September2012notesacademiclectures Contents1Relativisticwaveequations111TheKleinGordonequation112ModeexpansionofsolutionsoftheKGequation213S ID: 501178
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DIRACANDMAJORANAFERMIONSP.J.MuldersNikhefandDepartmentofPhysicsandAstronomy,FacultyofSciences,VUUniversity,1081HVAmsterdam,theNetherlandsE-mail:mulders@few.vu.nl September2012(notesacademiclectures) Contents1Relativisticwaveequations11.1TheKlein-Gordonequation.................................11.2ModeexpansionofsolutionsoftheKGequation.....................21.3SymmetriesoftheKlein-Gordonequation.........................22ThePoincareGroup42.1TheLorentzgroup......................................42.2ThegeneratorsofthePoincaregroup............................53TheDiracequation63.1TheLorentzgroupandSL(2C)..............................63.2Spin1=2representationsoftheLorentzgroup.......................73.3Generalrepresentationsof\rmatricesandDiracspinors.................83.4Planewavesolutions.....................................114Classicallagrangian eldtheory164.1Euler-Lagrangeequations..................................164.2Lagrangiansforspin0and1/2 elds............................175Quantizationof elds195.1Therealscalar eld.....................................195.2Thecomplexscalar eld...................................205.3TheDirac eld........................................216Discretesymmetries236.1Parity.............................................236.2Chargeconjugation......................................246.3Timereversal.........................................257Thestandardmodel287.1Thestartingpoint:SU(2)W\nU(1)Y............................287.2FamilymixingintheHiggssectorandneutrinomasses..................331 2ReferencesAsmostdirectlyrelatedbookstothesenotes,IrefertothebookofSrednicki[1]andRyder[2].OthertextbooksofQuantumFieldTheorythatareusefularegiveninrefs[3-6].ThenotesarepartofthelecturenotesforthecourseonQuantumFieldTheory(nextfullcoursescheduledNovember2012throughJanuary2013).1.M.Srednicki,QuantumFieldTheory,CambridgeUniversityPress,2007.2.L.H.Ryder,QuantumFieldTheory,CambridgeUniversityPress,1985.3.M.E.PeskinandD.V.Schroeder,AnintroductiontoQuantumFieldTheory,Addison-Wesly,1995.4.M.Veltman,Diagrammatica,CambridgeUniversityPress,1994.5.S.Weinberg,Thequantumtheoryof elds;Vol.I:Foundations,CambridgeUniversityPress,1995;Vol.II:ModernApplications,CambridgeUniversityPress,1996.6.C.ItzyksonandJ.-B.Zuber,QuantumFieldTheory,McGraw-Hill,1980. Chapter1Relativisticwaveequations1.1TheKlein-GordonequationWelookattheKlein-Gordon(KG)equationtodescribeafreespinlessparticlewithmassMwitha' eld' +M2 (r;t)=2 @t2 r2+M2 (r;t)=0(1.1)Althoughitisstraightforwardto ndthesolutionsofthisequation,namelyplanewavescharacterizedbyawavenumberk k(r;t)=exp( ik0t+ikr)(1.2)with(k0)2=k2+M2,theinterpretationofthisequationasasingle-particleequationinwhich isacomplexwavefunctionposesproblemsbecausetheenergyspectrumisnotboundedfrombelowandtheprobabilityisnotpositivede nite.Theenergyspectrumisnotboundedfrombelow:consideringtheabovestationaryplanewavesolutions,oneobtainsk0=p k2+M2=Ek(1.3)i.e.therearesolutionswithnegativeenergy.Probabilityisnotpositive:inquantummechanicsonehastheprobabilityandprobabilitycurrent= (1.4)j= i 2M( r (r ) ) i 2M $r :(1.5)Theysatisfythecontinuityequation,@ @t= rj(1.6)whichfollowsdirectlyfromtheSchrodingerequation.Thiscontinuityequationcanbewrittendowncovariantlyusingthecomponents(;j)ofthefour-currentjj=0(1.7)Therefore,relativisticallythedensityisnotascalarquantity,butratherthezerocomponentofafourvector.TheappropriatecurrentcorrespondingtotheKGequation(seeExcercise2.2)isj=i $ or(;j)=i $0 ; i $r (1.8)1 Relativisticwaveequations2Itiseasytoseethatthiscurrentisconservedif (and )satisfytheKGequation.TheKGequation,however,isasecondorderequationand and@ =@tcanbe xedarbitrarilyatagiventime.Thisleadstotheexistenceofnegativedensities.Theseproblemsarerelatedandhavetodowiththeexistenceofparticlesandantiparticles,forwhichweneedtheinterpretationof itselfasanoperator,ratherthanasawavefunction.Thisoperatorhasallpossiblesolutionsinitmultipliedwithcreation(andannihilation)operators.Atthatpointthedependenceonpositionrandtimetisjustadependenceonnumbers/parametersonwhichtheoperatordepends,justasthedependenceontimewasinordinaryquantummechanics.Then,therearenolongerfundamentalobjectionstomixupspaceandtime,whichiswhatLorentztransformationsdo.And,itissimplyamatterofbeingcarefulto ndaconsistent(covariant)theory.1.2ModeexpansionofsolutionsoftheKGequationBeforequantizing elds,havingtheKGequationasaspace-timesymmetric(classical)equation,wewantthemostgeneralsolution.Forthiswenotethatanarbitrarysolutionforthe eld canalwaysbewrittenasasuperpositionofplanewaves, (x)=Zd4k (2)42(k2 M2)e ikx~ (k)(1.9)with(inprinciplecomplex)coecients~ (k).Theintegrationoverk-modesclearlyiscovariantandrestrictedtothe`mass'-shell(asrequiredbyEq.1.1).Itispossibletorewriteitasanintegrationoverpositiveenergiesonlybutthisgivestwoterms(usetheresultofexercise2.3), (x)=Zd3k (2)32Eke ikx~ (Ekk)+eikx~ ( Ek k)(1.10)Introducing~ (Ekk)a(k)and~ ( Ek k)b(k)onehas (x)=Zd3k (2)32Ek e ikxa(k)+eikxb(k)= +(x)+ (x)(1.11)InEqs1.10and1.11onehaselimatedk0andinbothequationskx=Ekt kx.Thecoecientsa(k)andb(k)aretheamplitudesofthetwoindependentsolutions(two,afterrestrictingtheenergiestobepositive).Theyarereferredtoasmodeandanti-modeamplitudes(orbecauseoftheiroriginpositiveandnegativeenergymodes).Thechoiceofaandballowsaneasierdistinctionbetweenthecasesthat isreal(a=b)orcomplex(aandbareindependentamplitudes).1.3SymmetriesoftheKlein-GordonequationWewillexplicitlydiscusstheexampleofadiscretesymmetry,forwhichweconsiderspaceinversion,i.e.changingthesignofthespatialcoordinates,whichimplies(x)=(t;x)!(t; x)(~x)(1.12)TransformingeverywhereintheKGequationx!~xoneobtains~~+M2 (~x)=0(1.13)Sinceab=~a~b,itiseasytoseethat +M2 (~x)=0(1.14) Relativisticwaveequations3implyingthatforeachsolution (x)thereexistsacorrespondingsolutionwiththesameenergy, P(x) (~x)(Pforparity).Itiseasytoshowthat P(x)= (~x)=Zd3k (2)32Ek e ik~xa(k)+eik~xb(k)=Zd3k (2)32Eke i~kxa(k)+ei~kxb(k)=Zd3k (2)32Ek e ikxa( k)+eikxb( k)(1.15)orsinceonecande ne P(x)Zd3k (2)32Ek e ikxaP(k)+eikxbP(k)(1.16)onehasforthemodeamplitudesaP(k)=a( k)andbP(k)=b( k).Thisshowshowparitytrans-formsk-modesinto kmodes.AnothersymmetryisfoundbycomplexconjugatingtheKGequation.Itistrivialtoseethat +M2 (x)=0(1.17)showingthatwitheachsolutionthereisacorrespondingchargeconjugatedsolution C(x)= (x).Intermsofmodesonehas C(x)= (x)=Zd3k (2)32Ek e ikxb(k)+eikxa(k)Zd3k (2)32Ek e ikxaC(k)+eikxbC(k)(1.18)i.e.forthemodeamplitudesaC(k)=b(k)andbC(k)=a(k).Forthereal eldonehasaC(k)=a(k).Thisshowshowchargeconjugationtransforms'particle'modesinto'antiparticle'modesandviceversa. Chapter2ThePoincareGroup2.1TheLorentzgroupSpinhasbeenintroducedasarepresentationoftherotationgroupSU(2)withoutworryingmuchabouttherestofthesymmetriesoftheworld.Weconsideredthegeneratorsandlookedforrepresentationsin nitedimensionalspaces,e.g.