New phenomenological tests for supersymmetric theories with SU(5)-like unification

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New phenomenological tests for supersymmetric

theories with SU(5)-like unification

Yannick Stoll

To cite this version:

Yannick Stoll. New phenomenological tests for supersymmetric theories with SU(5)-like unifica-tion. High Energy Physics - Phenomenology [hep-ph]. Grenoble Alpes, 2015. English. �NNT : 2015GREAY053�. �tel-01292577�

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THÈSE

Pourobtenirlegradede

DOCTEURDEL’UNIVERSITÉDEGRENOBLEALPES

Spécialité:Physiquethéorique

Arrêtéministériel:7août2006

Présentéepar

Yann

ickSTOLL

ThèsedirigéeparFawziBOUDJEMA etcodirigéeparBjörnHERRMANN

préparée au sein du Laboratoire d’Annecy-Le-Vieux de Physique Théorique

etdel’EcoleDoctoraledePhysiquedeGrenoble

Nouveauxtests phénoméno

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lesthéor

iessupersymétr

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iondetypeSU(5)

Thèsesoutenuepubliquementle25Septembre2015, devantlejurycomposéde:

Jean-LoïcKNEUR

Directeurderecherche,L2C-Montpellier,Président

FarvahNazilaMAHMOUDI

Maîtredeconférences,UniversitédeLyon,Rapporteur

SabineKRAML

Directricederecherche,LPSC-Grenoble,Examinatrice

StéphaneLAVIGNAC

Chargéderecherche,CEA-Saclay,Examinateur

FawziBOUDJEMA

Directeurderecherche,LAPTh,Directeurdethèse

BjörnHERRMANN

Maîtredeconférences,UniversitéSavoieMontBlanc-LAPTh,Co-Directeurde

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Résumé

Danscettethèse,nousexploronslesconséquencesauxbassesénergiesdes modèlessupersymétriques degrandeuniﬁcation(SUSY-GUT)baséssurlegroupedeLieSU(5). Ces modèlesconstituentles prototypesdes modèlesSUSY-GUTet,entantquetels,ontreçubeaucoupd’attentionaucoursdes dernièresdécennies.Enparticulier,l’uniﬁcationquark-leptondanslesecteurdeYukawaaétél’objet denombreusesétudes.

L’originalitédenotreapprocheestderesterconfinéausecteurdessquarkshauts,oùnousavons montréqu’unerelationdesymétrieinduitepar SU(5)apparait,ànotreconnaissancejusqu’àprésent nonétudiée,etbienmoinsdépendantedelaréalisationparticulièredumodèlequel’unificat ionquark-lepton.Eneffet,nousavonsdémontréqu’un modèlesupersymétriqueéquipéd’unesymétriedejauge SU(5)impliquede manièregénéraleuncouplagetrilinéairedanslesecteurdessquarkshautsAU

symétriquei.e.AU = ATUàl’échelle GUT. Cetterelationrestebienpréservéeparleséquationsdu

groupederenormalisationdurantsonévolutionjusqu’àl’échelleduTeV,laissantainsiuneempreinte delasymétriedegrandeuniﬁcationdanslespectresupersymétriqueauxbassesénergies.Celaétant établie,nouspouvonsconstruireunesériedetestsphénoménologiquessurlesspectresSUSYdebasses énergies,potentiellementaccessiblesauLHC.Laplupartdecestestsconsistentàdéterminerune certainerelationparmilesobservablesdebassesénergies.

NousavonsréussiàexhiberdetellesrelationsdanstroissortesdespectreSUSY,àsavoirles spectres"SUSYlourds"danslesquelsl’échelledebrisuredelaSUSYestbienplusélevéequel’échelle du TeV,lesspectresde"SUSYnaturels"oùlatroisièmegénérationdescalaireestlégèreetenfin lesspectresde"SUSYTop-Charm"danslesquelsseulelapremièregénérationdescalaireestgardée lourde. Nousavonsdéveloppéune méthodegénérique,baséesurdestestsstatistiquesfréquentistes reposantsurlecalculd’unevaleurp,pourquantifieravecquellesignificativitédesécartsàcesrelations decontraintepeuventêtredétectés,danschacundestroiscas mentionnésplushaut.Typiquement, pourchacundestests,environO(10−100)événementssontnécessairesdanschacundesprocessus considérés,pourétablirunécartde50%desrelationstestsàunniveaudesignificativitéde3σ.Notons quedestechniquesexpérimentalestellesquel’identificationdesjetscharmésoulapolarimétriedutop, jouentunrôleessentieldansla miseenœuvredecestestsauxcollisionneursactuels,telqueleLHC. Lasecondepartiedecettethèseestconsacréeàuneanalysenumériqueplusglobale,baséesurles techniquesbayésiennesdecomparaisonde modèlesetapplicableàtoutessortesdespectresquelque soitleurshiérarchies. Danscecontexte,nousavonsdéveloppéune méthodenumériquebaséesurun algorithmedesimulationde MonteCarloparChainede Markov.Cettealgorithmepermetdetester, étantdonnéuncertainspectre,avecquellesignificativitécelui-cipointeounonversunesymétrie SU(5)àhauteénergie.Ànouveau,nousavonsconsidéréplusieurscas,selonlaquantitéd’information quipourraitêtrecollectéauLHCsurlespectredessquarkshauts.Noustrouvonsquetypiquement,ces informationsnecontraignentpassuffisammentlespectrepourpouvoirtireruneconclusionquantàla présenced’unesymétrieSU(5)àhauteénergie,exceptionfaitedecertainscas,oùlesincertitudesdes observablessontréduitesàdesvaleursdel’ordredeO(1%)et/oudesspectresloind’êtresymétriques sousSU(5)sontconsidérés.Danscettedernièresituation,cetteanalysesembledéfavoriserlaprésence d’unesymétrieSU(5)àl’échelleGUTavecuneévidencefaible.

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Abstract

Inthisthesis,weexploretheconsequencesatlowenergiesofsupersymmetricgranduniﬁedtheories (SUSY-GUT)basedontheLiegroupSU(5).Thesemodelsconstitutetheprototypeofminima lSUSY-GUTandassuch,havebeenintensivelystudiedinthelastdecades.Inparticular,thequark-lepton uniﬁcationintheYukawasectorhasreceivedalotofattention.

Theoriginalityofourapproachistostaywithintheup-squarksector,wherewehaveshownthat asofarunnoticedSU(5)inducedsymmetryrelationemerges,wayless modeldependentthantheone aforementioned.Indeed,wedemonstratethatasupersymmetric model,equippedwithaSU(5)gauge symmetrygenericallyimpliesthattheup-squarktrilineartermAU issymmetricatthe GUTscale,

namelythatAU= ATU. Thisrelationisonly mildlyspoiledbytherenormalizationgrouprunning,

andremainswellpreservedattheTeVscale,leavingthusaSU(5)footprintinthelowscaleSUSY spectrum. ThisallowstobuildaseriesofphenomenologicalSU(5)testsonlowscaleSUSYspectra, potentiallyaccessibleattheLHC. Mostofthesetestsgenericallyconsistindeterminingacertain relationamonglowscaleobservables.

WeexhibitsuchrelationsforthreedifferentsortofSUSYspectra,namely"HeavySUSY"spectra whichfeatureaSUSYbreakingscalewellabovetheTeVscale,"NaturalSUSY"spectrawhichfeature alightthirdscalargenerationand"TopCharmSUSY"spectrainwhichonlythefirstscalargeneration iskeptheavy. Wehavedevelopedageneric method,basedonap-valuefrequentiststatisticaltestto quantifytowhichsignificanceonecanassessadeparturefromtheselowscaleconstraintsrelations,in thethreeabove-mentionedtypesofSUSYspectra. Typically,foreachofthetests,O(10−100)events arenecessaryintheprocessesconsidered,toassessadepartureof50%at3σlevelofsignificance.It shouldbe mentionedthatexperimentaltechniquessuchascharmtaggingortop-polarimetryplaya crucialroletoapplythesetestsatcurrentcolliders,suchastheLHC.

Thesecondpartofthisthesisisdevotedtoa moreglobalnumericalanalysis,basedonBayesian modelcomparison methods,andapplicabletoanykindofspectrumwhateveritshierarchy.Inthis context, wehavedevelopedanumerical methodbasedona Markov Chain Monte Carloalgorithm allowingtotest,givenanobservedlowenergyspectrum, with whichsigniﬁcanceitpointsornot towardahighscaleSU(5)dynamic. Again,wehaveconsideredseveralcasesdependingontheamount ofinformationthatwillbecollectedonthesquarkspectrumattheLHC. Weﬁndthattypically,the constrainingpowerofthese would-becollectedinformationistoolowtodrawanyconclusionona possibleSU(5)−likehighscaledynamic,exceptincertaincases,whentheuncertaintiesattachedto theobservablesconsideredarereducedtoO(1%)and/orwhenspectrafarfrombeingSU(5)-symmetric areconsidered.Inthislastcase,thisanalysisseemstodisfavorthepresenceofa GUTscaleSU(5) symmetrywithweakevidence.

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2.3 Superspacesandsuperﬁelds... 33 2.3.1 Grassmanncoordinates ... 33 2.3.2 Superspacesandsuperﬁelds ... 34 2.3.3 SUSYlagrangiansinasuperspace ... 37 2.4 BreakingSupersymmetry. ... 40 2.4.1 BreakingSUSYspontaneously. ... 40 2.4.2 SoftSUSYbreaking. ... 41 2.5 The MSSM... 42 2.5.1 R-parity ... 43 2.5.2 MSSMsoftparameters ... 44 2.5.3 Aglanceatthe MSSM massspectrum... 46

3 TheSUSY-SU(5)GUT modelorthearttostickto minimality 49 3.1 Introduction ... 49

3.2 IntroductiontoSU(5)andﬁeldcontent ... 51

3.2.1 TheGlashow-Georgi model:... 51

3.2.2 TheYukawasector ... 52

3.2.3 SpontaneousSU(5)symmetrybreaking... 53

3.2.4 TheSUSYSU(5) model ... 55

3.3 Physicalconsequences ... 56

3.3.1 Chargequantization ... 57

3.3.2 Protondecay ... 58 3

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II NewtestsforSU(5)-likeup-squarks 61

4 TastingtheSU(5)natureofsupersymmetryattheLHC 63

4.1 Introduction ... 63

4.2 Theup-typesquarksectorinSU(5)theories ... 65

4.2.1 Theup-typesquark massspectrum ... 65

4.2.2 ThesoftSUSYbreakingsectorofSU(5)theories:theupsectorcase67 4.2.3 QualitativediscussionoftheRGEsintheup-squarksector... 67

