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OXFORDMASTERSERIESINPHYSICS
TheOxfordMasterSeriesisdesignedforfinalyearundergraduateandbeginninggraduatestudentsinphysicsand relateddisciplines.Ithasbeendrivenbyaperceivedgapintheliteraturetoday.Whilebasicundergraduatephysics textsoftenshowlittleornoconnectionwiththehugeexplosionofresearchoverthelasttwodecades,moreadvanced andspecializedtextstendtoberatherdauntingforstudents.Inthisseries,alltopicsandtheirconsequencesare treatedatasimplelevel,whilepointerstorecentdevelopmentsareprovidedatvariousstages.Theemphasisison clearphysicalprincipleslikesymmetry,quantummechanics,andelectromagnetismwhichunderliethewholeof physics.Atthesametime,thesubjectsarerelatedtorealmeasurementsandtotheexperimentaltechniquesand devicescurrentlyusedbyphysicistsinacademeandindustry.Booksinthisseriesarewrittenascoursebooks,and includeampletutorialmaterial,examples,illustrations,revisionpoints,andproblemsets.Theycanlikewisebeused aspreparationforstudentsstartingadoctorateinphysicsandrelatedfields,orforrecentgraduatesstartingresearch inoneofthesefieldsinindustry.
CONDENSEDMATTERPHYSICS
1.M.T.Dove: Structureanddynamics:anatomicviewofmaterials
2.J.Singleton: Bandtheoryandelectronicpropertiesofsolids
3.A.M.Fox: Opticalpropertiesofsolids,secondedition
4.S.J.Blundell: Magnetismincondensedmatter
5.J.F.Annett: Superconductivity,superfluids,andcondensates
6.R.A.L.Jones: Softcondensedmatter
17.S.Tautz: Surfacesofcondensedmatter
18.H.Bruus: Theoreticalmicrofluidics
19.C.L.Dennis,J.F.Gregg: Theartofspintronics:anintroduction
21.T.T.Heikkila: Thephysicsofnanoelectronics:transportandfluctuationphenomenaatlowtemperatures
22.M.Geoghegan,G.Hadziioannou: Polymerelectronics
25.R.Soto: Kinetictheoryandtransportphenomena
ATOMIC,OPTICAL,ANDLASERPHYSICS
7.C.J.Foot: Atomicphysics
8.G.A.Brooker: Modernclassicaloptics
9.S.M.Hooker,C.E.Webb: Laserphysics
15.A.M.Fox: Quantumoptics:anintroduction
16.S.M.Barnett: Quantuminformation
23.P.Blood: Quantumconfinedlaserdevices
PARTICLEPHYSICS,ASTROPHYSICS,ANDCOSMOLOGY
10.D.H.Perkins: Particleastrophysics,secondedition
11.Ta-PeiCheng: Relativity,gravitationandcosmology,secondedition
24.G.Barr,R.Devenish,R.Walczak,T.Weidberg: ParticlephysicsintheLHCera
STATISTICAL,COMPUTATIONAL,ANDTHEORETICALPHYSICS
12.M.Maggiore: Amodernintroductiontoquantumfieldtheory
13.W.Krauth: Statisticalmechanics:algorithmsandcomputations
14.J.P.Sethna: Statisticalmechanics:entropy,orderparameters,andcomplexity
20.S.N.Dorogovtsev: Lecturesoncomplexnetworks
Conceptsof ElementaryParticlePhysics
MichaelE.Peskin SLAC,StanfordUniversity
(versionofApril2,2019) CLARENDONPRESS . OXFORD
Preface
Thisisatextbookofelementaryparticlephysics,intendedforstudents whohaveasecureknowledgeofspecialrelativityandhavecompleted anundergraduatecourseinquantummechanics.
Particlephysicshasnowreachedtheendofamajorstageinitsdevelopment.Theprimaryforcesthatactwithintheatomicnucleus,the strongandweakinteractions,nowhaveafundamentaldescription,with equationsthataresimilarinformtoMaxwell’sequations.Theseforces aresummarizedinacompactmathematicaldescription,calledtheStandardModelofparticlephysics.Thepurposeofthisbookistoexplain whattheStandardModelisandhowitsvariousingredientsarerequired bytheresultsofelementaryparticleexperiments.
Increasingly,thereisagapbetweenthestudyofelementaryparticles andotherareasofphysicalscience.Whileotherareasofphysicsseemto applydirectlytomaterialsscience,modernelectronics,andevenbiology, particlephysicsdescribesanincreasinglyremoteregimeofverysmall distances.Physicistsinotherareasareputoffbythesheersizeand expenseofelementaryparticleexperiments,andbytheesoterictermsby whichparticlephysicistsexplainthemselves.Particlephysicsisbound upwithrelativisticquantumfieldtheory,ahighlytechnicalsubject,and thisaddstothedifficultyofunderstandingit.
Still,thereismuchtoappreciateinparticlephysicsifitcanbemade accessible.Particlephysicscontainsideasofgreatbeauty.Itreveals someofthemostdeepandsurprisingideasinphysicsthroughdirect connectionsbetweentheoryandexperimentalresults.Inthistextbook, IattempttopresentparticlephysicsandtheStandardModelinaway thatbringsthekeyideasforward.Ihopethatitwillgivestudentsan entrywayintothissubject,andwillhelpothersgainabetterunderstandingoftheintellectualvalueofourrecentdiscoveries.
