SUSY SU(5) Model with Flavor Group Delta(54)

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SUSY SU(5) Model with Flavor Group Delta(54)
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Escobar,Jesus A
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Doctorate ( Ph.D.)
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University of Florida
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Physics
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Ramond, Pierre
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Fry, James N
Thorn, Charles B
Korytov, Andrey
Berkovich, Alexander

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flavor -- group -- neutrino -- particle -- physics
Physics -- Dissertations, Academic -- UF
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Physics thesis, Ph.D.
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Abstract:
We design a supersymmetric SU(5) GUT model using Delta(54), a finite non-abelian subgroup of SU(3)_f. Heavy right handed neutrinos are introduced which transform as three-dimensional representation of our chosen family group. We produced novel possible Yukawa matrices for the quarks and the neutral lepton sectors that fit all currently available data. The model successfully reproduces the Yukawa matrices and thus the mass hierarchical structures of the Standard Model and the CKM mixing matrix. It provides predictions for the light neutrinos with a normal hierarchy and masses such that m?,1 ? 5 ? 10^(?3) eV , m?,2 ? 1 ? 10^(?2) eV , and m?,3 ? 5 ? 10^(?2) eV. We also provide predictions for masses of the heavy neutrinos, and corrections to the tri-bimaximal matrix that fit within experimental limits, e.g., a reactor angle of ?7.31^o .
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by Jesus A Escobar.
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Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Ramond, Pierre.

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SUSY SU (5)MODELWITHFLAVORGROUP (54) By JESUSALEJANDROESCOBAR ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c r 2011JesusAlejandroEscobar 2

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Tomyparents,whohavesacricedmuchformyeducation.Iwou ldalsoliketodedicate thisworkinparttomywife. 3

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ACKNOWLEDGMENTS IwouldliketothankPierreRamondforthemanyusefulandinf ormativediscussions thathavehelpedmakethisprojectthesuccessitis.Iwoulda lsoliketothankChristoph Luhn,whohastimeandtimeagainreadovermyworkandgivenme manygood questionsandadvice.SpecialthankstoChrisPankow,whoha shelpedmegureout thetechnicalsideofwritingthisdissertation. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................7 LISTOFFIGURES .....................................8 ABSTRACT .........................................9 CHAPTER 1INTRODUCTION ...................................10 2THESTANDARDMODEL ..............................13 2.1SymmetriesandForceCarriers ........................13 2.2ParticleContent ................................14 2.3TheYukawaSector ...............................16 2.4ElectroweakBreakingandMasses ......................17 2.4.1TheHiggsMechanism .........................18 2.4.2Masses .................................20 3BEYONDTHESTANDARDMODEL ........................22 3.1UnicationatLargeScales ..........................22 3.2SM:RunningofMasses ............................24 3.3ABriefHistoryofNeutrinoMasses ......................25 3.4Theory:NeutrinoMasses ...........................26 3.4.1TheSee-SawMechanism .......................28 3.4.2TheTri-BimaximalMatrix ........................29 3.4.3NeutrinoOscillations ..........................30 3.5MinimalSupersymmetricStandardModel ..................32 3.5.1TheSuperpotential ...........................34 3.5.2MSSM:RunningofMasses ......................36 3.6RemarksonModelBuilding ..........................36 4THEGUTGROUP SU (5) ..............................39 4.1TheGroupTheory ...............................39 4.2ParticleContent ................................41 4.3InvariantMassTerms .............................44 4.4TheGeorgi-JarlskogMechanism .......................45 4.5Problemswith SU (5) ..............................46 5YUKAWAMATRICESFROMPHENOMENOLOGY ................48 5.1YukawaMatrices:TheQuarkSector .....................49 5

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5.2YukawaMatricies:NeutralLeptonSector ..................53 5.3RemarksonFlavorSymmetry .........................56 6FLAVORTHROUGHFINITEGROUPS ......................58 6.1TheFroggatt-NielsenMechanism .......................58 6.2TheFiniteGroup (6 n 2 ) ............................62 6.2.1PresentationandClasses .......................62 6.2.2CharacterTable .............................63 6.3FlavorSymmetry (54) ............................64 6.3.1CharacterTableandKroneckerProducts ...............64 6.3.2ProducingYukawaMatrices ......................65 6.3.3ExamplesofYukawaMatriceswith (54) ..............69 7THESUPERSYMMETRIC SU (5) (54)MODEL ................72 7.1ProducingYukawaSectorPhenomenologywith (54) ...........72 7.1.1 (54)asaFlavorGroup ........................73 7.1.2AQuarkSectorToyModel .......................75 7.1.3ChoosingFlavonRepresentations ..................81 7.2MatterandFlavonContent ..........................81 7.3VacuumValues .................................83 7.4QuarkYukawas .................................85 7.5NeutrinoMasses ................................87 7.6PhenomenologicalResults ..........................89 APPENDIX AADDITIONAL SU (5)CONTENT ..........................94 A.1GeneratorsforNon-FundamentalRepresentations .............94 A.2ChargeOperatorsforNon-FundamentalRepresentations .........94 BCLEBSCH-GORDANCOEFFICIENTSFOR (54) ................96 LISTOFREFERENCES ..................................116 BIOGRAPHICALSKETCH ................................119 6

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LISTOFTABLES Table page 2-1GaugedsymmetriesoftheSM ...........................14 2-2ParticleContentoftheSM ..............................15 2-3GenerationsoftheSMandparticlemasses ....................20 3-1GaugedsymmetriesoftheMSSM .........................33 3-2ParticleContentoftheMSSM ............................33 6-1Finitegroupswiththree-dimensionalrepresentations ...............58 6-2Thecharactertablesof (6 n 2 ) ...........................64 6-3Thecharactertableof (54) ............................65 6-4Hypotheticalavonsandtheirvevs .........................65 6-5Yukawamatricesfortree-levelcoupling ......................69 6-6Yukawamatricesfromonethree-dimensionalavon ...............70 6-7Yukawamatricesfromonetwo-dimensionalavon ................71 7-1Toymodelmatterandavoncontent ........................76 7-2Fieldcontentandchargesforthesupersymmetric SU (5) n (54)model ....84 7

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LISTOFFIGURES Figure page 3-1RunningofcouplingsforbothSMandMSSM,one-loopcontr ibutions ......23 3-2RunningofSMmassesfordown-typequarksandchargedlep tons .......24 3-3RunningofMSSMmassesfordown-typequarksandchargedl eptons .....37 6-1CompletelistofKroneckerProductsfor (54). ..................66 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SUSY SU (5)MODELWITHFLAVORGROUP (54) By JesusAlejandroEscobar August2011 Chair:PierreRamondMajor:PhysicsWedesignasupersymmetric SU (5)GUTmodelusing (54),anitenon-abelian subgroupof SU (3) f .Heavyrighthandedneutrinosareintroducedwhichtransfo rm asthree-dimensionalrepresentationofourchosenfamilyg roup.Weproducednovel possibleYukawamatricesforthequarksandtheneutrallept onsectorsthattall currentlyavailabledata.Themodelsuccessfullyreproduc estheYukawamatricesand thusthemasshierarchicalstructuresoftheStandardModel andtheCKMmixingmatrix. Itprovidespredictionsforthelightneutrinoswithanorma lhierarchyandmassessuch that m ,1 5 10 3 eV m ,2 1 10 2 eV ,and m ,3 5 10 2 eV .Wealsoprovide predictionsformassesoftheheavyneutrinos,andcorrecti onstothetri-bimaximal matrixthattwithinexperimentallimits,e.g.,areactora ngleof 7.31 o 9

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CHAPTER1 INTRODUCTION CurrentlytheStandardModelcontainsnineteenparameters thatneedtobeput inbyhand.Evenso,ithashadremarkablepredictivepower,w ithalmostallofits predictionsveried.Thesoleexceptionhasbeenthepredic tionofthescalarparticle calledtheHiggsbosonwhichtheLHCwillverylikelyndinth enearfuture. Evenwithallitssuccesses,weknowthattheStandardModel( SM)asitstandsis notthecompletestory.Forexample,thereissignicantevi denceofpossibleunication ofthegaugecouplingsthatimpliestheunicationofthethr eeSMsymmetries.Also, moreimportanttothispaper,areexperimentsliketheSuper KamiokandeandSudbury NeutrinoObservatory(SNO)thathaveshownconclusivelyth atneutrinooscillations arerealandthusneutrinoshavemass.Thereistheissueofth elackofviolationsof P and CP symmetriesfromthestrongforceindicatingthatthereisso memechanism preventingthese,perhapstheaxion? Here,wewillprimarilyfocusonthequestionoftheoriginso fthethreeGenerations (Families)intheSM.AtitscoretheSMhasseeminglythreeco piesofthesame particles,withthedifferenceslyingsolelyonmasses.Byc opieswemeanthatthere arethreeparticleswhichhavethesamequantumnumbersunde rthegaugedgroups. Forexample,therearethreeparticlesthatonecouldcallth eelectronbutbecausethese havedifferentmasseswerecognizethemtobedifferentpart icles:electron,muonand tau.Thestoryisrepeatedforeveryothermatterparticlein theSM,andsowerealize thattherearethreefamiliesorgenerations(inpresentlan guage).Onecanalsosaythat theparticlescomeindifferent“avors”,alanguageoftena ssociatedwithaselectionof, say,ice-cream. Theoriginofthisavorstructure,i.e.,theoriginofmassh ierarchies,remainsa questionthathasstilleludedphysicists.Veryfewcluesfo rthisstructurecanbefound inthecurrentunderstandingoftheSMbysimplylookingatth equarksectordata. 10

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Datathatcomesinthewayofthequark-mixingmatrix,(alsok nownasthe U ckm or Cabibbo-Kobayashi-Maskawa(CKM)matrix),andquarkmasse s.Theoriginofthe matrixisthedifferencebetweenthemassbasisandtheweakf orcebasis,whichcannot existsimultaneously. Theprocessofputtingparticlesintheirmassbasiswouldbe simpleifitwere notoftheweakinteraction,whichallowsforcommunication ofthetwoofthequark's left-handedstates.Ifwechoosetobeinthemassbasis,thew eakinteractionwill encodethedifferencesinthediagonalizingprocedureofth eup-typeanddown-type quarksintheformof U ckm .Experimentalresultsforthe U ckm showthatthemixingangles arequitesmallandthatthereisoneCPviolatingphase 1 Mathematically,thequarkmasses(eigenvaluesofthemassb asis)andthe U ckm shouldbeindependentquantities.However,Gattoetal.[ 17 ]foundthatcertainzeros inthemassmatricesorYukawamatricescanconnectthesetwo otherwiseseparate concepts.SothatonecanpredicttheCabibboanglefromrati osofthedown-typequark masses.Theconclusionwastherstpredictionsoftheangle sintheCKMmatrix.Itwas alsorealizedthattoprotectthesezerossymmetrieswerene eded,butwhich? Progressinneutrinophysicshasyieldedcluestothequesti onoftheresponsible symmetry.Neutrinooscillationcanonlybepossibleinthec ontextofmassiveneutrinos, whichrequirestheadditionofrighthandedneutrinos.Ifth eserighthandedneutrinosare heavy,thenwearefreetousewhatisknownasthesee-sawmech anismandproduce quitesmallmassesfortheobservedneutrinos[ 28 31 39 ].Asadirectconsequence, andtheneedtoputtheseneutrinosinthemassbasis,onends thattheweak-basis 1 ThediscreteCPsymmetrycouldbeviolatedifthequark-mixi ngmatrixhasa complexphase.BothKoboyashiandMaskawa(CKM)showedthat fortheretobea complexphasetheremustbealeastthreegenerations.Theev idenceofCPviolating phasecamefromK-mesonoscillationswhichledthemtopredi ctthreegenerationsand sonewquarks.Theywonthenobleprizein2008forthiswork. 11

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neutrinosareactuallyasuper-positionofmass-basisneut rinos.Sothatanobserver makingameasurementadistanceawayfromaneutrinosourcew illobservethat neutrinososcillate,i.e.,changefromoneeigen-basisneu trinotoanother. Aspreviouslymentionedcurrentexperimentaldataclearly demonstratesthat neutrinooscillationdoesoccurandasaconsequencetherea remassiveneutrinos. ConsequentlythedatademonstratesthattheMNSPmatrixapp roximatelytakesona formcalledthetri-bimaximalmatrix[ 22 ].Thestructureofthismatrix,unliketheCKM matrixshowsveryremarkablepatterns,wheretheanglesreq uiredarelarge.The sizessuggestthatanaturalexplanationisathree-dimensi onalrepresentationofsome symmetry.Sothatonemayconcludethatweareseekingsymmet rieswhichcontain thesethree-dimensionalrepresentations. Inthispaperwewillreviewinsomedetailthebackgroundinf ormationrequired tounderstandthatmodelwehaveproduced.Wewillelaborate andmotivatekey mechanismsthatareimportanttoamodelbuilderwhoistryin gtounderstandtheavor problem.Afterwards,wepresentasymmetrygroup, (6 n 2 )andaspecialcasewhere n =3,asaproposedgroupforsolvingtheavorproblem[ 14 ].Anotherchapterwill containnewmaterialwithrespecttotheYukawamatrices,i. e.,wepresentboundson theYukawamatricesforthequarksandmakingchoicesforthe leptons.Producing Yukawamatriceswillprovidesomeofthebackbonetothemode lwepropose.Thenal chapterwillbethefullmodelthattriestoexplainSMmasses andhierarchicalstructures aswellasmakepredictionsforneutrinomassesandreactora ngle[ 13 ]. 12

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CHAPTER2 THESTANDARDMODEL SincewefullyintendtoexpandwhatisreferredtoastheStan dardModelof physics,asomewhatincompletedescriptionofitisinorder .Ourgoalhereistoreview theStandardModelandintroducenotationthatwillbecommo nlyusedthroughout.We beginbyrstdiscussingthesymmetriesoftheSMfollowedby listingtheparticle content.Wethenproceedtodiscusssomeofthenineteenpara meterswehave mentionedattheintroduction. 2.1SymmetriesandForceCarriers Inphysicsonendsthatnaturemakesuseofthemathematical descriptionof symmetry.Themostclearestexampleisinthedescriptionof threeofthefourknown forcesofnature.Whereonecanusethelanguageofalocal(ga uged)symmetryto fullydescribeexistenceandeffectofaforcecarrier.Part iclesthatarenotforcecarriers aretheneitherneutralorchargedunderthatsymmetry,mean ingtheyfallundera representationofthesymmetry. 1 Thisthenpinsdowntheirinteractionswithnotonlythe forcecarriersbutwithotherparticles.Sothatasimpledes criptionofasymmetryhas hugeimplicationsonthedynamicsofatheory. Table 2-1 containsthelocalsymmetriesoftheSM,thecorrespondingg eneratorsof thesymmetry,thelabelsusedforthelocalspace-timeforce carriers,andthecoupling constantsthatwillbeusedthroughout. Themattercontent,thosethatarenotforcecarriers,willf allunderrepresentations ofoneormoreoftheeldsgiveninTable 2-1 .Weshouldmentionthatitisconceivable thatthereareparticlesthatdonotfallunderrepresentati onsofthesymmetriesfoundin 1 Wewilloftensaythataeldisa“representation”ofasymmet rytoincludea U (1) symmetryeventhoughonewouldmorereadilysaytheeldis“c harged”underthe U (1). 13

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Table2-1.GaugedsymmetriesoftheSM. SymmetryFieldGeneratorCouplingconstantSU (3) G a ( x ) a g 3 SU (2) W a ( x ) a g 2 U (1) B ( x ) g 1 Thesuperscripts a areindicesassociatedwithadjointrepresentationofthes ymmetry group.For SU (3)wehave a =1,2,...,8andfor SU (2)wehave a =1,2,3. theTable 2-1 .Apossibilitythatmaybethecasefordarkmatter,andalsof ormassive righthandedneutrinos.WeshalldiscussthisindetailinCh apter 3 Finally,theauthorwillrefertheinterestedreadertoothe rsourcesforthedetailsof theSMlagrangian,e.g.,in[ 32 36 ]. 2.2ParticleContent ItispossibletounderstandtheparticlecontentoftheSMby identifyingtheir transformationpropertiesunderthesymmetriesintroduce dinthelastsection.Wedo soinTable 2-2 ,wherewelistthefermionicparticlecontentfoundinbotht heSMandits supersymmetricextension,i.e.,theMinimalSupersymmetr icStandardModel(MSSM). Supersymmetry,asweshalldiscusslater,doublestheparti clecontentofthetheoryby introducingthescalarpartnersofthefermions. Thetheoryischiralmeaningthatweidentifyhandednessand treatleft-handed particlesasdifferenttothosethatareright-handed.Expl icitlywemeanthatthe right-handedparticlesasthesingletsunderthe SU (2)symmetrywhiledoublets areleft-handed.Thepossibilityofachiraltheoryisadire ctresultofthespace-time symmetry SO (3,1)whichisisomorphicto SU (2) SU (2).However,itispossible, withoutanylosses,torewritetheright-handedparticlesi nleft-handednotation.This simpliesnotationandbyextensionthelagrangianagreatd eal,andsothisisthe approachwetake. 14

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Table2-2.ParticleContentoftheSM. ParticleTypeParticleGaugeReps. Leptonweakdoublet L e ( 1 c 2 ) 1 Leptonweaksinglet e ( 1 c 1 ) 2 Quarkweakdoublet Q u d ( 3 c 2 ) 1 = 3 Quarkweaksinglet u ( 3 c 1 ) 4 = 3 Quarkweaksinglet d ( 3 c 1 ) 2 = 3 UpHiggsweakdoublet H h + h 0 ( 1 c 2 ) 1 SMparticleslistedwiththeircorrespondingtransformati onpropertiesunderthegauged symmetries.Therighthandedparticlesarewritteninlefthandednotation.Numbers withasuperscript c areunderstoodtoberepresentationsof SU (3). AlookatTable 2-2 willrevealthatwehavechosenthehyperchargeassignments so thatwemayusethefollowingdenitionforcharge 2 Q = I 3 + 1 2 Y ,(2–1) where Q isthechargeoperator, I 3 isthethirdcomponentofisospin,and Y isthe electroweakhyperchargeassignment.Thiswillbecomemore importantwhenwe discusselectroweaksymmetrybreaking. Finally,weshouldmentionthatinteractionsarepresented inalagrangianasterms thataresingletsunderallsymmetries.Theseincludespace -timesymmetriesandthe gaugedSMsymmetriesofthelastsection.Higherorderedope rators,i.e.,effective 2 Inliteratureonegenerallyndsoneoftwowaystoassignhyp ercharge.Oneis introducedhere,theanotherwouldbe Q = I 3 + Y 15

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terms,inheritthesesymmetriesandsomustalsobeinvarian tunderallsymmetries discussed.Afactthatplaysacentralroleinfuturesection s. 2.3TheYukawaSector Thequestionofmassisourcentralproblem,theSMencodesin formationregarding massesinwhatisknownastheYukawaSector.Thesectoriscom posedofthree separateinteractionterms: L yukawa = Q i Y uij u j H + Q i Y dij d j H + L i Y eij e j H ,(2–2) with H = i 2 H ,(2–3) whereby H wemeanthecomplexconjugateoftheeldandwehavepurposel y suppressedallgaugeindices.The i and j subscriptslabelthethreegenerationsofthe SMtobediscussedinthenextsection. IngeneraltherearenorelationshipsbetweentheYukawamat ricesandthey correspondtotwenty-sevenfreeparameters.Mostofthesep arametersarenotphysical andcanbereducedtothirteen(9massesand3anglesand1phas e)becauseonecan diagonalizeintoeachparticles'typemassbasis. Fourofthethirteenphysicalparametersarethreeanglesan donephase.These areleftoverfromthediagonalizationinthemassbasis,whi chintheYukawasectors doeslittle.Butweakinteractions,wheretheentriesofany oneoftheleft-handed two-dimensionalrepresentationsmixwitheachother,prod uceamixingmatrixthat representstheangulardifferencesbetweenthediagonaliz ations. Thematrixwiththesefourparametersthatappearsinweakin teractionsiscalled thequarkmixingmatrix,ormorecommonlytheCKMmatrix, U ckm .Oneparametrization, wheretheanglesareexplicitlydened,is 16

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U ckm = 0BBBB@ c 12 c 13 s 12 c 13 s 13 e i s 12 c 23 c 12 s 23 s 13 e i c 12 c 23 s 12 s 23 s 13 e i s 23 c 13 s 12 s 23 c 12 c 23 s 13 e i c 12 s 23 s 12 c 23 s 13 e i c 23 c 13 1CCCCA ,(2–4) where c ij standsfor cos ( ij ),and 12 15.3635 o 23 2.8214 o 13 30.958 o 81.8699[ 29 ].Itshouldbenotedthatanglesarequitesmall,speciallyw hen comparedtotheanglesfoundintheleptonmixingmatrix(tob ediscussedshortly). Thesmallangle 12 ,oftenwritteninradiansas .225,allowsforparametrization oftheCKMmatrixtoordersof calledtheWolfenstein'sparametrization[ 38 ].TheCKM alaWolfensteinto O ( 3 )is 0BBBB@ 1 2 2 A 3 ( i ) 1 2 2 A 2 A 3 (1 i ) A 2 1 1CCCCA ,(2–5) where =.2253 0.0007istheCabibboangle, A =.808 +0.022 0.015 0.132 +0.022 0.014 ,and 0.341 0.013with (1 2 = 2), (1 2 = 2)[ 29 ].Oneseesexplicitlyfromthe abovethattheCKMmatrixisinfactclosetobeingunity.Howe ver,thesignicanceofthe aboveisthepossibilitythatonemayparametrizethematrix withasinglevariable ,all theotherstreatedmoreasnumericalfactors.Thusatheoryt hatcanexplaingenerations andthequarkmixingmatrixshouldbeabletoaccountfortheo rderedstructurein of theCKMmatrixandexplainthespecialsignicanceofthisan gle. 2.4ElectroweakBreakingandMasses Theprevioussectiondiscussedinsomedetailthelagrangia ntermsthatare ultimatelyresponsiblefortherebeingmasses.However,as writteninEquation( 2–2 ),will notgiveparticlesmassesbutsimplydescribeinteractions betweenthreedifferentelds. Fortheretobemasstermsforthequarksandleptonstheeldk nowsastheHiggseld mustobtainavacuumexpectationvalue(vev). 17