=2inatwo-dimensional(spin1/2)case.InthissectionweconsiderthePoincaregroup,consistingoftheLorentzgroupandtranslations.TheLorentztransformationsaredividedintorotationsandboosts.Rotationsaroundthez-axisaregivenbyR(';^z)=exp(i'J3),in nitesimallygivenbyR(';^z)I+i'J3.Thus8���������:V00V10V20V309���������;=8���������:10000cos'sin'00 sin'cos'000019���������;8���������:V0V1V2V39���������; !J3=8���������:000000 i00i0000009���������;(2.1)Boostsalongthez-directionaregivenbyB(;^z)=exp( iK3),in nitesimallygivenbyB(;^z)I iK3.Thus8���������:V00V10V20V309���������;=8���������:cosh00sinh01000010sinh00cosh9���������;8���������:V0V1V2V39���������; !K3=8���������:000i00000000i0009���������;(2.2)Theparameterrunsfrom 11.Notethatthevelocity =v=v=candtheLorentzcontractionfactor\r=(1 2) 12correspondingtotheboostarerelatedtoas\r=cosh \r=sinh.Usingtheseexplicittransformations,wehavefoundthegeneratorsofrotations,J=(J1;J2;J3),andthoseoftheboosts,K=(K1;K2;K3),whichsatisfythecommutationrelations(check!)!)Ji;Jj]=iijkJkkJi;Kj]=iijkKkkKi;Kj]= iijkJkThe rsttwosetsofcommutationrelationsexhibittherotationalbehaviorofJandKasvectorsinE(3)underrotations.Fromthecommutationrelationsoneseesthattheboosts(pureLorentztransformations)donotformagroup,sincethegeneratorsKdonotformaclosedalgebra.Thecommutatoroftwoboostsindi erentdirections(e.g.thedi erenceof rstperformingaboostinthey-directionandthereafterinthex-directionandtheboostsinreversedorder)containsarotation(intheexamplearoundthez-axis).ThisistheoriginoftheThomasprecession.4 ThePoincareGroup52.2ThegeneratorsofthePoincaregroupForthefullPoincaregroup,includingthetranslations,writingthegeneratorP=(H=c;P)intermsoftheHamiltonianandthethree-momentumoperators,oneobtainssPi;Pj]=[Pi;H]=[Ji;H]=00Ji;Jj]=iijkJkkJi;Pj]=iijkPkkJi;Kj]=iijkKkkKi;H]=iPiiKi;Kj]= iijkJk=c22Ki;Pj]=iijH=c2(2.3)Wehaveherereinstatedc,becauseonethenseesthatbylettingc!1thecommutationrelationsoftheGalileigroup,knownfromnon-relativisticquantummechanicsareobtained.Inthatcaseboostsandrotationsdecouple!ExercisesExercise3.7(optional)OnemightwonderifitisactuallypossibletowritedownasetofoperatorsthatgeneratethePoincaretransformations,consistentwiththe(canonical)commutationrelationsofaquantumtheory.Thisispossibleforasingleparticle.Dothisbyshowingthatthesetofoperators,H=p p2c2+m2c4P=pJ=rp+sK=1 2c2(rH+Hr) tp+ps H+mc2satisfythecommutationrelationsofthePoincaregroupiftheposition,momentumandspinoperatorssatisfythecanonicalcommutationrelations,[ri;pj]=iijand[si;sj]=iijksk;theothersvanish,,ri;rj]=[pi;pj]=[ri;sj]=[pi;sj]=0.Hint:fortheHamiltonian,show rsttheoperatoridentity[r;f(p)]=irpf(p);ifyoudon'twanttodothisingeneral,youmightjustcheckrelationsinvolvingJorKbytakingsome(relevant)explicitcomponents.Comment:extendingthistomoreparticlesisahighlynon-trivialprocedure,butitcanbedone,althoughthepresenceofaninteractiontermV(r1r2)inevitablyleadstointeractiontermsintheboostoperators.Thesedonotmatterinthenon-relativisticlimit(c!1),that'swhymany-particlenon-relativisticquantummechanicsis'easy'. Chapter3TheDiracequation3.1TheLorentzgroupandSL(2;C)InsteadofthegeneratorsJandKofthehomogeneousLorentztransformationswecanusethe(hermitean)combinationsA=1 2(J+iK)(3.1)B=1 2(J iK)(3.2)whichsatisfythecommutationrelationssAi;Aj]=iijkAk(3.3))Bi;Bj]=iijkBk(3.4))Ai;Bj]=0(3.5)ThisshowsthattheLiealgebraoftheLorentzgroupisidenticaltothatofSU(2)\nSU(2).Thistellsushowto ndtherepresentationsofthegroup.Theywillbelabeledbytwoangularmomentacorre-spondingtotheAandBgroups,respectively,(j;j0).Specialcasesarethefollowingrepresentations:TypeI:(j;0)K= iJ(B=0)(3.6)TypeII:(0;j)K=iJ(A=0)(3.7)Fromtheconsiderationsabove,italsofollowsdirectlythattheLorentzgroupishomeomorphicwiththegroupSL(2C),similarlyasthehomeomorphismbetweenSO(3)andSU(2).ThegroupSL(2C)isthegroupofcomplex22matriceswithdeterminantone.ItissimplyconnectedandformsthecoveringgroupofL+.ThematricesinSL(2C)canbewrittenasaproductofaunitarymatrixUandahermiteanmatrixHM=expi 2'exp1 2=U(')H()U(') H()(3.8)with=^nand'='^n,wherewerestrict(for xed^n)theparameters0'2and01.Withthischoiceofparameter-spacestheplusandminussignsareactuallyrelevant.Theypreciselycorrespondtothetwotypesofrepresentationsthatwehaveseenbefore,becomingthede ningrepresentationsofSL(2C):TypeI(denotedM):J= 2K= i 2(3.9)TypeII(denoted M):J= 2K=+i 2(3.10)6 TheDiracequation7Letusinvestigatethede ning(two-dimensional)representationsofSL(2C).Onede nesspinorsandtransformingsimilarlyunderunitaryrotations(Uy=U 1 U(Uy) 1=U)!U;!U;(3.11)U(')=exp(i'=2)butdi erentlyunderhermiteanboosts(Hy=H H(Hy) 1=H 1),namely !H;! H;(3.12)H()=exp(=2) H()=exp( =2)Consideringandasspinstatesintherest-frame,onecanuseaboosttotheframewithmomentump.ChoosingtheboostparameterssuchthatE=M\r=Mcosh()andp=M \r^n=Msinh()^ntheboostisgivenbyH(p)=exp 2=cosh 2+^nsinh 2=M+E+p p 2M(E+M)(3.13)(exercise4.2).AlsousefulistherelationH2(p)=~p=M=(E+p)=M,whereweusedthesetsoffouroperatorsde nedby(1)~(1 )(3.14)satisfyingTr(~)= 2gandTr()=2g=2(thematrices,thus,arenotcovariant!).3.2Spin1=2representationsoftheLorentzgroupBothrepresentations(01 2)and(1 20)ofSL(2C)aresuitableforrepresentingspin1/2particles.Therepresentations(01 2)and(1 20),furthermore,areinequivalent,i.e.theycannotbeconnectedbyaunitarytransformation.WithintheLorentzgroup,theycanbeconnected,butbyatransformationbelongingtotheclassP .Underparityonehas(01 2) !(1 20)(3.15)Innatureparityturns(often)outtobeagoodquantumnumberforelementaryparticlestates.Forthespin1/2representationsofthePoincaregroupincludingparitywe,therefore,mustcombinetherep-resentations,i.e.considerthefourcomponentspinorthattransformsunderaLorentztransformationasu=8��:9��; !8��:M()00 M()9��;8��:9��;(3.16)where M()=M() 1.Foraparticleatrestonlyangularmomentumisimportantandwecanchoose(0;m)=(0;m)= m,thewell-knowntwo-componentspinorforaspin1/2particle.TakingM()=H(p),theboostinEq.3.13,weobtainforthetwocomponentsofuwhichwewillrefertoaschiralrightandchiralleftcomponents,u(p;m)=8�����:uRuL9�����;=8�����:H(p)00 H(p)9�����;8�����: m m9�����;(3.17)withH(p)=E+M+p p 2M(E+M)(3.18)H(p)=E+M p p 2M(E+M)=H 1(p)(3.19) TheDiracequation8Itisstraightforwardtoeliminate mandobtainthefollowingconstraintonthecomponentsofu8�����:0H2(p)H 2(p)09�����;8�����:uRuL9�����;=8�����:uRuL9�����;(3.20)orexplicitlyinthesocalledWeylrepresentation8�����: ME+pE p M9�����;8�����:uRuL9�����;=0;;(3.21)whichisanexplicitrealizationofthe(momentumspace)Diracequation,whichingeneralisalinearequationinp(p\r M)u(p)(/p M)u(p)=0(3.22)where\rare44matricescalledtheDiracmatrices1Asinsection2wecanuseP=i@asarepresentationforthemomenta(translationoperators)infunctionspace.ThisleadstotheDiracequationfor (x)=u(p)e ipxincoordinatespace,(i\r M) (x)=0(3.23)whichisacovariant(linear) rstorderdi erentialequation.ItisofaformthatwealsoplayedwithinExercise2.7.Thegeneralrequirementsforthe\rmatricesarethuseasilyobtained.Applying(i\r+M)fromtheleftgives \r\r+M2 (x)=0(3.24)Sinceissymmetric,thiscanberewritten1 2f\r;\rg+M2 (x)=0(3.25)Toachievealsothattheenergy-momentumrelationp2=M2issatis edforu(p),onemustrequirethatfor (x)theKlein-Gordonrelation 2+M2 (x)=0isvalidforeachcomponentseparately).FromthisoneobtainstheCli ordalgebrafortheDiracmatrices,f\r;\rg=2g(3.26)suppressingontheRHStheidentitymatrixinDiracspace.TheexplicitrealizationappearinginEq.3.21isknownastheWeylrepresentation.Wewilldiscussanotherexplicitrealizationofthisalgebrainthenextsection.3.3Generalrepresentationsof\rmatricesandDiracspinorsThegeneralalgebrafortheDiracmatricesisf\r;\rg=2g(3.27)Twooftenusedexplicitrepresentationsarethefollowing2:Thestandardrepresentation:\r0=3\n1=8��:100 19��;;\rk=i2\nk=8��:0k k09��;;(3.28) 1Wede neforafourvectorathecontraction/a=a\r2Weusei\njwithbothandbeingthestandard22Paulimatrices. TheDiracequation9TheWeyl(orchiral)representation:\r0=1\n1=8��:01109��;;\rk= i2\nk=8��:0 kk09��;(3.29)Di erentrepresentationscanberelatedtoeachotherbyunitarytransformations,\r !S\rS 1(3.30) !S :(3.31)WenotethattheexplicitmatrixtakingusfromtheWeylrepresentationtothestandardrepresentation,(\r)S:R:=S(\r)W:R:S 1,isS=1 p 28��:111 19��;(3.32)Forallrepresentationsonehas\ry=\r0\r\r0(3.33)andanadjointspinorde nedby = y\r0(3.34)Anothermatrixwhichisoftenusedis\r5de nedas\r5=i\r0\r1\r2\r3= i\r0\r1\r2\r3=i 4!\r\r\r\r(3.35)Itsatis esf\r5;\rg=0andexplicitlyonehas(\r5)S:R:=1\n1=8��:01109��;(\r5)W:R:=3\n1=8��:100 19��;(3.36)ForinstanceintheWeylrepresentation(butvalidgenerally),itiseasytoseethatPR=L=1 2(1\r5)(3.37)areprojectionoperators,thatprojectoutchiralright/leftstates,whichinthecaseoftheWeylrepresentationarejusttheupperandlowercomponents.LorentzinvarianceTheLorentztransformationscanalsobewrittenintermsofDiracmatrices.Forexample,therotationandboostgeneratorsinWeylrepresentationinEqs3.9and3.10arerepresentedbymatricesSJ3=S12=1 28��:30039��;=i 44\r1;\r2]=i 2\r1\r2K3=S30=1 28��: i300i39��;=i 44\r3;\r0]=i 2\r3\r0andingeneralonehasthetransformation !L =exp i 2!S ;(3.38)withS=1 2=i 44\r;\r].Wenotethat !L and ! L 1,whileL 1\rL=\r.ThelatterassuresLorentzinvarianceoftheDiracequation(seeExercise4.4) TheDiracequation10ParityThereareanumberofsymmetriesintheDiracequation,e.g.parity.Itiseasytoconvinceoneselfthatif (x)isasolutionoftheDiracequation,(i/ M) (x)=0(3.39)onecanapplyspaceinversion,x=(t;x)!~x=(t; x)andviaafewmanipulationsobtainagaintheDiracequation(i/ M) p(x)=0(3.40)butwith p(x)p\r0 (~x)(Exercise4.7).Notethatwehave(asexpected)explicitlyinWeylrepresentationinDiracspace =8��:9��;P ! p=\r0 =8��:9��;(3.41)ChargeconjugationTheexistenceofpositiveandnegativeenergysolutionsimpliesanothersymmetryintheDiracequa-tion.Thissymmetrydoesnotchangethespin1/2nature,butitdoes,forinstance,reversethechargeoftheparticle.Aswithparitywelookforatransformation,calledchargeconjugation,thatbrings ! c,whichisagainasolutionoftheDiracequation.Startingwith(i/ M) (x)=0wenotethatbyhermiteanconjugatingandtransposingtheDiracequationoneobtains i\rT+M T(x)=0(3.42)InanyrepresentationamatrixCexist,suchthatC\rTC 1= \r(3.43)e.g.(C)S:R:= i\r2\r0=i1\n2=8��:0i2i209��;=8��:009��;(3.44)(C)W:R:= i\r2\r0=i3\n2=8��:i200 i29��;=8��:00 9��;(3.45)Thuswe ndbacktheDiracequation,(i/ M) c(x)=0(3.46)withthesolution c(x)=cC T(x)=cC\r0 (x)(3.47)wherecisanarbitrary(unobservable)phase,usuallytobetakenunity.Notethatthelaststep(relating cand isvalidinrepresentationswhere\r0isreal.SomepropertiesofCareC 1=CyandCT= C.Onehas c= TC 1.InS.R.(orW.R.)andallrepresentationsconnectedviaareal(uptoanoverallphase)matrixSCisrealandonehasC 1=Cy=CT= Cand[C;\r5]=0.Thelatterimpliesthattheconjugateofaright-handedspinor, cR,isaleft-handedspinor.Explicitly,inWeylrepresentationwe ndinDiracspace =8��:9��;C ! c=C T=8��: 9��;(3.48) TheDiracequation113.4PlanewavesolutionsForafreemassiveparticle,thebestrepresentationtodescribeparticlesatrestisthestandardrepre-sentation,inwhich\r0isdiagonal(seediscussionofnegativeenergystatesinsection4.1).TheexplicitDiracequationinthestandardrepresentationreads8������:i@ @t Mir ir i@ @t M9������; (x)=0(3.49)Lookingforpositiveenergysolutions/exp( iEt)one ndstwosolutions, (x)=us(p)e ipx,withE=Ep=+p p2+M2,whereusatis es8�����:Ep M pp (Ep+M)9�����;u(p)=0,(/p M)u(p)=0(3.50)Therearealsotwonegativeenergysolutions, (x)=vs(p)eipx,wherevsatis es8�����: (Ep+M)p p(Ep M)9�����;v(p)=0,(/p+M)v(p)=0(3.51)Explicitsolutionsinthestandardrepresentationareu(p;s)=p Ep+M8������: sp Ep+M s9������;;v(p;s)=p Ep+M8������:p Ep+M s s9������;(3.52)where saretwoindependent(s=)two-componentspinors..Notethatthespinorsinthenegativeenergymodes(antiparticles)couldbetwodi erentspinors.Choosing = (theequivalentspin1/2conjugaterepresentation),thespinorssatisfyCuT(p;s)=v(p;s)andCvT(p;s)=u(p;s).Thesolutionsarenormalizedtou(p;s)u(p;s0)=2Mss0v(p;s)v(p;s0)= 2Mss0(3.53)u(p;s)v(p;s0)=v(p;s)u(p;s0)=0(3.54)uy(p;s)u(p;s0)=vy(p;s)v(p;s0)=2Epss0(3.55)Anarbitraryspin1/2 eldcanbeexpandedintheindependentsolutions.Afterseparatingpositiveandnegativeenergysolutionsasdoneinthecaseofthescalar eldonehas (x)=XsZd3k (2)32Ek u(k;s)e ikxb(k;s)+v(k;s)eikxd(k;s)(3.56)Itisstraightforwardto ndprojectionoperatorsforthepositiveandnegativeenergystatesP+=Xsu(p;s)u(p;s) 2M=/p+M 2M(3.57)P = Xsv(p;s)v(p;s) 2M= /p+M 2M(3.58)Inordertoprojectoutspinstates,thespinpolarizationvectorintherestframeisthestartingpoint.Inthatframeisaspacelikeunitvectors=(0^s).Inanarbitraryframeonehassp=0ands(p)cane.g.beobtainedbyaLorentztransformation.ItiseasytocheckthatPs=1\r5/s 2=1 28��:1^s001^s9��;(3.59) TheDiracequation12(thelastequalityintherestframeandinstandardrepresentation)projectsoutspin1=2states(checkthisintherestframefor^s=^z).NotethatforsolutionsofthemasslessDiracequation/p =0.Therefore,\r5/p =0butalso/p\r5 = \r5/p =0.Thismeansthatinthesolutionspaceformasslessfermionsthechiralitystates, R=LPR=L areindependentsolutions.Inprinciplemasslessfermionscanbedescribedbytwo-componentspinors.ThechiralityprojectionoperatorsinEq.3.37replacethespinprojectionoperatorswhicharenotde ned(bylackofarestframe).Explicitexamplesofspinorsareusefultoillustratespineigenstates,helicitystates,chirality,etc.Forinstancewiththez-axisasspinquantizationaxis,onehasinstandardrepresentation:u(p;+1=2)=1 p E+M8���������:E+M0p3p1+ip29���������;;u(p; 1=2)=1 p E+M8���������:0E+Mp1 ip2 p39���������;(3.60)v(p;+1=2)=1 p E+M8���������:p1 ip2 p30E+M9���������;;v(p; 1=2)=1 p E+M8���������:p3p1+ip2E+M09���������;(3.61)Helicitystates(palong^z)inStandardrepresentationare:u p;=+1 2=8���������:p