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Fundamental aspets of Flavor and Higgs bosonPhenomenology at Colliders from supersymmetri andnon-supersymmetri extensions of the Standard ModelDavid López ValDepartament d'Estrutura i Constituents de la MatèriaInstitut de Ciènies del CosmosUniversitat de BarelonaPhD Advisors:Prof. Dr. Joan Solà PeraaulaProf. Dr. Jaume Guash IngladaMay 2010

Fundamental aspets of Flavor and Higgs bosonPhenomenology at Colliders from supersymmetri andnon-supersymmetri extensions of the Standard ModelMemòria presentada perDavid López Valper optar al grau deDotor en FísiaPrograma O�ial de Postgrau en FísiaLínia de Reera: Físia de Partíules i GravitaióTrienni 2008-2010Departament d'Estrutura i Constituents de la MatèriaInstitut de Ciènies del CosmosUniversitat de BarelonaAquesta Tesi Dotoral ha estat dirigida perProf. Dr. Joan Solà PeraaulaCatedràti del Departament d'Estrutura i Constituents de la MatèriaUniversitat de Barelona;amb la odire

ió deProf. Dr. Jaume Guash IngladaProfessor Agregat del Departament de Físia FonamentalUniversitat de Barelona.Aquesta tesi va ser defensada el dia 28 de Juny de 2010 a la Sala de Graus Eduard Fontserè de la Faultatde Físia de la Universitat de Barelona davant del tribunal onstituït per:Prof. Dr. Franiso del Águila Jiménez(Departamento de Físia Teória y del Cosmos & CAFPE, Universidad de Granada)Prof. Dra. María José Herrero Solans(Departamento de Físia Teória, Universidad Autónoma de Madrid)Prof. Dr. Georg Weiglein(Deutshes Elektronen-Synhrotron, Hamburg)

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Brunz vorael bleix voraçde matinadaEmili Rosales, Els dies i TuEmprendo la tarea(imposible, si es que hay algo imposible)de raionalizar, interpretar,[reonstruir y desandaraquellas fábulas y hehizosque graias a ti fueron realidadJosé Hierro, Cuaderno de Nueva YorkA terra, sulle pietredella stalla 'è la plaenta,il sa

o vuoto della nostra attesaErri de Lu

a, In nome della madreHimmlisher, als jene blitzenden Sterne,dünken uns die unendlihen Augen,die die Naht in uns geö�netNovalis, Hymnen an die NahtOù suis-je ? Soyer fran ! Ne me déguisez rien !En quel lieu, dans quel site, viens-je de hoir,Monsieur, omme un aérolithe ?Edmond Rostand, Cyrano de Bergera

ResumEn aquesta Tesi hem estudiat diversos aspetes fenomenològis, tant en l'àmbit de la Físia de Saborom de la Físia dels bosons de Higgs més enllà del Model Estàndard, dins del ontext del LHC i delsfuturs ol.lisionadors lineals. Per una banda, ens hem onentrat en les intera

ions de besanvi de sabormitjançades pels gluinos, hargino/neutralinos i els bosons de Higgs arregats en el Model Estàndard Super-simètri Mínim (MSSM), així om en llur subseqüent impate en la produ

ió de parelles de quarks pesatselètriament neutres (tc, bs) en ol.lisions protó-protó. En els règims més favorables, les se

ions e�aesper a ambdós anals poden elevar-se �ns a l'entorn d'1 pb; la qual osa representa aproximadament 105su

essos per ada 100 fb-1 de lluminositat integrada i, en el as partiular del anal t, un fator ∼ 107 perdamunt de les predi

ions establertes pel Model Estàndard � fortament suprimides om a onseqüènia delmeanisme de Glashow-Iliopoulos-Maiani (GIM). Per una altra banda, hem analitzat el rol fenomenològidels autoaoblaments trilineals de bosons de Higgs en el model general (no supersimètri) de Dos Dobletsde Higgs (2HDM); hem mostrat om, de resultes de la seva intensitat, potenialment molt elevada, se'npoden derivar efetes molt signi�atius sobre diversos anals de produ

ió de bosons de Higgs en futursol.lisionadors lineals, tant a nivell d'arbre (per exemple, un inrement substanial de la se

ió e�aç perals proessos e+e− → 3H i e+e− → 2H + X �ns a ∼ 1 pb); om a través d'efetes quàntis (amb orre-ions radiatives de �ns a δr ∼ ±50% sobre les se

ions e�aes de proessos om ara e+e− → hA0, hZ0,on h ≡ h0,H0). En el MSSM, i donada la naturalesa purament gauge dels autoaoblaments trilineals, talsefetes no es manifesten. Hem identi�at, per tant, una varietat d'empremtes araterístiques (i altamentdistintives d'ambdós models) que podrien ser perfetament visibles en ol.lisionadors amb un rang d'energiade l'ordre del TeV i que, en as de ser mai detetades, onstituïrien no tan sols una sòlida evidènia de NovaFísia, sinó també uns observables fortament sensibles a l'estrutura del model subjaent.AbstratIn this Thesis we have studied several phenomenologial aspets of Flavor and Higgs boson Physisbeyond the Standard Model at the LHC and the future linear olliders. On the one hand we have on-entrated on the �avor-hanging interations mediated by gluinos, hargino/neutralinos and harged Higgsbosons in the Minimal Supersymmetri Standard Model (MSSM), and their impat on the prodution ofeletrially-neutral heavy-quark pairs (tc, bs) in proton-proton ollisions. In the most favorable regimes,the ross-setions for both hannels may furnish up to barely 1 pb; this implies O(105) events per 100fb-1 of integrated luminosity and, in the partiular ase of t, a fator ∼ 107 above the SM expetations �whih are largely suppressed as a result of the Glashow-Iliopoulos-Maiani (GIM) mehanism. On the otherhand, we have analysed the phenomenologial role of trilinear Higgs boson self-ouplings in the general(non-supersymmetri) Two-Higgs-Doublet Model (2HDM). We have shown that their potentially large en-hanements may stamp signi�ant imprints on several Higgs-boson prodution mehanisms at future linearolliders, both at the tree level (e.g. in the form of a sizable boost of the ross-setions orresponding to

e+e− → 3H and e+e− → 2H + X whih may limb up to ∼ 1 pb); and through quantum e�ets (withrelative radiative orretions up to δr ∼ ±50% for proesses like e+e−→ hA0, hZ0 with h ≡ h0,H0). In theMSSM, and owing to the purely gauge nature of the triple Higgs boson self-ouplings, none of the lattere�ets beome manifest. All in all, we have identi�ed a manifold of harateristi signatures (highly distin-tive of both models), whih ould be well feasible at the TeV-lass ollider failities and, if ever observed,might onstitute not only a solid evidene of New Physis, but also a sensitive probe to the struture of theunderlying model.

This Thesis is based on the following publiations:Researh artiles• [FGLVS08℄ G. Ferrera, J. Guash, D. López-Val and J. Solà, Triple Higgs boson prodution in thelinear ollider, Phys. Lett. B659 (2007) 297-307, arXiv:0707.3162 [hep-ph℄.• [LVGS07b℄ D. López-Val, J. Guash and J. Solà , Single top-quark prodution by diret supersymmetri�avor-hanging interations at the LHC, JHEP 0712 (2007) 054, arXiv:0710.0587 [hep-ph℄.• [BGLVS08℄ S. Béjar, J. Guash, D. López-Val and J. Solà, FCNC-indued heavy-quark events at theLHC from Supersymmetry, Phys. Lett. B668 (2008) 364-372, arXiv:0805.0973 [hep-ph℄.• [HLVS09℄ R. N. Hodgkinson, D. López-Val and J. Solà, Higgs-boson pair prodution through gauge bo-son fusion at linear olliders within the general 2HDM, Phys. Lett. B673 (2009) 47-56, arXiv:0901.2257[hep-ph℄.• [BLVS09℄ N. Bernal, D. López-Val and J. Solà, Single Higgs-boson prodution through gamma-gammasattering within the general 2HDM, Phys. Lett. B677 (2009) 39-47, arXiv:0903.4978 [hep-ph℄.• [LVS10a℄ D. López-Val and J. Solà, Neutral Higgs-pair prodution at Linear Colliders within thegeneral 2HDM: Quantum e�ets and triple Higgs boson self-interations, Phys. Rev. D81 (2010)033003, arXiv:0908.2898 [hep-ph℄.• [LVSB10℄ D. López-Val, J. Solà and N. Bernal, Quantum e�ets on Higgs-strahlung events at LinearColliders within the general 2HDM, arXiv:1003.4312 [hep-ph℄, Phys. Rev. D81 (2010) 113005.Conferene proeedings• [LVGS07a℄ D. López-Val, J. Guash and J. Solà , Prodution of single top-quark �nal states at the LHCfrom supersymmetri FCNC interations, Proeedings of Siene RADCOR 2007 042, arXiv:0710.0587[hep-ph℄.• [FGLVS07℄ G. Ferrera, J. Guash, D. López-Val and J. Solà , Triple Higgs boson prodution atthe ILC within a generi Two-Higgs-Doublet Model, Proeedings of Siene RADCOR 2007 043,arXiv:0801.3907 [hep-ph℄.• [LVS10b℄ D. López-Val and J. Solà , Neutral Higgs-pair Prodution at one-loop from a Generi 2HDM,Proeedings of Siene RADCOR 2009 045, arXiv:1001.0473 [hep-ph℄.• [SLV10℄ J. Solà and D. López-Val, Neutral Higgs boson pair prodution at Linear Colliders, Fortshr.Phys. 58 (2010) 660.