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Ofcourse,fortheretobeinteractionwiththeHiggseldmea nsthattheHiggsmust bechargedundertheappropriatesymmetries.SincetheHigg smustdevelopavacuum valueonemustcompromisethesesymmetriesorinotherwords theymustbebroken. ThelinkbetweengivingavevtotheHiggsandbreakingitssym metriesissodirect thatwewillusethetermbreakingtosimultaneouslymeanbot handsoavoidfuture confusion.2.4.1TheHiggsMechanism Foraeldtoobtainavevmeansthatthecongurationofitspo tentialmustbein suchawaythatthelowestenergystateavailabletoitisnontrivial.Typicalscalarelds liketheHiggshaveapotentialwithselfinteractiontermst hatarequadraticandquartic only(invariantundersymmetriesandrenormalizableaswel l): V ( H y H )= m 2 h H y H + h ( H y H ) 2 ,(2–6) wherethesigninfrontof m 2 h isnegativeforreasonstobediscussedpresently. Thesignsofatypicalpotentialofthisformaretakensuchth attheenergiesare boundedfrombelowatlargeenergies,thusthequarticterm( h )ispositive.Thesignin frontofthemasssquaredtermontheotherhandcouldbeeithe rpositiveornegative. Apositivemasssquaredterm( m 2 h ),thoughphysical,wouldmeanthattheminimum ofthepotentialenergywouldcorrespondtoatriviallowest energyHiggsconguration. AnegativemasstermasfoundinEquation( 2–6 )correspondstoalowestenergystate inwhichtheHiggsisnolongertrivial.Anontrivialloweste nergystateiswhatweare seeking.Wewillnotdiscussanydynamicalwayofmakingthem asstermnegative,so wewillprimarilybetalkingaboutspontaneousbreaking. Attheminimumofthepotential,orthevacuumofthetheory,o fEquation( 2–6 )we wishtodenethestateoftheHiggstotaketheform 18

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H 0B@ 0 v + h 1CA ,(2–7) whereweidentifythevalue v asthevacuumvalueoftheeld.Itshouldbenotedthat thischoiceguaranteesthatourchoiceofchargeoperatoras givenbyEquation( 2–1 )is preserved,adirectconsequenceofbreakingsymmetry. Thus,wemaynowspeakofsymmetrybreaking.Itistruethatat lowenergyscales wemaybehappytodescribetheforcesofnatureas SU (3) U (1) EM ,wherethe SU (3) isthesamelistedinTable 2-1 and U (1) EM isthedescriptionofelectromagnetism. 3 This wouldmeanthatsomehowwehave SU (3) SU (2) U (1) 7! SU (3) U (1) EM ,(2–8) orsimplynotethat SU (2) U (1) 7! U (1) EM .(2–9) Suggestingthatthecombination SU (2) U (1)getsbrokentoasinglesymmetry.Thisis whatismeantbyelectroweakunication,i.e.,electromagn etismisuniedwiththeweak force(responsiblefordecaysofnuclei).Asdiscussedatth ebeginningweexpected suchabreakingofsymmetrytooccur,andasmentionedabove, therightchoiceof vevfortheHiggsexactlybreakstherightamountofsymmetri eswhilepreservingthe electromagneticcharge Q orinotherwords U (1) EM .Wewillnotgointothedetailsofthe 3 Baryonicmatterexistsbecauseofatlowenergies SU (3)boundstatesincludethe stablecongurationsofthreequarkssuchastheprotonandn eutron.Today,alow energydescriptionofthe SU (3)force,alsoknownasQCD,isnottotallyunderstood duetothenon-perturbativenatureofthetheoryatscalesbe low QCD 220 MeV [ 32 ]. 19

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Table2-3.GenerationsoftheSMandparticlemasses. GenerationIGenerationIIGenerationIII e m e =? m =? m =? e m e =.511 MeV m =105.66 MeV m =1.778 GeV u m u =1.5 to 3 MeVc m c =1.25 0.09 GeVt m t =174.2 3.3 GeV d m d =3 to 7 MeVs m s =95 25 MeVb m b =4.20 0.07 GeV Particlesareorganizedaccordingtotheirgeneration.Mas sesarealsolistedtocurrent experimentalvalues[ 29 ]. breakingprocesshere,butwewilldiscusstheresultsofbot hsymmetrybreakingand massproduction,whichcollectivelyiswhatwecalltheHigg sMechanism[ 12 23 ]. 2.4.2Masses WediscussedinsomedetailtheYukawasectorinSection 2.3 andwenotedthat thissectorisultimatelyresponsibleformassesofthequar ks.TheHiggsmechanism doesthisbymodifyingEquation( 2–2 )to L yukawa 7! u i m uij u j + d i m dij d j + e i m eij e j (2–10) by v Y u 7! m u v Y d 7! m d v Y e 7! m e ,(2–11) wherewechoosetoignoreinteractionswiththeHiggseld h ,presumablytheeldwe aresearchingforintheLargeHadronCollider(LHC).Asment ioned,themassmatrices m willeachbediagonalizedintheirrespectivesectorsandth uswewillhaveeach particleintheirmassbasis.Experimentallyonecanmeasur ethesenineparameters andproduceTable 2-3 .Themassesfoundinthetablearethosewewishtoreproduce alongwithmakingpredictionsforthemassesoftheneutrino s. Finally,becauseofthesymmetrybroken,viatheHiggsMecha nism,forcecarriers alsoreceivemasses.Thisisprimarilyduetothefactthatth epresenceofalocal symmetryimpliesamodicationtothekinetictermsofthee ldswhichinvolve 20

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derivatives.Thesederivativesaremodiedtowhatarecall edcovariantderivatives whichnowincludeinteractiontermswiththeforcecarriers ,e.g., D H =( @ i g 2 W a a i g 1 Y 2 B ) H ,(2–12) where g 2 istheweakeldcouplingconstant, g 1 isthehyperchagecouplingconstant, and a arethegeneratorsof SU (2)(foundinTable 2-1 ).Itshouldbeevidentthatthe combinationofthekinetictermandtheacquisitionofavevb ytheHiggsproduces massesforsomeoftheforcecarriers,namelythe W + W Z Theseeldsrepresentalinearcombinationofeldspresent edinTable 2-1 ,i.e.,for thersttwoeldswehave W = 1 p 2 W 1 i W 2 .(2–13) The Z alongwithelectromagneticpotentialeld A canbewrittenas 0B@ Z A 1CA = 0B@ cos ( W ) sin ( W ) sin ( W ) cos ( W ) 1CA 0B@ W 3 B 1CA ,(2–14) where W iscalledtheWeinbergangleanddenedinEquation( 2–15 ).Thus wehavethreemassiveforcecarriers,whichrepresenttheth reebrokensymmetries viatheHiggsmechanism,andonemasslessforcecarrier( A )whichweknowtobe electromagnetism. Finally,weshouldlistsomeconstantsthatareadirectresu ltofsymmetrybreaking: cos ( W )= g 2 p g 2 1 + g 2 2 m W = vg 2 2 m Z = v p g 2 1 + g 2 2 2 e = g 1 g 2 p g 2 1 + g 2 2 (2–15) where W istheWeinbergangle, m w =80.398 Mev isthemassoftheweakforce W ( x ), m Z =91.188 Mev isthemassofthe Z ( x )boson,and e istheelectricchargeof anelectron[ 29 ]. 21

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CHAPTER3 BEYONDTHESTANDARDMODEL AspresentedinChapter 2 theSMwouldbeconsideredincompletebecause thereareseveralkeyissueswiththetheorythatleadustobe lievethatitisnotthe fundamentaltheoryphysicistarelookingfor.Forstarters itfailstoincludegravity,since theSMiscomposedofrenormalizableeldcontent.Gravity, ontheotherhand,is notarenormalizabletheoryandsoasaconsequencecannotbe placedinthesame description. 1 TheinabilityofreconcilegravitywithQuantumFieldTheor y(QFT)isour rstcluethatourunderstandingisincomplete. Inthischapterweinvestigatefurthercluesthatwillbecom eimportanttoouravor discussion.Fortherstsectionwefollowcluesthatleadsu stopossibleunicationof forcesathigh-energyscales.Thesecluesareduetotherunn ingofthegaugegroup couplingconstants.Wethenproceedtoneutrinophysicswhe reweareledunderstand thattheremaybesomesymmetrythatisresponsiblefortheob servedmassesinthe SM.Finallyweendthechapterwithabriefintroductiontosu persymmetry.Itisthe solutiontotheproblemofcontrollingtherunawayultravio letcorrectionstothemassof theHiggsparticle,knownasthehierarchyproblem. 3.1UnicationatLargeScales QFTpredictsthatcouplingconstantsarefunctionsofenerg yscale.Calculations showthatthecouplingsoftheforcestendtocross,orunify, atarelativelysmallarea atenergiesofabout 10 15 GeV ,seeFigure 3-1A .Supersymmetry,discussedin Section 3.5 ,makestheareasmalleranddenestheenergyofunicationt obe 2 10 16 GeV ,seeFigure 3-1B .Weusedthewordunicationbecauseatthese high-energypointsthereexiststhepossibilitythattheth reeforcesmaycoalescetoa 1 Itshouldberememberedthatthedescriptionofmatterisqua ntummechanicalwhile thatofgravityisclassically.Einstein'sequationrelate sthesetwoconceptsandsoasis mustbothbedescribedviaquantummechanics. 22

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ASM BMSSM Figure3-1.Runningofcouplingsforboth A )SMand B )MSSM,one-loopcontributions. Notethatthereisanareawhereunicationseemstotakeplac enear energiesof10 15 GeV forSMand2 10 16 forMSSM. singlelargergaugesymmetrythataccommodatesallthreefo rces.Thus,unication meansthatthreeforcesoriginatefromoneforce,providing thereasonwhyonlyone couplingwouldbeneeded. Anothermotivationfortherebeingaunieddescriptionoft heforcesisthe quantizationofelectriccharges.Atthelevelofthestanda rdmodel,chargeissimply anoperatordenedbythegeneratorsoftheSM,seeEquation( 2–1 ).However,a generalfeatureofunicationisanexplanationtothequant izationbythetraceless conditionofthegeneratorsofagroup[ 18 19 36 ]. WiththeseclueswemayseekoutanencompassingLiegrouptha tisatleast rank-4(therankoftheSM),whichturnsouttoproducenineca ndidates,[ 19 ].Twoof thesecandidatesadmitcomplexrepresentationsandonlyon eofthemcanexplain quantizationofchargesforbothleptonsandquarks[ 19 ].Thegroupis SU (5),butthis isnottosaythatitstheonlypossiblepathtotheSM.Othergr oups,like,e.g., SO (10) canembedandbreaktoproduce SU (5)orelsecandirectlyreproducetheSMwithout havingtoevertoproduce SU (5),e.g.,Pati-Salamlikemodels[ 30 ]. 23

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A m e and m d B m and m s C m and m b D m e m d E m m s F m m b Figure3-2.RunningofSMmassesfordown-typequarksandcha rgedleptons. A C :We comparethemassesofdown-typequarksandchargedleptonsf oreach familyasthemassesareplottedasfunctionofenergyscale. D F :Theratio ofthemassesaregivenasfunctionofenergyscale. 3.2SM:RunningofMasses ItispossibletorunthemassesoftheSMtolargeenergies[ 2 – 4 9 25 ].Recall, thatintheSMmassesoriginatefromthebreakingoftheHiggs atthescaleofaround 100 GeV .Sowhenwesaywearerunningmasses,whatwearetrulydoingi srunning thesizeofthediagonalizedYukawacouplingsmultipliedby thevevoftheHiggs.The runningmayprovidecluesastowhethersomeunifyingscheme maymanifestitself.In thediscussionthatfollowswemakenotethattheplotsofthe runningarealltoone-loop orderandsoadmitsomeerror. AlookatFigure 3-2 willrevealthreewellchosenplotsfortherunningofthemas ses ofthedown-typequarksandthechargedleptons.Eachplotco mparesbothtypeof 24

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particlesintheircorrespondingfamily,withsomesuggest ivefeatures.FirstFigure 3-2C whichistherunningofthe m and m b ,suggeststhattheremaybeunicationofthese masses.ThisfactisshownmoreclearlyintheFigure 3-2F wherewehaveplottedthe ratioofthesetwo.Thusatunicationscale, m m b 1.(3–1) SimilarrelationscanbeconcludedfromFigures 3-2D and 3-2E : m e m d 1 3 m m s 3,(3–2) wheresomeexplanationisneededfortherstrelation.Onem aywonderwhywe havechosenthat m e = m d 1 = 3insteadof1 = 4.WeshallseeinChapter 4 ,involving adescriptionof SU (5),thatthereisasimplemechanismforreproducing m = m b and m = m s exactlybutwith m e = m d 1 = 3. 2 3.3ABriefHistoryofNeutrinoMasses TheSMdoesnotadmitmassesfortheneutrinosforseveralrea sons.First,aDirac typeofmasswouldrequiretheadditionofatypeofneutrinon otobservedasofyet. Second,aMajoranatypeofmasscontributionisforbiddenby theelectroweaksymmetry. Also,aneffectiveMajoranamasswouldeitherviolatelepto nnumber( L ),baryonminus leptonnumber( B L ),orwouldviolaterelativeleptonnumber[ 32 ].Infact,foralong timeitwasthoughtthatthemassesoftheneutrinoswereexac tlyzero. Beforediscussingtheoreticalaspectsinvolvedinneutrin omasses,letusrst introduceaverybriefhistoryofneutrinophysics. (1930)WolfgangPaulipredictstheexistenceofa“neutron” inanattemptto preservetheconservationofmassandmomentum,whichother wiseseemedtobe violatedinbetadecay. 2 RememberthatforFigure 3-2 weonlyincludeuptoone-loopcontributions. 25

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(1934)EnricoFermicoinedthename“neutrino”. (1956)Inwhatisknowntodayasthe“Cowan-ReinesNeutrinoE xperiment” ClydeCowan,FrederickReines,F.B.Harrison,H.W.Kruse,a ndA.D.McGuire, publishtheirconrmationofneutrinodetection.Theproce ssofdetectioninvolved inverse-betadecaywheredetectionwastriggeredbyadelay ofdetectionof gammasfromneutronabsorptionandgammadetectionfrompos itron-electron annihilation. (1962)ThesecondavorwasshowntoexistbyLeonM.Lederman ,Melvin SchwartzandJackSteinberger. (1968)The“HomesteakExperiment”showsadecitofabouttw o-thirdsinthe predictedsolarneutrinoux.Theproblemwascalledthe“so larneutrinoproblem”. Therstindicationofneutrinooscillations. (1975)Thetauleptondiscoveredandthirdneutrinohypothe sized. (1988)Severalexperimentsconclusivelyshowthatneutrin ososcillate. (2000)3 rd avordiscoveredbyDonutCollaborationatFermilab. (2002)DatagatheredbySNOexperimentleadP.F.Harrison,D .H.Perkinsand W.G.Scotttopostulatethatthelepton-mixingmatrixtwha tisknowasthe tri-bimaximalmixingscheme[ 22 ]. 3.4Theory:NeutrinoMasses Wementionedthatthepossibilityofneutrinomassesrequir esfromusthe introductionofright-handedneutrinos.Suchaneutrinoca nproduceaDiracmass termoftheform L i Y n ij N j H i j =1,2,3.(3–3) ForsuchatermtobeinvariantunderSMsymmetries,oneneeds that N j ( 1 c 1 ) 0 ,(3–4) containingnoSMcharges.Weassumedthatthenumberofsuchn eutrinosisthesame numberasthegenerations,i.e.,three.Wewillmaketheassu mptionfortheduration 26

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ofthepaper.(InChapter 7 ourmodelhasallthreesuchneutrinosuniedundera three-dimensionalrepresentationofsomeavorsymmetry. ) Weexpectthatsuchaparticlesas N abovebeneutralor“sterile”undertheSM. 3 TheneutrinobeingneutraltotheSMposesseveralpotential problems,withtherst beingthatthereisnodirectwaytointeractwithsuchaparti cle.Soitsexistenceis circumstantialatbestwithonlyindirectevidencebywayof neutrinomasses. Thedeterminationofmass,however,hasbeenquiteachallen ge.Thedifculty becomesevenmoreclearwhenoneconsidersthattheveryexis tenceoftheneutrino haditselfbeenindoubtwithonlyrelativelyrecentlyindir ectevidenceoftheexistenceof amassforneutrinoshasbeendetermined[ 29 ],seeSection 3.4.3 Anotherimportantfact,directlyrelatedtheparticlebein gneutralundertheSM,is thepossibilitytohaveaMajoranamassterm: N i M n ij N j .(3–5) Onoccasionwewillwrite M n as M Y maj .Theconstant M willbetheoverallscale oftheheaviestMajoranamass.ThepossibilityofhavingaMa joranamasswill haveconsequencesastothesizeofthemeasuredneutrinomas ses,thiswillbe discussedinSection 3.4.1 .Itshouldbementioned,andwewillreturntothislater, thatEquation( 3–3 )allowsforanothersourceoflepton-numberviolation,whi ch willmeanthatleptogenesisdataandmodelscanhavesomethi ngtosayaboutboth Equation( 3–3 )andEquation( 3–5 )[ 5 – 8 ]. 3 Thetermsterilethoughcorrectisoftenassociatedinliter aturewithaverylight right-handedneutrino.Asweshallsoonsee,thereisnoreas onwhythisneutrinocannot beheavy.So,toavoidconfusionweshalladopttheterminolo gyofaddingtheword “heavy”asinheavyright-handedorsimplyheavyneutrinoin steadof“sterile”whenwe meantosaylargemassneutrino.Fortheremainderofthepape rpaperwewillalmost exclusivelyconsidertherighthandedneutrinostobeofthe heavysort. 27

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3.4.1TheSee-SawMechanism WhenonewritesdowntheYukawatermEquation( 3–3 )itisdifculttounderstand whythemassesofneutrinosaresosmall.Fundamentallyther eisverylittledifferent fromthestructureofthechargedleptonYukawatermwithmas sescomingfromthevev oftheHiggs.Sosomeexplanationmustbefoundtoexplainthe largedifferencesofthe massesfromneutrinosandthoseofthechargedleptons. Todoso,wemakeuseofthepossibilitythatMajorananeutrin omassescanbe large.Theneutrinosectorlookslike i vY n ij N j + N i M n ij N j ,(3–6) wherewehavetakenthelibertyofapplyingelectroweaksymm etrybreaking.Both contributionscanbeputtogetherinamassmatrix T N T 0B@ 0 vY n vY n T M Y maj 1CA 0B@ N 1CA ,(3–7) with 0 beingathreebythreezeromatrix.Itisimportanttoremembe rthatthe 0 term isprotectedbythesymmetriesoftheSM,andthatthe M islargerthanthevevof theHiggs v .TheresulttheniscalledtheSee-SawMechanism[ 28 31 39 ].Forin themechanismoneisleftwithaverymassiveandverylightse tofneutrinosafter diagonalizationofEquation( 3–7 ). Thelightneutrinoapproximationis: Y v 2 M Y n Y maj 1 Y nT .(3–8) Itisthematrixabovethatproducesthelightmasses,andone canseethatbecause theoverallscaleisdened v 2 = M .Sowehaveanexplanationastowhythesizethe neutrinosaresosmall,theyaresuppressedbythemassofthe heavyneutrinos. 28

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Whatislefttodonowisdiagonalize Y : Y = U m U T ,(3–9) wherewewillidentify U asthetri-bimaximalmatrixinSection 3.4.3 .The U mnsp matrix howeverwillbeidentiedas[ 32 ]: U mnsp = U e U ,(3–10) wherethematrix U e istheresultofdiagonalizationofthechargedleptonmassm atrix. 3.4.2TheTri-BimaximalMatrix In2002experimentalevidencefromtheSNOExperimentledHa rrisonetal.[ 22 ]to suggestthetri-bimaximalmatrix: U mnsp U tri-bi = 0BBBBBB@ r 2 3 1 p 3 0 1 p 6 1 p 3 1 p 2 1 p 6 1 p 3 1 p 2 1CCCCCCA ,(3–11) asanapproximationtothelepton-mixingmatrix.Theabovec orrespondstoanglesof 13 =0, =35.26 o atm = 45 o ,(3–12) whichareapproximatelyclosetotheexperimentalresultso f j 13 j < 11.4 o 34.34 +1.35 1.22 o ,36.8 o < atm < 53.2 o (3–13) asfoundin[ 29 ].Oneofthepurposesofanymodel,includingtheonefoundin this paper,thattriestoexplainavormustincludepredictions forthereactorangle 13 .All theotheranglesmusttexperimentalresultswithalltheco rrectionsincluded.Thuswe shalltaketheanglesinEquation( 3–12 )asthestartingpointandprovidecorrectionsto themprovidedbyEquation( 3–10 ). 29

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Weendbymentioningthattheanglesarelargerthatthosefou ndintheCKM matrix.TherealsodoesnotseemtobeGattolikerelations[ 17 ](discussedinChapter 5 ) whichrelatestheeigenvaluesofmassestotheangles.Themo stnaturalexplanation thatwouldexplainEquation( 5–19 )wouldbethree-dimensionalrepresentationsofsome symmetry.Asweshalldiscusslater,thesymmetryweseekwou ldbeanite(discrete) symmetryandwillthenhavetobeasubgroupofeither SO (3)or SU (3). 3.4.3NeutrinoOscillations Theresolutiontothe“solarneutrinoproblem”wasthatanyn eutrinospecieswill undergoachangeoroscillationbetweenthedifferentpossi bleneutrinospecies.The problembecomessomewhatdifculttounderstandifoneonly considersneutrinosinthe contextoftheirmassbasis. Whenneutrinos,e.g.,thoseproducedinthesun,arecreated viabetadecayor someweakprocess,theejectedneutrinoisnotreallyinthem assbasis.Instead,what wendisthattheejectedneutrinoisactuallyasuperpositi onofvariousmassstates,a consequenceofthefactthatwecannotmeasurethemassesofa nyindividualneutrino. Consequentlywhenwedetectandmeasureaneutrino,sayonee jectedfromthesun, becauseofquantummechanics,theneutrinowillchangeitss tateovertime. Thereasonwhythereevenwasa“solarneutrinoproblem”isap urelyquantum mechanical.Asanexplicitexample,takenfrom[ 32 ],ofhowoscillationsworkwetake thecaseofjusttwoavorswhichcanthenbeextendedtothree avors. Firstweestablishthestatesofbothbasis,i.e.,weakandma ssbasis: j i j i i = e , i =1,2, j i = U mnsp i j i i ,(3–14) and U mnsp = 0B@ cos( 12 )sin( 12 ) sin( 12 )cos( 12 ) 1CA ,(3–15) 30