E+M0p E M09���������;;u p;= 1 2=8���������:0p E+M0 p E M9���������;(3.62)v p;=+1 2=8���������:0 p E M0p E+M9���������;;v p;= 1 2=8���������:p E M0p E+M09���������;(3.63)BywritingthehelicitystatesinWeylrepresentationitiseasytoprojectoutrighthanded(uppercomponents)andlefthanded(lowercomponents).One ndsforthehelicitystatesu(p;)andv(p;)inWeylrepresentation:u(p;+)=1 p 28���������:p E+M+p E M0p E+M p E M09���������;;u(p; )=1 p 28���������:0p E+M p E M0p E+M+p E M9���������;v(p;+)=1 p 28���������:0p E+M p E M0 p E+M p E M9���������;;v(p; )=1 p 28���������:p E+M+p E M0 p E+M+p E M09���������;NotethatforhelicitystatesCuT(p;)= v(p;)andCvT(p;)=u(p;).IntroducingtheWeylhelicityspinorsUR=L(p;),UR(p;+) p 2E=8���������:10009���������;UR(p; ) p 2E=8���������:01009���������;UL(p;+) p 2E=8���������:00109���������;UL(p; ) p 2E=8���������:00019���������; TheDiracequation13wegetu(p;=+1=2)=p 1 2UR(p;+)+UL(p;+)u(p;= 1=2)=p 1 2UL(p; )+UR(p; )v(p;=+1=2)= p 1 2UL(p; )+UR(p; )v(p;= 1=2)=p 1 2UR(p;+) UL(p;+)where=p E+M p E M 2p E=M p E(p E+M+p E M)M E=)M 2Ewhichvanishesintheultra-relativisticlimitE Morinthemasslesscase.Athighenergypositivehelicityfermions(=+1=2)areinessencerighthanded,whilethenegativehelicityfermions(= 1=2)aremostlylefthanded.Foramasslessfermionright-andleft-handedsolutionscoincidewithhelicitystates.u p;=+1 2=v p;= 1 2=UR(p;+)andu p;= 1 2= v p;=+1 2=UL(p; )ExerciseInthischapterwehaveusedtworepresentations(StandardandWeyl)forthegammamatrices,basedonf\r;\rg=2gandanticommutingwith\r5=i\r0\r1\r2\r3,thusf\r5;\rg=0.Thehermiteanconjugatematricesobey\ry=\r0\r\r0andthetransposedmatricesarefoundusingCde nedasC\rTC 1= \r.WehaveStandardRepresentation\r0=3\n1=8��:100 19��;\rj=i2\nj=8��:0j j09��;\r5=1\n1=8��:01109��;C= i\r2\r0=i1\n2=8��:009��;C\r0=2\n2=8��:0 09��;Kj=i 2\r3\r0= i 21j=1 28��:0 ijij09��;WeylRepresentation\r0=1\n1=8��:01109��;\rj= i2\nj=8��:0 jj09��;\r5=3\n1=8��:100 19��;C= i\r2\r0=i3\n2=8��:00 9��;C\r0= 2\n2=8��:0 09��;Kj=i 2\r3\r0= i 23j=1 28��: ij00ij9��;Constructthesesamematricesin1+1dimensionanddiscusstheimplicationsforparity,chirality,particle-antiparticle,....(solution)Thegeneralrelationsaresimilarexceptfor\r5,whichnowisgivenby\r5=\r0\r1.Explicitlyonehas2-dimensionalmatricesgivenby TheDiracequation14StandardRepresentation\r0=3=8��:100 19��;\r1=i2=8��:01 109��;\r5=1=8��:01109��;C= \r1= i2=8��:0 1109��;C\r0=1=8��:01109��;K1=i 2\r3\r0= i 2\r5= i 21=1 28��:0 i i09��;WeylRepresentation\r0=1=8��:01109��;\rj= i2=8��:0 1109��;\r5=3=8��:100 19��;C= \r1=i2=8��:01 109��;C\r0=3=8��:100 19��;K1=i 2\r1\r0= i 2\r5= i 23=1 28��: i00i9��;Notethatthereisnogeneratorforrotationsin1+1dimensionandonlyoneboost.Excercise(a)ProveEq.3.13,H(p)=exp 2=M+E+p p 2M(E+M)whereE=M\r=Mcosh( )andp=M \r^n=Msinh( )^n.Forthisyouneedtoexpresscosh(=2)andsinh(=2)intermsofenergyandmomentum.AnothersimplecheckthatyoucanperformisthattheRHSindeedalsoisequaltoH2(p)=exp( )=(E+p)=M=~p=M:(b)ThefullboostoperatorinDiracspacecanbewrittenasexp(iK).WiththeexplicitmatricesinExercise4.1onee.g.immediatelyreproducestheresultinWeylrepresentation(Eq.3.17),exp(iK)=1 p 2M(E+M)8��:E+M+p00E+M p9��;GivethefullboostoperatorinStandardrepresentation.Checkthattheexplicitboostoperatorsappliedtoarest-framespinorimmediatelygivetheexplicitspinorsstartingwithEq.3.60.(solution)exp(iK)=1 p 2M(E+M)8��:E+MppE+M9��;(b)Constructforbothrepresentationstheexplicitboostoperatorsin1+1dimensionandgivetheexplicitspinorsu(jp)andv(jp)withp=+p E2 M2,discussingtheirnature(suchasparity,chirality,particle-antiparticle).(solution)TheboostoperatorinStandardRepresentationexp(iK)=1 p 2M(E+M)8��:E+MppE+M9��; TheDiracequation15andinWeylRepresentationexp(iK)=1 p 2M(E+M)8��:E+M+p00E+M p9��;givingStandardRepresentationspinorsu(jp)=8��:p E+Mp E M9��;v(jp)=8��:p E Mp E+M9��;andWeylRepresentationspinorsu(jp)=1 p 28��:p E+Mp E Mp E+Mp E M9��;v(jp)=1 p 28��:p E M+p E+Mp E M p E+M9��;TheWeylRepresentationresultsshowthatin1+1dimensionforM=0,right-handedfermionsareright-moversandleft-handedfermionsareleft-movers(ratherthanspeci chelicitystatesin3+1dimensions).ExcerciseApplyspace-inversion,x!~x,totheDiracequationandusethistoshowthatthespinor p(x)=\r0 (~x),where~x=(t; x)isalsoasolutionoftheDiracequation. Chapter4Classicallagrangian eldtheory4.1Euler-LagrangeequationsInclassical eldtheoryoneproceedsincompleteanalogytoclassicalmechanicsbutusingfunctionsdependingonspaceandtime(classical elds,thinkforinstanceofatemperatureordensitydistributionorofanelectromagnetic eld).ConsideralagrangiandensityLwhichdependsonthesefunctions,theirderivativesandpossiblyontheposition,L( (x);@ (x);x)andanactionSS ]=Zt2t1dtL=Zdtd3xL( (x);@ (x))=ZRd4xL( (x);@ (x))(4.1)HereRindicatesaspace-timevolumeboundedby(R3;t1)andR3;t2),alsoindicatedby@R(amoregeneralvolumeinfour-dimensionalspace-timewithsomeboundary@Rcanalsobeconsidered).Variationsintheactioncancomefromthecoordinatesorthe elds,indicatedasx0=x+x(4.2) 0(x)= (x)+ (x)(4.3)orcombined 0(x0)= (x)+ (x)(4.4)with (x)= (x)+( )x.TheresultingvariationoftheactionisS=ZRd4x0L( 0;@ 0;x0) ZRd4xL( ;@ ;x)(4.5)Thechangeinvariablesx!x0intheintegrationvolumeinvolvesasurfacevariationoftheformZ@RdLxNoteforthespeci cchoiceofthesurfaceforconstanttimest1andt2Z@Rd:::=Z(R3;t2)d3x::: Z(R3;t1)d3x::::(4.6)Furthermorethevariations and@ contributetoS,giving1S=ZRd4xL ( )( )+L +Z@RdLx=ZRd4xL L ( ) +Z@RdL ( ) +Lx(4.7) 1Takingafunctionalderivative,indicatedwithF[ ]= shouldposenoproblems.Wewillcomebacktoitinabitmoreformalwayinsection9.2.16 Classicallagrangian eldtheory17Withforthesituationofclassical eldsallvariationsofthe eldsandcoordinatesatthesurfacevanishing,thesecondtermisirrelevant.Theintegrandofthe rsttermmustvanish,leadingtotheEuler-Lagrangeequations,L ( )=L (4.8)4.2Lagrangiansforspin0and1/2 eldsByanappropriatechoiceoflagrangiandensitytheequationsofmotiondiscussedinpreviouschaptersforthescalar eld(spin0),theDirac eld(spin1/2)andthevector eld(spin1)canbefound.Thescalar eldItisstraightforwardtoderivetheequationsofmotionforarealscalar eld fromthelagrangiandensities,L=1 2 @ 1 2M2 2(4.9)= 1 2 +M2 ;(4.10)whichdi eronlybysurfaceterms,leadingto(2+M2) (x)=0(4.11)Forthecomplexscalar eldoneconventionallyusesL= M2 (4.12)= +M2 ;(4.13)whichcanbeconsideredasthesumofthelagrangiandensitiesfortworealscalar elds 1and 2with =( 1+i 2)=p 2.Oneeasilyobtains(2+M2) (x)=0(4.14)(2+M2) (x)=0(4.15)TheDirac eldTheappropriatelagrangianfromwhichtoderivetheequationsofmotionisL=i 2 $/@ M =i 2 !/@ i 2 /@ M (4.16)= (i/ M) ;(4.17)wherethesecondlineisnotsymmetricbutintheactiononlydi ersfromthesymmetricversionbyasurfaceterm(partialintegration).Usingthevariationsin (inthesymmetricform),L ( )= i 2\r L =i 2!/@ M ;oneobtainsimmediatelyi!