4.3 StabilityofthesoftsectorundertheRGﬂow: quantitativediscussion ... 69

4.3.1 Setup ... 69

4.3.2 Flavorconstraints ... 70

4.3.3 Results ... 74

4.3.4 Stabilityofthesoftsector:theM2 Q=MU2case ... 76

4.4 StrategiesandtoolsfortestingtheSU(5)hypothesisattheTeVscale .. 77

4.4.1 Eﬀectiveﬁeldstheories... 78

4.4.2 Mass-insertionapproximation ... 82

4.4.3 Statisticaltools ... 84

4.5 CaseI:HeavySUSY ... 86

4.5.1 SU(5)testthroughsingletoppolarimetry. ... 88

4.5.2 ExistingLHCsearches. ... 89

4.6 CaseII:NaturalSUSY ... 90

4.6.1 Them˜t1,2> mW˜> mB˜case ... 91

4.6.2 Them_W˜> m_t˜_1,2> m_B˜case ... 94

4.7 CaseIII:Top-charmSUSY ... 96

4.7.1 EﬀectiveLagrangian ... 97

4.7.2 SU(5)testthroughHiggsproduction... 98

4.8 Conclusion ...102

5 A Bayesiananalysis 105 5.1 TheBayesianContext ...105

5.1.1 Limitationsofthefrequentistapproach ...106

5.1.2 TheBayestheorem,priorsandposteriors,orthedoomofthe"false idolofobjectivity"...107

5.1.3 Bayesfactorsand modelcomparison ...108

5.2 Numericalstrategy...110

5.2.1 Deﬁnitionofthe models ...111

5.2.2 A MCMCalgorithm ...111

5.2.3 Tools ...116 5.3 Results ...117 5.3.1 Referencescenarios ...117 5.3.2 CaseI ...118 5.3.3 CaseII ...121 5.3.4 CaseIII ...122 Appendices 133 Appendix A Conventions 133 A.1 PauliandDirac matrices. ...133

A.2 Weylspinors. ...133

A.3 Diracand Majoranaspinors ...134

Appendix B Thedipoleformfactors 135

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Remerc

iements

Entoutpremierlieu,jetiensàchaleureusementremercier mondirecteurdethèse,BjörnHerrmann pourm’avoirsoutenulorsdelapréparationdecettethèse.Cesconseilsquotidiensontétéinestimables à mesyeuxet m’ontpermisdebeaucoupprogresserlorsdecestroisansderecherche.

JetiensaussiàremercierFawziBoudjema,pour m’avoiraccueilliauseindesonlaboratoireetavoir servidedirecteurdethèseoﬃciel.

JevoudraisremercierFarvahNazila Mahmoudiet WernerPorod,touslesdeuxchercheursdestature internationaleàl’emploidutempsextrêmementchargéetqui malgrétout,ontacceptédeprendrele rôlederapporteurde mathèse.Leursremarquesontgrandementcontribuéàaméliorerlaqualitéde ce manuscrit. Merciàeux.

JesouhaiteraisaussiremercierDiegoGuadagnoliquiaacceptédeparrainer mathèseet m’asoutenu toutaulongde monséjourauLAPTh.

Biensûr,jeproﬁtedecettetribunepouraﬃrmeravecforcequenotrelaboratoirefermeraitsansdoute sesportesenunesemainesansnotreéquipeadministrativedechoc.Celle-ciestemportéeparDominique Turcentantqueresponsableadministrative,VéroniqueJonneryàlacomptabilitéetVirginie Malaval ausecrétariat.Jen’aipaslaplaceicipourcommenceràdécrireàquelpointleursdévouements,leurs professionnalismes,sontessentielsaubonfonctionnementdulaboratoire.Je mecontenteraidoncde leurdireungrand merci.

Jevoudraisaussiremercier MathieuGauthier-Lafaye,lemagiciendulaboratoirecommejel’ainommé, quiconstituelaclédevouteduserviceinformatiqueduLAPTh.Sadisponibilité,sonsérieux,ses conseilsetsa maitrisedu mondedulogiciellibre m’ontétégrandementutileunnombreincalculable defois.

JevoudraisaussiremercierAndreasGoudelis,aujourd’huipost-docàVienne,quiaétéunvéritableami pour moi,quasimentungrandfrère,lorsde mesdeuxpremièresannéesdethèse.C’estunepersonne quej’apprécieénormémentetaveclaquellej’espèresincèrementmainteniruncontactau-delàdumonde delarecherche.Enparticulier, merciàluide m’avoirhébergélorsqueje mesuisretrouvéquasi-SDF unesemaineavantdepartirenséjouràHambourg.J’aiaussiunepenséeparticulièrepoursachèreet tendre,AoifeBharucha,CRà Marseille.Jenepeuxqueleursouhaitertoutlebonheurdumondepour lasuite.

Jetiensaussiàremerciertouslesdoctorantsaveclesquelsj’aipupasserdebons momentstelque Mathieu Vanicat, MathieuBoudaud, VincentBizouard, VivianPoulin, Yoann Genolini...Enpart ic-ulier,jesouhaiteboncourageàceuxquirestent,laroutevaêtrelonguelesgars, maisjesaisquevous yarriverai.

Jesouhaiteraisaussiremercier ma mère,NadiaStoll,qui m’asoutenude manièreindéfectibledepuis toujourset m’apermitde mener mathèseàbien.Sonamour,sonsoutienquotidienontétécrucial danslabonneréussitede mesétudes,cettethèseluiestdédiée.

Enﬁn,jesouhaiteraisremerciertouslesamischèreà moncœur.Enparticulier,SinanetNicolasqui mènenttouslesdeuxdebrillantescarrièresenmathématiques.Nousnoussommesconstruitsensemble etils m’ontapportéénormément,àtelpointquenospersonnalitésrespectivessontirrémédiablement liées.Euxaussiontjouéunrôledéterminantdanslaréussitede mathèse, merci.Parm ilesautresco-pains,jesouhaiteraisciterpêle-mêle:Jean-Loup,Sylvie,Sébastien,Ariane,Éric,Sabine,Élie,Caroline, Odile,Rozenn,Vincent,Samuel.Cettethèseleurestaussidédiée.

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Introduct

ion

Itisfairtosaythat modernparticlephysicshasstartedwiththeRutherfordexperimentsconducted in1909byHans Geiger,Ernest MarsdenandErnestRutherford. Thesescatteringexperimentshave showntheexistenceoftheatomicnucleus,thepositivelychargedpartoftheatom. Mostimportantly, thisexperimenthasleadin1912totheplanetary model:atomsarecomposedofaverydense,very heavy,positivelychargednucleuswhosesizeiscompletelynegligibleinfrontoftheelectroniccloud. Hence,thishasleadtotheconclusionthat mattersis mostlycomposedofvoid.

Thesecondfoundingexperiment mightbethediscoveringbyCarlDavidAndersonin1932ofthe positron,theantimatterelectroncounterpart,theorizedfouryearsearlierbyPaulDirac.Finally,the thirdexperimentIwouldliketo mentionedisthediscoveryoftheHiggsbosonin2012bytheLHC’s ATLASandCMSexperiments. ThisdiscoveryhasputaﬁnaldottotheStandard Model,thetheory inwhichareinterpreted mostoftheresultsinparticlephysicstoday. However,oneknowsthatthisis nottheendofthestoryandthattheStandard Model(SM)isnotanultimatetheoryofnature,failing forexampletoinclude massiveneutrinosorsuitabledark mattercandidates.

Atthetimetheselinesarewritten(July2015),theLHChasjustrestartedtorecorddataatits nominalenergy. Onecansaythat,currently,thesituationisquiteinquiring,eventhoughnotyet worrying.Indeed,newphysicshavebeenwaitedforawhilenow,andthesimplestextensionsofthe Standard Modelhavebeenput moreand moreunderpressure.

Amongthebest motivatedframeworkstoextendtheStandard Model,thereisSupersymmetry (SUSY),whichpostulatestheexistenceofanewtypeofspace-timesymmetrybetweenbosons,which areintegerspinparticles,andfermions, whicharehalf-integerspinparticles. However ,thenon-observationoflightsuperpartnersclearlyindicatesthatSUSY,ifrealizedinnature, mustbeabroken symmetry.Inanycase,thesimplestSUSYextensionoftheSM,the MinimalSupersymmetricStandard Model(MSSM)isofspecialinterestforphenomenologists,resolvingmostofthequestionsunanswered bytheSM.

Ontheotherhand,thelackofexperimentalsignalsmakesthattheoretically,SUSYremainspoorly understood. Crucially,one missesaclearunderstandingofthe mechanismresponsib leforthesponta-neousbreakingofSUSY.Toparameterizethisignorance,awayaroundhavebeenfoundbyintroducing alot(aboutahundred)of"soft"-inasensthatwillbeprecisedlater-parameters. However,thepriceto paytoapplysuchprocedureisacleardropofpredictabilityofthetheory,unlessabreakingmechanism isimposed,thesofttermsareseenasfreeparameters.

Onecuretothepointraisedabove mightbetoembedtheSM’sLiegaugegroupintoalarger one,forexampleSU(5),SO(10)orE(6). Such Grand Unified Theory(GUT)allowstounifythe threefundamentalforcesoftheSMintoasingleone,throughtheunificationoftheirrespectivegauge couplings. Beside, whena GUTiscombinedwithSUSY,fittingthelowenergygaugecouplingsto their measuredvaluesiseasierandsuppressionofprotondecaycanbeobtainedtoareasonablelevel, phenomenonotherwisegenericallypredictedby GUTtheories. Furthermore,inaSUSYcontext,the softtermsgetunifiedaswell,thusreducingthehundredfreesoftparametersneededinthe MSSMto onlyafewindependenthighscaleterms.

ThepresentthesisisinterestedinthesimplestofsuchSUSY/GUTtheory,theonebasedonthe LiegroupSU(5). ThisSUSY/GUThavebeenthesubjectof multiplestudiesduringthelastdecades, inparticularthequark-leptonuniﬁcationhavereceivedalotofattention.Indeed,thisSUSY/GUT predictsthatYukawacouplingsinthequarksandleptonssectorsunifyatascaleM ∼O(1016_)GeV_,

calledtheGUTscale.Evolvingthesedowntoascaleaccessibletoexperimentswiththerenormalization group’stechniques,onecanthenscrutinizetheirlowscalevaluestoseeifthesearecompatiblewitha

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possiblehighscaleuniﬁcation.Inthesimplest model,theconclusionisquitepessimistic.Indeed,this uniﬁcationseemstoworkonlyforthethirdgenerationandeventhere,extra-carehavetobetaken, introducingforexamplethresholdcorrectionsoriginatingfromthedynamicabovetheGUTscale.

Ontheotherhand,theoriginalityofthisthesisistoinvestigatethelowscaleconsequencesof thistypeofuniﬁcationnotonYukawacouplingsbutonsoftparameters,restrictingourstudytothe up-squarkssector.Indeed,wehaveshownthataso-farunstudiedfeatureofSU(5)SUSY/GUTsis thattheup-squarkstrilinearcouplingissymmetricattheGUTscale. Thissymmetryrelationiswell preservedbytherenormalizationgroupequations,theasymmetryintroducedbytherunningremaining smallattheTeVscale. Hence,iftheGUTscaleissymmetricunderSU(5),weexpectthatafootprint shouldremaininthelowscaleup-squarksspectrum,allowingtobuildaseriesofphenomenological tests.

