DYNAMICSOFTHESTANDARDMODEL
Describingthefundamentaltheoryofparticlephysicsanditsapplications,this bookprovidesadetailedaccountoftheStandardModel,focusingontechniques thatcanproduceinformationaboutrealobservedphenomena.
ThebookbeginswithapedagogicaccountoftheStandardModel,introducing essentialtechniquessuchaseffectivefieldtheoryandpath-integralmethods.Itthen focusesontheuseoftheStandardModelinthecalculationofphysicalproperties ofparticles.Rigorousmethodsareemphasized,butotherusefulmodelsarealso described.
Thissecondeditionhasbeenupdatedtoincluderecenttheoreticalandexperimentaladvances,suchasthediscoveryoftheHiggsboson.Anewchapteris devotedtothetheoreticalandexperimentalunderstandingofneutrinos,andmajor advancesin CP violationandelectroweakphysicshavebeengivenamoderntreatment.Thisbookisvaluabletograduatestudentsandresearchersinparticlephysics, nuclearphysicsandrelatedfields.
This title, first published in 2014, has been reissued as an Open Access publication on Cambridge Core.
JohnF.Donoghue isDistinguishedProfessorintheDepartmentofPhysics, UniversityofMassachusetts.Hisresearchspansparticlephysics,quantumfield theoryandgeneralrelativity.HeisaFellowoftheAmericanPhysicalSociety.
EugeneGolowich isEmeritusProfessorintheDepartmentofPhysics, UniversityofMassachusetts.Hisresearchhasfocusedonparticletheoryandphenomenology.HeisaFellowoftheAmericanPhysicalSocietyandisarecipientof theCollegeOutstandingTeacherawardfromtheUniversityofMassachusetts.
BarryR.Holstein isEmeritusProfessorintheDepartmentofPhysics, UniversityofMassachusetts.Hisresearchisintheoverlapareaofparticleand nucleartheory.AFellowoftheAmericanPhysicalSociety,heisalsotheeditor of AnnualReviewsofNuclearandParticleScience andisalongtimeconsulting editorofthe AmericanJournalofPhysics.
CAMBRIDGEMONOGRAPHSONPARTICLEPHYSICS,NUCLEAR PHYSICSANDCOSMOLOGY
GeneralEditors:T.Ericson,P.V.Landshoff
Availabletitlesinthisseries:
3.E.LeaderandE.Predazzi: AnIntroductiontoGaugeTheoriesandModernParticlePhysics, Volume1:ElectroweakInteractions,the‘NewParticles’andthePartonModel
4.E.LeaderandE.Predazzi: AnIntroductiontoGaugeTheoriesandModernParticlePhysics, Volume2:CP-Violation,QCDandHardProcesses
6.H.GrosseandA.Martin: ParticlePhysicsandtheSchrödingerEquation
7.B.Andersson: TheLundModel
8.R.K.Ellis,W.J.StirlingandB.R.Webber: QCDandColliderPhysics
10.A.V.ManoharandM.B.Wise: HeavyQuarkPhysics
11.R.Frühwirth,M.Regler,R.K.Bock,H.GroteandD.Notz: DataAnalysisTechniquesfor High-EnergyPhysics,Secondedition
12.D.Green: ThePhysicsofParticleDetectors
13.V.N.GribovandJ.Nyiri: QuantumElectrodynamics
14.K.Winter(ed.): NeutrinoPhysics,Secondedition
15.E.Leader: SpininParticlePhysics
16.J.D.Walecka: ElectronScatteringforNuclearandNucleonStructure
17.S.Narison: QCDasaTheoryofHadrons
18.J.F.LetessierandJ.Rafelski: HadronsandQuark-GluonPlasma
19.ADonnachie,H.G.Dosch,P.V.LandshoffandO.Nachtmann: PomeronPhysicsandQCD
20.A.Hofmann: ThePhysicsofSynchrotronRadiation
21.J.B.KogutandM.A.Stephanov: ThePhasesofQuantumChromodynamics
22.D.Green: High PT PhysicsatHadronColliders
23.K.Yagi,T.HatsudaandY.Miake: Quark-GluonPlasma
24.D.M.BrinkandR.A.Broglia: NuclearSuperfluidity
25.F.E.Close,A.DonnachieandG.Shaw: ElectromagneticInteractionsandHadronicStructure
26.C.GrupenandB.A.Schwartz: ParticleDetectors,Secondedition
27.V.Gribov: StrongInteractionsofHadronsatHighEnergies
28.I.I.BigiandA.I.Sanda: CPViolation,Secondedition
29.P.JaranowskiandA.Królak: AnalysisofGravitational-WaveData
30.B.L.Ioffe,V.S.FadinandL.N.Lipatov: QuantumChromodynamics:Perturbativeand NonperturbativeAspects
31.J.M.Cornwall,J.PapavassiliouandD.Binosi: ThePinchTechniqueanditsApplicationsto Non-AbelianGaugeTheories
32.J.Collins: FoundationsofPerturbativeQCD
33.Y.V.KovchegovandE.Levin: QuantumChromodynamicsatHighEnergy
34.J.RakandM.J.Tannenbaum: High-pT PhysicsintheHeavyIonEra
35.J.F.Donoghue,E.GolowichandB.R.Holstein: DynamicsoftheStandardModel, Secondedition
DYNAMICSOFTHESTANDARD MODEL
secondedition
JOHNF.DONOGHUE
UniversityofMassachusetts
EUGENEGOLOWICH
UniversityofMassachusetts
BARRYR.HOLSTEIN
UniversityofMassachusetts
Shaftesbury Road, Cambridge CB2 8EA, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA
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Information on this title: www.cambridge.org/9781009291002
DOI: 10.1017/9781009291033
© John F. Donoghue, Eugene Golowich and Barry R. Holstein 2022