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whereneutrinoslabeledwith1,2areinthemassbasisandtho selabeledbyGreek lettersareintheweakbasis.Wenotethattheleptonmixingm atrixprovidesthe connectionbetweenthetwobasis.However,theaboveisonly trueatthetimeof particlecreationandweareinterestedtowhathappenswhen theparticletraverses somedistanceandthendetected. Quantummechanicsprovidestheanswerbygivingushowtopro pagateaparticle intimegivenaHamiltonian.Forourproblemaneutrinobarel yinteractsso j i ( t ) i = U ( t ) ij j j (0) i U ( t )= 0B@ e i E 1 t 0 0 e i E 2 t 1CA E i = q P 2 i + m 2 i .(3–16) Sowendthat j ( t ) i = U mnsp i U ( t ) ij U mnsp yj j (0) i .(3–17) Ifanelectronneutrinowasemitted,e.g.,fromthesun,then theprobabilityofthat electronconvertingtoamuontypeneutrinowouldbegivenby jh j e ( t ) ij 2 =sin 2 ( 12 )sin 2 ( E 2 t ),(3–18) with E = E 2 E 1 m 2 21 2 P ,with P 1 P 2 = P ,and m 2 21 = m 2 2 m 2 1 .(3–19) wherewehaveusedtherelativisticapproximationtotheene rgyforsmallmasses. Finally,wemaywriteEquation( 3–18 )as jh j e ( t ) ij 2 =sin 2 ( 12 )sin 2 ( 2 L t ), 2 L m 2 21 4 P ,(3–20) 31

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asfoundin[ 32 ].Atthedetectorlevel,onecanonlydeterminethevalueoft hemass squareddifferencesandnotthemassesdirectly,providing somedifculty.Todaywe knowthevalueofthemasssquareddifferencestobe[ 29 ]: m 2 21 7.59 +.19 .21 10 5 eV 2 j m 2 23 j 2.43 .13 10 3 eV 2 ,(3–21) wherebecausewedonotknowthesignofthelatterdifference wedonotknowfor certainthetypeofhierarchy,i.e.,whethernormalorinver tedhierarchy. 3.5MinimalSupersymmetricStandardModel AlmostimmediatelyafterthedevelopmentoftheHiggsmecha nism,itwasrealized thatthemassoftheHiggsisdifculttounderstandathigh-e nergies.Thereason beingthatthemattercontent,primarilyfermions,wouldco ntributetermslinearand negativeinthecut-off,agreatproblem.Aneatresolutiont otheproblemisprovidedby supersymmetry,whichrequiresthattherebeequalnumberof bothfermionsandscalars inthetheory.Thehierarchyproblem,asitisknown,canbere solvedbecausescalars contributethesamecut-offbutwithapositivesign,thusca ncellingoutthelargeleading orderdivergences[ 27 ]. Theparticlecontentofsuchanewtheory,onewithsupersymm etry,requiresan expansionofparticlecontentthatwouldotherwisenotbere quired.FortheSMwith addedsupersymmetryonedoublestheparticlecontent.Weca nseethedoublingby lookingatTable 3-1 whereoriginallywehadonlyscalargaugeparticlesbutnowi nclude theirsupersymmetricfermionicpartnerscalledgauginos. 4 ThemattercontentfoundinTable 3-2 showsasimilarstory.However,withone importantdifference,werequiretheinclusionofmorethan oneHiggsparticle.Such 4 Oneaddsthesufx“-ino”tothenameofaSMscalareldwhenna mingits supersymetricpartner. 32

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Table3-1.GaugedsymmetriesoftheMSSM. SymmetryFieldGauginosGeneratorCouplingconstantSU (3) G a ( x ) f G a ( x ) a g 3 SU (2) W a ( x ) f W a ( x ) a g 2 U (1) B ( x ) f B ( x ) g 1 Thesuperscripts a areindicesassociatedwithadjointrepresentationofthes ymmetry group.For SU (3)wehave a =1,2,...,8andfor SU (2)wehave a =1,2,3. Table3-2.ParticleContentoftheMSSM. ParticleTypeParticleSparticleGaugerepresetantion Leptonweakdoublet L e eL e e e ( 1 c 2 ) 1 Leptonweaksinglet e ee ( 1 c 1 ) 2 Quarkweakdoublet Q u d e Q e u e d ( 3 c 2 ) 1 = 3 Quarkweaksinglet u e u ( 3 c 1 ) 4 = 3 Quarkweaksinglet d e d ( 3 c 1 ) 2 = 3 UpHiggsweakdoublet H u h + u h 0 u e H u e h + u e h 0 u ( 1 c 2 ) 1 DownHiggsweakdoublet H d h 0 d h d e H d e h 0 d e h d ( 1 c 2 ) 1 MSSMparticlesincludingtheHiggs.Thesuperpartnersofea chSMsparticleisalso included.Note,thatnotallsuperpartnersarescalars.aninclusioncouldreadilybeunderstoodbytherequirement thatsupersymmetrybean exactsymmetryofourtheorycoupledwiththefactthatanewt ypeofpotentialcalledthe superpotentialcanonlybeholomorphicoranalytic,seeSec tion 3.5.1 .Theotherand moreevidentwayinwhichonerealizesthatneedforanotherH iggsisbyrequiringthe cancellationofanomalies. 33

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TheHiggsasintroducedinTable 2-2 isascalarparticle,whilenowweincludea fermionicpartnercalledtheHiggsino( e H ).Theintroductionofanewfermionparticle meansonemustincludeittotheanomalycalculation.Indoin gsoonerealizesthe theorycannotbeanomalyfreeunlessoneaddsasimilarparti clewithanopposite hypercharge,andsowemustincludeanothersetofHiggseld s. Onecouldaskifsupersymmetryisrequired,andthehonestan swerwouldbeno. Therehasbeennodirectevidence,eitherway,toitsnecessi ty.However,itprovidesa veryusefulandelegantexplanationtothehierarchyproble m.Wesayuseful,because thesuperpotential,tobediscussedwillturnouttoencodei nformationaboutthe Yukawa-sectorelegantly[ 27 ]. Soweassumethatsupersymmetryisasymmetryofnatureinthi spaper.Flavor breakingshouldoccurwellabovethesupersymmetrybreakin gscale,takenhereto betheusuallyaccepted1 TeV range[ 27 ].Weshallnotthendiscussthebreakingof supersymmetrysinceitlieswellbelowthescaleofinterest 3.5.1TheSuperpotential AlookatTable 3-2 willshowthesameparticlesfoundintheregularSM,butwe haveincludedthesupersymmetricpartnersofeachparticle whicharelabeledwitha tildesymbol.Inoccasionswherethereisnochanceofambigu itywewilldropthetilde forthesakeofclarity,e.g.,whenwewritewhatiscalledthe superpotential: W MSSM = Q i Y uij u j H u + Q i Y dij d j H d + L i Y eij e j H d + L i Y eij N j H u (3–22) + N i M n ij N j + H u H d Alleldsabovearescalarswithnoconjugationpresent,i.e .,thesuperpotentialis holomorphicoranalytic.Wehavealsotakenthelibertyofad dingneutrinomasses, includingMajoranatermsdiscussedinanearliersection.T heonlynewtermthatmust 34

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beaddedbyhandisthe term.TheinteractiontermbetweenthetwoHiggsisallowed bySMsymmetriesandsosuchatermmustbeincluded. 5 Ithappensthatthesuperpotentiallooksalmostexactlylik etheYukawaterms foundinEquation( 2–2 ).Bynomeansisthisanaccidentsinceitprovidesmasses forthefermionsaswellasscalars,andwiththeaideoftheau xiliaryeld D A ,denes completelythefullpotentialofthescalarelds. Forexample,themassesofthefermionsaredeterminedby L fermion mass 1 2 W ab a b + W ab a b W ab = @ 2 W @ a @ b ,(3–23) andthelabels a b arenotfamilylabelsbutrunthroughthedifferenteldsint hetheory. Wehavechosentonamethegeneralfermionswith andscalarswith .Asforthe potentialforthescalarelds,onemaywriteitas V ( ) = F a F a + 1 2 X A D A D A F a = @ W @ a D A = X a g a T A a .(3–24) Ingeneraltheelds(scalarsandfermions)arechargedunde rsomegaugegroup. Thegeneratorsofthesesymmetriesaregivenby T A andthesumovertheindex A is toincludeallthesymmetriesunderwhichaparticulareldm aybecharged.Sothe contributionsdueto D A aremerelyareectionofthesymmetriesofthegaugegroup. Weseethatthesuperpotentialisanicecompactwaytostorea greatdealof informationaboutthetheory.Wewillthen,inourmodelbuil ding,primarilyfocusour 5 However,thereisnothinginthetheorythatregulatesthesi zeorthescaleofthis term,butitisknownthatitssizeshouldbeattheorderofthe weakscale O (10 2 GeV ) [ 27 ].Here,inthispaper,weshallnotconcernourselveswithth istermandwilltherefore ignoreitinlatersections. 35

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attentiontocreatingasuperpotentialthatrecreatesthem assesandmixingmatrixofthe SM.3.5.2MSSM:RunningofMasses ThoughitmaynotbeasurprisethattheMSSMmassesalsorun,i tshouldbe understoodthatthemorethandoublingthecontentoftheSMs houldadjustthese runnings.Wesawtheeffectofthisonthecouplings,theuni cationwasevenmore profoundlyevidentthanintheSM,seeFigure 3-1 .Hereweshowthatthemassesrun forthemostpartinlinewiththeSM,(uptoone-loopcontribu tions). Theonlymajordifferenceistherunningofthebottom-quark andthetau-lepton seemtosuggestthattheirmassesconvergeathigherenergie s,butthetendencyfor convergenceisstillpresent. 3.6RemarksonModelBuilding Sofar,thechapterhassummarizedevidenceaboutthepossib lelargeenergy pictureoftheSM.Sincewehavecoveredagreatdealofmateri al,itmaybehelpfulto listsomeoftheconclusionsestablishedinprevioussectio ns.Theseinturnproducea listofpostulatesthatwillguideourmodelbuildinginfutu resections,makingitallthe moreimportanttolisttheseinaclearfashion. 1.Therunningofthecouplingshintatpossibleunicationo fthegaugedsymmetries. Thepossibilityinturnhassuggested SU (5)whichisthesimplestpossibleGUT candidate. 2.Massesalsorun,andtheysuggestpossiblemassrelations .Wedidnotshow,but mentionedthatthesecanbereproducedinthecontextofan SU (5)theory. 3.NeutrinosintheSMaremasslessleptons.Howeverwelearn edtwoimportant factsduetoneutrinooscillationdata: 3.1.Hintsthatneutrinosaremassive.Wearethereforeledt othepossibilityof addingrighthandedneutrinostotheSMinordertoaccountfo rmasses. 3.2.Theentriesoflepton-mixingmatrix U mnsp arequitelargecomparedtothose foundin U ckm .Thiscoupledtotheneedtohavebreakingofavorexplained byhigh-orderedinteractionswithavons(tobediscussedl ater)leadsusto believethatanitesymmetryisresponsibleforavor. 36

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A m e and m d B m and m s C m and m b D m e m d E m m s F m m b Figure3-3.RunningofMSSMmassesfordown-typequarksandc hargedleptons. A C : Wecomparethemassesofdown-typequarksandchargedlepton sforeach familyasthemassesareplottedasfunctionofenergyscale. D F :Theratio ofthemassesaregivenasfunctionofenergyscale. 4.RighthandedneutrinoscannotonlyproduceDiractypemas ses,butcouldalso produceMajoranatype.Thiscould,viatheSee-SawMechanis m,explainwhy neutrinomasseshavebeenelusive. 5.Themasshierarchyproblemcanbesolvedviasupersymmetr y,andinturn producesseveralresultsandsimplications: 5.1.Itmakesatighterpredictionfortheunicationscale: 2 10 16 GeV 5.2.RequirestwoHiggsinsteadofone: H u and H d byanomalycancellation. 5.3.ItdoublestheparticlecontentoftheSM.5.4.Formalismallowsforcertainsimplications,ourmost importantexample beingthesuperpotentialwhichallowstheencodingofYukaw aterms. 5.5.Wewillassumethatthescaleofsupersymmetrybreaking isataround1 TeV 37

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6.Intermsofscales,fromlowestenergytohighest:Wehave rstelectroweak breaking,thensupersymmetrybreaking,then SU (5)breaking.Itisatthisscale, betweentheGUTandPlanckscale( M p 10 19 GeV )thatweshallndavor breaking.Itisattheavorbreakingscalethatourmodelwil lproducemassesand mixingangles. Theabovelistprovidesthefoundationofourmodelbuilding .Somematerial mentionedabovewasnotdiscussedindetailandbutitshallb einsubsequentchapters. 38

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CHAPTER4 THEGUTGROUP SU (5) Wepostulatethatavorbreakingistooccuratenergieshigh erthanunication scaleandsowemustchooseaGUTtheoryforourmodel.InChapt er 3 wehave selectedthe SU (5)GUTtheoryandsowediscussherethemeritsandissuesofu sing SU (5)asagaugegroup.Wenoticedthattherenormalizationgro upequationsathigh energieshintatpossiblebottom-typeandchargedleptonma ssunicationwhichcould easilybeexplainedinthecontextofthisgaugegroup.Howev er,notallisgoodforthis group,andthereareimportantissueslikethedoublet-trip letproblemwhichrequire theintroductionofheavyHiggslikeparticlestocontrolth epredicteddecayrateofthe proton.Fornow,however,letusfocusonthegrouptheoretic alstructureof SU (5)then discussthephysics. 4.1TheGroupTheory Thegroup SU (5)ispartofafamilyofgroupslabeledunder SU ( N )forthespecial casewhere N =5.Beingso,thenweknowthatithas N 2 1generators.The 1is therebecausethegeneratorsshouldbetraceless.Wewillfo cusonthevedimensional representationsalsoknownasthefundamentalrepresentat ionandsowewillwrite theseout. FirstnotethatthePauliMatrices i asmentionedaredenedtobe 1 = 0B@ 0110 1CA 2 = 0B@ 0 i i 0 1CA 3 = 0B@ 100 1 1CA ,(4–1) whereitshouldbeunderstoodthatthesuperscriptofthePau liMatricescanonlytake values1,2,3.NexttheGell-MannMatrices a are 39

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1 = 0BBBB@ 010100000 1CCCCA 2 = 0BBBB@ 0 i 0 i 00 000 1CCCCA 3 = 0BBBB@ 100 010000 1CCCCA 4 = 0BBBB@ 001000100 1CCCCA (4–2) 5 = 0BBBB@ 00 i 000 i 00 1CCCCA 6 = 0BBBB@ 000001010 1CCCCA 7 = 0BBBB@ 00000 i 0 i 0 1CCCCA 8 = 1 p 3 0BBBB@ 10001000 2 1CCCCA andforthiscasethesuperscriptsareunderstoodtogofrom1 ...8,[ 18 36 ].Finally,we shalldenesomevectorandmatrixquantitiesthatwillbeco meuseful: ˆ v 1 = 0BBBB@ 100 1CCCCA ,ˆ v 3 = 0BBBB@ 010 1CCCCA ,ˆ v 5 = 0BBBB@ 001 1CCCCA 0 = 0BBBB@ 000 1CCCCA ,(4–3) withthefactthatthesuperscriptofˆ v canonlytakevaluesof1,3,5.Thezeromatrix 0 willhavedimensionsthatcanbedeterminedbycontext,i.e. ,byitsplacement. Now,wemaywritethegeneratorsinanefcientmanner,webeg inwith[ 36 ] T a = 1 2 0BBBB@ a 00 0 T 0 T 0 1CCCCA T 8+ i = 1 2 0BBBB@ 0 00 0 T 0 T i 1CCCCA ,(4–4) T 12 = 1 p 15 1 1 1 3 2 3 2 ,(4–5) T 12+ i = 1 2 0BBBB@ 0 ˆ v i 0 ˆ v i T 0 T 0 1CCCCA T 13+ i = 1 2 0BBBB@ 0 i ˆ v i 0 i ˆ v i T 0 T 0 1CCCCA ,(4–6) 40

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T 18+ i = 1 2 0BBBB@ 0 0 ˆ v i 0 T ˆ v i T 0 1CCCCA T 19+ i = 1 2 0BBBB@ 0 0 i ˆ v i 0 T i ˆ v i T 0 1CCCCA .(4–7) Fromthesegeneratorsonecanalsoobtainthegeneratorsfor theadjointrepresentation. Infact,wecanproducegeneratorsforotherrepresentation saswellsolongasweknow theKroneckerproductsofthegroup,whichiswhatwewilllis tnow(thetrivialcasesare purposelyleftout)[ 18 33 36 ]. 5 n 5 = 15 S 10 A 5 n 5 = 15 S 10 A 10 n 10 = 5 50 S 45 A 5 n 5 = 1 24 5 n 10 = 5 45 5 n 10 = 10 40 (4–8) Thoughwecouldincludehigherproductswewillseethatforo urpurposesit's unnecessary.Theonlyotherproductthatwillbecomeimport antisthefactthat 45 n 45 1 Theaboveproductscanbeveriedwithseveraltechniques,e .g.,iftakingthe productofthesamerepresentationoneshouldhavesymmetri candanti-symmetric quantities,therequirementofaproductbeingtraceless.T hedefaulttechniqueisthe Young-Tableauxschemewhichisguaranteedtosucceedaslon gasoneiscarefuland playsbytherules[ 18 ]. 4.2ParticleContent Withsomeofthegrouptheoryunderstoodit'snowpossibleto describethematter contentoftheSMinthecontextof SU (5)GUTmodel.TheSMcanbemadetotinto representationsof SU (5)quiteneatly[ 18 19 33 ]: 41

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L + d 1 c ,2 1 + 3 c ,1 2 = 3 = 5 (4–9) Q + d + e 3 c 2 1 = 3 + 3 c 1 4 = 3 + 1 c 1 2 += 10 Fromabovenotethatleptonsandquarkstogethercombineoru nifytomakeafull representationsoftheGUTsymmetry. TheHiggsrequiretheadditionofapossiblyheavierpartner inordertotintothe SU (5)scheme, H u + H c u 1 c 2 1 + 3 c 1 2 = 3 = 5 (4–10) H d + H c d 1 c 2 1 + 3 c 1 2 = 3 = 5 ,(4–11) moreimportantlythisadditionalHiggspartnerwillnotpro ducesingletYukawaterms sowillnotplayadirectroleinmasses. 1 Forbrevitywewillsimplycallthe 5 andthe 5 containingtheHiggsasjust H u and H d WecannowidentifytheforcecarriersoftheSMintheGUTtheo ryaswellasnew forcecarriersthatarenotintheSM.Sincewehavealreadypr oducedallthegenerators oftheGUTsymmetry,itisamatterofsimplyidentifyinggene ratorsoftheSMthatoccur withingtheGUTgenerators. Twoseparatediagonalblockscontainthesegeneratorswher ewemaynicelytthe gaugeeldsoftheSM.Theoff-diagonalblocksmustthenbeid entiedasentirelynew anddistinctelds.Explicitly,iftheeldunder SU (5)isgivenby F A ( x )thenwehavethat [ 36 ] 1 Itwillhoweverplayaroleinprotondecayandmusttherefore bemassive,see Section 4.5 42

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F A T A = 0BB@ A a d 2 1 p 15 a d B a i j d W j i 2 + 3 2 p 15 j i B 1CCA ,(4–12) with f a d g =1,2,3and f i j g =1,2.Wemustmentionthattheneweldscouldbe understoodas a i = 1 p 2 X a Y a ,and j d = 1 p 2 X d Y d .(4–13) TheSMeldsareclearlyvisibleinEquation( 4–12 )andassuchweknowthatforelds underthefundamentalrepresentationthetopdiagonalbloc kattacksthoseeldsthat arechargedundertheSM SU (3)and U (1).Thelowerblockattacksthosecharged underthe SU (2)and U (1).Noticethatthesumofthe U (1)chargesshouldvanish,a consequenceofthetracelessconditionwhichultimatelyle adstothequantizationof electriccharge. WekeepinmindthattheEquation( 4–12 )istrueforthefundamentalrepresentation, i.e., 5 .Toobtainsimilarconclusionsforthecomplex-conjugates onesimplydenesthe T 0 A = T Weremindthereaderthatitwasacombinationofhypercharge assignmentwiththe thirdcomponentofweakisospinoperatorsthatproducethee lectricchargeoperator,viz. Equation( 2–1 ).AttheleveloftheSMthehyperchargewasangivenviaacomb ination ofexperimentallydeducedquantitiesandtheoreticalcons iderations.Thatistosaythat attheleveloftheSMthereisnotheoreticalreason,outside experimentalconstraints,for thehyperchargeassignments. ThepicturechangesinthecontextofaGUTtheory,wherether equirementof inclusivenessoftheSMinalargerstructureallowsforathe oreticalpredictionforthese assignments.Ingeneral,extendingaGUTtothelevelofthee lectroweakbreaking,one cannothelpbutrealizethatchargeisquantized.Inthecase of SU (5)wendthatthe electricchargeoperatorofaparticleinthefundamentalre presentationcanbewrittenas 43

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Q 5 = T 11 +2 5 3 1 = 2 T 12 ,(4–14) withasubscript5toremindusitisthechargedoperatorfort hefundamentalrepresentation. Weshallextendthisoperatortootherrepresentationsfort hesakeofcompletionin Appendix A 4.3InvariantMassTerms ItispossibletoconstructYukawatermsEquation( 2–2 )orEquation( 3–22 )using themattercontentandHiggsrepresentationsof SU (5).Sinceourmodelwillbeofthe supersymmetrictypewefocusonEquation( 3–22 )andwend Q u H u ( 10 n 10 ) n 5 Q d H d 10 n 5 n 5 L e H d 5 n 10 n 5 ,(4–15) noticingthatthecontentinparenthesisofthethirdequati oncouldbeunderstoodasthe transposeofthatinthesecond.AsyourecallEquation( 4–8 )allowustodeterminehow theproductsformrepresentationsof SU (5)andsowendthattheaboveareindeed invariantterms. ItisalsopossibletoamendYukawatermsbecausebothproduc ts 10 n 10 and 10 n 5 canproduce 45 dimensionalrepresentations,whiletheformercanproduce anadditional 50 .Wewillnotdiscusstheadditionofa 50 Higgsbecausewewillliketobeeconomical inthenumberofparticlesweshalladdtoourmodel. 2 The 45 Higgscanplayakey roleinourmodelbyallowingforanexplanationtomassrelat ionsEquation( 3–1 )and Equation( 3–2 ),explainedindetailbelow. 2 Itisalsotruethattheydonotcontributecolorinvariantte rms[ 36 ]. 44