Contents1 Sumari 12 Introdution 7I Theoretial framework 113 Aspets of the Standard Model of Partile Physis 133.1 Gauge struture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Partile ontent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Flavor dynamis and the GIM mehanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 The SM Higgs mehanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 The Higgs mehanism and Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Hints for the existene of the Higgs boson . . . or something else? . . . . . . . . . . . . . . . . . 224 The Two-Higgs-Doublet Model 274.1 Higgs physis beyond the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.1 Custodial Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.2 Absene of tree-level FCNC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1.3 Unitarity bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 The 2HDM: Phenomenologial motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Building bloks and theoretial struture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Interations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.5 Theoretial and Phenomenologial onstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 The Minimal Supersymmetri Standard Model 415.1 A brief insight into Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.1.1 Fundamental onepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.1.2 Phenomenologial Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 The MSSM: onstrution and �eld ontent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.3 The MSSM Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4 Interations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.5 The partile spetrum of the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.5.1 The Higgs setor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.5.2 The SM setor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.5.3 The Chargino/neutralino setor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.5.4 The gluino setor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.5.5 The squark setor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.6 Theoretial and Phenomenologial onstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 55ix

6 Aspets of Renormalization 596.1 Regularization and Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.2 The On-shell Sheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.3 Renormalization of the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.3.1 Masses and �elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.3.2 Coupling onstants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.4 Renormalization of the Higgs setor within the 2HDM . . . . . . . . . . . . . . . . . . . . . . 666.5 Renormalization of the Higgs setor within the MSSM . . . . . . . . . . . . . . . . . . . . . . 756.6 Loop-orreted propagators and �nite WF normalization fators . . . . . . . . . . . . . . . . 766.7 The αeff approximation in the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78II Aspets of Flavor Physis 817 Supersymmetri Flavor Changing Neutral Currents at the LHC: general features 837.1 Phenomenologial overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.2 Formalism: FCNC interations in the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . 867.3 Relevant onstraints from low-energy Flavor Physis . . . . . . . . . . . . . . . . . . . . . . . 898 Single top-quark prodution by Strong and Eletroweak Supersymmetri Flavor Chang-ing interations at the LHC 938.1 Computation proedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938.2 Numerial analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.2.1 Standard Model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.2.2 SUSY-QCD ontributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.2.3 SUSY-EW ontributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.3 Disussion and onluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119 FCNC heavy-quark events through Supersymmetri Eletroweak Higgs boson deays 1159.1 Higgs boson deays into neutral heavy quark-pairs from SUSY-EW interations . . . . . . . . 1159.2 Framework for the numerial analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229.3 Numerial analysis of the bs prodution rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249.4 Numerial analysis of the tc prodution rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289.5 Disussion and onluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130III Aspets of Higgs boson Physis 13310 Leading-order results on double and triple Higgs boson prodution at Linear Colliders13510.1 Introdutory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13510.2 Higgs boson prodution at Linear Colliders: a onise review . . . . . . . . . . . . . . . . . . 13610.3 Exlusive Higgs boson-pair prodution at leading-order . . . . . . . . . . . . . . . . . . . . . . 13910.4 Triple Higgs boson prodution at leading-order . . . . . . . . . . . . . . . . . . . . . . . . . . 14310.5 Double Higgs boson prodution from weak gauge boson fusion . . . . . . . . . . . . . . . . . . 14710.5.1 A general desription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14710.5.2 Non-resonant double Higgs boson prodution . . . . . . . . . . . . . . . . . . . . . . . 14910.5.3 Resonant double Higgs boson prodution . . . . . . . . . . . . . . . . . . . . . . . . . 15210.6 Disussion and onluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15511 Quantum e�ets on the neutral Higgs boson-pair prodution within the general 2HDM15911.1 Introdutory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15911.2 Neutral Higgs boson-pair prodution at one loop: analytial details . . . . . . . . . . . . . . . 16011.3 Neutral Higgs-pair prodution at one loop: numerial analysis . . . . . . . . . . . . . . . . . . 16411.3.1 Computational setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16411.3.2 Light CP-even/ CP-odd hannel: e+e−→ h0A0 . . . . . . . . . . . . . . . . . . . . . . 16611.3.3 Heavy CP-even/ CP-odd hannel: e+e−→ H0A0 . . . . . . . . . . . . . . . . . . . . . 17711.4 Disussion and onluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

12 Quantum e�ets on Higgs-strahlung events from a generi 2HDM 19312.1 Introdutory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19312.2 Higgs-strahlung events at one loop: analytial details . . . . . . . . . . . . . . . . . . . . . . . 19412.3 Higgs-strahlung events at one loop: numerial analysis . . . . . . . . . . . . . . . . . . . . . . 19912.4 Disussion and onluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20613 Loop-indued single Higgs boson prodution through photon fusion within the general2HDM 21513.1 A glimpse at γγ olliders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21513.2 Loop-indued γγh interations within the 2HDM: general features and omputational setup . 21713.3 Single Higgs boson prodution in a γγ ollider . . . . . . . . . . . . . . . . . . . . . . . . . . . 21813.4 Single Higgs boson prodution through virtual γγ fusion . . . . . . . . . . . . . . . . . . . . . 22513.5 Non-standard gauge/Yukawa oupling e�ets . . . . . . . . . . . . . . . . . . . . . . . . . . . 22613.6 Disussion and onluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22814 Conlusions 23315 Outlook 237Appendix A: One-loop ounterterms for the general 2HDM 241Bibliography 249Agraïments 271

Herewith we list down the aronyms that we will most frequently use throughout this Thesis.• C.L. : Con�dene level• CLIC: Compat International Linear Collider• EW: Eletroweak• EWSB: Eletroweak Symmetry Breaking• FCNC: Flavor-hanging Neutral Current• GIM: Glashow-Iliopoulos-Maiani• GUT: Grand Uni�ation Theory• ILC: International Linear Collider• IR: Infrared• LH: Left-handed• LHC: Large Hadron Collider• lina: linear ollider (generi)• MSSM: Minimal Supersymmetri Standard Model• RH: Right-handed• SM: Standard Model• SSB: Spontaneous Symmetry Breaking• SUGRA: Supergravity• SUSY: Supersymmetry• UV: Ultraviolet• VEV: Vauum Expetation Value• WF: Wave funtion• 2HDM: Two-Higgs-Doublet Model• 2HX: Inlusive Higgs boson-pair prodution

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Capı́tol 1SumariConèixer la naturalesa dels onstituents fonamentals de la matèria, i omprendre les lleis que governenel seu omportament: aquesta és, tal vegada, la reera més ambiiosa i apassionant que l'ésser humà potemprendre. I, sense ap mena de dubte, una força motriu onstant en aquest desig tan nostre d'aprofundir enl'arquitetura més íntima del món que ens envolta. Aquesta reera ha esperonat molts esperits inquiets enuna llarga travessia (potser inaabable, qui sap); des del onepte primigeni d'tomo, enunyat originalmentper Demòrit d'Abdera entorn de l'any 450 a.C.; �ns a la desripió moderna que ens aporta la Físia dePartíules. Una desripió, aquesta darrera, ben plena d'intuïions brillants i de sòlides erteses, sens dubte,però no pas lliure de múltiples interrogants oberts. Uns interrogants que no han deixat de sembrar la rutaque travessa la vastitud d'esales de la matèria; això és, les esales de distània � o d'energia, si es prefereix� que separen la nostra mesura humana dels veritables on�ns de la petitesa. La dualitat ona-orpusleva il.lustrar la bitàora dels nostres navil·lis atuals: a

elerant partíules a energies su�ientment elevadespodem sondejar distànies ada vegada més petites. L'energia posa de relleu aquells graus de llibertat queno es manifesten en la realitzaió quotidiana del món físi més que d'una manera vaga (efetiva, en diem),a redós de multitud de simetries trenades, de realitzaions partiulars de tot l'espai de jo ofert per lateoria, motiu últim pel qual el nostre món, lluny de semblar-nos irreduïble i simètri, desplega vistosamentel paisatge de la omplexitat.L'anomenat Model Estàndard onstitueix, a dia d'avui, el paradigma teòri de la Físia de les PartíulesElementals. En el llenguatge modern de la teoria quàntia de amps, el Model Estàndard es pot formulard'una manera senzilla i elegant, en termes d'un prinipi d'invariània gauge sota el grup de Lie SUC(3) ⊗SUL(2)⊗ UY (1). En els darrers (i més que apassionants) trenta anys de reera en el amp de la Físia dePartíules s'ha explorat exhaustivament la fenomenologia a l'entorn de l'esala eletrofeble (MZ ∼ 90GeV),i la frontera del nostre oneixement s'ha situat ja al llindar del TeV. I tot això s'ha fet sense detetar(gairebé) ap disrepània signi�ativa respete de les predi

ions establertes pel Model Estàndard1, unateoria que ompta amb validaions experimentals d'una extraordinària exatitud. Tanmateix, ja de bellantuvi es va reonèixer que el Model Estàndard no onstituïa una teoria última de les intera

ions de laNatura, senzillament perquè no era apaç d'aomodar una desripió onsistent de la gravitaió. El ModelEstàndard, per tant, sabem que no ens proporiona una desripió �dedigna dels proessos físis que hande tenir llo a l'esala de Plank, MP ∼ 1019GeV, l'energia a partir del qual la naturalesa quàntia de lagravetat hauria de omençar a manifestar-se. En la terminologia atual assumim que el Model Estàndardés una realitzaió efetiva de baixes energies, vàlida �ns a una erta esala ultraviolada Λ, d'una teoria mésfonamental. De manera naïf podríem pensar que Λ ∼MP , la qual osa voldria dir que el Model Estàndardseria apaç de desriure tota la Físia del desert d'energies entre l'esala eletrofeble i l'esala de Plank,sense haver de donar ompte de nous graus de llibertat, nous nivells estruturals o noves intera

ions. Aixòés possible, per què no, però planteja algunes inomoditats teòriques (en partiular, problemes de jerarquies1És reoneguda l'existènia d'algunes tensions entre ertes mesures experimentals i les orresponents predi

ions teòriques;per exemple, el moment magnèti anòmal del muó, gµ − 2, el valor experimental del qual disrepa en aproximadament 3.4σ deles determinaions teòriques més aurades [A+08b℄. Per altra banda els fenòmens d'osil.laió del sabor dels neutrins obliguena dotar aquestes partíules d'una massa, la qual osa implia, neessàriament, haver d'extendre parialment la formulaiómínima del Model Estàndard. Cap d'aquestes irumstànies, tanmateix, posa en entredit la solidesa de la teoria.1