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ToLincolnWolfenstein
I–4Symmetriesandnearsymmetries11
On-shellrenormalizationoftheelectriccharge35
Electricchargeasarunningcouplingconstant37
II–2QuantumChromodynamics40
SU (3)gaugesymmetry40
QCD tooneloop45
Asymptoticfreedomandrenormalizationgroup51
II–3Electroweakinteractions57
Weakisospinandweakhyperchargeassignments57
SU (2)L × U(1)Y gauge-invariantlagrangian60
Spontaneoussymmetrybreaking62
Electroweakcurrents63
II–4Fermionmixing66
Diagonalizationofmassmatrices66
Quarkmixing68
Neutrinomixing70
Quark CP violationandrephasinginvariants72
IIISymmetriesandanomalies 76
III–1SymmetriesoftheStandardModel76
III–2Pathintegralsandsymmetries79
Thegeneratingfunctional80
Noether’stheoremandpathintegrals81
III–3The U (1)axialanomaly82
Diagrammaticanalysis84
Path-integralanalysis88
III–4Classicalscaleinvarianceandthetraceanomaly95
III–5Chiralanomaliesandvacuumstructure98
The θ vacuum99
The θ term101
Connectionwithchiralrotations102
III–6Baryon-andlepton-numberviolationintheStandardModel103
IVIntroductiontoeffectivefieldtheory 106
IV–1Effectivelagrangiansandthesigmamodel106
Representationsofthesigmamodel107
Representationindependence109
IV–2Integratingoutheavyfields111
Thedecouplingtheorem111
Integratingoutheavyfieldsattreelevel112
Matchingthesigmamodelattreelevel114
IV–3Loopsandrenormalization115
IV–4Generalfeaturesofeffectivefieldtheory119 Effectivelagrangiansandsymmetries120
Powercountingandloops121
Weinberg’spower-countingtheorem122 Thelimitsofaneffectivefieldtheory123
IV–5Symmetrybreaking124
IV–6Matrixelementsofcurrents126
Matrixelementsandtheeffectiveaction127
IV–7Effectivefieldtheoryofregionsofasinglefield129
IV–8Effectivelagrangiansin QED
IV–9EffectivelagrangiansasprobesofNewPhysics138 VChargedleptons
V–3The τ lepton163
Exclusiveleptonicdecays164
Exclusivesemileptonicdecays164 Inclusivesemileptonicdecays167 Someapplicationsof τ decays168
VI–1Neutrinomass173
EquivalenceofheavyMajoranamasstoadimension-five operator175
VI–2Leptonmixing176
VI–3Theoryofneutrinooscillations179 Oscillationsinvacuum179 Oscillationsinmatter:MSWeffect181 CP violation184
VI–4Neutrinophenomenology185
Solarandreactorneutrinos: θ12 and m2 21
Atmosphericandacceleratorneutrinos: θ23 and | m2 32 |
Short-baselinestudies: θ13
VI–5TestingfortheMajorananatureofneutrinos192
VI–6Leptogenesis195
VI–7Numberoflightneutrinospecies197
Studiesatthe Z0 peak197
Astrophysicaldata197
VIIEffectivefieldtheoryforlow-energy QCD
VII–1 QCD atlowenergies200
Vacuumexpectationvaluesandmasses201
Quarkmassratios202
Pionleptonicdecay,radiativecorrections,and Fπ
VII–2Chiralperturbationtheorytooneloop209
Theorder E 4 lagrangian210
Therenormalizationprogram211
VII–3Thenatureofchiralpredictions215
Thepionformfactor215
Rareprocesses219
Pion–pionscattering223
VII–4Thephysicsbehindthe QCD chirallagrangian225
VII–5TheWess–Zumino–Wittenanomalyaction228
VII–6Theaxialanomalyand π 0 → γγ 233
VIIIWeakinteractionsofkaons 237
VIII–1Leptonicandsemileptonicprocesses237
VIII–2Thenonleptonicweakinteraction240
VIII–3Matchingto QCD atshortdistance242
Short-distanceoperatorbasis242
Perturbativeanalysis243
Renormalization-groupanalysis245
VIII–4The I = 1/2rule249
Phenomenology249
Chirallagrangiananalysis252
Vacuumsaturation253
Nonleptoniclatticematrixelements254
VIII–5Rarekaondecays255
IXMassmixingand CP violation 260
IX–1 K 0 –K 0 mixing260 CP-conservingmixing263
IX–2Thephenomenologyofkaon CP violation266
IX–3Kaon CP violationintheStandardModel269
Analysisof | |
Analysisof | |
Chiralanalysisof ( / )EWP
IX–4Thestrong CP problem273
Theparameter θ 273
Connectionswiththeneutronelectricdipolemoment275
XThe N 1 c expansion
X–1Thenatureofthelarge Nc limit278
X–2Spectroscopyinthelarge Nc limit280
X–3Goldstonebosonsandtheaxialanomaly283
X–4The OZI rule285
X–5Chirallagrangians287
XIPhenomenologicalmodels 291
XI–1Quantumnumbersof QQ and Q3 states291
Hadronicflavor–spinstatevectors291 Quarkspatialwavefunctions295 Interpolatingfields297
XI–2Potentialmodel298 Basicingredients298 Mesons300 Baryons301
Colordependenceoftheinterquarkpotential302
XI–3Bagmodel303 Staticcavity304
Spherical-cavityapproximation304
Gluonsinabag307
Thequark–gluoninteraction308 XI–4Skyrmemodel308
Sine–Gordonsoliton309
Chiral SU (2)soliton310 TheSkyrmesoliton312 Quantizationandwavefunctions314
XI–5 QCD sumrules318
Correlators319
Operator-productexpansion321