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4.4TheGeorgi-JarlskogMechanism InSection 3.2 wediscussedthatthereappearstobeapproximatemassrelat ions: Equation( 3–1 )andEquation( 3–2 ).Whatisimportantistheformerrelationshipwhere thereissomeunicationoccurringbetweenthecharged-lep tonsandthedown-type quarks.IfweappealtotheGUTgroup SU (5)thereappearsanaturalexplanation. Themasstermsforbotharedeterminedbythesameinvariantq uantity;theproduct 10 n 5 n 5 .Theleptonsmasswouldthenbeobtainedbysimplytakingthe transpose ofthatforthedown-typequark. Thesimpletransposition,however,cannotexplainEquatio n( 3–2 ).Thereis,though, apossibleexplanationinthecontextof SU (5)bymakinguseofthe 45 Higgs.The explanationgoesbythenameoftheGeorgi-Jarlksog(GJ)mec hanism[ 20 36 ]. 3 TobeginwenoticethatfromEquation( 4–8 )wehave 5 n 10 = 5 45 ,and 10 n 10 = 5 50 S 45 A ,(4–16) andthat 45 n 45 1 .Thatwouldmeanthatwemayhavemasstermsifweaddtwo extraHiggselds: H 45 u 45 ,and/or H 45 d 45 .(4–17) Weincludedthepossibilityofhavingaup-typemasscontrib utionfroma 45 Higgsbut duetoalackofanyinterestingmassrelationsweignoreinth ispossibility.Itshouldbe notedthatifanomalycancellationisapointofconcerninmo delbuildingoneshould includebothnewtypesofHiggs. Toseehowonecanproducemassmatricesfromthe 45 Higgsoneonlyneedsto understanditsvev: 3 Themechanismcanalsobeappliedinthecontextof SO (10). 45

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h H 45 b ,5 a i = v 45 b a 4 4 a b 4 a b =1,...,4.(4–18) Thersttermgivescontributiontoonlythedown-typequark masses.Whilebothterms contributetothechargedleptons,producinganegativethr eevalue.Eitherwaythe negativethreewouldsimplybeapre-factor,whiletheYukaw atermwouldbeidentical forbothsincetheyoriginatefromthesameinvariantterm.I fthenweincludethe contributionsduetothe 5 Higgs: Y d = Y 5 + Y 45 ,(4–19) Y e = Y T 5 3 Y T 45 Asitstandshowever,theabovecannotfullyexplaintherela tionshipsweareseekingto explain.TodosoassumethattheYukawamatricesare Y 5 0BBBB@ 0 A B 0 1 1CCCCA ,and Y 45 = 0BBBB@ 00 0 C 0 1CCCCA ,(4–20) wheretheterms aresuchthattheydonotchangeappreciablytheeigenvalues ofthe Yukawamatrices.Withtheaddedconstraintthat A B << C andthenplacingtheabove intoEquation( 4–19 )followedbydiagonalizationonerealizeswecangainthere lations weareseeking. 4.5Problemswith SU (5) Asmentionedintheintroductionofthischapter,notallthe problemswith SU (5) unicationaresettled.Becauseanyconceptofunicationp lacestheleptonsandquarks inthesamerepresentationtherewillbeforcecarriersthat canpotentiallychangea quarkintoalepton.Inotherwords,therearedecaypathsfor quarksintoleptonswhich foraprotonwouldmeandecays[ 36 ].Onecouldcontrolthispossibilitybymakingthe 46

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bosonsresponsibleforthedecaysquiteheavy,anaturalout comeofGUTbreaking. However,evenmoreproblematicisthedoublet-tripletprob lem. RecallEquation( 4–10 )whereeachHiggssectorhasbedividedintoacolorHiggs (triplet)andaweakHiggs(doublet).Theproblemisthatthe tripletHiggswillalso interactwithmattercontent,andsoprovideapossiblepath forprotondecay.Toxthis onecouldalsomakethemassesofthetripletsveryheavy,how everhowtodothiswhile keepingthedoubletHiggslightisamajorproblem.Atthelev elofan SU (5)theorythe problemhasnotbeenresolved,atthelevelof SO (10)theproblemhasbeenpartially resolvedinthecontextofsupersymmetryviatheDimopoulos -Wilczekmechanism[ 11 ] wherebythecorrectchoiceinthevevofa 45 representationcanproduceveryheavy massesforthetripletHiggsandkeepthemassesofthelightH iggssmall. Althoughacomplete SU (5)supersymmetricmodelwilldiscussthisissueindetail wewillnotspendfurthertimediscussingthedoublet-tripl etproblem.Wewishtoshow thatunderthechoiceofaspecicavorgroupitispossiblet oproduceallcurrent dataandmakepredictionsfortheneutrinosector.Thedoubl et-tripletproblem,though important,haslittleimportanceintermsofourgoal. 47

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CHAPTER5 YUKAWAMATRICESFROMPHENOMENOLOGY ThegoalistoproducephenomenologicallycorrectYukawama tricesforthe quarksector,atthesametimeproduceviableneutrinomasse s,andtorstorder thetri-bimaximalleptonmixingmatrix.Asforthechargedl eptons,thechoiceofan SU (5)GUTwillautomaticallyproduceaYukawafromthedown-qu arksector.Thefocus ofthissectionisthenthephenomenologyinvolvedineachma ttersectorandhowto consolidatethedataintomassmatrices. Weshouldrememberthatthemotivationofthissectionwasin spiredbyapaperby Gattoetal.[ 17 ]wheretheywereabletoobtainarelationshipbetweenquark -masses andtheCabibboangle,nowcalledtheGattoRelation: r m d m s .(5–1) Thiswasdoneunderthecontextoftwoavors,asymmetrythat prohibiteda(1,1)entry whilehavingtheoff-diagonalmassentriesofthedownquark equalandmuchsmaller thanthe(3,3)entry.Later,theideaofaddingsymmetries(i ncludinghorizontalorfamily types)wasextendedtothreegenerationswithotherzerotex turestructuressuchas (1,3)and(3,1),e.g.,[ 15 21 34 37 ]. Suchtexturerelationscanleadtorelations[ 35 ] U ckm ub U ckm cb = r m u m c U ckm td U ckm ts = r m d m s jU ckm us j r m d m s e r m u m c ,(5–2) whereingeneralthelasttermcontainsaphasewhichcanbere latedtotheCPviolating phase.NotethatthelasttermisalsousedtodenetheCabibb oangle,andwillplaya centralroleinourdiscussionforobtainingtheYukawasint hequarksector. 48

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5.1YukawaMatrices:TheQuarkSector Herewefocusontherstsectorthatcontainsallinformatio nregardingthequarks. Currently,experimentsallowsforonlytwobutimportantpi ecesofdata.Thesecomein theformoftheapproximatemassesforthequarksandthequar k-mixingmatrix, U ckm MotivatedbytheGattoRelation,Equation( 5–1 ),wemayextrapolatemassdata totheunicationscalewhereonecanparametrizeallmasses intermsoftheCabibbo angle c .225,producingthehierarchicalstructure m e m O ( 4,5c ), m m O ( 2c ), m m b ,(5–3) m d m b O ( 4,5c ), m s m b O ( 2c ), m b m t O ( 3c ),(5–4) m u m t O ( 8c ), m c m t O ( 4c ).(5–5) Includedaboveistherelationbetweenmassofthetaulepton andbottomquark (discussedinanearlierchapter)whichareapproximatelye qualandtheintra-family hierarchy,boththelastrelationsinEquation( 5–3 )andEquation( 5–4 )respectively. AsdiscussedinChapter 4 ,thechoiceofan SU (5)modelwillguaranteethelepton massesanddown-typequarkmassesareinfactrelatedandsoe nsurethatthemass ofthetauandbottomsquarkareidenticalabovetheGUTscale .Sothatwithan SU (5) modelwewilltrytoreproduce,intheformofeigenvaluesoft woYukawamatrices,allthe informationfoundinEquation( 5–3 )-( 5–5 ). ThelastexperimentalpieceofdataatourdisposalistheCKM matrix.Asalready mentioned,itisamixingmatrixcomposedroughlyoutofdiff erencesinanglesthat occursfromdiagonalizingtheYukawamatricesofbothquark sectors.Theinformation containedtheretothirdorderapproximationis(fromEquat ion( 2–5 )) 49

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U ckm O 0BBBB@ 1 3 1 A 2 3 A 2 1 1CCCCA .(5–6) Recallthatitiscomposedonlyoftheleft-handeddiagonali zingmatricesoftheSVD procedure[ 32 ].Soduetotheverynatureofitsoriginsthereisalimitinho wmuch informationwecanderiveaboutthestructureofthequarkYu kawas.Nevertheless,there arecluesastothetexturestructuresandifweaddtotheseth enecessaryeigenvalues requiredwecanlimitthepossiblechoicesforYukawamatric es[ 21 34 35 ]. Tobegin,letsmakeuseofthesuccessoftheGattoRelation,w eassumethatwe maywritethetoptwo-dimensionaldiagonalblockas 0B@ 0 a a c 1CA ,(5–7) byignoringpre-factorsandonlylookingatthepowers,andw eknowthat a > c .The abovewouldcorrespondtoobtainingthemassmatrixforeith ertheupandcharmorthe downandstrangequarks.Inthecaseofthedownandstrangequ ark,theGattoRelation requiresthat a = c +1,whilefortheformer a = c +2.Thus,wemaywritethat Y u = y t 0BBBB@ 0 c +2 c +2 c 1 1CCCCA Y d = y b 0BBBB@ 0 c 0 +1 c 0 +1 c 0 1 1CCCCA ,(5–8) bytakingthelibertyoffactoringoutanoverallscaledueto eithertopandbottom quarks.Wewishnottoincludeanovertheoverallscaleatthi spoint,andsowemakea redenitionsuchthat Y (2 = 3) = 1 y t Y u Y ( 1 = 3) = 1 y b Y d ,(5–9) 50

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andforawhileforgetabouttheoverallscale.Atthispoint, ifwemakefurtherthe assumptionthatthethirdcolumnandthirdrowofeachmatrix willnotcontribute profoundly,atleastnottoaffecttheeigenvaluesoftwo-di mensionalblock,wemay ndtheeigenvaluesandrewriteEquation( 5–5 )andEquation( 5–4 )as m c m t 4 = c m u m t = 8 m d m b 2 = c 0 m d m b = 4 ,(5–10) inotherwordsforvaluesof c =4and c 0 =2weregainthehierarchicalstructurewe wrotedownearlier.Allthis,withthesolerequirementsimp osedbytheGattoRelation. However,nowwemustaskthequestionabouttheoff-diagonal terms,i.e.,thethirdrow andthirdcolumnmustbelledin.Theonlyconditionsofarim posedonthemisthatthey shouldnotdestroythehierarchywehavedetermined. Tollintheseunknownvalueswetaketheapproachoutlineda ndprescribedby [ 35 ]wheretheyusenotationusedin[ 21 ].ThemethodusesEulerdecompositionwith andtheassumptionthattheanglesarequitesmall.Byrotati ngtheYukawamatrices theythencanregaintheCKMmatrixinageneralformasfuncti onsofEulerangles.First Y (2 = 3) diagonal = U (2 = 3) ( u 12 u 13 u 23 ) Y (2 = 3) V (2 = 3) ( 0 u 23 0 u 13 0 u 12 ),(5–11) Y ( 1 = 3) diagonal = U ( 1 = 3) ( d 12 d 13 d 23 ) Y ( 1 = 3) V ( 1 = 3) ( 0 d 23 0 d 13 0 d 12 ),(5–12) wherewehavethat U ( 12 13 23 )= 0BBBB@ 1 s 12 0 s 12 10 001 1CCCCA 0BBBB@ 10 s 13 010 s 13 01 1CCCCA 0BBBB@ 10001 s 23 0 s 23 1 1CCCCA ,(5–13) with s ij = sin ( ij )andsimilarlytrueforthe V matrixbutwiththeproductsinreverseorder of U .Because U ckm = U (2 = 3) y U ( 1 = 3) wehavethat 51

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U ckm = 0BBBB@ 1( s d 12 s u 12 )+ s u 13 ( s d 23 s u 23 )( s d 13 s u 13 ) s u 12 ( s d 23 s u 23 ) ( s d 12 s u 12 ) s d 13 ( s d 23 s u 23 )1( s d 23 s u 13 )+ s u 12 ( s d 13 s u 13 ) ( s d 13 s u 13 )+ s d 12 ( s d 23 s u 23 ) ( s d 23 s u 13 ) s d 12 ( s d 13 s u 13 )1 1CCCCA (5–14) However,tondthesineoftheseangleswemustrelatethemto theentriesoftheir correspondingYukawas.Werewritetheseequationsforconv eniencebasedonour Yukawamatrices: s 23 Y 23 + Y 32 Y 22 s 0 23 Y 32 + Y 23 Y 22 s 13 e Y 13 s 0 13 e Y 31 s 12 e Y 12 e Y 22 s 0 12 e Y 21 e Y 22 (5–15) with e Y 12 = Y 12 Y 13 s 0 23 e Y 21 = Y 21 Y 31 s 23 e Y 13 = Y 13 + Y 12 s 0 23 e Y 31 = Y 31 + Y 21 s 23 e Y 22 = Y 22 Y 23 Y 32 (5–16) Takingalltheaboveandconstrainingtheentriestominimal lysatisfytheCKM matrix,wendthatattheveryminimumwewouldneed Y (2 = 3) O 0BBBB@ 0 6 4 6 4 2 4 2 1 1CCCCA Y ( 1 = 3) O 0BBBB@ 0 3 3 3 2 2 1 0 1 1CCCCA ,(5–17) assumingthatcoefcientsareof O (1).Itshouldbenotedthattheaboveisabit misleading,atleastoneofthe(2,3)positionsmustbe 2 .Now,themodelbuilding willhavetosatisfythehardtextureconstraintsgivenabov eandfallwithinthelimits placed. 52

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LateronweshallseethatwewillnotbeusingtheCabibboangl easourexpansion parameterforthewholemodelbutinstead .20.Thisinturnisthevalueofthe parameterresultingfromusingtheFNmechanism.Weshallse emoredetailsaboutthis factlater. 5.2YukawaMatricies:NeutralLeptonSector Unicationvia SU (5)willautomaticallyproduceinformationaboutthecharg ed leptonsoncethedown-quarkYukawasareknown.Sowewillonl yconcentrateonboth theneutralleptonsandheavyneutrinos. Intermsofexperimentaldatatheleptonsectordoesnotshar ethesamerichness asthequarksector,butwedohaveavailabletoustwokeypiec esofdata. 1 .First, experimentalresultshavegivenusthemasssquareddiffere nces(reproducedherefrom Equation( 3–21 ))[ 29 ] m 2 21 7.59 +.19 .21 10 5 eV 2 j m 2 23 j 2.43 .13 10 3 eV 2 Thesecondrelationdoesnotallowustodeterminetheexacth ierarchicalstructure. Nevertheless,ausefulconstraintthatcanbederivedfromt heaboveis 29.56 j m 2 23 j m 2 21 34.68,(5–18) theaveragevaluebeing 32.02. 1 Cosmologicaldataalsoprovidelimitsonthesumofneutrino massesandthesizeof themostmassiveneutrino[ 1 ] X m i < (.17 2.0) eV i =1,2,3,.04 < m ,heaviest < (.07 .70) eV 53

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Thesecondpieceofexperimentaldatacomesintheformofthe leptonmixing matrix U mnsp ,whichweshallassumetobeapproximatelythetri-bimaxima lmatrix,which werewritehere U tri-bi = 0BBBBBB@ r 2 3 1 p 3 0 1 p 6 1 p 3 1 p 2 1 p 6 1 p 3 1 p 2 1CCCCCCA .(5–19) Thesee-sawmechanismrequirestheexistenceoftheregular neutrallepton Yukawamatrix Y (0) andaninvertibleMajoranamatrix Y maj [ 28 31 39 ].Thesetogether areneededforthelightneutrinomassapproximationof Y v 2 M Y (0) Y maj 1 Y (0) T ,(5–20) whichwediagonalizeby U mnsp ,i.e., Y = U mnsp m U mnsp T .(5–21) Thediagonalterm m willcontainthreedifferenteigenvalues(masses)andafte r selectingtheseeigenvalueswecanproducethelightneutri nomatrixfromthe tri-bimaximalmatrix: m = 0BBBB@ m 1 m 2 m 3 1CCCCA )Y = 0BBBB@ 1 2 2 2 3 1 + 2 3 2 1 + 2 3 3 1CCCCA ,(5–22) inwhichwehavethat 1 = 1 6 (4 m 1 +2 m 2 ), 2 = 1 6 ( 2 m 1 +2 m 2 ), 3 = 1 6 ( m 1 +2 m 2 +3 m 3 ).(5–23) 54

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Thustheeigenvaluesasfunctionsofentriesof Y aregivenas m 1 = 1 2 m 2 = 1 +2 2 m 3 =2 3 1 2 .(5–24) WhatwereallycareaboutintermsofndingYukawamatricesi sthemasssquared differences: m 2 12 =3 2 (2 1 + 2 ) > 0.(5–25) Itsimportanttonotethatbecausedatashowsthat m 2 12 > 0impliesthat 2 6 =0.Sowe mayactuallyset 01 = 1 2 and 03 = 3 2 .(5–26) and Y 0 = 0BBBB@ 01 11 1 03 01 03 +1 1 01 03 +1 03 1CCCCA ,(5–27) Themasssquareddifferencerelationshipthenbecomes j m 2 32 j m 2 12 = 1 3 j 4 0 2 3 +4 03 01 +4 03 +2 01 +3 j 2 01 +1 .(5–28) Thetwosimplestchoiceswecanmakearewith 03 =0and 01 =0.Therstchoice wouldmeanthataviablesolutionthattsdata29.56 34.68requiresthat .511 < 01 < .510.Whileforthesecondcasewendeither5.31 < 03 < 5.70or4.31 < 03 < 4.69. GiventheCabibboangleis .225meansthat 1 4.44.Thisissuggestiveand itmeanswecouldpossiblyparametrizetheabovewiththeCab ibboangle.Withthatsaid weseektond,usingourparameter 55

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Y 0 / 0BBBB@ 0 1 +1 +1 1 1CCCCA ,(5–29) 5.3RemarksonFlavorSymmetry FromdatawehaveattemptedandproducedYukawamatriceswhi chwewillseek tobeconsistentwith.Someofthestructures,specicallyt hetexturezerostructureof thequarksaremotivatedmainlybyGattoRelations. 2 Afactwhichimpliesadirecttie betweentheeigenvaluesoftheYukawamatricesandrotation angles.Interestingstillis thatthequark-mixingmatrixisidentitytolowestorderand canbeextendedfromitto anyorderintheCabibboangle[ 38 ].Thesetwoideas,themassrelationandthesize oftheCabibboangle,mustthenimplythatindividualdiagon alizationmatricesofeach quark-sectoraretoleadingorderidentity. Ifweextendtheseconclusionstotheleptonsector,andcoup ledwiththestrong possibilityofaGUT,weareledtounderstandthatthestruct urewendinthelepton-mixing matrixmustbeprimarilyduetoneutrinodiagonalization.S othatdeviationsfrom Equation( 5–19 )mustbesuppliedprimarilybytheleptondiagonalization( whichcouldby GJmechanismberelatedtothequarks).Thereisalsothefact thatnosuchGattolike massrelationcanbefoundfortheleptons.Theleptonsseemt obeindicatingthatthe anglesshouldbeindependentfromtheireigenvalues. Soweseefromquarksectorsthatsymmetriesmaydictatetext urezeros,bysome veryspecicavor,orhorizontalsymmetry,orbysomelessa mbitioussymmetry.The largeangleswhicharedecoupledfrommassesseemtoindicat ethattheneurino 2 Itisworthmentioningthattexturezerosareascaledepende ntquantities.Since mixinganglesshouldnotrunmuch,giventhattheyarediffer encesofangles,weexpect thattexturezerosataGUTscaleshouldbeapproximatelyzer oatlowerenergyscales. 56

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sectorisprobingsomenewsymmetry!ThestructureofEquati on( 5–19 )suggestthat aavorsymmetryisatwork,andthatthesymmetrymustcontai nathreedimensional representation. Thenextquestionisinregardstothenatureoftheavorsymm etry.Toanswerthis wewillinprincipletaketheminimalistapproach: 1.Thesymmetrymustbebroken,otherwiseitwilldictatethe exactstructureat theleveloftheSM.Abrokencontinuoussymmetrywhenbroken canandwill produceNambu-GoldstoneBosons,apossibilitywewouldlik etoavoid.Discrete symmetrieswhenbrokendonotproducesuchbosons,whichisw hywechooseto usethesetypeofsymmetries. 2.Wewishtocontainthreedimensionalrepresentations,an dunifyatleasttwoof thethreequarkfamilies.Todosowemustthenhaveanon-Abel iansymmetry. OtherwiseforAbeliansymmetiresanyrepresentationcould bediagonalizedandso thefamilieswouldbeindependentofeachother. 3.Discretesymmetriescontainaxedsetofrepresentation s.Wewillbeinterestedin thosethatcontainthree-dimensionalrepresentations. Sowearemotivatedtolookatdiscretegroupswiththree-dim ensionalrepresentations asthesourceofavorsymmetry.Wewilltakealookattheposs ibilitiesandchooseone suchgroupinChapter 6 57

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CHAPTER6 FLAVORTHROUGHFINITEGROUPS Table6-1.Finitegroupswiththree-dimensionalrepresent ations. ContinuousgroupFinitesubgroup SO (3) D 2 n A 4 S 4 SU (3) (3 n 2 ) (6 n 2 ) P SL 2 (7) Listofnitegroups,subgroupsofeither SU (3)or SU (3). Wehavechosentouseanitegroupasouravorsymmetrythath asthree-dimensional representations.Table 6-1 containsthelistofnitesubgroupsofeither SO (3)or SU (3) whichonecouldexploreforavorphysics.Herewewilltryto answerthequestionof usinganitegroupasaavorgroup. WebeginbydiscussingtheFroggatt-Nielsen(FN)Mechanism whichwillrequire theintroductionofnewscalareldscalled“avons”.Aswes hallseetheseareelds thatbreakthenitesymmetrybypickingupvevsandasaresul twilldeterminethe entriesofYukawamatrices.Wetheninvestigatethedetails ofthesymmetry (6 n 2 ) andthespecialcasewhere n =3(ourchoiceforavorsymmetry).Weendbybriey demonstratinghowonecouldproduceYukawamatricesfrom (54). 6.1TheFroggatt-NielsenMechanism TheFroggatt-NielsenMechanism[ 16 ]beginswiththepremisethatthemassesof atheoryoriginatefromsomeeffectiveinteractionwithsca lareldswhichobtainvev. Addingscalareldstointeractionspossesaproblemofintr oducingnon-renormalizable termstotheoriginaltheory,becauseofthisweneedtointro ducesomemassscale M thatwillbepresumably,forperturbativereasons,muchlar gerthanthatofthevevofthe newscalarelds. Themechanismissimilartothefour-fermionorFermi-inter actionwhichisalow energyapproximationoftheweak-interactions.Onendsth atagaugeparticle,aweak 58