Thisthesisisorganizedasfollows:

1. Thechapter1isdedicatedtoapresentationoftheStandard Model.Itsgaugestructureaswell asits mattercontentarepresented. Also,theHiggs mechanism,responsiblefortheelectroweak symmetrybreakingaswellasthefermions mixing mechanismsareoutlined. Weconcludethis chapterbygivingsomehintsonwhywebelievethattheSMneedstobeextendedbya more completetheory.

2. Thechapter2givesanintroductiontosupersymmetry,focusingonthe MSSM. TheSUSY algebraandtheelegantformalismofsuperﬁeldsarepresented. Wethengivesomearguments tounderstand whySUSYspontaneousbreakingcannotbeachievedinaSUSYtheory whose breakingscaleliesneartheTeVscale,thus motivatingtheintroductionofasoftsector.Finally, the MSSMisdescribed,itsvirtuesandlimitationsexposed,andthethree mainparadigmsused to maintainthesuperpartnerscontributionstolowscaleobservablesoutlined.

3. Thechapter3presentstheSU(5)GUT,bothinanon-SUSYandinaSUSY-context. Aspecial attentionispaidtotheﬂavorsectorofthis GUT,inparticularoneshowsthatthesymmetry relationswhichholdintheYukawasectorpropagatetothesoftbreakingterms. Weconcludethis chapterbypresentingafewofthephysicalconsequencesthatderivefromthiskindofuniﬁcation. 4. Thechapter4containsthemaincontributionofthisthesis.Inthischapter,wehavebuildaseries ofphenonomenologicaltestsonup-typesquarksspectra,usingfrequentiststatisticalmethods.In particular,threetypesoftypicalSUSYspectraareconsidered,forwhichweprovideconstraint relationsonlowscaleobservablesderivingfromaSU(5)-like GUTsymmetry. Theserelations areonly midlyspoiled,shouldaSU(5)-likeGUTberealizedinnature.

5. Finally,inthechapter5,webuild moreglobalSU(5)tests,usingBayesian modelcomparison methods.Indeed,inthischapter,wecomputeBayesfactorsnumerically,using MarkovChains MonteCarlotechniques,todiscriminatethepossiblepresenceofremnantsofahighscale SU(5) symmetryintolowscalesquarksspectra. Wehencestartthischapterbyprovid inganintroduc-tiontoBayesianstatistics. Thenumericalstrategyisthenexhibitedandtheresultscommented.

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Introduct

ionenfrança

is

LaphysiquedesparticulesmoderneacommencéaveclesexpériencesdeRutherfords,conduitesen1909 par Hans Geiger,Ernest MarsdenetErnest Rutherford. Cesexpériencesdediﬀusionontdémontré l’existencedunoyauatomique,lapartiechargéepositivementdel’atome.Plusimportantpeut-être, cetteexpérienceaconduiten1912,àl’émergencedu modèleplanétaire:lesatomessontconstitués d’unnoyauchargépositivement,trèsdenseettrèslourd,dontlatailleestcomplètementnégligeable devantcelledunuageélectronique,enconséquencedequoila matièreestsurtoutcomposéedevide.

LadeuxièmeexpériencefondatricepourraitêtreladécouverteparCarl DavidAndersonen1932 dupositron,laparticuled’antimatièreassociéeàl’électron,théoriséequatreannéesplustôtparPaul Dirac.Finalement,latroisièmeexpériencequejevoudraisciterestladécouvertedubosondeHiggs en2012parlesexpériencesATLASetCMSduLHC.Cettedécouverteamisunpointﬁnalau Modèle Standard(MS),théoriedanslaquellesontinterprétéslaplupartdesrésultatsexpérimentauxau jour-d’hui. Cependant,noussavonsquel’histoirenes’arrêtepaslà,le modèlestandardn’étantpasune théorieultimedelanature,échouantparexampleàincluredestermesdemassepourlesneutrinosou àincluredesparticulesquipourraientêtredeboncandidatsde matièrenoire.

Au momentoùceslignessontécrites(Juillet2015),leLHCvientjustedereprendresaprise dedonnéeàsonénergienominale. Onpeutdirequepourl’instantlasituationestasseztroublante, bienquepasencoreinquiétante.Eneﬀet,lanouvellephysiquesefaitattendredepuisun moment maintenant,etlesextensionslesplussimplesdu MSsontdeplusenplussouspression.

Parmilescadresbien motivéspourétendrele MSsetrouvelasupersymétrie(SUSY),quipostule l’existenced’unnouveautypedesymétried’espace-tempsentrelesbosons,particulesauspinentier,et lesfermions,particulesauspindemi-entier.Cependant,lanon-observationdesuperpartenaireslégers indiqueclairementquelaSUSY,sielleestréaliséedanslanature,sedoitd’êtreunesymétriebrisée. Danstouslescas,l’extensionlaplussimpledu MScompatibleaveclaSUSY,lemodè lestandardsuper-symétrique minimal(MSSM)continud’êtred’unegrandeaidepourlesphénoménologistes,résolvant laplupartdesquestionslaisséessansréponseparle MS.

D’unautrecôté,lemanquedesignauxexpérimentauxfontquelaSUSYrestethéoriquementencore assezmalcomprise.Essentiellement,unecompréhensionclairedumécanismeresponsabledelabrisure delaSUSYnous manque.Aﬁndeparamétrercetteignorance,unesolutionaététrouvéequiconsiste àintroduirebeaucoup(environunecentaine)deparamètres«doux»1_{. Cependant}_,lepr_ixàpayer

pourappliquercetteprocédureestunenettebaissedelaprédictibilitédelathéorie,à moinsqu’un mécanismedebrisurenesoitimposé,cesparamètresdouxétantvuscommelibres.

Unesolutionaupointsoulevéplushautpeutconsisteràplongerlegroupedejaugedu MSdansun groupeplusgrand,parexampleSU(5),SO(10)ouE(6).Unetellethéoriedegrandeunification(GUT) permetd’unifierlestroisforcesfondamentalesdu MSenuneseule,autraversdel’unificationdeleurs couplagesdejaugerespectifs.Deplus,lorsqu’uneGUTestsupersymétrisée,ilestplusfaciled’ajuster lescouplagesdejaugedebasseénergieàleursvaleurs mesuréesexpérimentalement.Ilestalorsaussi plusfaciled’abaisserletauxdedésintégrationduprotonàunniveauacceptable,phénomèneautrement généralementpréditparlesthéoriesdegrandeunification.Enoutre,dansuncontextesupersymétrique, lestermesdouxdeviennentaussiunifiés,réduisantainsiàtrèshauteénergie,lacentainedeparamètre du MSSMàseulementquelquestermesindépendants.

Cettethèses’intéresseàlaplussimpledesthéoriesSUSY/GUT,cellebaséesurlegroupedejauge SU(5).Cettethéorieaétél’objetdenombreusesétudesdurantlesdernièresdécennies,enparticulier

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l’unificationquark-leptonareçuebeaucoupd’attention.Eneffet,cesthéoriesSUSY/GUTprédisentque lescouplagesdeYukawadesleptonsetdesquarkss’unifient,àuneéchelledel’ordredeM ∼O(1016₎

GeV,appeléel’échelleGUT.Enfaisantévoluercescouplagesàdeséchellesaccessiblesauxexpériences, vialestechniquesdugroupederenormalisation,ilestalorspossibledetesterleurscompatibilitésavec unepossibleunificationàhauteénergieenscrutantleursvaleursàbasseénergie.Danslecasleplus simple,laconclusionestclairementnégative.Eneffet,cetteunificationnesemblefonctionnerquepour latroisièmegénérationet, mêmedanscecas,desprécautionssupplémentairesdoiventêtreprises,par exampleenintroduisantdescorrectionsdeseuilsquipourraientnaitredeladynamiqueaudessusde l’échelleGUT.

D’unautrecôté,l’originalitédecettethèseestd’investirlesconséquencesàbasseénergiedece typed’unificationnonplusdanslesecteurdeYukawa, maisdanslesecteurdoux,enrestantconfiné ausecteurdessquarkshauts.Eneffet,nousavonsmontréqu’unecaractéristiquejusquelànonétudiée deces modèlesestquelecouplagetrilinéairedessquarkshautsestsymétriqueàl’échelleGUT.Cette relationdesymétrieestbienpréservéeparleséquationsdugroupederenormalisation,l’asymétrie introduiteparl’évolutiondescouplagesrestantpetiteàl’échelleduTeV.Sil’échelleGUTestgouvernée parunesymétrieSU(5),ons’attenddoncàcequ’uneempreintedecettesymétriepuissesubsister danslespectredessquarkshautsàl’échelleduTeV,autorisantainsilaconstructiond’unesériede testsphénoménologiques.

Cettethèseestorganiséedela manièresuivante:

1.Lechapitre1estdédiéàlaprésentationdu ModèleStandard.Sastructuredejaugeainsique sonspectrede matièresontprésentés.Deplus,le mécanismedeHiggs,responsabledelabrisure desymétrieélectrofaible,etles mécanismesde mélangedesfermionssontexposés.Cechapitre seconclutsurunediscussiondesprincipauxindicesquinouspousseàpenserquele MSabesoin d’êtreétenduparunethéoriepluscomplète.

2.Lechapitre2constitueuneintroductionàlasupersymétrie,enseconcentrantsurle MSSM. L’algèbresupersymétriqueainsiquel’élégantformalismedessuper-champssontprésentés.Des argumentssontensuitedonnéspermettantdecomprendrepourquoilabrisurespontanéedela SUSYnepeutêtreachevéedanslesecteurvisible,motivantainsil’introductiond’unsecteurdoux. Finalement,le MSSMestdécrit,sesvertusetlimitationsexposées,etlestroisgrandsparadigmes utiliséspour maintenirsouscontrôlelescontributionsdessuper-partenairesauxobservablesde basseénergieébauchés.

3.Lechapitre3présentelathéorie GUTSU(5),àlafoisdansuncontexteSUSYetnon-SUSY. Uneattentionparticulièreestdévouéeausecteurdelasaveur,enparticulierilest montréque lesrelationsdesymétriequiexistentdanslesecteurdeYukawasepropagentauxtermesdoux. Cechapitreseconclutparuneprésentationdequelquesunesdesconséquencesphysiquesdece typed’uniﬁcation.

4.Lechapitre4contientlacontributionprincipaledecettethèse. Danscechapitre,nousavons construitunesériedetestsphénoménologiquessurlesspectresdesquarkshauts,enutilisantdes méthodesstatistiquesfréquentistes.Enparticulier,troistypesdespectreSUSYsontconsidérés, pourlesquelsnousfournissonsdesrelationsdecontraintesurlesobservablesdebasseénergie, relationsquinedevraientêtrequepeudétruitessiunethéorieGUTdetypeSU(5)estréalisée danslanature.

5. Finalement,danslechapitre5,nousconstruisonsdestestsSU(5)plusglobaux,enutilisantdes méthodesbayésiennesdecomparaisonde modèles.Eneﬀet,danscechapitre,nouscalculonsdes facteursdeBayesnumériquement,enutilisantdesalgorithmesde MonteCarloparChainede Markov,aﬁndediscriminerlaprésenceéventuellederémanentsd’unesymétrie SU(5)dansles spectresdesquarkshautsàbasseénergie.Nouscommençonsdonccechapitreendonnantune introductionauxstatistiquesbayésiennes.Lastratégienumériqueutiliséeestensuiteprésentée etlesrésultatscommentés.