/ M =0(4.18) Classicallagrangian eldtheory18andsimilarlyfromthevariationwithrespectto i /+M=0(4.19)Itisoftenusefultolinktothetwo-componentspinorsandwhichwestartedwithinchapter4,orequivalentlyseparatethe eldintoright-andlefthandedones.Inthatcaseone ndstriviallyL=1 2 Ri$/@ R+1 2 Li$/@ L M( R L+ L R)(4.20)showinge.g.thatthelagrangianseparatesintotwoindependentpartsforM=0.Usingthetwo-spinorsand,themasstermintheDiraclagrangian4.20isgivenbyLM(Dirac)= M y+y(4.21)Thereexistsanotherpossibilitytowritedownamasstermwithonlyonekindof elds,namelyLM(Majorana)=+1 2 My MT(4.22)Thesetwotermsareeachothersconjugate2Attheleveloftheequationsofmotiononehas,usingthePaulimatricesand~inEq.3.14,foraDiracfermioni()=MDandi(~)=MD:(4.23)WiththemassterminEq.4.22thisbecomesforMajoranafermionsi()=Mori(~)= M;andi(~)=M(4.24)inwhichthemasscanbecomplex.Squaringgives(2 jM2)=(2 jM2)=0.Itisalsopossibletointroducea'real'(four-component)spinorsatisfyingc=ofwhichtheleftpartcoincideswith LL= L=8��:09��;)8��:9��;(4.25)forwhich0=0.Wenotethat cL( L)c=C LT=8��:09��;=R(4.26)SincethekineticterminLseparatesnaturallyinleftandrightparts(orand),itisintheabsenceofaDiracmasstermpossibletointroducealagrangianinwhichonlyleft elds Land cLappearoronecanworkwith'real'spinorsorMajoranaspinorsL=1 2 Li$/@ L 1 2 M cL L+M L cL(4.27)=1 4 i$/ 1 2 M RL+M LR(4.28)ThisgivesanexpressionwithamasstermthatisactuallyofthesameformastheDiracmassterm,butnotethefactor1/2ascomparedtotheDiraclagrangian,whichcomesbecauseweinessenceuse'real'spinors.TheMajoranacaseisinfactmoregeneral,sincealagrangianwithbothDiracandMajoranamasstermscanberewrittenasthesumoftwoMajoranalagrangiansafterrede ningthe elds(Seee.g.PeshkinandSchroederorExercise12.7). 2ThistermiswrittendownwithbeingananticommutingGrassmannnumberforwhich = ;( )= = ;andthus( )= = .ThereasonsforGrassmannvariableswillbecomeclearinthenextchapter. Chapter5Quantizationof elds5.1Therealscalar eldWehaveexpandedthe(classical) eldinplanewavesolutions,whichwehavesplitintopositiveandnegativeenergypieceswith(complex)coecientsa(k)anda(k)multiplyingthem.Thequantizationofthe eldisachievedbyquantizingthecoecientsintheFourierexpansion,e.g.therealscalar eld (x)becomes (x)=Zd3k (2)32Eka(k)e ikx+ay(k)eikx(5.1)wheretheFouriercoecientsa(k)anday(k)arenowoperators.Notethatwewilloftenwritea(k)oray(k),butoneneedstorealizethatinthatcasek0=Ek=p k2+M2.Thecanonicalmomentumbecomes(x)= (x)= i 2Zd3k (2)3a(k)e ikx ay(k)eikx(5.2)Itiseasytocheckthattheseequationscanbeinverted(seeExercise2.4fortheclassical eld)a(k)=Zd3xeikxi$0 (x)(5.3)ay(k0)=Zd3x (x)i$0e ik0x(5.4)Itisstraightforwardtoprovethattheequaltimecommutationrelationsbetween (x)and(x0)areequivalentwith'harmonicoscillator-like'commutationrelationsbetweena(k)anday(k0),i.e.. (x)(x0)]x0=x00=i3(x x0)andd (x); (x0)]x0=x00=[(x)(x0)]x0=x00=0(5.5)isequivalentwithha(k);ay(k0)]=(2)32Ek3(k k0)andda(k);a(k0)]=[ay(k);ay(k0)]=0(5.6)19 Quantizationof elds20ThehamiltoniancanberewrittenintermsofanumberoperatorN(k)=N(k)=ay(k)a(k),whichrepresentsthe'numberofparticles'withmomentumkH=Zd3x1 2(0 )2+1 2(r )2+1 2M2 2=Zd3k (2)32EkEk 2 ay(k)a(k)+a(k)ay(k)(5.7)=Zd3k (2)32EkEkN(k)+Evac(5.8)wherethenecessitytocommutea(k)ay(k)(asinthecaseofthequantummechanicscase)leadstoazero-pointenergy,in eldtheoryalsoreferredtoasvacuumenergyEvac=1 2VZd3k (2)3Ek(5.9)whereV=(2)33(0)isthespace-volume.Thistermwillbeadressedbelow.ForthemomentumoperatoronehasPi=Zd3x0i(x)=1 2Zd3x@0 @i =Zd3k (2)32EkkiN(k)(5.10)wherethevacuumcontributiondisappearsbecauseofrotationalsymmetry.Justasinthecaseoftheharmonicoscillatoritisessential(axiom)thatthereexistsagroundstate0ithatisannihilatedbya(k),a(k)0i=0.5.2Thecomplexscalar eldInspiteofthesimilaritywiththecaseofthereal eld,wewillconsideritasarepetitionofthequantizationprocedure,extendingitwiththechargeoperatorandtheintroductionofparticleandantiparticleoperators.The eldsatis estheKlein-Gordonequationandthedensitycurrent(U(1)transformations)andtheenergy-momentumtensorarej=i $ ;(5.11)= Lg(5.12)Thequantized eldsarewrittenas (x)=Zd3k (2)32Eka(k)e ikx+by(k)eikx(5.13) y(x)=Zd3k (2)32Ekb(k)e ikx+ay(k)eikx(5.14)andsatisfytheequaltimecommutationrelation(onlynonzeroones)) (x);@0 y(y)]x0=y0=i3(x y)(5.15)whichisequivalenttotherelations(onlynonzeroones))a(k);ay(k0)]=[b(k);by(k0)]=(2)32Ek3(k k0)(5.16) Quantizationof elds21ThehamiltonianisasbeforegivenbythenormalorderedexpressionH=Zd3x:00(x):=Zd3k (2)32EkEkay(k)a(k)+b(k)by(k)=Zd3k (2)32EkEkay(k)a(k)+by(k)b(k)(5.17)i.e.particles(createdbyay)andantiparticles(createdbyby)withthesamemomentumcontributeequallytotheenergy.Alsothechargeoperatorrequiresnormalordering(inordertogivethevacuumeigenvaluezero),Q=iZd3x y0 0 y (x)=Zd3k (2)32Ekay(k)a(k) b(k)by(k)=Zd3k (2)32Ekay(k)a(k) by(k)b(k)(5.18)Thecommutatorof and yisasforthereal eldgivenbyy (x); y(y)]=i(x y)(5.19)5.3TheDirac eldFromthelagrangiandensityL=i 2 \r$ M ;(5.20)theconserveddensityandenergy-momentumcurrentsareeasilyobtained,j= \r ;(5.21)=i 2 \r$ i 2 $/@ M g(5.22)Thecanonicalmomentumandthehamiltonianaregivenby(x)=L (x)=i y(x)(5.23)H(x)=00(x)= i 2 \ri$i +M =i \r00 =i y0 ;(5.24)wherethelastlineisobtainedbyusingtheDiracequation.Thequantized eldsarewritten (x)=XsZd3k (2)32Ekb(k;s)u(k;s)e ikx+dy(k;s)v(k;s)eikx(5.25) (x)=XsZd3k (2)32Ekby(k;s)u(k;s)eikx+d(k;s)v(k;s)e ikx(5.26) Quantizationof elds22Intermsoftheoperatorsforthebanddquantathehamiltonianandchargeoperatorsare(omittingmostlythespinsummationintherestofthissection)H=Zd3x y(x)i@0 (x):(5.27)=Zd3k (2)32EkEkby(k)b(k) d(k)dy(k):(5.28)Q=Zd3x y (5.29)=Zd3k (2)32Ekby(k)b(k)+d(k)dy(k)(5.30)whichseemstocauseproblemsastheantiparticles(d-quanta)contributenegativelytotheenergyandthechargesofparticles(b-quanta)andantiparticles(d-quanta)arethesame.Thesolutionistheintroductionofanticommutationrelations,fb(k;s);by(k0;s0)g=fd(k;s);dy(k0;s0)g=(2)32Ek3(k k0)ss0(5.31)Notethatachievingnormalordering,i.e.interchangingcreationandannihilationoperators,thenleadstoadditionalminussignsandH=Zd3k (2)32EkEkby(k)b(k)+dy(k)d(k)(5.32)Q=Zd3k (2)32Ekby(k)b(k) dy(k)d(k)(5.33) Chapter6DiscretesymmetriesInthischapterwediscussthediscretesymmetries,parity(P),timereversal(T)andchargeconjugation(C).TheconsequencesofP,TandCforclassicalquantitiesisshowninthetable1.6.1ParityTheparityoperatortransformsx=(t;r) !