsand Gravity R.PenroseandW.RindlerSpinorsandSpace-TimeVolume1:Two-SpinorCalculusand RelativisticFields† R.PenroseandW.RindlerSpinorsandSpace-TimeVolume2:SpinorandTwistorMethods in Space-TimeGeometry† S.PokorskiGaugeFieldTheories,2nd edition† J.PolchinskiStringTheoryVolume1:AnIntroduction totheBosonicString J.PolchinskiStringTheoryVolume2:SuperstringTheoryandBeyond J.C.PolkinghorneModelsofHighEnergyProcesses† V.N.PopovFunctionalIntegralsandCollectiveExcitations† L.V.ProkhorovandS.V.ShabanovHamiltonianMechanicsofGaugeSystems R.J.RiversPathIntegralMethodsinQuantumFieldTheory† R.G.RobertsTheStructureoftheProton:DeepInelasticScattering† C.RovelliQuantumGravity† W.C.SaslawGravitationalPhysicsofStellarandGalacticSystems† R.N.SenCausality,MeasurementTheoryandtheDifferentiableStructureofSpace-Time M.ShifmanandA.YungSupersymmetricSolitons H.Stephani,D.Kramer,M.MacCallum,C.HoenselaersandE.HerltExactSolutionsofEinstein’s FieldEquations,2nd edition† J.StewartAdvancedGeneralRelativity† J.C.TaylorGaugeTheoriesofWeakInteractions† T.ThiemannModernCanonicalQuantumGeneralRelativity D.J.TomsTheSchwingerActionPrincipleandEffectiveAction A.VilenkinandE.P.S.ShellardCosmicStringsandOtherTopological Defects† R.S.WardandR.O.Wells,JrTwistor GeometryandFieldTheory† E.J.WeinbergClassicalSolutionsinQuantumFieldTheory:SolitonsandInstantonsinHigh EnergyPhysics J.R.WilsonandG.J.MathewsRelativisticNumericalHydrodynamics † Issuedasapaperback Downlo Classical Solutions in Quantum Field Theory Solitons and Instantons in High Energy Physics ERICK J. WEINBERG Columbia University cambridge university press Cambridge,NewYork,Melbourne,Madrid,CapeTown, Singapore,S˜aoPaulo,Delhi,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress, NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9780521114639 (cid:2)c E.J.Weinberg2012 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2012 PrintedintheUnitedKingdomattheUniversityPress,Cambridge A catalog record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Weinberg,ErickJ. Classicalsolutionsinquantumfieldtheory:solitonsandinstantonsinhigh energyphysics/ErickJ.Weinberg. p. cm.–(Cambridgemonographsonmathematicalphysics) Includesbibliographicalreferencesandindex. ISBN978-0-521-11463-9(hardback) 1. Quantumtheory–Mathematics. I. Title. QC174.17.M35W45 2012 530.12–dc23 2012015503 ISBN978-0-521-11463-9Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. To Carolyn, Michael, and Cate Downlo Contents Preface page xiii 1 Introduction 1 1.1 Overview 1 1.2 Conventions 3 2 One-dimensional solitons 6 2.1 Kinks 6 2.2 Quantizing about the kink 13 2.3 Zero modes and collective coordinates 22 2.4 Fermions and fermion zero modes 24 2.5 Kinks in more spacetime dimensions 27 2.6 Multikink dynamics 29 2.7 The sine-Gordon–massive Thirring model equivalence 34 3 Solitons in more dimensions—Vortices and strings 38 3.1 First attempt—global vortices 38 3.2 Derrick’s theorem 42 3.3 Gauged vortices 44 3.4 Multivortex solutions 47 3.5 Quantization and zero modes 49 3.6 Adding fermions 52 4 Some topology 57 4.1 Vacuum manifolds 57 4.2 Homotopy and the fundamental group π (M) 58 1 4.3 Fundamental groups of Lie groups 61 4.4 Vortices and homotopy 64 4.5 Some illustrative vortex examples 68 4.6 Higher homotopy groups 74 4.7 Some results for higher homotopy groups 77 5 Magnetic monopoles with U(1) charges 81 5.1 Magnetic monopoles in electromagnetism 81 5.2 The ’t Hooft–Polyakov monopole 89 x Contents 5.3 Another gauge, another viewpoint 94 5.4 Solutions with higher magnetic charge 96 5.5 Zero modes and dyons 97 5.6 Spin from isospin, fermions from bosons 100 5.7 Fermions and monopoles 104 6 Magnetic monopoles in larger gauge groups 108 6.1 Larger gauge groups—the external view 108 6.2 Larger gauge groups—topology 115 6.2.1 SU(3) broken to SU(2)×U(1) 115 6.2.2 A Z monopole 119 2 6.2.3 A light doubly charged monopole 120 6.2.4 Electroweak monopoles? 121 6.3 Monopoles in grand unified theories 121 6.3.1 SU(5) monopoles 122 6.3.2 SO(10) monopoles 124 6.4 Chromodyons 125 6.5 The Callan–Rubakov effect 128 7 Cosmological implications and experimental bounds 130 7.1 Brief overview of big bang cosmology 130 7.2 Symmetry restoration and cosmological phase transitions 133 7.3 The Kibble mechanism 136 7.4 Gravitational and cosmological consequences of domain walls and strings 139 7.5 Evolution of the primordial monopole abundance 142 7.6 Observational bounds and the primordial monopole problem 145 8 BPS solitons, supersymmetry, and duality 149 8.1 The BPS limit as a limit of couplings 149 8.2 Energy bounds 151 8.3 Supersymmetry 155 8.4 Multisoliton solutions 160 8.5 The moduli space approximation 163 8.6 BPS monopoles in larger gauge groups 166 8.7 Montonen–Olive duality 172 9 Euclidean solutions 175 9.1 Tunneling in one dimension 175 9.2 WKB tunneling with many degrees of freedom 178 9.3 Path integral approach to tunneling: instantons 181 9.4 Path integral approach to tunneling: bounces 186 9.5 Field theory 190