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PartI

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Chapter1

TheStandard Mode

lo

fpart

ic

lephys

ics

1 .1 Introduct

ion

TheStandard Modelofparticlephysicsis,byitsabilitiestoaccuratelyﬁt mostoftheexperimental results,oneofthegreatesttheoreticalachievementsinparticlephysicsinthesecondhalfo fthetwen-tiethcentury.Itwasabletopredictnewparticlesbeforetheirobservationinexperiments,withfor someofthem,theirmasses,tosetorderinthejungleofnewboundstateswhichwerediscoveredinthe 1960’sbyintroducingthequarkmodelandtopredictthevalueoftheintrinsicmagneticmomentofthe electron. Thislastvaluehasturnedouttobeinagreementwiththemeasuredonetoanextraordinary accuracyoftheorderofO(10−13_).

Fromatheoreticalpointofview,theSMhasassessedtheimportanceofgaugesymmetries(see [1])inmodernphysicswhichnowﬁndapplicationswellbeyondthescopeofparticlephysics. However, despiteitssuccesses,theSMleavesseveralquestionsunanswered.Thishasleadthehighenergyphysics communitytobelievethatamorefundamentaltheoryinparticlephysicsshouldexistofwhichtheSM shouldbealowenergyeﬀectivetheory.

InthischapterwegiveageneraloverviewoftheSM.Insection1.2westartbypresentingthe theoreticalstructureoftheSMwithanemphasisonthefermions mixing mechanisms. Thiswillallow ustofixacertainnumberofnotationswhichwillbeusefulfortherestofthiswork. Section1.5is devotedtothepresentationoftheCabbibo Kobayashi Maskawa(CKM) matrix,whichdrivesflavor violationintheSM,andtheunitarytrianglefitwhichteststheconsistencyoftheCKM mechanism withdata.Insection1.6,wecommentonthequestionswhichareleftunansweredbytheSM.This willserveasatransitiontothenextchapter. Althoughitisaveryinsightfultopic,thehistorical constructionoftheSMisnotpresentedinthiswork,theinterestedreaderisinvitedtoreferto[2]and [3]aswellastothefirstchapterof[4].

1 .2 Theoret

ica

lframework

TheSMisaquantumﬁeldtheorywhichdescribesthreeoutofthefourfundamentalinteractionsof nature,namelytheelectromagneticinteraction,theweakinteraction,andthestronginteraction. The gravitationalinteractionisleftasideasitrequiresspecialcare. Quantumﬁeldtheory(QFT)allowsto properlyreconcilequantummechanicswithspecialrelativityandisbasedonanalyticalmechanicsusing theLagrangianformalism.ForacompletetourofQFT,seetheclassicalbookofPeskinandSchroeder [5]althoughotherreferencessuchas[6]or[7]shouldbe moreappropriateforgraduatestudents. Precisely,theSMisaYang-Millstheory(see[8])invariantunderthesemi-simplenon-abeliangauge groupGSM= SU(3)C×SU(2)L×U(1)Y. Thesubscripts(C,L,Y)haveno mathematical meaning

andrefertothephysicalcontentofthechargesofthefactors. ThefactorSU(3)Cisthegaugegroupof

QuantumChromodynamic(QCD)whichdescribesthestronginteractionresponsiblefor maintaining thequarksinsidethenucleonsbound. Tothequarksisassociatedanintrinsicquantumnumbercalled "color".Itwasﬁrstintroducedtopreservethespinstatistictheoremwhereincertainboundsstates, withoutcolor,severalquarks(whicharefermions)wouldbeinthesamequantumstate. Thefactor SU(2)L×U(1)YisassociatedtotheelectroweakinteractionwhichuniﬁestheweakandtheQuantum

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Electrodynamic(QED)interactions,thequantumversionofelectromagnetism. ThesubscriptLrefers tothe"weakisospin"introducedtoexplain whycertainpairsoffermionsintheSMhavesimilar quantumpropertiesundertheweakinteraction.Finally,Yreferstothe"weakhypercharge"introduced to maintaintheconsistencyoftheSManddeﬁnedthroughtheGell-Mann–Nishijimaformula:

Y

2=Q−TL3 (1.1)

with T3

LtheeigenvalueofthethirdgeneratorofSU(2)L(seeeqs(1.5)and(1.6))andQ theusual

electriccharge. ToeachfactorinGSMareassociatedspin-1bosonswhichtransmittheforcefromone

fermiontoanother. Theirnumberisequaltothedimensionofthegroupwhichis,forSU(N)factors: dim(SU(N))=_N(N−1)N (1.2) and1fortheabelianU(1)factor.InthecaseofQCD,wehave3(3−1)₂ =8bosons,calledgluons,and noted{Gi_}

{i=1..8}. Fortheelectroweakinteraction,threebosons{Wi}{i=1..3}areassociatedtothe

SU(2)factorandonebosonB totheU(1)factor.Ithastobenotedthataslongasthesymmetry groupremainsexact,thegaugebosonsaswellasthefermionsremain massless,a massterminthe Lagrangianbreakingexplicitlythegaugeinvariance. ThishasleadFrancoisEnglert, RobertBrout, PeterHiggs, Gerald Guralnik,CarlRichardHagenandThomasKibbleinthe mid-1960stoapplyto theSMa mechanisminheritedfromsolidstatephysics,theso-calledHiggs mechanism[9,10]1_.

Digressionontheparamagnetic → ferromagnetictransition Onecanhaveafeelingaboutthe physicsoftheHiggsmechanismbymakingaparallelwiththeferromagnetic/paramagnetictransitionin solidstatephysics.Letusconsideranelectricconductorwhichcanbethoughtofasaspinassembly. Ifthetemperatureofthesystemishighenoughthedirectionsofthediﬀerentspinsarerandomly distributed. The macroscopic magnetizationvector,whichisproportionaltothevectorialsumofthe spins,isnull. Thesystemissaidtobeintheparamagneticphase,thesolidbeingisotropthesymmetry groupisSO(3),thegroupof3Dspacerotations.Ifonedecreasesthetemperaturebelowacertain threshold,theCuriepointTC,a macroscopicdirectioninthesystemappearsamongthespins,the

magnetizationvectorisnotnullanymoreandthesystemissaidtobeintheferromagneticphase. The symmetrygroupoftheferromagneticphaseisSO(2)(subgroupofSO(3)),thesymmetrygroupofthe planerotationswhoseaxisissetbythe magnetizationdirection. Hence,duringtheparamagnetic→ ferromagnetictransition,thesymmetrygroupofthesystemhasbeenreducedfromSO(3)toSO(2). Asthisphasetransitiondidn’toccurduetoanexternalelement,forinstanceamagnet,butbecausethe temperaturei.e.theenergyofthesystemhasdecreased,itiscalledaspontaneoussymmetrybreaking.

Backtoparticlephysics,theHiggsmechanismdescribesthespontaneoussymmetrybreaking(SSB) oftheSU(2)L×U(1)YfactordowntoU(1)Q,thegaugegroupofQED.ToperformthisSSB,weneedto

introduceanextrascalar,theHiggsboson,whichplaysananalogousroletothemagnetizationvector, andreducesthesymmetrygroupofthesystemwhenpickingspeciﬁcdirectionsinthescalarpotential ofthetheory. ThisoccurswhenthesystemfallsinitsgroundstatewhichinaQFTframeworkiscalled thevacuum.Inthisprocess,thegaugebosons{W1,2_}acqu_ireanon-van_ish_ingmassandm_ixtog_ivethe

electricallychargedW+,−_phys_ica_lstates_{. Theothersgaugebosons{W}3_,B}m_ixtog_{ivethe mass}_ive

bosonZ0_,theth_irde_lectr_ica_l_lyneutra_lgaugebosono_{ftheweakinteract}_ion_{,andthephoton}_,thegauge

bosonof QED,whichremains masslessasthe Higgs mechanismpreservesanabeliangroup,U(1)Q,

toguaranteetheconservationoftheelectriccharge. Thegluonsremainalso masslessastheSU(3)C

factorofGSMisleftinvariantbytheHiggsmechanism. Thestates(W+,W−,Z0)havebeenobserved

atthe UA1and UA2experimentsattheCERNin1983withpropertieswhich matchthequantum numberspredictedbytheSM[11–14].Lastlyand,althoughfurtherinvestigationsareneeded,ascalar bosonhasbeendiscoveredin2012independentlybytheATLAS[15]andCMS[16]experimentswith propertieswhichseemto matchtotheHiggsbosonaspredictedbytheSMHiggs mechanism.

1_H_{istoryhasforgotten mosto}_{ftheauthorsintheusednamewh}_{ichonecancons}_{iderasnotveryfa}_ir_{. A}_lternat_ive_ly_,

ifonebelievesthatadiscoverybelongtoanybody,a moreradicalsolutioncouldbetouseanameforthe mechanismin whichnoauthorsappears.

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AfterthisqualitativedescriptionoftheStandard Model,wewanttoadoptnowamorequantitative approachbypresentingthedetailsofthetheory.Speciﬁcally,theSMLagrangiancanbesplitintofour parts:

LSM=LGauge+LMatter+LYukawa+LHiggs (1.3)

Inthefollowing,weshallhaveatourateachofthesesectors.

1 .3 The Gaugesector

Westartwiththegaugesectorwhichdescribesboththekinetictermsandself-interactionsofthegauge bosons. Thegaugesectorofthestandard modelcanbewrittenas:

LGauge=−1₂Tr(GµνGµν)−1₂Tr(WµνWµν)−1₂Tr(BµνBµν), (1.4)

wheretherank2tensorsGµν,WµνandBµνarecalledtheﬁeldstrengthtensorsofrespectivelythe

SU(3),SU(2),andU(1)factors. Thesecanbedecomposed,forinstanceinthecaseofSU(2),as:

Wµν=WµνiTi (1.5)

with {Ti},thegeneratorsofthegroup. Thegreekindices(µ,ν...)arelorentzindicesandindicate

howtheﬁeldstransformunderthegroupofspecialrelativityspacetimesymmetry,thePoincarégroup R1,3 _{SO(1,3)(see[17])wh}_{ichcanbereduced}_{,byapropercho}_iceo_fframe_{,totheLorentzgroup}

SO(1,3)2_{. Here}_{, wetaketheusua}_{l metr}_icconvent_{ionin QFT}_,η

µν= diag(+1,−1,−1,−1).. The

generatorscanbethoughtofas"basisvectors"ofthegroupencodinginﬁnitesimaltransformations deﬁnedbythefollowingcommutationrelations:

[Ti,Tj]=gcijkTk (1.6)

ThesecommutationrelationsformaLiealgebra3_.{c_ijk_}areca_l_{ledthestructureconstantso}_fthe

groupwith,inthecaseofSU(2)cijk= ijkthetotallyantisymmetricLevi-Civitatensordeﬁnedsuch

that123=+1.giscalledthecouplingconstantofthegroupandsetstheinteractionstrength. For

thethreefactorsofGSM,theﬁeldstrengthtensorscanbeexpressedintermofthegaugebosonsas:

Gi

µν=∂µGiν−∂νGiµ−g3fijkGjµGkµ i,j,k=1...8,

Wµνi=∂µWνi−∂νWµi−g1 ijkWµjWµk i,j,k=1..3,

Bµν=∂µBν−∂νBµ,

(1.7) with{fijk},thestructureconstantsofSU(3)deﬁneby Gell Mann[18]. Theindicesi,j,k...livein

theadjointrepresentationofthegroup4_{. Wewanttoemphas}_{izethatthestructureconstantsvan}_ish

inthecaseofanabeliangroupor,inequivalentterms,whentheirgaugebosonsareneutralunderthe group. Theconsequenceisthat,insuchacase,nogaugebosonsself-interactionverticesarepresent. ThisistypicallythecaseforU(1)YorinQEDforQ(1)Qforwhichstructureconstantsarezeroand

nophotonself-interactionsexist. Ontheotherhand,SU(3)andSU(2)beingnon-abelian,threeand fourpointselfinteractionsverticesarepresentamongthegluonsandW bosonsasillustratedinﬁgure 1.1.