Thepresentationofelementaryparticlephysicsinthisbookhasbeen shapedbymanyyearsofdiscussionwithexperimentalandtheoretical physicists.Particlephysicistsformaglobalcommunitythatbringstogethermanydifferentpointsofviewanddifferentnationalstyles.This diversityhasbeenakeysourceofnewideasthathavedriventhefield forward.Ithasalsobeenasourceofintuitivepicturesthatmakeitpossibletovisualizephysicalprocessesinthedistantandabstractdomainof thesubnuclearforces.Ihavetriedtobringasmanyofthesepicturesas possibleintomydiscussionhere.Myownwayofthinkingaboutparticle physicshasbeenshapedbymyconnectionwiththegreatlaboratories atCornellUniversityandSLAC.Iamindebtedtomanycolleaguesat
theselaboratoriesforcentralpartsofthedevelopmentgivenhere.
Ihavebeenremindedoftenduringthewritingofthisbookthatmany ofthegreatfiguresresponsiblefortheformulationoftheStandardModel havepassedontothatsymposiuminthebeyond.Inonlythepastfew years,wehavelostSidneyDrell,MartinPerl,RichardTaylor,Kenneth Wilson,and,mostrecently,BurtonRichter.Allofthesepeopleinfluencedmepersonallyandprofoundlyaffectedmythinkingaboutparticle physics.Itisachallengeforuswhofollowthemnotonlytofinishtheir workbutalsotoopennewchaptersinthedevelopmentoffundamental physics.Ihopethatthisbookwillprovideusefulbackgroundforthose whowishtodoso.
Thecoreofthispresentationwasdevelopedasasetoflecturesfor CERNsummerstudentsin1997;IthankLuisAlvarez-Gaum´eforthe invitationtopresenttheselectures.Ihavepresentedpartsofthismaterialatanumberofsummerschoolsandcourses,inparticular,thecourse onelementaryphysicsatthePerimeterScholarsInternationalprogram atthePerimeterInstitute.Mostrecently,Ihavepolishedthismaterial bymyteachingofthecoursePhysics152/252atStanfordUniversity.I amgratefultoPatriciaBurchatforgivingmethisopportunity,andfor muchadviceonteachingacourseatthislevel.Ithankthestudentsin allofthesecoursesfortheirpatiencewithpreliminaryversionsofthis bookandtheirattentiontoerrorstheycontained.IthankSonkeAdlung,HarrietKonishi,SalMoore,andtheirteamatOxfordUniversity Pressfortheirinterestinthisproject.IthankTimCohen,SergeDendas,ChristopherHill,SunghoonJung,AndrewLarkoski,AaronPierce, DanielSchroeder,BruceSchumm,andAndr´eDavidTinocoforvaluable commentsonthepresentation,andJongminYoonforanespeciallycarefulreadingofthemanuscript.Mostofall,Ithankmycolleaguesinthe SLACTheoryGroupfortheiradviceandcriticismthathasbenefited myunderstandingofelementaryparticlephysics.
MichaelE.Peskin Sunnyvale,CA August,2018
IPreliminariesandTools1
3RelativisticWaveEquations23
3.1TheKlein-Gordonequation
4TheHydrogenAtomandPositronium39
5TheQuarkModel49
5.1Thediscoveryofthehadrons
6DetectorsofElementaryParticles71
11QuantumChromodynamics165
12.2Thestructureofjets 187 Exercises 192
13QCDatHadronColliders195
13.1Hadronscatteringatlowmomentumtransfer 195 13.2Hadronscatteringatlargemomentumtransfer 200
13.3Jetstructureobservablesforhadroncollisions 205
13.4Thewidthofajetinhadron-hadroncollisions 206 13.5Productionofthetopquark 210 Exercises 212
14ChiralSymmetry215
14.1SymmetriesofQCDwithzeroquarkmasses 215 14.2Spontaneoussymmetrybreaking 217 14.3Goldstonebosons 221
14.4Propertiesof π mesonsasGoldstonebosons 222 Exercises 226
IIITheWeakInteraction229
15TheCurrent-CurrentModeloftheWeakInteraction231
15.1DevelopmentoftheV Atheoryoftheweakinteraction 232 15.2PredictionsoftheV Atheoryforleptons 233 15.3PredictionsoftheV Atheoryforpiondecay 241 15.4PredictionsoftheV Atheoryforneutrinoscattering 243 Exercises 246
16GaugeTheorieswithSpontaneousSymmetryBreaking249 16.1Fieldequationsforamassivephoton 249 16.2Modelfieldequationswithanon-Abeliangaugesymmetry 251 16.3TheGlashow-Salam-Weinbergelectroweakmodel 253 16.4Theneutralcurrentweakinteraction 258 Exercises 261
17The W and Z Bosons263
17.1Propertiesofthe W boson 263
17.2 W productionin pp collisions 266 17.3Propertiesofthe Z boson 268 17.4Precisiontestsoftheelectroweakmodel 269 Exercises 278
18QuarkMixingAnglesandWeakDecays281
18.1TheCabibbomixingangle 281
18.2QuarkandleptonmasstermsintheStandardModel 283
18.3Discretespace-timesymmetriesandtheStandardModel 285 18.4TheStandardModelofparticlephysics 287 18.5Quarkmixingincludingheavyquarks 289 Exercises 291
Introduction
Theaimofthisbookistodescribetheinteractionsofnaturethatact onelementaryparticlesatdistancesofthesizeofanatomicnucleus.
Atthistime,physicistsknowaboutfourdistinctfundamentalinteractions.Twoofthesearemacroscopic—gravityandelectromagnetism. GravityhasbeenknownsincethebeginningofhistoryandhasbeenunderstoodquantitativelysincethetimeofNewton.Electricalandmagneticphenomenahavealsobeenknownsinceancienttimes.Theunified theoryofelectromagnetismwasgivenitsdefinitiveformbyMaxwellin 1865.Throughallofthesedevelopments,therewasnosignthatthere couldbeadditionalfundamentalforces.Thesewouldappearonlywhen physicistscouldprobematteratverysmalldistances.