Sumari

Figura 1.1: Emplaçament del ol.lisionador LHC, al subsòl de Ginebra, Suïssa. Fotogra�a original de [htta℄.i de �ne-tuning); i, més enllà d'això, topa amb la mal�ança dels físis, po avesats al fet que la Natura deixiperdre 16 ordres de magnitud sense amagar-los-hi alguna sorpresa.Malgrat els seus formidables èxits, són enara molts els enigmes que romanen iresolts. Segurament elas més paradigmàti involura la naturalesa última del fenomen de Trenament Espontani de la SimetriaEletrofeble i de la orresponent generaió de la massa de les partíules. Per bé que l'anomenat meanis-me de Higgs aporta una elegant desripió d'aquest fenomen en el mar d'una teoria quàntia de ampspertorbativa, hom es veu forçat a postular l'existènia de, om a mínim, una partíula esalar fonamental(JP = 0+) que, a data d'avui, enara no s'ha on�rmat experimentalment. En aquest ontext, hom solfer al.lusió al prejudii teòri que les lleis fonamentals de la Natura no haurien de distingir la naturalesabosònia o fermiònia de les diferents partíules elementals, la qual osa empeny la hipòtesi que onsiderael Model Estàndard om la manifestaió de baixes energies d'una teoria subjaent, invariant sota transfor-maions de Supersimetria; no al dir que, amb l'adveniment de la nova generaió de grans ol.lisionadors,enapçalats pel Large Hadron Collider (LHC) al laboratori CERN (Ginebra) [htta℄, i que segueixen d'apropels projetes del International Linear Collider (ILC) [htt℄ o del Compat Linear Collider (CLIC) [hsS℄, lesperspetives són immillorables de ara a desvetllar noves laus que ens permetin seguir aprofundint en elsnombrosos enigmes que ens brinda la Físia de Partíules.Heus aquí el ontext en el qual s'emmara el treball que reollim en aquesta Tesi. Un treball amb elqual pretenem ontribuir a entendre millor erts fenòmens de la Físia de Partíules que són espeialmentaraterístis d'algunes de les més plausibles extensions pertorbatives del Model Estàndard. Més onreta-ment, ens entrarem en l'empremta deixada per sengles extensions supersimètriques i no-supersimètriquesdel Model Estàndard en els proessos de besanvi de sabor dels quarks pesats, per una banda; i en ladinàmia del setor de Higgs i de les seves intera

ions, per una altra.Els ontinguts d'aquesta Tesi s'artiulen bàsiament en tres grans blos. La primera Part s'enarregade revisar la fonamentaió teòria, mentre que les Parts II i III reullen els resultats originals del nostretreball de reera i fan referènia, respetivament, als àmbits de la Físia del Sabor i dels bosons de Higgsen ontextos més enllà del Model Estàndard. 2

Sumari

Figura 1.2: Representaió pitòria del futur oll.lisionador lineal ILC, a data d'avui enara en projete.Dibuix original de [httb℄.Obrim aquesta Tesi revisant alguns aspetes bàsis del Model Estàndard de la Físia de Partíules, posantèmfasi en aquells trets de la seva estrutura i de la seva fenomenologia que ontrasten amb els models extesossobre els quals treballarem en posteriors apítols. En partiular, parem esment en l'estrutura matemàtiade la teoria (en termes del prinipi d'invariània de gauge) i en els seus graus de llibertat. Tot seguit,explorem el meanisme de Glashow-Iliopoulos-Maiani, un element genuï del Model Estàndard i que dónaompte matemàtiament de la supressió d'aquells efetes que són regits per orrents neutres amb besanvide sabor (FCNC). Finalment, revisem també les motivaions físiques i la onstru

ió de l'anomenat setorde Higgs en el Model Estàndard bo i establint, d'aquesta manera, un mar de referènia per a la disussiódels setors de Higgs extesos que emprendrem més endavant.En el Capítol 4 revisem els fonaments teòris dels anomenats models de Doble Doblet de Higgs (del'anglès Two-Higgs-Doublet Models, o 2HDM) en la seva versió general (és a dir, no-supersimètria). Enprimer llo, introduïm una sèrie d'idees bàsiques sobre la Físia dels bosons de Higgs en ontextos més enllàdel Model Estàndard. En onret, donem una ullada a les restri

ions que, sobre aquestes teories exteses,imposen tant l'estrutura de sabor om les mesures de preisió amb què oneixem erts observables, perexemple els paràmetres eletrofebles. Similarment, introduïm les motivaions fenomenològiques que menenap a la onstru

ió del 2HDM i, seguidament, en desribim en detall la seva estrutura teòria, els seusonstituents i les seves intera

ions. Les restri

ions teòriques i experimentals que apliquen sobre el modelsón igualment disutides en profunditat.Una revisió similar es duu terme en el Capítol 5, aquest op fent referènia al Model Estàndard Super-simètri Mínim (MSSM). Un op havent de�nit formalment la Supersimetria i reunit algunes de les sevesmotivaions fenomenològiques més suggerents, en aquest apítol analitzem els diferents setors de partíu-les supersimètriques i les seves orresponents intera

ions, amb un èmfasi espeial en aquells aspetes quejugaran un paper espeialment signi�atiu en les posteriors anàlisis dutes a terme en aquesta Tesi. De lamateixa manera, les diferents restri

ions fenomenològiques es revisen auradament a la darrera part delapítol.El Capítol 6 es onentra en els aspetes de Renormalitzaió que jugaran un paper molt rellevant en ladarrera part de la Tesi. Un op revisades algunes idees i oneptes laus d'aquest àmbit, i ben establertstant el voabulari om la notaió adients, passem a desriure l'esquema de renormalizaió sobre la apa demasses (on-shell) que s'empra habitualment en els àluls de orre

ions radiatives en el setor eletrofebledel Model Estàndard. Seguidament, presentem un estudi exhaustiu de la renormalitzaió del setor de Higgs3

Sumarien els models 2HDM, inlosa una omparaió amb el orresponent paradigma supersimètri que de�neixel MSSM. Finalment, presentem també una dedu

ió detallada de les expressions que donen ompte delspropagadors renormalitzats de bosons de Higgs, així om dels orresponents fators de renormalitzaióde les seves funions d'ona; totes aquestes expressions són omunes, de fet, tant en el 2HDM om en elMSSM. Complementàriament, a l'Apèndix A inloem una llista dels ontratermes a 1 llaç orresponentsa les intera

ions genuïnes del 2HDM. Convé emfasitzar que, si bé la major part del ontingut d'aquestase

ió ha estat presentat, en una forma o una altra, en alguns treballs preedents disponibles a la literatura,aquesta és la primera vegada en què es duu a terme l'esforç de reunir tot aquest material i presentar-lo d'unamanera sistemàtia, amb una notaió uni�ada i un onjunt de onvenions ompletament autoonsistent.D'aquest punt ençà, omenem a presentar els resultats referits a l'anàlisi fenomenològia de diferentsproessos de produ

ió de partíules que, ja sigui en el ontext del LHC o bé dels futurs ol.lisionadorslineals, podrien ser rellevants de ara a desobrir (i, eventualment, poder aprofundir en) una possibledinàmia subjaent en l'estrutura oneguda del Model Estàndard. La Part II d'aquesta Tesi, sota eltítol de Aspetes de Físia del Sabor, engloba els Capítols 7, 8 i 9. En ells onsiderem la produ

ió deparelles de quarks pesats (tc, bs) a través d'intera

ions de tipus FCNC supersimètriques, om a exempleparadigmàti de om els efetes quàntis sobre proessos rars (en el sentit de fortament suprimits) sónapaços de omportar espetaulars inrements del nombre de su

essos que prediu el Model Estàndard, laqual osa podria traduir-se en evidènies signi�atives de Nova Físia. Més en onret, i després d'explorarels aspetes més formals de l'estrutura de sabor del MSSM, en el Capítol 8 presentem un estudi ompletde la produ

ió de parelles de quarks top i harm a través de proessos FCNC supersimètris diretes en elmar del ol.lisionador LHC. La se

ió e�aç total, σ(pp(gg)→ tc + tc), onsiderant la dispersió de gluonsom a anal dominant a nivell partòni, és alulada a 1 llaç (loop) en el MSSM més general. Aquest estudiextén els resultats obtinguts a partir dels efetes supersimètris a la intera

ió forta (SUSY-QCD), que vanser exposats primerament a la Ref. [GHPS06℄, i representa per tant el primer àlul omplet que inlouels efetes supersimètris eletrofebles (SUSY-EW). La nostra anàlisi de σ(pp(gg)→ tc + tc) en el MSSMomprèn igualment les importants restri

ions experimentals que es deriven dels observables de baixa energiaque involuren mesons B, tant en proessos de desintegraió radiativa (B → Xs γ) om també en proessosde mesla B0s −B0s. En els esenaris més favorables, les ontribuions SUSY-QCD poden donar llo a raonsde produ

ió de l'ordre de 105 su

essos per ada 100 fb-1 de lluminositat integrada. El nostre estudi posaaixí mateix de relleu l'existènia de regions en l'espai de paràmetres del MSSM on les orre

ions SUSY-EWesdevenen molt destaables. La qual osa representa un resultat ben rellevant, espeialment en esenarisen què, de manera simultània, els orresponents efetes SUSY-QCD es troben signi�ativament suprimits(e.g. en presènia de gluinos molt massius). En aquestes regions on les ontribuions SUSY-EW resultenafavorides, hom obté unes predi

ions per al nombre de su

essos ben apreiables, de l'ordre de 1000 peral mateix segment de lluminositat integrada. Atès que la produ

ió FCNC de quarks pesats de diferentssabors, om ara preisament tc o tc, és extremadament infreqüent en el Model Estàndard, la dete

ió d'unnombre signi�atiu d'aquests su

essos onstituïria una signatura de Nova Físia � de possible naturalesasupersimètria. De manera omplementària, en el Capítol 9 també analitzem un meanisme alternatiude produ

ió de parelles de quarks pesats a través de proessos no-estàndard de besanvi de sabor; aixòés, a través de la produ

ió, i la subseqüent desintegraió, d'un bosó de Higgs neutre h ≡ h0,H0,A0 delMSSM en parelles de quarks de diferents sabors, elètriament neutres, qq′ ≡ tc, bs. Aquesta anàlisi exténtambé alguns resultats prèviament onsiderats a la literatura (f. Ref. [BGS05,BGS06℄). En el nostre as,tanmateix, ens onentrem en la ontribuió SUSY-EW sobre el proés pp → h → qq′ i ompletem, aixímateix, els resultats del apítol anterior tot inloent els efetes SUSY-QCD i SUSY-EW sobre el anal deprodu

ió direta pp(gg) → bs + bs. D'aquesta manera portem a terme, per primer op a la literatura,un estudi omprensiu de les predi

ions MSSM per als ritmes de produ

ió d'estats �nals tc, bs induïtsper transiions de sabor no-estàndard. Els resultats evidenien una gran omplementarietat entre aquestsmeanismes, això és, entre la produ

ió direta i la mitjançada per la desintegraió d'un bosó de Higgs, idibuixen un ampli mapa de regions d'interès en l'espai de paràmetres del MSSM, en què aquests proessoses veuen molt afavorits i despleguen tot el seu potenial interès des del punt de vista teòri. La qual osa,en darrera instània, a

entua el seu aràter distintiu en omparaió amb la dinàmia inherent en el ModelEstàndard.Finalment, a la Part III desplaem la nostra atenió envers el amp de la Físia dels bosons de Higgs.Aquest serà el tema entral dels apítols restants de la tesi, al llarg dels quals disutirem diversos proessos4