2_{TheLorentzgroupSO(1,3)isdeﬁnedasthegroupo}_{f matr}_iceswh_{ichleavethe metr}_icinvar_ianti_.e_{.Λ∈SO(1,3):}

η=ΛηΛT_.

3_An_icev_iewo_fth_{is mach}_{inerycanbedeve}_lopedi_{fwerememberthat}_{,inthecontexto}_fd_iﬀerent_ia_lgeometry_,aL_ie

groupisisomorphictoa manifold. Thegeneratorsofthegroupthenreallyformabasisofvectorsonthetangential spaceofthis manifold.

4_{That meansthatagaugetrans}_format_{ionactsontheﬁe}_{ldstrengthtensor}_{,forinstanceinthecaseo}_{f SU(3),as}_:

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Figure1.1–Threeandfourpointgluonselfinteractions. Theverticesarisesduetothenonabelian natureofSU(3)i.e.theseareproportionaltothestructureconstantsfabc.

1 .4 The mattersector

ThemattersectoroftheSMdescribesthe24spin1/2fermionsformingthematterandtheirassociate anti-particles. Thesecanbegatheredintwofamiliesdependingontheirsensitivitytothestrong interaction. Thefermionssensitivetothestronginteractionarecalledquarksandtheirelectriccharges arequantiﬁedtobeeither +2

3e(the"up-type"quarks)or-13e(the"down-type"quarks)withethe

elementaryelectriccharge. Ontheotherhand,leptonsarenotsensitivetothestronginteractionand canbesplitintwosub-familiesdependingontheirelectriccharges,theelectricallyneutralleptons beingcalled"neutrinos"andtheelectricallychargedbeingcalled"chargedleptons".

Inthecontextofgrouprepresentationtheory,wecanrestricttheLorentzgroup mentionedabove tothesubgroupofLorentztransformationspreservingboththeorientationofspaceandthedirection oftime,notedSO(1,3)↑_{,theorthochronousproperLorentzgroup}5_{. Onecanshowthatthereis}

anisomorphismbetweenthefundamentalrepresentationofSO(1,3)↑_andad_{irectproducto}_ftwo

fundamentalrepresentationsofSU(2),i.e.SO(1,3)↑_{∼SU(2)⊗SU(2).Th}_{issuggeststhattheferm}_ions

oftheSMcanbedecomposedasacombinationoftwo Weylspinorswhichare mergedinanobject calledaDiracspinorΨ. Moreprecisely,aDiracspinorcanbedecomposedas:

Ψ=PLΨL+PRΨR, (1.8)

wherePLandPRarechiralityprojectoroperatorsdeﬁnedas:

PL=1−γ₂5,PR=1+γ₂5 (1.9)

withγ5,theﬁfthgamma matrix(seeappendixA.1).

LetusconsiderafermiondescribedbyaDiracspinorΨandalsoconsideritsassociateanti-spinor ΨC_=CΨC−1_w_{ithC thechargecon}_{jugateoperator}_{. Wecandeﬁnethech}_ira_l_{ityoperatorhasthe}

operatorwhichprojectsthespinSoverthe momentumpdirectionh=S·_|p_p|.Inthe masslesslimit, onecanshowthatΨalwayshasanegativehelicityandisinapureΨLstate(hΨ=−1,Ψ=ΨL)and

ΨC_a_lwayshasapos_it_ivehe_l_ic_{ityandisinapureΨ}_R_state(hΨC_=+1,ΨC_=ΨC

R). Forthisreason,

ΨRiscalleda"right"-handedspinorandΨLa"left"-handedspinor.

IfnowDirac masstermsarepresentintheelectroweakbrokenphaseasitresultsfromtheHiggs mechanism,achirality mixingisintroducedthroughaYukawacouplingtotheH iggsboson(seesub-section1.4.1). Hence,particlesandanti-particlesareinasuperpositionofbothleft-handedandr ight-handedstates.Itturnsoutthattheweakinteractioncouplesexclusivelytotheleft-handedcomponent

5_HenceSO(1,3)↑_{isdeﬁnedas:Λ∈SO(1,3)}↑_i_fdet_(Λ)=+1andΛ0 0>0.

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ofΨ,thatiswhyasubscriptLisaddedtothegaugegroupnameSU(2)Landtheweakinteractionis

saidtobreak maximallytheparitysymmetry. Thisenforcesthefactthatleft-handedcomponentsare foldintwo-dimensionalrepresentationsofSU(2)Li.e.particlephysicistsaytheyformSU(2)doublets.

whereastheright-handedcomponentswillbefoldin1-dimensionalrepresentationsofSU(2)Li.e.they

formSU(2)singlets. Moregenerally,theﬁeldsoftheSMcanbeclassiﬁedin multipletsaccordingto whichrepresentationspacestheyarelivinginfortheSU(3),SU(2),andU(1)factors. Thenotation isthefollowing: weusea3-plets{nC,nL,Y}ΨwherenCandnLareboldintegersdenotingthed

i-mensionof SU(3)CandSU(2)Lrepresentations,Y beingthecommon multiplethyperchargevalue.

Forinstance,left-handedquarksformSU(2)doubletscomposedofanupandadown-quarkandare noted: Q0 mL= u 0 m d0 m L. (1.10)

Thesedoubletsareuniquelydeﬁnedbythe3-plet {3,2,1

3}QL indicatingthattheyl

iveina3-dimensionalrepresentationofSU(3)C(quarksexistinthreecolorstates),ina2-dimensiona

lrepre-sentationofSU(2)andtheupanddownquarksshareacommonhyperchargevalueofYQL=1₃. The

electricchargeofu0

mandd0mcanthenbeeasilydeducedfromeq(1.1). Here,thesuperscript0deﬁnes

thesestatesasweakeigenstatesi.e.theypossessdeﬁnitetransformationpropertiesunderSU(2)Land

thesubscriptm referstothefamilyindex.Indeed,itisanempiricalfactofnaturethateachfermion typeintheSM(chargedleptons:e−0_,neutr_inos_:ν0_,upquarks_:u0_{anddownquarks}_:d0_)ex_ist

inthreecopies(m=1,2,3)named"generation"or"family". Thefullcontentofthem-thfamilyof fermionscanthenbesummarizedasfollows:

Q0 mL= u 0 m d0 m L:{3,2, 1 3}QL L0mL= ν 0 m e−0 m _L:{1,2,−1}L0mL u0mR:{3,1,4₃}u0 mR d 0 mR:{3,1,−2₃}d0 mR e−0 mR:{1,1,−2}e−0 mR ν 0 mR:{1,1,0}ν0 mR (1.11)

Weemphasizethefactthatdespite wehaveintroducedright-handedneutrinos ν0

mR above,strictly

speaking,theydonotbelongtotheSMspectrumsincetheyarecompletelyneutralunderGSM. As

ν0

mRdoesnotinteractthroughanyofthegaugeinteractionoftheSMwiththerestofthespectrum,it

iscalledasterileparticle.Beside,currently,right-handedneutrinosshouldstillbeseenashypothetical particlesasnoexperimenthasallowedtodisentangletheexactspinorialnature(MajoranaorDirac, seesubsubsection1.4.2.2andappendixA)ofneutrinosuntilnow. Howevertheirexistence mightbe needed,dependingontheexactneutrinonature,assuggestedbytheexperimentswhichhaveproven thatneutrinoscanoscillatefromonefamilytoanotherandhence,ownanon-vanishing mass.

The matterpartoftheSMLagrangiancanthenbewritten: LMatter =

3

m=1

Q0mLi/DQ0mL+L0mLi/DL0mL+u0mRi/Du0mR+d0mRi/Dd0mR+e0mRi/De0mR+ν0mRi/DνmR0

(1.12) Notethat masstermsoftheformLmass =−mΨΨareabsolutelyforbiddenatthisstageasthese

areequaltoLmass=−m ΨLΨR+ΨRΨL usingthedecomposition(1.8)andthusviolateexplicitly

theSU(2)Linvariance.

AnimportantremarkisthatLmatter ownsaU(3)6globalﬂavorsymmetry.Forexample,aglobal

transformationoftheformUQL ∈U(3),QL→ UQLQLleavesLMatter invariant. Asthissymmetry

isnotimposedexplicitlywhenconstructingthetheory, wecallitanaccidentalsymmetry.LMatter

containsallkinetictermsforthefermionsbutalsofermion-gaugebosoninteractionsascanbeseenif wedevelopthegaugecovariantderivative /D=Dµ_γ_µ_{intheﬁrsttermo}_f(₁_.12)_:

Q0mLi/DQ0mL=iu0αmLd0αmL γµ× ∂µI+ig1Ti1Wµi+ig₃2T2Bµ δαβ+ig3Tαβi3 GiµI u 0 mLβ

d0

mLβ .(1.13)

HereIistheSU(2)identity matrixandα,βarethecolorindicesofthequarks.T1

i=1..3andTi=1.3 .8

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Figure1.2–Singlecomplexscalarﬁeldcase: ThescalarpotentialV with,ontheleftµ2_>0and_,on

therightµ2_{<0. Takenfrom[}_19]_.

matrices.ForadeﬁnitionofthePaulimatrices,seeappendixA.TheuniqueU(1)Ygeneratorissimply

equaltoT2_=IY

2. WeusetheusualEinsteinconventionwhichimpliesimplicitsumsoverallrepeated

indices.