ThefirstevidenceforadditionalinteractionsofnaturewasBequerel’s discoveryofradioactivityin1896.In1911,Rutherforddiscoveredthat theatomconsistsofelectronssurroundingaverytiny,positivelycharged nucleus.Asphysicistslearnedmoreaboutatomicstructure,itbecame increasinglyclearthattheknownmacroscopicforcesofnaturecouldnot givethefullexplanation.Bythemiddleofthe20thcentury,experiments hadrevealedaseriesofquestionsthatcouldnotberesolvedwithoutnew particlesandinteractions.Theseincluded:
• Whatisradioactivity?Whydosomeatomicnucleiemithighenergyparticles?Whatspecificreactionsareresponsible?What aretheparticlesthatareemittedinradioactivedecay?
• Whatholdstheatomicnucleustogether?Thenucleusismadeof positivelychargedprotonsandneutralneutrons.Electromagnetic forcesdestabilizethenucleus—asweseefromthefactthatheavy nucleiareunstablewithrespecttofission.Whatisthecounterbalancingattractiveforce?
• Whatareprotonsandneutronsmadeof?Theseparticleshave propertiesthatindicatethattheyarenotelementarypointlike particles.Whatgivesthemstructure?Whatkindsofparticlesare inside?
Experimentsdesignedtostudytheseissuesproducedmoreconfusion beforetheyproducedmoreunderstanding.Theprotonandtheneutron turnedouttobethefirstofhundredsofparticlesinteractingthrough thenuclearforce.Theelectronturnedouttobeonlyoneofthreeapparentlypointlikeparticleswithelectricchargebutnostronginteractions. Alloftheseparticleswereobservedtointeractwithoneanotherthrough awebofnew,short-rangedinteractions.Finally,asthe1960’sturnedto
Thesesimplequestionsgivethestarting pointfortheexplorationofsubnuclear physics.
Itisimportanttorememberthetheory ofparticlephysicsmustbestudiedtogetherwiththeunderstandingofhow experimentsaredoneandhowtheirresultsareinterpreted.
the1970’s,thenewinteractionsweresortedintotwobasicforces—called thestrongandtheweakinteraction—andsimplemathematicalexpressionsfortheseforceswereconstructed.Today,physicistsrefertothese expressionscollectivelyas“theStandardModelofparticlephysics”.
Sometimes,authorsorlecturerspresentthetableofelementaryparticlesoftheStandardModelandimplythatthisisallthereistothe story.Itisnot.Thewaythattheforcesofnatureactontheelementary particlesisbeautifulandintricate.Often,thetellingdetailsofthese interactionsshowupthroughremarkableaspectsofthedatawhenwe examineelementaryparticlebehaviorexperimentally.
Theseideaselicitarelatedquestion:Ofallthewaysthatnaturecould bebuilt,howdoweknowthattheStandardModelisthecorrectone? Itseemshardlypossiblethatwecouldpindowntheexactnatureofnew fundamentalinteractionsbeyondgravityandelectromagnetism.Allof thephenomenaassociatedwiththenewforcesoccuratdistancessmaller thananatomicnucleus,andinaregimewherebothspecialrelativity andquantummechanicsplayanessentialrole.
Inthisbook,Iwillexplaintheanswerstothesequestions.Itturns outthatthenewforceshavecommonpropertiesandcanbebuiltup fromsimpleingredients.Thepresenceoftheseingredientsisrevealed bywell-chosenexperiments.Thedynamicsofthenewinteractionsbecomesmoreclearathigherenergies.Withthebenefitofhindsight,we canbeginourstudytodaybystudyingthesedynamicalingredientsin theirsimplestform,workingouttheconsequencesoftheselaws,and comparingtheresultingformulaetodatafromhighenergyaccelerator experimentsthatillustratethecorrectnessoftheseformulaeinavery directway.
Ourquestforafundamentaltheoryofnatureisfarfromcomplete. Inthefinalchapterofthebook,Iwilldiscussanumberofissuesabout fundamentalforcesforwhichwestillhavenounderstanding.Itisalso possible,asweprobemoredeeplyintothestructureofnature,thatwe willuncovernewinteractionsthatworkatevensmallerdistancesthan thosecurrentlyexplored.But,atleast,onechapterofthestory,open since1896,isnowfinished.Ihopethat,workingthroughthisbook, youwillnotonlyunderstandhowtoworkwiththeunderlyingtheories describingthestrongandweakinteractions,butalsothatyouwillbe amazedatthewealthofevidencethatsupportstheconnectionofthese theoriestotherealworld.
ThebookisorganizedintothreeParts.PartIintroducesthebasic Outlineofthebook. materialsthatwewillusetoprobethenatureofnewforcesatshort distances.PartsIIandIIIusethisasafoundationtobuildupthe StandardModeltheoriesofthestrongandweakinteractions.
PartIbeginswithbasictheorythatunderliesthesubjectofparticle
PartI physics.Evenbeforeweattempttowritetheoriesofthesubnuclear forces,weexpectthatthosetheorieswillobeythelawsofquantum mechanicsandspecialrelativity.Iwillprovidesomemethodsforusing theseimportantprinciplestomakepredictionsabouttheoutcomeof elementaryparticlecollisions.