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particle,thatismassiveinteractsviatree-levelvertice swiththefermions.Sinceeach vertexisattree-levelweknowtheyarethemselvespartofar enormalizabletheory. Onecanmodelfermioninteractionsatlowenergiesbyparame trizingtheeffectofthe gaugeparticlebywayitsheavymass.Effectivelythefour-f ermioninteractionisnot renormalizabledimensionally,butonerecoversthecorrec tdimensionsofmassbythe massoftheheavygaugeparticle.Onethenmodelstheinterac tionas L fermi / g 2 M 2 w ,(6–1) where g and M w arethevertexcouplingandweakmassrespectively. MuchinthesamewaytheFNmechanismassumesthatamassterm, likee.g.,from Equation( 3–22 ): Q Y u u H u ,(6–2) (ignoringgenerationindices),canbeexplainedbyproduci ngaYukawamatrixfrom Y = Y a =1 ( a ) n a ,(6–3) withtheindex a labelingdifferentscalarsinourtheory(againignoringge neration indices).Thequantity n a representsthepossibilitythatanyscalarcanbepresentma ny times,solongasitrespectsanysymmetryimposedonthetheo ry.Assuch,weexpect thatany n a shouldbeanon-negativeintegerandnothingelse. IfweincludetheabovetoEquation( 6–2 )wewouldbeleftwiththeobvious non-renormalizableterm,andevenifthescalareldstakeo nvevstheYukawa matrixwouldobtainthewrongunitsformass.Toremedythisp roblemwelookat Equation( 6–1 )forguidanceandobtainthecorrectmassunitsbydividingb ythecorrect powerinmass.Thepowerofmasswillonceagainencodethefac tthatweareinan effectiveregimeandthatsomeunknowngaugeparticleismed iatingtheinteractions. 59

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Theresultthenisthatthethesescalareldswillbesuppres sedbyanoverallmass scale M whichweassumeisquitelarge,i.e., a M .(6–4) Thetheoryatlowenergiesisonceagainrenormalizablewhil ethetheorynearthemass scale M ,isoftheeffectivekind.Forthepurposesofourpaperwhene verascalarpicks upavev, h a i ,wewillsaythat h a i M P a P a > 0,(6–5) andsowewillusejustoneparameter ,and P a isthepower.Clearlyweareatliberty touseoneparameter,asmanydifferentscalarscouldpotent iallypickupvevswhichare ofdifferentsizes.Wesimplychoosetousepowersin toencodethesedifferences.We maketherestrictionthat P a 6 =0becausethoughpossible,itwillprovedifculttocontro l andrestrictinteractionwithascalareldwithsuchalarge vev. 1 Inmuchthesamewayaselectroweaksymmetrybreaking,wesay thatthe avorbreakingoccursatthemassscalebelow M (butabovetheGUT).Sowehave successfullyexplainedthelowenergytheorybyuseofandef fectivetheory.However, ourgoalistoeventuallywritedownaLagrangianandsowemus thaveanorganizing principlethatcontrolsinteractions. 1 Logicallysuchascalareld,i.e.,onewithavevofthemass M ,willtypicallybe associatedwiththebreakingofthegaugesymmetrythatmedi atestheseinteractions intherstplace. 60

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Todosowewillassumethepresenceofasingleavorsymmetry ofwhicheach scalareldwillbechargedunder.Thuswecanidentifysucha scalareldasaavonto distinguishthemforothers. 2 SotogetanideaofwhattheprogramthattheFNmechanismprod ucesletuswrite anassumedavorinvarianttermthatwillbesomeorderin as C ( r Q r u ) ij m Y a =1 a M n ( a ) Q i u j H u i j =1,2,.., n f ,(6–6) where m isthenumberofscalareldsincludedinthemodeland n f isthenumber ofavours.Thequantity C ( r Q r u ) ij isschematicandcanbeconsideredinessencea Clebsch-Gordan(CG)coefcientbecauseitencodesinforma tionabouthowaterm shouldbeasingletinavorspace,whilewehaveignoredthe avorindexofthescalar elds. 3 Ingeneralforanytermitwilldependontherepresentationo feithereld Q r Q and u r u DuetoEquation( 6–5 )itispossibletoknowwhatorderin contributionEquation( 6–6 ) contributestoaLagrangian: n ,suchthat n = m X a n a P a .(6–7) WhichexplainsourmotivationforwritingEquation( 6–5 ),itwillmakemodelbuilding easierbecauseweshallbeabletodeterminetheorderofacon tributionbyperforminga simpleaddition. 2 TherewillbescalareldsthatthoughobeyEquation( 6–5 )willbesingletsunder theavorsymmetry.These,obviously,willnotbecalledav onsbutcanneverthelessbe usefulinmodelbuilding. 3 Wewillsolidifyadenitionfor C ij laterbylookingataspecicavorsymmetry.For nowitmustbeleftpurposelyvagueforthesakeofargument. 61

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6.2TheFiniteGroup (6 n 2 ) Hereweexamineafamilyofnitegroupsthattogetherarelab eledby (6 n 2 ).Then fromthefamilyofnitegroupsweshalltrythesmallestinte restingcasethatcontains boththreeandtwodimensionalrepresentations: (54)forwhich n =3. 6.2.1PresentationandClasses Thedetailsofthisniteavorgroupcanbefoundin[ 14 ],whereallthestructuresfor allcasesof n arediscussedingeneral.Herewewillbrieydiscusstheres ultsgivenon thatpaper. Theavorgroupisisomorphictoasemi-directproductoftwo groups,onebeinga simpleno-Abeliangroupandtheotheradirectproductoftwo Abeliangroups: (6 n 2 ) ( Z n Z n ) o S 3 .(6–8) Soweshouldexpecttosee,becauseoftheabove,atleastfour elementsinthe presentation.Twothatcomefromthenon-Abelian S 3 (asecondandathird-order operator),andanothertwofromtheAbelianpart Z n Z n (two n -orderedoperators).The presentationisthen (6 n 2 ): a 3 = b 2 =( ab ) 2 = c n = d n =1(6–9) cd = dc ,(6–10) aca 1 = c 1 d 1 ada 1 = c bcb 1 = d 1 bdb 1 = c 1 (6–11) withclassstructurethatdependsonwhether n isdivisiblebythreeornot.Thetwo separatecasesare: ( i ) n 6 =3 Z .Forthiscasewehaveintotalvetypesofclasses 1 C 1 ,3 C ( ) 1 ,6 C ( ) 1 ,2 n 2 C 2 ,3 nC ( ) 3 (6–12) 62

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whichintotaladdupto1+( n 1)+ n 2 3 n +2 6 +1+ n distinctclassesrespectively.In termsoftheelementsofthegroupwendthat(forthenon-tri vialclasses): 3 C ( ) 1 = c d c 2 d c d 2 =1,2,..., n 1,(6–13) 6 C ( ) 1 = c d c d c d c d c d c d ,(6–14) where =0,1,..., n 1,butexcludingpossibilitiesgivenby: + =0mod( n ),2 =0mod( n ), 2 =0mod( n ).(6–15) 2 n 2 C 2 = ac z d y a 2 c y d z j z y =0,1,..., n 1 .(6–16) 3 nC ( ) 3 = bc + x d x a 2 bc d x abc x d j x =0,1,..., n 1 =0,1,..., n 1. (6–17) ( ii ) n =3 Z .Herewehaveintotalsixtypesofclasses 1 C 1 ,1 C ( ) 1 ,3 C ( ) 1 ,6 C ( ) 1 2 n 2 3 C ( ) 2 ,3 nC ( ) 3 ,(6–18) resultingin1+2+( n 3)+ n 2 3 n +6 6 +3+ n differentclasses. 1 C ( ) 1 = c d 2 = n 3 2 n 3 ,(6–19) 3 C ( ) 1 = c d c 2 d c d 2 6 = n 3 2 n 3 ,(6–20) 6 C ( ) 1 = c d c d c d c d c d c d ,(6–21) where =0,1,..., n 1,againexcludingpossibilitiesgivenbyEquation( 6–15 ). 2 n 2 3 C ( ) 2 = f ac y 3 x d y a 2 c y d y +3 x j y =0,1,..., n 1, x =0,1,..., n 3 3 g =0,1,2. (6–22) 3 nC ( ) 3 = bc + x d x a 2 bc d x abc x d j x =0,1,..., n 1 =0,1,..., n 1. (6–23) 6.2.2CharacterTable OnceagainthedetailsofobtainingtheCharacterTable 6-2 canbefoundin[ 14 ]. 63

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Table6-2.Thecharactertablesof (6 n 2 ). (a) n 6 =3 Z 1 C 1 3 C ( ) 1 6 C ( ) 1 2 n 2 C 2 3 nC ( ) 3 1 1 11111 1 2 1111 1 2 1 222 10 3 1 ( l ) 3 P p ( ) M p 0 l l P p ( ) M p 0 l l 0 l 3 2 ( l ) 3 P p ( ) M p 0 l l P p ( ) M p 0 l l 0 l 6 ( k l ) 6 P p s ( ) M p s k l P p s ( ) M p s k l 00 (b) n =3 Z 1 C 1 1 C ( ) 1 3 C ( ) 1 6 C ( ) 1 2 n 2 3 C ( ) 2 3 nC ( ) 3 1 1 111111 1 2 11111 1 2 1 2222 10 2 2 222 P r ( + ) r P r (2+ ) r 0 2 3 222 P r ( + ) r P r (1+ ) r 0 2 4 222 P r ( + ) r P r ( ) r 0 3 1 ( l ) 3 P p ( ) M p 0 l l P p ( ) M p 0 l l P p ( ) M p 0 l l 0 l 3 2 ( l ) 3 P p ( ) M p 0 l l P p ( ) M p 0 l l P p ( ) M p 0 l l 0 l 6 ( k l ) 6 P p s ( ) M p s k l P p s ( ) M p s k l P p s ( ) M p s k l 00 Charactertablesforthecases(a) n 6 =3 Z and(b) n =3 Z .Note and takeondifferent valuesdependingontheclass, =0,1,2, = n 3 2 n 3 p =0,1,2, s =0,1, r =1,2, = e 2 i = 3 = e 2 i = n and l k =1,2,..., n 1. 6.3FlavorSymmetry (54) Theavorgroupunderconsiderationisaspecialcaseof (6 n 2 ),where n =3.From [ 14 ]wemayobtainthecharactertable,Kroneckerproducts,and theClebsch-Gordan coefcients.Welistsomeresultshere,specicallythecha ractertablesandKronecker products.6.3.1CharacterTableandKroneckerProducts Thecharactertablerevealsarichstructurebehindthisgro up.Oneclearlysee thatthereareone,twoandthree-dimensionalrepresentati ons.Noticethatthe three-dimensionalrepresentationsarecomplex,wherethe conjugatesareindicated byabar. 64

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Table6-3.Thecharactertableof (54). n =31 C 1 1 C (1) 1 1 C (2) 1 6 C 1 6 C (0) 2 6 C (1) 2 6 C (2) 2 9 C (0) 3 9 C (1) 3 9 C (2) 3 1 1111111111 1 1 1111111 1 1 1 2 1 2222 1 1 1000 2 2 222 1 12 1000 2 3 222 1 1 12000 2 4 222 12 1 1000 3 1 33 3 2 00001 2 3 1 33 2 3 00001 !! 2 3 2 33 3 2 0000 1 2 3 2 33 2 3 0000 1 ! 2 Notethat = e 2 i 3 Table6-4.Hypotheticalavonsandtheirvevs. FieldRepresentationVevs 1 3 1 a 1 b 1 c 1 T 2 3 2 a 2 b 2 c 2 T 3 3 1 a 3 b 3 c 3 T 4 3 2 a 4 b 4 c 4 T 1 2 1 d 1 e 1 T 2 2 2 d 2 e 2 T 3 2 3 d 3 e 3 T 4 2 4 d 4 e 4 T 1 1 h Asetofavonsforthegroup (54)andtheirvevs.Thesecan,andsomewill,beusedto examinepossibleYukawamatrices. Finally,inordertobuildatheorywithinvariantquantitie sit'snecessarytoknowhow productsofrepresentationsbreakdownintoirreduciblere presentations.Tothisendwe listinFigure 6-1 acompletesetofKroneckerproductsfor (54): 6.3.2ProducingYukawaMatrices Wefocusinthissectiononsomeofthemathematicsusedtoexp lainhowanite groupcanbeusedwiththeFNtoproduceYukawamatricesbyusi ngasouravorgroup (54).TodosoweassumethataYukawatermisproducedfromtwo matterelds and ,alongthesamelinesofsay Q and u 65

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1 1 n 1 1 = 1 1 1 n 2 1 = 2 1 1 1 n 2 2 = 2 2 1 1 n 2 3 = 2 3 1 1 n 2 4 = 2 4 1 1 n 3 1 = 3 2 1 1 n 3 1 = 3 2 1 1 n 3 2 = 3 1 1 1 n 3 2 = 3 1 2 1 n 2 1 = ( 1 + 2 1 ) S + ( 1 1 ) A 2 1 n 2 2 = 2 3 + 2 4 2 1 n 2 3 = 2 2 + 2 4 2 1 n 2 4 = 2 2 + 2 3 2 1 n 3 1 = 3 1 + 3 2 2 1 n 3 1 = 3 1 + 3 2 2 1 n 3 2 = 3 1 + 3 2 2 1 n 3 2 = 3 1 + 3 2 2 2 n 2 2 = ( 1 + 2 2 ) S + ( 1 1 ) A 2 2 n 2 3 = 2 1 + 2 4 2 2 n 2 4 = 2 1 + 2 3 2 2 n 3 1 = 3 1 + 3 2 2 2 n 3 1 = 3 1 + 3 2 2 2 n 3 2 = 3 1 + 3 2 2 2 n 3 2 = 3 1 + 3 2 2 3 n 2 3 = ( 1 + 2 3 ) S + ( 1 1 ) A 2 3 n 2 4 = 2 1 + 2 2 2 3 n 3 1 = 3 1 + 3 2 2 3 n 3 1 = 3 1 + 3 2 2 3 n 3 2 = 3 1 + 3 2 2 3 n 3 2 = 3 1 + 3 2 2 4 n 2 4 = ( 1 + 2 4 ) S + ( 1 1 ) A 2 4 n 3 1 = 3 1 + 3 2 2 4 n 3 1 = 3 1 + 3 2 2 4 n 3 2 = 3 1 + 3 2 2 4 n 3 2 = 3 1 + 3 2 3 1 n 3 1 = 3 1 + 3 1 S + 3 2 A 3 1 n 3 1 = 1 + 2 1 + 2 2 + 2 3 + 2 4 3 1 n 3 2 = 3 1 + 3 2 + 3 2 3 1 n 3 2 = 1 1 + 2 1 + 2 2 + 2 3 + 2 4 3 2 n 3 2 = 3 1 + 3 1 S + 3 2 A 3 2 n 3 2 = 1 + 2 1 + 2 2 + 2 3 + 2 4 Figure6-1.CompletelistofKroneckerProductsfor (54). Wewouldliketoproduceisalistofproductswithmatterterm s .Schematically thiswouldmeanwewouldliketohaveinvariantstermsundert hefamilysymmetryofthe form Y j k l j k l ( ) ,(6–24) withthevariables j k l willbenon-negativeintegersandeachofthescalarsare denedinTable 6-4 66

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WecanbuildtablesofYukawamatricesfromtheabovebyrst ndingthedirect productofthemattertermswhichwillformseveralrepresen tations.Sothatifthematter eldstransformasrepresentationsoftheavorgroup,i.e. n 1 ,and n 2 thentheproductwouldtransformasanotherrepresentation n 3 ,( ) n 3 k .Wewritethis schematicallyas ( ) n 3 k = C ij k ( n 1 n 2 ; n 3 ) i j f n 1 n 2 n 3 g = 1 1 1 2 1 2 2 2 3 2 4 3 1 3 1 3 1 3 2 ,(6–25) wherethe C ij k ( n 1 n 2 ; n 3 )aretheClebsch-Gordan(CG)coefcientsfortheproduct thatproducesthe n 3 representation.Forthegroup (54)thesecanbeobtainedfrom generalgrouptheoreticalconsiderations,andusuallyare functionsofthecomplex number whichisdenedsuchthat 3 =1. ForthepurposeofmodelbuildingwewishtousetheseCGcoef cientstoformthe Yukawacouplings gY lk l k ,(6–26) wherethe g isthecouplingconstantoftheterm,usuallycomplexandcan absorboverall constantsthatcanappearinthe(CG)coefcients(wewillus uallythereforenotwrite theseoverallconstantsinthetablesthatfollow).SotheYu kawatermwillgenerallybe madeupofCGcoefcientsandthevacuumexpectationvalueso ftheavons. Finallyasanexample,letusdiscussanimportantgrouptheo reticalfeatureby workingoutexampleYukawaterm.SupposetheYukawatermisf ormatterelds 3 1 and 3 2 andaavon 2 ( 2 2 ).TheYukawacouplingsforsuchanarrangementcould bewritenas Y ij = C ij k ( 3 1 3 2 ; 2 2 ) C kl ( 2 2 2 2 ; 1 ) h 2 l i M .(6–27) Ifwealsohadasimilartermbutwithmatterelds 0 3 1 and 0 3 2 andaavon 2 67

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Y ij 0 = C ij k ( 3 1 3 2 ; 2 2 ) C kl ( 2 2 2 2 ; 1 ) h 2 l i M .(6–28) Itwouldbenaturaltoassumethatifweweretotakethecomple xconjugateof Y wewouldobtainthesameYukawacoupling Y 0 .Asitturnsoutthisconditionwouldbe equivalenttothefollowing,withtheassumtionthatthevac uumexpectationvalueisreal, C ij k ( 3 1 3 2 ; 2 2 ) C kl ( 2 2 2 2 ; 1 ) = C ij k ( 3 1 3 2 ; 2 2 ) C kl ( 2 2 2 2 ; 1 ).(6–29) Itcanbeshownbydirectcalculationthatthe C kl ( 2 2 2 2 ; 1 )areinfactreal,leavinguswith havingtoprovethat C ij k ( 3 1 3 2 ; 2 2 ) = C ij k ( 3 1 3 2 ; 2 2 ).(6–30) Afterdirectcalcuationitcanbefound,explicitlyforthec ase k =1,that C ij 1 ( 3 1 3 2 ; 2 2 )= 0BBBB@ 01000 2 00 1CCCCA C ij 1 ( 3 1 3 2 ; 2 2 )= 0BBBB@ 00 1000 2 0 1CCCCA .(6–31) ThisexplicitlyshowsthatEquation( 6–30 )cannotbesatised.Howeveritistruethat C ij k ( 3 1 3 2 ; 2 2 )= C ij k ( 3 1 3 2 ; 2 2 ) T .(6–32) Whichmeansthat C ij k ( 3 1 3 2 ; 2 2 )= C ij k ( 3 2 3 1 ; 2 2 ).(6–33) ThuswehavefoundthattheYukawatermsarenotingeneralher mitianasonemay havehadexpected. 68

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Table6-5.Yukawamatricesfortree-levelcoupling. ( ) Yukawamatrix 3 1 3 1 1 3 2 3 2 1 2664 100010001 3775 ( 2 r 2 r ) 1 0110 # ( 1 1 1 1 ) 1 1 Resultsfortree-levelcouplingsfor (54)symmetry.Notethat r =1,2,3,4. 6.3.3ExamplesofYukawaMatriceswith (54) WepresentheresomeexamplesofYukawacouplingsthatcanar iseduetothe symmetrygroup (54)startingwithTable 6-5 .Thetablesherearenotexhaustive, butareplacedhereemphasizetothereaderhowchoicesinbot htheavoncoupling andvevscanaffecttheresultsforYukawas.Wewishtoalsono tethatattree-levelitis difculttoproduceaoneinthe(3,3)positionusingthissym metry.Forthemodelwe designthiswillmotivatethefactthatwechoosea2 1avorstructureforthequarks. 69

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Table6-6.Yukawamatricesfromonethree-dimensionalavo n. ( ) avonYukawamatrix ( 3 1 3 1 ) 3 1 3 1 ( 3 2 3 2 ) 3 1 3 1 1 2664 a 1 00 0 b 1 0 00 c 1 3775 2664 0 c 1 b 1 c 1 0 a 1 b 1 a 1 0 3775 ( 3 1 3 1 ) 3 2 3 2 ( 3 2 3 2 ) 3 2 3 2 1 2664 0 c 2 b 2 c 2 0 a 2 b 2 a 2 0 3775 ( 3 1 3 2 ) 3 1 3 1 1 2664 0 c 1 b 1 c 1 0 a 1 b 1 a 1 0 3775 ( 3 1 3 2 ) 3 2 3 2 1 2664 a 2 00 0 b 2 0 00 c 2 3775 2664 0 c 2 b 2 c 2 0 a 2 b 2 a 2 0 3775 Notethat = e 2 i = 3 1 TakingthecomplexconjugateoftheseYukawatermswillprod uce theappropriateconjugatebutwiththeneedfortheappropri atesubstitutions,e.g., ( 3 1 n 3 1 ) 3 1 3 1 7! ( 3 1 n 3 1 ) 3 1 3 1 withtheentriesmodiedby a 1 7! a 3 70

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Table6-7.Yukawamatricesfromonetwo-dimensionalavon. ( ) avonYukawamatrix 3 1 3 1 2 1 2 1 3 2 3 2 2 1 2 1 2664 d 1 + 2 e 1 00 0 ( d 1 + e 1 ) 0 00 2 d 1 + e 1 3775 3 1 3 1 2 2 2 2 3 2 3 2 2 2 2 2 2664 0 2 e 2 d 2 d 2 0 e 2 e 2 2 d 2 0 3775 3 1 3 1 2 3 2 3 3 2 3 2 2 3 2 3 2664 0 d 3 e 3 2 e 3 0 2 d 3 d 3 e 3 0 3775 3 1 3 1 2 4 2 4 3 2 3 2 2 4 2 4 2664 0 d 4 e 4 e 4 0 d 4 d 4 e 4 0 3775 Notethat = e 2 i = 3 ,andalsothatthe2 4 representationdoesnotcontaintermswith 71