1.4.1 The Higgs mechanism

Inthissubsection,wegivedetailsontheabove mentionedHiggs mechanismwh ichgeneratesdynam-ically masstermsforthegaugebosonsandforthefermionsthrough,respectively,theHiggskinetic termsandtheYukawainteractionswiththeHiggsboson. TheHiggspartoftheSMLagrangianLHiggs

is:

LHiggs=(DµΦ)†DµΦ−V(Φ) (1.14)

where Φ iscomposedofoneelectricallychargedcomplexscalarφ+ _andonee_lectr_ica_l_lyneutra_lφ0

forminganSU(2)doubletΦ= φ_φ+0 transformingas{1,2,12}φ. Thegaugederivativetakestheform:

Dµ_{Φ= ∂}

µ+igTi1Wµi+igT2Bµ Φ (1.15)

astheHiggsdoubletisnotchargedunderSU(3)C. Thesimultaneousneedforrenormalizabilityand

SU(2)×U(1)gaugeinvariancerestrictsthescalarpotentialV totheform:

V(Φ)=+µ2_Φ†_Φ+λ(Φ†_Φ)2 ₍₁_.16)

withλ >0inordertoenforcevacuumstability. The minimumofthispotentialwilldependonthe signofµ2_asdep_{ictedinﬁgure1}_.2inthecaseo_fas_ing_lecomp_lexsca_larﬁe_ld_.

Weseethatinthecase µ2_{>0,V presentsastab}_{le m}_in_{imumatΦ=0whereasforµ}2_<0ana_l_l

setofdegenerate minimaappearsatanon-vanishingdistancefromtheorigin,allthose minimabeing relatedbyaSO(2)∼ U(1)symmetry. The Higgsdoubletcanberewrittenintermsof4realﬁelds {φ}i=1..4as:

Φ= φ_φ+₀ = √12(φ1+iφ2) 1

√

2(φ3+iφ4) (1.17)

andthescalarpotentialbecomes:

V(Φ)=1₂µ2 4 i=1 φ2i +1₄λ 4 i=1 φ2i 2 (1.18) Thespontaneoussymmetrybreakingoftheelectroweakphaseistriggeredwhentheneutralcomponents {φi}{i=3,4}fallinoneofthosedegenerate minimawhicharerelated,inthecaseofadoublet,bya

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SO(4)symmetry. Ina QFTcontextφiissaidtotakeanon-vanishingvacuumexpectationvalue

(VEV)asitdoesnotannihilatethevacuum|0 anymore, meaning 0|φi|0 =0. Wecanthenusethis

SO(4)symmetrytosetallbutoneVEVstozero 0|φi|0 =0,i=1,2,4and 0|φ3|0 =ν >0. With

thisin mind,thevalueatthese minimaofthescalardoubletΦandofthepotentialV are: Φ→ 0|Φ|0 ≡v=√1 2 0 ν V(Φ)→ V(v)=1₂µ2_ν2₊1 4λν4 (1.19) andthevalueofνisobtainedby minimizingthepotentialV(ν):

ν= −µ_λ2 (1.20)

LetushavealookattheactionofthegeneratorsofSU(2)L×U(1)YoncetheHiggsdoublethastaken

aVEV: T1 iv=τ₂i√1₂ 0_{ν =0,} Yv=I₂√1 2 0 ν =0. (1.21) Weseethattheactionofthegeneratorsof SU(2)L×U(1)Ydonotleavethevacuuminvariantanymore.

This meansthatthevacuumhasgotchargedunderthesegroupsandtheelectroweaksymmetryhas indeedbeenspontaneouslybroken. However,acombinationofthesegeneratorscorrespondingtothe electriccharge,Q =(T1

3+Y2)(see(1.1))remainsintactsinceQv =0. Thisenforcesthefactthat

theelectricchargeisconservedasitshouldbe,andthefollowingelectroweakspontaneoussymmetry breaking(EWSSB)patternSU(2)L×U(1)Y→ U(1)Q hasbeenestablished. However,ifacharged

componentofφwouldhavetakenaVEV,thiswouldnotbetrueanymoreandtheSU(2)L×U(1)Y

wouldhavebeenbrokentonothing. Whenspontaneouslybreakingagaugesymmetry,itisthus mandatorythatonlyﬁeldsneutralunderthesymmetrieswewanttopreserveinthelowenergytheory areallowedtotakeanon-vanishingVEV.

Ontheotherhand,thecomplexscalar φ3isexpectedtooscillatearoundits minimumduetoits

quantumnature. Wecanthusshiftitaroundvandwriteφ3= v+h0withh0thephysicalneutral

Higgsbosonwhichhasavanishing VEV 0|h0_|_{0 =0. Go}_{ingbacktothes}_ituat_{ionwhereΦisnot}

alignedontherealaxis,ausefulpolarparametrizationhasbeenintroducedby Kibbleandcanbe writtenas: Φ=√1 2e(i 3 i=1ζi(x)Li) 0 v+h0 (1.22)

withLi_be_ingthethreee_{lectroweakbrokengeneratorsT}1

1,T21,T31−Y andwehave

madethespace-timedependenceofζ(x)iexplicit.Supposedthatweweredealingwithaglobalsymmetry,thegauge

parametersζ(x)iwouldthencorrespondtotheso-called massless Goldstonebosonsandtherewould

beas manyasbrokengenerators,here3. However,preciselyduetothelocalnatureofthegauge symmetry,wecaneliminatetheseusingthefollowinggaugetransformation:

Φ→ Φ=e(−i iζiLi)Φ= 1_√

2 0

v+h0 (1.23)

calledtheunitarygauge. Fromnowon,unlessotherwisestated,wewillalwaysworkintheunitary gauge. Withthedisappearanceofthe Goldstonebosons, weseemtohavelostphysicaldegreesof freedominourtheory,howeverthesewillreemergesomewhereelseaswewillnowsee.Indeed,ifwe havealookattheHiggskinetictermineq(1.14)afterEWSSB,ittakestheform:

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d.o.f BeforeEWSSB AfterEWSSB

gaugebosons 8 11

Goldstonebosons 3 0

Higgsﬁeld 1 1

Figure1.3–Thedistributionofd.o.famongtheﬁeldsbeforeandafterEWSSB.TheGoldstonebosons havedisappearedintothegaugesbosons,thesehavebeen"eaten".

Wecanperformthefollowingchangeofbasis: W±=W1√∓iW2

2 , T1±=T11±T21 (1.25) toseethateq.(1.24)containstheterms:

g2 1ν2 4 W+µWµ−+ 1 2g12+g22 ν2 4 g1Wµ3−g2Bµ g2 1+g22 2 ≡M2 WW+µWµ−+M 2 Z 2ZµZµ, (1.26) andseethatthechargedgaugebosonsW±_haveacqu_iredanon-van_ish_ingmasstermM

W =g1₂ν.W±

aretheonlyphysicalstatesintheSM whichallowattreelevelto mediatethedecayofaparticle changingitsﬂavor. Aswewillseeinsubsection1.4.2,andasW±_ownane_lectr_iccharge_,suchﬂavor

transitionisinevitablyfollowedbyachangeoftheelectricchargeofthedecayingparticle. Forthis reason,wesaythattheW±_aretheon_lymed_iatorso_{fﬂavorchang}_{ingchargedcurrents(FCCC)inthe}

SM.AthirdneutralphysicalZbosonhasalsoappearedwitha masstermMZ=_cosθMW_W.Zisdeﬁned

asalinearcombinationoftheBµandWµ3bosons:

Zµ=g1W 3 µ−g2Bµ g2 1+g22 ≡−sinθWBµ+cosθWW 3 µ (1.27)

wherewehavedeﬁnedtheweak mixingangleθW as:

tanθW ≡g_g2 1→ sinθW = g2 gZ,cosθW = g1 gZ, (1.28)

andgZ= g12+g22theZ bosoncouplingconstant. ThecombinationorthogonaltoZ canthenbe

easilydeﬁnedasA =cosθWB+sinθWW3andisidentiﬁedasthephoton. Thephotonremains

masslessatallorderinperturbationtheoryasitsassociatedgaugegroup U(1)Qisleftunbrokenby

theEWSSB.Asstatedabove,itseemsthatphysicaldegreesoffreedomhavedisappearedduringthe EWSSB whenabsorbingthe Goldstonebosonsusingtheunitarygauge. Letushaveacloserlook andcountthenumberofdegreesoffreedom(d.o.f.) beforeandafterEWSSB.BeforeEWSSB, we had4masslessvectorsbosonsassociatedwiththe SU(2)L×U(1)Ygaugegroup. Asitisverywell

known, masslessvectorsbosonsonlyhavetwod.o.f.correspondingtotwotransversepolarizations. Theelectroweakbosonshencebringatotalof4×2=8d.o.f.tothetheory.

Ontheotherhand,therearethree massless Goldstonerealbosonscorrespondingtothethree brokengeneratorsandone masslessHiggsboson. Allthesefourﬁeldsownonlyoned.o.f.each. The totalnumberofd.o.f.beforeEWSSBishence8+3+1=12. AfterEWSSB,thenow massivethree eletroweakgaugebosonsW+_,W−_{,Zhave3degreeso}_{ffreedomeachcorrespond}_{ingtotwotransverse}

andonelongitudinalpolarizationstates. Wealsohavea masslessvectorphotonA withtwod.o.f. andone massiverealscalarhwithoned.o.f. Thetotalnumberofd.o.f. afterEWSSBisalso 12, howeverweseethatthed.o.f. whichhavedisappearedwhenabsorbingthethree Goldstonebosons havereappearedinthenowmassivegaugebosonsastransversepolarizations. Theacceptedexpression todesignatethisre-balancingisthatthe Goldstonebosonshavebeen"eaten"bythe massivegauge bosons. Table1.3summarizesthedistributionofd.o.f.amongtheﬁeldsbeforeandafterEWSSB.

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1.4.2 The Yukawasector

Asstatedabove,thefermionsoftheSMcanbesplitintotwogroups,thequarksandtheleptons, dependingontheirsensitivitytothestronginteraction. To matchexperimentalobservations,each grouphastobeduplicatedthreetimesintothreecopiesnamedfamilyorgeneration.Inthissubsection, wehaveacloserlookattheYukawasectorLYukawaofLSM. TheYukawasectorallowstogenerate

massesforthethreefamiliesofleptonsandquarksthroughcouplingtotheHiggsboson. Althoughthe Yukawasectorisdiscussedinitsfullgenerality,wepostponecommentsontheexperimentalvaluesof the massspectrumtosubsection1.6.4.