Inaddition,Iwilldescribethetypesofmatterinthetheoriesof strongandweakinteractions,thebasicelementaryparticlesthatinteractthroughtheseforces.Itturnsoutthattherearetwotypesof matterparticlesthatareelementaryatthelevelofourcurrentunderstanding.Ofthese,onetype,the leptons,areseeninourexperimentsas individualparticles.Therearesixknownleptons.Threehaveelectric charge:theelectron(e),themuon(µ),andthe τ lepton.Theotherthree arethe neutrinos,particlesthatareelectricallyneutralandextremely weaklyinteracting.Despitethis,theevidenceforneutrinosasordinary relativisticparticlesisverypersuasive;IwilldiscussthisinPartIII.
Matterparticlesoftheothertype,the quarks,arehiddenfromview. Quarksappearasconstituentsofparticlessuchasprotonsandneutrons thatinteractthroughthestronginteraction.Therearemanyknown stronglyinteractingparticles,collectivelycalled hadrons.Iwillexplain thepropertiesofthemostprominentones,andshowthattheyarenaturallyconsideredinfamilies.Ontheotherhand,noexperimenthas everseenanisolatedquark.ItisactuallyapredictionoftheStandard Modelthatquarkscanneverappearsingly.Thismakesitespecially challengingtolearntheirproperties.Onepieceofevidencethatthe descriptionofquarksintheStandardModeliscorrectisfoundfromthe factitgivesasimpleexplanationforthequantumnumbersofobserved hadronsandtheirassortmentintofamilies.Iwilldiscussthisalsoin PartI.Intheprocess,Iwillgivenamestothehadronsthatappear mostofteninexperiments,sothatwecandiscussexperimentalmethods moreconcretely.
Inarelativisticquantumtheory,forcesarealsoassociatedwithparticlesthatcanbethoughttotransmitthem.TheStandardModelcontains fourtypesofsuchparticles.Thesearethe photon,thecarrieroftheelectromagneticinteraction,the gluon,thecarriersofthestronginteraction, the W and Z bosons,thecarriersoftheweakinteraction,andthe Higgs boson,whichplaysamoresubtlerole.Youwillhavealreadyencounteredthephotoninyourstudyofquantummechanics.Iwillintroduce thegluoninPartIIandthe W , Z,andHiggsbosonsinPartIII.
Tounderstandexperimentalfindingsaboutelementaryparticles,we willneedtoknowatleastthebasicsofhowexperimentsonelementary particlesaredone,andwhatsortsofquantitiesdescribingtheirpropertiesaremeasureable.IwilldiscussthismaterialalsoinPartI.
PartIIbeginswithadiscussionofthemostimportantexperiments PartII thatgiveinsightintotheunderlyingcharacterofthestronginteraction. Onemightguessintuitivelythatthemostconvincingdataonthestrong interactioncomesfromthestudyofcollisionsofhadronswithother hadrons.Thatisincorrect.Theexperimentsthatweremostcrucialin understandingthenatureofstronginteractioninvolvedelectronscatteringfromprotonsandtheannihilationofelectronsandpositronsathigh energy.Thislatterprocesshasainitialstatewithnohadronsatall. IwillbeginPartIIwithadiscussionofthefeaturesoftheseprocesses athighenergy.Ouranalysiswillintroducetheconceptofthe currentcurrentinteraction,whichisanessentialpartofthephysicsofboththe
strongandweakinteractions.Then,throughaseriesofargumentsthat passbackandforthbetweentheoryandexperiment,wewillexplorethe natureofhadron-hadroncollisionsathighenergy,asrevealedtodayin experimentsattheLargeHadronCollider.
ThefinalchapterofPartIIpresentsourcurrentunderstandingofthe massesofquarks.Itmightseemthatitisstraightforwardtomeasure themassofaquark,butinfactthisquestionbringsinanumberof new,subtleconcepts.Thischapterintroducestheimportantideaof spontaneoussymmetrybreaking,andotherideasthatwillprovetobe essentialpartsofthetheoryoftheweakinteraction.
PartIIIpresentsthedescriptionoftheweakinteraction.HereIwill
beginfromaproposalforthenatureoftheweakinteractionthatuses theconceptofthecurrent-currentinteractionthathasalreadyproven itsworthinthedescriptionofthestronginteraction.Iwillpresent somequitecounterintuitive,andevenstartling,predictionsofthattheoryandshowthattheyareactuallyreproducedbyexperiment.From thisstartingpoint,againindialoguebetweentheoryandexperiment, wewillbuildupthefulltheory.Mydiscussionwillincludetheprecision studyofthecarriersoftheweakinteraction,the W and Z bosons,and thenewestingredientsinthistheory,themassesofneutrinosandthe propertiesoftheHiggsboson.
Thisisnotacompletetextbookofelementaryparticlephysics.In general,IwillconcentrateonthesimplestapplicationsoftheStandard Model,theapplicationsthatmaketheunderlyingstructureofthemodel mostclear.MostoftheprocessesthatIwillconsiderwillbestudiedin thelimitofveryhighenergies,wherethemathematicalanalysiscanbe simplifiedasmuchaspossible.Afulldiscussionofthesubjectwould coveramorecompletelistofreactions,includingsomewhosetheoretical analysisisquitecomplex.Suchafulltreatmentofparticlephysicsis beyondthescopeofthisbook.
Inparticular,manyaspectsofthetheoryofelementaryparticlescannotbeunderstoodwithoutadeepunderstandingofquantumfieldtheory.Thisbookwillexplainthoseaspectsofquantumfieldtheorythat areabsolutelynecessaryforthepresentation,butwillomitanysophisticateddiscussionofthissubject.Afulldescriptionofthepropertiesof elementaryparticlesneedsmore.