Masterequation323 Examples324
XIIBaryonproperties 330
XII–1Matrix-elementcomputations330
Flavorandspinmatrixelements330
Overlapsofspatialwavefunctions332
Connectiontomomentumeigenstates333
CalculationsintheSkyrmemodel336
XII–2Electroweakmatrixelements339
Magneticmoments339
Semileptonicmatrixelements341
XII–3Symmetrypropertiesandmasses343
Effectivelagrangiansforbaryons343
Baryonmasssplittingsandquarkmasses344
Goldberger–Treimanrelation347
Thenucleonsigmaterm348
Strangenessinthenucleon349
Quarksandnucleonspinstructure351
XII–4Nuclearweakprocesses355
Measurementof Vud 355
Thepseudoscalaraxialformfactor357
XII–5Hyperonsemileptonicdecay359
XII–6Nonleptonicdecay360 Phenomenology360 Lowest-orderchiralanalysis362
XIIIHadronspectroscopy 366
XIII–1Thecharmoniumandbottomoniumsystems366 Transitionsinquarkonium372
XIII–2Lightmesonsandbaryons376
SU (6)classificationofthelighthadrons376
Reggetrajectories379
SU (6)breakingeffects381
XIII–3Theheavy-quarklimit385
Heavy-flavoredhadronsinthequarkmodel385
Spectroscopyinthe mQ →∞ limit387
XIII–4Nonconventionalhadronstates390
Thefirstresonance– σ (440)391 Gluonia394
Additionalnonconventionalstates396
XIVWeakinteractionsofheavyquarks
XIV–1Heavy-quarkmass399
Runningquarkmass399
Thepolemassofaquark401
XIV–2Inclusivedecays403
Thespectatormodel403
Theheavy-quarkexpansion405 Thetopquark408
XIV–3Exclusivedecaysintheheavy-quarklimit409
Inclusivevs.exclusivemodelsfor b → ceνe 410
HeavyQuarkEffectiveTheoryandexclusivedecays411
XIV–4 B 0 B 0 and D 0 D 0 mixing416 B0 –B0 mixing416
mixing418
XIV–5Theunitaritytriangle420
XIV–6 CP violationin B -mesondecays421
CP-oddsignalsinducedbymixing421
Decaysto CP eigenstates423
Decaystonon-CP eigenstates426
Semileptonicasymmetries427
CP-oddsignalsnotinducedbymixing428
XIV–7Raredecaysof B mesons430
XV–1Introduction434
XV–2MassandcouplingsoftheHiggsboson435 Higgsmassterm436 Thenaturalnessproblem436 Higgscouplingconstants437
XV–3ProductionanddecayoftheHiggsboson442 Decay442 Production445
ComparisonofStandardModelexpectationswithLHCdata447
XV–4Higgscontributionstoelectroweakcorrections448 Thecorrections ρ and r 449
Custodialsymmetry450
XV–5ThequantumHiggspotentialandvacuumstability452
XV–6TwoHiggsdoublets454
XVITheelectroweaksector 458
XVI–1Neutralweakphenomenaatlowenergy458
Neutral-currenteffectivelagrangians459
Deep-inelasticneutrinoscatteringfromisoscalartargets461
Atomicparityviolationincesium462
PolarizedMøllerscattering463
XVI–2Measurementsatthe Z0 massscale464
Decaysof Z 0 intofermion–antifermionpairs466
Asymmetriesatthe Z 0 peak467
Definitionsoftheweakmixingangle469
XVI–3Some W ± properties471
Decaysof W± intofermions471
Triple-gaugecouplings472
XVI–4Thequantumelectroweaklagrangian474
Gaugefixingandghostfieldsintheelectroweaksector475
AsubsetofelectroweakFeynmanrules476
On-shelldeterminationofelectroweakparameters478
XVI–5Self-energiesofthemassivegaugebosons479
Thechargedgaugebosons W± 480
Theneutralgaugebosons Z 0 , γ 482
XVI–6Examplesofelectroweakradiativecorrections483 The
The
The Z→b b vertexcorrection488
PrecisiontestsandNewPhysics489
AppendixAFunctionalintegration 493
A–1Quantum-mechanicalformalism493
Path-integralpropagator493
Externalsources496
Thegeneratingfunctional497
A–2Theharmonicoscillator499
A–3Field-theoreticformalism502
Pathintegralswithfields502
Generatingfunctionalwithfields503
A–4Quadraticforms506
Backgroundfieldmethodtooneloop507
A–5Fermionfieldtheory509
A–6Gaugetheories512
Gaugefixing513
Ghostfields516
AppendixBAdvancedfield-theoreticmethods 520
B–1Theheatkernel520
B–2Chiralrenormalizationandbackgroundfields524
B–3PCACandthesoft-piontheorem529
B–4Matchingfieldswithdifferentsymmetry-transformation properties532
AppendixCUsefulformulae 535
C–1Numerics535
C–2Notationsandidentities535
C–3Decaylifetimesandcrosssections538
C–4Fielddimension541
C–5Mathematicsin d dimensions541
TheStandardModelisthebasisofourunderstandingofthefundamentalinteractions.Atthepresenttime,itremainsinexcellentagreementwithexperiment.It isclearthatanyfurtherprogressinthefieldwillneedtobuildonasolidunderstandingoftheStandardModel.Sincethefirsteditionwaswrittenin1992there havebeenmajordiscoveriesinneutrinophysics,in CP violation,thediscoveries ofthetopquarkandtheHiggsboson,andadramaticincreaseinprecisioninboth electroweakphysicsandin QCD.Wefeelthatthepresentisagoodmomentto updateourbook,astheStandardModelseemslargelycomplete.