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CHAPTER7 THESUPERSYMMETRIC SU (5) (54)MODEL Wearereadytoproduceaviablemodelthattswithalldataan dmakesuseofall mechanismswehavepresentedonpreviouschapters.However ,beforewemoveinto themodelweshoulddiscusshowtoputthephenomenologicalc onclusionsaboutthe Yukawasectorwiththe (54)avorsymmetry.Therstsectionservesthispurposeso thatanyreadercanunderstandthelogicusedinordertomake themodel. Subsequentsectionsservetoillustratethefeaturesofthe model.Wedosobyrst presentingallmatterandavoncontentofthe SU (5) (54)model,alongwithextra symmetriesthatwereneededtotphenomenology.Wethenpro ceedtolookdeeperin eachquarksectorandseehoweachYukawamatrixisconstruct edbyuseofourchosen symmetries.Weconcludewiththepredictionsmentionedint heabstract. 7.1ProducingYukawaSectorPhenomenologywith (54) Thefocusofthissectionistodescribeinsomedetailthestr ategytakentoproduce ourmodel.Webeginwithanattempttofamiliarizeourselves with (54)byhavinga quicklookatitssalientfeatures. Themodelmakesuseofasupersymmetric SU (5)GUTtheory.Which,ofcourse, hasadirectimpactonhowwebuildatheoryunderouravorgro up.Nowalthoughfor themostpartthechoiceofGUTissomewhatarbitrary,an SU (5)theoryhasamethodof unifyingthechargedleptonanddown-typequarkmassesinas impleelegantway.Our choicemeansthatwemustplacematterintospecicrepresen tationsunder SU (5)(as showninEquation( 4–9 ))[ 19 ],theseare: N 1 L d 5 Q u e 10 .(7–1) The L and Q arethe SU (2)weakdoubletsandtheremainderparticlesaretheright handedweaksinglets. 72

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7.1.1 (54) asaFlavorGroup AglanceattheCharacterTable 6-3 showsthatthegrouphasbothtwoand three-dimensionalrepresentations.Meaningthereareman yoptionsforassigning representationstothemattercontent.Althoughalloption scanbeexploredwewishto limitthem,andforan SU (5)theorythiscanbedonebyexaminingthemassofthetop quark. Theoriginsofitsmassisattree-level,sinceitsvalueseem sclosetothatof (vev)oftheHiggsparticle.Ensuringthisresultsatisfact orilyforthethree-dimensional representationisverydifcultifnotimpossibletodo.Tos eethatthisisindeedthecase, let'sforthemomentdescribewhatwouldhappenifweusedsuc hathree-dimensional representation. First,ourmodelassumesthatthetopquarkmasscomesfromth eproductoftwo ten-dimensionalrepresentationsof SU (5).Let'sassumethatunderouravorgroup the X 10 transformsasanyofthefourthree-dimensionalrepresenta tions,i.e., 3 1 3 2 3 1 3 2 .Thentheinteractiontermresponsibleformassproducesno singletsbut instead,schematically,adirectsumofthree-dimensional representations X X 3 1 3 1 S 3 2, A .(7–2) Thebarshouldbeunderstoodasthecomplexconjugateofwhic hever 3 takenfor X .In ordertogetasinglettermwemusthaveaavon whichtransformsaseithera 3 1 ora 3 1 dependingontherepresentationchosenforthe 10 sothatviatheFNmechanism g M X X ,(7–3) where g isacouplingconstant, M isthemassscaleforthemechanism,andwehave suppressedtheHiggs.Inordertoexplainthemassofthetopp roperlythevevof theavoneldmustbethesameorderasthemassscale,i.e., h i M .Interms ofmodelbuildingthisfactisdifculttoexplainanditcanb edifculttocontrolthe 73

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interactiontermsinvolving .Thesedifcultiesareenoughtomakeusavoidtheusethe three-dimensionalrepresentationof (54)todescribetheup-quarks. Wehavechoseninsteadtohavethetopquarkbeasingletunder theavor group,i.e., X 3 1.Whilethetworemainingavorstogetherformatwo-dimens ional representations( X 1 X 2 ) T 2 r r =1,2,3,4.Underthisschemewehaveanaturalway toexplainthemassofthetopquarkattree-level: X 3 X 3 H u .Sowetaketheapproachthat bothquarksectorscanbewritteninthesamefashionjustdes cribed.Ourmotivationfor thechoiceof 2 1 structureistwo-fold. First,ifwehadchoseninsteadthatthe 5 transformas 3 under (54)itwouldbe difculttocontrolthepowerin ofanyoneentryinaYukawamatrixwithoutthedanger ofproducingthatsamepowerinanother.Anissuewhenthatsa mepowerislowerthan thepowerrequired,wereferthereadertoAppendix B toconrm.Thesecondweaker reasonissimplythattheYukawasofbothquarksectorsaresi milarbyhavingstructures whicharecopaceticwiththeuseoftwo-dimensionalreprese ntations.Texturezero structuresthatoccurinbothquarksectorsareeasilyachie vableandcanbeunderstood ascomingfromthevevsofthetwo-dimensionalavon. Wesummarizeourchoiceforthe SU (5)mattercontentunder (54): ( 10 1 10 2 ) T 10 3 (54) 2 p 1 ,(7–4) ( 5 1 5 2 ) T 5 3 (54) 2 r 1 p r = f 1,2,3,4 g ( 1 1 1 2 1 3 ) T (54) 3 s or 3 s s =1,2, includedaboveisthecasewhere p = r .WenowinvestigatethetypeofYukawamatrices wecanproducebasedonourchoiceofrepresentations.Allth epossibilitiesforthe up-quarkanddown-quarkYukawasaresummarizedwithjusttw omatricesrespectively 74

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0BBBBBBBBBBBBBBBBB@ 2 p n 1 = 2 p2 p n 2 p = ( 2 p 1 ) S 1 1, A 1 n 2 p = 2 p 1 n 1 = 1 1CCCCCCCCCCCCCCCCCA ,and 0BBBBBBBBBBBBBBBBB@ 2 p n 1 = 2 p2 p n 2 r = 2 s 0 2 s ” 1 n 2 r = 2 r 1 n 1 = 1 1CCCCCCCCCCCCCCCCCA ,(7–5) where s 0 s ”= f 1,2,3,4 g .TheupYukawamustalwaysnecessarilybetheleftcase. Whileforthedownitmaybeeithertherightcasewhen p 6 = r 6 = s 0 6 = s ”,ortheleftwhen p = r RecallthatattheendofSection 6.3.2 itwasmentionedthatweshalltryto reproducethetexturestructureandconstraintsofEquatio n( 5–17 ).Inordertoshowhow thiscanbeaccomplishedwewillmakeuseofatoymodelthatus estwomatterelds, ,andtwoavons 1 and 2 .Thegoalistothenshowhowtoobtainthetexture structureweseekfrommatricesconstructedinthefashions hownbyEquation( 7–5 ). 7.1.2AQuarkSectorToyModel Westartwithnotationthatisusedinthistoymodelandthrou ghoutothersections fromnowon.Sofarwehavedecidedthattherepresentationso fthemattercontentwill besplitinto 2 1 avorrepresentationsforreasonsexplainedinthesection before.So inordertodistinguishmatterthattransformsasa 2 fromthatasa 1 ourconventionuses anunderlinefordoubletsandnosuchunderlineforsinglets ,e.g.,wecouldwriteforthe lefthandedquark SU (2)doublet Q 0B@ Q 1 Q 2 1CA 2 2 Q Q 3 1 ,(7–6) 75

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Table7-1.Toymodelmatterandavoncontent. Matter SU (5) (54) 5 2 r 1 10 2 p 1 Flavons, h vev i 1 ab T 12 s 2 cd T 12 s Notethat p r s = f 1,2,3,4 g itshouldbeunderstoodthatthesubscriptsareavorindice s.Ascanbeseenthe notationwillbecleanerthanusingsubscriptsorsuperscri ptstodenotethedifferences inrepresentations.Fortheavoneldsthevariable willbeusedfor 3 for 2 ,and foreitherthe 1 1 orthe 1 representations.Anysubscriptsfoundontheavonswillai din simplydistinguishingamongthem. Returningtoourtoymodel,weshallassumethatoureldssho uldtransformas showninTable 7-1 Thesecondavonwillbeusedforthecasewherewewanttoshow withclaritya quadraticterminavons.Forthepurposeofbrevitywewilll ookattheYukawaterm forthedown-typequarks,butwhenpossiblewewilldiscusst heup-typequarkYukawa aswell.ThereasonforlookingatthedownYukawaisthatitpr esentsthemostgeneric possibleschemesinceitallowsboththecasewhere p = r and p 6 = r AYukawamatrixforthedownquarkcanbebuiltfromavoninte ractionswith , .Schematicallythestructureofthemassmatrixistherefor e 0B@ 1CA ,(7–7) followingthesamepartitioningschemeasinEquation( 7–5 ).Withthealltheabove wenowlookatseveralcasesinvolvingdifferentchoicesfor relationshipsbetweenthe 76

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variables p r s .Ineachcasewelistthepossibleresultsandlabelthem,onl ygoingas highasquadraticinavonelds.Greeklettersnotprevious lydenedarejustcoupling constants,andmultiplesuchconstantsinfrontofatermind icatethereareanumber ofdifferentwaystogetaavorinvariant.Therstcasewher e p = r isthecaseforour modelandsowewillspendsometimepointingoutitsimportan tfeatures. p = r :OneshouldnoticethisistherstcaseofEquation( 7–5 ).Therearetwo possiblechoiceswecantakefortheavon;either p = r = s or p = r 6 = s p = r = s :Thetree-levelresultsallowforanon-zeroterminthe(3,3 )position, usefulinthecaseofthetopquark.However,thisisnottheon lyallowedcontribution,in allthezerothordercontributionsare O ( 0 ): + 7! 0BBBB@ 0 0 00 00 1CCCCA .(7–8) Forarealisticmodel,wewouldnotlikethe2 2locationsoccupiedatthisorder.To avoidtheseresults,weareleadtoconcludethat mustbechargedundersome symmetrythatforbidsit.Asforrstordercontributionsin avonswehave: O ( ): 1 + 1 + 0 1 7! 0BBBB@ a 0 b 0 b a 0 b 0 a 0 1CCCCA .(7–9) Thereadershouldnoticehowthevevscontributetotheentri esabove.Achoiceof a =0wouldmeanthatthe(1,1)zerocouldbeprotected.Fortheu p-quarkswecould insteadhave a = 2 and b =0inordertosatisfyourtextureconstraintwhilehopingth at symmetriesdisallowany2 2terms. AlookattheKroneckerproductsrevealsthatthesecond-ord erinavonscan producedoubletsandtwotypesofsinglets. 77

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O ( 2 ): ( , r ) 1 2 + 1 2 + 0 1 2 + 1 2 7! (7–10) 0BBBB@ bd bc + r ad ac ad + r bc ac bd 0 ac 0 bd ( ad + bc ) 1CCCCA The( , r )istherebecausetheassociatedtermcontainsthreediffer entwaysto obtainasinglet,hencethethreecouplings(seeAppendix B ).Wepointoutthatthereare twodifferentbutequivalentwaystoperformtheproductoft herstterm: ( 1 )( 2 )and( 1 2 )( ).(7–11) Sincetheyareequivalent,therewillbenoneedtodifferent iatebetweenthemandwe shallmakenoeffortinthefuturetodoso. Fortheup-quarks,ifweforthemomentassumeonlyoneavon, say 1 with b =0, weseethatwerespectthe(1,1)zerowhilethe(2,2)canbell edin.ViaFNmechanism weareallowedtohavethat a 2 sothatwecanproducethetexturesallowedin Equation( 5–17 ).Alookatourmodelconrmsthatiswhatwasdone. p = r 6 = s :Thetree-levelresultsshouldremainthesame.Difference fromtheresults abovelieinthattherearenopossiblerst-orderinteracti ons. O ( 0 ): + 7! 0BBBB@ 0 0 00 00 1CCCCA .(7–12) Thesecond-orderresultsfollowsmuchinthesamewayasthec asewhere p = r = s ; 78

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O ( 2 ): ( ) 1 2 + r 1 2 7! 0BBBB@ 0 ad + bc 0 bc + ad 00 00 r ( ad + bc ) 1CCCCA .(7–13) Onceagainthereisanambiguityabouthowtoperformtheprod uctoftherstterm. Directcalculationforallpossiblecasesshowsagainthatt heambiguityisirrelevant becauseeachproductisequivalent.Noticethatthereareon lytwocouplings,which showsthatthereareonlytwowaystoproducesingletsforthi scase. p 6 = r :NowwehavethesecondcaseofEquation( 7–5 ).Beforewegoontodiscuss thetwopossiblechoices,fromEquation( 7–5 ),weknowthat 2 p n 2 r = 2 s 0 2 s ” p 6 = r 6 = s 0 6 = s ”.(7–14) Theabovehasdirectimplicationsattree-levelsincenowth ereisonlyoneresultwecan haveandthatis O ( 0 ): 7! 0BBBB@ 00000000 1CCCCA .(7–15) Asfortherstorder,aavoncanonlytransformaseitherthe 2 s 0 orthe2 s ” .The specicswilldependontherepresentations,buttheresult swillbeinoneoffoursetsof possiblecombinationswhereineachsetonlyonematrixwoul dbechosen: 79

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O ( ): 1 7! 0BBBB@ a 00 0 b 0 000 1CCCCA or 0BBBB@ 0 b 0 a 00 000 1CCCCA 0BBBB@ a 00 0 b 0 000 1CCCCA or 0BBBB@ 0 a 0 b 00 000 1CCCCA ,(7–16) listingonlytwosetsforbrevityandtheothertwocanbeobta inedbyinterchanging a and b .The“or”isbecausetherearetwopossiblechoicesforrepre sentationof 1 ,atheme thatcontinuesatsecondorder: O ( 2 ): 1 2 + r 1 2 7! 0BBBB@ ac 00 0 bd 0 00 r ( ad + bc ) 1CCCCA or 0BBBB@ 0 bd 0 ac 00 00 r ( ad + bc ) 1CCCCA 0BBBB@ ac 00 0 bd 0 00 r ( ad + bc ) 1CCCCA or 0BBBB@ 0 ac 0 bd 00 00 r ( ad + bc ) 1CCCCA (7–17) wheretogettheothersetofmatricesoneneedsonlyintercha ngetherolesofthe ac termswith bd Weprovidedaglimpseintotheworkingsoftwo-dimensionalr epresentations. Althoughnotdiscussedaboveonecantellwhichentriesprov idetexturezerosbyclever choiceofvevs.Withanunderstandingofthetexturestructu rethat (54)canproduce, wearenowreadytodiscussourmodel. 80

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7.1.3ChoosingFlavonRepresentations Wehadmentionedinthebeginningofthesectionthatwewould lettheright handedneutrinostransformas 3 1 ofouravorgroup.Thechoiceissomewhatarbitrary, wecouldhaveeasilychosentherepresentation 3 1 3 2 ,or 3 2 .Regardless,theirCG coefcientsaresimilarenoughthatanychoicewoulddowith noclearadvantageofone overtheother. Asforthechoiceoftwo-dimensionalrepresentationsforth emattercontent,there issomearbitrarinesstothistoo.AlookatAppendix B ,focusingontheCGcoefcients, willrevealthatalltwo-dimensionalrepresentationsunde rthecase 2 r n 2 r havethesame result.Theonlyinterestingfeatureoccursonthe 2 p n 2 r with p 6 = r case.One,interms ofmodelbuilding,couldmakeuseofthefactthatsuchaprodu ctproducestwodifferent two-dimensionalrepresentations.Makingitpossibletoex ploitthisinacleverfashion, buttheauthorhasfoundthatusingthesametwo-dimensional representationthroughout requireslessavonsandsoasimplermodel. Finally,nowthatwehaveoptedtousethesame 2 forourmodel,whichoneshould beused?LookingatAppendix B showsthattakingtheproductof 2 1 3 1 producesCG coefcientsthatcontainpowersof = e 2 i = 3 .Thesameistrueforthecasesinvolving 2 2 and 2 3 withthesoleexceptionof 2 4 .Itshouldbepossibletoabsorbthe intocoupling constants,thusineffectwehavenorealadvantageofusingo nerepresentationover another.However,forthesakeofclarityandsimplicitywec hooseinsteadtouse 2 4 and avoidtheissuealtogether. 7.2MatterandFlavonContent Themodelhasasupersymmetricbackground,andweassumetha tweareabove unicationscaleof SU (5)GUT.Themattercontentfoundinthestandardmodeltsin to SU (5)representationsas X 10 5 N 1 .(7–18) 81

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Forreasonsdiscussedinwechosetohaveboth 5 andthe 10 intotwoandone-dimensional representationsbutkepttheheavyneutrinosasthree-dime nsional,i.e., ( X 1 X 2 ) T 2 4 X 3 1 ;( 1 2 ) T 2 4 3 1 ; N 3 1 .(7–19) Rememberthatthetopquarkmasswasmotivationforusingthe singletanddoublet structureforthe 10 .Asidefromtheseassignmentsthereareotherchargesthatw ehave giventheseelds,namelythe Z u 3 n Z d 2 n Z 2 charges.Thesuperscriptsindicatethatthese chargesareprimarilygiventothoseeldsthatcontainthat righthandedparticle. Aswewillsoonshow,thequarkandchargedleptonsectorsare populatedmainly bythreeextraelds: u 2 4 d 2 4 1 ,(7–20) Thesubscriptsremindusthattheseeldsarechargedundert hecyclicsymmetry( Z ) withthesameletterasitssuperscript. Ontheotherhand,theneutralleptonsectorisprimarilypop ulatedbyjusttwo three-dimensionalavons: 3 1 0 3 1 ,(7–21) onceagainindicatingtheappropriate (54)charge.ThenalingredientsaretheHiggs eldswhichincludesboththeveandforty-vedimensional representationsof SU (5). Wemaynowpresentthesuperpotential,butwithoutallthecl utterofcoupling constants, W model = W u + W d + W dirac + W majorana ,(7–22) where 82

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W u H u +( u ) H u + 2 d ( u ) H u +( u )( u ) H u + 2 d ( u )( u ) H u W d H d +( u ) H d + 2 d ( u ) H d + ( d ) H d +( u )( d ) H d +( d )( H 45 d ), W dirac NH u +( 0 ) NH u W majorana M 2 N N + 0 2 N N (7–23) Thevalue M istheMajoranamassscalethatistobedeterminedatalatert ime.We havelistedonlytermsthatcontributetolowestorderinthe irrespectivematrixentries. Theparenthesishavenobearingonhowtotakeproductsunder ouravorgroup,they merelyindicatethatdistincteldshavethecorrectcyclic chargestobeneutralunder thosecharges.Forasummaryoftheeldcontentandtheircha rgeslookatTable 7-2 ItshouldbestatedthatonTable 7-2 wecouldhaveincludedanothercyclic symmetry Z n 2 .Forthissymmetrythe N wouldbeoddandsowouldthe and 0 avons. Allothereldscouldinprincipleremainneutral.Themodel however,doesnotseemto requiretheextrasymmetryandsoweleavethissymmetryouto fthetable. Thenextthreesectionswillcontainsomeofthenerdetails ofourmodel.The rsttwosectionsincludealookatthevevsoftheneweldswe haveintroducedanda detailedlookonhoweachofthesuperpotentialtermspopula tetheirmatrices.Thelast sectionpresentsthenalresultsofourmodel.Thesephenom enologicalresultsinclude themassesforbothlightandheavyneutrinosaswellastheex pectedcorrectionstothe tri-bimaximalmatrix. 7.3VacuumValues Thevacuumexpectationvaluesoftheavoneldsgoas: 83

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Table7-2.Fieldcontentandchargesforthesupersymmetric SU (5) n (54)model. Matter SU (5) (54) Z u 3 Z d 2 Z 2 N 13 1 111 5 2 4 1 1,1 1,11,1 10 2 4 1 ,11,11,1 HiggsH u H d 5 5 1 1 1,11,11,1 H 45 u H 45 d 45 45 1 1 2 1,1 1, 1 Flavons, h vev i u a 1 0 T 12 4 2 11 d 0 a 02 T 12 4 1 11 b 1 b 1 0 T 1 3 1 111 0 b 0 1 00 T 1 3 1 1 11 Singlets, h vev i c 11 11 1 Notethat = e 2 i 3 andtherecouldbeanothersymmetry Z n 2 butitisfoundunnecessary. h i c h i ( a 1 a 2 ) T h i ( b 1 b 2 b 3 ) T .(7–24) Assaidintheintroduction,wemakeuseoftheFNmechanism,w hichmeansthateach avonvevwillbesuppressedbyaneffectivemassscale( M )ofsomegaugedinteraction atmuchhigherenergies.Thesuppressedvevsthenarepostul atedtogoas c M m +1 a 1 M 2 a 02 M b 1 M b 0 1 M n m 0, n > 0(7–25) where m and n areintegers.TheCabbiboangleisnotourexpansionparamet erforthe wholemodelbutinstead .20.Thereissomearbitrarinesstothis,theonlyconstrain t beingthat > .182,butwechosenitsstatedvaluesothatthemassrelation sare 84

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consistentatenergiesoftheGUTscaleof2 10 16 GeV anditsvaluemustremainclose totheCabibboangleifweexpectEquation( 5–17 )toremaintrue. Thevalueof m canbedeterminedfromtherelativesizeof v 5, d ,thevevofthe H d ,to that v 45, d of H 45 d bywayof n H 45 d / m v 45, d v 5, d .(7–26) For v 45, u weassumethat v 5, u v 45, u v 45, d .Theimplicitassumptionofaboveisthat v 45, d v 5, d ,otherwisewemayloseourperturbativepowerbyhavingasin gletwithavev thatisgreaterorequaltotheFNscale.Finallywemustmenti ontherelativesizeofthe v 5, u tothatof v 5, d ,weexpect cot ( ) v 5, d v 5, u /O ( 3 ),(7–27) whichwouldsatisfytheintra-familyrelationship m b = m t Asforthevalueof n ,itmaybedeterminedbythesizeofthebaryonasymmetryour modelpredictsfromleptogenesisconstraintsonthelighte stoftheheavyneutrinos, M 1 [ 5 – 8 ].Currentapproximateboundslimitthemassof M 1 > 10 8 GeV and,asweshall seeattheendofthissection,thislimitwillrestrictourpo ssiblechoicesfor n suchthat n =1,2,3. 7.4QuarkYukawas Thepurposeofthissectionistoexploreindetailtheresult swritteninEquation( 7–23 ) forthequarksectors.Welimitourinvestigationtodemonst ratingtheoriginsofall Yukawatexturesandthenecessarycouplingconstants.Each superpotentialcontains termsthatproducetheleadingordercontributiontotheirY ukawamatrix.Allother termsincludingthosewhichareof O ( 8 )andhigherfortheup-quarks,and O ( 5 )forthe down-quarks,willbeneglected. 85