Sumaride produ

ió de bosons de Higgs que podrien ser altament distintius d'una estrutura extesa del setorde Higgs, amb dos doblets de Higgs no-supersimètris. Primerament, en aquest Capítol 10 onsiderem laprodu

ió simultània de tres bosons de Higgs, tant en el Model de Doble Doblet de Higgs general (2HDM)om en el MSSM, en el ontext dels futurs ol.lisionadors lineals e+e−. En alulem les se

ions e�aes pera les diferents possibles ombinaions d'estats �nals, om ara H+ H− h0, H0 A0 h0, et, a l'ordre O(α3ew) iels omparem amb els proessos més tradiionals de produ

ió de parelles de bosons de Higgs. Mentre queles se

ions e�aes de produ

ió de parells de Higgs es mantenen dins del mateix ordre de magnitud tanten el 2HDM om en el MSSM, els nostres àluls posen de manifest que les se

ions e�aes de produ

iótres bosons de Higgs simultanis, amb valors màxims de l'ordre de 0.1 pb, se situen molt per sobre de lesseves equivalents en el MSSM, les quals no sobrepassen, típiament, el llindar dels 10−5 pb. I, naturalment,sempre d'aord amb les severes restri

ions d'unitarietat, que aoten la mida màxima dels autoaoblamentsde tres (3H) i quatre (4H) bosons de Higgs. De fet, en el 2HDM els proessos de produ

ió simultàniade tres bosons de Higgs podrien onstituïr un meanisme plenament ompetitiu, almenys per a energiesde entre de masses a l'entorn de 1 TeV. En darrera instània, l'origen d'aquest inrement tan remarabledels anals de tres bosons de Higgs en el as del 2HDM (indistintament dels detalls del model; és a dir,tant en el as de models de Tipus I om de Tipus II) rau en l'estrutura dels autoaoblaments 3H. Aquestsautoaoblaments, a diferènia del as supersimètri, no vénen �xats per la simetria de gauge, sinó per unaombinaió de masses, angles de mesla i autointera

ions, de resultes de la qual la seva intensitat pot sermolt elevada � tan sols limitada, a la pràtia, per les ondiions d'unitarietat.El medi extraordinàriament �net� que arateritza els ol.lisionadors lineals, lliures del senyal de fonsde QCD que és inherent en qualsevol màquina hadrònia (om és el as de l'LHC) hauria de permetreuna identi�aió e�aç d'aquest tipus de su

essos, la signatura predominant dels quals seria en formade 6 jets de quarks pesats. En as de ser observades, i atès que en el MSSM aquests proessos estanàmpliament desafavorits, les menionades empremtes onstituïrien una sòlida evidènia de l'existènia d'unsetor de Higgs extès de naturalesa no supersimètria. En un ontext similar, en aquest apítol presentemseguidament el àlul de les se

ions e�aes a l'ordre O(α4ew) per als proessos inlusius de produ

ió deparelles de bosons de Higgs, aompanyats de leptons, a través d'un meanisme de fusió de bosons de gauge,e+e− → V ∗V ∗ → 2H + X (V = W±, Z), on denotem H = h0,H0,A0,H±. Aquest tipus de proessos sóntambé molt sensibles als autoaoblaments trilineals de bosons de Higgs i, talment om en l'exemple anterior,poden jugar un paper important en la mesura d'aquests aoblaments i, per tant, en la reonstru

ió delpotenial de Higgs. A més, i donada la peuliar estrutura inemàtia inherent a aquests anals de fusió� per als quals les se

ions e�aes exhibeixen una dependènia logarítmia amb l'energia de entre demasses ∼ log(s/M2) � el nombre de su

essos predits esdevé partiularment ompetitiu en el segment mésalt d'energies. De fet, ja per a una energia de entre de masses de √s = 1 TeV, els esenaris més favorablesomporten se

ions e�aes de �ns a 1 pb, la qual osa implia de l'ordre de 105 su

essos per 500 fb-1 delluminositat integrada. En omparaió amb altres possibles meanismes de produ

ió de múltiples bosonsde Higgs en ol.lisionadors lineals, onloiem que la produ

ió inlusiva de parelles de bosons de Higgs através de la fusió de bosons de gauge pot onstituir la seva signatura dominant a energies per sobre de 1TeVi en una àmplia regió de l'espai de paràmetres del 2HDM, sense que una situaió equivalent pugui donar-seen el MSSM.El nostre següent pas és onentrar-nos en els efetes quàntis sobre la produ

ió de parelles de bosonsde Higgs neutres (h0A0, H0A0), de nou en el mar dels ol.lisionadors lineals, i dins dels models 2HDM. En elCapítol 11 duem a terme una anàlisi exhaustiva de les se

ions e�aes predites a 1 llaç, inloent el onjuntomplet de ontribuions a l'ordre O(α3ew) juntament amb les pees dominants a l'ordre O(α4ew). Els nostresresultats il.lustren l'existènia de regions en l'espai de paràmetres del 2HDM, bàsiament per a valors detanβ ∼ 1 i λ5 ∼ O(10) (amb λ5 < 0), on les orre

ions radiatives a les se

ions e�aes de produ

ióde parelles de bosons de Higgs, σ(e+e− → A0h0/A0H0), poden assolir omfortablement valors relatius de|δσ|/σ ∼ 50%. Aquest omportament es pot atribuir a la intensitat que els autoaoblaments trilineals debosons de Higgs poden assolir � limitats tan sols, reordem-ho, per les restri

ions d'unitarietat, la qual osaexpressa una propietat genuïna del 2HDM, ompletament difereniada dels ontextos supersimètris (om elas del MSSM). Les orre

ions depenen fortament del valor de λ5, mentre que no ho fan signi�ativamentde les masses dels bosons de Higgs i, enara menys, de la manera espeí�a om s'aoblen els bosons de Higgsi els quarks � això és, si s'empra un model 2HDM de tipus I o de tipus II. Cal destaar també om aquestsefetes quàntis són positius per a √s ≃ 500GeV, la qual osa es tradueix en un inrement signi�atiu delnombre de su

essos, preisament a l'entorn del valor �duial de l'energia de entre de masses iniial del5

Sumarifutur ILC. El nombre total de su

essos pot ser molt destaable, assolint típiament les diverses desenes defemtobarns o, en altres paraules, diversos milers de parelles de bosons de Higgs produïdes per ada 500 fb−1de lluminositat integrada. Per ontra, les orre

ions són negatives en el rang més alt d'energies (viz.√s ∼ 1TeV i per damunt d'aquest valor), manifestant-se, per tant, en forma d'una apreiable supressió delvalor de la se

ió e�aç respete de la predi

ió establerta a nivell làssi, i.e. a l'ordre O(α3ew). En resum,l'estudi detallat d'aquest anals de produ

ió de parells de bosons de Higgs neutres són molt sensibles, anivell quànti, als detalls de l'estrutura del model físi subjaent, i poden dons dur assoiades empremtesaltament distintives d'una setor de Higgs extès de tipus no-supersimètri. Finalment, de ara a il.lustrarl'abast d'aquests efetes en un ontext més ampli, establim una omparaió entre aquests anals exlusiusde produ

ió de parelles i els proessos e+e− → 2H +X i e+e− → 3H estudiats en el apítol anterior, totposant de relleu una gran omplementarietat entre aquests diferents meanismes, adasun dels quals es veuespeialment afavorit en diferents segments d'energia de entre de masses.El Capítol 12 es mou en un ontext molt proper; a saber, la produ

ió assoiada de bosons de Higgsneutres juntament amb bosons Z0 (h0Z0, H0Z0), també oneguts om a anals de radiaió de bosons deHiggs (Higgs-strahlung). Aquests proessos són de nou investigats en ol.lisionadors lineals, a 1 llaç, i dinsdel 2HDM. Els resultats posen de manifest, en primer llo, que les orre

ions a la funió d'ona dels ampsde Higgs externs són la font dominant d'efetes quàntis sobre aquest proés i esdevenen, al seu torn,numèriament molt remarables (i sempre negatives) � això és, δ σ/σ ∼ −20% / − 50%, independentmentde l'interval de √s onsiderat, i entrades predominantment (om en el as anterior) en les regions amb

tanβ ∼ 1 i λ5 moderada (sempre prenent λ5 < 0). Aquest omportament, un op més, es relaiona ambla intensitat dels autoaoblaments trilineals i mostra, de nou, el seu aràter altament distintiu quan elomparem amb proessos equivalents en models de naturalesa supersimètria. Per altra banda, i malgratque el signe negatiu d'aquestes orre

ions omporta una notable supressió de les orresponents se

ionse�aes, el nombre total de su

essos de Higgs-strahlung que predim es manté al nivell de les desenes defemtobarn, això és, de l'ordre de ∼ 103−104 su

essos per 500 fb-1. Donada la seva gran omplementarietat,argumentem om l'estudi ombinat dels anals de Higgs-strahlung (h0Z0, H0Z0) i de parelles de Higgs (h0A0,H0A0) pot jugar un paper molt rellevant en l'estudi de l'estrutura del setor de Higgs en els futurs ol-lisionadors lineals.La produ

ió d'un sol bosó de Higgs neutre h a través de ol.lisions γγ en el 2HDM és explorada �nalmenten el Capítol 13. Aquest proés, que neessàriament ha de tenir llo induït per efetes quàntis, s'analitzaa través de dos meanismes alternatius: per una banda, el proés γγ → h de dispersió de dos fotonsreals en un ol.lisionador fotó-fotó; per una altra, el meanisme més tradiional de fusió de fotons virtuals,e+e− → e+e−γ∗γ∗ → e+e− + h. Atès el tamany potenial dels autoaoblaments trilineals entre bosons deHiggs en el 2HDM, trobem que el nombre de su

essos pot experimentar un notable inrement respeted'aquest mateix proés en el Model Estàndard, en tant en quant els bosons de Higgs arregats no siguinexessivament massius. Per exemple, si MH± & 100GeV i, a més a més, Mh0 pren valors entre ∼ 115 i 200GeV, les se

ions e�aes se situen típiament al nivell de σγγ→h ∼ 0.1−1 pb i σ(e+e−→ e+e−h0) . 0.01 pb� en ambdós asos �ns a un fator d'ordre 10 per damunt de les predi

ions que resulten d'aquest mateixproés en el SM. Per altre antó, i malgrat que per a masses del bosó de Higgs arregat de MH± & 300GeVel nombre de su

essos esdevé virtualment insensible als efetes bastits pels autoaoblaments trilineals, hiresta una ontribuió romanent d'efetes no-estàndard força signi�atius, nasuts de la ombinaió delsaoblaments de Yukawa dels bosons de Higgs amb fermions, així om dels efetes quàntis deguts als bosonsde gauge, i que podrien ser igualment reveladors de l'existènia de Nova Físia.La Tesi es lou en el Capítol 14 amb la reopil.laió dels resultats fonamentals que hem obtingut i el sumaride les onlusions més signi�atives, així om amb una breu perspetiva d'algunes possibles ontinuaionsd'aquest treball.