Forstudentswhowouldliketostudyfurtherinparticlephysics,there aremanyexcellentreferenceswrittenfromdifferentandcomplementary pointsofview.Ihaveputalistofthemostusefultextsatthebeginning oftheReferences.
Aparticularlyusefulreferenceworkisthe ReviewofParticlePhysics assembledbytheParticleDataGroup(Patrignani etal. 2016).This volumecompilesthebasicpropertiesofallknownelementaryparticles andprovidesup-to-datereviewsofthemajortopicsinthissubject.All elementaryparticlemassesandotherphysicalquantitiesquotedinthis bookbutnotexplicitlyreferencedaretakenfromthesummarytables giveninthatsource.
SymmetriesofSpace-Time 2
Wedonothavecompletefreedominpostulatingnewlawsofnature.Any lawsthatwepostulateshouldbeconsistentwithwell-establishedsymmetriesandinvarianceprinciples.Ondistancescalessmallerthananatom, space-timeisinvariantwithrespecttotranslationsofspaceandtime. Space-timeisalsoinvariantwithrespecttorotationsandboosts,the symmetrytransformationsofspecialrelativity.Manyaspectsofexperimentsonelementaryparticlestesttheprinciplesofenergy-momentum conservation,rotationalinvariance,theconstancyofthespeedoflight, andthespecial-relativityrelationofmass,momentum,andenergy.So far,nodiscrepancyhasbeenseen.Soitmakessensetoapplythesepowerfulconstraintstoanyproposalforelementaryparticleinteractions.
Perhapsyouconsiderthisstatementtoostrong.Asweexplorenew realmsinphysics,wemightwelldiscoverthatthebasicprinciplesapplied inmorefamiliarsettingsarenolongervalid.Intheearly20thcentury, realcrisesbroughtonbytheunderstandingofatomsandlightforced physiciststoabandonNewtonianspace-timeinfavorofthatofEinstein andMinkowski,andtoabandontheprinciplesofclassicalmechanicsin favoroftheverydifferenttoolsofquantummechanics.Bysettingrelativityandquantummechanicsasabsoluteprinciplestoberespectedin thesubnuclearworld,wearemakingaconservativechoiceoforientation.Therehavebeenmanysuggestionsofmoreradicalapproachesto formulatinglawsofelementaryparticles.Someofthesehaveevenledto newinsights:The bootstrap ofGeoffreyChew,inwhichthereisnofundamentalHamiltonian,isstillfindingnewapplicationsinquantumfield theory(Simmons-Duffin2017); stringtheory,whichradicallymodifies space-timestructure,isacandidatefortheoverallunificationofparticleinteractionswithquantumgravity(Zwiebach2004,Polchinski2005). However,themostsuccessfulroutestothetheoryofsubnuclearinteractionshavetakentranslationinvariance,specialrelativity,andstandard quantummechanicsasabsolutes.Inthisbook,Iwillmaketheassumptionthatspecialrelativityandquantummechanicsarecorrectinthe realmofelementaryparticleinteractions,andIwillusetheirprinciples inastrongwaytoorganizemyexplorationofelementaryparticleforces. Thisbeingso,itwillbeusefultoformulatetheconstraintsfromspacetimesymmetriesinsuchawaythatwecanapplythemeasily.We wouldliketousetheactualtransformationlawsassociatedwiththese symmetriesaslittleaspossible.Instead,weshouldformulatequestions insuchawaythattheanswersareexpressions invariant underspacetimesymmetries.Generally,therewillbeasmallandwell-constrained
Representationoftheenergyandmomentumofaparticlein4-vectornotation.
setofpossibleinvariants.Ifwearelucky,onlyoneofthesewillbe consistentwithexperiment.
2.1Relativisticparticlekinematics
Asafirststepinsimplifyingtheuseofconstraintsfromspecialrelativity,Iwilldiscussthekinematicsofparticleinteractions.Anyisolatedparticleischaracterizedbyanenergyandavectormomentum.In specialrelativity,theseareunifiedintoa4-vector.Iwillwriteenergymomentum4-vectorsinenergyunitsandnotatethemwithanindex µ =0, 1, 2, 3,
(2.1)
Iwillnowreviewaspectsoftheformalismofspecialrelativity.Probablyyouhaveseentheseformulaebeforeintermsofrulers,clocks,and movingtrains.Nowwewillneedtousetheminearnest,becauseelementaryparticlecollisionsgenerallyoccuratenergiesatwhichitisessential touserelativisticformulae.
Underaboostby v alongthe ˆ 3direction,theenergy-momentum 4-vectortransformsas p → p ,with
Itisconvenienttowritethisasamatrixtransformation
Inthisbook,unlessitisexplicitlyindicatedotherwise,repeatedindicesare summedover.Thisconventionisone ofEinstein’slesser,butstillmuchappreciated,innovations.
Inmultiplyingmatricesandvectorsinthisbook,Iwillusetheconventionthatrepeatedindicesaresummedover.Then,forexample,I willwrite(2.3)as
omittingtheexplicitsummationsignfortheindex ν.Lorentztrans-
formationsleaveinvarianttheMinkowskispacevectorproduct
Tokeeptrackoftheminussigninthisproduct,Iwillmakeuseof raisedandloweredLorentzindices.Lorentztransformationspreserve themetrictensor
Usingthismatrix,andthesummationconvention,wecanwrite(2.6)as
Alternatively,let q withaloweredindexbedefinedby
Theinvariantproductof p and q iswritten
Toformaninvariant,wealwayscombinearaisedindexwithalowered index.Astheequationsinthisbookbecomemorecomplex,wewill findthistrickveryusefulinkeepingtrackoftheMinkowskispaceminus signs.