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ThesuperpotentialcontributionsmakingtheupYukawamatr ixaregivenby W u H u + ( u ) H u + 2 d ( u ) H u + ( u )( u ) H u +(7–28) r 2 d ( u )( u ) H u Itshouldbestatedthatthe SU (5)algebrarequiresthatanycontributiontothe H 45 u fromthe 10 mustbeanti-symmetricinavorspace.Asaresulttheonlyan ti-symmetric combination( ) A producesa 1 1 .Sincetherearenoavon 1 1 singletsthereareno devastatinglowordercontributions,andtheonlycontribu tionsthatcansurvivewouldbe correctionstotheYukawamatrices,e.g.,thelowestorderc orrectionis: 3 u ( H 45 u ). InEquation( 7–28 )theGreekletters , r arecouplingswhichalsoaidin identifyingwhereeachtermcontributestotheupYukawamat rix: Y (2 = 3) 5 O 0BBBBBBBBBB@ 0 r 6 4 r 6 4 2 4 2 1 1CCCCCCCCCCA .(7–29) Thedown-quarksectorisbitmorecomplexforweincludeboth contributionsdueto theregularHiggs H d andthe H 45 d .Bothcontributionswillbeaddedtoproduceasingle Yukawamatrix,andsobelowweonlyincludethosetermsthata releadingintheirsum. PrimesonGreeklettersareforthecouplingsthatoccurinth iscase,andsotheterms wehaveare W d H d + 0 ( u ) H d + 0 2 d ( u ) H d + 0 0 ( d ) H d (7–30) +( r 0 r 0 0 )( u )( d ) H d + 0 ( d )( H 45 d ), 86

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with( r 0 r 0 0 )meaningthattherearetwowaystoproducesinglets,eachwi ththeirown couplings.Intermsof wehave Y ( 1 = 3) 5 O 0BBBBBBBBBB@ 0 r 0 3 0 4 r 0 0 3 0 0 2 0 0 01 1CCCCCCCCCCA Y ( 1 = 3) 45 O 0BBBBBBBBBB@ 0000 0 2 0 000 1CCCCCCCCCCA .(7–31) FinallywithalltheaboveresultsonecanconstructtheYuka wamatricesfromthewell knownresultsof SU (5)GUTmodels[ 20 36 ]: Y (2 = 3) = Y (2 = 3) 5 ,(7–32) Y ( 1 = 3) = Y ( 1 = 3) 5 + Y ( 1 = 3) 45 Y ( 1) = Y ( 1 = 3) T 5 3 Y ( 1 = 3) 45 7.5NeutrinoMasses Asimilarprocedureasoutlinedin[ 26 ]isfollowedhere.Wepostulatetheadditionof twonewtermstothesuperpotentialoftheMSSM: W = L H u Y (0) N + M N Y maj N .(7–33) TheMajoranatermalsocomeswithamassscale M whichwesupposecancome fromahigherenergyscale.Wedesignedthemodeltoproducet heabovewith theassumptionsthattheavorsof N togetherforma 3 1 underouravorgroup.To accomplishthetaskweemployedtheuseoftwothree-dimensi onalrepresentations and 0 ,whosedetailscanbefoundinTable 7-2 Ourmodel,Equation( 7–23 ),producestheDiracterm 87

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W dirac NH u +( 0 ) NH u ,(7–34) rewrittenhereforconvenience.TheresultingYukawamatri xis Y (0) 1 M 0BBBBBBBBBB@ 00 b 0 1 0 b 0 1 0 b 1 b 1 0 1CCCCCCCCCCA .(7–35) Fortheabovetherearecouplingconstantsnotincludedbeca usetheyareof O (1)and canbesimplyabsorbedbytheirrespectivevevs.Inprincipl e,itwouldbepossibletoget tri-bimaximalmixinginthecasethat O ( b 0 1 ) 6 = O ( b 1 ).However,ifthisisthecaseandit iscarriedthroughtotheMajoranamatrixthenthelightneut rinomatxi Y wouldcontain entriesthataresumsofvariouspowersin .Asomewhatsimplecalculationwillshow thatthisistrue. Incaseslikethese,itisdifculttodiagonalizeby U tri bi sinceeithercareful cancellationsareneededinthevariouspowersin orsomeexplanationforthe complexityofthecouplingconstantsshouldbegiven.Toavo idsuchacomplication fromarisingitisfoundbesttoassumethat O ( b 0 1 )= O ( b 1 ).Infact,itsfoundthatmuch moreelegantresultscanarisewhenoneassumesthat b 0 1 = b 1 andsothisisthe assumptionweshallmake. TheMajoranacontributionsterms,foundinEquation( 7–23 ),are W majorana M 2 N N + 0 2 N N .(7–36) TheMajoranamatrixisthen 88

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Y maj 2 n 0BBBB@ 1CCCCA + 0BBBB@ 0 00 00 0 0 0 0 1CCCCA .(7–37) TheunprimedGreekletterscorrespondtocouplingsforthe andprimedlettersfor 0 .Donotconfusetheseparametersforthosewrittendowninth equarksector.Just asbeforetheyarecouplingconstantsresultingfromthenum berofwaysonecanget asingletterm.Noticethatthevevsoftheavonsareinclude d,butfoundwithin 2 n by Equation( 7–25 ).Thebestchoicesfortheparametersaboveseemtobe = =0, = = 0 =1, j 0 j =.100 .004.(7–38) Theparameter 0 cancontrolthevalueofratioofthemasssquareddifference sfoundin Equation( 5–18 ).Thechoiceof j 0 j =.1producesexactlytheratioof32thattscurrent data. 7.6PhenomenologicalResults WehavesuccessfullyproducedYukawamatriceswithentries ofthesameorder aswehadsoughtinEquation( 5–17 ).Wehaveevenproducedasetofmatricesforthe neutrinosthattogetherproducealightneutrinomatrixtha tcanbediagonalizedbythe tri-bimaximalmatrix.Herewetakethingsastepfurtherand trytoreproducetheSM resultsandndvaluesforneutrinosector. TherststepistoreproducetheresultsoftheSMextrapolat edtotheenergy scaleof2 10 16 GeV [ 2 – 4 9 10 25 27 ].Wehaveseenthatforthequarksector, basedonoursuperpotentialterms,therearetenparameters tobedetermined.Oneof theseparametersiffoundtobeirrelevantandsoleftequalt oone(the(1,3)and(3,1) entriesoftheupYukawas).Wearethenleftwithninethatare chosensuchthatthey reproducemassesandtheCKMmatrixwhichmeansonlysevenco nstraintsandso twofreeparameters.Thelasttwoparametersarechosensuch thattheyatthesame 89

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timerespectthemassofthedown-quark(duetohigherorderc orrection)andalsot thelimitsoftheexperimentalresultsforthesolarangleof theleptonmixingmatrix.Our modelhassomesensitivitytothevaluesofthenalparamete rswhichexplainsthe errorsweplacedonthepredictedangles. Asfortheneutrinoswehavediscussedthesefreeparameters andbecauseof theconstrainsimposedbybothdataandthetri-bimaximalma trixwehaveonlyone parameter(whatwecalled 0 intheneutrinoanalysis). QuarkSector: Y (2 = 3) 0BBBB@ 01.1 6 4 1.1 6 4 1.8 2 4 1.8 2 1 1CCCCA m u v 5, u 0BBBB@ 2.7 8 2.3 4 1 1CCCCA ,(7–39) and Y ( 1 = 3) 0BBBB@ 0.5 3 .5 4 .3 3 .5 2 .6 2 .5 01 1CCCCA m d v 5, d 0BBBB@ .6 4 .5 2 1 1CCCCA .(7–40) DiagonalizationalsoreproducesaCKMmatrix( U ckm )consistentwithdataextrapolated totheGUTscale. LeptonSector: SU (5)with H 45 d guaranteesoursuccessfulreproductionofthe masses Y ( 1) 0BBBB@ 0 .3 3 .5 .5 3 1.5 2 0 .5 4 .6 2 1 1CCCCA m e v 5, d 0BBBB@ .2 4 1.5 2 1 1CCCCA .(7–41) Asfortheneutrinoswehavefoundthat 90

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Y (0) 2 n 0BBBB@ 001010110 1CCCCA Y maj 2 n 0BBBB@ 0 01 00212 1 1CCCCA j 0 j =.100 .004,(7–42) Usingthelightneutrinoapproximationandusing 0 = .100weobtain Y v 2 5, u 2 M 0BBBB@ 0 11+ 1+ 1 1CCCCA m v 2 5, u 2 M 0BBBB@ 2 2+ 1CCCCA ,(7–43) whereweremindthereaderthat m v isthelightneutrinomasses.Thevalueof issuch that .222for 0 = .100and .182for 0 =.100.Wepredictthatthemass scale M is M 3 10 15 GeV ,(7–44) avaluethatisatoneorderawayfromourGUTmodelscale.Resu ltsthatfolloware independentonthesignof 0 .Boththecorrectionstothetri-bimaximalmatrixandthe massesofthelightneutrinos(normalhierarchy)arepredic tedtobe j e i .825 j 1 i +.566 j 2 i .127 j 3 i m ,1 5 10 3 eV j i .474 j 1 i +.532 j 2 i .706 j 3 i m ,2 1 10 2 eV j i .329 j 1 i +.639 j 2 i +.702 j 3 i m ,3 5 10 2 eV ,(7–45) wherewewanttomakeitclearthat 91

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m ,2 m ,1 =2, m ,3 m ,1 =10,and X i m i =6.5 10 2 eV .(7–46) Whilewepredictthatthemassesfortheheavierneutrinosar e M heavy 2 n 0BBBB@ 9.7 10 12 GeV 2.2 10 14 GeV 3.4 10 14 GeV 1CCCCA ,(7–47) twomassesarenearlydegenerate.Asmentionedearlierthev alueof n couldbechosen suchthatthemassesareconsistentwithlimitsposedbylept ogenesisresponsiblefor thebaryonasymmetry[ 5 – 8 ], M 1 9.7 2 n 10 12 GeV > 10 8 GeV n =1,2,3.(7–48) Becausecorrectionsfortri-bimaximalmatrixareobtainfr omdiagonalizationofthe chargedleptonYukawa,caremustbetakensothattheangleso btainedarewellwithin experimentallimits[ 29 ]: j 13 j < 11.4 o 34.43 +1.35 1.22 o ,36.8 o < atm < 53.2 o .(7–49) Withtheaboveinmindwepredict(andpostdict)that 13 7.31 +0.60 1.75 o 34.46 +1.02 1.52 o atm 45.15 +0.04 0.10 o .(7–50) Weshouldmentionthatthereactorangle( 13 )issomewhatlarge.Theoriginforthisis the(1,3)positionofthechargedleptonYukawa,whichleads toarotationangle(from diagonalizingtheYukawa)“ 13 ”thatiscomparabletothe“ 12 ”rotationangle.Nowif wetrackthephasesbyfollowingtheguidelinesgivenin[ 24 ],whichprovidesmethods fordetermininghowmanyfreephasesthereareandwhereinth eYukawastheymay belocated.Wendthatthe(1,3)positionforthechargedlep tonYukawacouldhavea 92

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phase.Sothereactorangle,beingasumoftwocomparableang les(asstatedearlier) withaphasedifferencebetweenthem,couldbesuchthatinge neral0 o 13 7.31 o 93

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APPENDIXA ADDITIONAL SU (5)CONTENT Containedinthisappendixareextradetailsandpotentiall yusefulinformation involvingthegaugedsymmetry SU (5). A.1GeneratorsforNon-FundamentalRepresentations Thegeneratorsfortheconjugaterepresentationofthefund amentalaregivenas T andarerelatedtothefundamentalby T = T T ,or T ,(A–1) becausethegeneratorsthemselvesarehermitianoperators .Asfortheten-dimensional representationwendthat T 10 A B C D = A C T B D + T A C B D .(A–2) Fortheadjointrepresentationswehave T adj A D B C = A C T DB + T A C D B .(A–3) A.2ChargeOperatorsforNon-FundamentalRepresentations Asforthechargeoperatorswehave Q 5 = T 11 2 5 3 1 = 2 T 12 ,(A–4) Q 10 A B C D = A C T 11 BD + T 11 AC B D +2 5 3 1 = 2 A C T 12 BD + T 12 AC B D ,(A–5) and Q adj A D B C = A C Q 5 DB + Q 5 AC D B .(A–6) Asanexampleifweoperatetheelectromagneticchargeopera toroftheaboveonthe forcecarriers(whichareintheadjointrepresentation)we ndthechargesare 94

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Q adj F = 0BBBB@ 0 4 3 1 3 4 3 01 1 3 10 1CCCCA F ,(A–7) Theon-diagonaltermsareneutral,whichweshouldexpectfo rallthecolorforces,and thelinearcombinationthatresultsinthe A and Z potentials.Theoff-diagonalterms howeverpickupcharges,e.g.,the W eldsareofcoursechargedappropriatelyas indictatedabove.Wealsonoticethatthe X andthe Y arealsocharged,withcharges thatcorrespondexactlytowhatwouldbeneededforquark-le ptoninteractions. 95

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APPENDIXB CLEBSCH-GORDANCOEFFICIENTSFOR (54) Werstmustdeneavectorspaceofeachoftheirreduciblere presentations. Thesewilldemonstratehowavectortransformsunderthegen eratorsa,b,andcofthe irreduciblerepresentations. 3 1 : 0BBBB@ x 1 x 2 x 3 1CCCCA 7! 0BBBB@ x 2 x 3 x 1 1CCCCA a 0BBBB@ x 3 x 2 x 1 1CCCCA b 0BBBB@ x 1 2 x 2 x 3 1CCCCA c 3 1 : 0BBBB@ x 1 x 2 x 3 1CCCCA 7! 0BBBB@ x 2 x 3 x 1 1CCCCA a 0BBBB@ x 3 x 2 x 1 1CCCCA b 0BBBB@ 2 x 1 x 2 x 3 1CCCCA c 3 2 : 0BBBB@ x 1 x 2 x 3 1CCCCA 7! 0BBBB@ x 2 x 3 x 1 1CCCCA a 0BBBB@ x 3 x 2 x 1 1CCCCA b 0BBBB@ x 1 2 x 2 x 3 1CCCCA c 3 2 : 0BBBB@ x 1 x 2 x 3 1CCCCA 7! 0BBBB@ x 2 x 3 x 1 1CCCCA a 0BBBB@ x 3 x 2 x 1 1CCCCA b 0BBBB@ 2 x 1 x 2 x 3 1CCCCA c 96

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2 1 : 0B@ x 1 x 2 1CA 7! 0B@ x 1 2 x 2 1CA a 0B@ x 2 x 1 1CA b 0B@ x 1 x 2 1CA c 2 2 : 0B@ x 1 x 2 1CA 7! 0B@ x 1 2 x 2 1CA a 0B@ x 2 x 1 1CA b 0B@ 2 x 1 x 2 1CA c 2 3 : 0B@ x 1 x 2 1CA 7! 0B@ x 1 2 x 2 1CA a 0B@ x 2 x 1 1CA b 0B@ x 1 2 x 2 1CA c 2 4 : 0B@ x 1 x 2 1CA 7! 0B@ x 1 x 2 1CA a 0B@ x 2 x 1 1CA b 0B@ x 1 2 x 2 1CA c 1 1 : x 7! x a x b x c Withtheabovemappingsdeneditbecomespossiblendtheou tcomesoftakingthe productofanytworepresentations.Thelistbelowisnotexh austive,butweinclude thosethatareimportanttothispaper. x n y : 1 1 n 1 1 = 1 x n y = xy .(B–1) x n y : 1 1 n 2 r = 2 r r =1,2,3,4. Ifoneweretodirectlymultiplythesetworepresentationst henwewouldgetatwo dimensionalrepresentationswhodoesnttransformquiteri ghtunderthegenerator b xy 1 xy 2 7! xy 1 xy 2 b .(B–2) Transformationsundertheremaindergroupelementsarecor rect.Thuswewishto changethatof b whilekeepingtheothersthesame.Thismeansthatwewishto nd: 97

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Sa = aS (B–3) Sb 0 = bS Sc = cS where a = a 1 0 0 a 2 b = 0110 b 0 = 0 1 10 c = c 1 0 0 c 2 ,(B–4) where a 1 a 2 c 1 c 2 arealldependentonthetwodimensionalrepresentation2 r SatisfyingalltheconditionsEq.( B–3 )isthematrix S = 100 1 ,(B–5) whichmeansthatwehave x n y = xy 1 xy 2 .(B–6) x n y : 1 1 n 3 s = 3 s 0 ,( s s 0 )=(1,2),(2,1) Theeffectofmultiplyingby x isaswitchingofthesignswhenatransformationvia b occurs.Thusallwedoreallyischangeeithera3 1 7! 3 2 or3 2 7! 3 1 x n y = 0@ xy 1 xy 2 xy 3 1A .(B–7) x n y : 1 1 n 3 s = 3 s 0 ,( s s 0 )=(1,2),(2,1) Theeffectofmultiplyingby x isaswitchingofthesignswhenatransformationvia b occurs.Thusallwedoreallyischangeeithera 3 1 7! 3 2 or 3 2 7! 3 1 x n y = 0@ xy 1 xy 2 xy 3 1A .(B–8) x n y : 2 r n 2 r =( 1 2 r ) S ( 1 1 ) A x n y = 1 p 2 ( x 1 y 2 + x 2 y 1 ) x 2 y 2 x 1 y 1 S 1 p 2 ( x 1 y 2 x 2 y 1 ) A .(B–9) x n y : 2 1 n 2 2 = 2 3 2 4 x n y = x 2 y 2 x 1 y 1 x 1 y 2 x 2 y 1 .(B–10) 98

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x n y : 2 1 n 2 3 = 2 2 2 4 x n y = x 2 y 2 x 1 y 1 x 2 y 1 x 1 y 2 .(B–11) x n y : 2 1 n 2 4 = 2 2 2 3 x n y = x 1 y 2 x 2 y 1 x 1 y 1 x 2 y 2 .(B–12) x n y : 2 1 n 3 1 = 3 1 3 2 2 1 n 3 1 = 3 1 3 2 Theproductof2 1 and3 1( l ) isingeneralproducesasix-dimensionalrepresentation: 2 1 n 3 1( l ) =6 ( l l ) 0BBBBBB@ x 1 y 1 x 1 y 2 2 x 1 y 3 x 2 y 3 x 2 y 2 2 x 2 y 1 1CCCCCCA .(B–13) Whenwehavethat,for6 ( k l ) k + l =0 mod ( n )thenthematrixthatdiagonalizesall elementsofthegroupis S = 1e1 e 1 = 0@ 100010001 1A e = 0@ 001010100 1A .(B–14) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: x n y = 1 p 2 0@ x 1 y 1 + 2 x 2 y 1 x 1 y 2 + x 2 y 2 2 x 1 y 3 + x 2 y 3 1A 1 p 2 0@ x 1 y 1 2 x 2 y 1 x 1 y 2 x 2 y 2 2 x 1 y 3 x 2 y 3 1A .(B–15) x n y : 2 1 n 3 2 = 3 1 3 2 2 1 n 3 2 = 3 1 3 2 Theproductof2 1 and3 2( l ) isingeneralproducesasix-dimensionalrepresentation: 2 1 n 3 2( l ) =6 ( l l ) 0BBBBBB@ x 1 y 1 x 1 y 2 2 x 1 y 3 x 2 y 3 x 2 y 2 2 x 2 y 1 1CCCCCCA .(B–16) Whenwehavethat,for6 ( k l ) k + l =0 mod ( n )thenthematrixthatdiagonalizesall elementsofthegroupis 99

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S = 1e1 e 1 = 0@ 100010001 1A e = 0@ 001010100 1A .(B–17) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: x n y = 1 p 2 0@ x 1 y 1 2 x 2 y 1 x 1 y 2 x 2 y 2 2 x 1 y 3 x 2 y 3 1A 1 p 2 0@ x 1 y 1 + 2 x 2 y 1 x 1 y 2 + x 2 y 2 2 x 1 y 3 + x 2 y 3 1A (B–18) x n y : 2 2 n 2 3 = 2 1 2 4 x n y = x 2 y 2 x 1 y 1 x 1 y 2 x 2 y 1 .(B–19) x n y : 2 2 n 2 4 = 2 1 2 3 x n y = x 1 y 1 x 2 y 2 x 1 y 2 x 2 y 1 .(B–20) x n y : 2 2 n 3 1 = 3 1 3 2 Theproductof2 1 and3 1 isingeneralproducesasix-dimensionalrepresentation: 2 2 n 3 1 =6 ( 2 n 3 1, 2 n 3 +1) 0BBBBBB@ x 1 y 1 x 1 y 2 2 x 1 y 3 x 2 y 3 x 2 y 2 2 x 2 y 1 1CCCCCCA .(B–21) Whenwehavethat,for6 ( k l ) k =1and l =0 mod ( n )thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–22) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: x n y = 1 p 2 0@ x 1 y 2 + x 2 y 3 2 x 1 y 3 + 2 x 2 y 1 x 1 y 1 + x 2 y 2 1A 1 p 2 0@ x 1 y 2 x 2 y 3 2 x 1 y 3 2 x 2 y 1 x 1 y 1 x 2 y 2 1A (B–23) x n y : 2 2 n 3 1 = 3 1 3 2 Theproductof2 1 and 3 1 isingeneralproducesasix-dimensionalrepresentation: 100

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2 2 n 3 1 =6 ( 2 n 3 2, 2 n 3 +2) 0BBBBBB@ x 1 y 1 x 1 y 2 2 x 1 y 3 x 2 y 3 x 2 y 2 2 x 2 y 1 1CCCCCCA .(B–24) Whenwehavethat,for6 ( k l ) k =0 mod ( n )and l =1thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 001100010 1A e = 0@ 010100001 1A .(B–25) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: x n y = 1 p 2 0@ 2 x 1 y 3 + x 2 y 2 x 1 y 1 + x 2 y 3 x 1 y 2 + 2 x 2 y 1 1A 1 p 2 0@ 2 x 1 y 3 x 2 y 2 x 1 y 1 x 2 y 3 x 1 y 2 2 x 2 y 1 1A (B–26) x n y : 2 2 n 3 2 = 3 1 3 2 Theproductof2 1 and3 2 isingeneralproducesasix-dimensionalrepresentation: 2 2 n 3 2 =6 ( 2 n 3 1, 2 n 3 +1) 0BBBBBB@ x 1 y 1 x 1 y 2 2 x 1 y 3 x 2 y 3 x 2 y 2 2 x 2 y 1 1CCCCCCA .(B–27) Whenwehavethat,for6 ( k l ) k =1and l =0 mod ( n )thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–28) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: x n y = 1 p 2 0@ x 1 y 2 x 2 y 3 2 x 1 y 3 2 x 2 y 1 x 1 y 1 x 2 y 2 1A 1 p 2 0@ x 1 y 2 + x 2 y 3 2 x 1 y 3 + 2 x 2 y 1 x 1 y 1 + x 2 y 2 1A (B–29) x n y : 2 2 n 3 2 = 3 1 3 2 Theproductof2 2 and 3 2 isingeneralproducesasix-dimensionalrepresentation: 101