6

Chapter 2IntrodutionUnveiling the nature of the elementary onstituents of Matter and the ultimate laws that govern theirbehavior: this is perhaps the most ambitious and passioning enterprise that human beings an fae. And,ertainly, a onstant driving fore in the quest of mankind to unearth the intimate arhiteture of the worldsurrounding us. A quest that has lead many restless minds through a long-term (maybe endless) pathwayfrom the seminal onept of tomo � say, the unuttable, the indivisible partiles of matter �, whih was�rst oined by Demoritus of Abdera around the year 450 BCE; up to the ontemporary piture displayedby modern Partile Physis. The latter being endowed with its bright intuitions and solid ertainties, forsure, but also with its many open hallenges.For more than 40 years, the Standard Model (SM) of Strong and Eletroweak interations has furnisheda su

essful arena in whih to desribe the physis of Elementary Partiles. These fundamental intera-tions are understood as emerging from the exhange of spin-one bosoni fore arriers among the spin 1/2fermioni building-bloks that assemble the ordinary matter. In the modern language of Quantum FieldTheory, the above piture an be enapsulated into an elegant and simple formulation, via the priniple ofgauge invariane under the symmetry group SUc(3) ⊗ SUL(2) ⊗ UY (1). In the last (and more than fran-ti) thirty years of researh in this �eld, an exhaustive survey of the Phenomenology at the harateristiEletroweak energies (MZ ∼ 90GeV) has been undertaken, and the urrent frontier of our knowledge havebeen pushed up to the ∼ TeV range. Most signi�antly, (almost) no remarkable disrepanies have beenspotlighted between the theoretial preditions supplied by the SM and the experimental outomes 1: as amatter of fat, the latter is one of the best-tested models of ontemporary Physis. This su

essful areernotwithstanding, it was admitted from the very beginning that the SM ould not furnish a �nal theory of thefundamental interations, simply beause it annot embody a onsistent desription of gravitation. Henewe know that the SM does not provide a faithful portrait of the physial proesses taking plae at the Planksale MP ∼ 1019GeV, wherein the quantum features of gravity should emerge. In the urrent terminologyone assumes that the SM is an e�etive low-energy realization, presumably valid up to a ertain ultravioletuto� Λ, of a more fundamental theory. There is huge speulation about the prospet of suh a theory. Inpartiular, it is hoped that we will one day have a �Theory of Everything� whih will desribe all four foresof nature - eletromagneti, weak, strong and gravity. Naively, one might be tempted to think that perhaps

Λ ∼ MP , whih would mean that the SM enables to desribe the so-alled desert of energies between theEW sale and the Plank sale, with no need of introduing further degrees of freedom, deeper levels ofstruture or new interations. This is absolutely possible, though it involves a number of theoretial inom-modities (in partiular, hierarhy and �ne-tuning problems); and, most signi�antly, it onfronts the unbeliefof physiists, who are not used to Nature skipping 16 orders of magnitude without hidding any surprise.1The existene of tensions between ertain experimental measurements and the assoiated theoretial preditions is wellaknowledged; this is e.g. the ase of the the anomalous magneti moment of the muon, gµ − 2, whose experimental valuedeparts roughly 3.4σ with respet to the most a

urate theoretial determinations from the SM [A+08b℄. In another vein, thephenomenon of neutrino osillations enfores to partially extend the minimal SM so that it may ontain massive neutrinos.In any ase, none of these onerns do seriously question the robustness of the theory.7

IntrodutionIn spite of its formidable ahievements, therefore, a number of longstanding hallenges are still to beresolved. But perhaps the most paradigmati one onerns the ultimate nature of the Eletroweak SymmetryBreaking (EWSB) phenomenon and the related problem of the generation of the masses. Even though theHiggs mehanism provides an elegant desription of EWSB within a perturbative quantum �eld theoryframework, one is fored to postulate the existene of (at least) one salar (JP = 0+) fundamental building-blok of Nature, whose experimental on�rmation is onspiuously missing for the time being. In thisontext, it might be onjetured that the fundamental laws of Nature should exhibit a symmetry betweenfermioni and bosoni degrees of freedom � that is, a supersymmetry. Needless to say that the upominggeneration of TeV-lass olliders, headed by the startup of the operations at the Large Hadron Collider(LHC) at CERN, and followed by the ILC and the CLIC projets, will supply an invaluable oportunity todig deeper than ever into the many longstanding puzzles that remain unsettled in our present omprehensionof Partile Physis.This Thesis aims at ontributing to a better understanding of some partile prodution proesses whihare partiularly well-suited for the (momentous) task of probing New Physis senarios. Our analysiswill onern seleted aspets of Flavor and Higgs boson phenomenology within supersymmetri and non-supersymmetri extensions of the Standard Model, both at present and future olliders. More spei�ally,we shall be devoted to the following threefold endeavor:• Unraveling potentially visible signals of rare proesses from New Physis models, whih are preditedto be virtually absent within the SM ontext;• Spotlighting sizable boosts (or suppressions) from the expeted prodution rates of ertain proesses,either at the lassial or at the quantum level;• Identifying distintive behaviors that ould help to disentangle supersymmetri models from nonsuper-symmetri ones.The Thesis is organized in three main bloks of ontents. In Part I we mainly aim at surveying seletedaspets of the theoretial framework, whih are of relevane for the alulations that we disuss thereafter.Parts II and III, in turn, are devoted to present the original results of our researh work in the domains ofFlavor and Higgs boson physis, respetively.The Thesis opens with a review of some of the basi elements of the Standard Model of Partile Physis,with a speial emphasize in those aspets whih are more di�erentiative of the latter with respet to theextended models on whih we shall fous later on. In partiular, we shall dwell on the mathematialformulation of the SM (in terms of the gauge priniple) and its degrees of freedom. Besides, we shallalso review the Glashow-Iliopoulos-Maiani mehanism, a hallmark phenomenologial property of the SM,that a

ounts for the suppression of those e�ets whih are driven by Flavor-Changing Neutral Currents(FCNC). And �nally we will survey the phenomenologial motivations and the theoretial struture of theHiggs setor in the SM, so that we settle a proper benhmark for the analyses of more ompliated Higgssetors that we shall undertake further on.In Chapter 4 we review the theoretial settings of the general (non-supersymmetri) Two-Higgs-DoubletModel (2HDM). We begin by introduing some basi ideas about Higgs Physis beyond the SM, namelythe fundamental onstraints that emerge from the known �avor struture and the EW preision data; wethen introdue the phenomenologial motivations that lead to the 2HDM, and desribe in detail its buildingbloks, its theoretial struture and its pattern of interations.A similar review is provided in Chapter 5, now in regard to the Minimal Supersymmetri StandardModel (MSSM). After brie�y addressing the formal de�nition of Supersymmetry and one we have gatheredsome of its most suggestive phenomenologial motivations, we analyse the di�erent partile setors of themodel and the orresponding pattern of interations, with partiular emphasis in those whih shall play arelevant role at further stages of this Thesis. The assoiated onstraints are also reviewed in the last partof the Chapter.Chapter 6 is devoted to seleted aspets of Renormalization. To begin with, we settle a suitable groundby reviewing some elementary underlying onepts. We then move to the desription of the on-shell sheme,8

Introdutionwhih is usually employed for loop alulations in the Eletroweak setor of the SM. At this point we presentan exhaustive treatment of the renormalization of the Higgs setor within the 2HDM. A omparison withthe renormalization of the MSSM is also addressed. Finally, we also derive with some detail the expressionsfor the loop-orreted Higgs boson propagators and the �nite wave-funtion normalization fators, whihare ommon in both the MSSM and the 2HDM. Complementarily, in Appendix A we provide a ompletelist of the one-loop ounterterms for the genuine 2HDM interations. Although most part of the materialin this Setion has been presented elsewhere, this is perhaps the �rst time that it is gathered all togetherand further elaborated within a uni�ed set of onventions.One this extensive introdutory ontents are overed, from Chapter 7 onwards we onentrate on theanalysis of seleted phenomenologial aspets of Flavor (Part II) and Higgs boson physis (Part III) at thepresent and future ollider failities. More spei�ally, we shall fous on the genuine dynamial featuresgrounded on both the MSSM (as a perturbative, supersymmetri extension of the SM) and the 2HDM(as a perturbative, non-supersymmetri extension) and their potential impat on a number of produtionproesses involving i) heavy-quark; and ii) Higgs-boson �nal states. We unveil highly distintive phenomeno-logial features whih, at the end of the day, might enable to i) eluidate the existene of physis beyond theSM; and ii) disentangle the preise nature of the underlying non-standard dynamis, viz. to di�erentiatesupersymmetri from non-supersymmetri settings.Herewith we quote the omplete list of proesses that we address within this Thesis:i) In Chapter 8: the FCNC-indued heavy-quark pair prodution, pp(gg) → tc, bs (tc = tc, tc ; bs =bs, bs), driven by diret �avor-hanging interations within the MSSM, at the leading-order SUSY-QCD O(α4s) and SUSY-EW O(α2s α2ew).ii) In Chapter 9: the FCNC-indued heavy-quark pair prodution, pp→ h→ tc, bs, at the leading-orderO(α2s α2ew) within the MSSM.iii) In Chapter 10: the triple Higgs-boson prodution e+e− → 3H, at the leading order O(α3ew). Theavailable �nal states here are hH+H−, h0H0A0, hhA0, with h ≡ h0,H0,A0.iv) In Chapter 10: the inlusive Higgs boson pair prodution via gauge boson fusion at the leading orderO(α4ew), e+e− → V ∗V ∗ → 2H +X , where X ≡ e+e−, ν̄e νe and V = W±,Z0.v) In Chapter 11: the exlusive neutral Higgs boson pair prodution, e+e− → 2h, at order O(α3ew) andleading O(α4ew).vi) In Chapter 12: the so-alled Higgs-strahlung hannels, e+e− → hZ0, at order O(α3ew) and leadingO(α4ew).vii) In Chapter 13: the single Higgs boson prodution through the sattering of a real photon pair, γγ → hat the leading order O(α3ew); and a virtual one, e+e−→ γ∗γ∗ → h, at the leading order O(α5ew).Finally, we lose our Thesis by summarizing our main ahievements in Chapter 14, leaving Chapter 15to delineate some future perspetives of the present work.