1.4.2.1 Quarks

Westartwiththequarks. Theyaresixtypes(orﬂavors)ofquarksintheSMwhichexistinthree diﬀerentcolorseach. Asseeninsection1.4,eachfamilyiscomposedofonequarkofelectriccharge2

3e

andofonequarkofelectriccharge−1

3e. Theirleft-handed(resp:right-handed)componentsforming

SU(2)doublets(resp:singlets).Thelightestfamilyofquarksconstitutesthecomponentsoftheproton andtheneutronwhichformtheatomicnucleus. Forthisreason,thequarksofthefirstfamilyare simplynamedup"u"anddown"d"quarks. Thefourheavierflavorsarenamedforhistoricalreasons strange"s",charm"c",bottom"b"andtop"t"quarks. Thesecanonlybeproducedinhigherenergy processesforexamplewhenaprotonofcosmicrayshitsthehighatmosphereorwhentwoprotons collideinahadroncollider.Indeed,averyspecificfeatureof"QCD"named"confinement" makes thatquarkscanneverbeseenasfreestatesandcanonlybedetectedasboundstatescalledhadrons themselvescolorless. Thedown-typestrangeandtheup-typecharmquarksformthesecondfamilyof theSM.Finally,thedown-typebottomandtheup-typetopquarksformthethirdandheavierfamily oftheSM.ThequarksYukawasectorcanbewrittenas:

−Lquarks_Yukawa=yu

m,nQ0mLΦu0nR+ymd,nQ0mLΦd0nR+h.c. (1.29)

withΦ=iσ2Φ†,theconjugateHiggsﬁeldtransformingas{1,2,−1₂}_Φneededtowriteinvariantterms

oftheformQ0mLΦu0nR+h.cintheup-sector6.yu,ydare3×3arbitraryYukawamatricesdetermining

thestrengthofthecouplingtotheHiggsboson. TheYukawa matricesarecompletelyarbitraryand donothavetobeHermitian,realnordiagonal. Theyintroduce mostoftheSMfreeparametersand break mostoftheU(3)6_fam_i_lysymmetr_ieso_ftheresto_fL_matter_,on_lytwog_loba_lU(1)symmetr_ies_,

thetotalbaryonandleptonnumbersareaccidentallyconserved. AfterEWSSB,Lquarks_Yukawabecomes: −Lquarks_Yukawa=u0 mLyumn ν+h 0 √ 2 u0nR+d 0 mLymnd ν+h 0 √ 2 d0nR+h.c (1.30) whereDirac masstermshavebeengeneratedfortheupanddownquarks:

Mmnu =ymnu √ν₂, Mmnd =ydmn√ν₂ (1.31)

Inordertoidentifythephysicalstates,thosewhichwillactuallypropagate,itisstillneededtoperform achangeofbasistobringthemassmatricestodiagonalforms. Astheleftandrighthandedcomponents oftheﬁeldstransformindependently,wehavetoperformseparatetransformationsonthem.Sincethe electroweaksymmetryisnowbroken,wealsoneedtoperformindependenttransformationsontheup anddowncomponents,e.g.:

u0L=VuLuL, u0R=VuRuR, d0L=VdLdL, dR0=VdRdR, (1.32)

where wehaveswitchedtoindicefreenotationsandsimplynote u=(u,c,t)T_,d=(d_,s_,b)T_{. We}

emphasizethatallthe matricesineq.(1.32) mustbeunitaryinordertopreservethecanonicalform

6_Letusbeexp_l_ic_it_,i_{f wecons}_{iderthehypercharge wehaveY Q}

L = −13andY(uR) =43or wehavetoimpose

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ofthekinetictermsintoLmatter. Thischangeofbasisrotatesthemassmatricestodiagonalform,e.g.: V† uLMuVuR≡MDu=diag(mu,mc,mt) V_d†_LMd_V dR≡MDd=diag(md,ms,mb). (1.33) withmftherealandpositivephysical massofthefermionf. ForthisreasonthestatesΨ,without

thesuperscript0,arecalledthe masseigenstatesofthetheory. Nowthatwehaveswitchedtothe massbasis,westillhavetoscrutinizetheconsequencesofthechangeofbasis(1.32)ontherestofthe Lagrangian. TheHiggsandgaugepartremaininvariantastheydonotdependonthefermionsﬁelds. However,ifwerewriteLMatter intermsofthephysicalweakbosonsW±,onecanshowwithalittle

bitofalgebrathatthefollowingtermisincludedintheSMLagrangian: LSM − g1

2√2u0γµ(1−γ5)d0Wµ++h.c =−g1

2√2um(Vu†LVdLγµ(1−γ5))mndn Wµ++h.c

(1.34) where wehaveswitchedtothe massbasisinthelastlineofeq. (1.34). Weseethatthisterm induceschargedﬂavortransitionsbetweentheupanddownsectorsthroughcouplingtotheW bosons. Furthermore,iftheupanddownquarksarenotalignedinﬂavorspace,thatisifVuL=VdL,thecoupling

ofthesetransitionsissetbytheunitary matrixVCKM = Vu†LVdL calledthe Cabibbo-Kobayash

i-Maskawa(CKM) matrix(see[20])andsection1.5. 1.4.2.2 Leptons

Wenowdiscussbrieﬂytheleptonic Yukawasector. Asstatedinsection 1.4,thethreefamiliesof leptonsintheSMcanbesplitintwogroupsaccordingtotheirelectriccharges. Chargedleptonse−

m

ownanelectricchargeequalto−ewhereasneutrinosareelectricallyneutral.Left-handedcomponents ofchargedleptonsarefoldinthedown-partofSU(2)doublets(T1

3(e−Lm)=−12)whereaslefthanded

componentsofneutrinosarefoldintheup-partofSU(2)doublets(T1

3(νLm)=+1₂),theirright-handed

componentsbeingagainSU(2)singlets. Thelightestchargedleptonbelongingtotheﬁrstleptonic familyissimplytheusualelectronnotede−_{,the med}_iatoro_fchem_ica_lreact_ions_{. Thesecondlepton}

isnamedthe muonandnotedµ−_{. The muonhasaspart}_icu_lar_{itythatitisab}_letotrave_lonlong

distancesduetoitslongproperlifetimeofτ(µ−_{) =2.}_{2µs. Oppos}_itetothat_,theth_irdlepton_,the

tauonnotedτ−_{hasa muchsma}_l_lerl_i_fet_imeo_fτ(τ−_{) =2.}_9×10−1_{psandthusdecays much more}

quickly. Ontheotherhand,theneutrinosarenamedaftertheirchargedleptonsSU(2)counterparts. Hencethethreeneutrinosaretheelectronic,muonicandtauicneutrinos,notedrespectivelyνe,νµand

ντ. TheleptonicYukawaLagrangianisthen:

Lleptons_Yukawa=ye_L0

LΦe0R+yνLL0ΦνnR0 +h.c. (1.35)

wherewehavedeﬁnethefollowingvectorsinﬂavorspacee0_=(e0_,µ0_,τ0₎T_,ν0_=(ν0

e,νµ0,ντ0)T. As

inthequarksector,Lleptons_Yukawagenerates Dirac masstermsfortheleptonsandtheneutrinosoncethe neutralcomponentofΦhastakenaVEVν:

Me_=ye_√ν

2, Mν=yν ν √

2 (1.36)

Onecanthenperformunitaryrotationsontheﬁeldstobringthe mass matricestodiagonalform: V_Le†MeVRe≡MDe=diag(me,mµ,mτ) VLν†MνVRν≡MDν=diag(mν1,mν2,mν3) (1.37)

whichdeﬁnethe masseigenstatebasisforthechargedleptonse0

L,R=VLe,ReL,Randfortheneutrinos

ν0

L,R=VLν,R(ν1,ν2,ν3)TL,R. Wehavelabeledthethreeneutrino masseigenstatesν1,ν2andν3because,

oppositetothequarkcase,themixingintheleptonicsectorissigniﬁcantlyhigher,eachmasseigenstate ν1,ν2andν3beingtrulyasuperpositionofallthreeweakeigenstatesνe,νµandντ. Thischangeof

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mismatchintherotationsoftheleft-handedleptonsandneutrinos. This mismatchisparameterized bytheunitaryPontecorvo-Maki-Nakagawa-Sakata(PMNS) matrixVP MNS=V_Lν†V_Le(see[21]). Note

thatthePMNS matrixisdiﬀerentfromunityonlyinthecasewhere masstermsfortheneutrinosare present.Indeed,ifnoneutrinos masstermsarepresentintheLagrangianonecanadjustVν

L freely

andsetVν

L=VLe→ VP MNS=✶.

Acaveatonneutrino masses: Inthisparagraph, we wanttopointoutthattheapproach we haveconsidereduntilhere-withneutrino Dirac masstermsgeneratedviathe Higgs mechanism-, althoughhavingthevirtueofsymmetrizingthequarksandleptonicsectorsaskseriousnaturalness problems.Indeed,the massesoftheneutrinosbeingextremelysmall( _i=1,_2,₃mi< 0.49eV @95

CLfromcosmology[22]), Dirac masstermswouldimplyextremelyhighﬁnetuningofthe Yukawa couplings. However,itispossibletoconstruct modelsbeyondtheSMinwhichthesetinyneutrinos massesarisenaturallythankstothepresenceofoneorseveralheavyright-handedsterileneutrinos NR. Tothisend,neutrinosneedtobeviewedas Majoranasparticles. A4-components Majorana

spinorΨM _isaﬁe_ldinwh_ichthele_{ft-handedcomponentisthechargecon}_jugateo_fther_ighthanded

onei.e. ΨM

L,R= C(ΨMR,L)T. Thisimplythata Majoranaparticleshouldbeitsownanti-particle

i.e.ν= νc_{= Cν}T_andshou_{ldnotcarryanyconservedquantumnumbers}_{. Or}_,th_{isisindeedthe}

caseoftheneutrinosintheSM.Severalexperimentsaretryingtodisentanglethe Dirac/Majorana natureoftheneutrinobyseeking,forexample,forneutrinolessdoublebetadecays0νββinwhicha neutrinoannihilateswithitsantineutrinotonothing(see[23]forareview).Intheseso-called"seesaw" mechanismssmallneutrinos massesarethengeneratedwhenintegratingoutright-handedneutrinos NR. Thisresultsinnon-renormalizable,dimensionﬁveoperatorsoftheform:

δLM

mass=1₂gv 2

ΛνLTC†νL+h.c (1.38)

whereΛisthecut-oﬀoftheeﬀectiveoperatorΛ∼MNR whichsuppressthelightneutrinos masses.

However,asthisthesisisfocusingonthehadronicsector,thesequestions willnotbefurther investigatedandwerefertoreferences[4],[24]foratourofseesaw models.

1 .5 Afocusonthe CKM matr

ix

Asstatedinsubsubsection1.4.2.1,theCKMmatrixarisesfromamismatchbetweenthefermiongauge andYukawainteractionsi.e.betweentheweakandmasseigenstatesbasis.Furthermore,afterEWSSB, a mismatchappearsbetweentheupanddownsectorandtheCKMcanbeseenasthecompositionof therotationsinthesetwosectorsi.e.VCKM=Vu†LVdL,onlyleft-handedcomponentsbeinginvolved

duetothechiralnatureoftheelectroweaktheory.Itisthusaunitary3by3complex matrixwhich setsthestrengthofFCCCprocessesintheSM.Inthissection,wewanttogive moredetailsonthe CKM matrixasitplaysaprominentroleforﬂavorphysicsintheSMandbeyond. Letusstartby countingthenumberofd.o.f.inVCKM. AgeneralF×F complex matrixhas2F2independents

parameters. Outofthose2F2_parameters_{,onecansubtractF}2_{parametersduetoun}_itarycond_it_ions

(V_CKM† VCKM)mn=δmn. Wecanfurtherremove2F−1parametersthankstoaredeﬁnitionofphases

ontheupanddownquarks. TheseareofthetypeuL→ KuLuLwithKuL=diag eiφ1,eiφ2,eiφ3 . We

arethusleftwithF2_{−(2F−1)=(F−1)}2F=3_{−→4independentrea}_{lparametersintheCKM matr}_ix_.

These4realparameterscanbefurthersplitin3mixingangles θijand1CPviolatingphaseδCKM.