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2 2 n 3 2 =6 ( 2 n 3 2, 2 n 3 +2) 0BBBBBB@ x 1 y 1 x 1 y 2 2 x 1 y 3 x 2 y 3 x 2 y 2 2 x 2 y 1 1CCCCCCA .(B–30) Whenwehavethat,for6 ( k l ) k =0 mod ( n )and l =1thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 001100010 1A e = 0@ 010100001 1A .(B–31) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: x n y = 1 p 2 0@ 2 x 1 y 3 x 2 y 2 x 1 y 1 x 2 y 3 x 1 y 2 2 x 2 y 1 1A 1 p 2 0@ 2 x 1 y 3 + x 2 y 2 x 1 y 1 + x 2 y 3 x 1 y 2 + 2 x 2 y 1 1A (B–32) x n y : 2 3 n 2 4 = 2 1 2 2 x n y = x 1 y 2 x 2 y 1 x 1 y 1 x 2 y 2 .(B–33) x n y : 2 3 n 3 1 = 3 1 3 2 Theproductof2 3 and3 1 isingeneralproducesasix-dimensionalrepresentation: 2 3 n 3 1 =6 ( n 3 1, n 3 +1) 0BBBBBB@ x 1 y 1 x 1 y 2 2 x 1 y 3 x 2 y 3 x 2 y 2 2 x 2 y 1 1CCCCCCA .(B–34) Whenwehavethat,for6 ( k l ) k =0 mod ( n )and l =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 001100010 1A e = 0@ 010100001 1A .(B–35) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 102

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x n y = 1 p 2 0@ 2 x 1 y 3 + x 2 y 2 x 1 y 1 + x 2 y 3 x 1 y 2 + 2 x 2 y 1 1A 1 p 2 0@ 2 x 1 y 3 x 2 y 2 x 1 y 1 x 2 y 3 x 1 y 2 2 x 2 y 1 1A (B–36) x n y : 2 3 n 3 1 = 3 1 3 2 Theproductof2 3 and 3 1 isingeneralproducesasix-dimensionalrepresentation: 2 3 n 3 1 =6 ( n 3 2, n 3 +2) 0BBBBBB@ x 1 y 1 x 1 y 2 2 x 1 y 3 x 2 y 3 x 2 y 2 2 x 2 y 1 1CCCCCCA .(B–37) Whenwehavethat,for6 ( k l ) l =0 mod ( n )and k =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–38) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: x n y = 1 p 2 0@ x 1 y 2 + x 2 y 3 2 x 1 y 3 + 2 x 2 y 1 x 1 y 1 + x 2 y 2 1A 1 p 2 0@ x 1 y 2 x 2 y 3 2 x 1 y 3 2 x 2 y 1 x 1 y 1 x 2 y 2 1A (B–39) x n y : 2 3 n 3 2 = 3 1 3 2 Theproductof2 3 and3 2 isingeneralproducesasix-dimensionalrepresentation: 2 3 n 3 2 =6 ( n 3 1, n 3 +1) 0BBBBBB@ x 1 y 1 x 1 y 2 2 x 1 y 3 x 2 y 3 x 2 y 2 2 x 2 y 1 1CCCCCCA .(B–40) Whenwehavethat,for6 ( k l ) k =0 mod ( n )and l =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 001100010 1A e = 0@ 010100001 1A .(B–41) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 103

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x n y = 1 p 2 0@ 2 x 1 y 3 x 2 y 2 x 1 y 1 x 2 y 3 x 1 y 2 2 x 2 y 1 1A 1 p 2 0@ 2 x 1 y 3 + x 2 y 2 x 1 y 1 + x 2 y 3 x 1 y 2 + 2 x 2 y 1 1A (B–42) x n y : 2 3 n 3 2 = 3 1 3 2 Theproductof2 3 and 3 2 isingeneralproducesasix-dimensionalrepresentation: 2 3 n 3 2 =6 ( n 3 2, n 3 +2) 0BBBBBB@ x 1 y 1 x 1 y 2 2 x 1 y 3 x 2 y 3 x 2 y 2 2 x 2 y 1 1CCCCCCA .(B–43) Whenwehavethat,for6 ( k l ) l =0 mod ( n )and k =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–44) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: x n y = 1 p 2 0@ x 1 y 2 x 2 y 3 2 x 1 y 3 2 x 2 y 1 x 1 y 1 x 2 y 2 1A 1 p 2 0@ x 1 y 2 + x 2 y 3 2 x 1 y 3 + 2 x 2 y 1 x 1 y 1 + x 2 y 2 1A (B–45) x n y : 2 4 n 3 1 = 3 1 3 2 Theproductof2 4 and3 1 isingeneralproducesasix-dimensionalrepresentation: 2 4 n 3 1 =6 ( n 3 1, n 3 +1) 0BBBBBB@ x 1 y 1 x 1 y 2 x 1 y 3 x 2 y 3 x 2 y 2 x 2 y 1 1CCCCCCA .(B–46) Whenwehavethat,for6 ( k l ) k =0 mod ( n )and l =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 001100010 1A e = 0@ 010100001 1A .(B–47) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 104

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x n y = 1 p 2 0@ x 1 y 3 + x 2 y 2 x 1 y 1 + x 2 y 3 x 1 y 2 + x 2 y 1 1A 1 p 2 0@ x 1 y 3 x 2 y 2 x 1 y 1 x 2 y 3 x 1 y 2 x 2 y 1 1A .(B–48) x n y : 2 4 n 3 1 = 3 1 3 2 Theproductof2 4 and 3 1 isingeneralproducesasix-dimensionalrepresentation: 2 4 n 3 1 =6 ( n 3 2, n 3 +2) 0BBBBBB@ x 1 y 1 x 1 y 2 x 1 y 3 x 2 y 3 x 2 y 2 x 2 y 1 1CCCCCCA .(B–49) Whenwehavethat,for6 ( k l ) l =0 mod ( n )and k =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–50) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: x n y = 1 p 2 0@ x 1 y 2 + x 2 y 3 x 1 y 3 + x 2 y 1 x 1 y 1 + x 2 y 2 1A 1 p 2 0@ x 1 y 2 x 2 y 3 x 1 y 3 x 2 y 1 x 1 y 1 x 2 y 2 1A .(B–51) x n y : 2 4 n 3 2 = 3 1 3 2 Theproductof2 4 and3 2 isingeneralproducesasix-dimensionalrepresentation: 2 4 n 3 2 =6 ( n 3 1, n 3 +1) 0BBBBBB@ x 1 y 1 x 1 y 2 x 1 y 3 x 2 y 3 x 2 y 2 x 2 y 1 1CCCCCCA .(B–52) Whenwehavethat,for6 ( k l ) k =0 mod ( n )and l =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 001100010 1A e = 0@ 010100001 1A .(B–53) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 105

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x n y = 1 p 2 0@ x 1 y 3 x 2 y 2 x 1 y 1 x 2 y 3 x 1 y 2 x 2 y 1 1A 1 p 2 0@ x 1 y 3 + x 2 y 2 x 1 y 1 + x 2 y 3 x 1 y 2 + x 2 y 1 1A (B–54) x n y : 2 4 n 3 2 = 3 1 3 2 Theproductof2 4 and 3 2 isingeneralproducesasix-dimensionalrepresentation: 2 4 n 3 2 =6 ( n 3 2, n 3 +2) 0BBBBBB@ x 1 y 1 x 1 y 2 x 1 y 3 x 2 y 3 x 2 y 2 x 2 y 1 1CCCCCCA .(B–55) Whenwehavethat,for6 ( k l ) l =0 mod ( n )and k =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–56) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: x n y = 1 p 2 0@ x 1 y 2 x 2 y 3 x 1 y 3 x 2 y 1 x 1 y 1 x 2 y 2 1A 1 p 2 0@ x 1 y 2 + x 2 y 3 x 1 y 3 + x 2 y 1 x 1 y 1 + x 2 y 2 1A (B–57) x n y : 3 1 n 3 1 =( 3 1 3 1 ) S ( 3 2 ) A Theproductisdenedtoresultin: 3 1 n 3 1 = 3 1 6 (2,0) ,(B–58) where 3 1 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A ,6 (2,0) 0BBBBBB@ x 1 y 2 x 2 y 3 x 3 y 1 x 3 y 2 x 2 y 1 x 1 y 3 1CCCCCCA ,(B–59) Whenwehavethat,for6 ( k l ) l =0 mod ( n )and k =2thenthematrixthat diagonalizesallelementsofthegroupis 106

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S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–60) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 6 (2,0) 7! 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A .(B–61) Thustheendresultisthat x n y = 24 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 35 S 24 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A 35 A .(B–62) x n y : 3 1 n 3 1 = 1 2 1 2 2 2 3 2 4 Theproductisdenedtoresultin: 3 1 n 3 1 = 3 1(0) 6 (2,2) ,(B–63) where 3 1(0) 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A ,6 (2,2) 0BBBBBB@ x 1 y 2 x 2 y 3 x 3 y 1 x 3 y 2 x 2 y 1 x 1 y 3 1CCCCCCA .(B–64) Therstrepresentationabovecanbediagonalizedinusingt hefollowingmatrix s = 0@ 1111 2 !! 2 1 1A .(B–65) Thisresultsinthefollowingdecomposition 3 1(0) 7! 1 2 1 1 p 3 ( x 1 y 1 + x 2 y 2 + x 3 y 3 ) 1 p 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 .(B–66) For6 (2,2) thematrixthatdiagonalizesallelementsofthisrepresent ationis 107

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S = 0BBBBBB@ 1 2 000 0001 2 000 2 1 ! 2 1 000 000111111000 1CCCCCCA .(B–67) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations:6 (2,2) 7! 2 2 2 3 2 4 1 p 3 x 1 y 2 + 2 x 2 y 3 + x 3 y 1 x 3 y 2 + 2 x 2 y 1 + x 1 y 3 1 p 3 x 2 y 1 + 2 x 3 y 2 + x 1 y 3 x 2 y 3 + 2 x 1 y 2 + x 3 y 1 1 p 3 x 3 y 2 + x 2 y 1 + x 1 y 3 x 2 y 3 + x 1 y 2 + x 3 y 1 .(B–68) Thustheendresultisthat x n y = 1 p 3 ( x 1 y 1 + x 2 y 2 + x 3 y 3 ) 1 p 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 (B–69) 1 p 3 x 1 y 2 + 2 x 2 y 3 + x 3 y 1 x 3 y 2 + 2 x 2 y 1 + x 1 y 3 1 p 3 x 2 y 1 + 2 x 3 y 2 + x 1 y 3 x 2 y 3 + 2 x 1 y 2 + x 3 y 1 1 p 3 x 3 y 2 + x 2 y 1 + x 1 y 3 x 2 y 3 + x 1 y 2 + x 3 y 1 x n y : 3 1 n 3 2 = 3 2 3 2 3 1 Theproductisdenedtoresultin: 3 1 n 3 2 = 3 2 6 (2,0) ,(B–70) where 3 2 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A ,6 (2,0) 0BBBBBB@ x 1 y 2 x 2 y 3 x 3 y 1 x 3 y 2 x 2 y 1 x 1 y 3 1CCCCCCA ,(B–71) Whenwehavethat,for6 ( k l ) l =0 mod ( n )and k =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–72) 108

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Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 6 (2,0) 7! 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A .(B–73) Thustheendresultisthat x n y = 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A .(B–74) x n y : 3 1 n 3 2 = 1 1 2 1 2 2 2 3 2 4 Theproductisdenedtoresultin: 3 1 n 3 2 = 3 2(0) 6 (2,2) ,(B–75) where 3 2(0) 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A ,6 (2,2) 0BBBBBB@ x 1 y 2 x 2 y 3 x 3 y 1 x 3 y 2 x 2 y 1 x 1 y 3 1CCCCCCA .(B–76) Therstrepresentationabovecanbediagonalizedinusingt hefollowingmatrix s = 0@ 1111 2 ! 2 1 1A .(B–77) Thisresultsinthefollowingdecomposition 3 2(0) 7! 1 2 1 1 p 3 ( x 1 y 1 + x 2 y 2 + x 3 y 3 ) 1 p 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 x 1 y 1 2 x 2 y 2 x 3 y 3 .(B–78) For6 (2,2) thematrixthatdiagonalizesallelementsofthisrepresent ationis S = 0BBBBBB@ 1 2 000 0001 2 000 2 1 ! 2 1 000 000111111000 1CCCCCCA .(B–79) 109

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Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 6 (2,2) 7! 2 2 2 3 2 4 1 p 3 x 1 y 2 + 2 x 2 y 3 + x 3 y 1 x 3 y 2 2 x 2 y 1 x 1 y 3 (B–80) 1 p 3 x 2 y 1 2 x 3 y 2 x 1 y 3 x 2 y 3 + 2 x 1 y 2 + x 3 y 1 1 p 3 x 3 y 2 x 2 y 1 x 1 y 3 x 2 y 3 + x 1 y 2 + x 3 y 1 Thustheendresultisthat x n y = 1 p 3 ( x 1 y 1 + x 2 y 2 + x 3 y 3 ) 1 p 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 x 1 y 1 2 x 2 y 2 x 3 y 3 (B–81) 1 p 3 x 1 y 2 + 2 x 2 y 3 + x 3 y 1 x 3 y 2 2 x 2 y 1 x 1 y 3 1 p 3 x 2 y 1 2 x 3 y 2 x 1 y 3 x 2 y 3 + 2 x 1 y 2 + x 3 y 1 1 p 3 x 3 y 2 x 2 y 1 x 1 y 3 x 2 y 3 + x 1 y 2 + x 3 y 1 x n y : 3 1 n 3 1 =( 3 1 3 1 ) S ( 3 2 ) A Theproductisdenedtoresultin: 3 1 n 3 1 =3 1 6 (1,0) ,(B–82) where 3 1 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A ,6 (1,0) 0BBBBBB@ x 1 y 2 x 2 y 3 x 3 y 1 x 3 y 2 x 2 y 1 x 1 y 3 1CCCCCCA ,(B–83) Whenwehavethat,for6 ( k l ) l =0 mod ( n )and k =1thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–84) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 110

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6 (1,0) 7! 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A .(B–85) Thustheendresultisthat x n y = 24 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 35 S 24 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A 35 A .(B–86) x n y : 3 1 n 3 2 = 1 1 2 1 2 2 2 3 2 4 Theproductisdenedtoresultin: 3 1 n 3 2 = 3 2(0) 6 (1,1) ,(B–87) where 3 2(0) 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A ,6 (1,1) 0BBBBBB@ x 1 y 2 x 2 y 3 x 3 y 1 x 3 y 2 x 2 y 1 x 1 y 3 1CCCCCCA .(B–88) Therstrepresentationabovecanbediagonalizedinusingt hefollowingmatrix s = 0@ 1111 2 ! 2 1 1A .(B–89) Thisresultsinthefollowingdecomposition 3 2(0) 7! 1 2 1 1 p 3 ( x 1 y 1 + x 2 y 2 + x 3 y 3 ) 1 p 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 x 1 y 1 2 x 2 y 2 x 3 y 3 .(B–90) For6 (1,1) thematrixthatdiagonalizesallelementsofthisrepresent ationis S = 0BBBBBB@ 1 2 000 0001 2 000 2 1 ! 2 1 000 000111111000 1CCCCCCA .(B–91) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 111

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6 (1,1) 7! 2 2 2 3 2 4 1 p 3 x 1 y 2 + 2 x 2 y 3 + x 3 y 1 x 3 y 2 2 x 2 y 1 x 1 y 3 (B–92) 1 p 3 x 2 y 1 2 x 3 y 2 x 1 y 3 x 2 y 3 + 2 x 1 y 2 + x 3 y 1 1 p 3 x 3 y 2 x 2 y 1 x 1 y 3 x 2 y 3 + x 1 y 2 + x 3 y 1 Thustheendresultisthat x n y = 1 p 3 ( x 1 y 1 + x 2 y 2 + x 3 y 3 ) 1 p 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 x 1 y 1 2 x 2 y 2 x 3 y 3 (B–93) 1 p 3 x 1 y 2 + 2 x 2 y 3 + x 3 y 1 x 3 y 2 2 x 2 y 1 x 1 y 3 1 p 3 x 2 y 1 2 x 3 y 2 x 1 y 3 x 2 y 3 + 2 x 1 y 2 + x 3 y 1 1 p 3 x 3 y 2 x 2 y 1 x 1 y 3 x 2 y 3 + x 1 y 2 + x 3 y 1 x n y : 3 1 n 3 2 = 3 2 3 2 3 1 Theproductisdenedtoresultin: 3 1 n 3 2 =3 2 6 (2,0) ,(B–94) where 3 2 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A ,6 (2,0) 0BBBBBB@ x 1 y 2 x 2 y 3 x 3 y 1 x 3 y 2 x 2 y 1 x 1 y 3 1CCCCCCA ,(B–95) Whenwehavethat,for6 ( k l ) l =0 mod ( n )and k =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–96) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 6 (2,0) 7! 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A .(B–97) 112

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Thustheendresultisthat x n y = 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A .(B–98) x n y : 3 2 n 3 2 =( 3 1 3 1 ) S ( 3 2 ) A Theproductisdenedtoresultin: 3 2 n 3 2 = 3 1 6 (2,0) ,(B–99) where 3 1 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A ,6 (2,0) 0BBBBBB@ x 1 y 2 x 2 y 3 x 3 y 1 x 3 y 2 x 2 y 1 x 1 y 3 1CCCCCCA ,(B–100) Whenwehavethat,for6 ( k l ) l =0 mod ( n )and k =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–101) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 6 (2,0) 7! 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A .(B–102) Thustheendresultisthat x n y = 24 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 35 S 24 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A 35 A .(B–103) x n y : 3 2 n 3 2 = 1 2 1 2 2 2 3 2 4 Theproductisdenedtoresultin: 3 2 n 3 2 = 3 1(0) 6 (2,2) ,(B–104) where 113

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3 1(0) 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A ,6 (2,2) 0BBBBBB@ x 1 y 2 x 2 y 3 x 3 y 1 x 3 y 2 x 2 y 1 x 1 y 3 1CCCCCCA .(B–105) Therstrepresentationabovecanbediagonalizedinusingt hefollowingmatrix s = 0@ 1111 2 !! 2 1 1A .(B–106) Thisresultsinthefollowingdecomposition 3 1(0) 7! 1 2 1 1 p 3 ( x 1 y 1 + x 2 y 2 + x 3 y 3 ) 1 p 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 .(B–107) For6 (2,2) thematrixthatdiagonalizesallelementsofthisrepresent ationis S = 0BBBBBB@ 1 2 000 0001 2 000 2 1 ! 2 1 000 000111111000 1CCCCCCA .(B–108) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations:6 (2,2) 7! 2 2 2 3 2 4 1 p 3 x 1 y 2 + 2 x 2 y 3 + x 3 y 1 x 3 y 2 + 2 x 2 y 1 + x 1 y 3 1 p 3 x 2 y 1 + 2 x 3 y 2 + x 1 y 3 x 2 y 3 + 2 x 1 y 2 + x 3 y 1 1 p 3 x 3 y 2 + x 2 y 1 + x 1 y 3 x 2 y 3 + x 1 y 2 + x 3 y 1 .(B–109) Thustheendresultisthat x n y = 1 p 3 ( x 1 y 1 + x 2 y 2 + x 3 y 3 ) 1 p 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 (B–110) 1 p 3 x 1 y 2 + 2 x 2 y 3 + x 3 y 1 x 3 y 2 + 2 x 2 y 1 + x 1 y 3 1 p 3 x 2 y 1 + 2 x 3 y 2 + x 1 y 3 x 2 y 3 + 2 x 1 y 2 + x 3 y 1 1 p 3 x 3 y 2 + x 2 y 1 + x 1 y 3 x 2 y 3 + x 1 y 2 + x 3 y 1 114

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x n y : 3 2 n 3 2 =( 3 1 3 1 ) S ( 3 2 ) A Theproductisdenedtoresultin: 3 2 n 3 2 =3 1 6 (1,0) ,(B–111) where 3 1 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A ,6 (1,0) 0BBBBBB@ x 1 y 2 x 2 y 3 x 3 y 1 x 3 y 2 x 2 y 1 x 1 y 3 1CCCCCCA ,(B–112) Whenwehavethat,for6 ( k l ) l =0 mod ( n )and k =2thenthematrixthat diagonalizesallelementsofthegroupis S = aea e a = 0@ 010001100 1A e = 0@ 100001010 1A .(B–113) Theendresultisofapplyingthisdiagonalizingmatrixunto thevectorspaceistwo three-dimensionalrepresentations: 6 (1,0) 7! 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A .(B–114) Thustheendresultisthat x n y = 24 0@ x 1 y 1 x 2 y 2 x 3 y 3 1A 1 p 2 0@ x 2 y 3 + x 3 y 2 x 3 y 1 + x 1 y 3 x 1 y 2 + x 2 y 1 1A 35 S 24 1 p 2 0@ x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 1A 35 A .(B–115) 115

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BIOGRAPHICALSKETCH JesusAlejandroEscobarwasborninLaCiudaddeGuatemalain thecountryof Guatemala.Theoldestchildinhisfamilyofthreechildren. Thefamilymovedtothe UnitedStateswithhimandhisyoungerbrotherwhenhewasaro undtwoyearsold. TheymovedtothecityofBoston,Massachusettsandlivedthe reforapproximately elevenyears.ThenaltwoyearsofhistimeinBostonwereato neoftheleading schoolscalledBostonLatinAcademy.Atabouttheeighthgra dehisfamilymovedto Miami,Floridawheretheycurrentlyreside. Forcollege,asaconsequenceofthemoveandthankstoBright Futures,he attendedtheUniversityofFlorida(UF).Herehegothisbach elor'sdegreeinaerospace engineeringwithaminorinphysicsin2005.Hewasadmittedt otheDepartmentof PhysicsatUFthatsameyearwhichledtohisMasterofScience degreein2007. Hemarriedin2010toawonderfulwomannamedYasminwhohehad courtedfor approximately2years.Hisplansaftercompletinghisgradu ation,Summer2011withhis Ph.D.inPhysics,istonallymoveinwithYasmininthestate ofWashingtonandndan academicjobtocontinuehisPhysicscareer. 119