Theopportunitytoreviseourbookatthistimehasalsoenabledustosurveythe progresssincethefirsteditionwenttoprint.Besidestheexperimentaldiscoveries thathavetakenplaceduringthesetwodecades,wehavebeenimpressedbythe increaseintheoreticalsophistication.Manyofthetopicswhichwerenovelatthe timeofthefirsteditionhavenowbeenextensivelydeveloped.Perturbativetreatmentshaveprogressedtohigherordersandnewtechniqueshavebeendeveloped. Tocoverallofthesecompletelywouldrequiretheexpansionofmanychapters intobook-lengthtreatments.Indeed,inmanycases,entirenewbooksdedicatedto specializedtopicshavebeenpublished.1 Ourrevisionismeantasacoherentpedagogicintroductiontothesetopics,providingthereaderwiththebasicbackground topursuemoredetailedstudieswhenappropriate.
TherehasalsobeengreatprogressonthepossibleNewPhysicswhichcould emergebeyondtheStandardModel–darkmatteranddarkenergy,grandunification,supersymmetry,extradimensions,etc.Weareatamomentwherethisphysics couldemergeinthenextroundofexperimentsattheLargeHadronCollider(LHC) aswellasinprecisionmeasurementsattheintensityfrontier.Welookforwardwith greatanticipationtothenewdiscoveriesofthenextdecade.
1 Forexample,see[BaP99,Be00,BiS00,EISW03,FuS04,Gr04,IoFL10,La10,Ma04,MaW07,Co11]
Wethankourcolleaguesandstudentsforfeedbackaboutthefirsteditionofthis book.Alistoferrataforthesecondeditionwillbemaintainedatthehomepage ofJohnDonoghueattheUniversityofMassachusetts,Amherst.Weencourage readerswhofindanymistakesinthiseditiontosubmitthemtoProfessorDonoghue atdonoghue@physics.umass.edu.
Fromtheprefacetothefirstedition
TheStandardModellagrangian LSM embodiesourknowledgeofthestrongand electroweakinteractions.Itcontainsasfundamentaldegreesoffreedomthespin one-halfquarksandleptons,thespinonegaugebosons,andthespinzeroHiggs fields.Symmetryplaysthecentralroleindeterminingitsdynamicalstructure.The lagrangianexhibitsinvarianceunder SU(3) gaugetransformationsforthestrong interactionsandunder SU(2) × U(1) gaugetransformationsfortheelectroweak interactions.Despitethepresenceof(alltoo)manyinputparameters,itisamathematicalconstructionofconsiderablepredictivepower.
Therearebooksavailablewhichdescribeindetailtheconstructionof LSM and itsquantization,andwhichdealwithaspectsofsymmetrybreaking.Wefeltthe needforabookdescribingthenextsteps,how LSM isconnectedtotheobservable physicsoftherealworld.Thereareaconsiderablevarietyoftechniques,ofdifferingrigor,whichareusedbyparticlephysiciststoaccomplishthis.Wepresenthere thosewhichhavebecomeindispensabletools.Inaddition,weattempttoconvey theinsightsand‘conventionalwisdom’whichhavebeendevelopedthroughoutthe field.Thisbookcanonlybeanintroductiontotherichescontainedinthesubject, hopefullyprovidingafoundationandamotivationforfurtherexplorationbyits readers.
Inwritingthebook,wehavebecomealltoopainfullyawarethateachtopic, indeedeachspecificreaction,hasanextensiveliteratureandphenomenology,and thatthereisalimitationtothedepththatcanbepresentedcompactly.Weemphasizeapplications,notfundamentals,ofquantumfieldtheory.Proofsofformaltopicslikerenormalizabilityorthequantizationofgaugefieldsarelefttootherbooks, asisthetopicofpartonphenomenology.Inaddition,thestudybycomputerof latticefieldtheoryisanextensiveandrapidlychangingdiscipline,whichwedo notattempttocover.Althoughitwouldbetemptingtodiscusssomeofthemany stimulatingideas,amongthemsupersymmetry,grandunification,andstringtheory,whichattempttodescribephysicsbeyondtheStandardModel,limitationsof spacepreventusfromdoingso.