~xx=(t; r)(6.1)Wewillconsiderthetransformationpropertiesforafermion eld (x),writing (x) !Pop (x)P 1op=PA (~x) p(x)(6.2)wherePistheintrinsicparityofthe eldandAisa44matrixactinginthespinorspace.Both pand satisfytheDiracequation.WecandetermineA,startingwiththeDiracequationfor (x),(i\r M) (x)=0Table6.1:ThebehaviorofclassicalquantitiesunderP,T,andC quantityPTC tt-ttr-rrrx~xx ~xxEEEEp-p-ppp~p~ppLL-LLss-ss=s^p 23 Discretesymmetries24Afterparitytransformingxto~xtheDiracequationbecomesaftersomemanipulationsi\r~ M (~x)=0(i~\r M) (~x)=0 i\ry M (~x)=0(i\r M)\r0 (~x)=0(6.3)Therefore\r0 (~x)isagainasolutionoftheDiracequationandwehave p(x)=\r0 (~x)(6.4)Itisstraightforwardtoapplythistotheexplicit eldoperator (x)using\r0u(k;m)=u(~k;m);\r0u(k;)=u(~k; )(6.5)\r0v(k;m)= v(~k;m);\r0v(k;)= v(~k; )(6.6)Checkthisforthestandardrepresentation;notethatifhelicityisusedinsteadofthez-componentofthespinm,theaboveoperationreversesthesignof(whichdoesdependonthesignofp3,however!).Theresultis p(x)=Pop (x)P 1op=P\r0 (~x)(6.7)=XZd3k (2)32EkPb(k;)\r0u(k;)e ik~x+dy(k;)\r0v(k;)eik~x=XZd3k (2)32EkPhb(k;)u(~k; )e i~kx dy(k;)v(~k; )ei~kxi=XZd3~k (2)32E~kPhb(k;)u(~k; )e i~kx dy(k;)v(~k; )ei~kxi=XZd3k (2)32EkPhb(~k; )u(k;)e ikx dy(~k; )v(k;)eikxi(6.8)FromthisoneseesimmediatelythatPopb(k;)P 1op=Pb(~k; )(6.9)Popd(k;)P 1op= Pd(~k; )(6.10)i.e.choosingPisreal(P=1)particleandantiparticlehaveoppositeparity.InthesamewayastheFermion eld,onecanalsoconsiderthescalar eldandvector elds.Forthescalar eldwehaveseen (x) !Pop (x)P 1op=P (~x)(6.11)andforthevector eldA(x) !PopA(x)P 1op= A(~x)(6.12)Thelatterbehaviorofthevector eldwillbediscussedfurtherbelow.6.2ChargeconjugationWehavealreadyseentheparticle-antiparticlesymmetrywithunderwhatwewillcallchargeconju-gationthebehavior (x) ! c(x)=CC T(x)(6.13) Discretesymmetries25thelatterbeingalsoasolutionoftheDiracequation.TheactiononthespinorsgivesCuT(k;)=v(k;)(6.14)CvT(k;)= u(k;)(6.15)(whereonemustbeawareofthechoiceofspinorsmadeintheexpansion,asdiscussedinsection4).Therefore c(x)=Cop (x)C 1op=CC T(x)(6.16)=XZd3k (2)32EkCd(k;)CvT(k;)e ikx+by(k;)CuT(k;)eikx=XZd3k (2)32EkCd(k;)u(k;)e ikx by(k;)v(k;)eikx(6.17)ThisshowsthatCopb(k;)C 1op=Cd(k;)(6.18)Copd(k;)C 1op= Cb(k;)(6.19)6.3TimereversalThetimereversaloperatortransformsx=(t;r) ! ~x x=( t;r)(6.20)Wewillagainconsiderthetransformationpropertiesforafermion eld (x),writing (x) !Top (x)T 1op=TA ( ~x) t( ~x)(6.21)whereAisa44matrixactinginthespinorspace.Astimereversalwilltransform'bra'into'ket',Top i=h t=( ti),itisantilinear1.NormconservationrequiresToptobeanti-unitary2.Foraquantized eldonehasTopfk(x)bkT 1op=fk(x)TopbkT 1opi.e.to nd t( ~x)thatisasolutionoftheDiracequation,westartwiththecomplexconjugatedDiracequationfor ((i\r) M) (x)=0The(time-reversed)Diracequationbecomes, (i\r)~ M ( ~x)=0(i~\r M) ( ~x)=0 i\rT M ( ~x)=0 iC 1\rC@ M ( ~x)=0 i(\r5C) 1\r\r5C@ M ( ~x)=0(i\r M)\r5C ( ~x)=0(6.22)Therefore\r5C ( ~x)isagainasolutionofthe(ordinary)Diracequationandwecanchoose(phaseisconvention) t(x)=\r5C ( ~x)(6.23) 1AisantilinearifA(j i+j i)=Aj i+Aj i.2Anantilinearoperatorisanti-unitaryifAy=A 1.OnehashA jA i=h j i=hA jA i=h jAyA i=h j i. Discretesymmetries26Table6.2:Thetransformationpropertiesofphysicalstatesforparticles(a)andantiparticles(a). statePTC ap;ija p iha p;jjap;iap;ija p iha p;jjap;i Inthestandardrepresentationi\r5C=i2anditisstraightforwardtoapplythistotheexplicit eldoperator (x)using\r5Cu(k;)=u(~k;)(6.24)\r5Cv(k;)=v(~k;)(6.25)(checkthisforthestandardrepresentationandmakeproperuseofhelicitystatesfor p).Theresultis t(x)=Top (x)T 1op=iT\r5C ( ~x)(6.26)=XZd3k (2)32EkTb(k;)i\r5Cu(k;)eik~x+dy(k;)i\r5Cv(k;)e ik~x=XZd3k (2)32EkThb(k;)u(~k;)ei~kx+dy(k;)v(~k;)e i~kxi=XZd3~k (2)32EkThb(k;)u(~k;)ei~kx+dy(k;)v(~k;)e i~kxi=XZd3k (2)32EkThb(~k;)u(k;)eikx+dy(~k;)v(k;)e ikxi(6.27)FromthisoneobtainsTopb(k;)T 1op=Tb(~k;)(6.28)Topd(k;)T 1op=Td(~k;)(6.29)Intable2thebehaviorofparticlestatesunderthevarioustransformationshasbeensummarized.Notethatapplyingananti-unitarytransformationsuchasToponemusttakeforthematrixelementthecomplexconjugate.ThereforeonehashAki=hAby(k)0i=hATyTby(k)TyT0i=hAtby(~k)0i=h0b(~k)Ati=h~kAtiExercise8.4Startwiththeexpressionforthefermionquantum eld (x).OnecanrewritethewavefunctionsintermsoftherealWeylhelicityspinorsUR=L(k;).Showthatonecanexpress intermsofrealspinors (x)=1(x)+i2(x)oftheformi(x)=XZd3k (2)32Ekui(k;)hbi(k;)e ikx+byi(k;)eikxiGivetheexpressionsforthecreationoperatorsb(k;)anddy(k;)intermsofb1(k;)andb2(k;)andthetwospinorsu(k;)intermsofthe(real)Weylhelicityspinors(seeChapter4). Discretesymmetries27(solution)Wecanexpressthe eldintermsoftheWeylspinorsas (x)=Zd3k (2)32Ekp 1 2(UR(k;+)b(k;+)e ikx+UL(k; )b(k; )e ikx UL(k; )dy(k;+)eikx+UR(k;+)dy(k; )eikx)+(UL(k;+)b(k;+)e ikx+UR(k; )b(k; )e ikx+UR(k; )dy(k;+)eikx+UL(k;+)dy(k; )eikx)re-orderedas (x)=Zd3k (2)32Ekp 1 2(UR(k;+)b(k;+)e ikx+dy(k; )eikx+UL(k; )b(k; )e ikx dy(k;+)eikx)+(UL(k;+)b(k;+)e ikx+dy(k; )eikx+UR(k; )b(k; )e ikx dy(k;+)eikx)=Zd3k (2)32Ek(u(k;+)b(k;+)e ikx+dy(k; )eikx+u(k; )b(k; )e ikx dy(k;+)eikx)withu(k;+)=p 1 2UR(k;+)+UL(k;+)u(k; )=p 1 2UL(k; )+UR(k; )Introducingthecreationandannihilationoperatorsb(k;+)=b1(k;+)+ib2(k;+)b(k; )=b1(k; )+ib2(k; )d(k; )=b1(k;+) ib2(k;+)d(k;+)= b1(k; )+ib2(k; )one ndstherequestedform. Chapter7Thestandardmodel7.1Thestartingpoint:SU(2)W\nU(1)YSymmetryplaysanessentialroleinthestandardmodelthatdescribestheelementaryparticles,thequarks(up,down,etc.),theleptons(elektrons,muons,neutrinos,etc.)andthegaugebosonsresponsibleforthestrong,electromagneticandweakforces.Inthestandardmodelonestartswithaverysimplebasiclagrangianfor(massless)fermionswhichexhibitsmoresymmetrythanobservedinnature.Byintroducinggauge eldsandbreakingthesymmetryamorecomplexlagrangianisobtained,thatgivesagooddescriptionofthephysicalworld.Theprocedure,however,impliescertainnontrivialrelationsbetweenmassesandmixinganglesthatcanbetestedexperimentallyandsofarareinexcellentagreementwithexperiment.Thelagrangianfortheleptonsconsistsofthreefamilieseachcontaininganelementaryfermion(electrone ,muon ortau ),itscorrespondingneutrino(eand )andtheirantiparticles.Astheyaremassless,left-andrighthandedparticles, R=L=1 2(1\r5) decouple.Fortheneutrinoonlyalefthandedparticle(andrighthandedantiparticle)exist.ThusL(f)=i eR/@eR+i eL/@eL+i eL/@eL+(; )(7.1)Oneintroducesa(weak)SU(2)WsymmetryunderwhicheRformsasinglet,whilethelefthandedparticlesformadoublet,i.e.L=8��:L0L 9��;=8��:eeL9��;withTW=1 2andT3W=+1=2 1=2andR=R =eRwithTW=0andT3W=0ThusthebasiclagrangiandensityisaDiraclagrangianwithmassless(independentleft-andright-handedspecies)L(f)=i L/@L+i R/@R;(7.2)whichhasanSU(2)Wsymmetryundertransformationsei~ TW,explicitlyLSU(2)W !ei~ ~ =2L;(7.3)RSU(2)W !R:(7.4)OnenotesthatthechargesoftheleptonscanbeobtainedasQ=T3W 1=2forlefthandedparticlesandQ=T3W 1forrighthandedparticles.Thedi erencebetweenchargeand3-componentofisospiniscalledweakhyperchargeandonewritesQ=T3W+YW 2(7.5)28 Thestandardmodel29TheweakhyperchargeYWisanoperatorthatgeneratesaU(1)Ysymmetrywithforthelefthandedandrighthandedparticlesdi erenthypercharges,YW(L)= 1andYW(R)= 2.Theparticlestransformaccordingtoei YW2,explicitlyLU(1)Y !e i =2L;(7.6)RU(1)Y !e i R:(7.7)NexttheSU(2)W\nU(1)Ysymmetryismadeintoalocalsymmetryintroducinggauge elds~WandBinthecovariantderivativeD= ig~W~TW ig0BYW=2,explicitlyDL=L i 2g~W~ L+i 2g0BL;(7.8)DR=R+ig0BR;(7.9)where~WisatripletofgaugebosonswithTW=1,T3W=1or0andYW=0(thusQ=T3W)andBisasingletunderSU(2)W(TW=T3W=0)andalsohasYW=0.PuttingthisinleadstoL(f)=L(f1)+L(f2)(7.10)L(f1)=i