TherearemanywaystoparameterizetheCKMmatrixbutherewefollowtheonegivenbytheParticle DataGroup[25]: ‘ VCKM≡  V_Vud_cd V_Vus_cs V_Vub_cb Vtd Vts Vtb   =  1 0 0_{0 c}₂₃ _s₂₃ 0 −s12 c23     c13 0 s13e −iδ 0 1 0 −s13eiδ 0 c13    _−sc12₁₂ s_c12₁₂ 0₀ 0 0 1   =   c12c13 s12c13 s13e −iδ −s12c23−c12s23s13eiδ c12c23−s12s23s13eiδ s23c13 s12s23−c12c23s13eiδ −c12s23−s12c23−s12c23s13eiδ c23s13   (1.39)

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withcij=cos(θij)andsij=sin(θij). The magnitudeoftheelementsoftheCKM matrixhavebeen

measuredthroughprocessesinvolvingdiﬀerentﬂavortransitionsandareequalto: |Vij|=  0._0.9742 0._{226 0.}226 0._{973 0.}0036₀₄₂ 0.0087 0.041 0.9991   ₍₁_.40)

whichimpliess12∼0.23,whiles13<<s23<<s12. Theimportantpointtonoticeisthattheelements

modulusdecreaseofoneorderof magnitudeeachtimewetakeonestepawayfromthediagonal. This suggeststouse4parametersλ,A,ρ,ηdeﬁnedas:

s12=λ= |Vus|

|Vud|2+|Vus|2,s23=Aλ

2_=λVcb

Vus ,

s13eiδ=Vub∗=Aλ3(ρ+iη)= Aλ

3_(ρ+iη)√_1−A2_λ4

√

1−λ2_[1−A2_λ4_(ρ+iη)_]

(1.41) towriteanexpansionofVCKM,usingλasexpansionparameter.Followingthisprocedure

,aparame-terizationhasbeenproposedby Wolfenstein[26],andweendupwith,atO(λ3_):

VCKM=    1− λ2 2 λ Aλ3(ρ−iη) −λ 1−λ2 2 Aλ2

Aλ3_{(1−ρ−iη) −Aλ}2 ₁

 

+O(λ4) (1.42) Moreover, wecansafelycheckthat VCKM staysunitaryatallordersinthe Wolfensteinexpansion,

theCPviolationphaseδCKMbeingassociatedwithηthroughtanδ=η_ρ. Therearevariouspossible

conventionsfordeﬁningthe CPphaseinVCKM however,ρ+iη≡ −V

∗ ubVud

V∗

cbVcd isphase-convent

ion-independent.

Itisespeciallyimportantforphenomenologytogetaccurate measuresoftheelementsofVCKM

andtotestitsfeatures,especiallyitsunitarity.Inparticular,wewanttocheckifsumsofthetype

jVijVkj∗reallyvanishfori= k. Usingthe Wolfensteinparameterization,theserelationstakefor

exampletheform: V†_V

31=V ∗

ubVud+Vcb∗Vcd+Vtb∗Vtd

∼ 1−λ₂2 Aλ3_{(ρ+i+η)−Aλ}3_+Aλ3_{(1−ρ−iη)}

∼Aλ3_{(ρ+iη)−Aλ}3_+Aλ3_{(1−ρ−iη)=0}

(1.43) but,|V∗

cbVcd|∼Aλ3isknownwithgreataccuracy,sothatitcanbeusedtonormalizedeq.(1.43):

V†_V 31

Aλ3 ∼ρ+iη−1+1−ρ−iη (1.44)

andeq.(1.44)canberepresentedbyanormalizedtriangleinthecomplexplane(ρ,η)withthevertices at(0,0);(1,0)and(ρ,η). Onecanthenapplydifferentconstraintsfromflavorobservablestofitthe Wolfensteinparametersandtocheckifthesumoftheanglesarereallyequalto 180◦_i_.etocheck

thatunitarityisindeedsatisfied. Forexample,thelengthofthetrianglescanbeobtainedfromCP conservingobservables,asintheb→ udecayratewhereastheanglesareobtainedfromCPviolating effectssuchasneutral mesons mixing. Figure1.4presentsaglobalfitonthetriangle(1.43)withall availableflavorconstraintsincludedhere@95%C.L.Theredhashedblobindicatesthebestfitvalues ofρandηat68%C.L.Thebestfitvaluesofthe Wolfensteinparametersarethus:

A=0.810+0.018

−0.024 λ=0.22548+0.−0.0006800034

ρ=0.1453+0.0133

−0.0073 η=0.343+0.−0.011012 (1.45)

Weseeonfig. 1.4thatallflavorobservablesarecompatiblewiththeSMCKM mechanismi.e. no significantdeparturefromunitarityisobserveduntilnow.

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γ α α d m ∆ K ε K ε s m ∆ & d m ∆ ub V β sin 2 (excl. at CL >β 0 <.95) 0 sol. w/ cos 2 α β γ

ρ

0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0

η

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ex cl ud ed a re a ha s CL > 0 .9 5 Winter 14 CKM_f_i_t_t_e_r

Figure1.4–Unitaritytriangletest. Allflavorobservablesaregivenat95%C.L.Theredhashedregion representsthe68%C.Lfittedvalueofthe Wolfensteinparametersρandη.PlottakenfromtheCKM fittergroup[27],

H

f

http://CKMfitter.in2p3.fr

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1 .6 L

im

itat

ionsandopenquest

ions

ThereismuchmoretosayabouttheSM,howeverforthesakeofconcision,westophereandmoveon bygivingsomedetailsonafewofthereasonsthatpushthehighenergyphysicscommunitytothink thatdespiteallitsvirtues,theSMcannotbeanultimatetheoryofnature,ifanysuchultimatetheory doesexistatall.

1.6.1 Thehierarchyproblem

Westartwiththefamous"gaugehierarchyproblem".Beyonditsformalpresentationonwhichwewill shedlightbelow,itisofcrucialimportancetounderstandwhatthegaugehierarchyproblemreallyis about. Asstatedintheintroduction,theSMdoesnotaddressthequestionofthequantiﬁcationof gravity. Thequantumeﬀectsofgravityshouldbecomerelevantat mostatascaleoftheorderofthe Planck massMP= _G_Nc ∼ O(1019)GeV. OnecansaythatthePlanck massisthe"natural"scale

associatedtogravity.IfonewantstopickanaturalscalefortheSM,agoodchoicemightbethevacuum expectationvalueν∼246GeVwhichdrivesthespontaneousbreakingoftheelectroweaksymmetry.In itsveryessence,thegaugehierarchyproblemconsistsinunderstandingwhytheelectroweakscaleis17 ordersofmagnitudesbelowthePlanckscale.Stateddiﬀerently,sinceMP∝√_G1_N,itisequivalenttoask

thequestionofthe-apparent-weaknessofthegravityinteractioncomparedtotheotherinteractions oftheStandard Model.

Moreformally,thegaugehierarchyproblemarisesbecausethe massoftheHiggsboson hreceives radiativecorrectionswhicharedrivennaturallytohugevaluesfromallparticlesitcoupled.Indeed,in thecurrentstate-of-the-art,the massoftheHiggsboson,asall massesin QuantumFieldTheory,is computedperturbatively. ThatmeansthephysicalHiggsmassm2

hisgivenbym2h=m2h0+ _i(∆m2_h)i

wherethesuccessivecorrections(∆m2

h)ihavetodecreaserapidlyfortheseriestoconverge. Now,ifwe

consideraDiracfermionfwhichcouplestotheHiggsbosonthroughaYukawainteraction−λfhff,

wecanshowviathediagram1.5thattheﬁrsttermsofthisperturbativeseriesaregivenby: m2 h=m2h0−|λf| 2 8π2 Λ2UV+Am2fln m 2 f Λ2 UV (1.46)

whereAisanirrelevantdimensionlessconstant,ΛUVthecutoﬀusedtoregularizedthetheoryandm2_h0

isthe"bare" mass.ΛUVhastobeseenastheenergyscaleatwhichtheSM,seenasaneﬀectiveﬁeld

theory,breaksdown. WeseethatwehavenaturallyΛUV∼MPiftheSMisbelievedtobevalidallthe

wayuptothePlanckscale.Indeed,weknowthatatthisscale,thequantumeﬀectsofgravitybecome nonnegligibleandtheSMhastobreakdown. HavingΛUV∼MPaskshoweverseriousnaturalness

problemsasweknownowthatthephysicalHiggs massismh≈125GeV.However,forthephysical

massm2

htobeofthatorder,weneedthebaremassm2h0tocanceloutthequadraticdivergences∝Λ2_UV

atanincrediblelevelofaccuracy(seeeq.(1.46)). Animportantpointtonoticeisthatthisproblem arisesbecausescalar massesarenot"protected"byanykindofsymmetriesintheSM.A massissaid tobeprotectedbyasymmetryif,inthelimitwhereitvanishes,asymmetryisrestoredinthetheory. Inthiscase,justwithdimensionalanalysis,onecanshowthatnoquadratictermsareallowedinthe perturbativeseries,alltermsbeingroughlyoftheformm2

0lnΛ

2 UV

m2

0 ensuringsmootherdivergences

andguaranteeingthatnoaccuratecancellationshappenedbetweenthesuccessiveparametersofthe series. Thisisexactlywhatoccurswhenafermionmassiscomputedasinthelimitmf→ 0,thechiral

symmetryisrestoredandthefemions massesareprotectedfromdangerousquadraticdivergences. Ontheotherhand,nosymmetryprotectingtheHiggs mass,thenaturehastopickcouplingsinthe perturbativeserieswhichareveryﬁne-tuned,anydeviationsfromthosehightunedcouplingsresulting inthe masstoblowup. Thiskindofextremelyaccuratecancellationseemshighlyunnatural,unless a morefundamentaltheoryofnatureisatwork,toexplainthatthosecouplingslooktunedinthelow energyeﬀectivetheoryasitwillbecomeclearlaterinthiswork. Finally,weconcludethissectionby pointedoutthatthegaugehierarchyproblemisnotreallyaproblemif,despitehighlyunlikely,the SMisanultimatetheoryofnature.Inthiscase,ΛUV→ ∞ andtheSMbeingarenormalizabletheory,

New phenomenological tests for supersymmetric theories with SU(5)-like unification

HAL Id: tel-01292577

https://tel.archives-ouvertes.fr/tel-01292577

New phenomenological tests for supersymmetric

theories with SU(5)-like unification

Yannick Stoll

To cite this version:

THÈSE

Yann

ickSTOLL

Nouveauxtests phénoméno

log

iques pour

lesthéor

iessupersymétr

iquesavecun

iﬁca-t

iondetypeSU(5)

Contents

Remerc

iements

Introduct

ion

Introduct

ionenfrança

is

PartI

Chapter1

TheStandard Mode

lo

fpart

ic

lephys

ics

1

.1 Introduct

ion

1

.2 Theoret

ica

lframework

1

.3 The Gaugesector

1

.4 The mattersector

1

.5 Afocusonthe CKM matr

ix

ρ

η

H

f

1

.6 L

im

itat

ionsandopenquest

ions