Althoughthisbookbeginsgently,wedoassumethatthereaderalreadyhassome familiaritywithquantumfieldtheory.Asanaidtothosewholackfamiliaritywith path-integralmethods,weincludeapresentation,inAppendixA,whichtreatsthis
subjectinanintroductorymanner.Inaddition,weassumeaknowledgeofthebasic phenomenologyofparticlephysics.
Wehaveconstructedthematerialtobeofusetoawidespectrumofreaders whoareinvolvedwiththephysicsofelementaryparticles.Certainlyitcontains materialofinteresttoboththeoristandexperimentalistalike.Giventhetrendto incorporatetheStandardModelinthestudyofnuclei,weexpectthebooktobe ofusetothenuclearphysicscommunityaswell.Eventhestudentbeingtrainedin themathematicsofstringtheorywouldbewelladvisedtolearntherolethatsigma modelsplayinparticletheory.
Thisisagoodplacetostresssomeconventionsemployedinthisbook.Chaptersareidentifiedwithromannumerals.Incross-referencingequations,weinclude thechapternumberifthereferencedequationisinachapterdifferentfromthe pointofcitation.TheMinkowskimetricis gμν = diag {1, 1, 1, 1}.Throughout,weusethenaturalunits = c = 1,andchoose e> 0sothattheelectronhaselectriccharge e .WeemployrationalizedHeaviside–Lorentzunits,and thefine-structureconstantisrelatedtothechargevia α = e 2 /4π .Thecouplingconstantsforthe SU(3)c × SU(2)L × U(1) gaugestructureoftheStandardModel aredenotedrespectivelyas g3 ,g2 ,g1 ,andweemploycoupling-constantphase conventionsanalogoustoelectromagnetismfortheotherabelianandnonabelian covariantderivativesoftheStandardModel.Thechiralprojectionoperatorforlefthandedmasslessspinone-halfparticlesis (1 + γ5 )/2,andinanalyzingsystemsin d dimensions,weemploytheparameter ≡ (4 d)/2.Whatismeantbythe‘Fermi constant’isdiscussedinSect.V–2.
Amherst,MA,2013
InputstotheStandardModel
ThisbookisabouttheStandardModelofelementaryparticlephysics.Ifweset thebeginningofthemoderneraofparticlephysicsin1947,theyearthepionwas discovered,thentheensuingyearsofresearchhaverevealedtheexistenceofaconsistent,self-containedlayerofreality.Theenergyrangewhichdefinesthislayerof realityextendsuptoabout1TeVor,intermsoflength,downtodistancesoforder 10 17 cm.TheStandardModelisafield-theoreticdescriptionofstrongandelectroweakinteractionsattheseenergies.Itrequirestheinputofasmanyas28independentparameters.1 TheseparametersarenotexplainedbytheStandardModel; theirpresenceimpliestheneedforanunderstandingofNatureatanevendeeper level.Nonetheless,processesdescribedbytheStandardModelpossessaremarkableinsulationfromsignalsofsuchNewPhysics.Althoughthestronginteractions remainacalculationalchallenge,theStandardModel(generalizedfromitsoriginal formtoincludeneutrinomass)wouldappeartohavesufficientcontenttodescribe allexistingdata.2 Thusfar,itisatheoreticalstructurewhichhasworkedsplendidly.
I–1Quarksandleptons
TheStandardModelisan SU(3) × SU(2) × U(1) gaugetheorywhichisspontaneouslybrokenbytheHiggspotential.TableI–1displaysmassdeterminations [RPP12]ofthe Z 0 and W ± gaugebosons,theHiggsboson H 0 ,andtheexisting masslimitonthephoton γ
IntheStandardModel,thefundamentalfermionicconstitutentsofmatterarethe quarksandtheleptons.Quarks,butnotleptons,engageinthestronginteractions asaconsequenceoftheircolorcharge.Eachquarkandleptonhasspinone-half.
1 Therearesixleptonmasses,sixquarkmasses,threegaugecouplingconstants,threequark-mixingangles andonecomplexphase,threeneutrino-mixinganglesandasmanyasthreecomplexphases,aHiggsmass andquarticcouplingconstant,andthe QCD vacuumangle.
2 Admittedly,atthistimethesourcesof darkmatter andof darkenergy areunknown.
TableI–1. Bosonmasses.
ParticleMass(GeV/c 2 )
γ< 1 × 10 27
W ±
Z 0
H 0
80 385 ± 0 015
91.1876 ± 0.0021
126.0 ± 0.4
Collectively,theydisplayconventionalFermi–Diracstatistics.Noattemptismade intheStandardModeleithertoexplainthevarietyandnumberofquarksandleptonsortocomputeanyoftheirproperties.Thatis,theseparticlesaretakenatthis levelastrulyelementary.Thisisnotunreasonable.Thereisnoexperimentalevidenceforquarkorleptoncompositeness,suchasexcitedstatesorformfactors associatedwithintrinsicstructure.