R\r(+ig0B)R+i L\r(+i 2g0B i 2g~W~ )LL(f2)= 1 4(~W ~W+g~W~W)2 1 4(B B)2Inordertobreakthesymmetrytothesymmetryofthephysicalworld,theU(1)Qsymmetry(generatedbythechargeoperator),acomplexHiggs eld =8��: + 09��;=8���:1 p 2(2+i1)1 p 2(4 i3)9���;(7.11)withTW=1=2andYW=1isintroduced,withthefollowinglagrangiandensityconsistingofasymmetrybreakingpieceandacouplingtothefermions,L(h)=L(h1)+L(h2)(7.12)whereL(h1)=(D )y(D ) m2 y ( y )2| {z } V( )L(h2)= Ge( L R+ R yL)andD =( i 2g~W~ i 2g0B) :(7.13)TheHiggspotentialV( )ischoosensuchthatitgivesrisetospontaneoussymmetrybreakingwith'y'= m2=2v2=2.Fortheclassical eldthechoice4=vismade,whichassureswiththechoiceofYWfortheHiggs eldthatQgeneratestheremainingU(1)symmetry.Usinglocalgaugeinvarianceifori=1,2and3maybeeliminated(thenecessarySU(2)Wrotationispreciselye i(x) ),leadingtotheparametrization (x)=1 p 28��:0v+h(x)9��;(7.14) Thestandardmodel30andD =8������: ig 2W1 iW2 p 2(v+h)1 p 2h+i 2gW3 g0B p 2(v+h)9������;(7.15)Uptocubicterms,thisleadstothelagrangianL(h1)=1 2(h)2+m2h2+g2v2 8(W1)2+(W2)2+v2 8 gW3 g0B2+:::(7.16)=1 2(h)2+m2h2+g2v2 8(W+)2+(W )2+(g2+g02)v2 8(Z)2+:::;(7.17)wherethequadraticallyappearinggauge eldsthatarefurthermoreeigenstatesofthechargeoperatorareW=1 p 2 W1iW2(7.18)Z=gW3 g0B p g2+g02cosWW3 sinWB(7.19)A=g0W3+gB p g2+g02sinWW3+cosWB(7.20)andcorrespondtothreemassiveparticle elds(WandZ0)andonemassless eld(photon\r)withM2W=g2v2 4(7.21)M2Z=g2v2 4cos2W=M2W cos2W(7.22)M2\r=0(7.23)Theweakmixingangleisrelatedtotheratioofcouplingconstants,g0=g=tanWThecouplingofthefermionstothephysicalgaugebosonsarecontainedinL(f1)givingL(f1)=i e\re+i e\re gsinW e\reA+g cosWsin2W eR\reR 1 2cos2W eL\reL+1 2 e\reZ+g p 2 e\reLW + eL\reW+(7.24)Fromthecouplingtothephoton,wecanreado e=gsinW=g0cosW(7.25)Thecouplingofelectronsormuonstotheirrespectiveneutrinos,forinstanceintheamplitudeforthedecayofthemuon mnmn m-m-e-e- ne ne-W = Thestandardmodel31isgivenby iM= g2 2( \rL) ig+::: k2+M2W( eL\re)ig2 8M2W( \r(1 \r5))| {z }(j()yL)( e\r(1 \r5)e)| {z }(j(e)L)(7.26)iGF p 2(j()yL)(j(e)L)(7.27)thegoodoldfour-pointinteractionintroducedbyFermitoexplaintheweakinteractions,i.e.onehastherelationGF p 2=g2 8M2W=e2 8M2Wsin2W=1 2v2(7.28)Inthiswaytheparametersgg0andvdetermineanumberofexperimentallymeasurablequantities,suchas1 =e2=41=127(7.29)GF=1166410 5GeV 2(7.30)sin2W=02312(7.31)MW=8040GeV(7.32)MZ=9119GeV(7.33)ThecouplingoftheZ0tofermionsisgivenby(g=cosW)\rmultipliedwithT3W1 2(1 \r5) sin2WQ1 2CV 1 2CA\r5(7.34)withCV=T3W 2sin2WQ;(7.35)CA=T3W(7.36)FromthiscouplingitisstraightforwardtocalculatethepartialwidthforZ0intoafermion-antifermionpair, (Z0!f f)=MZ 48g2 cos2W(C2V+C2A)(7.37)Fortheelectron,muonortau,leptonswithCV= 1=2+2sin2W 005andCA= 1=2wecalculate (e+e )785MeV(exp. e 83MeV).Foreachneutrinospecies(withCV=1/2andCA=1/2oneexpects ( )155MeV.Comparingthiswiththetotalwidthinto(invisible!)channels, invisible=480MeVoneseesthatthreefamiliesof(light)neutrinosareallowed.Actuallyincludingcorrectionscorrespondingtohigherorderdiagramstheagreementforthedecaywidthintoelectronscanbecalculatedmuchmoreaccuratelyandthenumberofallowed(light)neutrinosturnstobeevenclosertothree.ThemassesofthefermionsandthecouplingtotheHiggsparticlearecontainedinL(h2).WiththechoosenvacuumexpectationvaluefortheHiggs eld,oneobtainsL(h2)= Gev p 2( eLeR+ eReL) Ge p 2( eLeR+ eReL)h= me ee me v eeh:(7.38) 1Thevalueof deviatesfromtheknownvalue 1=137becauseofhigherordercontributions,givingrisetoarunningcouplingconstantafterrenormalizationofthe eldtheory. Thestandardmodel32 LdRnRneLeRLdLuRdnLuReRuReLRdLu Y+1/2-1/2+1-1Q = 0Q = +1Q = -1_WW0B0+WI3W Figure7.1:AppropriateYWandT3Wassign-mentsofquarks,leptons,theirantiparticlesandtheelectroweakgaugebosonsasappearingineachfamily.TheelectricchargeQisthen xed,Q=T3W+YW=2andconstantalongspeci cdiagonalsasindicatedinthe gure.Thepatternisactuallyintriguing,suggestinganunderlyinglargerunifyingsymmetrygroup,forwhichSU(5)orSO(10)areactuallynicecandidates.Wewillnotdiscussthisanyfur-therinthischapter.First,themassoftheelectroncomesfromthespontaneoussymmetrybreakingbutisnotpredicted(itisinthecouplingGe).ThecouplingtotheHiggsparticleisweakasthevalueforvcalculatede.g.fromtheMWmassisabout250GeV,i.e.me=visextremelysmall.Finallywewanttosaysomethingabouttheweakpropertiesofthequarks,asappearforinstanceinthedecayoftheneutronorthedecayofthe(quarkcontentuds), e- ne -Wdu n !pe e()d !ue e e- ne -Wus !pe e()s !ue eThequarksalsoturnoutto tintodoubletsofSU(2)Wforthelefthandedspeciesandintosingletsfortherighthandedquarks.AsshowninFig.7.1,thisrequiresparticularYW-T3Wassignmentstogetthechargesright.Acomplicationarisesforquarks(andaswewilldiscussinthenextsectioninmoredetailalsoforleptons)asitarenotthe'mass'eigenstatesthatappearintheweakisospindoubletsbutlinearcombinationsofthem,8��:ud09��;L8��:cs09��;L8��:tb09��;Lwhere8�����:d0s0b09�����;L=8�����:VudVusVubVcdVcsVcbVtdVtsVtb9�����;8�����:dsb9�����;L(7.39)ThismixingallowsallquarkswithT3W= 1=2todecayintoanupquark,butwithdi erentstrength.ComparingneutrondecayanddecayonecangetanestimateofthemixingparameterVusinthe Thestandardmodel33socalledCabibbo-Kobayashi-Maskawamixingmatrix.DecayofB-mesonscontainingb-quarksallowestimateofVub,etc.InprincipleonecomplexphaseisallowedinthemostgeneralformoftheCKMmatrix,whichcanaccountforthe(observed)CPviolationoftheweakinteractions.Thisisonlytrueifthemixingmatrixisatleastthree-dimensional,i.e.CPviolationrequiresthreegenerations.ThemagnitudesoftheentriesintheCKMmatrixarenicelyrepresentedusingthesocalledWolfensteinparametrizationV=8�����:1 1 223A( i) 1 1 222A3A(1 i) 2A19�����;+O(4)with0227,A082and022and034.TheimaginarypartigivesrisetoCPviolationindecaysof KandB-mesons(containingsandbquarks,respectively).7.2FamilymixingintheHiggssectorandneutrinomassesThequarksectorAllowingforthemostgeneral(Dirac)massgeneratingterminthelagrangianonestartswithL(h2;q)= QL dDR DRyd yQL QL cuUR URyu cyQL(7.40)whereweincludenowthethreelefthandedquarkdoubletsinQL,thethreerighthandedquarkswithcharge+2/3inURandthethreerighthandedquarkswithcharges 1=3inDR,eachofthesecontainingthethreefamilies,e.g. UR=8: uR cR tR9;.Theuanddarecomplexmatricesinthe33familyspace.TheHiggs eldisstilllimitedtoonecomplexdoublet.NotethatweneedtheconjugateHiggs eldtogetaU(1)Ysingletinthecaseofthecharge+2/3quarks,forwhichweneedtheappropriateweakisospindoublet c=8��: 0 9��;=1 p 28��:v+h09��;Forthe(squared)complexmatriceswecan ndpositiveeigenvaluesuyu=VuG2uVyuanddyd=VdG2dVyd(7.41)whereVuandVdareunitarymatrices,allowingustowriteu=VuGuWyuandd=VdGdWyd(7.42)withGuandGdbeingrealandpositiveandWuandWdbeingdi erentunitarymatrices.ThusonehasL(h2;q)=) DLVdMdWydDR DRWdMdVydDL ULVuMuWyuUR URWuMuVyuUL(7.43)withMu=Guv=p 2(diagonalmatrixcontainingmumcandmt)andMd=Gdv=p 