AparticularlyimportantLorentzinvariantisthesquareofaLorentz vector,
Beinganinvariant,thisquantityisindependentofthestateofmotion oftheparticle.Intherestframe
IwilluseraisedandloweredLorentz indicestokeeptrackoftheminus signintheMinkowskivectorproduct.Pleasepayattentiontothepositionofindices—raisedorlowered— throughoutthisbook.
Iwilldefinethemassofaparticleasitsrest-frameenergy
ThemassofaparticleisaLorentzinvariantquantitythatcharacterizes thatparticleinanyreferenceframe. (mc 2) ≡ E0 (2.13)
Since p2 isaninvariant,theexpression
istrueinanyframeofreference.
Inthisbook,Iwillwriteparticlemomentaintwostandardways
where
Ep = c(|p|2 +(mc)2)1/2 ,β = |p|c Ep ,γ =(1 β2)1/2 . (2.16)
Especially,thesymbol Ep willalwaysbeusedinthisbooktorepresent thisstandardfunctionofmomentumandmass.Iwillrefertoa4-vector with E = Ep asbeing“onthemassshell”.
Toillustratetheseconventions,Iwillnowworkoutsomesimplebut importantexercisesinrelativistickinematics.Imaginethataparticleof mass M ,atrest,decaystotwolighterparticles,ofmasses m1 and m2.In thesimplestcase,bothparticleshavezeromass: m1 = m2 =0.Then, energy-momentumconservationdictatesthatthetwoparticleenergies
Definitionsofthequantities Ep, β, γ associatedwithrelativisticparticlemotion.
Thesekinematicformulaewillbeused veryofteninthisbook.
areequal,withthevalue Mc2/2.Then,ifthefinalparticlesmoveinthe ˆ 3direction,wecanwritetheir4-vectorsas
Thenextcase,whichwillappearoftenintheexperimentswewill consider,isthatwith m1 nonzerobut m2 =0.Intherestframeofthe originalparticle,themomentaofthetwofinalparticleswillbeequal andopposite.Withalittlealgebra,onecandetermine
(formotioninthe ˆ 3direction),where
Itiseasytocheckthattheseformulaesatisfytheconstraintsoftotalenergy-momentumconservationandthat p
satisfiesthemass-shell constraint(2.14).
Finally,wemightconsiderthegeneralcaseofnonzero m1 and m2 Here,ittakesalittlemorealgebratoarriveatthefinalformulae
with
and
wherethekinematic λ functionisdefinedby
Thesethreesetsofformulaeapplyequallywelltoreactionswithtwo particlesintheinitialstateandtwoparticlesinthefinalstate.Itis onlynecessarytoreplace Mc2 withthecenterofmassenergy ECM of thereaction.
2.2Naturalunits
Inthediscussionofthepreviouschapter,Ineededtointroducemany factorsof c inordertomakethetreatmentofenergy,momentum,and massmoreuniform.Thisisafactoflifeinthedescriptionofhigh energyparticles.Ideally,weshouldtakeadvantageoftheworldviewof relativitytopassseamlesslyamongtheseconcepts.Equallywell,our discussionsofparticledynamicswilltakeplaceinaregimeinwhich quantummechanicsplaysanessentialrole.Tomakethebestuseof
quantumconcepts,weshouldbeabletopasseasilybetweentheconcepts ofmomentumandwavenumber,orenergyandfrequency.
Tomakethesetransitionsmosteasily,Iwill,inthisbook,adopt naturalunits, h = c =1 (2.24)
Thatis,Iwillmeasuremomentumandmassinenergyunits,andIwill measuredistancesandtimesininverseunitsofenergy.Forconvenience
Theconventionsthatdefine natural units indiscussingelementaryparticlephysics,IwilltypicallyusetheenergyunitsMeVorGeV.Thiswilleliminateagreatdealofunnecessary baggagethatwewouldotherwiseneedtocarryaroundinourformulae.
Forexample,towritethemassoftheelectron,Iwillwrite not me =0 91 × 10 27gbutrather me =0 51MeV (2.25)
Anelectronwithamomentumoftheorderofitsrestenergyhas,accordingtotheHeisenberguncertaintyprinciple,apositionuncertainty
h mec =3 9 × 10 11 cm , (2.26) whichIwillequallywellwriteas
Naturalunitsmakeitveryintuitivetoestimateenergies,lengths,and timesintheregimeofelementaryparticlephysics.Forexample,the
Naturalunitsareusefulforestimation. lighteststronglyinteractingparticle,the π meson,hasamass mπc 2 =140MeV (2.28)
Thiscorrespondstoadistance h mπc =1 4 × 10 13 cm (2.29) andatime h mπ
Thesegive—withinafactor2orso—thesizeoftheprotonandthe
Thematerialinthisbookwillbeeasier tograspifyoumakeyourselfcomfortablewiththeuseofnaturalunits.This willbothsimplifyformulaeandsimplify manyestimatesofenergies,distances, andtimes. lifetimesoftypicalunstablehadrons.So,theuseof mπ givesagood firstestimateofalldimensionfulstronginteractionquantities.Toobtain anestimateinthedesiredunits—MeV,cm,sec—wewoulddecorate thesimpleexpression mπ withappropriatefactorsof¯h and c andthen evaluateasabove.
Itmaymakeyouuncomfortableatfirsttodiscardfactorsof¯h and c. Getusedtoit.Thatwillmakeitmucheasierforyoutoperformcalculationsofthesortthatwewilldointhisbook.Someusefulconversion factorsformovingbetweendistance,time,andenergyunitsaregivenin AppendixB.