Quarks
Therearesixquarks,whichfallintotwoclassesaccordingtotheirelectricalcharge Q.The u,c,t quarkshave Q = 2e/3andthe d,s,b quarkshave Q =− e/3, where e istheelectricchargeoftheproton.The u,c,t and d,s,b quarks areeigenstatesofthehamiltonian(‘masseigenstates’).However,becausetheyare believedtobepermanentlyconfinedentities,somethoughtmustgointoproperly definingquarkmass.Indeed,severaldistinctdefinitionsarecommonlyused.We deferadiscussionofthisissueandsimplynotethatthevaluesinTableI–2provide
FlavorMassa (GeV/c 2 )Charge I3
u(2.55+0 75 1 05 ) × 10 3 2e/31/20000
d(5.04+0 96 1 54 ) × 10 3 e/3 1/20000
s 0.105+0.025 0.035 e/30 1000
c 1.27+0.07 0.11 2e/300100
b 4.20+0 17 0 07 e/3000 10
t 173.4 ± 1.62e/300001
a The t -quarkmassisinferredfromtopquarkevents.Allothersaredeterminedin MS renormalization(cf.Sect.II–1)atscales mu,d,s (2GeV/c 2 ), mc (mc ) and mb (mb ) respectively.
TableI–2. Thequarks.
TableI–3. Theleptons.
FlavorMass(GeV/c 2 )Charge Le Lμ Lτ
νe < 0.2 × 10 8 0100
e 5 10998928(11) × 10 4 e100
νμ < 1.9 × 10 4 0010
μ 0 1056583715(35) e010
ντ < 0 01820001
τ 1 77682(16) e001
anoverviewofthequarkmassspectrum.Ausefulbenchmarkforquarkmassesis theenergyscale QCD ( severalhundredMeV)associatedwiththeconfinement phenomenon.Relativeto QCD ,the u,d,s quarksarelight,the b,t quarksare heavy,andthe c quarkhasintermediatemass.Thedynamicalbehavioroflight quarksisdescribedbythechiralsymmetryofmasslessparticles(cf.Chap.VI) whereasheavyquarksareconstrainedbytheso-calledHeavyQuarkEffective Theory(cf.Sect.XIII–3).Eachquarkissaidtoconstituteaseparate flavor,i.e. sixquarkflavorsexistinNature.The s,c,b,t quarkscarryrespectivelythe quantumnumbersofstrangeness(S ),charm(C ),bottomness(B ),andtopness(T ).
The u,d quarksobeyan SU (2)symmetry(isospin)andaredistinguishedbythe three-componentofisospin(I3 ).Theflavorquantumnumbersofeachquarkare displayedinTableI–2.
Leptons
Therearesixleptonswhichfallintotwocategoriesaccordingtotheirelectrical charge.Thechargedleptons e,μ,τ have Q =− e andtheneutrinos νe ,νμ ,ντ have Q = 0.Leptonsarealsoclassifiedintermsofthreeleptontypes:electron (νe ,e),muon (νμ ,μ),andtau (ντ ,τ).Thisfollowsfromthestructureofthecharged weakinteractions(cf.Sect.II–3)inwhichthesecharged-lepton/neutrinopairsare coupledto W ± gaugebosons.Associatedwitheachleptontypeisaleptonnumber Le ,Lμ ,Lτ .TableI–3summarizesleptonproperties.
Atthistime,thereisonlyincompleteknowledgeofneutrinomasses.Information onthemassparameters mνe ,mνμ ,mντ isobtainedfromtheirpresenceinvarious weaktransitionamplitudes.Forexample,thesinglebetadecayexperiment 3 H → 3 He + e + ν e issensitivetothemass mνe .Inlikemanner,oneconstrainsthemasses mνμ and mντ inprocessessuchas π + → μ+ + νμ and τ → 2π + π + + ντ respectively.ExistingboundsonthesemassesaredisplayedinTableI–3.
Itisknownexperimentallythatuponcreationtheneutrinos {να }≡ (νe .νμ ,ντ ) willnotpropagateindefinitelybutwillinsteadmixwitheachother.Thismeansthat thebasisofstates {να } cannotbeeigenstatesofthehamiltonian.Diagonalization oftheleptonichamiltonianiscarriedoutinSect.VI–2andyieldsthebasis {νi }≡ {ν1 ,ν2 ,ν3 } ofmasseigenstates.Informationontheneutrinomasseigenvalues m1 ,m2 ,m3 isobtainedfromneutrinooscillationexperimentsandcosmological studies.Oscillationexperiments(cf.Sects.VI–3,VI–4)aresensitivetosquaredmassdifferences.3 Throughoutthebook,weadheretothefollowingrelations,
Fromthecompilationin[RPP12],thesquared-massdifference | m2 32 | deduced fromthestudyofatmosphericandacceleratorneutrinosgives
whereasdatafromsolarandreactorneutrinosimplyasquared-massdifference roughly31timessmaller,
Thustheneutrinos ν1 and ν2 formaquasi-doublet.Onespeaksofa normal or inverted neutrinomassspectrum,respectively,forthecases4
Sincethelargestneutrinomass mlgst ,beit m2 or m3 ,cannotbelighterthanthe masssplittingofEq.(1.2),wehavethebound mlgst > 0.049eV.Finally,acombinationofcosmologicalinputscanbeemployedtoboundtheneutrinomasssum 3 i = 1 mi ,theprecisebounddependingonthechoseninputdataset.Inoneexample[deP etal.12],photometricredshiftsmeasuredfromalargegalaxysample,cosmicmicrowavebackground(CMB)dataandarecentdeterminationoftheHubble parameterareusedtoobtainthebound
1 + m2 + m3 < 0.26eV, (1.3a) whereasdatafromtheCMBcombinedwiththatfrombaryonacousticoscillations yields[Ad etal. (Planckcollab.)13]
AfurtherdiscussionoftheneutrinomassspectrumappearsinSect.VI–4.