9

Introdution

10

Part I

Theoretical framework

11

Chapter 3Aspets of the Standard Model of PartilePhysisThe so-alled Standard Model of Strong and Eletroweak interations is the urrent paradigm of PartilePhysis. It enables an a

urate understanding of (almost) all observed phenomena involving the knownElementary Partiles. Furthermore, it provides an amazing degree of a

ordane between preditions andexperiments. The model ombines some of the brightest onstrutions in Theoretial Physis of the last halfentury. On the one hand, the Glashow-Weinberg-Salam Eletroweak theory of the 1960s, whih desribes ina uni�ed manner the Eletromagneti and Weak interations between quarks and leptons [Gla61,Wei67,Sal℄.On the other hand, the strong interations of quarks and gluons are haraterized by means of QuantumChromodynamis (QCD) [GM64,Zwe,FGML73,Pol73℄, whose theoretial orpus was ereted in the 1970'supon the works of Gell-Mann, Gross, Wilzek and Politzer, among others. At this point we are yet to inludea third ingredient � the only one with no experimental on�rmation so far. That is, the mehanism by whihthe EW symmetry breaks down spontaneously and gives masses to all Elementary Partiles without spoilingthe gauge invariane of the theory. In the SM framework, this is ahieved by inluding a salar doublet�eld whih develops a non-zero vauum expetation value [Hig64b,Hig64a,GHK64,EB64,Kib67℄. In theSM we are thus requested to postulate the existene of an additional spinless partile, alled the Higgsboson. The only one remaining interation, this is gravitation, lies beyond the sope of the SM. Atually,a onsistent quantum �eld theoretial formulation of gravity is yet to be onstruted. Notwithstanding, asthe enter-of-mass energies used at present or at planned future olliders lie at ∼ 10TeV at the topmost,the e�ets due to gravitational interations should be by far negligible1. Therefore, the SM provides anexellent playground for a preise quantitative desription of the typial physial proesses taking plae atolliders.In this hapter we shall brie�y review some fundamental aspets of the SM, whih are of partiularinterest for us in view of the problems that we shall address further on in this Thesis. Standard textbooks[CL84,PS95,BM07℄ and pedagogial reviews [Hol06,Wil04,Lan09℄ are suggested to the interested reader fora omprehensive bakground material.

3.1 Gauge structureThe SM is based on the idea of symmetries. In partiular, it assumes that the �elds desribing its elementaryonstituents are arranged in representations of symmetry groups and that the orresponding ation of thetheory is invariant under the respetive symmetry transformations. More exatly, the omplete symmetrypattern of the SM onsists of i) an outer symmetry, the Poinaré group of spaetime transformations; and ii)an internal non-abelian symmetry group, whih is obtained through the produt SUC(3)⊗SUL(2)⊗UY (1).1A word of aution is mandatory: the gravitational e�ets will atually be irrelevant for ollider physis unless the existeneof extra dimensions e�etively pulls the Plank sale down to values within the TeV range.13

Aspets of the Standard Model of Partile PhysisIn partiular, SUC(3) is the olor gauge group and a

ounts for the strong interations among quarks andgluons in the framework of QCD. The produt SUL(2)⊗ UY (1), in turn, provides a desription of the EWinterations and, via the Higgs mehanism, it spontaneously breaks down to the abelian U(1) group thatharaterizes the quantum theory of Eletromagneti interations, that is QED.The gauge priniple enapsulates the very ore of the SM: in short, it assumes that the whole theoryis invariant under SUC(3) ⊗ SUL(2) ⊗ UY (1) transformations whih an be performed independently ateah point in spae and time. Sine simple derivative terms in the lassial Lagrangian involving salar,fermion and vetor �elds are not invariant under the foresaid transformations, one needs to introdue agauge ovariant derivative whih, in our onventions, shall be denoted as:Dµ ≡ ∂µ + i gsGaµ Fa + i g2 Waµ τa + i g1Bµ

Y

2. (3.1)Here Fa, τa and Y stand for the generators of the respetive symmetry groups SUC(3), SUL(2) and

UY (1). In partiular, τa = σa/2, where σa denote the standard Pauli matries ful�lling the orrespondingLie algebra [σa σb] = i ǫabc σc; ǫabc stands for the fully antisymmetri tensor. On the other hand Gaµ,Waµ andBµ are the so-alled gauge �elds, whih assess the right transformation properties of the ovariant derivativeating on a ertain multiplet. Finally, we have g3, g2 and g1 as the oupling onstants of the three di�erentinterations desribed by eah of the gauge groups.Let us onentrate on the EW setor of the theory. We notie that eah of the generalized harges,namely the isospin harges T a, a = 1 . . . 3 and the hyperharge Y , is assoiated with a vetor �eld: a tripletof vetor �elds W aµ a = 1 . . . 3, and a singlet �eld Bµ respetively. In order to generate suitable kineti termsfor the EW gauge boson �elds, one introdues the following �eld-strength tensors:

Waµν = ∂µWaν − ∂ν Waµ + g2 ǫabcWµ bWν c

Bµν = ∂µBν − ∂ν Bµ, (3.2)wherefrom the following gauge-�eld Lagrangian ensues:Lgauge = −

1

4WaµνW

µνa −

1

4Bµν B

µν , (3.3)This Lagrangian is manifestly gauge invariant under SUL(2) ⊗ UY (1) symmetry transformations. In turn,the gauge interations of the fermion �elds an be easily worked out by means of the gauge ovariantderivative (3.1), and generially yieldLfermion = i ψ γµDµ ψ. (3.4)

3.2 Particle contentThe fermioni setor of the SM onsists of spin-1/2 leptons (νe, νµ, ντ , e, µ, τ) and quarks (u, d, s, c, b, t)whih are arranged in a 3-generation struture. While the former are insensitive to strong interations, thelatter have a non-trivial SUC(3) harge (olor), and hene interat via gluon exhange. The partiles ofeah generation share ommon quantum numbers, albeit they have di�erent ouplings to the Higgs �eld,as we shall present later on. These fermioni building-bloks of Nature are seen to transform under thefundamental representation of SUC(3)⊗SUL(2)⊗UY (1), and may be lassi�ed by their weak isospin (T, T3)and their hyperharge (Y ). A Gell-Mann-Nishijima relation ties these basi quantum numbers to the (morefamiliar) eletri harge,Q = T3 +

Y

2, (3.5)in whih Q stems from the onserved harge of the U(1) symmetry group of Eletromagnetism.Most partiularly, left-handed fermions transform as SUL(2) doublets, where the upper omponentsorrespond to the neutrinos (νe, νµ, ντ ) and up-like quarks (u, c, t) respetively, while the lower omponentsare assoiated to the eletron-type leptons (e, µ, τ) and down-like quarks (d, s, b). In turn, right-handed14

3.3 Flavor dynamis and the GIM mehanismSUC(3) SUL(2) UY (1)

lL i

(νeL

eL

) (νµL

µL

) (ντ L

τL

) 1 2 -1eRi eR µR τR 1 1 -2qL i

(uL

d′L

) (cL

s′L

) (tL

b′L

) 3 2 13uRi uR cR tR 3 1 43dR i dR sR bR 3 1 − 23Table 3.1: Fermioni ontent of the SM and respetive behavior under the symmetry transformations ofthe orresponding gauge groups.fermions transform as SUL(2) singlets, with the only proviso that (massless) right-handed neutrinos areabsent in the SM framework. In addition to that the quarks � as olored partiles � settle in the fundamentalrepresentation of SUC(3) in olor spae. In Table 3.1 we summarize the fermion ontent of the SM andthe way these fermions behave under the di�erent symmetry transformations. Conerning the gauge setor,the spin-1 gauge bosons assoiated to eah group generator transform under the adjoint representationof that group. Consequently there are eight SUC(3) gauge bosons (the gluons), three SUL(2) bosons(W aµ , a = 1, 2, 3) and a single UY (1) boson, whih we all Bµ.So far so good, but experiments show that not all gauge bosons are massless. On the other hand, addingexpliit mass terms for these gauge bosons would break the gauge invariane of the SM. In Setion 3.4 wewill further elaborate on this most preeminent aspet.