Theintrinsicstrengthsofthebasicelementaryparticleinteractionsarenot apparentfromthesizeoftheireffect— orfromtheirnames.Hereisapreview.
Oneinterestingquantitytoputintonaturalunitsisthestrengthof theelectricchargeoftheelectronorproton.TheCoulombpotentialis giveninstandardnotationby
Iwilluseunitsforelectromagnetisminwhichalso
ThentheCoulombpotentialreads
Since r,innaturalunits,hasthedimensionsof(energy) 1,thevalueof theelectricchargemusthaveaforminwhichitisdimensionless.Indeed,
isadimensionlessnumber,calledthe
,withthe value
Therearetworemarkablethingsaboutthisequation.First,itissurprisingthatthereisadimensionlessnumber α thatcharacterizesthe strengthoftheelectromagneticinteraction.Second,thatnumberis small,signallingthattheelectromagneticinteractionisaweakinteraction.Oneofthegoalsofthisbookwillbetodeterminewhetherthe
strongandweaksubnuclearinteractionscanbecharacterizedinthesame way,andwhethertheseinteractions—lookingbeyondtheirnames—are intrinsicallystrongorweak.Iwilldiscussestimatesofthestrongand weakinteractioncouplingstrengthsatappropriatepointsinthecourse. Itwillturnoutthatthestronginteractionisweak,atleastwhenmeasuredunderthecorrectconditions.Itwillalsoturnoutthattheweak Grouptheoryplaysanimportantrole inquantummechanics,andthisimportanceextendstothestudyofelementaryparticlephysics.Youhave encounteredgrouptheoryconceptsin yourquantummechanicscourse,butit islikelythatthoseargumentsdidnot makeexplicitreferencetogrouptheoryconcepts.Inparticlephysics,we leanmuchmoreheavilyongrouptheory,andsoitisbesttodiscussthese conceptsformallyandgivethemtheir propernames.Please,then,studySections2.3and2.4carefully,especially ifyouareuncomfortablewithmathematicabstraction.Withcarefulreading,youwillseethattheconceptsI describegeneralizephysicalarguments thatarealreadyfamiliartoyou.
interactionisalsoweakindimensionlessterms.Itisweakerthanthe stronginteractions,butnotasweakaselectromagnetism.
2.3Alittletheoryofdiscretegroups
Grouptheoryisaveryimportanttoolforelementaryparticlephysics. Inthissectionandthenext,Iwillreviewhowgrouptheoryisusedin quantummechanics,andIwilldiscusssomepropertiesofgroupsthat wewillmeetinthisbook.Forthemostpart,thesesectionswillreview materialthatyouhaveseeninyourquantummechanicscourse.But, becausetherewillbemanyappealstogrouptheoryconceptsinthis book,itwillbebesttoputtheseconceptsclearlyinorder.Forthis reason,thesetwosectionswillberatherpreciseandformal.Thislevel ofprecisionwillpayoffasweusetheseideasinmanyexamples.
Inquantummechanics,wedealwithgroupsontwolevels.First, thereareabstractgroups.Inmathematics,a group isasetofelements G = {a,b,...} withamultiplicationlawdefined,sothat ab isdefinedand isanelementof G.Themultiplicationlawsatisfiesthethreeproperties Herearetheaxiomsthatdefinea group.
(1) Multiplicationisassociative: a(bc)=(ab)c.
(2) G containsan identityelement 1suchthat,foranyelementof G, 1a = a1= a
(3) Foreach a in G,thereisanotherelement a 1 suchthat aa 1 = a 1a =1.
Everysymmetryofnaturenormallyencounteredinphysicssatisfiesthese axiomsandisdescribedbyanabstractgroup.
Inquantummechanics,thebasicelementsarevectors(or,quantum states)inaHilbertspace.Symmetriesconvertoneofthesestatesto anotherbyaunitarytransformation.Thephysicsproblemweare TheactionofagroupontheHilbert spaceofstatesinquantummechanicsis describedthroughunitaryrepresentationsofthegroup.Thus,unitarygroup representationswillbeusedinmanyaspectsofthephysicsdiscussedinthis book. interestedinisdescribedbyaHamiltonian H whoseeigenvaluesgive theenergylevels.Asymmetryoftheproblemisimplementedbya unitarytransformation U.If[U ,H]=0,stateslinkedby U havethe sameenergy.
ThisrelationbetweensymmetriesoftheHamiltonianandunitaryoperatorsgivesspecialimportancetothefollowingconstruction:Forany group G withelements {a},wecanfindunitarymatrices Ua thatobey themultiplicationlawofthegroup.Thatis,if a,b,c areelementsof G with ab = c,thenthecorrespondingmatricesobey UaUb = Uc (2.36)
bymatrixmultiplication.Inparticular,theunitarymatrixcorrespondingto1isthematrix 1,andtheunitarymatrixcorrespondingto a 1 isthematrix U 1 = U †.Thesetofmatrices {Ua} iscalleda unitary matrixrepresentation ofthegroup G.Thegroup G isasymmetryofthe Hamiltonian H ifthisgrouphasaunitaryrepresentation {Ua} acting ontheHilbertspacesuchthat,forall a,[Ua,H]=0.