3 Onlytwoofthemassdifferencescanbeindependent,so m2 12 + m2 23 + m2 31 = 0.
4 Thereisalsothepossibilityofa quasi-degenerate neutrinomassspectrum(m1 m2 m3 ),whichcanbe thoughtofasalimitingcaseofboththenormalandinvertedcasesinwhichtheindividualmassesare sufficientlylargetodwarfthe | m2 32 | splitting.
Quarkandleptonnumbers
Individualquarkandleptonnumbersareknowntobenotconserved,butfordifferentreasonsandwithdifferentlevelsofnonconservation.IndividualquarknumberisnotconservedintheStandardModelduetothechargedweakinteractions (cf.Sect.II–3).Indeed,quarktransitionsofthetype qi → qj + W ± inducethe decaysofmostmesonandbaryonstatesandhaveledtothephenomenologyof FlavorPhysics.Individualleptonnumberisnotconserved,asevidencedbythe observed να ↔ νβ (α,β = e,μ,τ) oscillations.Thissourceofthisphenomenonis associatedwithnonzeroneutrinomasses.Thereiscurrentlynoadditionalevidence fortheviolationofindividualleptonnumberdespiteincreasinglysensitivelimits suchasthebranchingfractionBμ →e e e + < 1.0 × 10 12 .
Existingdataareconsistentwithconservationoftotalquarkandtotallepton number,e.g.theprotonlifetimebound τp > 2 1 × 1029 yr[RPP12]andthenuclear half-lifelimit t 0νββ 1/2 [136 Xe] > 1.6 × 1025 yr[Ac etal.(EXO-200collab.)11].These conservationlawsareempirical.Theyarenotrequiredasaconsequenceofany knowndynamicalprincipleandinfactareexpectedtobeviolatedbycertainnonperturbativeeffectswithintheStandardModel(associatedwithquantumtunneling betweentopologicallyinequivalentvacua–seeSect.III–6).
I–2Chiralfermions
Consideraworldinwhichquarksandleptonshavenomassatall.Atfirst,this wouldappeartobeasurprisingsupposition.Toanexperimentalist,massisthe mostpalpablepropertyaparticlehas.Itiswhy,say,amuonbehavesdifferently fromanelectroninthelaboratory.Nonetheless,themasslesslimitiswherethe StandardModelbegins.
Themasslesslimit
Let ψ(x) beasolutiontotheDiracequationforamasslessparticle, i/∂ψ = 0 (2.1)
Wecanmultiplythisequationfromtheleftby γ5 andusetheanticommutativityof γ5 with γ μ toobtainanothersolution, i/∂γ5 ψ = 0. (2.2)
Wesuperposethesesolutionstoformthecombinations ψL = 1 2 (1 + γ5 )ψ,ψR = 1 2 (1 γ5 )ψ, (2.3)
InputstotheStandardModel
where‘1’representstheunit4 × 4matrix.Thequantities ψL and ψR aresolutions ofdefinite chirality (i.e.handedness).Foramasslessparticlemovingwithprecise momentum,thesesolutionscorrespondrespectivelytothespinbeinganti-aligned (left-handed)andaligned(right-handed)relativetothemomentum.Inotherwords, chiralitycoincideswithhelicityforzero-massparticles.Thematrices L R = (1 ± γ5 )/2arechiralityprojectionoperators.Theyobeytheusualprojectionoperator conditionsunderaddition,
andundermultiplication,
Inthemasslesslimit,aparticle’schiralityisaLorentz-invariantconcept.For example,aparticlewhichisleft-handedtooneobserverwillappearleft-handedto allobservers.Thuschiralityisanaturallabeltouseformasslessfermions,anda collectionofsuchparticlesmaybecharacterizedaccordingtotheseparatenumbers ofleft-handedandright-handedparticles.
Itissimpletoincorporatechiralityintoalagrangianformalism.Thelagrangian foramasslessnoninteractingfermionis
orintermsofchiralfields,
Thelagrangians LL,R areinvariantundertheglobalchiralphasetransformations
wherethephases αL,R areconstantandreal-valuedbutotherwisearbitrary.AnticipatingthediscussionofNoether’stheoreminSect.I–4,wecanassociateconserved particle-numbercurrentdensities J μ L,R ,
withthisinvariance.Fromthesechiralcurrentdensities,wecanconstructthevector current V μ (x),
andtheaxial-vectorcurrent Aμ (x),
= J μ L J μ R . (2.12)
Chiralcharges QL,R aredefinedasspatialintegralsofthechiralchargedensities, QL,R (t) = d 3 xJ 0 L,R (x), (2.13)
andrepresentthenumberoperatorsforthechiralfields ψL,R .Theyaretimeindependentifthechiralcurrentsareconserved.Onecansimilarlydefinethevector charge Q andtheaxial-vectorcharge Q5 ,
= d 3 xV 0 (x),Q5 (t) = d 3 xA0 (x). (2.14)
Thevectorcharge Q isthetotalnumberoperator,
= QR + QL , (2.15) whereastheaxial-vectorchargeisthenumberoperatorforthedifference
5 = QL QR (2.16)
Thevectorcharge Q andaxial-vectorcharge Q5 simplycountthesumanddifference,respectively,oftheleft-handedandright-handedparticles.