3.3 Flavor dynamics and the GIM mechanismFlavors are repliations of states with idential quantum numbers [GP10℄. Flavor-hanging proesses on-stitute a hallmark phenomenologial feature of the SM and turn out to be of paramount importane in theanalysis of potential hints of New Physis. The �avor dynamis of the SM, this is to say, the pattern ofallowed transitions between partiles of di�erent �avors and the quantitative signi�ane of eah of thesetransition hannels, is ompletely settled in the Glashow-Weinberg-Salam theory of the EW interations,one the latter is onveniently supplemented with the 3-generation struture. The latter implies, amongother subtleties, an expliit generation mixing in the quark setor whih is parametrized in terms of theCabbibo-Kobaiashi-Maskawa (CKM) matrix [Cab63,KM73℄. Its origin an be traed bak to the diagonal-ization of the quark mass matries from the Higgs-fermion Yukawa ouplings that generate their respetivemass terms; the reason for this misalignment is that quarks with the same quantum numbers have di�erentmasses. Noteworthy, as long as one onsiders massless neutrinos, no suh mixing is present in the leptonisetor.To ease the disussion on the peuliar features of the �avor dynamis in the SM, it will be illustrativeto onsider the following e�etive formulation of the EW theory. Let us write down an e�etive Lagrangianfor the EW interations as:LEW = LCC + LNC , (3.6)where LCC denotes the (�avor-hanging) harged urrents (to wit, the proesses mediated by the interhangeof W± bosons); and LNC traks down the neutral urrents (the latter being triggered by the interhangeof a Z0 boson). In this e�etive language, an EW proess an be simply understood as the oupling of twoneutral urrents (J0µ) or two harged urrents (J±µ ), in suh a way that we an write:LCC ∼ Jµ +J−µ (3.7)LNC ∼ Jµ 0J0µ. (3.8)15

Aspets of the Standard Model of Partile PhysisLet us restrit ourselves for the moment to the light quark setor (u, d, s). In this simpli�ed setup ofthree light quarks, we shall parametrize the aforementioned quark generation mixing in terms of a singleparameter, the so-alled Cabibbo angle θC . Hene we get(u

d′

)=

(u

d cos θc + s sin θc

). (3.9)Having the above equations at hand, we may sort out the expliit expression that the harged (3.7) and theneutral (3.8) EW urrents shall take within this e�etive framework:

J+µ =g22ūγµ(1− γ5)d′

=g22ūγµ(1− γ5)d cos θc

︸ ︷︷ ︸∆S=0

+g22ūγµ(1− γ5)s sin θc

︸ ︷︷ ︸∆S=1

(3.10)J0µ ∼

g22ūγµu+

g22d̄′γµd

′

∼ g22ūγµu +

g22

(d̄γµd cos

2 θc + ss̄ sin2 θc)

︸ ︷︷ ︸∆S=0

+g22

(s̄γµd+ d̄γµs

)sin θc cos θc︸ ︷︷ ︸

∆S=1

, (3.11)where ∆S stems from the hange of the strangeness ontent. All in all, we onlude that the di�erent EWurrents, both those of harged and of neutral nature, might give rise to proesses in whih either i) thestrangeness is onserved, ∆S = 0; or ii) the strangeness hanges by one unit, ∆S = 1. The �rst possibilitywould orrespond to the usual transition between two states within the same SUL(2) multiplet, for instaneu → W+ d; whereas the seond one involves an expliit mixing between the 1st and 2nd generations, e.g.s →W− u. Experimentally, it is a well-known fat that weak harged-urrent proesses involving ∆S 6= 0are relatively suppressed, whilst the neutral-urrent interations are virtually absent. The former evidenean be omfortably embedded in the above desription of the harged urrents, as far as the measured valuefor the Cabibbo angle, sin θC = 0.22 implies a substantial depletion of the orresponding transition rates.However, and by the very same token, the absene of ∆S = 1 neutral-urrent interations is by no meansguaranteed in the present desription. Were this true, the EW setor of the SM would allow for diret(tree-level) �avor-hanging (∆S = 1) neutral-urrent interations, and would thus ome into great on�itwith the experimental grounds. This puzzle was takled in a lassial paper by Glashow, Iliopoulos andMaiani in 1970 [GIM70℄, in whih they suggested to extend the quark ontent of the SM � at that time,only the three lightest speies of quarks were known � with a fourth speies of quark, the harm quark,whih was predited to have the same pattern of ouplings and the same quantum numbers than the upquark. One the harm quark omes into the game, the doublet of physial quarks of the 2nd generation(c, s′) must be related to the orresponding gauge eigenstates, similarly to Eq. (3.8):

(c

s′

)=

(c

s cos θc − d sin θc

), (3.12)in suh a way that the rotation matrix that relates (d′, s′) and (d, s) reads

Vd′s′,ds =

(cos θC sin θC

− sin θC cos θC

), (3.13)ful�lling the ondition of unitarity: V †d′s′,ds = Vd′s′,ds. We note that, within this assumption, the neutral-16

3.3 Flavor dynamis and the GIM mehanismt

c

g

W

dl

dlt

c

g

t

dl

W

t

c

gc

dl

G

t

c

gc

dl

W

Figure 3.1: Set of Feynman diagrams desribing the FCNC deay of the top quark into a harm quarkand a gluon in the SM within the 'tHooft-Feynman gauge.urrent interation Lagrangian an be rewritten as:J0µ ∼

e

2 sW

(ūγµu+ d̄

′γµd′ + s̄′γµs

′)

∼ e2 sW

[ūγµu + (d̄γµd+ sγµs̄) cos

2 θc + (d̄γµd+ sγµs̄) sin2 θc

+ (s̄γµd+ d̄γµs− s̄γµd− d̄γµs) sin θc cos θc]

=e

2 sW

(ūγµu+ d̄γµd+ s̄γµs

)︸ ︷︷ ︸

∆S=0

. (3.14)From the above equation (3.14) it is manifest that the diret ∆S = 1 oupling yields identially zero. Inother words: there are no diret weak interations enabling the hange of the strangeness ontent of apartile without a onomitant hange of the eletri harge. At the prie of adding a 4th speies of quark,Glashow, Iliopoulos and Maiani managed to explain the large suppression of ∆S 6= 0 proesses withoutthe need of performing ad-ho �ne tunings of the parameters of the theory. It goes without saying thatthe experimental on�rmation of the harm quark at SLAC and BNL in 1974 [A+74b,A+74a℄ signi�ed abreakthrough in the development of Theoretial Partile Physis.The mathematial struture of the SM and, most partiularly, the unitarity of the CKM matrix � theequivalent objet to that of Eq. (3.13) for the general ase of three generations � naturally inorporates thisso-alled GIM mehanism and ensures a suitable suppression of the FCNC proesses. Noteworthy, the GIMmehanism also ontributes to the suppression of the FCNC interations indued by quantum e�ets. Letus take for instane a paradigmati FCNC deay hannel of the top quark into a harm quark and a gluon,t → cg [EHS91℄. This proess an only proeed at the quantum level through quark-mediated loops, asdepited in Figure 3.1. Let us onentrate e.g. on the triangle topologies. Simple power ounting argumentsand eduated guess provide the following approximate expression for the orresponding amplitude:

M(t→ cg) ∼ gs αew16π2

NC∑

i=d,s,b

(VCKM )ci (V∗CKM )ti f(m

2i ,M

2W ,m

2t ), (3.15)where the form fator f is dimensionless. A key observation here is that, if all the quarks in the loophad the same mass, the funtion f would then be the same for all i = d, s, b quark �avors. Therefore, theamplitude (3.15) would be proportional to∑i Vci V ∗ti , whih renders 0 as a result of unitarity. Consequently,

M(t→ cg) must be suppressed by fators of small quark masses. In partiular, beause of the massive Wboson propagator in the diagram, we expet a typial momentum �owing of ∼MW . Therefore, the massesof the fermion lines in the loop translate into O(m2i /M2W ) orretions, whih are either way very small � evenfor mi = mb. In the ase of the WF orretion topologies, instead, a logarithmi GIM suppression takesplae. This sort of suppressions, in whih loop-indued FCNC e�etive ouplings are severely restrained bysmall mass ratios, are the trademark �ngerprints of the GIM mehanism at the quantum level. Interestinglyenough, ertain dynamial properties of some extensions of the SM, suh as Supersymmetry, may be able tooverome suh GIM suppression and thus bring forward a manifest signature of an underlying New Physis.A dediated analysis of these senarios will be performed in Chapters 7-9 .17

Aspets of the Standard Model of Partile Physis3.4 The SM Higgs mechanismIt was already in the 1960's when the pioneering works by Higgs [Hig64b,Hig64a℄, Englert-Brout [EB64℄ andGuralnik-Hagen-Kibble [GHK64,Kib67℄ suggested the existene of a fundamental spinless building-blok ofNature 2, whose non-vanishing vauum expetation value ould explain the spontaneous breaking of theSU(2)L ⊗U(1)Y gauge group of the Eletroweak interations down to the U(1)em abelian symmetry groupof the Eletromagnetism. Suh a simple and elegant piture enapsulates the original proposals by Nambuand Jona-Lasinio, in whih they explored the possibility that partile masses were dynamially generatedthrough a symmetry breaking mehanism similar to the one that gives rise to Superondutivity. Theirseminal ideas hit the ground running for the modern understanding of Eletroweak Symmetry Breaking(EWSB).For one thing the Higgs mehanism is the only known strategy by whih one may embed the EWSBphenomenon into a (perturbative) quantum �eld theoretial desription. It is di�ult to overemphasizethat, to a great extent, the Higgs mehanism embodies the bakbone of the SM struture and that in itsabsene we would have to ope with an entirely di�erent oneption of the inner funtioning of the SM. Aswe will disuss hereafter, if no Higgs bosons existed below the TeV sale, weak interations would enter aperilous runaway regime at high energies, in whih ertain sattering ross setions (e.g. the WW ross-setion) would beome unphysially large � thus violating the unitarity of the sattering matrix. Moreover,in the absene of Higgs bosons the various partile masses ould not be generated onsistently � this is tosay, without spoiling the ultraviolet behavior of the theory at higher orders of perturbation theory. As amatter of fat, the ouplings of gauge bosons and hiral fermions to the Higgs boson (or bosons, if morethan one) play a ritial role to ensure that the overall desription of EWSB is de�ned in terms of a unitary,renormalizable QFT.Let us introdue a single omplex-valued salar �eld, transforming as an SUL(2) doublet with weakhyperharge Y = 1:

Φ =

(φ+

φ0

). (3.16)The dynamis of this �eld is determined at the lassial level by the Lagrangian

L = (DµΦ)† (Dµ Φ)− V (Φ), (3.17)with Dµ being the gauge ovariant derivative quoted in Eq. (3.1), through whih the salar �eld is minimallyoupled to the Eletroweak gauge bosons. Let us note that the Lagrangian (3.17) inludes a salar massterm and a quarti salar self-oupling, whih are enoded in the potentialV (Φ) = −µ2 Φ† Φ + λ

4(Φ† Φ)2 . (3.18)As a result, the ground state of the theory is suh that φ aquires a non-vanishing VEV (provided that

µ2 > 0). The latter is normalized as follows: 〈|Φ |〉 = 1√2v, with v = 2µ/√λ. Hene there is a spei�diretion in the isospin spae � this being singled out by the ground-state solution � whih spontaneouslybreaks the isospin invariane of the theory.The omplex isodoublet struture displayed in Eq. (3.16) an be split into 4 real (resp. 2 omplex)degrees of freedom in the following fashion:

Φ =

(φ+

v+H0+iχ0√2

), (3.19)where the omponents φ+,H0 and χ0 have vanishing VEV's and denote �utuations of the salar �eld aroundthe ground state. Owing to the gauge invariane of the Lagrangian, both the the φ+ and χ0 omponents ofthe Higg doublet an be set to zero upon an appropiate hoie of the gauge (the so-alled unitarity gauge),meaning that none of them arry any physial meaning � they a

ount for the Goldstone bosons whih are2See also Ref. [Ple09℄ for a very pedagogial approah to SM Higgs boson phenomenology at the LHC.18

3.4 The SM Higgs mehanisminherent to all senarios of SSB; in other words, the low-energy exitations assoiated to eah of the brokengenerators. In the partiular ase where these Goldstone bosons are gauged away, to wit φ+, χ0 = 0, we areleft with a simpli�ed salar struture;Φ =

(0

v+H0√2

), (3.20)in whih H0 stands for the only physial spinless �eld: the Higgs boson. It follows from the Lagrangianof Eq. (3.17) that the Higgs �eld gets a mass term whih an be written in terms of the µ parameter in(3.18), MH = µ√2. Or, if preferred, also as a funtion of the VEV v and the quarti Higgs self-oupling λ,

MH = v√λ/√

2. Notie, thereby, that the Higgs mass is a pure parameter, meaning that it is not alulablefrom �rst priniples within the SM framework. In addition, the ouplings to the gauge bosons via the kinetiterm of the Higgs �eld are responsible for generating the masses of the vetor bosons, as we an read o�from Eq. (3.17):Lgm =

1

2

(g22v)2

(W 21 +W22 ) +

v2

4(W 3µ , Bµ)

(g22 g1 g2

g1 g2 g21

)(W 3µ

Bµ

). (3.21)The bilinear term above may be diagonalized by rotating the �eldsW aµ , Bµ (the so-alled gauge-eigenstates,in terms of whih the gauge symmetry is manifest) into the orresponding mass eigenstates: these are, thephysial degrees of freedom (

W+µ

W−µ

)=

1√2

(1 −i1 i

) (W1µ

W2µ

), (3.22)

(Z0µ

γµ

)=

(cos θW − sin θWsin θW cos θW

) (W 3µ

Bµ

), (3.23)these are the familiar weak vetor bosons, W± and Z0, together with the photon �eld, γ. In the latterphysial basis the mass term of Eq. (3.21) beomes diagonal,

Lgm = M2W W+µ Wµ− +1

2(γµ,Zµ)

(0 0

0 M2Z

) (γµ

Zµ

). (3.24)The masses of the gauge bosons are anhored to the parameters in the Lagrangian by means of the followingrelations,

MW =1

2g2 v , MZ =

1

2v√g21 + g

22. (3.25)We de�ne the Weinberg angle θW through the relation:

sin2 θW = 1−M2WM2Z

. (3.26)In terms of the eletron harge−e and the abovementionedWeinberg angle, the Eletroweak gauge ouplingsg1, g2 an be rewritten as

e =g1g2√g21 + g

22

; g2 =e

sin θW; g1 =

e

cos θW. (3.27)Let us reall that e = √4παem, with αem being the �ne struture onstant, whose value at the EW saleyields αem(MZ) = 1/127.09. There is an additional parameter to be mentioned here; the Fermi onstant

GF = 1.16637(1) × 10−5GeV−2, whih is the 4-fermion oupling onstant in the e�etive Fermi theoryof weak interations, an be extrated with great preision upon the measure of the muon lifetime. Themathing of the Fermi theory with the SM desription at low energies (q2 ≪M2W ) enfores the identi�ationGF√

2=

e2

8 s2W M2W

. (3.28)19

Aspets of the Standard Model of Partile PhysisAll in all, the essential virtue of the Higgs mehanism beomes now transparent. Masses for the gaugebosons are generated while leaving the gauge invariane of the Lagrangian untouhed. This is preiselythe onept of Spontaneous Symmetry Breaking: the theory preserves the symmetry whereas the physialstates do not. Let us also reall that the photon �eld remains massless; indeed, Slavnov-Taylor identitiesasertain that it will remain massless to all orders in perturbation theory. This is preisely the signaturethat an abelian Uem(1) symmetry remains unbroken one the masses for the physial �elds are generated. Inturn, the masses of the gauge bosons an be traed bak to their longitudinal omponents: atually, in theunitary gauge the Goldstone bosons are absorbed into the aforementioned longitudinal degrees of freedomof the gauge �elds.In addition to that, Yukawa ouplings to fermions must be introdued in order to make the hargedfermions massive upon the SUL(2)-singlet bloks (f̄Lφ) fR + h.c. The aforementioned Yukawa setor isonveniently expressed in terms of the doublet �eld omponents of Eq. (3.19), for one family of leptons andquarks given byLHff = −gl (νLφ+lR + l̄R φ−νL + l̄Lφ0lR + l̄Rφ0∗ lL)

−gd (ūL φ+ dR + d̄R φ− uL + d̄L φ0 dR + d̄R φ0∗ dL)−gu (−ūR φ+ dL − d̄L φ− uR + ūRφ0uL + ūLφ0∗uR). (3.29)there φ− stands for the adjoint �eld of φ+. The fermion masses follow from the non-vanishing VEV of φ0,and hene the Yukawa oupling onstants gl,d,u are related to the masses of the harged fermions through

mf =gf v√

2=

√2MWg2

gf . (3.30)Inidentally, let us notie that no mass term for neutrinos an be generated through the urrent mehanism.3.5 The Higgs mechanism and UnitarityThe existene of the Higgs boson is, to a great extent, an uneludible requirement to assess a onsistentformulation of the SM as a quantum �eld theory. Its relevane is partiularly evident if we take into a

ountthat the presene of one or more suh �elds in the struture of the model is indispensable to guarantee theunitarity and the renormalizability of the theory. A paradigmati example of unitarity violation appearswhen onsidering the sattering of longitudinally polarized W± bosons. If no Higgs bosons exist belowthe TeV sale, the assoiated ross-setion would yield σ(W+LW−L → W+LW−L ) ∼ O( sM2

W

). The unitarityviolation of the sattering matrix beomes then manifest in this growing trend with the enter-of-mass energy,√s. The ompensation of the above power-like non-unitary behavior requires non-trivial anellations amongFeynman diagrams � and this is preisely the point at whih the interations mediated by the Higgs bosonsplay a fundamental role.There is a seond relevant aspet onerning unitarity, whih involves the size of the onstant (namelyenergy-independent) term that dominates the gauge-boson sattering at high-energies, one the ritialHiggs-gauge boson ouplings are introdued. The ondition of perturbative unitarity of the orrespondingS-matrix sets an upper bound on the abovementioned ontribution whih, on its turn, translates into alimit over the Higgs boson mass. This problem was �rst takled long ago in a pioneering work by Lee,Quik and Thaker (LQT) [LQT77b, LQT77a℄. To start with we observe that the sattering proess ofgauge bosons is dominated by their longitudinal omponents in the high-energy regime (namely, when suhgauge bosons are highly relativisti); on the other hand, in view of the Goldstone Equivalene Theoremsuh longitudinal omponents of the gauge bosons are equivalent to the Goldstone bosons up to orretionsof order O(MV√

s); hene it follows that the sattering amplitudes involving longitudinally-polarized gaugebosons may be approximated by the salar-salar interations in whih the longitudinal omponents of thegauge bosons (W±L ,Z0L), are replaed by their assoiated Goldstone bosons (G±, G0).Let us therefore onsider the sattering amplitude whih desribes a generi 4-salar proess S1S2 →

S3S4, and perform a partial-wave expansion;M(s, t, u) =

∞∑

l=0

Ml(s, t, u) , Ml(s, t, u) = 16 π (2l + 1)Pl (cos θ) al. (3.31)20

3.5 The Higgs mehanism and Unitaritywith l denoting the spin index of the orresponding partial wave and Pl denoting the Legendre Polynomials,whih satisfy∫ 1

−1dxPl(x)Pl′ (x) = δl l′ . (3.32)The di�erential ross setion for S1S2 → S3S4 is given by

dσ

dΩ=

1

64π2s|M(s, t, u)|2 (3.33)and, upon integration, one is left with the total ross setion

σ = Σ∞l=0 σl ; σl =16 π

s(2l + 1) |al|2. (3.34)The notion of unitarity is introdued preisely at this point; the requirement of onservation of probabilityimposes

1 = S† S = (1− iM†) (1 + iM) = 1 + i (M−M†) +MM†, (3.35)whereby one gets the usual formulation of the optial theorem:i (M−M†) = −M†M. (3.36)Using Eqs. (3.31)-(3.33) and exploiting the fat that the Legendre polynomials onstitute an orthogonalbasis, we an prove that the optial theorem enfores the real and imaginary parts of the sattering length

al to ful�ll:ℜ e (al)2 + ℑm (al)2 = |al|2 = ℑm(al) ∀l . (3.37)The above equation is nothing but the equation of a irle in the omplex plane [(ℜ e (al),ℑm(al))] withradius 12 and enter (0, 12 ),

(ℜ e (al))2 +(ℑm (al)−

1

2

)2=

1

4. (3.38)If Eq. (3.37) is meant to be valid, namely if we do assume ondition of tree-level unitarity as a onsistenyrequirement, it follows that there are no sattering lengths allowed for ℜ e(al) > 1/2 or ℜ e < −1/2. Thelatter relations an be simply rephrased as

|ℜ e(al)| <1

2∀l . (3.39)In turn, the sattering length al(s) an be extrated from Eq. (3.31), from whih we get

al(s) =1

32π

∫ 1

−1d(cos θ)Pl(cos θ)M(s, t, u). (3.40)Let us now apply these results to the proess W+L W−L → W+L W−L ; as we have argued, the above proessan essentially be mapped into G+G− → G+G− in the high-energy limit. In that regime, the dominantontribution to the sattering amplitude (3.31) omes from the energy-independent piee in the J = 0partial wave � whih is then not suppressed by inverse powers of s. Thereby we get

a0(G+G− → G+G−) = − αemM

2H

8M2W sin2 θW

+O(M2Hs

). (3.41)Plugging the unitarity requirement of Eq. (3.39), it follows that M2H < 4πv2, and hene we dedue the LQTbound,

M2LQT =4√

2 π

GF≃ (1.2TeV)2. (3.42)21

Aspets of the Standard Model of Partile PhysisThe fat that the leading term of the W+LW−L sattering am