Theseideasareeasiesttounderstandinthecontextofasmallset ofquantumstatesthatformafinite-dimensionalHilbertspace.The simplestexampleinvolvestheabstractgroupcalled Z2 thatcontains twoelements {1, 1} satisfyingthemultiplicationlaw
1 1=( 1)( 1)=11 ( 1)=( 1) 1=( 1) . (2.37)
Consider,then,aquantummechanicalsystemwithtwoparticles π+ and π .Definetheoperator C totransform Cπ+ = π,Cπ = π+ (2.38)
Theactionof C onthis2-dimensionalsubspaceisrepresentedbythe matrix
If H istheHamiltonianforthisquantum-mechanicalsystemand[C,H]= 0,thatwouldimplythatthemassesanddecayratesof π+ and π must beequal.OnthesameHilbertspace,wecandefinethetrivialoperation
Thisisrepresentedby
Theunitarymatrices {1,C} formaunitaryrepresentationofthegroup Z2.Ifthesematricescommutewith H,wesaythat H has Z2 symmetry. Wecandiscusstherelationof C to H anditseigenstateswithout makingexplicitreferencetothefactthattheunitarymatrix C represents agroup.However,usingthelanguageofgrouptheoryconnectsthis exampletoothersthatwemighthavestudied.Notallgroupsareas simpletounderstandas Z2,and,themorecomplicatedthegroup,the moreusefulthisconnectionis.
Agroup G iscalled Abelian if,forall a, b in G, ab = ba.Aunitarity representationofanAbeliangroup G consistsofunitarymatricesthat commutewithoneanother.Thismeansthattheycanbesimultaneously diagonalized.Theoperationofthegroupisthenreducedtosimple numbers.Intheexampleabove,thematrices(2.41)and(2.39)are AnAbeliangroupisdescribedbyits eigenstatesandtheireigenvalues.The eigenvaluesarepreciselywhatphysicistscallthe quantumnumbers ofa state.
diagonalizedinacommonbasis.Itisconventionaltouse C alsoasa symbolfortheeigenvalueof C ononeofitseigenstates.Inthiscase, theeigenstatesare
Because C 2 =1,operatingtwicewiththematrix C mustgivebackthe originalstate: CC |ψ = |ψ .Thismust,inparticular,betrueforan eigenstate.Thentheeigenvaluesof C canonlybe ±1.Wesaythatthe firststatein(2.42)has C =+1andthesecondhas C = 1. SymmetriesoftheHamiltonianmayinvolvetransformationsofspacetimecoordinates,suchasthespecialrelativitytransformationsdiscussed inSection2.1.Thesearecalled space-timesymmetries.Intheexamples liketheoneabove,thesymmetryrelatedifferentparticlesorquantum stateswithoutreferencetospace-time.Thesearecalled internalsymmetries.Agivenabstractgroupsuchas Z2 maydescribeaspace-time oraninternalsymmetry.
If G containstwoelements a,b thatdonotcommute, ab = ba,itis calleda non-Abelian group.If G isnon-Abelian,and {Ua} isaunitary representationof G,itisgenerallynotpossibletosimultaneouslydiagonalizealloftheunitarymatricesin {Ua}.However,byachangeof basis,wecanreducethesematricestoacommonblock-diagonalform
C =+1:[ π+ + π ]/√2
C = 1:[ π+ π ]/√2 . (2.42)
wheretheblocks U1,U2,U3, ··· areassmallaspossible.Theseminimalsizeunitarytransformationsrepresenting G arecalled irreducibleunitary representationsof G.Foranirreduciblerepresentation {Ui},thesizeof thematricesiscalledthe dimension di oftherepresentation.Thenotion ofirreduciblerepresentationsisprobablymorefamiliartoyouinthe contextofcontinuousgroups.Iwillputyourknowledgeoftherotation
Theconceptofan irreducible group representation.Manyphysicsproblems inquantummechanicsaresolvedby breakingupalargerHilbertspaceinto irreduciblerepresentationsofanappropriatesymmetrygroup. groupintothiscontextinthenextsection.
Itisastandardmathematicalproblemingrouptheorytoworkout thesetofirreduciblerepresentationsofagroup G thatareinequivalent byunitarytransformations.Itcanbeprovedthat,foradiscretegroup G with n elements,theinequivalentunitarytransformationssatisfy
AnexampleisgivenbythegroupofΠ3 ofpermutationsonthree elements.Wecanrepresentsuchapermutationastheresultoftransformingthesetoflabels[123]toasetoflabelsinanotherorder.With thisrepresentation,thegrouphas6elementsthatcanbewritten
{ [123] , [231] , [312] , [132] , [321] , [213] } (2.45)
Permutationsmultiply ab = c bycomposition,forexample,
[231] · [231]=[312]
[132] · [312]=[321] . (2.46)
Thatis,applyingthetwopermutationsinorder(righttoleft)givesthe resultingpermutationasshown.
The6permutationsin(2.45)canbeassociatedwith6statesina Hilbertspace.Inthisrepresentation,therepresentationmatricesare 6 × 6matriceswithentries0and1.Itcanbeshownthatthisisa reduciblerepresentation.Itcontainstwo1-dimensionalirreduciblerepresentations.Oneoftheseisthetrivialrepresentationthatmultiplies eachelementby1.Anotheristherepresentationthatmultipliesastate by+1foranevenorcyclicpermutation—thefirstthreeelementsof (2.45)—andmultipliesastateby 1foranoddpermutation—thelast threeelementsof(2.45).Thereisalsoone2-dimensionrepresentation, presentedinProblem2.3.Thesethreeirreduciblerepresentationstogethersatisfy(2.44).
2.4Alittletheoryofcontinuousgroups
Theconceptsreviewedintheprevioussectionextendtothesituation ofgroupswithacontinoussetofelements.Importantexamplesare thebasicspace-timesymmetries:thegroupofspatialtranslations,the groupofspatialrotations,andthegroupofLorentztransformations, whichincludesrotationsandboosts.