2(diagonalmatrixcontainingmdmsandmb).Onethenreadso thatstartingwiththefamilybasisasde nedviatheleftdoubletsthatthemasseigenstates(andstatescouplingtotheHiggs eld)involvetherighthandedstatesUmassR=WyuURandDmassR=WydDRandthelefthandedstatesUmassL=VyuULandDmassL=VydDL.WorkingwiththemasseigenstatesonesimplyseesthattheweakcurrentcouplingtotheWbecomes UL\rDL= UmassL\rVyuVdDmassL,i.e.theweakmasseigenstatesareD0L=DweakL=VyuVdDmassL=VCKMDmassL(7.44)theunitaryCKM-matrixintroducedaboveinanadhocway. Thestandardmodel34Theleptonsector(masslessneutrinos)ForaleptonsectorwithalagrangiandensityoftheformL(h2;`)= L eER ERye yL;(7.45)inwhichL=8��:NLEL9��;isaweakdoubletcontainingthethreefamiliesofneutrinos(NL)andchargedleptons(EL)andERisathree-familyweaksinglet,we ndmasslessneutrinos.Asbefore,onecanwritee=VeGeWyeandwe ndL(h2;`)=) Me ELVeWyeER ERWeVyeEL(7.46)withMe=Gev=p 2thediagonalmassmatrixwithmassesmemandm .Themass eldsEmassR=WyeEREmassL=VyeEL.Forthe(massless)neutrino eldswejustcanrede ne eldsintoNmassL=VyeNL,sincetheweakcurrentistheonlyplacewheretheyshowup.TheW-currentthenbecomes EL\rNL= EmassL\rNmassL,i.e.thereisnofamilymixingformasslessneutrinos.Theleptonsector(massiveDiracneutrinos)InprincipleamassiveDiracneutrinocouldbeaccountedforbyalagrangianofthetypeL(h2;`)= L eER ERye yL L cnNR NRyn cyL(7.47)withthreerighthandedneutrinosaddedtothepreviouscase,decouplingfromallknowninteractions.Againwecontinueasbeforenowwithmatricese=VeGeWyeandn=VnGnWyn,andobtainL(h2;`)=) ELVeMeWyeER ERWeMeVyeEL NLVnMnWynNR NRWnMnVynNL(7.48)Wenotethattherearemass eldsEmassR=WyeEREmassL=VyeELNmassL=VynNLandNmassR=WynNRandtheweakcurrentbecomes EL\rNL= EmassL\rVyeVnNmassL.WorkingwiththemasseigenstatesforthechargedleptonsweseethattheweakeigenstatesfortheneutrinosareNweakL=VyeNLwiththerelationtothemasseigenstatesforthelefthandedneutrinosgivenbyN0L=NweakL=VyeVnNmassL=UyPMNSNmassL(7.49)withUPMNS=VynVeknownasthePontecorvo-Maki-Nakagawa-Sakatamixingmatrix.Forneutrino'sthismatrixisparametrizedintermsofthreeanglesijwithcij=cosijandsij=sinijandoneangleUPMNS=8�����:1000c23s230 s23c239�����;8�����:c130s13ei010 s13ei0c139�����;8�����:c12s120 s12c1200019�����;(7.50)aparametrizationthatinprinciplealsocouldhavebeenusedforquarks.Inthiscase,itisparticularlyusefulbecause12isessentiallydeterminedbysolarneutrinooscillationsrequiringm212810 5eV2(conventionm2�m1),while23thenisdeterminedbyatmosphericneutrinooscillationsrequiringm223j2510 3eV2.ThemixingisintriguinglyclosetotheHarrison-Perkins-Scotttri-bimaximalmixingmatrixUHPS=8������:1000p 1=2 p 1=20p 1=2p 1=29������;8������:p 2=3p 1=30 p 1=3p 2=300019������;=8������:p 2=3p 1=30 p 1=6p 1=3 p 1=2 p 1=6p 1=3p 1=29������;(7.51) Thestandardmodel35Theleptonsector(massiveMajorana elds)AnevensimpleroptionthansterilerighthandedDiracneutrinos,istoaddinEq.7.46aMajoranamasstermforthe(lefthanded)neutrinomasseigenstates,Lmass;= 1 2 ML NcLNL+ML NLNcL(7.52)althoughthisoptionisnotattractiveasitviolatestheelectroweaksymmetry.Thewaytocircumventthisistointroduceasintheprevioussectionrighthandedneutrinos,withfortherighthandedsectoramasstermMRLmass;= 1 2 MR NRNcR+MR NcRNR(7.53)Inordertohavemorethanacompletelydecoupledsector,onemustfortheneutrinosaswellaschargedleptons,coupletheright-andlefthandedspeciesthroughDiracmasstermscomingfromthecouplingtotheHiggssectorasintheprevioussection.Thus(disregardingfamilystructure)onehastwoMajorananeutrinos,onebeingmassive.ForthechargedleptonstherecannotexistaMajoranamasstermasthiswouldbreaktheU(1)electromagneticsymmetry.Fortheleptons,theleft-andrighthandedspeciesthenjustformaDiracfermion.Fortheneutrinosector,themasslessandmassiveMajorananeutrinos,coupledbyaDiracmassterm,areequivalenttotwodecoupledMajorananeutrinos(seebelow).IftheMajoranamassMR MDoneactuallyobtainsinanaturalwayoneMajorananeutrinowithaverysmallmass.Thisiscalledthesee-sawmechanism(outlinedbelow).FortheselightMajorananeutrinosonehas,asabove,aunitarymatrixrelatingthemtotheweakeigenstates.AbsorptionofphasesinthestatesisnotpossibleforMajorananeutrinos,however,hencethemixingmatrixbecomesVPMNS=UPMNSKwithK=8������:ei 12000ei 2200019������;(7.54)containingthree(CP-violating)phases( 1 2and).Thesee-sawmechanismConsider(foronefamilyN=n)themostgeneralLorentzinvariantmasstermfortwoindependentMajoranaspinors,01and02(satisfyingc=andasdiscussedinchapter6,cL(L)c=RandcR=L).Weuseheretheprimesstartingwiththeweakeigenstates.Actually,itiseasytoseethatthisincorporatestheDiraccasebyconsideringthelefthandedpartof01andtherighthandedpartof02asaDiracspinor .Thus01=ncL+nL02=nR+ncR; =nR+nL(7.55)AsthemostgeneralmassterminthelagrangiandensitywehaveLmass= 1 2 ML ncLnL+ML nLncL 1 2 MR nRncR+MR ncRnR 1 2 MD ncLncR+MD nLnR 1 2 MD nRnL+MD ncRncL(7.56)= 1 28: ncL nR9;8��:MLMDMDMR9��;8��:nLncR9��;+h.c.(7.57)whichforMD=0isapureMajoranalagrangianandforML=MR=0andrealMDrepresentstheDiraccase.ThemassmatrixcanbewrittenasM=8��:MLMDei MDei MR9��;(7.58) Thestandardmodel36takingMLandMRrealandnon-negative.Thischoiceispossiblewithoutlossofgeneralitybecausethephasescanbeabsorbedinto01and02(realmustbereplacedbyhermiteanifoneincludesfamilies).Thisisamixingproblemwithasymmetric(complex)massmatrixleadingtotwo(real)masseigenstates.Thediagonalizationisanalogoustowhatwasdoneforthe-matricesandone ndsUMUT=M0witha(unitary)matrixU,whichimpliesUMyUy=UMUy=M0anda'normal'diagonalizationofthe(hermitean)matrixMMyU(MMy)Uy=M20(7.59)ThusoneobtainsfromMMy=8��:M2L+MD2MD MLe i +MRe+i MD MLe+i +MRe i M2R+MD29��;(7.60)theeigenvaluesM212=1 2"M2L+M2R+2MD2q (M2L M2R)2+4MD2(M2L+M2R+2MLMRcos(2 ))#(7.61)andweareleftwithtwodecoupledMajorana elds1and2,relatedvia8��:1L2L9��;=U8��:nLncR9��;8��:1R2R9��;=U8��:ncLnR9��;(7.62)foreachofwhichone ndsthelagrangiansL=1 4 ii$/i 1 2Mi ii(7.63)fori=1,2withrealmassesMi.ForthesituationML=0andMR MD(takingMDreal)one ndsM1M2D=MRandM2MRExerciseInthisexercisetwolimitsareinvestigatedforthetwo-Majoranacase.(a)CalculateforthespecialchoiceML=MR=0andMDreal,themasseigenvaluesandshowthatthemixingmatrixisU=1 p 211i iwhichenablesonetorewritetheDirac eldintermsofMajoranaspinors.Givetheexplicitexpressionsthatrelate and cwith1and2(solution)One ndsM1=M2=MD.Forbothleft-andrighthanded eldstherelationsbetween cand1and2arethesame, =1 p 2(1+i2); c=1 p 2(1 i2)(b)Amoreinterestingsituationis0=MLMDj MR,whichleadstothesocalledseesawmechanism.CalculatetheeigenvaluesML=0andMR=MX.Giventhatneutrinomassesareoftheorderof1/20eV,whatisthemassMXifwetakeforMDtheelectroweaksymmetry Thestandardmodel37breakingscalev(about250GeV).(solution)TheeigenvaluesareM1M2D=MXp 2andM2M.Foraneutrinomassoftheorderof1/20eV,andafermionmassoftheorderoftheelectroweakbreakingscaling250GeV,thisleadstoMX1015GeV.TherecouplingmatrixinthiscaseisU=icosS isinSsinScosSwithsinSMD=MX.TheweakcurrentcouplestonL=sinS2 icosS1,where1isthelightneutrino(mass)eigenstate.