Parity,timereversal,andchargeconjugation
ThefieldtransformationsofEq.(2.9)involveparameters αL,R whichcantakeon acontinuumofvalues.Inadditiontosuchcontinuousfieldmappings,oneoften encountersavarietyof discrete transformationsaswell.Letusconsidertheoperationsofparity
andoftimereversal
asdefinedbytheireffectsonspacetimecoordinates.Theeffectofdiscretetransformationsonafermionfield ψ(x) willbeimplementedbyaunitaryoperator P forparityandanantiunitaryoperator T fortimereversal.Intherepresentationof Diracmatricesusedinthisbook,wehave
Anadditionaloperationtypicallyconsideredinconjunctionwithparityandtime reversalisthatofchargeconjugation,themappingofmatterintoantimatter,
TableI–4. ResponseofDiracbilinearsto discretemappings.
CPT
S(x)S(xP )S(xT ) P(x) P(xP ) P(xT ) J μ (x)Jμ (xP )Jμ (xT ) J μ 5 (x) J5μ (xP )J5μ (xT ) T μν (x)Tμν (xP ) Tμν (xT )
where ψ T β ≡ ψ † α γ 0 αβ (α,β = 1,..., 4).Thespacetimecoordinatesoffield ψ(x) are unaffectedbychargeconjugation.
Inthestudyofdiscretetransformations,theresponseofthenormal-ordered Diracbilinears
S(x) =: ψ(x)ψ(x) : P(x) =: ψ(x)γ5 ψ(x) : J μ (x) =: ψ(x)γ μ ψ(x) : J μ 5 (x) =: ψ(x)γ μ γ5 ψ(x) : T μν (x) =: ψ(x)σ μν ψ(x) : (2.21)
isofspecialimportancetophysicalapplications.Theirtransformationproperties appearinTableI–4.Closeattentionshouldbepaidtheretothelocationofthe indicesintheserelations.Anotherexampleofafield’sresponsetothesediscrete transformationsisthatofthephoton Aμ (x),
CAμ (x)C 1 c =−Aμ (x),PAμ (x)P 1 = Aμ (xP ),
TAμ (x)T 1 c
BeginningwiththediscussionofNoether’stheoreminSect.1–4,weshallexplore thetopicofinvariancethroughoutmuchofthisbook.Itsufficestonoteherethat theStandardModel,beingatheorywhosedynamicalcontentisexpressedinterms ofhermitian,Lorentz-invariantlagrangiansoflocalquantumfields,isguaranteed tobeinvariantunderthecombinedoperation CPT .Interestingly,however,these discretetransformationsareindividuallysymmetryoperationsonlyofthestrong andelectromagneticinteractions,butnotofthefullelectroweaksector.Wesee alreadythepossibilityforsuchbehaviorintheoccurrenceofchiralfermions ψL,R , sinceparitymapsthefields ψL,R intoeachother,
Thusanyeffect,liketheweakinteraction,whichtreatsleft-handedandrighthandedfermionsdifferently,willleadinevitablytoparity-violatingphenomena.
I–3Fermionmass
Althoughthediscussionofchiralfermionsiscastinthelimitofzeromass,fermions inNaturedoinfacthavenonzeromassandwemustaccountforthis.Inalagrangian,amasstermwillappearasahermitian,Lorentz-invariantbilinearinthefields. Forfermionfields,theseconditionsallowrealizationsreferredtoas Dirac massand Majorana mass.5
Diracmass
TheDiracmasstermforfermionfields ψL,R involvesthebilinearcouplingoffields withoppositechirality
where ψ ≡ ψL + ψR and mD istheDiracmass.TheDiracmasstermisinvariantunderthephasetransformation ψ(x) → exp( iα)ψ(x) andthusdoesnot upsetconservationofthevectorcurrent V μ = ψγ μ ψ andthecorrespondingnumberfermionoperator Q ofEq.(2.15).AllfieldsintheStandardModel,savepossiblyfortheneutrinos,haveDiracmassesobtainedfromtheirinteractionwiththe Higgsfield(cf.Sects.II–3,II–4).Althoughright-handedneutrinoshavenocouplingstotheStandardModelgaugebosons,thereisnoprincipleprohibitingtheir interactionwiththeHiggsfieldandthusgeneratingneutrinoDiracmassesinthe samemannerastheotherparticles.
Majoranamass
AMajoranamasstermisonewhichviolatesfermionnumberbycouplingtwo fermions(ortwoantifermions).IntheMajoranaconstruction,useismadeofthe charge-conjugate fields,
where C isthecharge-conjugationoperator,obeying
IntheDiracrepresentationofgammamatrices(cf.App.C),onehas C = iγ 2 γ 0 . Someusefulidentitiesinvolving ψ c include 5 Wesuppressspacetimedependenceofthefieldsinthissection.
